Step | Hyp | Ref
| Expression |
1 | | caucvgpr.f |
. . . . . . 7
⊢ (𝜑 → 𝐹:N⟶Q) |
2 | | caucvgpr.cau |
. . . . . . 7
⊢ (𝜑 → ∀𝑛 ∈ N ∀𝑘 ∈ N (𝑛 <N
𝑘 → ((𝐹‘𝑛) <Q ((𝐹‘𝑘) +Q
(*Q‘[〈𝑛, 1𝑜〉]
~Q )) ∧ (𝐹‘𝑘) <Q ((𝐹‘𝑛) +Q
(*Q‘[〈𝑛, 1𝑜〉]
~Q ))))) |
3 | | caucvgpr.bnd |
. . . . . . 7
⊢ (𝜑 → ∀𝑗 ∈ N 𝐴 <Q (𝐹‘𝑗)) |
4 | | caucvgpr.lim |
. . . . . . 7
⊢ 𝐿 = 〈{𝑙 ∈ Q ∣ ∃𝑗 ∈ N (𝑙 +Q
(*Q‘[〈𝑗, 1𝑜〉]
~Q )) <Q (𝐹‘𝑗)}, {𝑢 ∈ Q ∣ ∃𝑗 ∈ N ((𝐹‘𝑗) +Q
(*Q‘[〈𝑗, 1𝑜〉]
~Q )) <Q 𝑢}〉 |
5 | 1, 2, 3, 4 | caucvgprlemcl 6774 |
. . . . . 6
⊢ (𝜑 → 𝐿 ∈ P) |
6 | | caucvgprlemladd.s |
. . . . . . 7
⊢ (𝜑 → 𝑆 ∈ Q) |
7 | | nqprlu 6645 |
. . . . . . 7
⊢ (𝑆 ∈ Q →
〈{𝑙 ∣ 𝑙 <Q
𝑆}, {𝑢 ∣ 𝑆 <Q 𝑢}〉 ∈
P) |
8 | 6, 7 | syl 14 |
. . . . . 6
⊢ (𝜑 → 〈{𝑙 ∣ 𝑙 <Q 𝑆}, {𝑢 ∣ 𝑆 <Q 𝑢}〉 ∈
P) |
9 | | df-iplp 6566 |
. . . . . . 7
⊢
+P = (𝑥 ∈ P, 𝑦 ∈ P ↦ 〈{𝑓 ∈ Q ∣
∃𝑔 ∈
Q ∃ℎ
∈ Q (𝑔
∈ (1st ‘𝑥) ∧ ℎ ∈ (1st ‘𝑦) ∧ 𝑓 = (𝑔 +Q ℎ))}, {𝑓 ∈ Q ∣ ∃𝑔 ∈ Q
∃ℎ ∈
Q (𝑔 ∈
(2nd ‘𝑥)
∧ ℎ ∈
(2nd ‘𝑦)
∧ 𝑓 = (𝑔 +Q
ℎ))}〉) |
10 | | addclnq 6473 |
. . . . . . 7
⊢ ((𝑔 ∈ Q ∧
ℎ ∈ Q)
→ (𝑔
+Q ℎ) ∈ Q) |
11 | 9, 10 | genpelvu 6611 |
. . . . . 6
⊢ ((𝐿 ∈ P ∧
〈{𝑙 ∣ 𝑙 <Q
𝑆}, {𝑢 ∣ 𝑆 <Q 𝑢}〉 ∈ P)
→ (𝑟 ∈
(2nd ‘(𝐿
+P 〈{𝑙 ∣ 𝑙 <Q 𝑆}, {𝑢 ∣ 𝑆 <Q 𝑢}〉)) ↔ ∃𝑠 ∈ (2nd
‘𝐿)∃𝑡 ∈ (2nd
‘〈{𝑙 ∣
𝑙
<Q 𝑆}, {𝑢 ∣ 𝑆 <Q 𝑢}〉)𝑟 = (𝑠 +Q 𝑡))) |
12 | 5, 8, 11 | syl2anc 391 |
. . . . 5
⊢ (𝜑 → (𝑟 ∈ (2nd ‘(𝐿 +P
〈{𝑙 ∣ 𝑙 <Q
𝑆}, {𝑢 ∣ 𝑆 <Q 𝑢}〉)) ↔ ∃𝑠 ∈ (2nd
‘𝐿)∃𝑡 ∈ (2nd
‘〈{𝑙 ∣
𝑙
<Q 𝑆}, {𝑢 ∣ 𝑆 <Q 𝑢}〉)𝑟 = (𝑠 +Q 𝑡))) |
13 | 12 | biimpa 280 |
. . . 4
⊢ ((𝜑 ∧ 𝑟 ∈ (2nd ‘(𝐿 +P
〈{𝑙 ∣ 𝑙 <Q
𝑆}, {𝑢 ∣ 𝑆 <Q 𝑢}〉))) → ∃𝑠 ∈ (2nd
‘𝐿)∃𝑡 ∈ (2nd
‘〈{𝑙 ∣
𝑙
<Q 𝑆}, {𝑢 ∣ 𝑆 <Q 𝑢}〉)𝑟 = (𝑠 +Q 𝑡)) |
14 | | breq2 3768 |
. . . . . . . . . . . . . . . 16
⊢ (𝑢 = 𝑠 → (((𝐹‘𝑗) +Q
(*Q‘[〈𝑗, 1𝑜〉]
~Q )) <Q 𝑢 ↔ ((𝐹‘𝑗) +Q
(*Q‘[〈𝑗, 1𝑜〉]
~Q )) <Q 𝑠)) |
15 | 14 | rexbidv 2327 |
. . . . . . . . . . . . . . 15
⊢ (𝑢 = 𝑠 → (∃𝑗 ∈ N ((𝐹‘𝑗) +Q
(*Q‘[〈𝑗, 1𝑜〉]
~Q )) <Q 𝑢 ↔ ∃𝑗 ∈ N ((𝐹‘𝑗) +Q
(*Q‘[〈𝑗, 1𝑜〉]
~Q )) <Q 𝑠)) |
16 | 4 | fveq2i 5181 |
. . . . . . . . . . . . . . . 16
⊢
(2nd ‘𝐿) = (2nd ‘〈{𝑙 ∈ Q ∣
∃𝑗 ∈
N (𝑙
+Q (*Q‘[〈𝑗, 1𝑜〉]
~Q )) <Q (𝐹‘𝑗)}, {𝑢 ∈ Q ∣ ∃𝑗 ∈ N ((𝐹‘𝑗) +Q
(*Q‘[〈𝑗, 1𝑜〉]
~Q )) <Q 𝑢}〉) |
17 | | nqex 6461 |
. . . . . . . . . . . . . . . . . 18
⊢
Q ∈ V |
18 | 17 | rabex 3901 |
