Step | Hyp | Ref
| Expression |
1 | | fzfi 12633 |
. . . . . 6
⊢
(0...𝐾) ∈
Fin |
2 | | fzfi 12633 |
. . . . . 6
⊢
(1...𝑁) ∈
Fin |
3 | | mapfi 8145 |
. . . . . 6
⊢
(((0...𝐾) ∈ Fin
∧ (1...𝑁) ∈ Fin)
→ ((0...𝐾)
↑𝑚 (1...𝑁)) ∈ Fin) |
4 | 1, 2, 3 | mp2an 704 |
. . . . 5
⊢
((0...𝐾)
↑𝑚 (1...𝑁)) ∈ Fin |
5 | | fzfi 12633 |
. . . . 5
⊢
(0...(𝑁 − 1))
∈ Fin |
6 | | mapfi 8145 |
. . . . 5
⊢
((((0...𝐾)
↑𝑚 (1...𝑁)) ∈ Fin ∧ (0...(𝑁 − 1)) ∈ Fin) → (((0...𝐾) ↑𝑚
(1...𝑁))
↑𝑚 (0...(𝑁 − 1))) ∈ Fin) |
7 | 4, 5, 6 | mp2an 704 |
. . . 4
⊢
(((0...𝐾)
↑𝑚 (1...𝑁)) ↑𝑚 (0...(𝑁 − 1))) ∈
Fin |
8 | 7 | a1i 11 |
. . 3
⊢ (𝜑 → (((0...𝐾) ↑𝑚 (1...𝑁)) ↑𝑚
(0...(𝑁 − 1))) ∈
Fin) |
9 | | 2z 11286 |
. . . 4
⊢ 2 ∈
ℤ |
10 | 9 | a1i 11 |
. . 3
⊢ (𝜑 → 2 ∈
ℤ) |
11 | | fzofi 12635 |
. . . . . . . 8
⊢
(0..^𝐾) ∈
Fin |
12 | | mapfi 8145 |
. . . . . . . 8
⊢
(((0..^𝐾) ∈ Fin
∧ (1...𝑁) ∈ Fin)
→ ((0..^𝐾)
↑𝑚 (1...𝑁)) ∈ Fin) |
13 | 11, 2, 12 | mp2an 704 |
. . . . . . 7
⊢
((0..^𝐾)
↑𝑚 (1...𝑁)) ∈ Fin |
14 | | mapfi 8145 |
. . . . . . . . 9
⊢
(((1...𝑁) ∈ Fin
∧ (1...𝑁) ∈ Fin)
→ ((1...𝑁)
↑𝑚 (1...𝑁)) ∈ Fin) |
15 | 2, 2, 14 | mp2an 704 |
. . . . . . . 8
⊢
((1...𝑁)
↑𝑚 (1...𝑁)) ∈ Fin |
16 | | f1of 6050 |
. . . . . . . . . 10
⊢ (𝑓:(1...𝑁)–1-1-onto→(1...𝑁) → 𝑓:(1...𝑁)⟶(1...𝑁)) |
17 | 16 | ss2abi 3637 |
. . . . . . . . 9
⊢ {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)} ⊆ {𝑓 ∣ 𝑓:(1...𝑁)⟶(1...𝑁)} |
18 | | ovex 6577 |
. . . . . . . . . 10
⊢
(1...𝑁) ∈
V |
19 | 18, 18 | mapval 7756 |
. . . . . . . . 9
⊢
((1...𝑁)
↑𝑚 (1...𝑁)) = {𝑓 ∣ 𝑓:(1...𝑁)⟶(1...𝑁)} |
20 | 17, 19 | sseqtr4i 3601 |
. . . . . . . 8
⊢ {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)} ⊆ ((1...𝑁) ↑𝑚 (1...𝑁)) |
21 | | ssfi 8065 |
. . . . . . . 8
⊢
((((1...𝑁)
↑𝑚 (1...𝑁)) ∈ Fin ∧ {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)} ⊆ ((1...𝑁) ↑𝑚 (1...𝑁))) → {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)} ∈ Fin) |
22 | 15, 20, 21 | mp2an 704 |
. . . . . . 7
⊢ {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)} ∈ Fin |
23 | | xpfi 8116 |
. . . . . . 7
⊢
((((0..^𝐾)
↑𝑚 (1...𝑁)) ∈ Fin ∧ {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)} ∈ Fin) → (((0..^𝐾) ↑𝑚
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∈ Fin) |
24 | 13, 22, 23 | mp2an 704 |
. . . . . 6
⊢
(((0..^𝐾)
↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∈ Fin |
25 | | fzfi 12633 |
. . . . . 6
⊢
(0...𝑁) ∈
Fin |
26 | | xpfi 8116 |
. . . . . 6
⊢
(((((0..^𝐾)
↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∈ Fin ∧ (0...𝑁) ∈ Fin) → ((((0..^𝐾) ↑𝑚
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∈ Fin) |
27 | 24, 25, 26 | mp2an 704 |
. . . . 5
⊢
((((0..^𝐾)
↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∈ Fin |
28 | | rabfi 8070 |
. . . . 5
⊢
(((((0..^𝐾)
↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∈ Fin → {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)))} ∈ Fin) |
29 | 27, 28 | ax-mp 5 |
. . . 4
⊢ {𝑡 ∈ ((((0..^𝐾) ↑𝑚
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)))} ∈ Fin |
30 | | hashcl 13009 |
. . . . 5
⊢ ({𝑡 ∈ ((((0..^𝐾) ↑𝑚
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)))} ∈ Fin → (#‘{𝑡 ∈ ((((0..^𝐾) ↑𝑚
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)))}) ∈
ℕ0) |
31 | 30 | nn0zd 11356 |
. . . 4
⊢ ({𝑡 ∈ ((((0..^𝐾) ↑𝑚
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)))} ∈ Fin → (#‘{𝑡 ∈ ((((0..^𝐾) ↑𝑚
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)))}) ∈ ℤ) |
32 | 29, 31 | mp1i 13 |
. . 3
⊢ ((𝜑 ∧ 𝑥 ∈ (((0...𝐾) ↑𝑚 (1...𝑁)) ↑𝑚
(0...(𝑁 − 1))))
→ (#‘{𝑡 ∈
((((0..^𝐾)
↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)))}) ∈ ℤ) |
33 | | dfrex2 2979 |
. . . . 5
⊢
(∃𝑡 ∈
((((0..^𝐾)
↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁))((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)) ↔ ¬ ∀𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ¬ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾))) |
34 | | nfv 1830 |
. . . . . 6
⊢
Ⅎ𝑡(𝜑 ∧ 𝑥 ∈ (((0...𝐾) ↑𝑚 (1...𝑁)) ↑𝑚
(0...(𝑁 −
1)))) |
35 | | nfcv 2751 |
. . . . . . 7
⊢
Ⅎ𝑡2 |
36 | | nfcv 2751 |
. . . . . . 7
⊢
Ⅎ𝑡
∥ |
37 | | nfcv 2751 |
. . . . . . . 8
⊢
Ⅎ𝑡# |
38 | | nfrab1 3099 |
. . . . . . . 8
⊢
Ⅎ𝑡{𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)))} |
39 | 37, 38 | nffv 6110 |
. . . . . . 7
⊢
Ⅎ𝑡(#‘{𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)))}) |
40 | 35, 36, 39 | nfbr 4629 |
. . . . . 6
⊢
Ⅎ𝑡2 ∥
(#‘{𝑡 ∈
((((0..^𝐾)
↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)))}) |
41 | | neq0 3889 |
. . . . . . . . . . . 12
⊢ (¬
{𝑡 ∈ ((((0..^𝐾) ↑𝑚
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} = ∅ ↔
∃𝑠 𝑠 ∈ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))}) |
42 | | iddvds 14833 |
. . . . . . . . . . . . . . . . 17
⊢ (2 ∈
ℤ → 2 ∥ 2) |
43 | 9, 42 | ax-mp 5 |
. . . . . . . . . . . . . . . 16
⊢ 2 ∥
2 |
44 | | vex 3176 |
. . . . . . . . . . . . . . . . . . 19
⊢ 𝑠 ∈ V |
45 | | hashsng 13020 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑠 ∈ V → (#‘{𝑠}) = 1) |
46 | 44, 45 | ax-mp 5 |
. . . . . . . . . . . . . . . . . 18
⊢
(#‘{𝑠}) =
1 |
47 | 46 | oveq2i 6560 |
. . . . . . . . . . . . . . . . 17
⊢ (1 +
(#‘{𝑠})) = (1 +
1) |
48 | | df-2 10956 |
. . . . . . . . . . . . . . . . 17
⊢ 2 = (1 +
1) |
49 | 47, 48 | eqtr4i 2635 |
. . . . . . . . . . . . . . . 16
⊢ (1 +
(#‘{𝑠})) =
2 |
50 | 43, 49 | breqtrri 4610 |
. . . . . . . . . . . . . . 15
⊢ 2 ∥
(1 + (#‘{𝑠})) |
51 | | rabfi 8070 |
. . . . . . . . . . . . . . . . . . . 20
⊢
(((((0..^𝐾)
↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∈ Fin → {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} ∈
Fin) |
52 | | diffi 8077 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ({𝑡 ∈ ((((0..^𝐾) ↑𝑚
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} ∈ Fin →
({𝑡 ∈ ((((0..^𝐾) ↑𝑚
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} ∖ {𝑠}) ∈ Fin) |
53 | 27, 51, 52 | mp2b 10 |
. . . . . . . . . . . . . . . . . . 19
⊢ ({𝑡 ∈ ((((0..^𝐾) ↑𝑚
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} ∖ {𝑠}) ∈ Fin |
54 | | snfi 7923 |
. . . . . . . . . . . . . . . . . . 19
⊢ {𝑠} ∈ Fin |
55 | | incom 3767 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (({𝑡 ∈ ((((0..^𝐾) ↑𝑚
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} ∖ {𝑠}) ∩ {𝑠}) = ({𝑠} ∩ ({𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} ∖ {𝑠})) |
56 | | disjdif 3992 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ({𝑠} ∩ ({𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} ∖ {𝑠})) = ∅ |
57 | 55, 56 | eqtri 2632 |
. . . . . . . . . . . . . . . . . . 19
⊢ (({𝑡 ∈ ((((0..^𝐾) ↑𝑚
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} ∖ {𝑠}) ∩ {𝑠}) = ∅ |
58 | | hashun 13032 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((({𝑡 ∈ ((((0..^𝐾) ↑𝑚
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} ∖ {𝑠}) ∈ Fin ∧ {𝑠} ∈ Fin ∧ (({𝑡 ∈ ((((0..^𝐾) ↑𝑚
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} ∖ {𝑠}) ∩ {𝑠}) = ∅) → (#‘(({𝑡 ∈ ((((0..^𝐾) ↑𝑚
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} ∖ {𝑠}) ∪ {𝑠})) = ((#‘({𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} ∖ {𝑠})) + (#‘{𝑠}))) |
59 | 53, 54, 57, 58 | mp3an 1416 |
. . . . . . . . . . . . . . . . . 18
⊢
(#‘(({𝑡 ∈
((((0..^𝐾)
↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} ∖ {𝑠}) ∪ {𝑠})) = ((#‘({𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} ∖ {𝑠})) + (#‘{𝑠})) |
60 | | difsnid 4282 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑠 ∈ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} → (({𝑡 ∈ ((((0..^𝐾) ↑𝑚
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} ∖ {𝑠}) ∪ {𝑠}) = {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))}) |
61 | 60 | fveq2d 6107 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑠 ∈ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} → (#‘(({𝑡 ∈ ((((0..^𝐾) ↑𝑚
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} ∖ {𝑠}) ∪ {𝑠})) = (#‘{𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))})) |
62 | 59, 61 | syl5eqr 2658 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑠 ∈ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} → ((#‘({𝑡 ∈ ((((0..^𝐾) ↑𝑚
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} ∖ {𝑠})) + (#‘{𝑠})) = (#‘{𝑡 ∈ ((((0..^𝐾) ↑𝑚
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))})) |
63 | 62 | adantl 481 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝜑 ∧ 𝑥 ∈ (((0...𝐾) ↑𝑚 (1...𝑁)) ↑𝑚
(0...(𝑁 − 1)))) ∧
∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)) ∧ 𝑠 ∈ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))}) →
((#‘({𝑡 ∈
((((0..^𝐾)
↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} ∖ {𝑠})) + (#‘{𝑠})) = (#‘{𝑡 ∈ ((((0..^𝐾) ↑𝑚
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))})) |
64 | | poimir.0 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝜑 → 𝑁 ∈ ℕ) |
65 | 64 | ad3antrrr 762 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((((𝜑 ∧ 𝑥 ∈ (((0...𝐾) ↑𝑚 (1...𝑁)) ↑𝑚
(0...(𝑁 − 1)))) ∧
∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)) ∧ 𝑠 ∈ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))}) → 𝑁 ∈
ℕ) |
66 | | fveq2 6103 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (𝑡 = 𝑢 → (2nd ‘𝑡) = (2nd ‘𝑢)) |
67 | 66 | breq2d 4595 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝑡 = 𝑢 → (𝑦 < (2nd ‘𝑡) ↔ 𝑦 < (2nd ‘𝑢))) |
68 | 67 | ifbid 4058 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑡 = 𝑢 → if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) = if(𝑦 < (2nd ‘𝑢), 𝑦, (𝑦 + 1))) |
69 | 68 | csbeq1d 3506 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑡 = 𝑢 → ⦋if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ⦋if(𝑦 < (2nd
‘𝑢), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) |
70 | | fveq2 6103 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (𝑡 = 𝑢 → (1st ‘𝑡) = (1st ‘𝑢)) |
71 | 70 | fveq2d 6107 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝑡 = 𝑢 → (1st
‘(1st ‘𝑡)) = (1st ‘(1st
‘𝑢))) |
72 | 70 | fveq2d 6107 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ (𝑡 = 𝑢 → (2nd
‘(1st ‘𝑡)) = (2nd ‘(1st
‘𝑢))) |
73 | 72 | imaeq1d 5384 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ (𝑡 = 𝑢 → ((2nd
‘(1st ‘𝑡)) “ (1...𝑗)) = ((2nd ‘(1st
‘𝑢)) “
(1...𝑗))) |
74 | 73 | xpeq1d 5062 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (𝑡 = 𝑢 → (((2nd
‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) = (((2nd
‘(1st ‘𝑢)) “ (1...𝑗)) × {1})) |
75 | 72 | imaeq1d 5384 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ (𝑡 = 𝑢 → ((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) = ((2nd ‘(1st
‘𝑢)) “ ((𝑗 + 1)...𝑁))) |
76 | 75 | xpeq1d 5062 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (𝑡 = 𝑢 → (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}) = (((2nd
‘(1st ‘𝑢)) “ ((𝑗 + 1)...𝑁)) × {0})) |
77 | 74, 76 | uneq12d 3730 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝑡 = 𝑢 → ((((2nd
‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})) = ((((2nd
‘(1st ‘𝑢)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑢)) “ ((𝑗 + 1)...𝑁)) × {0}))) |
78 | 71, 77 | oveq12d 6567 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑡 = 𝑢 → ((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((1st
‘(1st ‘𝑢)) ∘𝑓 +
((((2nd ‘(1st ‘𝑢)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑢)) “ ((𝑗 + 1)...𝑁)) × {0})))) |
79 | 78 | csbeq2dv 3944 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑡 = 𝑢 → ⦋if(𝑦 < (2nd ‘𝑢), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ⦋if(𝑦 < (2nd
‘𝑢), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑢)) ∘𝑓 +
((((2nd ‘(1st ‘𝑢)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑢)) “ ((𝑗 + 1)...𝑁)) × {0})))) |
80 | 69, 79 | eqtrd 2644 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑡 = 𝑢 → ⦋if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ⦋if(𝑦 < (2nd
‘𝑢), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑢)) ∘𝑓 +
((((2nd ‘(1st ‘𝑢)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑢)) “ ((𝑗 + 1)...𝑁)) × {0})))) |
81 | 80 | mpteq2dv 4673 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑡 = 𝑢 → (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑢), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑢)) ∘𝑓 +
((((2nd ‘(1st ‘𝑢)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑢)) “ ((𝑗 + 1)...𝑁)) × {0}))))) |
82 | | breq1 4586 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝑦 = 𝑤 → (𝑦 < (2nd ‘𝑢) ↔ 𝑤 < (2nd ‘𝑢))) |
83 | | id 22 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝑦 = 𝑤 → 𝑦 = 𝑤) |
84 | | oveq1 6556 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝑦 = 𝑤 → (𝑦 + 1) = (𝑤 + 1)) |
85 | 82, 83, 84 | ifbieq12d 4063 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑦 = 𝑤 → if(𝑦 < (2nd ‘𝑢), 𝑦, (𝑦 + 1)) = if(𝑤 < (2nd ‘𝑢), 𝑤, (𝑤 + 1))) |
86 | 85 | csbeq1d 3506 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑦 = 𝑤 → ⦋if(𝑦 < (2nd ‘𝑢), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑢)) ∘𝑓 +
((((2nd ‘(1st ‘𝑢)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑢)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ⦋if(𝑤 < (2nd
‘𝑢), 𝑤, (𝑤 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑢)) ∘𝑓 +
((((2nd ‘(1st ‘𝑢)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑢)) “ ((𝑗 + 1)...𝑁)) × {0})))) |
87 | | oveq2 6557 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ (𝑗 = 𝑖 → (1...𝑗) = (1...𝑖)) |
88 | 87 | imaeq2d 5385 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ (𝑗 = 𝑖 → ((2nd
‘(1st ‘𝑢)) “ (1...𝑗)) = ((2nd ‘(1st
‘𝑢)) “
(1...𝑖))) |
89 | 88 | xpeq1d 5062 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (𝑗 = 𝑖 → (((2nd
‘(1st ‘𝑢)) “ (1...𝑗)) × {1}) = (((2nd
‘(1st ‘𝑢)) “ (1...𝑖)) × {1})) |
90 | | oveq1 6556 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢ (𝑗 = 𝑖 → (𝑗 + 1) = (𝑖 + 1)) |
91 | 90 | oveq1d 6564 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ (𝑗 = 𝑖 → ((𝑗 + 1)...𝑁) = ((𝑖 + 1)...𝑁)) |
92 | 91 | imaeq2d 5385 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ (𝑗 = 𝑖 → ((2nd
‘(1st ‘𝑢)) “ ((𝑗 + 1)...𝑁)) = ((2nd ‘(1st
‘𝑢)) “ ((𝑖 + 1)...𝑁))) |
93 | 92 | xpeq1d 5062 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (𝑗 = 𝑖 → (((2nd
‘(1st ‘𝑢)) “ ((𝑗 + 1)...𝑁)) × {0}) = (((2nd
‘(1st ‘𝑢)) “ ((𝑖 + 1)...𝑁)) × {0})) |
94 | 89, 93 | uneq12d 3730 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝑗 = 𝑖 → ((((2nd
‘(1st ‘𝑢)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑢)) “ ((𝑗 + 1)...𝑁)) × {0})) = ((((2nd
‘(1st ‘𝑢)) “ (1...𝑖)) × {1}) ∪ (((2nd
‘(1st ‘𝑢)) “ ((𝑖 + 1)...𝑁)) × {0}))) |
95 | 94 | oveq2d 6565 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑗 = 𝑖 → ((1st
‘(1st ‘𝑢)) ∘𝑓 +
((((2nd ‘(1st ‘𝑢)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑢)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((1st
