Proof of Theorem csbresgVD
Step | Hyp | Ref
| Expression |
1 | | idn1 37811 |
. . . . . . . . 9
⊢ ( 𝐴 ∈ 𝑉 ▶ 𝐴 ∈ 𝑉 ) |
2 | | csbconstg 3512 |
. . . . . . . . 9
⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌V = V) |
3 | 1, 2 | e1a 37873 |
. . . . . . . 8
⊢ ( 𝐴 ∈ 𝑉 ▶ ⦋𝐴 / 𝑥⦌V = V ) |
4 | | xpeq2 5053 |
. . . . . . . 8
⊢
(⦋𝐴 /
𝑥⦌V = V
→ (⦋𝐴 /
𝑥⦌𝐶 × ⦋𝐴 / 𝑥⦌V) = (⦋𝐴 / 𝑥⦌𝐶 × V)) |
5 | 3, 4 | e1a 37873 |
. . . . . . 7
⊢ ( 𝐴 ∈ 𝑉 ▶ (⦋𝐴 / 𝑥⦌𝐶 × ⦋𝐴 / 𝑥⦌V) = (⦋𝐴 / 𝑥⦌𝐶 × V) ) |
6 | | csbxpgOLD 38075 |
. . . . . . . 8
⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌(𝐶 × V) = (⦋𝐴 / 𝑥⦌𝐶 × ⦋𝐴 / 𝑥⦌V)) |
7 | 1, 6 | e1a 37873 |
. . . . . . 7
⊢ ( 𝐴 ∈ 𝑉 ▶ ⦋𝐴 / 𝑥⦌(𝐶 × V) = (⦋𝐴 / 𝑥⦌𝐶 × ⦋𝐴 / 𝑥⦌V) ) |
8 | | eqeq2 2621 |
. . . . . . . 8
⊢
((⦋𝐴 /
𝑥⦌𝐶 × ⦋𝐴 / 𝑥⦌V) = (⦋𝐴 / 𝑥⦌𝐶 × V) → (⦋𝐴 / 𝑥⦌(𝐶 × V) = (⦋𝐴 / 𝑥⦌𝐶 × ⦋𝐴 / 𝑥⦌V) ↔ ⦋𝐴 / 𝑥⦌(𝐶 × V) = (⦋𝐴 / 𝑥⦌𝐶 × V))) |
9 | 8 | biimpd 218 |
. . . . . . 7
⊢
((⦋𝐴 /
𝑥⦌𝐶 × ⦋𝐴 / 𝑥⦌V) = (⦋𝐴 / 𝑥⦌𝐶 × V) → (⦋𝐴 / 𝑥⦌(𝐶 × V) = (⦋𝐴 / 𝑥⦌𝐶 × ⦋𝐴 / 𝑥⦌V) → ⦋𝐴 / 𝑥⦌(𝐶 × V) = (⦋𝐴 / 𝑥⦌𝐶 × V))) |
10 | 5, 7, 9 | e11 37934 |
. . . . . 6
⊢ ( 𝐴 ∈ 𝑉 ▶ ⦋𝐴 / 𝑥⦌(𝐶 × V) = (⦋𝐴 / 𝑥⦌𝐶 × V) ) |
11 | | ineq2 3770 |
. . . . . 6
⊢
(⦋𝐴 /
𝑥⦌(𝐶 × V) =
