Step | Hyp | Ref
| Expression |
1 | | gsumval3.b |
. . 3
⊢ 𝐵 = (Base‘𝐺) |
2 | | gsumval3.0 |
. . 3
⊢ 0 =
(0g‘𝐺) |
3 | | gsumval3.p |
. . 3
⊢ + =
(+g‘𝐺) |
4 | | eqid 2610 |
. . 3
⊢ {𝑧 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑧 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑧) = 𝑦)} = {𝑧 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑧 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑧) = 𝑦)} |
5 | | gsumval3a.w |
. . . . 5
⊢ 𝑊 = (𝐹 supp 0 ) |
6 | 5 | a1i 11 |
. . . 4
⊢ (𝜑 → 𝑊 = (𝐹 supp 0 )) |
7 | | gsumval3.f |
. . . . . . 7
⊢ (𝜑 → 𝐹:𝐴⟶𝐵) |
8 | | gsumval3.a |
. . . . . . 7
⊢ (𝜑 → 𝐴 ∈ 𝑉) |
9 | 7, 8 | jca 553 |
. . . . . 6
⊢ (𝜑 → (𝐹:𝐴⟶𝐵 ∧ 𝐴 ∈ 𝑉)) |
10 | | fex 6394 |
. . . . . 6
⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐴 ∈ 𝑉) → 𝐹 ∈ V) |
11 | 9, 10 | syl 17 |
. . . . 5
⊢ (𝜑 → 𝐹 ∈ V) |
12 | | fvex 6113 |
. . . . . 6
⊢
(0g‘𝐺) ∈ V |
13 | 2, 12 | eqeltri 2684 |
. . . . 5
⊢ 0 ∈
V |
14 | | suppimacnv 7193 |
. . . . 5
⊢ ((𝐹 ∈ V ∧ 0 ∈ V)
→ (𝐹 supp 0 ) = (◡𝐹 “ (V ∖ { 0 }))) |
15 | 11, 13, 14 | sylancl 693 |
. . . 4
⊢ (𝜑 → (𝐹 supp 0 ) = (◡𝐹 “ (V ∖ { 0 }))) |
16 | | gsumval3.g |
. . . . . . . 8
⊢ (𝜑 → 𝐺 ∈ Mnd) |
17 | 1, 2, 3, 4 | gsumvallem2 17195 |
. . . . . . . 8
⊢ (𝐺 ∈ Mnd → {𝑧 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑧 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑧) = 𝑦)} = { 0 }) |
18 | 16, 17 | syl 17 |
. . . . . . 7
⊢ (𝜑 → {𝑧 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑧 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑧) = 𝑦)} = { 0 }) |
19 | 18 | eqcomd 2616 |
. . . . . 6
⊢ (𝜑 → { 0 } = {𝑧 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑧 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑧) = 𝑦)}) |
20 | 19 | difeq2d 3690 |
. . . . 5
⊢ (𝜑 → (V ∖ { 0 }) = (V
∖ {𝑧 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑧 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑧) = 𝑦)})) |
21 | 20 | imaeq2d 5385 |
. . . 4
⊢ (𝜑 → (◡𝐹 “ (V ∖ { 0 })) = (◡𝐹 “ (V ∖ {𝑧 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑧 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑧) = 𝑦)}))) |
22 | 6, 15, 21 | 3eqtrd 2648 |
. . 3
⊢ (𝜑 → 𝑊 = (◡𝐹 “ (V ∖ {𝑧 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑧 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑧) = 𝑦)}))) |
23 | 1, 2, 3, 4, 22, 16, 8, 7 | gsumval 17094 |
. 2
⊢ (𝜑 → (𝐺 Σg 𝐹) = if(ran 𝐹 ⊆ {𝑧 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑧 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑧) = 𝑦)}, 0 , if(𝐴 ∈ ran ..., (℩𝑥∃𝑚∃𝑛 ∈ (ℤ≥‘𝑚)(𝐴 = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚( + , 𝐹)‘𝑛))), (℩𝑥∃𝑓(𝑓:(1...(#‘𝑊))–1-1-onto→𝑊 ∧ 𝑥 = (seq1( + , (𝐹 ∘ 𝑓))‘(#‘𝑊))))))) |
24 | | gsumval3a.n |
. . . 4
⊢ (𝜑 → 𝑊 ≠ ∅) |
25 | 18 | sseq2d 3596 |
. . . . . 6
⊢ (𝜑 → (ran 𝐹 ⊆ {𝑧 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑧 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑧) = 𝑦)} ↔ ran 𝐹 ⊆ { 0 })) |