. . . . . . . . . . . . . . . . 17
⊢ {𝑙 ∈ Q ∣
∃𝑗 ∈
N (𝑙
+Q (*Q‘[〈𝑗, 1𝑜〉]
~Q )) <Q (𝐹‘𝑗)} ∈ V |
19 | 17 | rabex 3901 |
. . . . . . . . . . . . . . . . 17
⊢ {𝑢 ∈ Q ∣
∃𝑗 ∈
N ((𝐹‘𝑗) +Q
(*Q‘[〈𝑗, 1𝑜〉]
~Q )) <Q 𝑢} ∈ V |
20 | 18, 19 | op2nd 5774 |
. . . . . . . . . . . . . . . 16
⊢
(2nd ‘〈{𝑙 ∈ Q ∣ ∃𝑗 ∈ N (𝑙 +Q
(*Q‘[〈𝑗, 1𝑜〉]
~Q )) <Q (𝐹‘𝑗)}, {𝑢 ∈ Q ∣ ∃𝑗 ∈ N ((𝐹‘𝑗) +Q
(*Q‘[〈𝑗, 1𝑜〉]
~Q )) <Q 𝑢}〉) = {𝑢 ∈ Q ∣ ∃𝑗 ∈ N ((𝐹‘𝑗) +Q
(*Q‘[〈𝑗, 1𝑜〉]
~Q )) <Q 𝑢} |
21 | 16, 20 | eqtri 2060 |
. . . . . . . . . . . . . . 15
⊢
(2nd ‘𝐿) = {𝑢 ∈ Q ∣ ∃𝑗 ∈ N ((𝐹‘𝑗) +Q
(*Q‘[〈𝑗, 1𝑜〉]
~Q )) <Q 𝑢} |
22 | 15, 21 | elrab2 2700 |
. . . . . . . . . . . . . 14
⊢ (𝑠 ∈ (2nd
‘𝐿) ↔ (𝑠 ∈ Q ∧
∃𝑗 ∈
N ((𝐹‘𝑗) +Q
(*Q‘[〈𝑗, 1𝑜〉]
~Q )) <Q 𝑠)) |
23 | 22 | biimpi 113 |
. . . . . . . . . . . . 13
⊢ (𝑠 ∈ (2nd
‘𝐿) → (𝑠 ∈ Q ∧
∃𝑗 ∈
N ((𝐹‘𝑗) +Q
(*Q‘[〈𝑗, 1𝑜〉]
~Q )) <Q 𝑠)) |
24 | 23 | adantr 261 |
. . . . . . . . . . . 12
⊢ ((𝑠 ∈ (2nd
‘𝐿) ∧ 𝑡 ∈ (2nd
‘〈{𝑙 ∣
𝑙
<Q 𝑆}, {𝑢 ∣ 𝑆 <Q 𝑢}〉)) → (𝑠 ∈ Q ∧
∃𝑗 ∈
N ((𝐹‘𝑗) +Q
(*Q‘[〈𝑗, 1𝑜〉]
~Q )) <Q 𝑠)) |
25 | 24 | adantl 262 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑟 ∈ (2nd ‘(𝐿 +P
〈{𝑙 ∣ 𝑙 <Q
𝑆}, {𝑢 ∣ 𝑆 <Q 𝑢}〉))) ∧ (𝑠 ∈ (2nd
‘𝐿) ∧ 𝑡 ∈ (2nd
‘〈{𝑙 ∣
𝑙
<Q 𝑆}, {𝑢 ∣ 𝑆 <Q 𝑢}〉))) → (𝑠 ∈ Q ∧
∃𝑗 ∈
N ((𝐹‘𝑗) +Q
(*Q‘[〈𝑗, 1𝑜〉]
~Q )) <Q 𝑠)) |
26 | 25 | adantr 261 |
. . . . . . . . . 10
⊢ ((((𝜑 ∧ 𝑟 ∈ (2nd ‘(𝐿 +P
〈{𝑙 ∣ 𝑙 <Q
𝑆}, {𝑢 ∣ 𝑆 <Q 𝑢}〉))) ∧ (𝑠 ∈ (2nd
‘𝐿) ∧ 𝑡 ∈ (2nd
‘〈{𝑙 ∣
𝑙
<Q 𝑆}, {𝑢 ∣ 𝑆 <Q 𝑢}〉))) ∧ 𝑟 = (𝑠 +Q 𝑡)) → (𝑠 ∈ Q ∧ ∃𝑗 ∈ N ((𝐹‘𝑗) +Q
(*Q‘[〈𝑗, 1𝑜〉]
~Q )) <Q 𝑠)) |
27 | 26 | simpld 105 |
. . . . . . . . 9
⊢ ((((𝜑 ∧ 𝑟 ∈ (2nd ‘(𝐿 +P
〈{𝑙 ∣ 𝑙 <Q
𝑆}, {𝑢 ∣ 𝑆 <Q 𝑢}〉))) ∧ (𝑠 ∈ (2nd
‘𝐿) ∧ 𝑡 ∈ (2nd
‘〈{𝑙 ∣
𝑙
<Q 𝑆}, {𝑢 ∣ 𝑆 <Q 𝑢}〉))) ∧ 𝑟 = (𝑠 +Q 𝑡)) → 𝑠 ∈ Q) |
28 | | vex 2560 |
. . . . . . . . . . . . . 14
⊢ 𝑡 ∈ V |
29 | | breq2 3768 |
. . . . . . . . . . . . . 14
⊢ (𝑢 = 𝑡 → (𝑆 <Q 𝑢 ↔ 𝑆 <Q 𝑡)) |
30 | | ltnqex 6647 |
. . . . . . . . . . . . . . 15
⊢ {𝑙 ∣ 𝑙 <Q 𝑆} ∈ V |
31 | | gtnqex 6648 |
. . . . . . . . . . . . . . 15
⊢ {𝑢 ∣ 𝑆 <Q 𝑢} ∈ V |
32 | 30, 31 | op2nd 5774 |
. . . . . . . . . . . . . 14
⊢
(2nd ‘〈{𝑙 ∣ 𝑙 <Q 𝑆}, {𝑢 ∣ 𝑆 <Q 𝑢}〉) = {𝑢 ∣ 𝑆 <Q 𝑢} |
33 | 28, 29, 32 | elab2 2690 |
. . . . . . . . . . . . 13
⊢ (𝑡 ∈ (2nd
‘〈{𝑙 ∣
𝑙
