Proof of Theorem brecop
Step | Hyp | Ref
| Expression |
1 | | brecop.1 |
. . . 4
⊢ ∼ ∈ V |
2 | | brecop.4 |
. . . 4
⊢ 𝐻 = ((𝐺 × 𝐺) / ∼ ) |
3 | 1, 2 | ecopqsi 6097 |
. . 3
⊢
((A ∈ 𝐺 ∧ B ∈ 𝐺) → [〈A, B〉]
∼
∈ 𝐻) |
4 | 1, 2 | ecopqsi 6097 |
. . 3
⊢ ((𝐶 ∈ 𝐺 ∧ 𝐷 ∈ 𝐺) → [〈𝐶, 𝐷〉] ∼ ∈ 𝐻) |
5 | | df-br 3756 |
. . . . 5
⊢
([〈A, B〉] ∼ ≤ [〈𝐶, 𝐷〉] ∼ ↔
〈[〈A, B〉] ∼ , [〈𝐶, 𝐷〉] ∼ 〉 ∈ ≤ ) |
6 | | brecop.5 |
. . . . . 6
⊢ ≤ =
{〈x, y〉 ∣ ((x ∈ 𝐻 ∧ y ∈ 𝐻) ∧ ∃z∃w∃v∃u((x = [〈z,
w〉] ∼ ∧ y =
[〈v, u〉] ∼ ) ∧ φ))} |
7 | 6 | eleq2i 2101 |
. . . . 5
⊢
(〈[〈A, B〉] ∼ , [〈𝐶, 𝐷〉] ∼ 〉 ∈ ≤ ↔
〈[〈A, B〉] ∼ , [〈𝐶, 𝐷〉] ∼ 〉 ∈ {〈x,
y〉 ∣ ((x ∈ 𝐻 ∧ y ∈ 𝐻) ∧ ∃z∃w∃v∃u((x = [〈z,
w〉] ∼ ∧ y =
[〈v, u〉] ∼ ) ∧ φ))}) |
8 | 5, 7 | bitri 173 |
. . . 4
⊢
([〈A, B〉] ∼ ≤ [〈𝐶, 𝐷〉] ∼ ↔
〈[〈A, B〉] ∼ , [〈𝐶, 𝐷〉] ∼ 〉 ∈ {〈x,
y〉 ∣ ((x ∈ 𝐻 ∧ y ∈ 𝐻) ∧ ∃z∃w∃v∃u((x = [〈z,
w〉] ∼ ∧ y =
[〈v, u〉] ∼ ) ∧ φ))}) |
9 | | eqeq1 2043 |
. . . . . . . 8
⊢ (x = [〈A,
B〉] ∼ → (x = [〈z,
w〉] ∼ ↔
[〈A, B〉] ∼ = [〈z, w〉]
∼
)) |
10 | 9 | anbi1d 438 |
. . . . . . 7
⊢ (x = [〈A,
B〉] ∼ → ((x = [〈z,
w〉] ∼ ∧ y =
[〈v, u〉] ∼ ) ↔
([〈A, B〉] ∼ = [〈z, w〉]
∼
∧ y =
[〈v, u〉] ∼
))) |
11 | 10 | anbi1d 438 |
. . . . . 6
⊢ (x = [〈A,
B〉] ∼ → (((x = [〈z,
w〉] ∼ ∧ y =
[〈v, u〉] ∼ ) ∧ φ) ↔
(([〈A, B〉] ∼ = [〈z, w〉]
∼
∧ y =
[〈v, u〉] ∼ ) ∧ φ))) |
12 | 11 | 4exbidv 1747 |
. . . . 5
⊢ (x = [〈A,
B〉] ∼ → (∃z∃w∃v∃u((x = [〈z,
w〉] ∼ ∧ y =
[〈v, u〉] ∼ ) ∧ φ) ↔
∃z∃w∃v∃u(([〈A,
B〉] ∼ = [〈z, w〉]
∼
∧ y =
[〈v, u〉] ∼ ) ∧ φ))) |
13 | | eqeq1 2043 |
. . . . . . . 8
⊢ (y = [〈𝐶, 𝐷〉] ∼ → (y = [〈v,
u〉] ∼ ↔ [〈𝐶, 𝐷〉] ∼ = [〈v, u〉]