(⦋𝐴 / 𝑥⦌𝐶 × V) → (⦋𝐴 / 𝑥⦌𝐵 ∩ ⦋𝐴 / 𝑥⦌(𝐶 × V)) = (⦋𝐴 / 𝑥⦌𝐵 ∩ (⦋𝐴 / 𝑥⦌𝐶 × V))) |
12 | 10, 11 | e1a 37873 |
. . . . 5
⊢ ( 𝐴 ∈ 𝑉 ▶ (⦋𝐴 / 𝑥⦌𝐵 ∩ ⦋𝐴 / 𝑥⦌(𝐶 × V)) = (⦋𝐴 / 𝑥⦌𝐵 ∩ (⦋𝐴 / 𝑥⦌𝐶 × V)) ) |
13 | | csbingOLD 38076 |
. . . . . 6
⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌(𝐵 ∩ (𝐶 × V)) = (⦋𝐴 / 𝑥⦌𝐵 ∩ ⦋𝐴 / 𝑥⦌(𝐶 × V))) |
14 | 1, 13 | e1a 37873 |
. . . . 5
⊢ ( 𝐴 ∈ 𝑉 ▶ ⦋𝐴 / 𝑥⦌(𝐵 ∩ (𝐶 × V)) = (⦋𝐴 / 𝑥⦌𝐵 ∩ ⦋𝐴 / 𝑥⦌(𝐶 × V)) ) |
15 | | eqeq2 2621 |
. . . . . 6
⊢
((⦋𝐴 /
𝑥⦌𝐵 ∩ ⦋𝐴 / 𝑥⦌(𝐶 × V)) = (⦋𝐴 / 𝑥⦌𝐵 ∩ (⦋𝐴 / 𝑥⦌𝐶 × V)) → (⦋𝐴 / 𝑥⦌(𝐵 ∩ (𝐶 × V)) = (⦋𝐴 / 𝑥⦌𝐵 ∩ ⦋𝐴 / 𝑥⦌(𝐶 × V)) ↔ ⦋𝐴 / 𝑥⦌(𝐵 ∩ (𝐶 × V)) = (⦋𝐴 / 𝑥⦌𝐵 ∩ (⦋𝐴 / 𝑥⦌𝐶 × V)))) |
16 | 15 | biimpd 218 |
. . . . 5
⊢
((⦋𝐴 /
𝑥⦌𝐵 ∩ ⦋𝐴 / 𝑥⦌(𝐶 × V)) = (⦋𝐴 / 𝑥⦌𝐵 ∩ (⦋𝐴 / 𝑥⦌𝐶 × V)) → (⦋𝐴 / 𝑥⦌(𝐵 ∩ (𝐶 × V)) = (⦋𝐴 / 𝑥⦌𝐵 ∩ ⦋𝐴 / 𝑥⦌(𝐶 × V)) → ⦋𝐴 / 𝑥⦌(𝐵 ∩ (𝐶 × V)) = (⦋𝐴 / 𝑥⦌𝐵 ∩ (⦋𝐴 / 𝑥⦌𝐶 × V)))) |
17 | 12, 14, 16 | e11 37934 |
. . . 4
⊢ ( 𝐴 ∈ 𝑉 ▶ ⦋𝐴 / 𝑥⦌(𝐵 ∩ (𝐶 × V)) = (⦋𝐴 / 𝑥⦌𝐵 ∩ (⦋𝐴 / 𝑥⦌𝐶 × V)) ) |
18 | | df-res 5050 |
. . . . . 6
⊢ (𝐵 ↾ 𝐶) = (𝐵 ∩ (𝐶 × V)) |
19 | 18 | ax-gen 1713 |
. . . . 5
⊢
∀𝑥(𝐵 ↾ 𝐶) = (𝐵 ∩ (𝐶 × V)) |
20 | | csbeq2gOLD 37786 |
. . . . 5
⊢ (𝐴 ∈ 𝑉 → (∀𝑥(𝐵 ↾ 𝐶) = (𝐵 ∩ (𝐶 × V)) → ⦋𝐴 / 𝑥⦌(𝐵 ↾ 𝐶) = ⦋𝐴 / 𝑥⦌(𝐵 ∩ (𝐶 × V)))) |