‘(1st ‘𝑢)) ∘𝑓 +
((((2nd ‘(1st ‘𝑢)) “ (1...𝑖)) × {1}) ∪ (((2nd
‘(1st ‘𝑢)) “ ((𝑖 + 1)...𝑁)) × {0})))) |
96 | 95 | cbvcsbv 3505 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢
⦋if(𝑤
< (2nd ‘𝑢), 𝑤, (𝑤 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑢)) ∘𝑓 +
((((2nd ‘(1st ‘𝑢)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑢)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ⦋if(𝑤 < (2nd
‘𝑢), 𝑤, (𝑤 + 1)) / 𝑖⦌((1st
‘(1st ‘𝑢)) ∘𝑓 +
((((2nd ‘(1st ‘𝑢)) “ (1...𝑖)) × {1}) ∪ (((2nd
‘(1st ‘𝑢)) “ ((𝑖 + 1)...𝑁)) × {0}))) |
97 | 86, 96 | syl6eq 2660 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑦 = 𝑤 → ⦋if(𝑦 < (2nd ‘𝑢), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑢)) ∘𝑓 +
((((2nd ‘(1st ‘𝑢)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑢)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ⦋if(𝑤 < (2nd
‘𝑢), 𝑤, (𝑤 + 1)) / 𝑖⦌((1st
‘(1st ‘𝑢)) ∘𝑓 +
((((2nd ‘(1st ‘𝑢)) “ (1...𝑖)) × {1}) ∪ (((2nd
‘(1st ‘𝑢)) “ ((𝑖 + 1)...𝑁)) × {0})))) |
98 | 97 | cbvmptv 4678 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑢), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑢)) ∘𝑓 +
((((2nd ‘(1st ‘𝑢)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑢)) “ ((𝑗 + 1)...𝑁)) × {0})))) = (𝑤 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑤 < (2nd
‘𝑢), 𝑤, (𝑤 + 1)) / 𝑖⦌((1st
‘(1st ‘𝑢)) ∘𝑓 +
((((2nd ‘(1st ‘𝑢)) “ (1...𝑖)) × {1}) ∪ (((2nd
‘(1st ‘𝑢)) “ ((𝑖 + 1)...𝑁)) × {0})))) |
99 | 81, 98 | syl6eq 2660 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑡 = 𝑢 → (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) = (𝑤 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑤 < (2nd
‘𝑢), 𝑤, (𝑤 + 1)) / 𝑖⦌((1st
‘(1st ‘𝑢)) ∘𝑓 +
((((2nd ‘(1st ‘𝑢)) “ (1...𝑖)) × {1}) ∪ (((2nd
‘(1st ‘𝑢)) “ ((𝑖 + 1)...𝑁)) × {0}))))) |
100 | 99 | eqeq2d 2620 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑡 = 𝑢 → (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ↔ 𝑥 = (𝑤 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑤 < (2nd
‘𝑢), 𝑤, (𝑤 + 1)) / 𝑖⦌((1st
‘(1st ‘𝑢)) ∘𝑓 +
((((2nd ‘(1st ‘𝑢)) “ (1...𝑖)) × {1}) ∪ (((2nd
‘(1st ‘𝑢)) “ ((𝑖 + 1)...𝑁)) × {0})))))) |
101 | 100 | cbvrabv 3172 |
. . . . . . . . . . . . . . . . . . 19
⊢ {𝑡 ∈ ((((0..^𝐾) ↑𝑚
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} = {𝑢 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑤 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑤 < (2nd
‘𝑢), 𝑤, (𝑤 + 1)) / 𝑖⦌((1st
‘(1st ‘𝑢)) ∘𝑓 +
((((2nd ‘(1st ‘𝑢)) “ (1...𝑖)) × {1}) ∪ (((2nd
‘(1st ‘𝑢)) “ ((𝑖 + 1)...𝑁)) × {0}))))} |
102 | | elmapi 7765 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑥 ∈ (((0...𝐾) ↑𝑚 (1...𝑁)) ↑𝑚
(0...(𝑁 − 1))) →
𝑥:(0...(𝑁 − 1))⟶((0...𝐾) ↑𝑚 (1...𝑁))) |
103 | 102 | ad3antlr 763 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((((𝜑 ∧ 𝑥 ∈ (((0...𝐾) ↑𝑚 (1...𝑁)) ↑𝑚
(0...(𝑁 − 1)))) ∧
∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)) ∧ 𝑠 ∈ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))}) → 𝑥:(0...(𝑁 − 1))⟶((0...𝐾) ↑𝑚 (1...𝑁))) |
104 | | simpr 476 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((((𝜑 ∧ 𝑥 ∈ (((0...𝐾) ↑𝑚 (1...𝑁)) ↑𝑚
(0...(𝑁 − 1)))) ∧
∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)) ∧ 𝑠 ∈ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))}) → 𝑠 ∈ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))}) |
105 | | simpl 472 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
((∃𝑝 ∈
ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾) → ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0) |
106 | 105 | ralimi 2936 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
(∀𝑛 ∈
(1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾) → ∀𝑛 ∈ (1...𝑁)∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0) |
107 | 106 | ad2antlr 759 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((((𝜑 ∧ 𝑥 ∈ (((0...𝐾) ↑𝑚 (1...𝑁)) ↑𝑚
(0...(𝑁 − 1)))) ∧
∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)) ∧ 𝑠 ∈ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))}) → ∀𝑛 ∈ (1...𝑁)∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0) |
108 | | fveq2 6103 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑛 = 𝑚 → (𝑝‘𝑛) = (𝑝‘𝑚)) |
109 | 108 | neeq1d 2841 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑛 = 𝑚 → ((𝑝‘𝑛) ≠ 0 ↔ (𝑝‘𝑚) ≠ 0)) |
110 | 109 | rexbidv 3034 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑛 = 𝑚 → (∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ↔ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑚) ≠ 0)) |
111 | | fveq1 6102 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑝 = 𝑞 → (𝑝‘𝑚) = (𝑞‘𝑚)) |
112 | 111 | neeq1d 2841 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑝 = 𝑞 → ((𝑝‘𝑚) ≠ 0 ↔ (𝑞‘𝑚) ≠ 0)) |
113 | 112 | cbvrexv 3148 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
(∃𝑝 ∈ ran
𝑥(𝑝‘𝑚) ≠ 0 ↔ ∃𝑞 ∈ ran 𝑥(𝑞‘𝑚) ≠ 0) |
114 | 110, 113 | syl6bb 275 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑛 = 𝑚 → (∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ↔ ∃𝑞 ∈ ran 𝑥(𝑞‘𝑚) ≠ 0)) |
115 | 114 | rspccva 3281 |
. . . . . . . . . . . . . . . . . . . 20
⊢
((∀𝑛 ∈
(1...𝑁)∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ 𝑚 ∈ (1...𝑁)) → ∃𝑞 ∈ ran 𝑥(𝑞‘𝑚) ≠ 0) |
116 | 107, 115 | sylan 487 |
. . . . . . . . . . . . . . . . . . 19
⊢
(((((𝜑 ∧ 𝑥 ∈ (((0...𝐾) ↑𝑚 (1...𝑁)) ↑𝑚
(0...(𝑁 − 1)))) ∧
∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)) ∧ 𝑠 ∈ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))}) ∧ 𝑚 ∈ (1...𝑁)) → ∃𝑞 ∈ ran 𝑥(𝑞‘𝑚) ≠ 0) |
117 | | simpr 476 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
((∃𝑝 ∈
ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾) → ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾) |
118 | 117 | ralimi 2936 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
(∀𝑛 ∈
(1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾) → ∀𝑛 ∈ (1...𝑁)∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾) |
119 | 118 | ad2antlr 759 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((((𝜑 ∧ 𝑥 ∈ (((0...𝐾) ↑𝑚 (1...𝑁)) ↑𝑚
(0...(𝑁 − 1)))) ∧
∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)) ∧ 𝑠 ∈ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))}) → ∀𝑛 ∈ (1...𝑁)∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾) |
120 | 108 | neeq1d 2841 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑛 = 𝑚 → ((𝑝‘𝑛) ≠ 𝐾 ↔ (𝑝‘𝑚) ≠ 𝐾)) |
121 | 120 | rexbidv 3034 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑛 = 𝑚 → (∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾 ↔ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑚) ≠ 𝐾)) |
122 | 111 | neeq1d 2841 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑝 = 𝑞 → ((𝑝‘𝑚) ≠ 𝐾 ↔ (𝑞‘𝑚) ≠ 𝐾)) |
123 | 122 | cbvrexv 3148 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
(∃𝑝 ∈ ran
𝑥(𝑝‘𝑚) ≠ 𝐾 ↔ ∃𝑞 ∈ ran 𝑥(𝑞‘𝑚) ≠ 𝐾) |
124 | 121, 123 | syl6bb 275 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑛 = 𝑚 → (∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾 ↔ ∃𝑞 ∈ ran 𝑥(𝑞‘𝑚) ≠ 𝐾)) |
125 | 124 | rspccva 3281 |
. . . . . . . . . . . . . . . . . . . 20
⊢
((∀𝑛 ∈
(1...𝑁)∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾 ∧ 𝑚 ∈ (1...𝑁)) → ∃𝑞 ∈ ran 𝑥(𝑞‘𝑚) ≠ 𝐾) |
126 | 119, 125 | sylan 487 |
. . . . . . . . . . . . . . . . . . 19
⊢
(((((𝜑 ∧ 𝑥 ∈ (((0...𝐾) ↑𝑚 (1...𝑁)) ↑𝑚
(0...(𝑁 − 1)))) ∧
∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)) ∧ 𝑠 ∈ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))}) ∧ 𝑚 ∈ (1...𝑁)) → ∃𝑞 ∈ ran 𝑥(𝑞‘𝑚) ≠ 𝐾) |
127 | 65, 101, 103, 104, 116, 126 | poimirlem22 32601 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝜑 ∧ 𝑥 ∈ (((0...𝐾) ↑𝑚 (1...𝑁)) ↑𝑚
(0...(𝑁 − 1)))) ∧
∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)) ∧ 𝑠 ∈ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))}) → ∃!𝑧 ∈ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))}𝑧 ≠ 𝑠) |
128 | | eldifsn 4260 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑧 ∈ ({𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} ∖ {𝑠}) ↔ (𝑧 ∈ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} ∧ 𝑧 ≠ 𝑠)) |
129 | 128 | eubii 2480 |
. . . . . . . . . . . . . . . . . . 19
⊢
(∃!𝑧 𝑧 ∈ ({𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} ∖ {𝑠}) ↔ ∃!𝑧(𝑧 ∈ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} ∧ 𝑧 ≠ 𝑠)) |
130 | 53 | elexi 3186 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ({𝑡 ∈ ((((0..^𝐾) ↑𝑚
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} ∖ {𝑠}) ∈ V |
131 | | euhash1 13069 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (({𝑡 ∈ ((((0..^𝐾) ↑𝑚
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} ∖ {𝑠}) ∈ V →
((#‘({𝑡 ∈
((((0..^𝐾)
↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} ∖ {𝑠})) = 1 ↔ ∃!𝑧 𝑧 ∈ ({𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} ∖ {𝑠}))) |
132 | 130, 131 | ax-mp 5 |
. . . . . . . . . . . . . . . . . . 19
⊢
((#‘({𝑡 ∈
((((0..^𝐾)
↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} ∖ {𝑠})) = 1 ↔ ∃!𝑧 𝑧 ∈ ({𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} ∖ {𝑠})) |
133 | | df-reu 2903 |
. . . . . . . . . . . . . . . . . . 19
⊢
(∃!𝑧 ∈
{𝑡 ∈ ((((0..^𝐾) ↑𝑚
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))}𝑧 ≠ 𝑠 ↔ ∃!𝑧(𝑧 ∈ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} ∧ 𝑧 ≠ 𝑠)) |
134 | 129, 132,
133 | 3bitr4ri 292 |
. . . . . . . . . . . . . . . . . 18
⊢
(∃!𝑧 ∈
{𝑡 ∈ ((((0..^𝐾) ↑𝑚
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))}𝑧 ≠ 𝑠 ↔ (#‘({𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} ∖ {𝑠})) = 1) |
135 | 127, 134 | sylib 207 |
. . . . . . . . . . . . . . . . 17
⊢ ((((𝜑 ∧ 𝑥 ∈ (((0...𝐾) ↑𝑚 (1...𝑁)) ↑𝑚
(0...(𝑁 − 1)))) ∧
∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)) ∧ 𝑠 ∈ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))}) → (#‘({𝑡 ∈ ((((0..^𝐾) ↑𝑚
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} ∖ {𝑠})) = 1) |
136 | 135 | oveq1d 6564 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝜑 ∧ 𝑥 ∈ (((0...𝐾) ↑𝑚 (1...𝑁)) ↑𝑚
(0...(𝑁 − 1)))) ∧
∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)) ∧ 𝑠 ∈ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))}) →
((#‘({𝑡 ∈
((((0..^𝐾)
↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} ∖ {𝑠})) + (#‘{𝑠})) = (1 + (#‘{𝑠}))) |
137 | 63, 136 | eqtr3d 2646 |
. . . . . . . . . . . . . . 15
⊢ ((((𝜑 ∧ 𝑥 ∈ (((0...𝐾) ↑𝑚 (1...𝑁)) ↑𝑚
(0...(𝑁 − 1)))) ∧
∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)) ∧ 𝑠 ∈ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))}) → (#‘{𝑡 ∈ ((((0..^𝐾) ↑𝑚
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))}) = (1 + (#‘{𝑠}))) |
138 | 50, 137 | syl5breqr 4621 |
. . . . . . . . . . . . . 14
⊢ ((((𝜑 ∧ 𝑥 ∈ (((0...𝐾) ↑𝑚 (1...𝑁)) ↑𝑚
(0...(𝑁 − 1)))) ∧
∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)) ∧ 𝑠 ∈ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))}) → 2 ∥
(#‘{𝑡 ∈
((((0..^𝐾)
↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))})) |
139 | 138 | ex 449 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑥 ∈ (((0...𝐾) ↑𝑚 (1...𝑁)) ↑𝑚
(0...(𝑁 − 1)))) ∧
∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)) → (𝑠 ∈ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} → 2 ∥
(#‘{𝑡 ∈
((((0..^𝐾)
↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))}))) |
140 | 139 | exlimdv 1848 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑥 ∈ (((0...𝐾) ↑𝑚 (1...𝑁)) ↑𝑚
(0...(𝑁 − 1)))) ∧
∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)) → (∃𝑠 𝑠 ∈ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} → 2 ∥
(#‘{𝑡 ∈
((((0..^𝐾)
↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))}))) |
141 | 41, 140 | syl5bi 231 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑥 ∈ (((0...𝐾) ↑𝑚 (1...𝑁)) ↑𝑚
(0...(𝑁 − 1)))) ∧
∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)) → (¬ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} = ∅ → 2
∥ (#‘{𝑡 ∈
((((0..^𝐾)
↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))}))) |
142 | | dvds0 14835 |
. . . . . . . . . . . . . 14
⊢ (2 ∈
ℤ → 2 ∥ 0) |
143 | 9, 142 | ax-mp 5 |
. . . . . . . . . . . . 13
⊢ 2 ∥
0 |
144 | | hash0 13019 |
. . . . . . . . . . . . 13
⊢
(#‘∅) = 0 |
145 | 143, 144 | breqtrri 4610 |
. . . . . . . . . . . 12
⊢ 2 ∥
(#‘∅) |
146 | | fveq2 6103 |
. . . . . . . . . . . 12
⊢ ({𝑡 ∈ ((((0..^𝐾) ↑𝑚
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} = ∅ →
(#‘{𝑡 ∈
((((0..^𝐾)
↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))}) =
(#‘∅)) |
147 | 145, 146 | syl5breqr 4621 |
. . . . . . . . . . 11
⊢ ({𝑡 ∈ ((((0..^𝐾) ↑𝑚
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} = ∅ → 2
∥ (#‘{𝑡 ∈
((((0..^𝐾)
↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))})) |
148 | 141, 147 | pm2.61d2 171 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑥 ∈ (((0...𝐾) ↑𝑚 (1...𝑁)) ↑𝑚
(0...(𝑁 − 1)))) ∧
∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)) → 2 ∥ (#‘{𝑡 ∈ ((((0..^𝐾) ↑𝑚
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))})) |
149 | 148 | ex 449 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ∈ (((0...𝐾) ↑𝑚 (1...𝑁)) ↑𝑚
(0...(𝑁 − 1))))
→ (∀𝑛 ∈
(1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾) → 2 ∥ (#‘{𝑡 ∈ ((((0..^𝐾) ↑𝑚
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))}))) |
150 | 149 | adantld 482 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ (((0...𝐾) ↑𝑚 (1...𝑁)) ↑𝑚
(0...(𝑁 − 1))))
→ (((0...(𝑁 −
1)) ⊆ ran (𝑝 ∈
ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)) → 2 ∥ (#‘{𝑡 ∈ ((((0..^𝐾) ↑𝑚
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))}))) |
151 | | iba 523 |
. . . . . . . . . . 11
⊢
(((0...(𝑁 −
1)) ⊆ ran (𝑝 ∈
ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)) → (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ↔ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾))))) |
152 | 151 | rabbidv 3164 |
. . . . . . . . . 10
⊢
(((0...(𝑁 −
1)) ⊆ ran (𝑝 ∈
ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)) → {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} = {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)))}) |
153 | 152 | fveq2d 6107 |
. . . . . . . . 9
⊢
(((0...(𝑁 −
1)) ⊆ ran (𝑝 ∈
ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)) → (#‘{𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))}) = (#‘{𝑡 ∈ ((((0..^𝐾) ↑𝑚
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)))})) |
154 | 153 | breq2d 4595 |
. . . . . . . 8
⊢
(((0...(𝑁 −
1)) ⊆ ran (𝑝 ∈
ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)) → (2 ∥ (#‘{𝑡 ∈ ((((0..^𝐾) ↑𝑚
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))}) ↔ 2 ∥
(#‘{𝑡 ∈
((((0..^𝐾)
↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)))}))) |
155 | 150, 154 | mpbidi 230 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ (((0...𝐾) ↑𝑚 (1...𝑁)) ↑𝑚
(0...(𝑁 − 1))))
→ (((0...(𝑁 −
1)) ⊆ ran (𝑝 ∈
ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)) → 2 ∥ (#‘{𝑡 ∈ ((((0..^𝐾) ↑𝑚
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)))}))) |
156 | 155 | a1d 25 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ (((0...𝐾) ↑𝑚 (1...𝑁)) ↑𝑚
(0...(𝑁 − 1))))
→ (𝑡 ∈
((((0..^𝐾)
↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) → (((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)) → 2 ∥ (#‘{𝑡 ∈ ((((0..^𝐾) ↑𝑚
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)))})))) |
157 | 34, 40, 156 | rexlimd 3008 |
. . . . 5
⊢ ((𝜑 ∧ 𝑥 ∈ (((0...𝐾) ↑𝑚 (1...𝑁)) ↑𝑚
(0...(𝑁 − 1))))
→ (∃𝑡 ∈
((((0..^𝐾)
↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁))((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)) → 2 ∥ (#‘{𝑡 ∈ ((((0..^𝐾) ↑𝑚
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)))}))) |
158 | 33, 157 | syl5bir 232 |
. . . 4
⊢ ((𝜑 ∧ 𝑥 ∈ (((0...𝐾) ↑𝑚 (1...𝑁)) ↑𝑚
(0...(𝑁 − 1))))
→ (¬ ∀𝑡
∈ ((((0..^𝐾)
↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ¬ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)) → 2 ∥ (#‘{𝑡 ∈ ((((0..^𝐾) ↑𝑚
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)))}))) |
159 | | simpr 476 |
. . . . . . . . 9
⊢ ((𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾))) → ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾))) |
160 | 159 | con3i 149 |
. . . . . . . 8
⊢ (¬
((0...(𝑁 − 1))
⊆ ran (𝑝 ∈ ran
𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)) → ¬ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)))) |
161 | 160 | ralimi 2936 |
. . . . . . 7
⊢
(∀𝑡 ∈
((((0..^𝐾)
↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ¬ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)) → ∀𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ¬ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)))) |
162 | | rabeq0 3911 |
. . . . . . 7
⊢ ({𝑡 ∈ ((((0..^𝐾) ↑𝑚
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)))} = ∅ ↔ ∀𝑡 ∈ ((((0..^𝐾) ↑𝑚
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ¬ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)))) |
163 | 161, 162 | sylibr 223 |
. . . . . 6
⊢
(∀𝑡 ∈
((((0..^𝐾)
↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ¬ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)) → {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)))} = ∅) |
164 | 163 | fveq2d 6107 |
. . . . 5
⊢
(∀𝑡 ∈
((((0..^𝐾)
↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ¬ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)) → (#‘{𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)))}) = (#‘∅)) |
165 | 145, 164 | syl5breqr 4621 |
. . . 4
⊢
(∀𝑡 ∈
((((0..^𝐾)
↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ¬ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)) → 2 ∥ (#‘{𝑡 ∈ ((((0..^𝐾) ↑𝑚
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)))})) |
166 | 158, 165 | pm2.61d2 171 |
. . 3
⊢ ((𝜑 ∧ 𝑥 ∈ (((0...𝐾) ↑𝑚 (1...𝑁)) ↑𝑚
(0...(𝑁 − 1))))
→ 2 ∥ (#‘{𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)))})) |
167 | 8, 10, 32, 166 | fsumdvds 14868 |
. 2
⊢ (𝜑 → 2 ∥ Σ𝑥 ∈ (((0...𝐾) ↑𝑚 (1...𝑁)) ↑𝑚
(0...(𝑁 −
1)))(#‘{𝑡 ∈
((((0..^𝐾)
↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)))})) |
168 | | rabfi 8070 |
. . . . 5
⊢
(((((0..^𝐾)
↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∈ Fin → {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶} ∈ Fin) |
169 | 27, 168 | ax-mp 5 |
. . . 4
⊢ {𝑡 ∈ ((((0..^𝐾) ↑𝑚
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶} ∈ Fin |
170 | | simp1 1054 |
. . . . . . 7
⊢
((∀𝑖 ∈
(0...(𝑁 −
1))∃𝑗 ∈
(0...(𝑁 − 1))𝑖 =
⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁) → ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶) |
171 | | sneq 4135 |
. . . . . . . . . . . . 13
⊢
((2nd ‘𝑡) = 𝑁 → {(2nd ‘𝑡)} = {𝑁}) |
172 | 171 | difeq2d 3690 |
. . . . . . . . . . . 12
⊢
((2nd ‘𝑡) = 𝑁 → ((0...𝑁) ∖ {(2nd ‘𝑡)}) = ((0...𝑁) ∖ {𝑁})) |
173 | | difun2 4000 |
. . . . . . . . . . . . 13
⊢
(((0...(𝑁 −
1)) ∪ {𝑁}) ∖
{𝑁}) = ((0...(𝑁 − 1)) ∖ {𝑁}) |
174 | 64 | nnnn0d 11228 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → 𝑁 ∈
ℕ0) |
175 | | nn0uz 11598 |
. . . . . . . . . . . . . . . . . 18
⊢
ℕ0 = (ℤ≥‘0) |
176 | 174, 175 | syl6eleq 2698 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → 𝑁 ∈
(ℤ≥‘0)) |
177 | | fzm1 12289 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑁 ∈
(ℤ≥‘0) → (𝑛 ∈ (0...𝑁) ↔ (𝑛 ∈ (0...(𝑁 − 1)) ∨ 𝑛 = 𝑁))) |
178 | 176, 177 | syl 17 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → (𝑛 ∈ (0...𝑁) ↔ (𝑛 ∈ (0...(𝑁 − 1)) ∨ 𝑛 = 𝑁))) |
179 | | elun 3715 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑛 ∈ ((0...(𝑁 − 1)) ∪ {𝑁}) ↔ (𝑛 ∈ (0...(𝑁 − 1)) ∨ 𝑛 ∈ {𝑁})) |
180 | | velsn 4141 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑛 ∈ {𝑁} ↔ 𝑛 = 𝑁) |
181 | 180 | orbi2i 540 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑛 ∈ (0...(𝑁 − 1)) ∨ 𝑛 ∈ {𝑁}) ↔ (𝑛 ∈ (0...(𝑁 − 1)) ∨ 𝑛 = 𝑁)) |
182 | 179, 181 | bitri 263 |
. . . . . . . . . . . . . . . 16
⊢ (𝑛 ∈ ((0...(𝑁 − 1)) ∪ {𝑁}) ↔ (𝑛 ∈ (0...(𝑁 − 1)) ∨ 𝑛 = 𝑁)) |
183 | 178, 182 | syl6bbr 277 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → (𝑛 ∈ (0...𝑁) ↔ 𝑛 ∈ ((0...(𝑁 − 1)) ∪ {𝑁}))) |
184 | 183 | eqrdv 2608 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → (0...𝑁) = ((0...(𝑁 − 1)) ∪ {𝑁})) |
185 | 184 | difeq1d 3689 |
. . . . . . . . . . . . 13
⊢ (𝜑 → ((0...𝑁) ∖ {𝑁}) = (((0...(𝑁 − 1)) ∪ {𝑁}) ∖ {𝑁})) |
186 | 64 | nnzd 11357 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → 𝑁 ∈ ℤ) |
187 | | uzid 11578 |
. . . . . . . . . . . . . . 15
⊢ (𝑁 ∈ ℤ → 𝑁 ∈
(ℤ≥‘𝑁)) |
188 | | uznfz 12292 |
. . . . . . . . . . . . . . 15
⊢ (𝑁 ∈
(ℤ≥‘𝑁) → ¬ 𝑁 ∈ (0...(𝑁 − 1))) |
189 | 186, 187,
188 | 3syl 18 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → ¬ 𝑁 ∈ (0...(𝑁 − 1))) |
190 | | disjsn 4192 |
. . . . . . . . . . . . . . 15
⊢
(((0...(𝑁 −
1)) ∩ {𝑁}) = ∅
↔ ¬ 𝑁 ∈
(0...(𝑁 −
1))) |
191 | | disj3 3973 |
. . . . . . . . . . . . . . 15
⊢
(((0...(𝑁 −
1)) ∩ {𝑁}) = ∅
↔ (0...(𝑁 − 1))
= ((0...(𝑁 − 1))
∖ {𝑁})) |
192 | 190, 191 | bitr3i 265 |
. . . . . . . . . . . . . 14
⊢ (¬
𝑁 ∈ (0...(𝑁 − 1)) ↔ (0...(𝑁 − 1)) = ((0...(𝑁 − 1)) ∖ {𝑁})) |
193 | 189, 192 | sylib 207 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (0...(𝑁 − 1)) = ((0...(𝑁 − 1)) ∖ {𝑁})) |
194 | 173, 185,
193 | 3eqtr4a 2670 |
. . . . . . . . . . . 12
⊢ (𝜑 → ((0...𝑁) ∖ {𝑁}) = (0...(𝑁 − 1))) |
195 | 172, 194 | sylan9eqr 2666 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ (2nd
‘𝑡) = 𝑁) → ((0...𝑁) ∖ {(2nd ‘𝑡)}) = (0...(𝑁 − 1))) |
196 | 195 | rexeqdv 3122 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (2nd
‘𝑡) = 𝑁) → (∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ↔ ∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶)) |
197 | 196 | biimprd 237 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (2nd
‘𝑡) = 𝑁) → (∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 → ∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶)) |
198 | 197 | ralimdv 2946 |
. . . . . . . 8
⊢ ((𝜑 ∧ (2nd
‘𝑡) = 𝑁) → (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 → ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶)) |
199 | 198 | expimpd 627 |
. . . . . . 7
⊢ (𝜑 → (((2nd
‘𝑡) = 𝑁 ∧ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶) → ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶)) |
200 | 170, 199 | sylan2i 685 |
. . . . . 6
⊢ (𝜑 → (((2nd
‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁)) → ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶)) |
201 | 200 | adantr 480 |
. . . . 5
⊢ ((𝜑 ∧ 𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁))) → (((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁)) → ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶)) |
202 | 201 | ss2rabdv 3646 |
. . . 4
⊢ (𝜑 → {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))} ⊆ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶}) |
203 | | hashssdif 13061 |
. . . 4
⊢ (({𝑡 ∈ ((((0..^𝐾) ↑𝑚
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶} ∈ Fin ∧ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))} ⊆ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶}) → (#‘({𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶} ∖ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))})) = ((#‘{𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶}) − (#‘{𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))}))) |
204 | 169, 202,
203 | sylancr 694 |
. . 3
⊢ (𝜑 → (#‘({𝑡 ∈ ((((0..^𝐾) ↑𝑚
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶} ∖ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))})) = ((#‘{𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶}) − (#‘{𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))}))) |
205 | 64 | adantr 480 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁))) → 𝑁 ∈ ℕ) |
206 | | poimirlem28.1 |
. . . . . . . . . 10
⊢ (𝑝 = ((1st ‘𝑠) ∘𝑓 +
((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) → 𝐵 = 𝐶) |
207 | | poimirlem28.2 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑝:(1...𝑁)⟶(0...𝐾)) → 𝐵 ∈ (0...𝑁)) |
208 | 207 | adantlr 747 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁))) ∧ 𝑝:(1...𝑁)⟶(0...𝐾)) → 𝐵 ∈ (0...𝑁)) |
209 | | xp1st 7089 |
. . . . . . . . . . . 12
⊢ (𝑡 ∈ ((((0..^𝐾) ↑𝑚
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) → (1st ‘𝑡) ∈ (((0..^𝐾) ↑𝑚
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})) |
210 | | xp1st 7089 |
. . . . . . . . . . . 12
⊢
((1st ‘𝑡) ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) → (1st
‘(1st ‘𝑡)) ∈ ((0..^𝐾) ↑𝑚 (1...𝑁))) |
211 | | elmapi 7765 |
. . . . . . . . . . . 12
⊢
((1st ‘(1st ‘𝑡)) ∈ ((0..^𝐾) ↑𝑚 (1...𝑁)) → (1st
‘(1st ‘𝑡)):(1...𝑁)⟶(0..^𝐾)) |
212 | 209, 210,
211 | 3syl 18 |
. . . . . . . . . . 11
⊢ (𝑡 ∈ ((((0..^𝐾) ↑𝑚
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) → (1st
‘(1st ‘𝑡)):(1...𝑁)⟶(0..^𝐾)) |
213 | 212 | adantl 481 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁))) → (1st
‘(1st ‘𝑡)):(1...𝑁)⟶(0..^𝐾)) |
214 | | xp2nd 7090 |
. . . . . . . . . . . . 13
⊢
((1st ‘𝑡) ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) → (2nd
‘(1st ‘𝑡)) ∈ {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) |
215 | | fvex 6113 |
. . . . . . . . . . . . . 14
⊢
(2nd ‘(1st ‘𝑡)) ∈ V |
216 | | f1oeq1 6040 |
. . . . . . . . . . . . . 14
⊢ (𝑓 = (2nd
‘(1st ‘𝑡)) → (𝑓:(1...𝑁)–1-1-onto→(1...𝑁) ↔ (2nd
‘(1st ‘𝑡)):(1...𝑁)–1-1-onto→(1...𝑁))) |
217 | 215, 216 | elab 3319 |
. . . . . . . . . . . . 13
⊢
((2nd ‘(1st ‘𝑡)) ∈ {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)} ↔ (2nd
‘(1st ‘𝑡)):(1...𝑁)–1-1-onto→(1...𝑁)) |
218 | 214, 217 | sylib 207 |
. . . . . . . . . . . 12
⊢
((1st ‘𝑡) ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) → (2nd
‘(1st ‘𝑡)):(1...𝑁)–1-1-onto→(1...𝑁)) |
219 | 209, 218 | syl 17 |
. . . . . . . . . . 11
⊢ (𝑡 ∈ ((((0..^𝐾) ↑𝑚
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) → (2nd
‘(1st ‘𝑡)):(1...𝑁)–1-1-onto→(1...𝑁)) |
220 | 219 | adantl 481 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁))) → (2nd
‘(1st ‘𝑡)):(1...𝑁)–1-1-onto→(1...𝑁)) |
221 | | xp2nd 7090 |
. . . . . . . . . . 11
⊢ (𝑡 ∈ ((((0..^𝐾) ↑𝑚
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) → (2nd ‘𝑡) ∈ (0...𝑁)) |
222 | 221 | adantl 481 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁))) → (2nd ‘𝑡) ∈ (0...𝑁)) |
223 | 205, 206,
208, 213, 220, 222 | poimirlem24 32603 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁))) → (∃𝑥 ∈ (((0...𝐾) ↑𝑚 (1...𝑁)) ↑𝑚
(0...(𝑁 − 1)))(𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑁) ≠ 0)) ↔ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋〈(1st
‘(1st ‘𝑡)), (2nd ‘(1st
‘𝑡))〉 / 𝑠⦌𝐶 ∧ ¬ ((2nd ‘𝑡) = 𝑁 ∧ (((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))))) |
224 | 209 | adantl 481 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁))) → (1st ‘𝑡) ∈ (((0..^𝐾) ↑𝑚
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})) |
225 | | 1st2nd2 7096 |
. . . . . . . . . . . . . . 15
⊢
((1st ‘𝑡) ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) → (1st ‘𝑡) = 〈(1st
‘(1st ‘𝑡)), (2nd ‘(1st
‘𝑡))〉) |
226 | 225 | csbeq1d 3506 |
. . . . . . . . . . . . . 14
⊢
((1st ‘𝑡) ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) → ⦋(1st
‘𝑡) / 𝑠⦌𝐶 = ⦋〈(1st
‘(1st ‘𝑡)), (2nd ‘(1st
‘𝑡))〉 / 𝑠⦌𝐶) |
227 | 226 | eqeq2d 2620 |
. . . . . . . . . . . . 13
⊢
((1st ‘𝑡) ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) → (𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ↔ 𝑖 = ⦋〈(1st
‘(1st ‘𝑡)), (2nd ‘(1st
‘𝑡))〉 / 𝑠⦌𝐶)) |
228 | 227 | rexbidv 3034 |
. . . . . . . . . . . 12
⊢
((1st ‘𝑡) ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) → (∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ↔ ∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋〈(1st
‘(1st ‘𝑡)), (2nd ‘(1st
‘𝑡))〉 / 𝑠⦌𝐶)) |
229 | 228 | ralbidv 2969 |
. . . . . . . . . . 11
⊢
((1st ‘𝑡) ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) → (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ↔ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋〈(1st
‘(1st ‘𝑡)), (2nd ‘(1st
‘𝑡))〉 / 𝑠⦌𝐶)) |
230 | 229 | anbi1d 737 |
. . . . . . . . . 10
⊢
((1st ‘𝑡) ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) → ((∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ¬ ((2nd ‘𝑡) = 𝑁 ∧ (((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))) ↔ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋〈(1st
‘(1st ‘𝑡)), (2nd ‘(1st
‘𝑡))〉 / 𝑠⦌𝐶 ∧ ¬ ((2nd ‘𝑡) = 𝑁 ∧ (((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))))) |
231 | 224, 230 | syl 17 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁))) → ((∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ¬ ((2nd ‘𝑡) = 𝑁 ∧ (((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))) ↔ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋〈(1st
‘(1st ‘𝑡)), (2nd ‘(1st
‘𝑡))〉 / 𝑠⦌𝐶 ∧ ¬ ((2nd ‘𝑡) = 𝑁 ∧ (((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))))) |
232 | 223, 231 | bitr4d 270 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁))) → (∃𝑥 ∈ (((0...𝐾) ↑𝑚 (1...𝑁)) ↑𝑚
(0...(𝑁 − 1)))(𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑁) ≠ 0)) ↔ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ¬ ((2nd ‘𝑡) = 𝑁 ∧ (((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))))) |
233 | | frn 5966 |
. . . . . . . . . . . . . . 15
⊢ (𝑥:(0...(𝑁 − 1))⟶((0...𝐾) ↑𝑚 (1...𝑁)) → ran 𝑥 ⊆ ((0...𝐾) ↑𝑚 (1...𝑁))) |
234 | 102, 233 | syl 17 |
. . . . . . . . . . . . . 14
⊢ (𝑥 ∈ (((0...𝐾) ↑𝑚 (1...𝑁)) ↑𝑚
(0...(𝑁 − 1))) →
ran 𝑥 ⊆ ((0...𝐾) ↑𝑚
(1...𝑁))) |
235 | 234 | anim2i 591 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑥 ∈ (((0...𝐾) ↑𝑚 (1...𝑁)) ↑𝑚
(0...(𝑁 − 1))))
→ (𝜑 ∧ ran 𝑥 ⊆ ((0...𝐾) ↑𝑚 (1...𝑁)))) |
236 | | dfss3 3558 |
. . . . . . . . . . . . . 14
⊢
((0...(𝑁 − 1))
⊆ ran (𝑝 ∈ ran
𝑥 ↦ 𝐵) ↔ ∀𝑛 ∈ (0...(𝑁 − 1))𝑛 ∈ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵)) |