<Q 𝑆}, {𝑢 ∣ 𝑆 <Q 𝑢}〉) ↔ 𝑆 <Q
𝑡) |
34 | | ltrelnq 6463 |
. . . . . . . . . . . . . 14
⊢
<Q ⊆ (Q ×
Q) |
35 | 34 | brel 4392 |
. . . . . . . . . . . . 13
⊢ (𝑆 <Q
𝑡 → (𝑆 ∈ Q ∧ 𝑡 ∈
Q)) |
36 | 33, 35 | sylbi 114 |
. . . . . . . . . . . 12
⊢ (𝑡 ∈ (2nd
‘〈{𝑙 ∣
𝑙
<Q 𝑆}, {𝑢 ∣ 𝑆 <Q 𝑢}〉) → (𝑆 ∈ Q ∧
𝑡 ∈
Q)) |
37 | 36 | simprd 107 |
. . . . . . . . . . 11
⊢ (𝑡 ∈ (2nd
‘〈{𝑙 ∣
𝑙
<Q 𝑆}, {𝑢 ∣ 𝑆 <Q 𝑢}〉) → 𝑡 ∈
Q) |
38 | 37 | ad2antll 460 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑟 ∈ (2nd ‘(𝐿 +P
〈{𝑙 ∣ 𝑙 <Q
𝑆}, {𝑢 ∣ 𝑆 <Q 𝑢}〉))) ∧ (𝑠 ∈ (2nd
‘𝐿) ∧ 𝑡 ∈ (2nd
‘〈{𝑙 ∣
𝑙
<Q 𝑆}, {𝑢 ∣ 𝑆 <Q 𝑢}〉))) → 𝑡 ∈
Q) |
39 | 38 | adantr 261 |
. . . . . . . . 9
⊢ ((((𝜑 ∧ 𝑟 ∈ (2nd ‘(𝐿 +P
〈{𝑙 ∣ 𝑙 <Q
𝑆}, {𝑢 ∣ 𝑆 <Q 𝑢}〉))) ∧ (𝑠 ∈ (2nd
‘𝐿) ∧ 𝑡 ∈ (2nd
‘〈{𝑙 ∣
𝑙
<Q 𝑆}, {𝑢 ∣ 𝑆 <Q 𝑢}〉))) ∧ 𝑟 = (𝑠 +Q 𝑡)) → 𝑡 ∈ Q) |
40 | | addclnq 6473 |
. . . . . . . . 9
⊢ ((𝑠 ∈ Q ∧
𝑡 ∈ Q)
→ (𝑠
+Q 𝑡) ∈ Q) |
41 | 27, 39, 40 | syl2anc 391 |
. . . . . . . 8
⊢ ((((𝜑 ∧ 𝑟 ∈ (2nd ‘(𝐿 +P
〈{𝑙 ∣ 𝑙 <Q
𝑆}, {𝑢 ∣ 𝑆 <Q 𝑢}〉))) ∧ (𝑠 ∈ (2nd
‘𝐿) ∧ 𝑡 ∈ (2nd
‘〈{𝑙 ∣
𝑙
<Q 𝑆}, {𝑢 ∣ 𝑆 <Q 𝑢}〉))) ∧ 𝑟 = (𝑠 +Q 𝑡)) → (𝑠 +Q 𝑡) ∈
Q) |
42 | | eleq1 2100 |
. . . . . . . . 9
⊢ (𝑟 = (𝑠 +Q 𝑡) → (𝑟 ∈ Q ↔ (𝑠 +Q
𝑡) ∈
Q)) |
43 | 42 | adantl 262 |
. . . . . . . 8
⊢ ((((𝜑 ∧ 𝑟 ∈ (2nd ‘(𝐿 +P
〈{𝑙 ∣ 𝑙 <Q
𝑆}, {𝑢 ∣ 𝑆 <Q 𝑢}〉))) ∧ (𝑠 ∈ (2nd
‘𝐿) ∧ 𝑡 ∈ (2nd
‘〈{𝑙 ∣
𝑙
<Q 𝑆}, {𝑢 ∣ 𝑆 <Q 𝑢}〉))) ∧ 𝑟 = (𝑠 +Q 𝑡)) → (𝑟 ∈ Q ↔ (𝑠 +Q
𝑡) ∈
Q)) |
44 | 41, 43 | mpbird 156 |
. . . . . . 7
⊢ ((((𝜑 ∧ 𝑟 ∈ (2nd ‘(𝐿 +P
〈{𝑙 ∣ 𝑙 <Q
𝑆}, {𝑢 ∣ 𝑆 <Q 𝑢}〉))) ∧ (𝑠 ∈ (2nd
‘𝐿) ∧ 𝑡 ∈ (2nd
‘〈{𝑙 ∣
𝑙
<Q 𝑆}, {𝑢 ∣ 𝑆 <Q 𝑢}〉))) ∧ 𝑟 = (𝑠 +Q 𝑡)) → 𝑟 ∈ Q) |
45 | 26 | simprd 107 |
. . . . . . . . . 10
⊢ ((((𝜑 ∧ 𝑟 ∈ (2nd ‘(𝐿 +P
〈{𝑙 ∣ 𝑙 <Q
𝑆}, {𝑢 ∣ 𝑆 <Q 𝑢}〉))) ∧ (𝑠 ∈ (2nd
‘𝐿) ∧ 𝑡 ∈ (2nd
‘〈{𝑙 ∣
𝑙
<Q 𝑆}, {𝑢 ∣ 𝑆 <Q 𝑢}〉))) ∧ 𝑟 = (𝑠 +Q 𝑡)) → ∃𝑗 ∈ N ((𝐹‘𝑗) +Q
(*Q‘[〈𝑗, 1𝑜〉]
~Q )) <Q 𝑠) |
46 | | fveq2 5178 |
. . . . . . . . . . . . 13
⊢ (𝑗 = 𝑚 → (𝐹‘𝑗) = (𝐹‘𝑚)) |
47 | | opeq1 3549 |
. . . . . . . . . . . . . . 15
⊢ (𝑗 = 𝑚 → 〈𝑗, 1𝑜〉 = 〈𝑚,
1𝑜〉) |
48 | 47 | eceq1d 6142 |
. . . . . . . . . . . . . 14
⊢ (𝑗 = 𝑚 → [〈𝑗, 1𝑜〉]
~Q = [〈𝑚, 1𝑜〉]
~Q ) |
49 | 48 | fveq2d 5182 |
. . . . . . . . . . . . 13
⊢ (𝑗 = 𝑚 →
(*Q‘[〈𝑗, 1𝑜〉]
~Q ) = (*Q‘[〈𝑚, 1𝑜〉]
~Q )) |
