Proof of Theorem resasplit
Step | Hyp | Ref
| Expression |
1 | | fnresdm 6000 |
. . . 4
⊢ (𝐹 Fn 𝐴 → (𝐹 ↾ 𝐴) = 𝐹) |
2 | | fnresdm 6000 |
. . . 4
⊢ (𝐺 Fn 𝐵 → (𝐺 ↾ 𝐵) = 𝐺) |
3 | | uneq12 3762 |
. . . 4
⊢ (((𝐹 ↾ 𝐴) = 𝐹 ∧ (𝐺 ↾ 𝐵) = 𝐺) → ((𝐹 ↾ 𝐴) ∪ (𝐺 ↾ 𝐵)) = (𝐹 ∪ 𝐺)) |
4 | 1, 2, 3 | syl2an 494 |
. . 3
⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵) → ((𝐹 ↾ 𝐴) ∪ (𝐺 ↾ 𝐵)) = (𝐹 ∪ 𝐺)) |
5 | 4 | 3adant3 1081 |
. 2
⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵 ∧ (𝐹 ↾ (𝐴 ∩ 𝐵)) = (𝐺 ↾ (𝐴 ∩ 𝐵))) → ((𝐹 ↾ 𝐴) ∪ (𝐺 ↾ 𝐵)) = (𝐹 ∪ 𝐺)) |
6 | | simp3 1063 |
. . . . . . 7
⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵 ∧ (𝐹 ↾ (𝐴 ∩ 𝐵)) = (𝐺 ↾ (𝐴 ∩ 𝐵))) → (𝐹 ↾ (𝐴 ∩ 𝐵)) = (𝐺 ↾ (𝐴 ∩ 𝐵))) |
7 | 6 | uneq1d 3766 |
. . . . . 6
⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵 ∧ (𝐹 ↾ (𝐴 ∩ 𝐵)) = (𝐺 ↾ (𝐴 ∩ 𝐵))) → ((𝐹 ↾ (𝐴 ∩ 𝐵)) ∪ (𝐺 ↾ (𝐵 ∖ 𝐴))) = ((𝐺 ↾ (𝐴 ∩ 𝐵)) ∪ (𝐺 ↾ (𝐵 ∖ 𝐴)))) |
8 | 7 | uneq2d 3767 |
. . . . 5
⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵 ∧ (𝐹 ↾ (𝐴 ∩ 𝐵)) = (𝐺 ↾ (𝐴 ∩ 𝐵))) → (((𝐹 ↾ (𝐴 ∩ 𝐵)) ∪ (𝐹 ↾ (𝐴 ∖ 𝐵))) ∪ ((𝐹 ↾ (𝐴 ∩ 𝐵)) ∪ (𝐺 ↾ (𝐵 ∖ 𝐴)))) = (((𝐹 ↾ (𝐴 ∩ 𝐵)) ∪ (𝐹 ↾ (𝐴 ∖ 𝐵))) ∪ ((𝐺 ↾ (𝐴 ∩ 𝐵)) ∪ (𝐺 ↾ (𝐵 ∖ 𝐴))))) |
9 | | inundif 4046 |
. . . . . . . 8
⊢ ((𝐴 ∩ 𝐵) ∪ (𝐴 ∖ 𝐵)) = 𝐴 |
10 | 9 | reseq2i 5393 |
. . . . . . 7
⊢ (𝐹 ↾ ((𝐴 ∩ 𝐵) ∪ (𝐴 ∖ 𝐵))) = (𝐹 ↾ 𝐴) |
11 | | resundi 5410 |
. . . . . . 7
⊢ (𝐹 ↾ ((𝐴 ∩ 𝐵) ∪ (𝐴 ∖ 𝐵))) = ((𝐹 ↾ (𝐴 ∩ 𝐵)) ∪ (𝐹 ↾ (𝐴 ∖ 𝐵))) |
12 | 10, 11 | eqtr3i 2646 |
. . . . . 6
⊢ (𝐹 ↾ 𝐴) = ((𝐹 ↾ (𝐴 ∩ 𝐵)) ∪ (𝐹 ↾ (𝐴 ∖ 𝐵))) |
13 | | incom 3805 |
. . . . . . . . . 10
⊢ (𝐴 ∩ 𝐵) = (𝐵 ∩ 𝐴) |
14 | 13 | uneq1i 3763 |
. . . . . . . . 9
⊢ ((𝐴 ∩ 𝐵) ∪ (𝐵 ∖ 𝐴)) = ((𝐵 ∩ 𝐴) ∪ (𝐵 ∖ 𝐴)) |
15 | | inundif 4046 |
. . . . . . . . 9
⊢ ((𝐵 ∩ 𝐴) ∪ (𝐵 ∖ 𝐴)) = 𝐵 |