237 | | vex 3176 |
. . . . . . . . . . . . . . . 16
⊢ 𝑛 ∈ V |
238 | | eqid 2610 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑝 ∈ ran 𝑥 ↦ 𝐵) = (𝑝 ∈ ran 𝑥 ↦ 𝐵) |
239 | 238 | elrnmpt 5293 |
. . . . . . . . . . . . . . . 16
⊢ (𝑛 ∈ V → (𝑛 ∈ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ↔ ∃𝑝 ∈ ran 𝑥 𝑛 = 𝐵)) |
240 | 237, 239 | ax-mp 5 |
. . . . . . . . . . . . . . 15
⊢ (𝑛 ∈ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ↔ ∃𝑝 ∈ ran 𝑥 𝑛 = 𝐵) |
241 | 240 | ralbii 2963 |
. . . . . . . . . . . . . 14
⊢
(∀𝑛 ∈
(0...(𝑁 − 1))𝑛 ∈ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ↔ ∀𝑛 ∈ (0...(𝑁 − 1))∃𝑝 ∈ ran 𝑥 𝑛 = 𝐵) |
242 | 236, 241 | sylbb 208 |
. . . . . . . . . . . . 13
⊢
((0...(𝑁 − 1))
⊆ ran (𝑝 ∈ ran
𝑥 ↦ 𝐵) → ∀𝑛 ∈ (0...(𝑁 − 1))∃𝑝 ∈ ran 𝑥 𝑛 = 𝐵) |
243 | | 1eluzge0 11608 |
. . . . . . . . . . . . . . . . 17
⊢ 1 ∈
(ℤ≥‘0) |
244 | | fzss1 12251 |
. . . . . . . . . . . . . . . . 17
⊢ (1 ∈
(ℤ≥‘0) → (1...(𝑁 − 1)) ⊆ (0...(𝑁 − 1))) |
245 | | ssralv 3629 |
. . . . . . . . . . . . . . . . 17
⊢
((1...(𝑁 − 1))
⊆ (0...(𝑁 − 1))
→ (∀𝑛 ∈
(0...(𝑁 −
1))∃𝑝 ∈ ran
𝑥 𝑛 = 𝐵 → ∀𝑛 ∈ (1...(𝑁 − 1))∃𝑝 ∈ ran 𝑥 𝑛 = 𝐵)) |
246 | 243, 244,
245 | mp2b 10 |
. . . . . . . . . . . . . . . 16
⊢
(∀𝑛 ∈
(0...(𝑁 −
1))∃𝑝 ∈ ran
𝑥 𝑛 = 𝐵 → ∀𝑛 ∈ (1...(𝑁 − 1))∃𝑝 ∈ ran 𝑥 𝑛 = 𝐵) |
247 | 64 | nncnd 10913 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝜑 → 𝑁 ∈ ℂ) |
248 | | npcan1 10334 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑁 ∈ ℂ → ((𝑁 − 1) + 1) = 𝑁) |
249 | 247, 248 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝜑 → ((𝑁 − 1) + 1) = 𝑁) |
250 | | peano2zm 11297 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑁 ∈ ℤ → (𝑁 − 1) ∈
ℤ) |
251 | 186, 250 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝜑 → (𝑁 − 1) ∈ ℤ) |
252 | | uzid 11578 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝑁 − 1) ∈ ℤ
→ (𝑁 − 1) ∈
(ℤ≥‘(𝑁 − 1))) |
253 | | peano2uz 11617 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝑁 − 1) ∈
(ℤ≥‘(𝑁 − 1)) → ((𝑁 − 1) + 1) ∈
(ℤ≥‘(𝑁 − 1))) |
254 | 251, 252,
253 | 3syl 18 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝜑 → ((𝑁 − 1) + 1) ∈
(ℤ≥‘(𝑁 − 1))) |
255 | 249, 254 | eqeltrrd 2689 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘(𝑁 − 1))) |
256 | | fzss2 12252 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑁 ∈
(ℤ≥‘(𝑁 − 1)) → (1...(𝑁 − 1)) ⊆ (1...𝑁)) |
257 | 255, 256 | syl 17 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝜑 → (1...(𝑁 − 1)) ⊆ (1...𝑁)) |
258 | 257 | sselda 3568 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ 𝑛 ∈ (1...(𝑁 − 1))) → 𝑛 ∈ (1...𝑁)) |
259 | 258 | adantlr 747 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝜑 ∧ ran 𝑥 ⊆ ((0...𝐾) ↑𝑚 (1...𝑁))) ∧ 𝑛 ∈ (1...(𝑁 − 1))) → 𝑛 ∈ (1...𝑁)) |
260 | | simplr 788 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (((𝜑 ∧ ran 𝑥 ⊆ ((0...𝐾) ↑𝑚 (1...𝑁))) ∧ 𝑛 ∈ (1...𝑁)) → ran 𝑥 ⊆ ((0...𝐾) ↑𝑚 (1...𝑁))) |
261 | | ssel2 3563 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((ran
𝑥 ⊆ ((0...𝐾) ↑𝑚
(1...𝑁)) ∧ 𝑝 ∈ ran 𝑥) → 𝑝 ∈ ((0...𝐾) ↑𝑚 (1...𝑁))) |
262 | | elmapi 7765 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑝 ∈ ((0...𝐾) ↑𝑚 (1...𝑁)) → 𝑝:(1...𝑁)⟶(0...𝐾)) |
263 | 261, 262 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((ran
𝑥 ⊆ ((0...𝐾) ↑𝑚
(1...𝑁)) ∧ 𝑝 ∈ ran 𝑥) → 𝑝:(1...𝑁)⟶(0...𝐾)) |
264 | 260, 263 | sylan 487 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((((𝜑 ∧ ran 𝑥 ⊆ ((0...𝐾) ↑𝑚 (1...𝑁))) ∧ 𝑛 ∈ (1...𝑁)) ∧ 𝑝 ∈ ran 𝑥) → 𝑝:(1...𝑁)⟶(0...𝐾)) |
265 | | poimirlem28.3 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝐾) ∧ (𝑝‘𝑛) = 0)) → 𝐵 < 𝑛) |
266 | | elfzelz 12213 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
⊢ (𝑛 ∈ (1...𝑁) → 𝑛 ∈ ℤ) |
267 | 266 | zred 11358 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
⊢ (𝑛 ∈ (1...𝑁) → 𝑛 ∈ ℝ) |
268 | 267 | ltnrd 10050 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢ (𝑛 ∈ (1...𝑁) → ¬ 𝑛 < 𝑛) |
269 | | breq1 4586 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
⊢ (𝑛 = 𝐵 → (𝑛 < 𝑛 ↔ 𝐵 < 𝑛)) |
270 | 269 | notbid 307 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢ (𝑛 = 𝐵 → (¬ 𝑛 < 𝑛 ↔ ¬ 𝐵 < 𝑛)) |
271 | 268, 270 | syl5ibcom 234 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ (𝑛 ∈ (1...𝑁) → (𝑛 = 𝐵 → ¬ 𝐵 < 𝑛)) |
272 | 271 | necon2ad 2797 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ (𝑛 ∈ (1...𝑁) → (𝐵 < 𝑛 → 𝑛 ≠ 𝐵)) |
273 | 272 | 3ad2ant1 1075 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ ((𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝐾) ∧ (𝑝‘𝑛) = 0) → (𝐵 < 𝑛 → 𝑛 ≠ 𝐵)) |
274 | 273 | adantl 481 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝐾) ∧ (𝑝‘𝑛) = 0)) → (𝐵 < 𝑛 → 𝑛 ≠ 𝐵)) |
275 | 265, 274 | mpd 15 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝐾) ∧ (𝑝‘𝑛) = 0)) → 𝑛 ≠ 𝐵) |
276 | 275 | 3exp2 1277 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝜑 → (𝑛 ∈ (1...𝑁) → (𝑝:(1...𝑁)⟶(0...𝐾) → ((𝑝‘𝑛) = 0 → 𝑛 ≠ 𝐵)))) |
277 | 276 | imp31 447 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (((𝜑 ∧ 𝑛 ∈ (1...𝑁)) ∧ 𝑝:(1...𝑁)⟶(0...𝐾)) → ((𝑝‘𝑛) = 0 → 𝑛 ≠ 𝐵)) |
278 | 277 | necon2d 2805 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (((𝜑 ∧ 𝑛 ∈ (1...𝑁)) ∧ 𝑝:(1...𝑁)⟶(0...𝐾)) → (𝑛 = 𝐵 → (𝑝‘𝑛) ≠ 0)) |
279 | 278 | adantllr 751 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((((𝜑 ∧ ran 𝑥 ⊆ ((0...𝐾) ↑𝑚 (1...𝑁))) ∧ 𝑛 ∈ (1...𝑁)) ∧ 𝑝:(1...𝑁)⟶(0...𝐾)) → (𝑛 = 𝐵 → (𝑝‘𝑛) ≠ 0)) |
280 | 264, 279 | syldan 486 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((((𝜑 ∧ ran 𝑥 ⊆ ((0...𝐾) ↑𝑚 (1...𝑁))) ∧ 𝑛 ∈ (1...𝑁)) ∧ 𝑝 ∈ ran 𝑥) → (𝑛 = 𝐵 → (𝑝‘𝑛) ≠ 0)) |
281 | 280 | reximdva 3000 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝜑 ∧ ran 𝑥 ⊆ ((0...𝐾) ↑𝑚 (1...𝑁))) ∧ 𝑛 ∈ (1...𝑁)) → (∃𝑝 ∈ ran 𝑥 𝑛 = 𝐵 → ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0)) |
282 | 259, 281 | syldan 486 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ ran 𝑥 ⊆ ((0...𝐾) ↑𝑚 (1...𝑁))) ∧ 𝑛 ∈ (1...(𝑁 − 1))) → (∃𝑝 ∈ ran 𝑥 𝑛 = 𝐵 → ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0)) |
283 | 282 | ralimdva 2945 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ ran 𝑥 ⊆ ((0...𝐾) ↑𝑚 (1...𝑁))) → (∀𝑛 ∈ (1...(𝑁 − 1))∃𝑝 ∈ ran 𝑥 𝑛 = 𝐵 → ∀𝑛 ∈ (1...(𝑁 − 1))∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0)) |
284 | 283 | imp 444 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ ran 𝑥 ⊆ ((0...𝐾) ↑𝑚 (1...𝑁))) ∧ ∀𝑛 ∈ (1...(𝑁 − 1))∃𝑝 ∈ ran 𝑥 𝑛 = 𝐵) → ∀𝑛 ∈ (1...(𝑁 − 1))∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0) |
285 | 246, 284 | sylan2 490 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ ran 𝑥 ⊆ ((0...𝐾) ↑𝑚 (1...𝑁))) ∧ ∀𝑛 ∈ (0...(𝑁 − 1))∃𝑝 ∈ ran 𝑥 𝑛 = 𝐵) → ∀𝑛 ∈ (1...(𝑁 − 1))∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0) |
286 | 285 | biantrurd 528 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ ran 𝑥 ⊆ ((0...𝐾) ↑𝑚 (1...𝑁))) ∧ ∀𝑛 ∈ (0...(𝑁 − 1))∃𝑝 ∈ ran 𝑥 𝑛 = 𝐵) → (∃𝑝 ∈ ran 𝑥(𝑝‘𝑁) ≠ 0 ↔ (∀𝑛 ∈ (1...(𝑁 − 1))∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑁) ≠ 0))) |
287 | | nnuz 11599 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ℕ =
(ℤ≥‘1) |
288 | 64, 287 | syl6eleq 2698 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝜑 → 𝑁 ∈
(ℤ≥‘1)) |
289 | | fzm1 12289 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑁 ∈
(ℤ≥‘1) → (𝑛 ∈ (1...𝑁) ↔ (𝑛 ∈ (1...(𝑁 − 1)) ∨ 𝑛 = 𝑁))) |
290 | 288, 289 | syl 17 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝜑 → (𝑛 ∈ (1...𝑁) ↔ (𝑛 ∈ (1...(𝑁 − 1)) ∨ 𝑛 = 𝑁))) |
291 | | elun 3715 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑛 ∈ ((1...(𝑁 − 1)) ∪ {𝑁}) ↔ (𝑛 ∈ (1...(𝑁 − 1)) ∨ 𝑛 ∈ {𝑁})) |
292 | 180 | orbi2i 540 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝑛 ∈ (1...(𝑁 − 1)) ∨ 𝑛 ∈ {𝑁}) ↔ (𝑛 ∈ (1...(𝑁 − 1)) ∨ 𝑛 = 𝑁)) |
293 | 291, 292 | bitri 263 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑛 ∈ ((1...(𝑁 − 1)) ∪ {𝑁}) ↔ (𝑛 ∈ (1...(𝑁 − 1)) ∨ 𝑛 = 𝑁)) |
294 | 290, 293 | syl6bbr 277 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 → (𝑛 ∈ (1...𝑁) ↔ 𝑛 ∈ ((1...(𝑁 − 1)) ∪ {𝑁}))) |
295 | 294 | eqrdv 2608 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → (1...𝑁) = ((1...(𝑁 − 1)) ∪ {𝑁})) |
296 | 295 | raleqdv 3121 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → (∀𝑛 ∈ (1...𝑁)∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ↔ ∀𝑛 ∈ ((1...(𝑁 − 1)) ∪ {𝑁})∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0)) |
297 | | ralunb 3756 |
. . . . . . . . . . . . . . . . 17
⊢
(∀𝑛 ∈
((1...(𝑁 − 1)) ∪
{𝑁})∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ↔ (∀𝑛 ∈ (1...(𝑁 − 1))∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∀𝑛 ∈ {𝑁}∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0)) |
298 | 296, 297 | syl6bb 275 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → (∀𝑛 ∈ (1...𝑁)∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ↔ (∀𝑛 ∈ (1...(𝑁 − 1))∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∀𝑛 ∈ {𝑁}∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0))) |
299 | | fveq2 6103 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑛 = 𝑁 → (𝑝‘𝑛) = (𝑝‘𝑁)) |
300 | 299 | neeq1d 2841 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑛 = 𝑁 → ((𝑝‘𝑛) ≠ 0 ↔ (𝑝‘𝑁) ≠ 0)) |
301 | 300 | rexbidv 3034 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑛 = 𝑁 → (∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ↔ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑁) ≠ 0)) |
302 | 301 | ralsng 4165 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑁 ∈ ℕ →
(∀𝑛 ∈ {𝑁}∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ↔ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑁) ≠ 0)) |
303 | 64, 302 | syl 17 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → (∀𝑛 ∈ {𝑁}∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ↔ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑁) ≠ 0)) |
304 | 303 | anbi2d 736 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → ((∀𝑛 ∈ (1...(𝑁 − 1))∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∀𝑛 ∈ {𝑁}∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0) ↔ (∀𝑛 ∈ (1...(𝑁 − 1))∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑁) ≠ 0))) |
305 | 298, 304 | bitrd 267 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → (∀𝑛 ∈ (1...𝑁)∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ↔ (∀𝑛 ∈ (1...(𝑁 − 1))∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑁) ≠ 0))) |
306 | 305 | ad2antrr 758 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ ran 𝑥 ⊆ ((0...𝐾) ↑𝑚 (1...𝑁))) ∧ ∀𝑛 ∈ (0...(𝑁 − 1))∃𝑝 ∈ ran 𝑥 𝑛 = 𝐵) → (∀𝑛 ∈ (1...𝑁)∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ↔ (∀𝑛 ∈ (1...(𝑁 − 1))∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑁) ≠ 0))) |
307 | | 0z 11265 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ 0 ∈
ℤ |
308 | | 1z 11284 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ 1 ∈
ℤ |
309 | | fzshftral 12297 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((0
∈ ℤ ∧ (𝑁
− 1) ∈ ℤ ∧ 1 ∈ ℤ) → (∀𝑛 ∈ (0...(𝑁 − 1))∃𝑝 ∈ ran 𝑥 𝑛 = 𝐵 ↔ ∀𝑚 ∈ ((0 + 1)...((𝑁 − 1) + 1))[(𝑚 − 1) / 𝑛]∃𝑝 ∈ ran 𝑥 𝑛 = 𝐵)) |
310 | 307, 308,
309 | mp3an13 1407 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝑁 − 1) ∈ ℤ
→ (∀𝑛 ∈
(0...(𝑁 −
1))∃𝑝 ∈ ran
𝑥 𝑛 = 𝐵 ↔ ∀𝑚 ∈ ((0 + 1)...((𝑁 − 1) + 1))[(𝑚 − 1) / 𝑛]∃𝑝 ∈ ran 𝑥 𝑛 = 𝐵)) |
311 | 186, 250,
310 | 3syl 18 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝜑 → (∀𝑛 ∈ (0...(𝑁 − 1))∃𝑝 ∈ ran 𝑥 𝑛 = 𝐵 ↔ ∀𝑚 ∈ ((0 + 1)...((𝑁 − 1) + 1))[(𝑚 − 1) / 𝑛]∃𝑝 ∈ ran 𝑥 𝑛 = 𝐵)) |
312 | | 0p1e1 11009 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (0 + 1) =
1 |
313 | 312 | a1i 11 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝜑 → (0 + 1) =
1) |
314 | 313, 249 | oveq12d 6567 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝜑 → ((0 + 1)...((𝑁 − 1) + 1)) = (1...𝑁)) |
315 | 314 | raleqdv 3121 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝜑 → (∀𝑚 ∈ ((0 + 1)...((𝑁 − 1) + 1))[(𝑚 − 1) / 𝑛]∃𝑝 ∈ ran 𝑥 𝑛 = 𝐵 ↔ ∀𝑚 ∈ (1...𝑁)[(𝑚 − 1) / 𝑛]∃𝑝 ∈ ran 𝑥 𝑛 = 𝐵)) |
316 | 311, 315 | bitrd 267 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝜑 → (∀𝑛 ∈ (0...(𝑁 − 1))∃𝑝 ∈ ran 𝑥 𝑛 = 𝐵 ↔ ∀𝑚 ∈ (1...𝑁)[(𝑚 − 1) / 𝑛]∃𝑝 ∈ ran 𝑥 𝑛 = 𝐵)) |
317 | | ovex 6577 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑚 − 1) ∈
V |
318 | | eqeq1 2614 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑛 = (𝑚 − 1) → (𝑛 = 𝐵 ↔ (𝑚 − 1) = 𝐵)) |
319 | 318 | rexbidv 3034 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑛 = (𝑚 − 1) → (∃𝑝 ∈ ran 𝑥 𝑛 = 𝐵 ↔ ∃𝑝 ∈ ran 𝑥(𝑚 − 1) = 𝐵)) |
320 | 317, 319 | sbcie 3437 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
([(𝑚 −
1) / 𝑛]∃𝑝 ∈ ran 𝑥 𝑛 = 𝐵 ↔ ∃𝑝 ∈ ran 𝑥(𝑚 − 1) = 𝐵) |
321 | 320 | ralbii 2963 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
(∀𝑚 ∈
(1...𝑁)[(𝑚 − 1) / 𝑛]∃𝑝 ∈ ran 𝑥 𝑛 = 𝐵 ↔ ∀𝑚 ∈ (1...𝑁)∃𝑝 ∈ ran 𝑥(𝑚 − 1) = 𝐵) |
322 | | oveq1 6556 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑚 = 𝑛 → (𝑚 − 1) = (𝑛 − 1)) |
323 | 322 | eqeq1d 2612 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑚 = 𝑛 → ((𝑚 − 1) = 𝐵 ↔ (𝑛 − 1) = 𝐵)) |
324 | 323 | rexbidv 3034 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑚 = 𝑛 → (∃𝑝 ∈ ran 𝑥(𝑚 − 1) = 𝐵 ↔ ∃𝑝 ∈ ran 𝑥(𝑛 − 1) = 𝐵)) |
325 | 324 | cbvralv 3147 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
(∀𝑚 ∈
(1...𝑁)∃𝑝 ∈ ran 𝑥(𝑚 − 1) = 𝐵 ↔ ∀𝑛 ∈ (1...𝑁)∃𝑝 ∈ ran 𝑥(𝑛 − 1) = 𝐵) |
326 | 321, 325 | bitri 263 |
. . . . . . . . . . . . . . . . . . . 20
⊢
(∀𝑚 ∈
(1...𝑁)[(𝑚 − 1) / 𝑛]∃𝑝 ∈ ran 𝑥 𝑛 = 𝐵 ↔ ∀𝑛 ∈ (1...𝑁)∃𝑝 ∈ ran 𝑥(𝑛 − 1) = 𝐵) |
327 | 316, 326 | syl6bb 275 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 → (∀𝑛 ∈ (0...(𝑁 − 1))∃𝑝 ∈ ran 𝑥 𝑛 = 𝐵 ↔ ∀𝑛 ∈ (1...𝑁)∃𝑝 ∈ ran 𝑥(𝑛 − 1) = 𝐵)) |
328 | 327 | biimpa 500 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ ∀𝑛 ∈ (0...(𝑁 − 1))∃𝑝 ∈ ran 𝑥 𝑛 = 𝐵) → ∀𝑛 ∈ (1...𝑁)∃𝑝 ∈ ran 𝑥(𝑛 − 1) = 𝐵) |
329 | 328 | adantlr 747 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ ran 𝑥 ⊆ ((0...𝐾) ↑𝑚 (1...𝑁))) ∧ ∀𝑛 ∈ (0...(𝑁 − 1))∃𝑝 ∈ ran 𝑥 𝑛 = 𝐵) → ∀𝑛 ∈ (1...𝑁)∃𝑝 ∈ ran 𝑥(𝑛 − 1) = 𝐵) |
330 | | poimirlem28.4 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝐾) ∧ (𝑝‘𝑛) = 𝐾)) → 𝐵 ≠ (𝑛 − 1)) |
331 | 330 | necomd 2837 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝐾) ∧ (𝑝‘𝑛) = 𝐾)) → (𝑛 − 1) ≠ 𝐵) |
332 | 331 | 3exp2 1277 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝜑 → (𝑛 ∈ (1...𝑁) → (𝑝:(1...𝑁)⟶(0...𝐾) → ((𝑝‘𝑛) = 𝐾 → (𝑛 − 1) ≠ 𝐵)))) |
333 | 332 | imp31 447 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (((𝜑 ∧ 𝑛 ∈ (1...𝑁)) ∧ 𝑝:(1...𝑁)⟶(0...𝐾)) → ((𝑝‘𝑛) = 𝐾 → (𝑛 − 1) ≠ 𝐵)) |
334 | 333 | necon2d 2805 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (((𝜑 ∧ 𝑛 ∈ (1...𝑁)) ∧ 𝑝:(1...𝑁)⟶(0...𝐾)) → ((𝑛 − 1) = 𝐵 → (𝑝‘𝑛) ≠ 𝐾)) |
335 | 334 | adantllr 751 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((((𝜑 ∧ ran 𝑥 ⊆ ((0...𝐾) ↑𝑚 (1...𝑁))) ∧ 𝑛 ∈ (1...𝑁)) ∧ 𝑝:(1...𝑁)⟶(0...𝐾)) → ((𝑛 − 1) = 𝐵 → (𝑝‘𝑛) ≠ 𝐾)) |