50 | 46, 49 | oveq12d 5530 |
. . . . . . . . . . . 12
⊢ (𝑗 = 𝑚 → ((𝐹‘𝑗) +Q
(*Q‘[〈𝑗, 1𝑜〉]
~Q )) = ((𝐹‘𝑚) +Q
(*Q‘[〈𝑚, 1𝑜〉]
~Q ))) |
51 | 50 | breq1d 3774 |
. . . . . . . . . . 11
⊢ (𝑗 = 𝑚 → (((𝐹‘𝑗) +Q
(*Q‘[〈𝑗, 1𝑜〉]
~Q )) <Q 𝑠 ↔ ((𝐹‘𝑚) +Q
(*Q‘[〈𝑚, 1𝑜〉]
~Q )) <Q 𝑠)) |
52 | 51 | cbvrexv 2534 |
. . . . . . . . . 10
⊢
(∃𝑗 ∈
N ((𝐹‘𝑗) +Q
(*Q‘[〈𝑗, 1𝑜〉]
~Q )) <Q 𝑠 ↔ ∃𝑚 ∈ N ((𝐹‘𝑚) +Q
(*Q‘[〈𝑚, 1𝑜〉]
~Q )) <Q 𝑠) |
53 | 45, 52 | sylib 127 |
. . . . . . . . 9
⊢ ((((𝜑 ∧ 𝑟 ∈ (2nd ‘(𝐿 +P
〈{𝑙 ∣ 𝑙 <Q
𝑆}, {𝑢 ∣ 𝑆 <Q 𝑢}〉))) ∧ (𝑠 ∈ (2nd
‘𝐿) ∧ 𝑡 ∈ (2nd
‘〈{𝑙 ∣
𝑙
<Q 𝑆}, {𝑢 ∣ 𝑆 <Q 𝑢}〉))) ∧ 𝑟 = (𝑠 +Q 𝑡)) → ∃𝑚 ∈ N ((𝐹‘𝑚) +Q
(*Q‘[〈𝑚, 1𝑜〉]
~Q )) <Q 𝑠) |
54 | 33 | biimpi 113 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑡 ∈ (2nd
‘〈{𝑙 ∣
𝑙
<Q 𝑆}, {𝑢 ∣ 𝑆 <Q 𝑢}〉) → 𝑆 <Q
𝑡) |
55 | 54 | ad2antll 460 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑟 ∈ (2nd ‘(𝐿 +P
〈{𝑙 ∣ 𝑙 <Q
𝑆}, {𝑢 ∣ 𝑆 <Q 𝑢}〉))) ∧ (𝑠 ∈ (2nd
‘𝐿) ∧ 𝑡 ∈ (2nd
‘〈{𝑙 ∣
𝑙
<Q 𝑆}, {𝑢 ∣ 𝑆 <Q 𝑢}〉))) → 𝑆 <Q
𝑡) |
56 | 55 | adantr 261 |
. . . . . . . . . . . . . . 15
⊢ ((((𝜑 ∧ 𝑟 ∈ (2nd ‘(𝐿 +P
〈{𝑙 ∣ 𝑙 <Q
𝑆}, {𝑢 ∣ 𝑆 <Q 𝑢}〉))) ∧ (𝑠 ∈ (2nd
‘𝐿) ∧ 𝑡 ∈ (2nd
‘〈{𝑙 ∣
𝑙
<Q 𝑆}, {𝑢 ∣ 𝑆 <Q 𝑢}〉))) ∧ 𝑟 = (𝑠 +Q 𝑡)) → 𝑆 <Q 𝑡) |
57 | 56 | ad2antrr 457 |
. . . . . . . . . . . . . 14
⊢
((((((𝜑 ∧ 𝑟 ∈ (2nd
‘(𝐿
+P 〈{𝑙 ∣ 𝑙 <Q 𝑆}, {𝑢 ∣ 𝑆 <Q 𝑢}〉))) ∧ (𝑠 ∈ (2nd
‘𝐿) ∧ 𝑡 ∈ (2nd
‘〈{𝑙 ∣
𝑙
<Q 𝑆}, {𝑢 ∣ 𝑆 <Q 𝑢}〉))) ∧ 𝑟 = (𝑠 +Q 𝑡)) ∧ 𝑚 ∈ N) ∧ ((𝐹‘𝑚) +Q
(*Q‘[〈𝑚, 1𝑜〉]
~Q )) <Q 𝑠) → 𝑆 <Q 𝑡) |
58 | 6 | ad5antr 465 |
. . . . . . . . . . . . . . 15
⊢
((((((𝜑 ∧ 𝑟 ∈ (2nd
‘(𝐿
+P 〈{𝑙 ∣ 𝑙 <Q 𝑆}, {𝑢 ∣ 𝑆 <Q 𝑢}〉))) ∧ (𝑠 ∈ (2nd
‘𝐿) ∧ 𝑡 ∈ (2nd
‘〈{𝑙 ∣
𝑙
<Q 𝑆}, {𝑢 ∣ 𝑆 <Q 𝑢}〉))) ∧ 𝑟 = (𝑠 +Q 𝑡)) ∧ 𝑚 ∈ N) ∧ ((𝐹‘𝑚) +Q
(*Q‘[〈𝑚, 1𝑜〉]
~Q )) <Q 𝑠) → 𝑆 ∈ Q) |
59 | 39 | ad2antrr 457 |
. . . . . . . . . . . . . . 15
⊢
((((((𝜑 ∧ 𝑟 ∈ (2nd
‘(𝐿
+P 〈{𝑙 ∣ 𝑙 <Q 𝑆}, {𝑢 ∣ 𝑆 <Q 𝑢}〉))) ∧ (𝑠 ∈ (2nd
‘𝐿) ∧ 𝑡 ∈ (2nd
‘〈{𝑙 ∣
𝑙
<Q 𝑆}, {𝑢 ∣ 𝑆 <Q 𝑢}〉))) ∧ 𝑟 = (𝑠 +Q 𝑡)) ∧ 𝑚 ∈ N) ∧ ((𝐹‘𝑚) +Q
(*Q‘[〈𝑚, 1𝑜〉]
~Q )) <Q 𝑠) → 𝑡 ∈ Q) |
60 | 1 | ad5antr 465 |
. . . . . . . . . . . . . . . . 17
⊢
((((((𝜑 ∧ 𝑟 ∈ (2nd
‘(𝐿
+P 〈{𝑙 ∣ 𝑙 <Q 𝑆}, {𝑢 ∣ 𝑆 <Q 𝑢}〉))) ∧ (𝑠 ∈ (2nd
‘𝐿) ∧ 𝑡 ∈ (2nd
‘〈{𝑙 ∣
𝑙
<Q 𝑆}, {𝑢 ∣ 𝑆 <Q 𝑢}〉))) ∧ 𝑟 = (𝑠 +Q 𝑡)) ∧ 𝑚 ∈ N) ∧ ((𝐹‘𝑚) +Q
