Step | Hyp | Ref
| Expression |
1 | | eqid 2738 |
. . . . . 6
⊢ ran
(𝑟 ∈ 𝑅, 𝑠 ∈ 𝑆 ↦ (𝑟 × 𝑠)) = ran (𝑟 ∈ 𝑅, 𝑠 ∈ 𝑆 ↦ (𝑟 × 𝑠)) |
2 | 1 | txval 22315 |
. . . . 5
⊢ ((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) → (𝑅 ×t 𝑆) = (topGen‘ran (𝑟 ∈ 𝑅, 𝑠 ∈ 𝑆 ↦ (𝑟 × 𝑠)))) |
3 | 2 | adantr 484 |
. . . 4
⊢ (((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) → (𝑅 ×t 𝑆) = (topGen‘ran (𝑟 ∈ 𝑅, 𝑠 ∈ 𝑆 ↦ (𝑟 × 𝑠)))) |
4 | 3 | oveq1d 7185 |
. . 3
⊢ (((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) → ((𝑅 ×t 𝑆) ↾t (𝐴 × 𝐵)) = ((topGen‘ran (𝑟 ∈ 𝑅, 𝑠 ∈ 𝑆 ↦ (𝑟 × 𝑠))) ↾t (𝐴 × 𝐵))) |
5 | 1 | txbasex 22317 |
. . . 4
⊢ ((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) → ran (𝑟 ∈ 𝑅, 𝑠 ∈ 𝑆 ↦ (𝑟 × 𝑠)) ∈ V) |
6 | | xpexg 7491 |
. . . 4
⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌) → (𝐴 × 𝐵) ∈ V) |
7 | | tgrest 21910 |
. . . 4
⊢ ((ran
(𝑟 ∈ 𝑅, 𝑠 ∈ 𝑆 ↦ (𝑟 × 𝑠)) ∈ V ∧ (𝐴 × 𝐵) ∈ V) → (topGen‘(ran (𝑟 ∈ 𝑅, 𝑠 ∈ 𝑆 ↦ (𝑟 × 𝑠)) ↾t (𝐴 × 𝐵))) = ((topGen‘ran (𝑟 ∈ 𝑅, 𝑠 ∈ 𝑆 ↦ (𝑟 × 𝑠))) ↾t (𝐴 × 𝐵))) |
8 | 5, 6, 7 | syl2an 599 |
. . 3
⊢ (((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) → (topGen‘(ran (𝑟 ∈ 𝑅, 𝑠 ∈ 𝑆 ↦ (𝑟 × 𝑠)) ↾t (𝐴 × 𝐵))) = ((topGen‘ran (𝑟 ∈ 𝑅, 𝑠 ∈ 𝑆 ↦ (𝑟 × 𝑠))) ↾t (𝐴 × 𝐵))) |
9 | | elrest 16804 |
. . . . . . . 8
⊢ ((ran
(𝑟 ∈ 𝑅, 𝑠 ∈ 𝑆 ↦ (𝑟 × 𝑠)) ∈ V ∧ (𝐴 × 𝐵) ∈ V) → (𝑥 ∈ (ran (𝑟 ∈ 𝑅, 𝑠 ∈ 𝑆 ↦ (𝑟 × 𝑠)) ↾t (𝐴 × 𝐵)) ↔ ∃𝑤 ∈ ran (𝑟 ∈ 𝑅, 𝑠 ∈ 𝑆 ↦ (𝑟 × 𝑠))𝑥 = (𝑤 ∩ (𝐴 × 𝐵)))) |
10 | 5, 6, 9 | syl2an 599 |
. . . . . . 7
⊢ (((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) → (𝑥 ∈ (ran (𝑟 ∈ 𝑅, 𝑠 ∈ 𝑆 ↦ (𝑟 × 𝑠)) ↾t (𝐴 × 𝐵)) ↔ ∃𝑤 ∈ ran (𝑟 ∈ 𝑅, 𝑠 ∈ 𝑆 ↦ (𝑟 × 𝑠))𝑥 = (𝑤 ∩ (𝐴 × 𝐵)))) |
11 | | vex 3402 |
. . . . . . . . . . 11
⊢ 𝑟 ∈ V |
12 | 11 | inex1 5185 |
. . . . . . . . . 10
⊢ (𝑟 ∩ 𝐴) ∈ V |
13 | 12 | a1i 11 |