∼
)) |
14 | 13 | anbi2d 437 |
. . . . . . 7
⊢ (y = [〈𝐶, 𝐷〉] ∼ →
(([〈A, B〉] ∼ = [〈z, w〉]
∼
∧ y =
[〈v, u〉] ∼ ) ↔
([〈A, B〉] ∼ = [〈z, w〉]
∼
∧ [〈𝐶, 𝐷〉] ∼ = [〈v, u〉]
∼
))) |
15 | 14 | anbi1d 438 |
. . . . . 6
⊢ (y = [〈𝐶, 𝐷〉] ∼ →
((([〈A, B〉] ∼ = [〈z, w〉]
∼
∧ y =
[〈v, u〉] ∼ ) ∧ φ) ↔
(([〈A, B〉] ∼ = [〈z, w〉]
∼
∧ [〈𝐶, 𝐷〉] ∼ = [〈v, u〉]
∼
) ∧ φ))) |
16 | 15 | 4exbidv 1747 |
. . . . 5
⊢ (y = [〈𝐶, 𝐷〉] ∼ → (∃z∃w∃v∃u(([〈A,
B〉] ∼ = [〈z, w〉]
∼
∧ y =
[〈v, u〉] ∼ ) ∧ φ) ↔
∃z∃w∃v∃u(([〈A,
B〉] ∼ = [〈z, w〉]
∼
∧ [〈𝐶, 𝐷〉] ∼ = [〈v, u〉]
∼
) ∧ φ))) |
17 | 12, 16 | opelopab2 3998 |
. . . 4
⊢
(([〈A, B〉] ∼ ∈ 𝐻 ∧
[〈𝐶, 𝐷〉] ∼ ∈ 𝐻) → (〈[〈A, B〉]
∼
, [〈𝐶, 𝐷〉] ∼ 〉 ∈ {〈x,
y〉 ∣ ((x ∈ 𝐻 ∧ y ∈ 𝐻) ∧ ∃z∃w∃v∃u((x = [〈z,
w〉] ∼ ∧ y =
[〈v, u〉] ∼ ) ∧ φ))}
↔ ∃z∃w∃v∃u(([〈A,
B〉] ∼ = [〈z, w〉]
∼
∧ [〈𝐶, 𝐷〉] ∼ = [〈v, u〉]
∼
) ∧ φ))) |
18 | 8, 17 | syl5bb 181 |
. . 3
⊢
(([〈A, B〉] ∼ ∈ 𝐻 ∧
[〈𝐶, 𝐷〉] ∼ ∈ 𝐻) → ([〈A, B〉]
∼
≤
[〈𝐶, 𝐷〉] ∼ ↔ ∃z∃w∃v∃u(([〈A,
B〉] ∼ = [〈z, w〉]
∼
∧ [〈𝐶, 𝐷〉] ∼ = [〈v, u〉]
∼
) ∧ φ))) |
19 | 3, 4, 18 | syl2an 273 |
. 2
⊢
(((A ∈ 𝐺 ∧ B ∈ 𝐺) ∧ (𝐶 ∈ 𝐺 ∧ 𝐷 ∈ 𝐺)) → ([〈A, B〉]
∼
≤
[〈𝐶, 𝐷〉] ∼ ↔ ∃z∃w∃v∃u(([〈A,
B〉] ∼ = [〈z, w〉]
∼
∧ [〈𝐶, 𝐷〉] ∼ = [〈v, u〉]
∼
) ∧ φ))) |
20 | | opeq12 3542 |
. . . . . 6
⊢
((z = A ∧ w = B) →
〈z, w〉 = 〈A, B〉) |
21 | 20 | eceq1d 6078 |
. . . . 5
⊢
((z = A ∧ w = B) →
[〈z, w〉] ∼ = [〈A, B〉]
∼
) |
22 | | opeq12 3542 |
. . . . . 6
⊢
((v = 𝐶 ∧ u = 𝐷) → 〈v, u〉 =
〈𝐶, 𝐷〉) |
23 | 22 | eceq1d 6078 |
. . . . 5
⊢
((v = 𝐶 ∧ u = 𝐷) → [〈v, u〉]
∼
= [〈𝐶, 𝐷〉] ∼ ) |
24 | 21, 23 | anim12i 321 |
. . . 4
⊢