336 | 264, 335 | syldan 486 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((((𝜑 ∧ ran 𝑥 ⊆ ((0...𝐾) ↑𝑚 (1...𝑁))) ∧ 𝑛 ∈ (1...𝑁)) ∧ 𝑝 ∈ ran 𝑥) → ((𝑛 − 1) = 𝐵 → (𝑝‘𝑛) ≠ 𝐾)) |
337 | 336 | reximdva 3000 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝜑 ∧ ran 𝑥 ⊆ ((0...𝐾) ↑𝑚 (1...𝑁))) ∧ 𝑛 ∈ (1...𝑁)) → (∃𝑝 ∈ ran 𝑥(𝑛 − 1) = 𝐵 → ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)) |
338 | 337 | ralimdva 2945 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ ran 𝑥 ⊆ ((0...𝐾) ↑𝑚 (1...𝑁))) → (∀𝑛 ∈ (1...𝑁)∃𝑝 ∈ ran 𝑥(𝑛 − 1) = 𝐵 → ∀𝑛 ∈ (1...𝑁)∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)) |
339 | 338 | imp 444 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ ran 𝑥 ⊆ ((0...𝐾) ↑𝑚 (1...𝑁))) ∧ ∀𝑛 ∈ (1...𝑁)∃𝑝 ∈ ran 𝑥(𝑛 − 1) = 𝐵) → ∀𝑛 ∈ (1...𝑁)∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾) |
340 | 329, 339 | syldan 486 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ ran 𝑥 ⊆ ((0...𝐾) ↑𝑚 (1...𝑁))) ∧ ∀𝑛 ∈ (0...(𝑁 − 1))∃𝑝 ∈ ran 𝑥 𝑛 = 𝐵) → ∀𝑛 ∈ (1...𝑁)∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾) |
341 | 340 | biantrud 527 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ ran 𝑥 ⊆ ((0...𝐾) ↑𝑚 (1...𝑁))) ∧ ∀𝑛 ∈ (0...(𝑁 − 1))∃𝑝 ∈ ran 𝑥 𝑛 = 𝐵) → (∀𝑛 ∈ (1...𝑁)∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ↔ (∀𝑛 ∈ (1...𝑁)∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∀𝑛 ∈ (1...𝑁)∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾))) |
342 | | r19.26 3046 |
. . . . . . . . . . . . . . 15
⊢
(∀𝑛 ∈
(1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾) ↔ (∀𝑛 ∈ (1...𝑁)∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∀𝑛 ∈ (1...𝑁)∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)) |
343 | 341, 342 | syl6bbr 277 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ ran 𝑥 ⊆ ((0...𝐾) ↑𝑚 (1...𝑁))) ∧ ∀𝑛 ∈ (0...(𝑁 − 1))∃𝑝 ∈ ran 𝑥 𝑛 = 𝐵) → (∀𝑛 ∈ (1...𝑁)∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ↔ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾))) |
344 | 286, 306,
343 | 3bitr2d 295 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ ran 𝑥 ⊆ ((0...𝐾) ↑𝑚 (1...𝑁))) ∧ ∀𝑛 ∈ (0...(𝑁 − 1))∃𝑝 ∈ ran 𝑥 𝑛 = 𝐵) → (∃𝑝 ∈ ran 𝑥(𝑝‘𝑁) ≠ 0 ↔ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾))) |
345 | 235, 242,
344 | syl2an 493 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑥 ∈ (((0...𝐾) ↑𝑚 (1...𝑁)) ↑𝑚
(0...(𝑁 − 1)))) ∧
(0...(𝑁 − 1)) ⊆
ran (𝑝 ∈ ran 𝑥 ↦ 𝐵)) → (∃𝑝 ∈ ran 𝑥(𝑝‘𝑁) ≠ 0 ↔ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾))) |
346 | 345 | pm5.32da 671 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑥 ∈ (((0...𝐾) ↑𝑚 (1...𝑁)) ↑𝑚
(0...(𝑁 − 1))))
→ (((0...(𝑁 −
1)) ⊆ ran (𝑝 ∈
ran 𝑥 ↦ 𝐵) ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑁) ≠ 0) ↔ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)))) |
347 | 346 | anbi2d 736 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑥 ∈ (((0...𝐾) ↑𝑚 (1...𝑁)) ↑𝑚
(0...(𝑁 − 1))))
→ ((𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑁) ≠ 0)) ↔ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾))))) |
348 | 347 | rexbidva 3031 |
. . . . . . . . 9
⊢ (𝜑 → (∃𝑥 ∈ (((0...𝐾) ↑𝑚 (1...𝑁)) ↑𝑚
(0...(𝑁 − 1)))(𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑁) ≠ 0)) ↔ ∃𝑥 ∈ (((0...𝐾) ↑𝑚 (1...𝑁)) ↑𝑚
(0...(𝑁 − 1)))(𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾))))) |
349 | 348 | adantr 480 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁))) → (∃𝑥 ∈ (((0...𝐾) ↑𝑚 (1...𝑁)) ↑𝑚
(0...(𝑁 − 1)))(𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑁) ≠ 0)) ↔ ∃𝑥 ∈ (((0...𝐾) ↑𝑚 (1...𝑁)) ↑𝑚
(0...(𝑁 − 1)))(𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾))))) |
350 | 194 | rexeqdv 3122 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 → (∃𝑗 ∈ ((0...𝑁) ∖ {𝑁})𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ↔ ∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶)) |
351 | 350 | biimpd 218 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → (∃𝑗 ∈ ((0...𝑁) ∖ {𝑁})𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 → ∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶)) |
352 | 351 | ralimdv 2946 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑁})𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 → ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶)) |
353 | 172 | rexeqdv 3122 |
. . . . . . . . . . . . . . . . . . 19
⊢
((2nd ‘𝑡) = 𝑁 → (∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ↔ ∃𝑗 ∈ ((0...𝑁) ∖ {𝑁})𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶)) |
354 | 353 | ralbidv 2969 |
. . . . . . . . . . . . . . . . . 18
⊢
((2nd ‘𝑡) = 𝑁 → (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ↔ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑁})𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶)) |
355 | 354 | imbi1d 330 |
. . . . . . . . . . . . . . . . 17
⊢
((2nd ‘𝑡) = 𝑁 → ((∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 → ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶) ↔ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑁})𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 → ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶))) |
356 | 352, 355 | syl5ibrcom 236 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → ((2nd
‘𝑡) = 𝑁 → (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 → ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶))) |
357 | 356 | com23 84 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 → ((2nd ‘𝑡) = 𝑁 → ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶))) |
358 | 357 | imp 444 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶) → ((2nd ‘𝑡) = 𝑁 → ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶)) |
359 | 358 | adantrd 483 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶) → (((2nd ‘𝑡) = 𝑁 ∧ (((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁)) → ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶)) |
360 | 359 | pm4.71rd 665 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶) → (((2nd ‘𝑡) = 𝑁 ∧ (((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁)) ↔ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((2nd ‘𝑡) = 𝑁 ∧ (((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))))) |
361 | | an12 834 |
. . . . . . . . . . . . 13
⊢
((∀𝑖 ∈
(0...(𝑁 −
1))∃𝑗 ∈
(0...(𝑁 − 1))𝑖 =
⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ((2nd ‘𝑡) = 𝑁 ∧ (((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))) ↔ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ (((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁)))) |
362 | | 3anass 1035 |
. . . . . . . . . . . . . 14
⊢
((∀𝑖 ∈
(0...(𝑁 −
1))∃𝑗 ∈
(0...(𝑁 − 1))𝑖 =
⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁) ↔ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ (((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))) |
363 | 362 | anbi2i 726 |
. . . . . . . . . . . . 13
⊢
(((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁)) ↔ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ (((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁)))) |
364 | 361, 363 | bitr4i 266 |
. . . . . . . . . . . 12
⊢
((∀𝑖 ∈
(0...(𝑁 −
1))∃𝑗 ∈
(0...(𝑁 − 1))𝑖 =
⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ((2nd ‘𝑡) = 𝑁 ∧ (((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))) ↔ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))) |
365 | 360, 364 | syl6bb 275 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶) → (((2nd ‘𝑡) = 𝑁 ∧ (((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁)) ↔ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁)))) |
366 | 365 | notbid 307 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶) → (¬ ((2nd
‘𝑡) = 𝑁 ∧ (((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁)) ↔ ¬ ((2nd
‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁)))) |
367 | 366 | pm5.32da 671 |
. . . . . . . . 9
⊢ (𝜑 → ((∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ¬ ((2nd ‘𝑡) = 𝑁 ∧ (((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))) ↔ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ¬ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))))) |
368 | 367 | adantr 480 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁))) → ((∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ¬ ((2nd ‘𝑡) = 𝑁 ∧ (((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))) ↔ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ¬ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))))) |
369 | 232, 349,
368 | 3bitr3d 297 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁))) → (∃𝑥 ∈ (((0...𝐾) ↑𝑚 (1...𝑁)) ↑𝑚
(0...(𝑁 − 1)))(𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾))) ↔ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ¬ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))))) |
370 | 369 | rabbidva 3163 |
. . . . . 6
⊢ (𝜑 → {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ∃𝑥 ∈ (((0...𝐾) ↑𝑚 (1...𝑁)) ↑𝑚
(0...(𝑁 − 1)))(𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)))} = {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ¬ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁)))}) |
371 | | iunrab 4503 |
. . . . . 6
⊢ ∪ 𝑥 ∈ (((0...𝐾) ↑𝑚 (1...𝑁)) ↑𝑚
(0...(𝑁 − 1))){𝑡 ∈ ((((0..^𝐾) ↑𝑚
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)))} = {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ∃𝑥 ∈ (((0...𝐾) ↑𝑚 (1...𝑁)) ↑𝑚
(0...(𝑁 − 1)))(𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)))} |
372 | | difrab 3860 |
. . . . . 6
⊢ ({𝑡 ∈ ((((0..^𝐾) ↑𝑚
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶} ∖ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))}) = {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ¬ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁)))} |
373 | 370, 371,
372 | 3eqtr4g 2669 |
. . . . 5
⊢ (𝜑 → ∪ 𝑥 ∈ (((0...𝐾) ↑𝑚 (1...𝑁)) ↑𝑚
(0...(𝑁 − 1))){𝑡 ∈ ((((0..^𝐾) ↑𝑚
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)))} = ({𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶} ∖ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))})) |
374 | 373 | fveq2d 6107 |
. . . 4
⊢ (𝜑 → (#‘∪ 𝑥 ∈ (((0...𝐾) ↑𝑚 (1...𝑁)) ↑𝑚
(0...(𝑁 − 1))){𝑡 ∈ ((((0..^𝐾) ↑𝑚
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)))}) = (#‘({𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶} ∖ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))}))) |
375 | 27, 28 | mp1i 13 |
. . . . 5
⊢ ((𝜑 ∧ 𝑥 ∈ (((0...𝐾) ↑𝑚 (1...𝑁)) ↑𝑚
(0...(𝑁 − 1))))
→ {𝑡 ∈
((((0..^𝐾)
↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)))} ∈ Fin) |
376 | | simpl 472 |
. . . . . . . . . . . 12
⊢ ((𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾))) → 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))) |
377 | 376 | a1i 11 |
. . . . . . . . . . 11
⊢ (𝑡 ∈ ((((0..^𝐾) ↑𝑚
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) → ((𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾))) → 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))))) |
378 | 377 | ss2rabi 3647 |
. . . . . . . . . 10
⊢ {𝑡 ∈ ((((0..^𝐾) ↑𝑚
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)))} ⊆ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} |
379 | 378 | sseli 3564 |
. . . . . . . . 9
⊢ (𝑠 ∈ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)))} → 𝑠 ∈ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))}) |
380 | | fveq2 6103 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑡 = 𝑠 → (2nd ‘𝑡) = (2nd ‘𝑠)) |
381 | 380 | breq2d 4595 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑡 = 𝑠 → (𝑦 < (2nd ‘𝑡) ↔ 𝑦 < (2nd ‘𝑠))) |
382 | 381 | ifbid 4058 |
. . . . . . . . . . . . . . . 16
⊢ (𝑡 = 𝑠 → if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) = if(𝑦 < (2nd ‘𝑠), 𝑦, (𝑦 + 1))) |
383 | 382 | csbeq1d 3506 |
. . . . . . . . . . . . . . 15
⊢ (𝑡 = 𝑠 → ⦋if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ⦋if(𝑦 < (2nd
‘𝑠), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) |
384 | | fveq2 6103 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑡 = 𝑠 → (1st ‘𝑡) = (1st ‘𝑠)) |
385 | 384 | fveq2d 6107 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑡 = 𝑠 → (1st
‘(1st ‘𝑡)) = (1st ‘(1st
‘𝑠))) |
386 | 384 | fveq2d 6107 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑡 = 𝑠 → (2nd
‘(1st ‘𝑡)) = (2nd ‘(1st
‘𝑠))) |
387 | 386 | imaeq1d 5384 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑡 = 𝑠 → ((2nd
‘(1st ‘𝑡)) “ (1...𝑗)) = ((2nd ‘(1st
‘𝑠)) “
(1...𝑗))) |
388 | 387 | xpeq1d 5062 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑡 = 𝑠 → (((2nd
‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) = (((2nd
‘(1st ‘𝑠)) “ (1...𝑗)) × {1})) |
389 | 386 | imaeq1d 5384 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑡 = 𝑠 → ((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) = ((2nd ‘(1st
‘𝑠)) “ ((𝑗 + 1)...𝑁))) |
390 | 389 | xpeq1d 5062 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑡 = 𝑠 → (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}) = (((2nd
‘(1st ‘𝑠)) “ ((𝑗 + 1)...𝑁)) × {0})) |
391 | 388, 390 | uneq12d 3730 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑡 = 𝑠 → ((((2nd
‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})) = ((((2nd
‘(1st ‘𝑠)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑠)) “ ((𝑗 + 1)...𝑁)) × {0}))) |
392 | 385, 391 | oveq12d 6567 |
. . . . . . . . . . . . . . . 16
⊢ (𝑡 = 𝑠 → ((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((1st
‘(1st ‘𝑠)) ∘𝑓 +
((((2nd ‘(1st ‘𝑠)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑠)) “ ((𝑗 + 1)...𝑁)) × {0})))) |
393 | 392 | csbeq2dv 3944 |
. . . . . . . . . . . . . . 15
⊢ (𝑡 = 𝑠 → ⦋if(𝑦 < (2nd ‘𝑠), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ⦋if(𝑦 < (2nd
‘𝑠), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑠)) ∘𝑓 +
((((2nd ‘(1st ‘𝑠)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑠)) “ ((𝑗 + 1)...𝑁)) × {0})))) |
394 | 383, 393 | eqtrd 2644 |
. . . . . . . . . . . . . 14
⊢ (𝑡 = 𝑠 → ⦋if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ⦋if(𝑦 < (2nd
‘𝑠), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑠)) ∘𝑓 +
((((2nd ‘(1st ‘𝑠)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑠)) “ ((𝑗 + 1)...𝑁)) × {0})))) |
395 | 394 | mpteq2dv 4673 |
. . . . . . . . . . . . 13
⊢ (𝑡 = 𝑠 → (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑠), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑠)) ∘𝑓 +
((((2nd ‘(1st ‘𝑠)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑠)) “ ((𝑗 + 1)...𝑁)) × {0}))))) |
396 | 395 | eqeq2d 2620 |
. . . . . . . . . . . 12
⊢ (𝑡 = 𝑠 → (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ↔ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑠), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑠)) ∘𝑓 +