26 | 5 | a1i 11 |
. . . . . . . 8
⊢ ((𝜑 ∧ ran 𝐹 ⊆ { 0 }) → 𝑊 = (𝐹 supp 0 )) |
27 | 9 | adantr 480 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ ran 𝐹 ⊆ { 0 }) → (𝐹:𝐴⟶𝐵 ∧ 𝐴 ∈ 𝑉)) |
28 | 27, 10 | syl 17 |
. . . . . . . . 9
⊢ ((𝜑 ∧ ran 𝐹 ⊆ { 0 }) → 𝐹 ∈ V) |
29 | 28, 13, 14 | sylancl 693 |
. . . . . . . 8
⊢ ((𝜑 ∧ ran 𝐹 ⊆ { 0 }) → (𝐹 supp 0 ) = (◡𝐹 “ (V ∖ { 0 }))) |
30 | | ffn 5958 |
. . . . . . . . . . . 12
⊢ (𝐹:𝐴⟶𝐵 → 𝐹 Fn 𝐴) |
31 | 7, 30 | syl 17 |
. . . . . . . . . . 11
⊢ (𝜑 → 𝐹 Fn 𝐴) |
32 | 31 | adantr 480 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ ran 𝐹 ⊆ { 0 }) → 𝐹 Fn 𝐴) |
33 | | simpr 476 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ ran 𝐹 ⊆ { 0 }) → ran 𝐹 ⊆ { 0 }) |
34 | | df-f 5808 |
. . . . . . . . . 10
⊢ (𝐹:𝐴⟶{ 0 } ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ { 0 })) |
35 | 32, 33, 34 | sylanbrc 695 |
. . . . . . . . 9
⊢ ((𝜑 ∧ ran 𝐹 ⊆ { 0 }) → 𝐹:𝐴⟶{ 0 }) |
36 | | disjdif 3992 |
. . . . . . . . 9
⊢ ({ 0 } ∩ (V
∖ { 0 })) =
∅ |
37 | | fimacnvdisj 5996 |
. . . . . . . . 9
⊢ ((𝐹:𝐴⟶{ 0 } ∧ ({ 0 } ∩ (V
∖ { 0 })) = ∅) →
(◡𝐹 “ (V ∖ { 0 })) =
∅) |
38 | 35, 36, 37 | sylancl 693 |
. . . . . . . 8
⊢ ((𝜑 ∧ ran 𝐹 ⊆ { 0 }) → (◡𝐹 “ (V ∖ { 0 })) =
∅) |
39 | 26, 29, 38 | 3eqtrd 2648 |
. . . . . . 7
⊢ ((𝜑 ∧ ran 𝐹 ⊆ { 0 }) → 𝑊 = ∅) |
40 | 39 | ex 449 |
. . . . . 6
⊢ (𝜑 → (ran 𝐹 ⊆ { 0 } → 𝑊 = ∅)) |
41 | 25, 40 | sylbid 229 |
. . . . 5
⊢ (𝜑 → (ran 𝐹 ⊆ {𝑧 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑧 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑧) = 𝑦)} → 𝑊 = ∅)) |
42 | 41 | necon3ad 2795 |
. . . 4
⊢ (𝜑 → (𝑊 ≠ ∅ → ¬ ran 𝐹 ⊆ {𝑧 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑧 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑧) = 𝑦)})) |
43 | 24, 42 | mpd 15 |
. . 3
⊢ (𝜑 → ¬ ran 𝐹 ⊆ {𝑧 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑧 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑧) = 𝑦)}) |
44 | 43 | iffalsed 4047 |
. 2
⊢ (𝜑 → if(ran 𝐹 ⊆ {𝑧 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑧 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑧) = 𝑦)}, 0 , if(𝐴 ∈ ran ..., (℩𝑥∃𝑚∃𝑛 ∈ (ℤ≥‘𝑚)(𝐴 = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚( + , 𝐹)‘𝑛))), (℩𝑥∃𝑓(𝑓:(1...(#‘𝑊))–1-1-onto→𝑊 ∧ 𝑥 = (seq1( + , (𝐹 ∘ 𝑓))‘(#‘𝑊)))))) = if(𝐴 ∈ ran ..., (℩𝑥∃𝑚∃𝑛 ∈ (ℤ≥‘𝑚)(𝐴 = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚( + , 𝐹)‘𝑛))), (℩𝑥∃𝑓(𝑓:(1...(#‘𝑊))–1-1-onto→𝑊 ∧ 𝑥 = (seq1( + , (𝐹 ∘ 𝑓))‘(#‘𝑊)))))) |
45 | | gsumval3a.i |
. . 3
⊢ (𝜑 → ¬ 𝐴 ∈ ran ...) |
46 | 45 | iffalsed 4047 |
. 2
⊢ (𝜑 → if(𝐴 ∈ ran ..., (℩𝑥∃𝑚∃𝑛 ∈ (ℤ≥‘𝑚)(𝐴 = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚( + , 𝐹)‘𝑛))), (℩𝑥∃𝑓(𝑓:(1...(#‘𝑊))–1-1-onto→𝑊 ∧ 𝑥 = (seq1( + , (𝐹 ∘ 𝑓))‘(#‘𝑊))))) = (℩𝑥∃𝑓(𝑓:(1...(#‘𝑊))–1-1-onto→𝑊 ∧ 𝑥 = (seq1( + , (𝐹 ∘ 𝑓))‘(#‘𝑊))))) |
47 | 23, 44, 46 | 3eqtrd 2648 |
1
⊢ (𝜑 → (𝐺 Σg 𝐹) = (℩𝑥∃𝑓(𝑓:(1...(#‘𝑊))–1-1-onto→𝑊 ∧ 𝑥 = (seq1( + , (𝐹 ∘ 𝑓))‘(#‘𝑊))))) |