(*Q‘[〈𝑚, 1𝑜〉]
~Q )) <Q 𝑠) → 𝐹:N⟶Q) |
61 | | simplr 482 |
. . . . . . . . . . . . . . . . 17
⊢
((((((𝜑 ∧ 𝑟 ∈ (2nd
‘(𝐿
+P 〈{𝑙 ∣ 𝑙 <Q 𝑆}, {𝑢 ∣ 𝑆 <Q 𝑢}〉))) ∧ (𝑠 ∈ (2nd
‘𝐿) ∧ 𝑡 ∈ (2nd
‘〈{𝑙 ∣
𝑙
<Q 𝑆}, {𝑢 ∣ 𝑆 <Q 𝑢}〉))) ∧ 𝑟 = (𝑠 +Q 𝑡)) ∧ 𝑚 ∈ N) ∧ ((𝐹‘𝑚) +Q
(*Q‘[〈𝑚, 1𝑜〉]
~Q )) <Q 𝑠) → 𝑚 ∈ N) |
62 | 60, 61 | ffvelrnd 5303 |
. . . . . . . . . . . . . . . 16
⊢
((((((𝜑 ∧ 𝑟 ∈ (2nd
‘(𝐿
+P 〈{𝑙 ∣ 𝑙 <Q 𝑆}, {𝑢 ∣ 𝑆 <Q 𝑢}〉))) ∧ (𝑠 ∈ (2nd
‘𝐿) ∧ 𝑡 ∈ (2nd
‘〈{𝑙 ∣
𝑙
<Q 𝑆}, {𝑢 ∣ 𝑆 <Q 𝑢}〉))) ∧ 𝑟 = (𝑠 +Q 𝑡)) ∧ 𝑚 ∈ N) ∧ ((𝐹‘𝑚) +Q
(*Q‘[〈𝑚, 1𝑜〉]
~Q )) <Q 𝑠) → (𝐹‘𝑚) ∈ Q) |
63 | | nnnq 6520 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑚 ∈ N →
[〈𝑚,
1𝑜〉] ~Q ∈
Q) |
64 | | recclnq 6490 |
. . . . . . . . . . . . . . . . 17
⊢
([〈𝑚,
1𝑜〉] ~Q ∈ Q
→ (*Q‘[〈𝑚, 1𝑜〉]
~Q ) ∈ Q) |
65 | 61, 63, 64 | 3syl 17 |
. . . . . . . . . . . . . . . 16
⊢
((((((𝜑 ∧ 𝑟 ∈ (2nd
‘(𝐿
+P 〈{𝑙 ∣ 𝑙 <Q 𝑆}, {𝑢 ∣ 𝑆 <Q 𝑢}〉))) ∧ (𝑠 ∈ (2nd
‘𝐿) ∧ 𝑡 ∈ (2nd
‘〈{𝑙 ∣
𝑙
<Q 𝑆}, {𝑢 ∣ 𝑆 <Q 𝑢}〉))) ∧ 𝑟 = (𝑠 +Q 𝑡)) ∧ 𝑚 ∈ N) ∧ ((𝐹‘𝑚) +Q
(*Q‘[〈𝑚, 1𝑜〉]
~Q )) <Q 𝑠) →
(*Q‘[〈𝑚, 1𝑜〉]
~Q ) ∈ Q) |
66 | | addclnq 6473 |
. . . . . . . . . . . . . . . 16
⊢ (((𝐹‘𝑚) ∈ Q ∧
(*Q‘[〈𝑚, 1𝑜〉]
~Q ) ∈ Q) → ((𝐹‘𝑚) +Q
(*Q‘[〈𝑚, 1𝑜〉]
~Q )) ∈ Q) |
67 | 62, 65, 66 | syl2anc 391 |
. . . . . . . . . . . . . . 15
⊢
((((((𝜑 ∧ 𝑟 ∈ (2nd
‘(𝐿
+P 〈{𝑙 ∣ 𝑙 <Q 𝑆}, {𝑢 ∣ 𝑆 <Q 𝑢}〉))) ∧ (𝑠 ∈ (2nd
‘𝐿) ∧ 𝑡 ∈ (2nd
‘〈{𝑙 ∣
𝑙
<Q 𝑆}, {𝑢 ∣ 𝑆 <Q 𝑢}〉))) ∧ 𝑟 = (𝑠 +Q 𝑡)) ∧ 𝑚 ∈ N) ∧ ((𝐹‘𝑚) +Q
(*Q‘[〈𝑚, 1𝑜〉]
~Q )) <Q 𝑠) → ((𝐹‘𝑚) +Q
(*Q‘[〈𝑚, 1𝑜〉]
~Q )) ∈ Q) |
68 | | ltanqg 6498 |
. . . . . . . . . . . . . . 15
⊢ ((𝑆 ∈ Q ∧
𝑡 ∈ Q
∧ ((𝐹‘𝑚) +Q
(*Q‘[〈𝑚, 1𝑜〉]
~Q )) ∈ Q) → (𝑆 <Q
𝑡 ↔ (((𝐹‘𝑚) +Q
(*Q‘[〈𝑚, 1𝑜〉]
~Q )) +Q 𝑆) <Q (((𝐹‘𝑚) +Q
(*Q‘[〈𝑚, 1𝑜〉]
~Q )) +Q 𝑡))) |
69 | 58, 59, 67, 68 | syl3anc 1135 |
. . . . . . . . . . . . . 14
⊢
((((((𝜑 ∧ 𝑟 ∈ (2nd
‘(𝐿
+P 〈{𝑙 ∣ 𝑙 <Q 𝑆}, {𝑢 ∣ 𝑆 <Q 𝑢}〉))) ∧ (𝑠 ∈ (2nd
‘𝐿) ∧ 𝑡 ∈ (2nd
‘〈{𝑙 ∣
𝑙
<Q 𝑆}, {𝑢 ∣ 𝑆 <Q 𝑢}〉))) ∧ 𝑟 = (𝑠 +Q 𝑡)) ∧ 𝑚 ∈ N) ∧ ((𝐹‘𝑚) +Q
(*Q‘[〈𝑚, 1𝑜〉]
~Q )) <Q 𝑠) → (𝑆 <Q 𝑡 ↔ (((𝐹‘𝑚) +Q
(*Q‘[〈𝑚, 1𝑜〉]
~Q )) +Q 𝑆) <Q (((𝐹‘𝑚) +Q
(*Q‘[〈𝑚, 1𝑜〉]
~Q )) +Q 𝑡))) |
70 | 57, 69 | mpbid 135 |
. . . . . . . . . . . . 13
⊢
((((((𝜑 ∧ 𝑟 ∈ (2nd
‘(𝐿
+P 〈{𝑙 ∣ 𝑙 <Q 𝑆}, {𝑢 ∣ 𝑆 <Q 𝑢}〉))) ∧ (𝑠 ∈ (2nd
‘𝐿) ∧ 𝑡 ∈ (2nd