((((2nd ‘(1st ‘𝑠)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑠)) “ ((𝑗 + 1)...𝑁)) × {0})))))) |
397 | | eqcom 2617 |
. . . . . . . . . . . 12
⊢ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑠), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑠)) ∘𝑓 +
((((2nd ‘(1st ‘𝑠)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑠)) “ ((𝑗 + 1)...𝑁)) × {0})))) ↔ (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑠), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑠)) ∘𝑓 +
((((2nd ‘(1st ‘𝑠)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑠)) “ ((𝑗 + 1)...𝑁)) × {0})))) = 𝑥) |
398 | 396, 397 | syl6bb 275 |
. . . . . . . . . . 11
⊢ (𝑡 = 𝑠 → (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ↔ (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑠), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑠)) ∘𝑓 +
((((2nd ‘(1st ‘𝑠)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑠)) “ ((𝑗 + 1)...𝑁)) × {0})))) = 𝑥)) |
399 | 398 | elrab 3331 |
. . . . . . . . . 10
⊢ (𝑠 ∈ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} ↔ (𝑠 ∈ ((((0..^𝐾) ↑𝑚
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∧ (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑠), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑠)) ∘𝑓 +
((((2nd ‘(1st ‘𝑠)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑠)) “ ((𝑗 + 1)...𝑁)) × {0})))) = 𝑥)) |
400 | 399 | simprbi 479 |
. . . . . . . . 9
⊢ (𝑠 ∈ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} → (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑠), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑠)) ∘𝑓 +
((((2nd ‘(1st ‘𝑠)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑠)) “ ((𝑗 + 1)...𝑁)) × {0})))) = 𝑥) |
401 | 379, 400 | syl 17 |
. . . . . . . 8
⊢ (𝑠 ∈ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)))} → (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑠), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑠)) ∘𝑓 +
((((2nd ‘(1st ‘𝑠)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑠)) “ ((𝑗 + 1)...𝑁)) × {0})))) = 𝑥) |
402 | 401 | rgen 2906 |
. . . . . . 7
⊢
∀𝑠 ∈
{𝑡 ∈ ((((0..^𝐾) ↑𝑚
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)))} (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑠), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑠)) ∘𝑓 +
((((2nd ‘(1st ‘𝑠)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑠)) “ ((𝑗 + 1)...𝑁)) × {0})))) = 𝑥 |
403 | 402 | rgenw 2908 |
. . . . . 6
⊢
∀𝑥 ∈
(((0...𝐾)
↑𝑚 (1...𝑁)) ↑𝑚 (0...(𝑁 − 1)))∀𝑠 ∈ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)))} (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑠), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑠)) ∘𝑓 +
((((2nd ‘(1st ‘𝑠)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑠)) “ ((𝑗 + 1)...𝑁)) × {0})))) = 𝑥 |
404 | | invdisj 4571 |
. . . . . 6
⊢
(∀𝑥 ∈
(((0...𝐾)
↑𝑚 (1...𝑁)) ↑𝑚 (0...(𝑁 − 1)))∀𝑠 ∈ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)))} (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑠), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑠)) ∘𝑓 +
((((2nd ‘(1st ‘𝑠)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑠)) “ ((𝑗 + 1)...𝑁)) × {0})))) = 𝑥 → Disj 𝑥 ∈ (((0...𝐾) ↑𝑚 (1...𝑁)) ↑𝑚
(0...(𝑁 − 1))){𝑡 ∈ ((((0..^𝐾) ↑𝑚
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)))}) |
405 | 403, 404 | mp1i 13 |
. . . . 5
⊢ (𝜑 → Disj 𝑥 ∈ (((0...𝐾) ↑𝑚 (1...𝑁)) ↑𝑚
(0...(𝑁 − 1))){𝑡 ∈ ((((0..^𝐾) ↑𝑚
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)))}) |
406 | 8, 375, 405 | hashiun 14395 |
. . . 4
⊢ (𝜑 → (#‘∪ 𝑥 ∈ (((0...𝐾) ↑𝑚 (1...𝑁)) ↑𝑚
(0...(𝑁 − 1))){𝑡 ∈ ((((0..^𝐾) ↑𝑚
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)))}) = Σ𝑥 ∈ (((0...𝐾) ↑𝑚 (1...𝑁)) ↑𝑚
(0...(𝑁 −
1)))(#‘{𝑡 ∈
((((0..^𝐾)
↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)))})) |
407 | 374, 406 | eqtr3d 2646 |
. . 3
⊢ (𝜑 → (#‘({𝑡 ∈ ((((0..^𝐾) ↑𝑚
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶} ∖ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))})) = Σ𝑥 ∈ (((0...𝐾) ↑𝑚 (1...𝑁)) ↑𝑚
(0...(𝑁 −
1)))(#‘{𝑡 ∈
((((0..^𝐾)
↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)))})) |
408 | | fo1st 7079 |
. . . . . . . . . . . . 13
⊢
1st :V–onto→V |
409 | | fofun 6029 |
. . . . . . . . . . . . 13
⊢
(1st :V–onto→V → Fun 1st ) |
410 | 408, 409 | ax-mp 5 |
. . . . . . . . . . . 12
⊢ Fun
1st |
411 | | ssv 3588 |
. . . . . . . . . . . . 13
⊢ {𝑡 ∈ ((((0..^𝐾) ↑𝑚
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))} ⊆ V |
412 | | fof 6028 |
. . . . . . . . . . . . . . 15
⊢
(1st :V–onto→V → 1st
:V⟶V) |
413 | 408, 412 | ax-mp 5 |
. . . . . . . . . . . . . 14
⊢
1st :V⟶V |
414 | 413 | fdmi 5965 |
. . . . . . . . . . . . 13
⊢ dom
1st = V |
415 | 411, 414 | sseqtr4i 3601 |
. . . . . . . . . . . 12
⊢ {𝑡 ∈ ((((0..^𝐾) ↑𝑚
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))} ⊆ dom
1st |
416 | | fores 6037 |
. . . . . . . . . . . 12
⊢ ((Fun
1st ∧ {𝑡
∈ ((((0..^𝐾)
↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))} ⊆ dom 1st ) →
(1st ↾ {𝑡
∈ ((((0..^𝐾)
↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))}):{𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))}–onto→(1st “ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))})) |
417 | 410, 415,
416 | mp2an 704 |
. . . . . . . . . . 11
⊢
(1st ↾ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))}):{𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))}–onto→(1st “ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))}) |
418 | | fveq2 6103 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑡 = 𝑥 → (2nd ‘𝑡) = (2nd ‘𝑥)) |
419 | 418 | eqeq1d 2612 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑡 = 𝑥 → ((2nd ‘𝑡) = 𝑁 ↔ (2nd ‘𝑥) = 𝑁)) |
420 | | fveq2 6103 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑡 = 𝑥 → (1st ‘𝑡) = (1st ‘𝑥)) |
421 | 420 | csbeq1d 3506 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑡 = 𝑥 → ⦋(1st
‘𝑡) / 𝑠⦌𝐶 = ⦋(1st
‘𝑥) / 𝑠⦌𝐶) |
422 | 421 | eqeq2d 2620 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑡 = 𝑥 → (𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ↔ 𝑖 = ⦋(1st
‘𝑥) / 𝑠⦌𝐶)) |
423 | 422 | rexbidv 3034 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑡 = 𝑥 → (∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ↔ ∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑥) / 𝑠⦌𝐶)) |
424 | 423 | ralbidv 2969 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑡 = 𝑥 → (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ↔ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑥) / 𝑠⦌𝐶)) |
425 | 420 | fveq2d 6107 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑡 = 𝑥 → (1st
‘(1st ‘𝑡)) = (1st ‘(1st
‘𝑥))) |
426 | 425 | fveq1d 6105 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑡 = 𝑥 → ((1st
‘(1st ‘𝑡))‘𝑁) = ((1st ‘(1st
‘𝑥))‘𝑁)) |
427 | 426 | eqeq1d 2612 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑡 = 𝑥 → (((1st
‘(1st ‘𝑡))‘𝑁) = 0 ↔ ((1st
‘(1st ‘𝑥))‘𝑁) = 0)) |
428 | 420 | fveq2d 6107 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑡 = 𝑥 → (2nd
‘(1st ‘𝑡)) = (2nd ‘(1st
‘𝑥))) |
429 | 428 | fveq1d 6105 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑡 = 𝑥 → ((2nd
‘(1st ‘𝑡))‘𝑁) = ((2nd ‘(1st
‘𝑥))‘𝑁)) |
430 | 429 | eqeq1d 2612 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑡 = 𝑥 → (((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁 ↔ ((2nd
‘(1st ‘𝑥))‘𝑁) = 𝑁)) |
431 | 424, 427,
430 | 3anbi123d 1391 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑡 = 𝑥 → ((∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁) ↔ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑥) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑥))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑥))‘𝑁) = 𝑁))) |
432 | 419, 431 | anbi12d 743 |
. . . . . . . . . . . . . . . 16
⊢ (𝑡 = 𝑥 → (((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁)) ↔ ((2nd ‘𝑥) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑥) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑥))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑥))‘𝑁) = 𝑁)))) |
433 | 432 | rexrab 3337 |
. . . . . . . . . . . . . . 15
⊢
(∃𝑥 ∈
{𝑡 ∈ ((((0..^𝐾) ↑𝑚
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))} (1st ‘𝑥) = 𝑠 ↔ ∃𝑥 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁))(((2nd ‘𝑥) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑥) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑥))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑥))‘𝑁) = 𝑁)) ∧ (1st ‘𝑥) = 𝑠)) |
434 | | xp1st 7089 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑥 ∈ ((((0..^𝐾) ↑𝑚
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) → (1st ‘𝑥) ∈ (((0..^𝐾) ↑𝑚
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})) |
435 | 434 | anim1i 590 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑥 ∈ ((((0..^𝐾) ↑𝑚
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑥) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑥))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑥))‘𝑁) = 𝑁)) → ((1st ‘𝑥) ∈ (((0..^𝐾) ↑𝑚
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑥) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑥))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑥))‘𝑁) = 𝑁))) |
436 | | eleq1 2676 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
((1st ‘𝑥) = 𝑠 → ((1st ‘𝑥) ∈ (((0..^𝐾) ↑𝑚
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ↔ 𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}))) |
437 | | csbeq1a 3508 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (𝑠 = (1st ‘𝑥) → 𝐶 = ⦋(1st
‘𝑥) / 𝑠⦌𝐶) |
438 | 437 | eqcoms 2618 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢
((1st ‘𝑥) = 𝑠 → 𝐶 = ⦋(1st
‘𝑥) / 𝑠⦌𝐶) |
439 | 438 | eqcomd 2616 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢
((1st ‘𝑥) = 𝑠 → ⦋(1st
‘𝑥) / 𝑠⦌𝐶 = 𝐶) |
440 | 439 | eqeq2d 2620 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢
((1st ‘𝑥) = 𝑠 → (𝑖 = ⦋(1st
‘𝑥) / 𝑠⦌𝐶 ↔ 𝑖 = 𝐶)) |
441 | 440 | rexbidv 3034 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢
((1st ‘𝑥) = 𝑠 → (∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑥) / 𝑠⦌𝐶 ↔ ∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶)) |
442 | 441 | ralbidv 2969 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
((1st ‘𝑥) = 𝑠 → (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑥) / 𝑠⦌𝐶 ↔ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶)) |
443 | | fveq2 6103 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢
((1st ‘𝑥) = 𝑠 → (1st
‘(1st ‘𝑥)) = (1st ‘𝑠)) |
444 | 443 | fveq1d 6105 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢
((1st ‘𝑥) = 𝑠 → ((1st
‘(1st ‘𝑥))‘𝑁) = ((1st ‘𝑠)‘𝑁)) |
445 | 444 | eqeq1d 2612 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
((1st ‘𝑥) = 𝑠 → (((1st
‘(1st ‘𝑥))‘𝑁) = 0 ↔ ((1st ‘𝑠)‘𝑁) = 0)) |
446 | | fveq2 6103 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢
((1st ‘𝑥) = 𝑠 → (2nd
‘(1st ‘𝑥)) = (2nd ‘𝑠)) |
447 | 446 | fveq1d 6105 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢
((1st ‘𝑥) = 𝑠 → ((2nd
‘(1st ‘𝑥))‘𝑁) = ((2nd ‘𝑠)‘𝑁)) |
448 | 447 | eqeq1d 2612 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
((1st ‘𝑥) = 𝑠 → (((2nd
‘(1st ‘𝑥))‘𝑁) = 𝑁 ↔ ((2nd ‘𝑠)‘𝑁) = 𝑁)) |
449 | 442, 445,
448 | 3anbi123d 1391 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
((1st ‘𝑥) = 𝑠 → ((∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑥) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑥))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑥))‘𝑁) = 𝑁) ↔ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶 ∧ ((1st ‘𝑠)‘𝑁) = 0 ∧ ((2nd ‘𝑠)‘𝑁) = 𝑁))) |
450 | 436, 449 | anbi12d 743 |
. . . . . . . . . . . . . . . . . . . 20
⊢
((1st ‘𝑥) = 𝑠 → (((1st ‘𝑥) ∈ (((0..^𝐾) ↑𝑚
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑥) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑥))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑥))‘𝑁) = 𝑁)) ↔ (𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶 ∧ ((1st ‘𝑠)‘𝑁) = 0 ∧ ((2nd ‘𝑠)‘𝑁) = 𝑁)))) |
451 | 435, 450 | syl5ibcom 234 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑥 ∈ ((((0..^𝐾) ↑𝑚
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑥) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑥))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑥))‘𝑁) = 𝑁)) → ((1st ‘𝑥) = 𝑠 → (𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶 ∧ ((1st ‘𝑠)‘𝑁) = 0 ∧ ((2nd ‘𝑠)‘𝑁) = 𝑁)))) |
452 | 451 | adantrl 748 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑥 ∈ ((((0..^𝐾) ↑𝑚
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∧ ((2nd ‘𝑥) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑥) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑥))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑥))‘𝑁) = 𝑁))) → ((1st ‘𝑥) = 𝑠 → (𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶 ∧ ((1st ‘𝑠)‘𝑁) = 0 ∧ ((2nd ‘𝑠)‘𝑁) = 𝑁)))) |
453 | 452 | expimpd 627 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑥 ∈ ((((0..^𝐾) ↑𝑚
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) → ((((2nd ‘𝑥) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑥) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑥))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑥))‘𝑁) = 𝑁)) ∧ (1st ‘𝑥) = 𝑠) → (𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶 ∧ ((1st ‘𝑠)‘𝑁) = 0 ∧ ((2nd ‘𝑠)‘𝑁) = 𝑁)))) |
454 | 453 | rexlimiv 3009 |
. . . . . . . . . . . . . . . 16
⊢
(∃𝑥 ∈
((((0..^𝐾)
↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁))(((2nd ‘𝑥) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑥) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑥))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑥))‘𝑁) = 𝑁)) ∧ (1st ‘𝑥) = 𝑠) → (𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶 ∧ ((1st ‘𝑠)‘𝑁) = 0 ∧ ((2nd ‘𝑠)‘𝑁) = 𝑁))) |
455 | | nn0fz0 12306 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑁 ∈ ℕ0
↔ 𝑁 ∈ (0...𝑁)) |
456 | 174, 455 | sylib 207 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝜑 → 𝑁 ∈ (0...𝑁)) |
457 | | opelxpi 5072 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ 𝑁 ∈ (0...𝑁)) → 〈𝑠, 𝑁〉 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁))) |
458 | 456, 457 | sylan2 490 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ 𝜑) → 〈𝑠, 𝑁〉 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁))) |
459 | 458 | ancoms 468 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})) → 〈𝑠, 𝑁〉 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁))) |
460 | | opelxp2 5075 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
(〈𝑠, 𝑁〉 ∈ ((((0..^𝐾) ↑𝑚
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) → 𝑁 ∈ (0...𝑁)) |
461 | | op2ndg 7072 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝑠 ∈ V ∧ 𝑁 ∈ (0...𝑁)) → (2nd ‘〈𝑠, 𝑁〉) = 𝑁) |
