Step | Hyp | Ref
| Expression |
1 | | ltexprlem.1 |
. . . . 5
⊢ 𝐶 = 〈{x ∈
Q ∣ ∃y(y ∈ (2nd ‘A) ∧ (y +Q x) ∈
(1st ‘B))}, {x ∈
Q ∣ ∃y(y ∈ (1st ‘A) ∧ (y +Q x) ∈
(2nd ‘B))}〉 |
2 | 1 | ltexprlemell 6572 |
. . . 4
⊢ (𝑞 ∈ (1st ‘𝐶) ↔ (𝑞 ∈
Q ∧ ∃y(y ∈
(2nd ‘A) ∧ (y
+Q 𝑞)
∈ (1st ‘B)))) |
3 | 2 | simprbi 260 |
. . 3
⊢ (𝑞 ∈ (1st ‘𝐶) → ∃y(y ∈
(2nd ‘A) ∧ (y
+Q 𝑞)
∈ (1st ‘B))) |
4 | | 19.42v 1783 |
. . . . . . . 8
⊢ (∃y(A<P B ∧ (𝑞 ∈ Q ∧ (y ∈ (2nd ‘A) ∧ (y +Q 𝑞) ∈
(1st ‘B)))) ↔
(A<P B ∧ ∃y(𝑞 ∈ Q ∧ (y ∈ (2nd ‘A) ∧ (y +Q 𝑞) ∈
(1st ‘B))))) |
5 | | 19.42v 1783 |
. . . . . . . . 9
⊢ (∃y(𝑞 ∈ Q ∧ (y ∈ (2nd ‘A) ∧ (y +Q 𝑞) ∈
(1st ‘B))) ↔ (𝑞 ∈ Q ∧ ∃y(y ∈ (2nd ‘A) ∧ (y +Q 𝑞) ∈
(1st ‘B)))) |
6 | 5 | anbi2i 430 |
. . . . . . . 8
⊢
((A<P
B ∧ ∃y(𝑞 ∈ Q ∧ (y ∈ (2nd ‘A) ∧ (y +Q 𝑞) ∈
(1st ‘B)))) ↔
(A<P B ∧ (𝑞 ∈ Q ∧ ∃y(y ∈ (2nd ‘A) ∧ (y +Q 𝑞) ∈
(1st ‘B))))) |
7 | 4, 6 | bitri 173 |
. . . . . . 7
⊢ (∃y(A<P B ∧ (𝑞 ∈ Q ∧ (y ∈ (2nd ‘A) ∧ (y +Q 𝑞) ∈
(1st ‘B)))) ↔
(A<P B ∧ (𝑞 ∈ Q ∧ ∃y(y ∈ (2nd ‘A) ∧ (y +Q 𝑞) ∈
(1st ‘B))))) |
8 | | ltrelpr 6488 |
. . . . . . . . . . . . . 14
⊢
<P ⊆ (P ×
P) |
9 | 8 | brel 4335 |
. . . . . . . . . . . . 13
⊢ (A<P B → (A
∈ P ∧ B ∈ P)) |
10 | 9 | simprd 107 |
. . . . . . . . . . . 12
⊢ (A<P B → B ∈ P) |
11 | | prop 6458 |
. . . . . . . . . . . . 13
⊢ (B ∈
P → 〈(1st ‘B), (2nd ‘B)〉 ∈
P) |
12 | | prnmaxl 6471 |
. . . . . . . . . . . . 13
⊢
((〈(1st ‘B),
(2nd ‘B)〉 ∈ P ∧ (y
+Q 𝑞)
∈ (1st ‘B)) → ∃𝑠 ∈ (1st ‘B)(y
+Q 𝑞)
<Q 𝑠) |
13 | 11, 12 | sylan 267 |
. . . . . . . . . . . 12
⊢
((B ∈ P ∧ (y
+Q 𝑞)
∈ (1st ‘B)) → ∃𝑠 ∈ (1st ‘B)(y
+Q 𝑞)
<Q 𝑠) |
14 | 10, 13 | sylan 267 |