‘〈{𝑙 ∣
𝑙
<Q 𝑆}, {𝑢 ∣ 𝑆 <Q 𝑢}〉))) ∧ 𝑟 = (𝑠 +Q 𝑡)) ∧ 𝑚 ∈ N) ∧ ((𝐹‘𝑚) +Q
(*Q‘[〈𝑚, 1𝑜〉]
~Q )) <Q 𝑠) → (((𝐹‘𝑚) +Q
(*Q‘[〈𝑚, 1𝑜〉]
~Q )) +Q 𝑆) <Q (((𝐹‘𝑚) +Q
(*Q‘[〈𝑚, 1𝑜〉]
~Q )) +Q 𝑡)) |
71 | | simpr 103 |
. . . . . . . . . . . . . 14
⊢
((((((𝜑 ∧ 𝑟 ∈ (2nd
‘(𝐿
+P 〈{𝑙 ∣ 𝑙 <Q 𝑆}, {𝑢 ∣ 𝑆 <Q 𝑢}〉))) ∧ (𝑠 ∈ (2nd
‘𝐿) ∧ 𝑡 ∈ (2nd
‘〈{𝑙 ∣
𝑙
<Q 𝑆}, {𝑢 ∣ 𝑆 <Q 𝑢}〉))) ∧ 𝑟 = (𝑠 +Q 𝑡)) ∧ 𝑚 ∈ N) ∧ ((𝐹‘𝑚) +Q
(*Q‘[〈𝑚, 1𝑜〉]
~Q )) <Q 𝑠) → ((𝐹‘𝑚) +Q
(*Q‘[〈𝑚, 1𝑜〉]
~Q )) <Q 𝑠) |
72 | | ltanqg 6498 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑧 ∈ Q ∧
𝑤 ∈ Q
∧ 𝑣 ∈
Q) → (𝑧
<Q 𝑤 ↔ (𝑣 +Q 𝑧) <Q
(𝑣
+Q 𝑤))) |
73 | 72 | adantl 262 |
. . . . . . . . . . . . . . 15
⊢
(((((((𝜑 ∧ 𝑟 ∈ (2nd
‘(𝐿
+P 〈{𝑙 ∣ 𝑙 <Q 𝑆}, {𝑢 ∣ 𝑆 <Q 𝑢}〉))) ∧ (𝑠 ∈ (2nd
‘𝐿) ∧ 𝑡 ∈ (2nd
‘〈{𝑙 ∣
𝑙
<Q 𝑆}, {𝑢 ∣ 𝑆 <Q 𝑢}〉))) ∧ 𝑟 = (𝑠 +Q 𝑡)) ∧ 𝑚 ∈ N) ∧ ((𝐹‘𝑚) +Q
(*Q‘[〈𝑚, 1𝑜〉]
~Q )) <Q 𝑠) ∧ (𝑧 ∈ Q ∧ 𝑤 ∈ Q ∧
𝑣 ∈ Q))
→ (𝑧
<Q 𝑤 ↔ (𝑣 +Q 𝑧) <Q
(𝑣
+Q 𝑤))) |
74 | 27 | ad2antrr 457 |
. . . . . . . . . . . . . . 15
⊢
((((((𝜑 ∧ 𝑟 ∈ (2nd
‘(𝐿
+P 〈{𝑙 ∣ 𝑙 <Q 𝑆}, {𝑢 ∣ 𝑆 <Q 𝑢}〉))) ∧ (𝑠 ∈ (2nd
‘𝐿) ∧ 𝑡 ∈ (2nd
‘〈{𝑙 ∣
𝑙
<Q 𝑆}, {𝑢 ∣ 𝑆 <Q 𝑢}〉))) ∧ 𝑟 = (𝑠 +Q 𝑡)) ∧ 𝑚 ∈ N) ∧ ((𝐹‘𝑚) +Q
(*Q‘[〈𝑚, 1𝑜〉]
~Q )) <Q 𝑠) → 𝑠 ∈ Q) |
75 | | addcomnqg 6479 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑧 ∈ Q ∧
𝑤 ∈ Q)
→ (𝑧
+Q 𝑤) = (𝑤 +Q 𝑧)) |
76 | 75 | adantl 262 |
. . . . . . . . . . . . . . 15
⊢
(((((((𝜑 ∧ 𝑟 ∈ (2nd
‘(𝐿
+P 〈{𝑙 ∣ 𝑙 <Q 𝑆}, {𝑢 ∣ 𝑆 <Q 𝑢}〉))) ∧ (𝑠 ∈ (2nd
‘𝐿) ∧ 𝑡 ∈ (2nd
‘〈{𝑙 ∣
𝑙
<Q 𝑆}, {𝑢 ∣ 𝑆 <Q 𝑢}〉))) ∧ 𝑟 = (𝑠 +Q 𝑡)) ∧ 𝑚 ∈ N) ∧ ((𝐹‘𝑚) +Q
(*Q‘[〈𝑚, 1𝑜〉]
~Q )) <Q 𝑠) ∧ (𝑧 ∈ Q ∧ 𝑤 ∈ Q)) →
(𝑧
+Q 𝑤) = (𝑤 +Q 𝑧)) |
77 | 73, 67, 74, 59, 76 | caovord2d 5670 |
. . . . . . . . . . . . . 14
⊢
((((((𝜑 ∧ 𝑟 ∈ (2nd
‘(𝐿
+P 〈{𝑙 ∣ 𝑙 <Q 𝑆}, {𝑢 ∣ 𝑆 <Q 𝑢}〉))) ∧ (𝑠 ∈ (2nd
‘𝐿) ∧ 𝑡 ∈ (2nd
‘〈{𝑙 ∣
𝑙
<Q 𝑆}, {𝑢 ∣ 𝑆 <Q 𝑢}〉))) ∧ 𝑟 = (𝑠 +Q 𝑡)) ∧ 𝑚 ∈ N) ∧ ((𝐹‘𝑚) +Q
(*Q‘[〈𝑚, 1𝑜〉]
~Q )) <Q 𝑠) → (((𝐹‘𝑚) +Q
(*Q‘[〈𝑚, 1𝑜〉]
~Q )) <Q 𝑠 ↔ (((𝐹‘𝑚) +Q
(*Q‘[〈𝑚, 1𝑜〉]
~Q )) +Q 𝑡) <Q (𝑠 +Q
𝑡))) |
78 | 71, 77 | mpbid 135 |
. . . . . . . . . . . . 13
⊢
((((((𝜑 ∧ 𝑟 ∈ (2nd
‘(𝐿
+P 〈{𝑙 ∣ 𝑙 <Q 𝑆}, {𝑢 ∣ 𝑆 <Q 𝑢}〉))) ∧ (𝑠 ∈ (2nd
‘𝐿) ∧ 𝑡 ∈ (2nd
‘〈{𝑙 ∣
𝑙
<Q 𝑆}, {𝑢 ∣ 𝑆 <Q 𝑢}〉))) ∧ 𝑟 = (𝑠 +Q 𝑡)) ∧ 𝑚 ∈ N) ∧ ((𝐹‘𝑚) +Q
(*Q‘[〈𝑚, 1𝑜〉]