462 | 461 | biantrurd 528 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝑠 ∈ V ∧ 𝑁 ∈ (0...𝑁)) → ((∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘〈𝑠, 𝑁〉) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘〈𝑠, 𝑁〉))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘〈𝑠, 𝑁〉))‘𝑁) = 𝑁) ↔ ((2nd ‘〈𝑠, 𝑁〉) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘〈𝑠, 𝑁〉) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘〈𝑠, 𝑁〉))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘〈𝑠, 𝑁〉))‘𝑁) = 𝑁)))) |
463 | | op1stg 7071 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ ((𝑠 ∈ V ∧ 𝑁 ∈ (0...𝑁)) → (1st ‘〈𝑠, 𝑁〉) = 𝑠) |
464 | | csbeq1a 3508 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ (𝑠 = (1st
‘〈𝑠, 𝑁〉) → 𝐶 = ⦋(1st
‘〈𝑠, 𝑁〉) / 𝑠⦌𝐶) |
465 | 464 | eqcoms 2618 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢
((1st ‘〈𝑠, 𝑁〉) = 𝑠 → 𝐶 = ⦋(1st
‘〈𝑠, 𝑁〉) / 𝑠⦌𝐶) |
466 | 465 | eqcomd 2616 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢
((1st ‘〈𝑠, 𝑁〉) = 𝑠 → ⦋(1st
‘〈𝑠, 𝑁〉) / 𝑠⦌𝐶 = 𝐶) |
467 | 463, 466 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝑠 ∈ V ∧ 𝑁 ∈ (0...𝑁)) → ⦋(1st
‘〈𝑠, 𝑁〉) / 𝑠⦌𝐶 = 𝐶) |
468 | 467 | eqeq2d 2620 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝑠 ∈ V ∧ 𝑁 ∈ (0...𝑁)) → (𝑖 = ⦋(1st
‘〈𝑠, 𝑁〉) / 𝑠⦌𝐶 ↔ 𝑖 = 𝐶)) |
469 | 468 | rexbidv 3034 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝑠 ∈ V ∧ 𝑁 ∈ (0...𝑁)) → (∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘〈𝑠, 𝑁〉) / 𝑠⦌𝐶 ↔ ∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶)) |
470 | 469 | ralbidv 2969 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝑠 ∈ V ∧ 𝑁 ∈ (0...𝑁)) → (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘〈𝑠, 𝑁〉) / 𝑠⦌𝐶 ↔ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶)) |
471 | 463 | fveq2d 6107 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝑠 ∈ V ∧ 𝑁 ∈ (0...𝑁)) → (1st
‘(1st ‘〈𝑠, 𝑁〉)) = (1st ‘𝑠)) |
472 | 471 | fveq1d 6105 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝑠 ∈ V ∧ 𝑁 ∈ (0...𝑁)) → ((1st
‘(1st ‘〈𝑠, 𝑁〉))‘𝑁) = ((1st ‘𝑠)‘𝑁)) |
473 | 472 | eqeq1d 2612 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝑠 ∈ V ∧ 𝑁 ∈ (0...𝑁)) → (((1st
‘(1st ‘〈𝑠, 𝑁〉))‘𝑁) = 0 ↔ ((1st ‘𝑠)‘𝑁) = 0)) |
474 | 463 | fveq2d 6107 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝑠 ∈ V ∧ 𝑁 ∈ (0...𝑁)) → (2nd
‘(1st ‘〈𝑠, 𝑁〉)) = (2nd ‘𝑠)) |
475 | 474 | fveq1d 6105 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝑠 ∈ V ∧ 𝑁 ∈ (0...𝑁)) → ((2nd
‘(1st ‘〈𝑠, 𝑁〉))‘𝑁) = ((2nd ‘𝑠)‘𝑁)) |
476 | 475 | eqeq1d 2612 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝑠 ∈ V ∧ 𝑁 ∈ (0...𝑁)) → (((2nd
‘(1st ‘〈𝑠, 𝑁〉))‘𝑁) = 𝑁 ↔ ((2nd ‘𝑠)‘𝑁) = 𝑁)) |
477 | 470, 473,
476 | 3anbi123d 1391 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝑠 ∈ V ∧ 𝑁 ∈ (0...𝑁)) → ((∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘〈𝑠, 𝑁〉) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘〈𝑠, 𝑁〉))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘〈𝑠, 𝑁〉))‘𝑁) = 𝑁) ↔ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶 ∧ ((1st ‘𝑠)‘𝑁) = 0 ∧ ((2nd ‘𝑠)‘𝑁) = 𝑁))) |
478 | 463 | biantrud 527 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝑠 ∈ V ∧ 𝑁 ∈ (0...𝑁)) → (((2nd
‘〈𝑠, 𝑁〉) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘〈𝑠, 𝑁〉) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘〈𝑠, 𝑁〉))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘〈𝑠, 𝑁〉))‘𝑁) = 𝑁)) ↔ (((2nd
‘〈𝑠, 𝑁〉) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘〈𝑠, 𝑁〉) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘〈𝑠, 𝑁〉))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘〈𝑠, 𝑁〉))‘𝑁) = 𝑁)) ∧ (1st ‘〈𝑠, 𝑁〉) = 𝑠))) |
479 | 462, 477,
478 | 3bitr3d 297 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝑠 ∈ V ∧ 𝑁 ∈ (0...𝑁)) → ((∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶 ∧ ((1st ‘𝑠)‘𝑁) = 0 ∧ ((2nd ‘𝑠)‘𝑁) = 𝑁) ↔ (((2nd
‘〈𝑠, 𝑁〉) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘〈𝑠, 𝑁〉) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘〈𝑠, 𝑁〉))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘〈𝑠, 𝑁〉))‘𝑁) = 𝑁)) ∧ (1st ‘〈𝑠, 𝑁〉) = 𝑠))) |
480 | 44, 460, 479 | sylancr 694 |
. . . . . . . . . . . . . . . . . . . 20
⊢
(〈𝑠, 𝑁〉 ∈ ((((0..^𝐾) ↑𝑚
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) → ((∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶 ∧ ((1st ‘𝑠)‘𝑁) = 0 ∧ ((2nd ‘𝑠)‘𝑁) = 𝑁) ↔ (((2nd
‘〈𝑠, 𝑁〉) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘〈𝑠, 𝑁〉) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘〈𝑠, 𝑁〉))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘〈𝑠, 𝑁〉))‘𝑁) = 𝑁)) ∧ (1st ‘〈𝑠, 𝑁〉) = 𝑠))) |
481 | 480 | biimpa 500 |
. . . . . . . . . . . . . . . . . . 19
⊢
((〈𝑠, 𝑁〉 ∈ ((((0..^𝐾) ↑𝑚
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶 ∧ ((1st ‘𝑠)‘𝑁) = 0 ∧ ((2nd ‘𝑠)‘𝑁) = 𝑁)) → (((2nd
‘〈𝑠, 𝑁〉) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘〈𝑠, 𝑁〉) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘〈𝑠, 𝑁〉))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘〈𝑠, 𝑁〉))‘𝑁) = 𝑁)) ∧ (1st ‘〈𝑠, 𝑁〉) = 𝑠)) |
482 | | fveq2 6103 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑥 = 〈𝑠, 𝑁〉 → (2nd ‘𝑥) = (2nd
‘〈𝑠, 𝑁〉)) |
483 | 482 | eqeq1d 2612 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑥 = 〈𝑠, 𝑁〉 → ((2nd ‘𝑥) = 𝑁 ↔ (2nd ‘〈𝑠, 𝑁〉) = 𝑁)) |
484 | | fveq2 6103 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (𝑥 = 〈𝑠, 𝑁〉 → (1st ‘𝑥) = (1st
‘〈𝑠, 𝑁〉)) |
485 | 484 | csbeq1d 3506 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝑥 = 〈𝑠, 𝑁〉 → ⦋(1st
‘𝑥) / 𝑠⦌𝐶 = ⦋(1st
‘〈𝑠, 𝑁〉) / 𝑠⦌𝐶) |
486 | 485 | eqeq2d 2620 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑥 = 〈𝑠, 𝑁〉 → (𝑖 = ⦋(1st
‘𝑥) / 𝑠⦌𝐶 ↔ 𝑖 = ⦋(1st
‘〈𝑠, 𝑁〉) / 𝑠⦌𝐶)) |
487 | 486 | rexbidv 3034 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑥 = 〈𝑠, 𝑁〉 → (∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑥) / 𝑠⦌𝐶 ↔ ∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘〈𝑠, 𝑁〉) / 𝑠⦌𝐶)) |
488 | 487 | ralbidv 2969 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑥 = 〈𝑠, 𝑁〉 → (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑥) / 𝑠⦌𝐶 ↔ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘〈𝑠, 𝑁〉) / 𝑠⦌𝐶)) |
489 | 484 | fveq2d 6107 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑥 = 〈𝑠, 𝑁〉 → (1st
‘(1st ‘𝑥)) = (1st ‘(1st
‘〈𝑠, 𝑁〉))) |
490 | 489 | fveq1d 6105 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑥 = 〈𝑠, 𝑁〉 → ((1st
‘(1st ‘𝑥))‘𝑁) = ((1st ‘(1st
‘〈𝑠, 𝑁〉))‘𝑁)) |
491 | 490 | eqeq1d 2612 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑥 = 〈𝑠, 𝑁〉 → (((1st
‘(1st ‘𝑥))‘𝑁) = 0 ↔ ((1st
‘(1st ‘〈𝑠, 𝑁〉))‘𝑁) = 0)) |
492 | 484 | fveq2d 6107 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑥 = 〈𝑠, 𝑁〉 → (2nd
‘(1st ‘𝑥)) = (2nd ‘(1st
‘〈𝑠, 𝑁〉))) |
493 | 492 | fveq1d 6105 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑥 = 〈𝑠, 𝑁〉 → ((2nd
‘(1st ‘𝑥))‘𝑁) = ((2nd ‘(1st
‘〈𝑠, 𝑁〉))‘𝑁)) |
494 | 493 | eqeq1d 2612 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑥 = 〈𝑠, 𝑁〉 → (((2nd
‘(1st ‘𝑥))‘𝑁) = 𝑁 ↔ ((2nd
‘(1st ‘〈𝑠, 𝑁〉))‘𝑁) = 𝑁)) |
495 | 488, 491,
494 | 3anbi123d 1391 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑥 = 〈𝑠, 𝑁〉 → ((∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑥) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑥))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑥))‘𝑁) = 𝑁) ↔ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘〈𝑠, 𝑁〉) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘〈𝑠, 𝑁〉))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘〈𝑠, 𝑁〉))‘𝑁) = 𝑁))) |
496 | 483, 495 | anbi12d 743 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑥 = 〈𝑠, 𝑁〉 → (((2nd ‘𝑥) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑥) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑥))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑥))‘𝑁) = 𝑁)) ↔ ((2nd
‘〈𝑠, 𝑁〉) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘〈𝑠, 𝑁〉) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘〈𝑠, 𝑁〉))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘〈𝑠, 𝑁〉))‘𝑁) = 𝑁)))) |
497 | 484 | eqeq1d 2612 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑥 = 〈𝑠, 𝑁〉 → ((1st ‘𝑥) = 𝑠 ↔ (1st ‘〈𝑠, 𝑁〉) = 𝑠)) |
498 | 496, 497 | anbi12d 743 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑥 = 〈𝑠, 𝑁〉 → ((((2nd
‘𝑥) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑥) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑥))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑥))‘𝑁) = 𝑁)) ∧ (1st ‘𝑥) = 𝑠) ↔ (((2nd ‘〈𝑠, 𝑁〉) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘〈𝑠, 𝑁〉) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘〈𝑠, 𝑁〉))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘〈𝑠, 𝑁〉))‘𝑁) = 𝑁)) ∧ (1st ‘〈𝑠, 𝑁〉) = 𝑠))) |
499 | 498 | rspcev 3282 |
. . . . . . . . . . . . . . . . . . 19
⊢
((〈𝑠, 𝑁〉 ∈ ((((0..^𝐾) ↑𝑚
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∧ (((2nd
‘〈𝑠, 𝑁〉) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘〈𝑠, 𝑁〉) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘〈𝑠, 𝑁〉))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘〈𝑠, 𝑁〉))‘𝑁) = 𝑁)) ∧ (1st ‘〈𝑠, 𝑁〉) = 𝑠)) → ∃𝑥 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁))(((2nd ‘𝑥) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑥) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑥))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑥))‘𝑁) = 𝑁)) ∧ (1st ‘𝑥) = 𝑠)) |
500 | 481, 499 | syldan 486 |
. . . . . . . . . . . . . . . . . 18
⊢
((〈𝑠, 𝑁〉 ∈ ((((0..^𝐾) ↑𝑚
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶 ∧ ((1st ‘𝑠)‘𝑁) = 0 ∧ ((2nd ‘𝑠)‘𝑁) = 𝑁)) → ∃𝑥 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁))(((2nd ‘𝑥) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑥) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑥))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑥))‘𝑁) = 𝑁)) ∧ (1st ‘𝑥) = 𝑠)) |
501 | 459, 500 | sylan 487 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})) ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶 ∧ ((1st ‘𝑠)‘𝑁) = 0 ∧ ((2nd ‘𝑠)‘𝑁) = 𝑁)) → ∃𝑥 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁))(((2nd ‘𝑥) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑥) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑥))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑥))‘𝑁) = 𝑁)) ∧ (1st ‘𝑥) = 𝑠)) |
502 | 501 | expl 646 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → ((𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶 ∧ ((1st ‘𝑠)‘𝑁) = 0 ∧ ((2nd ‘𝑠)‘𝑁) = 𝑁)) → ∃𝑥 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁))(((2nd ‘𝑥) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑥) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑥))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑥))‘𝑁) = 𝑁)) ∧ (1st ‘𝑥) = 𝑠))) |
503 | 454, 502 | impbid2 215 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → (∃𝑥 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁))(((2nd ‘𝑥) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑥) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑥))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑥))‘𝑁) = 𝑁)) ∧ (1st ‘𝑥) = 𝑠) ↔ (𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶 ∧ ((1st ‘𝑠)‘𝑁) = 0 ∧ ((2nd ‘𝑠)‘𝑁) = 𝑁)))) |
504 | 433, 503 | syl5bb 271 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → (∃𝑥 ∈ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))} (1st ‘𝑥) = 𝑠 ↔ (𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶 ∧ ((1st ‘𝑠)‘𝑁) = 0 ∧ ((2nd ‘𝑠)‘𝑁) = 𝑁)))) |
505 | 504 | abbidv 2728 |
. . . . . . . . . . . . 13
⊢ (𝜑 → {𝑠 ∣ ∃𝑥 ∈ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))} (1st ‘𝑥) = 𝑠} = {𝑠 ∣ (𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶 ∧ ((1st ‘𝑠)‘𝑁) = 0 ∧ ((2nd ‘𝑠)‘𝑁) = 𝑁))}) |
506 | | dfimafn 6155 |
. . . . . . . . . . . . . . 15
⊢ ((Fun
1st ∧ {𝑡
∈ ((((0..^𝐾)
↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))} ⊆ dom 1st ) →
(1st “ {𝑡
∈ ((((0..^𝐾)
↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))}) = {𝑦 ∣ ∃𝑥 ∈ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))} (1st ‘𝑥) = 𝑦}) |
507 | 410, 415,
506 | mp2an 704 |
. . . . . . . . . . . . . 14
⊢
(1st “ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))}) = {𝑦 ∣ ∃𝑥 ∈ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))} (1st ‘𝑥) = 𝑦} |
508 | | nfv 1830 |
. . . . . . . . . . . . . . . . . 18
⊢
Ⅎ𝑠(2nd ‘𝑡) = 𝑁 |
509 | | nfcv 2751 |
. . . . . . . . . . . . . . . . . . . 20
⊢
Ⅎ𝑠(0...(𝑁 − 1)) |
510 | | nfcsb1v 3515 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
Ⅎ𝑠⦋(1st ‘𝑡) / 𝑠⦌𝐶 |
511 | 510 | nfeq2 2766 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
Ⅎ𝑠 𝑖 =
⦋(1st ‘𝑡) / 𝑠⦌𝐶 |
512 | 509, 511 | nfrex 2990 |
. . . . . . . . . . . . . . . . . . . 20
⊢
Ⅎ𝑠∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 |
513 | 509, 512 | nfral 2929 |
. . . . . . . . . . . . . . . . . . 19
⊢
Ⅎ𝑠∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 |
514 | | nfv 1830 |
. . . . . . . . . . . . . . . . . . 19
⊢
Ⅎ𝑠((1st ‘(1st
‘𝑡))‘𝑁) = 0 |
515 | | nfv 1830 |
. . . . . . . . . . . . . . . . . . 19
⊢
Ⅎ𝑠((2nd ‘(1st
‘𝑡))‘𝑁) = 𝑁 |
516 | 513, 514,
515 | nf3an 1819 |
. . . . . . . . . . . . . . . . . 18
⊢
Ⅎ𝑠(∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁) |
517 | 508, 516 | nfan 1816 |
. . . . . . . . . . . . . . . . 17
⊢
Ⅎ𝑠((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁)) |
518 | | nfcv 2751 |
. . . . . . . . . . . . . . . . 17
⊢
Ⅎ𝑠((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) |
519 | 517, 518 | nfrab 3100 |
. . . . . . . . . . . . . . . 16
⊢
Ⅎ𝑠{𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))} |
520 | | nfv 1830 |
. . . . . . . . . . . . . . . 16
⊢
Ⅎ𝑠(1st ‘𝑥) = 𝑦 |
521 | 519, 520 | nfrex 2990 |
. . . . . . . . . . . . . . 15
⊢
Ⅎ𝑠∃𝑥 ∈ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))} (1st ‘𝑥) = 𝑦 |
522 | | nfv 1830 |
. . . . . . . . . . . . . . 15
⊢
Ⅎ𝑦∃𝑥 ∈ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))} (1st ‘𝑥) = 𝑠 |
523 | | eqeq2 2621 |
. . . . . . . . . . . . . . . 16
⊢ (𝑦 = 𝑠 → ((1st ‘𝑥) = 𝑦 ↔ (1st ‘𝑥) = 𝑠)) |
524 | 523 | rexbidv 3034 |
. . . . . . . . . . . . . . 15
⊢ (𝑦 = 𝑠 → (∃𝑥 ∈ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))} (1st ‘𝑥) = 𝑦 ↔ ∃𝑥 ∈ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))} (1st ‘𝑥) = 𝑠)) |
525 | 521, 522,
524 | cbvab 2733 |
. . . . . . . . . . . . . 14
⊢ {𝑦 ∣ ∃𝑥 ∈ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))} (1st ‘𝑥) = 𝑦} = {𝑠 ∣ ∃𝑥 ∈ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))} (1st ‘𝑥) = 𝑠} |
526 | 507, 525 | eqtri 2632 |
. . . . . . . . . . . . 13
⊢
(1st “ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))}) = {𝑠 ∣ ∃𝑥 ∈ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))} (1st ‘𝑥) = 𝑠} |