. . . . . . . . . . 11
⊢
((A<P
B ∧
(y +Q 𝑞) ∈ (1st ‘B)) → ∃𝑠 ∈ (1st ‘B)(y
+Q 𝑞)
<Q 𝑠) |
15 | 14 | adantrl 447 |
. . . . . . . . . 10
⊢
((A<P
B ∧
(y ∈
(2nd ‘A) ∧ (y
+Q 𝑞)
∈ (1st ‘B))) → ∃𝑠 ∈
(1st ‘B)(y +Q 𝑞) <Q 𝑠) |
16 | 15 | adantrl 447 |
. . . . . . . . 9
⊢
((A<P
B ∧ (𝑞 ∈ Q ∧ (y ∈ (2nd ‘A) ∧ (y +Q 𝑞) ∈
(1st ‘B)))) → ∃𝑠 ∈
(1st ‘B)(y +Q 𝑞) <Q 𝑠) |
17 | 9 | simpld 105 |
. . . . . . . . . . . . . . 15
⊢ (A<P B → A ∈ P) |
18 | 17 | ad2antrr 457 |
. . . . . . . . . . . . . 14
⊢
(((A<P
B ∧ (𝑞 ∈ Q ∧ (y ∈ (2nd ‘A) ∧ (y +Q 𝑞) ∈
(1st ‘B)))) ∧ (𝑠 ∈
(1st ‘B) ∧ (y
+Q 𝑞)
<Q 𝑠)) → A ∈
P) |
19 | | simplrr 488 |
. . . . . . . . . . . . . . 15
⊢
(((A<P
B ∧ (𝑞 ∈ Q ∧ (y ∈ (2nd ‘A) ∧ (y +Q 𝑞) ∈
(1st ‘B)))) ∧ (𝑠 ∈
(1st ‘B) ∧ (y
+Q 𝑞)
<Q 𝑠)) → (y ∈
(2nd ‘A) ∧ (y
+Q 𝑞)
∈ (1st ‘B))) |
20 | 19 | simpld 105 |
. . . . . . . . . . . . . 14
⊢
(((A<P
B ∧ (𝑞 ∈ Q ∧ (y ∈ (2nd ‘A) ∧ (y +Q 𝑞) ∈
(1st ‘B)))) ∧ (𝑠 ∈
(1st ‘B) ∧ (y
+Q 𝑞)
<Q 𝑠)) → y ∈
(2nd ‘A)) |
21 | | prop 6458 |
. . . . . . . . . . . . . . 15
⊢ (A ∈
P → 〈(1st ‘A), (2nd ‘A)〉 ∈
P) |
22 | | elprnqu 6465 |
. . . . . . . . . . . . . . 15
⊢
((〈(1st ‘A),
(2nd ‘A)〉 ∈ P ∧ y ∈ (2nd ‘A)) → y
∈ Q) |
23 | 21, 22 | sylan 267 |
. . . . . . . . . . . . . 14
⊢
((A ∈ P ∧ y ∈ (2nd ‘A)) → y
∈ Q) |
24 | 18, 20, 23 | syl2anc 391 |
. . . . . . . . . . . . 13
⊢
(((A<P
B ∧ (𝑞 ∈ Q ∧ (y ∈ (2nd ‘A) ∧ (y +Q 𝑞) ∈
(1st ‘B)))) ∧ (𝑠 ∈
(1st ‘B) ∧ (y
+Q 𝑞)
<Q 𝑠)) → y ∈
Q) |
25 | | simplrl 487 |
. . . . . . . . . . . . 13
⊢
(((A<P
B ∧ (𝑞 ∈ Q ∧ (y ∈ (2nd ‘A) ∧ (y +Q 𝑞) ∈
(1st ‘B)))) ∧ (𝑠 ∈
(1st ‘B) ∧ (y
+Q 𝑞)
<Q 𝑠)) → 𝑞 ∈
Q) |
26 | | ltaddnq 6390 |