. . . . . . . . 9
⊢ ((((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) ∧ 𝑟 ∈ 𝑅) → (𝑟 ∩ 𝐴) ∈ V) |
14 | | elrest 16804 |
. . . . . . . . . 10
⊢ ((𝑅 ∈ 𝑉 ∧ 𝐴 ∈ 𝑋) → (𝑢 ∈ (𝑅 ↾t 𝐴) ↔ ∃𝑟 ∈ 𝑅 𝑢 = (𝑟 ∩ 𝐴))) |
15 | 14 | ad2ant2r 747 |
. . . . . . . . 9
⊢ (((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) → (𝑢 ∈ (𝑅 ↾t 𝐴) ↔ ∃𝑟 ∈ 𝑅 𝑢 = (𝑟 ∩ 𝐴))) |
16 | | xpeq1 5539 |
. . . . . . . . . . . 12
⊢ (𝑢 = (𝑟 ∩ 𝐴) → (𝑢 × 𝑣) = ((𝑟 ∩ 𝐴) × 𝑣)) |
17 | 16 | eqeq2d 2749 |
. . . . . . . . . . 11
⊢ (𝑢 = (𝑟 ∩ 𝐴) → (𝑥 = (𝑢 × 𝑣) ↔ 𝑥 = ((𝑟 ∩ 𝐴) × 𝑣))) |
18 | 17 | rexbidv 3207 |
. . . . . . . . . 10
⊢ (𝑢 = (𝑟 ∩ 𝐴) → (∃𝑣 ∈ (𝑆 ↾t 𝐵)𝑥 = (𝑢 × 𝑣) ↔ ∃𝑣 ∈ (𝑆 ↾t 𝐵)𝑥 = ((𝑟 ∩ 𝐴) × 𝑣))) |
19 | | vex 3402 |
. . . . . . . . . . . . 13
⊢ 𝑠 ∈ V |
20 | 19 | inex1 5185 |
. . . . . . . . . . . 12
⊢ (𝑠 ∩ 𝐵) ∈ V |
21 | 20 | a1i 11 |
. . . . . . . . . . 11
⊢ ((((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) ∧ 𝑠 ∈ 𝑆) → (𝑠 ∩ 𝐵) ∈ V) |
22 | | elrest 16804 |
. . . . . . . . . . . 12
⊢ ((𝑆 ∈ 𝑊 ∧ 𝐵 ∈ 𝑌) → (𝑣 ∈ (𝑆 ↾t 𝐵) ↔ ∃𝑠 ∈ 𝑆 𝑣 = (𝑠 ∩ 𝐵))) |
23 | 22 | ad2ant2l 746 |
. . . . . . . . . . 11
⊢ (((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) → (𝑣 ∈ (𝑆 ↾t 𝐵) ↔ ∃𝑠 ∈ 𝑆 𝑣 = (𝑠 ∩ 𝐵))) |
24 | | xpeq2 5546 |
. . . . . . . . . . . . 13
⊢ (𝑣 = (𝑠 ∩ 𝐵) → ((𝑟 ∩ 𝐴) × 𝑣) = ((𝑟 ∩ 𝐴) × (𝑠 ∩ 𝐵))) |
25 | 24 | eqeq2d 2749 |
. . . . . . . . . . . 12
⊢ (𝑣 = (𝑠 ∩ 𝐵) → (𝑥 = ((𝑟 ∩ 𝐴) × 𝑣) ↔ 𝑥 = ((𝑟 ∩ 𝐴) × (𝑠 ∩ 𝐵)))) |
26 | 25 | adantl 485 |
. . . . . . . . . . 11
⊢ ((((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) ∧ 𝑣 = (𝑠 ∩ 𝐵)) → (𝑥 = ((𝑟 ∩ 𝐴) × 𝑣) ↔ 𝑥 = ((𝑟 ∩ 𝐴) × (𝑠 ∩ 𝐵)))) |
27 | 21, 23, 26 | rexxfr2d 5278 |
. . . . . . . . . 10
⊢ (((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) → (∃𝑣 ∈ (𝑆 ↾t 𝐵)𝑥 = ((𝑟 ∩ 𝐴) × 𝑣) ↔ ∃𝑠 ∈ 𝑆 𝑥 = ((𝑟 ∩ 𝐴) × (𝑠 ∩ 𝐵)))) |