527 | | df-rab 2905 |
. . . . . . . . . . . . 13
⊢ {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶 ∧ ((1st ‘𝑠)‘𝑁) = 0 ∧ ((2nd ‘𝑠)‘𝑁) = 𝑁)} = {𝑠 ∣ (𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶 ∧ ((1st ‘𝑠)‘𝑁) = 0 ∧ ((2nd ‘𝑠)‘𝑁) = 𝑁))} |
528 | 505, 526,
527 | 3eqtr4g 2669 |
. . . . . . . . . . . 12
⊢ (𝜑 → (1st “
{𝑡 ∈ ((((0..^𝐾) ↑𝑚
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))}) = {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶 ∧ ((1st ‘𝑠)‘𝑁) = 0 ∧ ((2nd ‘𝑠)‘𝑁) = 𝑁)}) |
529 | | foeq3 6026 |
. . . . . . . . . . . 12
⊢
((1st “ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))}) = {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶 ∧ ((1st ‘𝑠)‘𝑁) = 0 ∧ ((2nd ‘𝑠)‘𝑁) = 𝑁)} → ((1st ↾ {𝑡 ∈ ((((0..^𝐾) ↑𝑚
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))}):{𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))}–onto→(1st “ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))}) ↔ (1st ↾ {𝑡 ∈ ((((0..^𝐾) ↑𝑚
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))}):{𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))}–onto→{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶 ∧ ((1st ‘𝑠)‘𝑁) = 0 ∧ ((2nd ‘𝑠)‘𝑁) = 𝑁)})) |
530 | 528, 529 | syl 17 |
. . . . . . . . . . 11
⊢ (𝜑 → ((1st ↾
{𝑡 ∈ ((((0..^𝐾) ↑𝑚
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))}):{𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))}–onto→(1st “ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))}) ↔ (1st ↾ {𝑡 ∈ ((((0..^𝐾) ↑𝑚
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))}):{𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))}–onto→{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶 ∧ ((1st ‘𝑠)‘𝑁) = 0 ∧ ((2nd ‘𝑠)‘𝑁) = 𝑁)})) |
531 | 417, 530 | mpbii 222 |
. . . . . . . . . 10
⊢ (𝜑 → (1st ↾
{𝑡 ∈ ((((0..^𝐾) ↑𝑚
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))}):{𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))}–onto→{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶 ∧ ((1st ‘𝑠)‘𝑁) = 0 ∧ ((2nd ‘𝑠)‘𝑁) = 𝑁)}) |
532 | | fof 6028 |
. . . . . . . . . 10
⊢
((1st ↾ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))}):{𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))}–onto→{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶 ∧ ((1st ‘𝑠)‘𝑁) = 0 ∧ ((2nd ‘𝑠)‘𝑁) = 𝑁)} → (1st ↾ {𝑡 ∈ ((((0..^𝐾) ↑𝑚
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))}):{𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))}⟶{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶 ∧ ((1st ‘𝑠)‘𝑁) = 0 ∧ ((2nd ‘𝑠)‘𝑁) = 𝑁)}) |
533 | 531, 532 | syl 17 |
. . . . . . . . 9
⊢ (𝜑 → (1st ↾
{𝑡 ∈ ((((0..^𝐾) ↑𝑚
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))}):{𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))}⟶{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶 ∧ ((1st ‘𝑠)‘𝑁) = 0 ∧ ((2nd ‘𝑠)‘𝑁) = 𝑁)}) |
534 | | fvres 6117 |
. . . . . . . . . . . 12
⊢ (𝑥 ∈ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))} → ((1st ↾ {𝑡 ∈ ((((0..^𝐾) ↑𝑚
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))})‘𝑥) = (1st ‘𝑥)) |
535 | | fvres 6117 |
. . . . . . . . . . . 12
⊢ (𝑦 ∈ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))} → ((1st ↾ {𝑡 ∈ ((((0..^𝐾) ↑𝑚
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))})‘𝑦) = (1st ‘𝑦)) |
536 | 534, 535 | eqeqan12d 2626 |
. . . . . . . . . . 11
⊢ ((𝑥 ∈ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))} ∧ 𝑦 ∈ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))}) → (((1st ↾ {𝑡 ∈ ((((0..^𝐾) ↑𝑚
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))})‘𝑥) = ((1st ↾ {𝑡 ∈ ((((0..^𝐾) ↑𝑚
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))})‘𝑦) ↔ (1st ‘𝑥) = (1st ‘𝑦))) |
537 | | simpl 472 |
. . . . . . . . . . . . . . . 16
⊢
(((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁)) → (2nd ‘𝑡) = 𝑁) |
538 | 537 | a1i 11 |
. . . . . . . . . . . . . . 15
⊢ (𝑡 ∈ ((((0..^𝐾) ↑𝑚
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) → (((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁)) → (2nd ‘𝑡) = 𝑁)) |
539 | 538 | ss2rabi 3647 |
. . . . . . . . . . . . . 14
⊢ {𝑡 ∈ ((((0..^𝐾) ↑𝑚
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))} ⊆ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (2nd ‘𝑡) = 𝑁} |
540 | 539 | sseli 3564 |
. . . . . . . . . . . . 13
⊢ (𝑥 ∈ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))} → 𝑥 ∈ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (2nd ‘𝑡) = 𝑁}) |
541 | 419 | elrab 3331 |
. . . . . . . . . . . . 13
⊢ (𝑥 ∈ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (2nd ‘𝑡) = 𝑁} ↔ (𝑥 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∧ (2nd ‘𝑥) = 𝑁)) |
542 | 540, 541 | sylib 207 |
. . . . . . . . . . . 12
⊢ (𝑥 ∈ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))} → (𝑥 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∧ (2nd ‘𝑥) = 𝑁)) |
543 | 539 | sseli 3564 |
. . . . . . . . . . . . 13
⊢ (𝑦 ∈ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))} → 𝑦 ∈ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (2nd ‘𝑡) = 𝑁}) |
544 | | fveq2 6103 |
. . . . . . . . . . . . . . 15
⊢ (𝑡 = 𝑦 → (2nd ‘𝑡) = (2nd ‘𝑦)) |
545 | 544 | eqeq1d 2612 |
. . . . . . . . . . . . . 14
⊢ (𝑡 = 𝑦 → ((2nd ‘𝑡) = 𝑁 ↔ (2nd ‘𝑦) = 𝑁)) |
546 | 545 | elrab 3331 |
. . . . . . . . . . . . 13
⊢ (𝑦 ∈ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (2nd ‘𝑡) = 𝑁} ↔ (𝑦 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∧ (2nd ‘𝑦) = 𝑁)) |
547 | 543, 546 | sylib 207 |
. . . . . . . . . . . 12
⊢ (𝑦 ∈ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))} → (𝑦 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∧ (2nd ‘𝑦) = 𝑁)) |
548 | | eqtr3 2631 |
. . . . . . . . . . . . . 14
⊢
(((2nd ‘𝑥) = 𝑁 ∧ (2nd ‘𝑦) = 𝑁) → (2nd ‘𝑥) = (2nd ‘𝑦)) |
549 | | xpopth 7098 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑥 ∈ ((((0..^𝐾) ↑𝑚
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∧ 𝑦 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁))) → (((1st ‘𝑥) = (1st ‘𝑦) ∧ (2nd
‘𝑥) = (2nd
‘𝑦)) ↔ 𝑥 = 𝑦)) |
550 | 549 | biimpd 218 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑥 ∈ ((((0..^𝐾) ↑𝑚
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∧ 𝑦 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁))) → (((1st ‘𝑥) = (1st ‘𝑦) ∧ (2nd
‘𝑥) = (2nd
‘𝑦)) → 𝑥 = 𝑦)) |
551 | 550 | ancomsd 469 |
. . . . . . . . . . . . . . 15
⊢ ((𝑥 ∈ ((((0..^𝐾) ↑𝑚
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∧ 𝑦 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁))) → (((2nd ‘𝑥) = (2nd ‘𝑦) ∧ (1st
‘𝑥) = (1st
‘𝑦)) → 𝑥 = 𝑦)) |
552 | 551 | expdimp 452 |
. . . . . . . . . . . . . 14
⊢ (((𝑥 ∈ ((((0..^𝐾) ↑𝑚
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∧ 𝑦 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁))) ∧ (2nd ‘𝑥) = (2nd ‘𝑦)) → ((1st
‘𝑥) = (1st
‘𝑦) → 𝑥 = 𝑦)) |
553 | 548, 552 | sylan2 490 |
. . . . . . . . . . . . 13
⊢ (((𝑥 ∈ ((((0..^𝐾) ↑𝑚
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∧ 𝑦 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁))) ∧ ((2nd ‘𝑥) = 𝑁 ∧ (2nd ‘𝑦) = 𝑁)) → ((1st ‘𝑥) = (1st ‘𝑦) → 𝑥 = 𝑦)) |
554 | 553 | an4s 865 |
. . . . . . . . . . . 12
⊢ (((𝑥 ∈ ((((0..^𝐾) ↑𝑚
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∧ (2nd ‘𝑥) = 𝑁) ∧ (𝑦 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∧ (2nd ‘𝑦) = 𝑁)) → ((1st ‘𝑥) = (1st ‘𝑦) → 𝑥 = 𝑦)) |
555 | 542, 547,
554 | syl2an 493 |
. . . . . . . . . . 11
⊢ ((𝑥 ∈ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))} ∧ 𝑦 ∈ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))}) → ((1st ‘𝑥) = (1st ‘𝑦) → 𝑥 = 𝑦)) |
556 | 536, 555 | sylbid 229 |
. . . . . . . . . 10
⊢ ((𝑥 ∈ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))} ∧ 𝑦 ∈ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))}) → (((1st ↾ {𝑡 ∈ ((((0..^𝐾) ↑𝑚
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))})‘𝑥) = ((1st ↾ {𝑡 ∈ ((((0..^𝐾) ↑𝑚
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))})‘𝑦) → 𝑥 = 𝑦)) |
557 | 556 | rgen2a 2960 |
. . . . . . . . 9
⊢
∀𝑥 ∈
{𝑡 ∈ ((((0..^𝐾) ↑𝑚
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))}∀𝑦 ∈ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))} (((1st ↾ {𝑡 ∈ ((((0..^𝐾) ↑𝑚
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))})‘𝑥) = ((1st ↾ {𝑡 ∈ ((((0..^𝐾) ↑𝑚
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))})‘𝑦) → 𝑥 = 𝑦) |
558 | 533, 557 | jctir 559 |
. . . . . . . 8
⊢ (𝜑 → ((1st ↾
{𝑡 ∈ ((((0..^𝐾) ↑𝑚
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))}):{𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))}⟶{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶 ∧ ((1st ‘𝑠)‘𝑁) = 0 ∧ ((2nd ‘𝑠)‘𝑁) = 𝑁)} ∧ ∀𝑥 ∈ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))}∀𝑦 ∈ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))} (((1st ↾ {𝑡 ∈ ((((0..^𝐾) ↑𝑚
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))})‘𝑥) = ((1st ↾ {𝑡 ∈ ((((0..^𝐾) ↑𝑚
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))})‘𝑦) → 𝑥 = 𝑦))) |
559 | | dff13 6416 |
. . . . . . . 8
⊢
((1st ↾ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))}):{𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))}–1-1→{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶 ∧ ((1st ‘𝑠)‘𝑁) = 0 ∧ ((2nd ‘𝑠)‘𝑁) = 𝑁)} ↔ ((1st ↾ {𝑡 ∈ ((((0..^𝐾) ↑𝑚
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))}):{𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))}⟶{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶 ∧ ((1st ‘𝑠)‘𝑁) = 0 ∧ ((2nd ‘𝑠)‘𝑁) = 𝑁)} ∧ ∀𝑥 ∈ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))}∀𝑦 ∈ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))} (((1st ↾ {𝑡 ∈ ((((0..^𝐾) ↑𝑚
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))})‘𝑥) = ((1st ↾ {𝑡 ∈ ((((0..^𝐾) ↑𝑚
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))})‘𝑦) → 𝑥 = 𝑦))) |
560 | 558, 559 | sylibr 223 |
. . . . . . 7
⊢ (𝜑 → (1st ↾
{𝑡 ∈ ((((0..^𝐾) ↑𝑚
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))}):{𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))}–1-1→{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶 ∧ ((1st ‘𝑠)‘𝑁) = 0 ∧ ((2nd ‘𝑠)‘𝑁) = 𝑁)}) |
561 | | df-f1o 5811 |
. . . . . . 7
⊢
((1st ↾ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))}):{𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))}–1-1-onto→{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶 ∧ ((1st ‘𝑠)‘𝑁) = 0 ∧ ((2nd ‘𝑠)‘𝑁) = 𝑁)} ↔ ((1st ↾ {𝑡 ∈ ((((0..^𝐾) ↑𝑚
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))}):{𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))}–1-1→{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶 ∧ ((1st ‘𝑠)‘𝑁) = 0 ∧ ((2nd ‘𝑠)‘𝑁) = 𝑁)} ∧ (1st ↾ {𝑡 ∈ ((((0..^𝐾) ↑𝑚
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))}):{𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))}–onto→{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶 ∧ ((1st ‘𝑠)‘𝑁) = 0 ∧ ((2nd ‘𝑠)‘𝑁) = 𝑁)})) |
562 | 560, 531,
561 | sylanbrc 695 |
. . . . . 6
⊢ (𝜑 → (1st ↾
{𝑡 ∈ ((((0..^𝐾) ↑𝑚
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))}):{𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))}–1-1-onto→{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶 ∧ ((1st ‘𝑠)‘𝑁) = 0 ∧ ((2nd ‘𝑠)‘𝑁) = 𝑁)}) |
563 | | rabfi 8070 |
. . . . . . . . 9
⊢
(((((0..^𝐾)
↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∈ Fin → {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))} ∈ Fin) |
564 | 27, 563 | ax-mp 5 |
. . . . . . . 8
⊢ {𝑡 ∈ ((((0..^𝐾) ↑𝑚
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))} ∈ Fin |
565 | 564 | elexi 3186 |
. . . . . . 7
⊢ {𝑡 ∈ ((((0..^𝐾) ↑𝑚
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))} ∈ V |
566 | 565 | f1oen 7862 |
. . . . . 6
⊢
((1st ↾ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))}):{𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))}–1-1-onto→{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶 ∧ ((1st ‘𝑠)‘𝑁) = 0 ∧ ((2nd ‘𝑠)‘𝑁) = 𝑁)} → {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))} ≈ {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶 ∧ ((1st ‘𝑠)‘𝑁) = 0 ∧ ((2nd ‘𝑠)‘𝑁) = 𝑁)}) |
567 | 562, 566 | syl 17 |
. . . . 5
⊢ (𝜑 → {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))} ≈ {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶 ∧ ((1st ‘𝑠)‘𝑁) = 0 ∧ ((2nd ‘𝑠)‘𝑁) = 𝑁)}) |
568 | | rabfi 8070 |
. . . . . . 7
⊢
((((0..^𝐾)
↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∈ Fin → {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶 ∧ ((1st ‘𝑠)‘𝑁) = 0 ∧ ((2nd ‘𝑠)‘𝑁) = 𝑁)} ∈ Fin) |
569 | 24, 568 | ax-mp 5 |
. . . . . 6
⊢ {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶 ∧ ((1st ‘𝑠)‘𝑁) = 0 ∧ ((2nd ‘𝑠)‘𝑁) = 𝑁)} ∈ Fin |
570 | | hashen 12997 |
. . . . . 6
⊢ (({𝑡 ∈ ((((0..^𝐾) ↑𝑚
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))} ∈ Fin ∧ {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶 ∧ ((1st ‘𝑠)‘𝑁) = 0 ∧ ((2nd ‘𝑠)‘𝑁) = 𝑁)} ∈ Fin) → ((#‘{𝑡 ∈ ((((0..^𝐾) ↑𝑚
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))}) = (#‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶 ∧ ((1st ‘𝑠)‘𝑁) = 0 ∧ ((2nd ‘𝑠)‘𝑁) = 𝑁)}) ↔ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))} ≈ {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶 ∧ ((1st ‘𝑠)‘𝑁) = 0 ∧ ((2nd ‘𝑠)‘𝑁) = 𝑁)})) |
571 | 564, 569,
570 | mp2an 704 |
. . . . 5
⊢
((#‘{𝑡 ∈
((((0..^𝐾)
↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))}) = (#‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶 ∧ ((1st ‘𝑠)‘𝑁) = 0 ∧ ((2nd ‘𝑠)‘𝑁) = 𝑁)}) ↔ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))} ≈ {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶 ∧ ((1st ‘𝑠)‘𝑁) = 0 ∧ ((2nd ‘𝑠)‘𝑁) = 𝑁)}) |
572 | 567, 571 | sylibr 223 |
. . . 4
⊢ (𝜑 → (#‘{𝑡 ∈ ((((0..^𝐾) ↑𝑚
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))}) = (#‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶 ∧ ((1st ‘𝑠)‘𝑁) = 0 ∧ ((2nd ‘𝑠)‘𝑁) = 𝑁)})) |
573 | 572 | oveq2d 6565 |
. . 3
⊢ (𝜑 → ((#‘{𝑡 ∈ ((((0..^𝐾) ↑𝑚
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶}) − (#‘{𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))})) = ((#‘{𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶}) − (#‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶 ∧ ((1st ‘𝑠)‘𝑁) = 0 ∧ ((2nd ‘𝑠)‘𝑁) = 𝑁)}))) |
574 | 204, 407,
573 | 3eqtr3d 2652 |
. 2
⊢ (𝜑 → Σ𝑥 ∈ (((0...𝐾) ↑𝑚 (1...𝑁)) ↑𝑚
(0...(𝑁 −
1)))(#‘{𝑡 ∈
((((0..^𝐾)
↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)))}) = ((#‘{𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶}) − (#‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶 ∧ ((1st ‘𝑠)‘𝑁) = 0 ∧ ((2nd ‘𝑠)‘𝑁) = 𝑁)}))) |
575 | 167, 574 | breqtrd 4609 |
1
⊢ (𝜑 → 2 ∥ ((#‘{𝑡 ∈ ((((0..^𝐾) ↑𝑚
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶}) − (#‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶 ∧ ((1st ‘𝑠)‘𝑁) = 0 ∧ ((2nd ‘𝑠)‘𝑁) = 𝑁)}))) |