. . . . . . . . . . . . 13
⊢
((y ∈ Q ∧ 𝑞
∈ Q) → y <Q (y +Q 𝑞)) |
27 | 24, 25, 26 | syl2anc 391 |
. . . . . . . . . . . 12
⊢
(((A<P
B ∧ (𝑞 ∈ Q ∧ (y ∈ (2nd ‘A) ∧ (y +Q 𝑞) ∈
(1st ‘B)))) ∧ (𝑠 ∈
(1st ‘B) ∧ (y
+Q 𝑞)
<Q 𝑠)) → y <Q (y +Q 𝑞)) |
28 | | simprr 484 |
. . . . . . . . . . . 12
⊢
(((A<P
B ∧ (𝑞 ∈ Q ∧ (y ∈ (2nd ‘A) ∧ (y +Q 𝑞) ∈
(1st ‘B)))) ∧ (𝑠 ∈
(1st ‘B) ∧ (y
+Q 𝑞)
<Q 𝑠)) → (y +Q 𝑞) <Q 𝑠) |
29 | | ltsonq 6382 |
. . . . . . . . . . . . 13
⊢
<Q Or Q |
30 | | ltrelnq 6349 |
. . . . . . . . . . . . 13
⊢
<Q ⊆ (Q ×
Q) |
31 | 29, 30 | sotri 4663 |
. . . . . . . . . . . 12
⊢
((y <Q
(y +Q 𝑞) ∧ (y
+Q 𝑞)
<Q 𝑠) → y <Q 𝑠) |
32 | 27, 28, 31 | syl2anc 391 |
. . . . . . . . . . 11
⊢
(((A<P
B ∧ (𝑞 ∈ Q ∧ (y ∈ (2nd ‘A) ∧ (y +Q 𝑞) ∈
(1st ‘B)))) ∧ (𝑠 ∈
(1st ‘B) ∧ (y
+Q 𝑞)
<Q 𝑠)) → y <Q 𝑠) |
33 | 10 | ad2antrr 457 |
. . . . . . . . . . . . 13
⊢
(((A<P
B ∧ (𝑞 ∈ Q ∧ (y ∈ (2nd ‘A) ∧ (y +Q 𝑞) ∈
(1st ‘B)))) ∧ (𝑠 ∈
(1st ‘B) ∧ (y
+Q 𝑞)
<Q 𝑠)) → B ∈
P) |
34 | | simprl 483 |
. . . . . . . . . . . . 13
⊢
(((A<P
B ∧ (𝑞 ∈ Q ∧ (y ∈ (2nd ‘A) ∧ (y +Q 𝑞) ∈
(1st ‘B)))) ∧ (𝑠 ∈
(1st ‘B) ∧ (y
+Q 𝑞)
<Q 𝑠)) → 𝑠 ∈
(1st ‘B)) |
35 | | elprnql 6464 |
. . . . . . . . . . . . . 14
⊢
((〈(1st ‘B),
(2nd ‘B)〉 ∈ P ∧ 𝑠
∈ (1st ‘B)) → 𝑠 ∈
Q) |
36 | 11, 35 | sylan 267 |
. . . . . . . . . . . . 13
⊢
((B ∈ P ∧ 𝑠
∈ (1st ‘B)) → 𝑠 ∈
Q) |
37 | 33, 34, 36 | syl2anc 391 |
. . . . . . . . . . . 12
⊢
(((A<P
B ∧ (𝑞 ∈ Q ∧ (y ∈ (2nd ‘A) ∧ (y +Q 𝑞) ∈
(1st ‘B)))) ∧ (𝑠 ∈
(1st ‘B) ∧ (y
+Q 𝑞)
<Q 𝑠)) → 𝑠 ∈
Q) |
38 | | ltexnqq 6391 |
. . . . . . . . . . . 12
⊢
((y ∈ Q ∧ 𝑠
∈ Q) → (y <Q 𝑠 ↔ ∃𝑟 ∈
Q (y
+Q 𝑟)
= 𝑠)) |