~Q )) <Q 𝑠) → (((𝐹‘𝑚) +Q
(*Q‘[〈𝑚, 1𝑜〉]
~Q )) +Q 𝑡) <Q (𝑠 +Q
𝑡)) |
79 | | ltsonq 6496 |
. . . . . . . . . . . . . 14
⊢
<Q Or Q |
80 | 79, 34 | sotri 4720 |
. . . . . . . . . . . . 13
⊢
(((((𝐹‘𝑚) +Q
(*Q‘[〈𝑚, 1𝑜〉]
~Q )) +Q 𝑆) <Q (((𝐹‘𝑚) +Q
(*Q‘[〈𝑚, 1𝑜〉]
~Q )) +Q 𝑡) ∧ (((𝐹‘𝑚) +Q
(*Q‘[〈𝑚, 1𝑜〉]
~Q )) +Q 𝑡) <Q (𝑠 +Q
𝑡)) → (((𝐹‘𝑚) +Q
(*Q‘[〈𝑚, 1𝑜〉]
~Q )) +Q 𝑆) <Q (𝑠 +Q
𝑡)) |
81 | 70, 78, 80 | syl2anc 391 |
. . . . . . . . . . . 12
⊢
((((((𝜑 ∧ 𝑟 ∈ (2nd
‘(𝐿
+P 〈{𝑙 ∣ 𝑙 <Q 𝑆}, {𝑢 ∣ 𝑆 <Q 𝑢}〉))) ∧ (𝑠 ∈ (2nd
‘𝐿) ∧ 𝑡 ∈ (2nd
‘〈{𝑙 ∣
𝑙
<Q 𝑆}, {𝑢 ∣ 𝑆 <Q 𝑢}〉))) ∧ 𝑟 = (𝑠 +Q 𝑡)) ∧ 𝑚 ∈ N) ∧ ((𝐹‘𝑚) +Q
(*Q‘[〈𝑚, 1𝑜〉]
~Q )) <Q 𝑠) → (((𝐹‘𝑚) +Q
(*Q‘[〈𝑚, 1𝑜〉]
~Q )) +Q 𝑆) <Q (𝑠 +Q
𝑡)) |
82 | | simpllr 486 |
. . . . . . . . . . . 12
⊢
((((((𝜑 ∧ 𝑟 ∈ (2nd
‘(𝐿
+P 〈{𝑙 ∣ 𝑙 <Q 𝑆}, {𝑢 ∣ 𝑆 <Q 𝑢}〉))) ∧ (𝑠 ∈ (2nd
‘𝐿) ∧ 𝑡 ∈ (2nd
‘〈{𝑙 ∣
𝑙
<Q 𝑆}, {𝑢 ∣ 𝑆 <Q 𝑢}〉))) ∧ 𝑟 = (𝑠 +Q 𝑡)) ∧ 𝑚 ∈ N) ∧ ((𝐹‘𝑚) +Q
(*Q‘[〈𝑚, 1𝑜〉]
~Q )) <Q 𝑠) → 𝑟 = (𝑠 +Q 𝑡)) |
83 | 81, 82 | breqtrrd 3790 |
. . . . . . . . . . 11
⊢
((((((𝜑 ∧ 𝑟 ∈ (2nd
‘(𝐿
+P 〈{𝑙 ∣ 𝑙 <Q 𝑆}, {𝑢 ∣ 𝑆 <Q 𝑢}〉))) ∧ (𝑠 ∈ (2nd
‘𝐿) ∧ 𝑡 ∈ (2nd
‘〈{𝑙 ∣
𝑙
<Q 𝑆}, {𝑢 ∣ 𝑆 <Q 𝑢}〉))) ∧ 𝑟 = (𝑠 +Q 𝑡)) ∧ 𝑚 ∈ N) ∧ ((𝐹‘𝑚) +Q
(*Q‘[〈𝑚, 1𝑜〉]
~Q )) <Q 𝑠) → (((𝐹‘𝑚) +Q
(*Q‘[〈𝑚, 1𝑜〉]
~Q )) +Q 𝑆) <Q 𝑟) |
84 | 83 | ex 108 |
. . . . . . . . . 10
⊢
(((((𝜑 ∧ 𝑟 ∈ (2nd
‘(𝐿
+P 〈{𝑙 ∣ 𝑙 <Q 𝑆}, {𝑢 ∣ 𝑆 <Q 𝑢}〉))) ∧ (𝑠 ∈ (2nd
‘𝐿) ∧ 𝑡 ∈ (2nd
‘〈{𝑙 ∣
𝑙
<Q 𝑆}, {𝑢 ∣ 𝑆 <Q 𝑢}〉))) ∧ 𝑟 = (𝑠 +Q 𝑡)) ∧ 𝑚 ∈ N) → (((𝐹‘𝑚) +Q
(*Q‘[〈𝑚, 1𝑜〉]
~Q )) <Q 𝑠 → (((𝐹‘𝑚) +Q
(*Q‘[〈𝑚, 1𝑜〉]
~Q )) +Q 𝑆) <Q 𝑟)) |
85 | 84 | reximdva 2421 |
. . . . . . . . 9
⊢ ((((𝜑 ∧ 𝑟 ∈ (2nd ‘(𝐿 +P
〈{𝑙 ∣ 𝑙 <Q
𝑆}, {𝑢 ∣ 𝑆 <Q 𝑢}〉))) ∧ (𝑠 ∈ (2nd
‘𝐿) ∧ 𝑡 ∈ (2nd
‘〈{𝑙 ∣
𝑙
<Q 𝑆}, {𝑢 ∣ 𝑆 <Q 𝑢}〉))) ∧ 𝑟 = (𝑠 +Q 𝑡)) → (∃𝑚 ∈ N ((𝐹‘𝑚) +Q
(*Q‘[〈𝑚, 1𝑜〉]
~Q )) <Q 𝑠 → ∃𝑚 ∈ N (((𝐹‘𝑚) +Q
(*Q‘[〈𝑚, 1𝑜〉]
~Q )) +Q 𝑆) <Q 𝑟)) |
86 | 53, 85 | mpd 13 |
. . . . . . . 8
⊢ ((((𝜑 ∧ 𝑟 ∈ (2nd ‘(𝐿 +P
〈{𝑙 ∣ 𝑙 <Q
𝑆}, {𝑢 ∣ 𝑆 <Q 𝑢}〉))) ∧ (𝑠 ∈ (2nd
‘𝐿) ∧ 𝑡 ∈ (2nd
‘〈{𝑙 ∣
𝑙
<Q 𝑆}, {𝑢 ∣ 𝑆 <Q 𝑢}〉))) ∧ 𝑟 = (𝑠 +Q 𝑡)) → ∃𝑚 ∈ N (((𝐹‘𝑚) +Q
(*Q‘[〈𝑚, 1𝑜〉]
~Q )) +Q 𝑆) <Q 𝑟) |
87 | 50 | oveq1d 5527 |
. . . . . . . . . 10
⊢ (𝑗 = 𝑚 → (((𝐹‘𝑗) +Q
(*Q‘[〈𝑗, 1𝑜〉]
~Q )) +Q 𝑆) = (((𝐹‘𝑚) +Q
(*Q‘[〈𝑚, 1𝑜〉]
~Q )) +Q 𝑆)) |
88 | 87 | breq1d 3774 |