(((z = A ∧ w = B) ∧ (v = 𝐶 ∧ u = 𝐷)) → ([〈z, w〉]
∼
= [〈A, B〉] ∼ ∧ [〈v,
u〉] ∼ = [〈𝐶, 𝐷〉] ∼ )) |
25 | | opelxpi 4319 |
. . . . . . . 8
⊢
((A ∈ 𝐺 ∧ B ∈ 𝐺) → 〈A, B〉 ∈ (𝐺 × 𝐺)) |
26 | | opelxp 4317 |
. . . . . . . . 9
⊢
(〈z, w〉 ∈ (𝐺 × 𝐺) ↔ (z ∈ 𝐺 ∧ w ∈ 𝐺)) |
27 | | brecop.2 |
. . . . . . . . . . 11
⊢ ∼ Er
(𝐺 × 𝐺) |
28 | 27 | a1i 9 |
. . . . . . . . . 10
⊢
([〈z, w〉] ∼ = [〈A, B〉]
∼
→ ∼ Er (𝐺 × 𝐺)) |
29 | | id 19 |
. . . . . . . . . 10
⊢
([〈z, w〉] ∼ = [〈A, B〉]
∼
→ [〈z, w〉] ∼ = [〈A, B〉]
∼
) |
30 | 28, 29 | ereldm 6085 |
. . . . . . . . 9
⊢
([〈z, w〉] ∼ = [〈A, B〉]
∼
→ (〈z, w〉 ∈ (𝐺 × 𝐺) ↔ 〈A, B〉 ∈ (𝐺 × 𝐺))) |
31 | 26, 30 | syl5bbr 183 |
. . . . . . . 8
⊢
([〈z, w〉] ∼ = [〈A, B〉]
∼
→ ((z ∈ 𝐺 ∧ w ∈ 𝐺) ↔ 〈A, B〉 ∈ (𝐺 × 𝐺))) |
32 | 25, 31 | syl5ibr 145 |
. . . . . . 7
⊢
([〈z, w〉] ∼ = [〈A, B〉]
∼
→ ((A ∈ 𝐺 ∧ B ∈ 𝐺) → (z ∈ 𝐺 ∧ w ∈ 𝐺))) |
33 | | opelxpi 4319 |
. . . . . . . 8
⊢ ((𝐶 ∈ 𝐺 ∧ 𝐷 ∈ 𝐺) → 〈𝐶, 𝐷〉 ∈
(𝐺 × 𝐺)) |
34 | | opelxp 4317 |
. . . . . . . . 9
⊢
(〈v, u〉 ∈ (𝐺 × 𝐺) ↔ (v ∈ 𝐺 ∧ u ∈ 𝐺)) |
35 | 27 | a1i 9 |
. . . . . . . . . 10
⊢
([〈v, u〉] ∼ = [〈𝐶, 𝐷〉] ∼ → ∼ Er
(𝐺 × 𝐺)) |
36 | | id 19 |
. . . . . . . . . 10
⊢
([〈v, u〉] ∼ = [〈𝐶, 𝐷〉] ∼ →
[〈v, u〉] ∼ = [〈𝐶, 𝐷〉] ∼ ) |
37 | 35, 36 | ereldm 6085 |
. . . . . . . . 9
⊢
([〈v, u〉] ∼ = [〈𝐶, 𝐷〉] ∼ →
(〈v, u〉 ∈ (𝐺 × 𝐺) ↔ 〈𝐶, 𝐷〉 ∈
(𝐺 × 𝐺))) |
38 | 34, 37 | syl5bbr 183 |
. . . . . . . 8
⊢
([〈v, u〉] ∼ = [〈𝐶, 𝐷〉] ∼ → ((v ∈ 𝐺 ∧ u ∈ 𝐺) ↔ 〈𝐶, 𝐷〉 ∈
(𝐺 × 𝐺))) |
39 | 33, 38 | syl5ibr 145 |
. . . . . . 7
⊢
([〈v, u〉] ∼ = [〈𝐶, 𝐷〉] ∼ → ((𝐶 ∈ 𝐺 ∧ 𝐷 ∈ 𝐺) → (v ∈ 𝐺 ∧ u ∈ 𝐺))) |