39 | 24, 37, 38 | syl2anc 391 |
. . . . . . . . . . 11
⊢
(((A<P
B ∧ (𝑞 ∈ Q ∧ (y ∈ (2nd ‘A) ∧ (y +Q 𝑞) ∈
(1st ‘B)))) ∧ (𝑠 ∈
(1st ‘B) ∧ (y
+Q 𝑞)
<Q 𝑠)) → (y <Q 𝑠 ↔ ∃𝑟 ∈
Q (y
+Q 𝑟)
= 𝑠)) |
40 | 32, 39 | mpbid 135 |
. . . . . . . . . 10
⊢
(((A<P
B ∧ (𝑞 ∈ Q ∧ (y ∈ (2nd ‘A) ∧ (y +Q 𝑞) ∈
(1st ‘B)))) ∧ (𝑠 ∈
(1st ‘B) ∧ (y
+Q 𝑞)
<Q 𝑠)) → ∃𝑟 ∈
Q (y
+Q 𝑟)
= 𝑠) |
41 | | simplrr 488 |
. . . . . . . . . . . . . . 15
⊢
((((A<P
B ∧ (𝑞 ∈ Q ∧ (y ∈ (2nd ‘A) ∧ (y +Q 𝑞) ∈
(1st ‘B)))) ∧ (𝑠 ∈
(1st ‘B) ∧ (y
+Q 𝑞)
<Q 𝑠)) ∧ (𝑟 ∈ Q ∧ (y
+Q 𝑟)
= 𝑠)) → (y +Q 𝑞) <Q 𝑠) |
42 | | simprr 484 |
. . . . . . . . . . . . . . 15
⊢
((((A<P
B ∧ (𝑞 ∈ Q ∧ (y ∈ (2nd ‘A) ∧ (y +Q 𝑞) ∈
(1st ‘B)))) ∧ (𝑠 ∈
(1st ‘B) ∧ (y
+Q 𝑞)
<Q 𝑠)) ∧ (𝑟 ∈ Q ∧ (y
+Q 𝑟)
= 𝑠)) → (y +Q 𝑟) = 𝑠) |
43 | 41, 42 | breqtrrd 3781 |
. . . . . . . . . . . . . 14
⊢
((((A<P
B ∧ (𝑞 ∈ Q ∧ (y ∈ (2nd ‘A) ∧ (y +Q 𝑞) ∈
(1st ‘B)))) ∧ (𝑠 ∈
(1st ‘B) ∧ (y
+Q 𝑞)
<Q 𝑠)) ∧ (𝑟 ∈ Q ∧ (y
+Q 𝑟)
= 𝑠)) → (y +Q 𝑞) <Q (y +Q 𝑟)) |
44 | 25 | adantr 261 |
. . . . . . . . . . . . . . 15
⊢
((((A<P
B ∧ (𝑞 ∈ Q ∧ (y ∈ (2nd ‘A) ∧ (y +Q 𝑞) ∈
(1st ‘B)))) ∧ (𝑠 ∈
(1st ‘B) ∧ (y
+Q 𝑞)
<Q 𝑠)) ∧ (𝑟 ∈ Q ∧ (y
+Q 𝑟)
= 𝑠)) → 𝑞 ∈
Q) |
45 | | simprl 483 |
. . . . . . . . . . . . . . 15
⊢
((((A<P
B ∧ (𝑞 ∈ Q ∧ (y ∈ (2nd ‘A) ∧ (y +Q 𝑞) ∈
(1st ‘B)))) ∧ (𝑠 ∈
(1st ‘B) ∧ (y
+Q 𝑞)
<Q 𝑠)) ∧ (𝑟 ∈ Q ∧ (y
+Q 𝑟)
= 𝑠)) → 𝑟 ∈
Q) |
46 | 24 | adantr 261 |
. . . . . . . . . . . . . . 15
⊢
((((A<P
B ∧ (𝑞 ∈ Q ∧ (y ∈ (2nd ‘A) ∧ (y +Q 𝑞) ∈
(1st ‘B)))) ∧ (𝑠 ∈
(1st ‘B) ∧ (y
+Q 𝑞)
<Q 𝑠)) ∧ (𝑟 ∈ Q ∧ (y
+Q 𝑟)
= 𝑠)) → y ∈
Q) |
47 | | ltanqg 6384 |
. . . . . . . . . . . . . . 15
⊢ ((𝑞 ∈ Q ∧ 𝑟