. . . . . . . . 9
⊢ (𝑗 = 𝑚 → ((((𝐹‘𝑗) +Q
(*Q‘[〈𝑗, 1𝑜〉]
~Q )) +Q 𝑆) <Q 𝑟 ↔ (((𝐹‘𝑚) +Q
(*Q‘[〈𝑚, 1𝑜〉]
~Q )) +Q 𝑆) <Q 𝑟)) |
89 | 88 | cbvrexv 2534 |
. . . . . . . 8
⊢
(∃𝑗 ∈
N (((𝐹‘𝑗) +Q
(*Q‘[〈𝑗, 1𝑜〉]
~Q )) +Q 𝑆) <Q 𝑟 ↔ ∃𝑚 ∈ N (((𝐹‘𝑚) +Q
(*Q‘[〈𝑚, 1𝑜〉]
~Q )) +Q 𝑆) <Q 𝑟) |
90 | 86, 89 | sylibr 137 |
. . . . . . 7
⊢ ((((𝜑 ∧ 𝑟 ∈ (2nd ‘(𝐿 +P
〈{𝑙 ∣ 𝑙 <Q
𝑆}, {𝑢 ∣ 𝑆 <Q 𝑢}〉))) ∧ (𝑠 ∈ (2nd
‘𝐿) ∧ 𝑡 ∈ (2nd
‘〈{𝑙 ∣
𝑙
<Q 𝑆}, {𝑢 ∣ 𝑆 <Q 𝑢}〉))) ∧ 𝑟 = (𝑠 +Q 𝑡)) → ∃𝑗 ∈ N (((𝐹‘𝑗) +Q
(*Q‘[〈𝑗, 1𝑜〉]
~Q )) +Q 𝑆) <Q 𝑟) |
91 | | breq2 3768 |
. . . . . . . . 9
⊢ (𝑢 = 𝑟 → ((((𝐹‘𝑗) +Q
(*Q‘[〈𝑗, 1𝑜〉]
~Q )) +Q 𝑆) <Q 𝑢 ↔ (((𝐹‘𝑗) +Q
(*Q‘[〈𝑗, 1𝑜〉]
~Q )) +Q 𝑆) <Q 𝑟)) |
92 | 91 | rexbidv 2327 |
. . . . . . . 8
⊢ (𝑢 = 𝑟 → (∃𝑗 ∈ N (((𝐹‘𝑗) +Q
(*Q‘[〈𝑗, 1𝑜〉]
~Q )) +Q 𝑆) <Q 𝑢 ↔ ∃𝑗 ∈ N (((𝐹‘𝑗) +Q
(*Q‘[〈𝑗, 1𝑜〉]
~Q )) +Q 𝑆) <Q 𝑟)) |
93 | 92 | elrab 2698 |
. . . . . . 7
⊢ (𝑟 ∈ {𝑢 ∈ Q ∣ ∃𝑗 ∈ N (((𝐹‘𝑗) +Q
(*Q‘[〈𝑗, 1𝑜〉]
~Q )) +Q 𝑆) <Q 𝑢} ↔ (𝑟 ∈ Q ∧ ∃𝑗 ∈ N (((𝐹‘𝑗) +Q
(*Q‘[〈𝑗, 1𝑜〉]
~Q )) +Q 𝑆) <Q 𝑟)) |
94 | 44, 90, 93 | sylanbrc 394 |
. . . . . 6
⊢ ((((𝜑 ∧ 𝑟 ∈ (2nd ‘(𝐿 +P
〈{𝑙 ∣ 𝑙 <Q
𝑆}, {𝑢 ∣ 𝑆 <Q 𝑢}〉))) ∧ (𝑠 ∈ (2nd
‘𝐿) ∧ 𝑡 ∈ (2nd
‘〈{𝑙 ∣
𝑙
<Q 𝑆}, {𝑢 ∣ 𝑆 <Q 𝑢}〉))) ∧ 𝑟 = (𝑠 +Q 𝑡)) → 𝑟 ∈ {𝑢 ∈ Q ∣ ∃𝑗 ∈ N (((𝐹‘𝑗) +Q
(*Q‘[〈𝑗, 1𝑜〉]
~Q )) +Q 𝑆) <Q 𝑢}) |
95 | 94 | ex 108 |
. . . . 5
⊢ (((𝜑 ∧ 𝑟 ∈ (2nd ‘(𝐿 +P
〈{𝑙 ∣ 𝑙 <Q
𝑆}, {𝑢 ∣ 𝑆 <Q 𝑢}〉))) ∧ (𝑠 ∈ (2nd
‘𝐿) ∧ 𝑡 ∈ (2nd
‘〈{𝑙 ∣
𝑙
<Q 𝑆}, {𝑢 ∣ 𝑆 <Q 𝑢}〉))) → (𝑟 = (𝑠 +Q 𝑡) → 𝑟 ∈ {𝑢 ∈ Q ∣ ∃𝑗 ∈ N (((𝐹‘𝑗) +Q
(*Q‘[〈𝑗, 1𝑜〉]
~Q )) +Q 𝑆) <Q 𝑢})) |
96 | 95 | rexlimdvva 2440 |
. . . 4
⊢ ((𝜑 ∧ 𝑟 ∈ (2nd ‘(𝐿 +P
〈{𝑙 ∣ 𝑙 <Q
𝑆}, {𝑢 ∣ 𝑆 <Q 𝑢}〉))) → (∃𝑠 ∈ (2nd
‘𝐿)∃𝑡 ∈ (2nd
‘〈{𝑙 ∣
𝑙
<Q 𝑆}, {𝑢 ∣ 𝑆 <Q 𝑢}〉)𝑟 = (𝑠 +Q 𝑡) → 𝑟 ∈ {𝑢 ∈ Q ∣ ∃𝑗 ∈ N (((𝐹‘𝑗) +Q
(*Q‘[〈𝑗, 1𝑜〉]
~Q )) +Q 𝑆) <Q 𝑢})) |
97 | 13, 96 | mpd 13 |
. . 3
⊢ ((𝜑 ∧ 𝑟 ∈ (2nd ‘(𝐿 +P
〈{𝑙 ∣ 𝑙 <Q
𝑆}, {𝑢 ∣ 𝑆 <Q 𝑢}〉))) → 𝑟 ∈ {𝑢 ∈ Q ∣ ∃𝑗 ∈ N (((𝐹‘𝑗) +Q
(*Q‘[〈𝑗, 1𝑜〉]
~Q )) +Q 𝑆) <Q 𝑢}) |
98 | 97 | ex 108 |
. 2
⊢ (𝜑 → (𝑟 ∈ (2nd ‘(𝐿 +P
〈{𝑙 ∣ 𝑙 <Q
𝑆}, {𝑢 ∣ 𝑆 <Q 𝑢}〉)) → 𝑟 ∈ {𝑢 ∈ Q ∣ ∃𝑗 ∈ N (((𝐹‘𝑗) +Q
(*Q‘[〈𝑗, 1𝑜〉]
~Q )) +Q 𝑆) <Q 𝑢})) |
99 | 98 | ssrdv 2951 |
1
⊢ (𝜑 → (2nd
‘(𝐿
+P 〈{𝑙 ∣ 𝑙 <Q 𝑆}, {𝑢 ∣ 𝑆 <Q 𝑢}〉)) ⊆ {𝑢 ∈ Q ∣
∃𝑗 ∈
N (((𝐹‘𝑗) +Q
(*Q‘[〈𝑗, 1𝑜〉]
~Q )) +Q 𝑆) <Q 𝑢}) |