28 | 18, 27 | sylan9bbr 514 |
. . . . . . . . 9
⊢ ((((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) ∧ 𝑢 = (𝑟 ∩ 𝐴)) → (∃𝑣 ∈ (𝑆 ↾t 𝐵)𝑥 = (𝑢 × 𝑣) ↔ ∃𝑠 ∈ 𝑆 𝑥 = ((𝑟 ∩ 𝐴) × (𝑠 ∩ 𝐵)))) |
29 | 13, 15, 28 | rexxfr2d 5278 |
. . . . . . . 8
⊢ (((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) → (∃𝑢 ∈ (𝑅 ↾t 𝐴)∃𝑣 ∈ (𝑆 ↾t 𝐵)𝑥 = (𝑢 × 𝑣) ↔ ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 𝑥 = ((𝑟 ∩ 𝐴) × (𝑠 ∩ 𝐵)))) |
30 | 11, 19 | xpex 7494 |
. . . . . . . . . 10
⊢ (𝑟 × 𝑠) ∈ V |
31 | 30 | rgen2w 3066 |
. . . . . . . . 9
⊢
∀𝑟 ∈
𝑅 ∀𝑠 ∈ 𝑆 (𝑟 × 𝑠) ∈ V |
32 | | eqid 2738 |
. . . . . . . . . 10
⊢ (𝑟 ∈ 𝑅, 𝑠 ∈ 𝑆 ↦ (𝑟 × 𝑠)) = (𝑟 ∈ 𝑅, 𝑠 ∈ 𝑆 ↦ (𝑟 × 𝑠)) |
33 | | ineq1 4096 |
. . . . . . . . . . . 12
⊢ (𝑤 = (𝑟 × 𝑠) → (𝑤 ∩ (𝐴 × 𝐵)) = ((𝑟 × 𝑠) ∩ (𝐴 × 𝐵))) |
34 | | inxp 5675 |
. . . . . . . . . . . 12
⊢ ((𝑟 × 𝑠) ∩ (𝐴 × 𝐵)) = ((𝑟 ∩ 𝐴) × (𝑠 ∩ 𝐵)) |
35 | 33, 34 | eqtrdi 2789 |
. . . . . . . . . . 11
⊢ (𝑤 = (𝑟 × 𝑠) → (𝑤 ∩ (𝐴 × 𝐵)) = ((𝑟 ∩ 𝐴) × (𝑠 ∩ 𝐵))) |
36 | 35 | eqeq2d 2749 |
. . . . . . . . . 10
⊢ (𝑤 = (𝑟 × 𝑠) → (𝑥 = (𝑤 ∩ (𝐴 × 𝐵)) ↔ 𝑥 = ((𝑟 ∩ 𝐴) × (𝑠 ∩ 𝐵)))) |
37 | 32, 36 | rexrnmpo 7305 |
. . . . . . . . 9
⊢
(∀𝑟 ∈
𝑅 ∀𝑠 ∈ 𝑆 (𝑟 × 𝑠) ∈ V → (∃𝑤 ∈ ran (𝑟 ∈ 𝑅, 𝑠 ∈ 𝑆 ↦ (𝑟 × 𝑠))𝑥 = (𝑤 ∩ (𝐴 × 𝐵)) ↔ ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 𝑥 = ((𝑟 ∩ 𝐴) × (𝑠 ∩ 𝐵)))) |
38 | 31, 37 | ax-mp 5 |
. . . . . . . 8
⊢
(∃𝑤 ∈ ran
(𝑟 ∈ 𝑅, 𝑠 ∈ 𝑆 ↦ (𝑟 × 𝑠))𝑥 = (𝑤 ∩ (𝐴 × 𝐵)) ↔ ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 𝑥 = ((𝑟 ∩ 𝐴) × (𝑠 ∩ 𝐵))) |
39 | 29, 38 | bitr4di 292 |
. . . . . . 7
⊢ (((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) → (∃𝑢 ∈ (𝑅 ↾t 𝐴)∃𝑣 ∈ (𝑆 ↾t 𝐵)𝑥 = (𝑢 × 𝑣) ↔ ∃𝑤 ∈ ran (𝑟 ∈ 𝑅, 𝑠 ∈ 𝑆 ↦ (𝑟 × 𝑠))𝑥 = (𝑤 ∩ (𝐴 × 𝐵)))) |
40 | 10, 39 | bitr4d 285 |
. . . . . 6