∈ Q ∧ y ∈ Q) → (𝑞 <Q 𝑟 ↔ (y +Q 𝑞) <Q (y +Q 𝑟))) |
48 | 44, 45, 46, 47 | syl3anc 1134 |
. . . . . . . . . . . . . 14
⊢
((((A<P
B ∧ (𝑞 ∈ Q ∧ (y ∈ (2nd ‘A) ∧ (y +Q 𝑞) ∈
(1st ‘B)))) ∧ (𝑠 ∈
(1st ‘B) ∧ (y
+Q 𝑞)
<Q 𝑠)) ∧ (𝑟 ∈ Q ∧ (y
+Q 𝑟)
= 𝑠)) → (𝑞 <Q 𝑟 ↔ (y +Q 𝑞) <Q (y +Q 𝑟))) |
49 | 43, 48 | mpbird 156 |
. . . . . . . . . . . . 13
⊢
((((A<P
B ∧ (𝑞 ∈ Q ∧ (y ∈ (2nd ‘A) ∧ (y +Q 𝑞) ∈
(1st ‘B)))) ∧ (𝑠 ∈
(1st ‘B) ∧ (y
+Q 𝑞)
<Q 𝑠)) ∧ (𝑟 ∈ Q ∧ (y
+Q 𝑟)
= 𝑠)) → 𝑞 <Q 𝑟) |
50 | 20 | adantr 261 |
. . . . . . . . . . . . . 14
⊢
((((A<P
B ∧ (𝑞 ∈ Q ∧ (y ∈ (2nd ‘A) ∧ (y +Q 𝑞) ∈
(1st ‘B)))) ∧ (𝑠 ∈
(1st ‘B) ∧ (y
+Q 𝑞)
<Q 𝑠)) ∧ (𝑟 ∈ Q ∧ (y
+Q 𝑟)
= 𝑠)) → y ∈
(2nd ‘A)) |
51 | | simplrl 487 |
. . . . . . . . . . . . . . 15
⊢
((((A<P
B ∧ (𝑞 ∈ Q ∧ (y ∈ (2nd ‘A) ∧ (y +Q 𝑞) ∈
(1st ‘B)))) ∧ (𝑠 ∈
(1st ‘B) ∧ (y
+Q 𝑞)
<Q 𝑠)) ∧ (𝑟 ∈ Q ∧ (y
+Q 𝑟)
= 𝑠)) → 𝑠 ∈
(1st ‘B)) |
52 | 42, 51 | eqeltrd 2111 |
. . . . . . . . . . . . . 14
⊢
((((A<P
B ∧ (𝑞 ∈ Q ∧ (y ∈ (2nd ‘A) ∧ (y +Q 𝑞) ∈
(1st ‘B)))) ∧ (𝑠 ∈
(1st ‘B) ∧ (y
+Q 𝑞)
<Q 𝑠)) ∧ (𝑟 ∈ Q ∧ (y
+Q 𝑟)
= 𝑠)) → (y +Q 𝑟) ∈
(1st ‘B)) |
53 | 50, 52 | jca 290 |
. . . . . . . . . . . . 13
⊢
((((A<P
B ∧ (𝑞 ∈ Q ∧ (y ∈ (2nd ‘A) ∧ (y +Q 𝑞) ∈
(1st ‘B)))) ∧ (𝑠 ∈
(1st ‘B) ∧ (y
+Q 𝑞)
<Q 𝑠)) ∧ (𝑟 ∈ Q ∧ (y
+Q 𝑟)
= 𝑠)) → (y ∈
(2nd ‘A) ∧ (y
+Q 𝑟)
∈ (1st ‘B))) |
54 | 49, 45, 53 | jca32 293 |
. . . . . . . . . . . 12
⊢
((((A<P
B ∧ (𝑞 ∈ Q ∧ (y ∈ (2nd ‘A) ∧ (y +Q 𝑞) ∈
(1st ‘B)))) ∧ (𝑠 ∈
(1st ‘B) ∧ (y
+Q 𝑞)
<Q 𝑠)) ∧ (𝑟 ∈ Q ∧ (y
+Q 𝑟)
= 𝑠)) → (𝑞 <Q 𝑟 ∧
(𝑟 ∈ Q ∧ (y ∈ (2nd ‘A) ∧ (y +Q 𝑟) ∈
(1st ‘B))))) |
55 | 54 | expr 357 |
. . . . . . . . . . 11
⊢
((((A<P
B ∧ (𝑞 ∈ Q ∧ (y ∈ (2nd ‘A) ∧ (y +Q 𝑞) ∈