16 | 14, 15 | eqtri 2644 |
. . . . . . . 8
⊢ ((𝐴 ∩ 𝐵) ∪ (𝐵 ∖ 𝐴)) = 𝐵 |
17 | 16 | reseq2i 5393 |
. . . . . . 7
⊢ (𝐺 ↾ ((𝐴 ∩ 𝐵) ∪ (𝐵 ∖ 𝐴))) = (𝐺 ↾ 𝐵) |
18 | | resundi 5410 |
. . . . . . 7
⊢ (𝐺 ↾ ((𝐴 ∩ 𝐵) ∪ (𝐵 ∖ 𝐴))) = ((𝐺 ↾ (𝐴 ∩ 𝐵)) ∪ (𝐺 ↾ (𝐵 ∖ 𝐴))) |
19 | 17, 18 | eqtr3i 2646 |
. . . . . 6
⊢ (𝐺 ↾ 𝐵) = ((𝐺 ↾ (𝐴 ∩ 𝐵)) ∪ (𝐺 ↾ (𝐵 ∖ 𝐴))) |
20 | 12, 19 | uneq12i 3765 |
. . . . 5
⊢ ((𝐹 ↾ 𝐴) ∪ (𝐺 ↾ 𝐵)) = (((𝐹 ↾ (𝐴 ∩ 𝐵)) ∪ (𝐹 ↾ (𝐴 ∖ 𝐵))) ∪ ((𝐺 ↾ (𝐴 ∩ 𝐵)) ∪ (𝐺 ↾ (𝐵 ∖ 𝐴)))) |
21 | 8, 20 | syl6reqr 2675 |
. . . 4
⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵 ∧ (𝐹 ↾ (𝐴 ∩ 𝐵)) = (𝐺 ↾ (𝐴 ∩ 𝐵))) → ((𝐹 ↾ 𝐴) ∪ (𝐺 ↾ 𝐵)) = (((𝐹 ↾ (𝐴 ∩ 𝐵)) ∪ (𝐹 ↾ (𝐴 ∖ 𝐵))) ∪ ((𝐹 ↾ (𝐴 ∩ 𝐵)) ∪ (𝐺 ↾ (𝐵 ∖ 𝐴))))) |
22 | | un4 3773 |
. . . 4
⊢ (((𝐹 ↾ (𝐴 ∩ 𝐵)) ∪ (𝐹 ↾ (𝐴 ∖ 𝐵))) ∪ ((𝐹 ↾ (𝐴 ∩ 𝐵)) ∪ (𝐺 ↾ (𝐵 ∖ 𝐴)))) = (((𝐹 ↾ (𝐴 ∩ 𝐵)) ∪ (𝐹 ↾ (𝐴 ∩ 𝐵))) ∪ ((𝐹 ↾ (𝐴 ∖ 𝐵)) ∪ (𝐺 ↾ (𝐵 ∖ 𝐴)))) |
23 | 21, 22 | syl6eq 2672 |
. . 3
⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵 ∧ (𝐹 ↾ (𝐴 ∩ 𝐵)) = (𝐺 ↾ (𝐴 ∩ 𝐵))) → ((𝐹 ↾ 𝐴) ∪ (𝐺 ↾ 𝐵)) = (((𝐹 ↾ (𝐴 ∩ 𝐵)) ∪ (𝐹 ↾ (𝐴 ∩ 𝐵))) ∪ ((𝐹 ↾ (𝐴 ∖ 𝐵)) ∪ (𝐺 ↾ (𝐵 ∖ 𝐴))))) |
24 | | unidm 3756 |
. . . 4
⊢ ((𝐹 ↾ (𝐴 ∩ 𝐵)) ∪ (𝐹 ↾ (𝐴 ∩ 𝐵))) = (𝐹 ↾ (𝐴 ∩ 𝐵)) |
25 | 24 | uneq1i 3763 |
. . 3
⊢ (((𝐹 ↾ (𝐴 ∩ 𝐵)) ∪ (𝐹 ↾ (𝐴 ∩ 𝐵))) ∪ ((𝐹 ↾ (𝐴 ∖ 𝐵)) ∪ (𝐺 ↾ (𝐵 ∖ 𝐴)))) = ((𝐹 ↾ (𝐴 ∩ 𝐵)) ∪ ((𝐹 ↾ (𝐴 ∖ 𝐵)) ∪ (𝐺 ↾ (𝐵 ∖ 𝐴)))) |
26 | 23, 25 | syl6eq 2672 |
. 2
⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵 ∧ (𝐹 ↾ (𝐴 ∩ 𝐵)) = (𝐺 ↾ (𝐴 ∩ 𝐵))) → ((𝐹 ↾ 𝐴) ∪ (𝐺 ↾ 𝐵)) = ((𝐹 ↾ (𝐴 ∩ 𝐵)) ∪ ((𝐹 ↾ (𝐴 ∖ 𝐵)) ∪ (𝐺 ↾ (𝐵 ∖ 𝐴))))) |
27 | 5, 26 | eqtr3d 2658 |
1
⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵 ∧ (𝐹 ↾ (𝐴 ∩ 𝐵)) = (𝐺 ↾ (𝐴 ∩ 𝐵))) → (𝐹 ∪ 𝐺) = ((𝐹 ↾ (𝐴 ∩ 𝐵)) ∪ ((𝐹 ↾ (𝐴 ∖ 𝐵)) ∪ (𝐺 ↾ (𝐵 ∖ 𝐴))))) |