⊢ (((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) → (𝑥 ∈ (ran (𝑟 ∈ 𝑅, 𝑠 ∈ 𝑆 ↦ (𝑟 × 𝑠)) ↾t (𝐴 × 𝐵)) ↔ ∃𝑢 ∈ (𝑅 ↾t 𝐴)∃𝑣 ∈ (𝑆 ↾t 𝐵)𝑥 = (𝑢 × 𝑣))) |
41 | 40 | abbi2dv 2869 |
. . . . 5
⊢ (((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) → (ran (𝑟 ∈ 𝑅, 𝑠 ∈ 𝑆 ↦ (𝑟 × 𝑠)) ↾t (𝐴 × 𝐵)) = {𝑥 ∣ ∃𝑢 ∈ (𝑅 ↾t 𝐴)∃𝑣 ∈ (𝑆 ↾t 𝐵)𝑥 = (𝑢 × 𝑣)}) |
42 | | eqid 2738 |
. . . . . 6
⊢ (𝑢 ∈ (𝑅 ↾t 𝐴), 𝑣 ∈ (𝑆 ↾t 𝐵) ↦ (𝑢 × 𝑣)) = (𝑢 ∈ (𝑅 ↾t 𝐴), 𝑣 ∈ (𝑆 ↾t 𝐵) ↦ (𝑢 × 𝑣)) |
43 | 42 | rnmpo 7299 |
. . . . 5
⊢ ran
(𝑢 ∈ (𝑅 ↾t 𝐴), 𝑣 ∈ (𝑆 ↾t 𝐵) ↦ (𝑢 × 𝑣)) = {𝑥 ∣ ∃𝑢 ∈ (𝑅 ↾t 𝐴)∃𝑣 ∈ (𝑆 ↾t 𝐵)𝑥 = (𝑢 × 𝑣)} |
44 | 41, 43 | eqtr4di 2791 |
. . . 4
⊢ (((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) → (ran (𝑟 ∈ 𝑅, 𝑠 ∈ 𝑆 ↦ (𝑟 × 𝑠)) ↾t (𝐴 × 𝐵)) = ran (𝑢 ∈ (𝑅 ↾t 𝐴), 𝑣 ∈ (𝑆 ↾t 𝐵) ↦ (𝑢 × 𝑣))) |
45 | 44 | fveq2d 6678 |
. . 3
⊢ (((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) → (topGen‘(ran (𝑟 ∈ 𝑅, 𝑠 ∈ 𝑆 ↦ (𝑟 × 𝑠)) ↾t (𝐴 × 𝐵))) = (topGen‘ran (𝑢 ∈ (𝑅 ↾t 𝐴), 𝑣 ∈ (𝑆 ↾t 𝐵) ↦ (𝑢 × 𝑣)))) |
46 | 4, 8, 45 | 3eqtr2d 2779 |
. 2
⊢ (((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) → ((𝑅 ×t 𝑆) ↾t (𝐴 × 𝐵)) = (topGen‘ran (𝑢 ∈ (𝑅 ↾t 𝐴), 𝑣 ∈ (𝑆 ↾t 𝐵) ↦ (𝑢 × 𝑣)))) |
47 | | ovex 7203 |
. . 3
⊢ (𝑅 ↾t 𝐴) ∈ V |
48 | | ovex 7203 |
. . 3
⊢ (𝑆 ↾t 𝐵) ∈ V |
49 | | eqid 2738 |
. . . 4
⊢ ran
(𝑢 ∈ (𝑅 ↾t 𝐴), 𝑣 ∈ (𝑆 ↾t 𝐵) ↦ (𝑢 × 𝑣)) = ran (𝑢 ∈ (𝑅 ↾t 𝐴), 𝑣 ∈ (𝑆 ↾t 𝐵) ↦ (𝑢 × 𝑣)) |
50 | 49 | txval 22315 |
. . 3
⊢ (((𝑅 ↾t 𝐴) ∈ V ∧ (𝑆 ↾t 𝐵) ∈ V) → ((𝑅 ↾t 𝐴) ×t (𝑆 ↾t 𝐵)) = (topGen‘ran (𝑢 ∈ (𝑅 ↾t 𝐴), 𝑣 ∈ (𝑆 ↾t 𝐵) ↦ (𝑢 × 𝑣)))) |
51 | 47, 48, 50 | mp2an 692 |
. 2
⊢ ((𝑅 ↾t 𝐴) ×t (𝑆 ↾t 𝐵)) = (topGen‘ran (𝑢 ∈ (𝑅 ↾t 𝐴), 𝑣 ∈ (𝑆 ↾t 𝐵) ↦ (𝑢 × 𝑣))) |
52 | 46, 51 | eqtr4di 2791 |
1
⊢ (((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) → ((𝑅 ×t 𝑆) ↾t (𝐴 × 𝐵)) = ((𝑅 ↾t 𝐴) ×t (𝑆 ↾t 𝐵))) |