21 | 1, 19, 20 | e10 37940 |
. . . 4
⊢ ( 𝐴 ∈ 𝑉 ▶ ⦋𝐴 / 𝑥⦌(𝐵 ↾ 𝐶) = ⦋𝐴 / 𝑥⦌(𝐵 ∩ (𝐶 × V)) ) |
22 | | eqeq2 2621 |
. . . . 5
⊢
(⦋𝐴 /
𝑥⦌(𝐵 ∩ (𝐶 × V)) = (⦋𝐴 / 𝑥⦌𝐵 ∩ (⦋𝐴 / 𝑥⦌𝐶 × V)) → (⦋𝐴 / 𝑥⦌(𝐵 ↾ 𝐶) = ⦋𝐴 / 𝑥⦌(𝐵 ∩ (𝐶 × V)) ↔ ⦋𝐴 / 𝑥⦌(𝐵 ↾ 𝐶) = (⦋𝐴 / 𝑥⦌𝐵 ∩ (⦋𝐴 / 𝑥⦌𝐶 × V)))) |
23 | 22 | biimpd 218 |
. . . 4
⊢
(⦋𝐴 /
𝑥⦌(𝐵 ∩ (𝐶 × V)) = (⦋𝐴 / 𝑥⦌𝐵 ∩ (⦋𝐴 / 𝑥⦌𝐶 × V)) → (⦋𝐴 / 𝑥⦌(𝐵 ↾ 𝐶) = ⦋𝐴 / 𝑥⦌(𝐵 ∩ (𝐶 × V)) → ⦋𝐴 / 𝑥⦌(𝐵 ↾ 𝐶) = (⦋𝐴 / 𝑥⦌𝐵 ∩ (⦋𝐴 / 𝑥⦌𝐶 × V)))) |
24 | 17, 21, 23 | e11 37934 |
. . 3
⊢ ( 𝐴 ∈ 𝑉 ▶ ⦋𝐴 / 𝑥⦌(𝐵 ↾ 𝐶) = (⦋𝐴 / 𝑥⦌𝐵 ∩ (⦋𝐴 / 𝑥⦌𝐶 × V)) ) |
25 | | df-res 5050 |
. . 3
⊢
(⦋𝐴 /
𝑥⦌𝐵 ↾ ⦋𝐴 / 𝑥⦌𝐶) = (⦋𝐴 / 𝑥⦌𝐵 ∩ (⦋𝐴 / 𝑥⦌𝐶 × V)) |
26 | | eqeq2 2621 |
. . . 4
⊢
((⦋𝐴 /
𝑥⦌𝐵 ↾ ⦋𝐴 / 𝑥⦌𝐶) = (⦋𝐴 / 𝑥⦌𝐵 ∩ (⦋𝐴 / 𝑥⦌𝐶 × V)) → (⦋𝐴 / 𝑥⦌(𝐵 ↾ 𝐶) = (⦋𝐴 / 𝑥⦌𝐵 ↾ ⦋𝐴 / 𝑥⦌𝐶) ↔ ⦋𝐴 / 𝑥⦌(𝐵 ↾ 𝐶) = (⦋𝐴 / 𝑥⦌𝐵 ∩ (⦋𝐴 / 𝑥⦌𝐶 × V)))) |
27 | 26 | biimprcd 239 |
. . 3
⊢
(⦋𝐴 /
𝑥⦌(𝐵 ↾ 𝐶) = (⦋𝐴 / 𝑥⦌𝐵 ∩ (⦋𝐴 / 𝑥⦌𝐶 × V)) → ((⦋𝐴 / 𝑥⦌𝐵 ↾ ⦋𝐴 / 𝑥⦌𝐶) = (⦋𝐴 / 𝑥⦌𝐵 ∩ (⦋𝐴 / 𝑥⦌𝐶 × V)) → ⦋𝐴 / 𝑥⦌(𝐵 ↾ 𝐶) = (⦋𝐴 / 𝑥⦌𝐵 ↾ ⦋𝐴 / 𝑥⦌𝐶))) |
28 | 24, 25, 27 | e10 37940 |
. 2
⊢ ( 𝐴 ∈ 𝑉 ▶ ⦋𝐴 / 𝑥⦌(𝐵 ↾ 𝐶) = (⦋𝐴 / 𝑥⦌𝐵 ↾ ⦋𝐴 / 𝑥⦌𝐶) ) |
29 | 28 | in1 37808 |
1
⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌(𝐵 ↾ 𝐶) = (⦋𝐴 / 𝑥⦌𝐵 ↾ ⦋𝐴 / 𝑥⦌𝐶)) |