(1st ‘B)))) ∧ (𝑠 ∈
(1st ‘B) ∧ (y
+Q 𝑞)
<Q 𝑠)) ∧ 𝑟 ∈ Q) → ((y +Q 𝑟) = 𝑠 → (𝑞 <Q 𝑟 ∧
(𝑟 ∈ Q ∧ (y ∈ (2nd ‘A) ∧ (y +Q 𝑟) ∈
(1st ‘B)))))) |
56 | 55 | reximdva 2415 |
. . . . . . . . . 10
⊢
(((A<P
B ∧ (𝑞 ∈ Q ∧ (y ∈ (2nd ‘A) ∧ (y +Q 𝑞) ∈
(1st ‘B)))) ∧ (𝑠 ∈
(1st ‘B) ∧ (y
+Q 𝑞)
<Q 𝑠)) → (∃𝑟 ∈
Q (y
+Q 𝑟)
= 𝑠 → ∃𝑟 ∈
Q (𝑞
<Q 𝑟 ∧ (𝑟 ∈ Q ∧ (y ∈ (2nd ‘A) ∧ (y +Q 𝑟) ∈
(1st ‘B)))))) |
57 | 40, 56 | mpd 13 |
. . . . . . . . 9
⊢
(((A<P
B ∧ (𝑞 ∈ Q ∧ (y ∈ (2nd ‘A) ∧ (y +Q 𝑞) ∈
(1st ‘B)))) ∧ (𝑠 ∈
(1st ‘B) ∧ (y
+Q 𝑞)
<Q 𝑠)) → ∃𝑟 ∈
Q (𝑞
<Q 𝑟 ∧ (𝑟 ∈ Q ∧ (y ∈ (2nd ‘A) ∧ (y +Q 𝑟) ∈
(1st ‘B))))) |
58 | 16, 57 | rexlimddv 2431 |
. . . . . . . 8
⊢
((A<P
B ∧ (𝑞 ∈ Q ∧ (y ∈ (2nd ‘A) ∧ (y +Q 𝑞) ∈
(1st ‘B)))) → ∃𝑟 ∈
Q (𝑞
<Q 𝑟 ∧ (𝑟 ∈ Q ∧ (y ∈ (2nd ‘A) ∧ (y +Q 𝑟) ∈
(1st ‘B))))) |
59 | 58 | eximi 1488 |
. . . . . . 7
⊢ (∃y(A<P B ∧ (𝑞 ∈ Q ∧ (y ∈ (2nd ‘A) ∧ (y +Q 𝑞) ∈
(1st ‘B)))) → ∃y∃𝑟 ∈
Q (𝑞
<Q 𝑟 ∧ (𝑟 ∈ Q ∧ (y ∈ (2nd ‘A) ∧ (y +Q 𝑟) ∈
(1st ‘B))))) |
60 | 7, 59 | sylbir 125 |
. . . . . 6
⊢
((A<P
B ∧ (𝑞 ∈ Q ∧ ∃y(y ∈ (2nd ‘A) ∧ (y +Q 𝑞) ∈
(1st ‘B)))) → ∃y∃𝑟 ∈
Q (𝑞
<Q 𝑟 ∧ (𝑟 ∈ Q ∧ (y ∈ (2nd ‘A) ∧ (y +Q 𝑟) ∈
(1st ‘B))))) |
61 | | rexcom4 2571 |
. . . . . 6
⊢ (∃𝑟 ∈
Q ∃y(𝑞
<Q 𝑟 ∧ (𝑟 ∈ Q ∧ (y ∈ (2nd ‘A) ∧ (y +Q 𝑟) ∈
(1st ‘B)))) ↔ ∃y∃𝑟 ∈
Q (𝑞
<Q 𝑟 ∧ (𝑟 ∈ Q ∧ (y ∈ (2nd ‘A) ∧ (y +Q 𝑟) ∈
(1st ‘B))))) |
62 | 60, 61 | sylibr 137 |
. . . . 5
⊢
((A<P
B ∧ (𝑞 ∈ Q ∧ ∃y(y ∈ (2nd ‘A) ∧ (y +Q 𝑞) ∈
(1st ‘B)))) → ∃𝑟 ∈
Q ∃y(𝑞
<Q 𝑟 ∧ (𝑟 ∈ Q ∧ (y ∈ (2nd ‘A) ∧ (y +Q 𝑟) ∈
(1st ‘B))))) |
63 | | 19.42v 1783 |
. . . . . . 7
⊢ (∃y(𝑞 <Q
𝑟 ∧ (𝑟 ∈
Q ∧ (y ∈
(2nd ‘A) ∧ (y
+Q 𝑟)
∈ (1st ‘B)))) ↔ (𝑞 <Q 𝑟 ∧