40 | 32, 39 | im2anan9 530 |
. . . . . 6
⊢
(([〈z, w〉] ∼ = [〈A, B〉]
∼
∧ [〈v, u〉]
∼
= [〈𝐶, 𝐷〉] ∼ ) →
(((A ∈
𝐺 ∧ B ∈ 𝐺) ∧ (𝐶 ∈ 𝐺 ∧ 𝐷 ∈ 𝐺)) → ((z ∈ 𝐺 ∧ w ∈ 𝐺) ∧
(v ∈
𝐺 ∧ u ∈ 𝐺)))) |
41 | | brecop.6 |
. . . . . . . . 9
⊢
((((z ∈ 𝐺 ∧ w ∈ 𝐺) ∧ (A ∈ 𝐺 ∧ B ∈ 𝐺)) ∧ ((v ∈ 𝐺 ∧ u ∈ 𝐺) ∧ (𝐶 ∈ 𝐺 ∧ 𝐷 ∈ 𝐺))) → (([〈z, w〉]
∼
= [〈A, B〉] ∼ ∧ [〈v,
u〉] ∼ = [〈𝐶, 𝐷〉] ∼ ) → (φ ↔ ψ))) |
42 | 41 | an4s 522 |
. . . . . . . 8
⊢
((((z ∈ 𝐺 ∧ w ∈ 𝐺) ∧ (v ∈ 𝐺 ∧ u ∈ 𝐺)) ∧ ((A ∈ 𝐺 ∧ B ∈ 𝐺) ∧ (𝐶 ∈ 𝐺 ∧ 𝐷 ∈ 𝐺))) → (([〈z, w〉]
∼
= [〈A, B〉] ∼ ∧ [〈v,
u〉] ∼ = [〈𝐶, 𝐷〉] ∼ ) → (φ ↔ ψ))) |
43 | 42 | ex 108 |
. . . . . . 7
⊢
(((z ∈ 𝐺 ∧ w ∈ 𝐺) ∧ (v ∈ 𝐺 ∧ u ∈ 𝐺)) → (((A ∈ 𝐺 ∧ B ∈ 𝐺) ∧ (𝐶 ∈ 𝐺 ∧ 𝐷 ∈ 𝐺)) → (([〈z, w〉]
∼
= [〈A, B〉] ∼ ∧ [〈v,
u〉] ∼ = [〈𝐶, 𝐷〉] ∼ ) → (φ ↔ ψ)))) |
44 | 43 | com13 74 |
. . . . . 6
⊢
(([〈z, w〉] ∼ = [〈A, B〉]
∼
∧ [〈v, u〉]
∼
= [〈𝐶, 𝐷〉] ∼ ) →
(((A ∈
𝐺 ∧ B ∈ 𝐺) ∧ (𝐶 ∈ 𝐺 ∧ 𝐷 ∈ 𝐺)) → (((z ∈ 𝐺 ∧ w ∈ 𝐺) ∧
(v ∈
𝐺 ∧ u ∈ 𝐺)) → (φ ↔ ψ)))) |
45 | 40, 44 | mpdd 36 |
. . . . 5
⊢
(([〈z, w〉] ∼ = [〈A, B〉]
∼
∧ [〈v, u〉]
∼
= [〈𝐶, 𝐷〉] ∼ ) →
(((A ∈
𝐺 ∧ B ∈ 𝐺) ∧ (𝐶 ∈ 𝐺 ∧ 𝐷 ∈ 𝐺)) → (φ ↔ ψ))) |
46 | 45 | pm5.74d 171 |
. . . 4
⊢
(([〈z, w〉] ∼ = [〈A, B〉]
∼
∧ [〈v, u〉]
∼
= [〈𝐶, 𝐷〉] ∼ ) →
((((A ∈
𝐺 ∧ B ∈ 𝐺) ∧ (𝐶 ∈ 𝐺 ∧ 𝐷 ∈ 𝐺)) → φ) ↔ (((A ∈ 𝐺 ∧ B ∈ 𝐺) ∧ (𝐶 ∈ 𝐺 ∧ 𝐷 ∈ 𝐺)) → ψ))) |
47 | 24, 46 | cgsex4g 2585 |
. . 3
⊢
(((A ∈ 𝐺 ∧ B ∈ 𝐺) ∧ (𝐶 ∈ 𝐺 ∧ 𝐷 ∈ 𝐺)) → (∃z∃w∃v∃u(([〈z,
w〉] ∼ = [〈A, B〉]
∼
∧ [〈v, u〉]
∼
= [〈𝐶, 𝐷〉] ∼ ) ∧ (((A ∈ 𝐺 ∧ B ∈ 𝐺) ∧ (𝐶 ∈ 𝐺 ∧ 𝐷 ∈ 𝐺)) → φ)) ↔ (((A ∈ 𝐺 ∧ B ∈ 𝐺) ∧ (𝐶 ∈ 𝐺 ∧ 𝐷 ∈ 𝐺)) → ψ))) |
48 | | eqcom 2039 |
. . . . . . 7
⊢
([〈A, B〉] ∼ = [〈z, w〉]
∼
↔ [〈z, w〉] ∼ = [〈A, B〉]
∼
) |
49 | | eqcom 2039 |
. . . . . . 7
⊢
([〈𝐶, 𝐷〉] ∼ = [〈v, u〉]
∼
↔ [〈v, u〉] ∼ = [〈𝐶, 𝐷〉] ∼ ) |
50 | 48, 49 | anbi12i 433 |
. . . . . 6
⊢
(([〈A, B〉] ∼ = [〈z, w〉]
∼
∧ [〈𝐶, 𝐷〉] ∼ = [〈v, u〉]
∼
) ↔ ([〈z, w〉] ∼ = [〈A, B〉]
∼
∧ [〈v, u〉]
∼
= [〈𝐶, 𝐷〉] ∼ )) |
51 | 50 | a1i 9 |
. . . . 5
⊢
(((A ∈ 𝐺 ∧ B ∈ 𝐺) ∧ (𝐶 ∈ 𝐺 ∧ 𝐷 ∈ 𝐺)) → (([〈A, B〉]
∼
= [〈z, w〉] ∼ ∧ [〈𝐶, 𝐷〉] ∼ = [〈v, u〉]
∼
) ↔ ([〈z, w〉] ∼ = [〈A, B〉]
∼
∧ [〈v, u〉]
∼
= [〈𝐶, 𝐷〉] ∼
))) |
52 | | biimt 230 |
. . . . 5
⊢
(((A ∈ 𝐺 ∧ B ∈ 𝐺) ∧ (𝐶 ∈ 𝐺 ∧ 𝐷 ∈ 𝐺)) → (φ ↔ (((A ∈ 𝐺 ∧ B ∈ 𝐺) ∧ (𝐶 ∈ 𝐺 ∧ 𝐷 ∈ 𝐺)) → φ))) |
53 | 51, 52 | anbi12d 442 |
. . . 4
⊢
(((A ∈ 𝐺 ∧ B ∈ 𝐺) ∧ (𝐶 ∈ 𝐺 ∧ 𝐷 ∈ 𝐺)) → ((([〈A, B〉]
∼
= [〈z, w〉] ∼ ∧ [〈𝐶, 𝐷〉] ∼ = [〈v, u〉]
∼
) ∧ φ)
↔ (([〈z, w〉] ∼ = [〈A, B〉]
∼
∧ [〈v, u〉]
∼
= [〈𝐶, 𝐷〉] ∼ ) ∧ (((A ∈ 𝐺 ∧ B ∈ 𝐺) ∧ (𝐶 ∈ 𝐺 ∧ 𝐷 ∈ 𝐺)) → φ)))) |
54 | 53 | 4exbidv 1747 |
. . 3
⊢
(((A ∈ 𝐺 ∧ B ∈ 𝐺) ∧ (𝐶 ∈ 𝐺 ∧ 𝐷 ∈ 𝐺)) → (∃z∃w∃v∃u(([〈A,
B〉] ∼ = [〈z, w〉]
∼
∧ [〈𝐶, 𝐷〉] ∼ = [〈v, u〉]
∼
) ∧ φ)
↔ ∃z∃w∃v∃u(([〈z,
w〉] ∼ = [〈A, B〉]
∼
∧ [〈v, u〉]
∼
= [〈𝐶, 𝐷〉] ∼ ) ∧ (((A ∈ 𝐺 ∧ B ∈ 𝐺) ∧ (𝐶 ∈ 𝐺 ∧ 𝐷 ∈ 𝐺)) → φ)))) |
55 | | biimt 230 |
. . 3
⊢
(((A ∈ 𝐺 ∧ B ∈ 𝐺) ∧ (𝐶 ∈ 𝐺 ∧ 𝐷 ∈ 𝐺)) → (ψ ↔ (((A ∈ 𝐺 ∧ B ∈ 𝐺) ∧ (𝐶 ∈ 𝐺 ∧ 𝐷 ∈ 𝐺)) → ψ))) |
56 | 47, 54, 55 | 3bitr4d 209 |
. 2
⊢
(((A ∈ 𝐺 ∧ B ∈ 𝐺) ∧ (𝐶 ∈ 𝐺 ∧ 𝐷 ∈ 𝐺)) → (∃z∃w∃v∃u(([〈A,
B〉] ∼ = [〈z, w〉]
∼
∧ [〈𝐶, 𝐷〉] ∼ = [〈v, u〉]
∼
) ∧ φ)
↔ ψ)) |
57 | 19, 56 | bitrd 177 |
1
⊢
(((A ∈ 𝐺 ∧ B ∈ 𝐺) ∧ (𝐶 ∈ 𝐺 ∧ 𝐷 ∈ 𝐺)) → ([〈A, B〉]
∼
≤
[〈𝐶, 𝐷〉] ∼ ↔ ψ)) |