∃y(𝑟 ∈ Q ∧ (y ∈ (2nd ‘A) ∧ (y +Q 𝑟) ∈
(1st ‘B))))) |
64 | | 19.42v 1783 |
. . . . . . . 8
⊢ (∃y(𝑟 ∈ Q ∧ (y ∈ (2nd ‘A) ∧ (y +Q 𝑟) ∈
(1st ‘B))) ↔ (𝑟 ∈ Q ∧ ∃y(y ∈ (2nd ‘A) ∧ (y +Q 𝑟) ∈
(1st ‘B)))) |
65 | 64 | anbi2i 430 |
. . . . . . 7
⊢ ((𝑞 <Q
𝑟 ∧ ∃y(𝑟
∈ Q ∧ (y ∈ (2nd ‘A) ∧ (y +Q 𝑟) ∈
(1st ‘B)))) ↔ (𝑞 <Q
𝑟 ∧ (𝑟 ∈
Q ∧ ∃y(y ∈
(2nd ‘A) ∧ (y
+Q 𝑟)
∈ (1st ‘B))))) |
66 | 63, 65 | bitri 173 |
. . . . . 6
⊢ (∃y(𝑞 <Q
𝑟 ∧ (𝑟 ∈
Q ∧ (y ∈
(2nd ‘A) ∧ (y
+Q 𝑟)
∈ (1st ‘B)))) ↔ (𝑞 <Q 𝑟 ∧
(𝑟 ∈ Q ∧ ∃y(y ∈ (2nd ‘A) ∧ (y +Q 𝑟) ∈
(1st ‘B))))) |
67 | 66 | rexbii 2325 |
. . . . 5
⊢ (∃𝑟 ∈
Q ∃y(𝑞
<Q 𝑟 ∧ (𝑟 ∈ Q ∧ (y ∈ (2nd ‘A) ∧ (y +Q 𝑟) ∈
(1st ‘B)))) ↔ ∃𝑟 ∈
Q (𝑞
<Q 𝑟 ∧ (𝑟 ∈ Q ∧ ∃y(y ∈ (2nd ‘A) ∧ (y +Q 𝑟) ∈
(1st ‘B))))) |
68 | 62, 67 | sylib 127 |
. . . 4
⊢
((A<P
B ∧ (𝑞 ∈ Q ∧ ∃y(y ∈ (2nd ‘A) ∧ (y +Q 𝑞) ∈
(1st ‘B)))) → ∃𝑟 ∈
Q (𝑞
<Q 𝑟 ∧ (𝑟 ∈ Q ∧ ∃y(y ∈ (2nd ‘A) ∧ (y +Q 𝑟) ∈
(1st ‘B))))) |
69 | 1 | ltexprlemell 6572 |
. . . . . 6
⊢ (𝑟 ∈ (1st ‘𝐶) ↔ (𝑟 ∈
Q ∧ ∃y(y ∈
(2nd ‘A) ∧ (y
+Q 𝑟)
∈ (1st ‘B)))) |
70 | 69 | anbi2i 430 |
. . . . 5
⊢ ((𝑞 <Q
𝑟 ∧ 𝑟
∈ (1st ‘𝐶)) ↔ (𝑞 <Q 𝑟 ∧
(𝑟 ∈ Q ∧ ∃y(y ∈ (2nd ‘A) ∧ (y +Q 𝑟) ∈
(1st ‘B))))) |
71 | 70 | rexbii 2325 |
. . . 4
⊢ (∃𝑟 ∈
Q (𝑞
<Q 𝑟 ∧ 𝑟 ∈ (1st ‘𝐶)) ↔ ∃𝑟 ∈
Q (𝑞
<Q 𝑟 ∧ (𝑟 ∈ Q ∧ ∃y(y ∈ (2nd ‘A) ∧ (y +Q 𝑟) ∈
(1st ‘B))))) |
72 | 68, 71 | sylibr 137 |
. . 3
⊢
((A<P
B ∧ (𝑞 ∈ Q ∧ ∃y(y ∈ (2nd ‘A) ∧ (y +Q 𝑞) ∈
(1st ‘B)))) → ∃𝑟 ∈
Q (𝑞
<Q 𝑟 ∧ 𝑟 ∈ (1st ‘𝐶))) |
73 | 3, 72 | sylanr2 385 |
. 2
⊢
((A<P
B ∧ (𝑞 ∈ Q ∧ 𝑞
∈ (1st ‘𝐶))) → ∃𝑟 ∈
Q (𝑞
<Q 𝑟 ∧ 𝑟 ∈ (1st ‘𝐶))) |
74 | 73 | 3impb 1099 |
1
⊢
((A<P
B ∧ 𝑞 ∈ Q ∧ 𝑞
∈ (1st ‘𝐶)) → ∃𝑟 ∈
Q (𝑞
<Q 𝑟 ∧ 𝑟 ∈ (1st ‘𝐶))) |