Step | Hyp | Ref
| Expression |
1 | | simpl 486 |
. . . . . 6
⊢ ((𝑎 = 𝐴 ∧ 𝑏 = 𝐵) → 𝑎 = 𝐴) |
2 | 1 | neeq1d 3000 |
. . . . 5
⊢ ((𝑎 = 𝐴 ∧ 𝑏 = 𝐵) → (𝑎 ≠ 𝐶 ↔ 𝐴 ≠ 𝐶)) |
3 | | simpr 488 |
. . . . . 6
⊢ ((𝑎 = 𝐴 ∧ 𝑏 = 𝐵) → 𝑏 = 𝐵) |
4 | 3 | neeq1d 3000 |
. . . . 5
⊢ ((𝑎 = 𝐴 ∧ 𝑏 = 𝐵) → (𝑏 ≠ 𝐶 ↔ 𝐵 ≠ 𝐶)) |
5 | 3 | oveq2d 7229 |
. . . . . . 7
⊢ ((𝑎 = 𝐴 ∧ 𝑏 = 𝐵) → (𝐶𝐼𝑏) = (𝐶𝐼𝐵)) |
6 | 1, 5 | eleq12d 2832 |
. . . . . 6
⊢ ((𝑎 = 𝐴 ∧ 𝑏 = 𝐵) → (𝑎 ∈ (𝐶𝐼𝑏) ↔ 𝐴 ∈ (𝐶𝐼𝐵))) |
7 | 1 | oveq2d 7229 |
. . . . . . 7
⊢ ((𝑎 = 𝐴 ∧ 𝑏 = 𝐵) → (𝐶𝐼𝑎) = (𝐶𝐼𝐴)) |
8 | 3, 7 | eleq12d 2832 |
. . . . . 6
⊢ ((𝑎 = 𝐴 ∧ 𝑏 = 𝐵) → (𝑏 ∈ (𝐶𝐼𝑎) ↔ 𝐵 ∈ (𝐶𝐼𝐴))) |
9 | 6, 8 | orbi12d 919 |
. . . . 5
⊢ ((𝑎 = 𝐴 ∧ 𝑏 = 𝐵) → ((𝑎 ∈ (𝐶𝐼𝑏) ∨ 𝑏 ∈ (𝐶𝐼𝑎)) ↔ (𝐴 ∈ (𝐶𝐼𝐵) ∨ 𝐵 ∈ (𝐶𝐼𝐴)))) |
10 | 2, 4, 9 | 3anbi123d 1438 |
. . . 4
⊢ ((𝑎 = 𝐴 ∧ 𝑏 = 𝐵) → ((𝑎 ≠ 𝐶 ∧ 𝑏 ≠ 𝐶 ∧ (𝑎 ∈ (𝐶𝐼𝑏) ∨ 𝑏 ∈ (𝐶𝐼𝑎))) ↔ (𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶 ∧ (𝐴 ∈ (𝐶𝐼𝐵) ∨ 𝐵 ∈ (𝐶𝐼𝐴))))) |
11 | | eqid 2737 |
. . . 4
⊢
{〈𝑎, 𝑏〉 ∣ ((𝑎 ∈ 𝑃 ∧ 𝑏 ∈ 𝑃) ∧ (𝑎 ≠ 𝐶 ∧ 𝑏 ≠ 𝐶 ∧ (𝑎 ∈ (𝐶𝐼𝑏) ∨ 𝑏 ∈ (𝐶𝐼𝑎))))} = {〈𝑎, 𝑏〉 ∣ ((𝑎 ∈ 𝑃 ∧ 𝑏 ∈ 𝑃) ∧ (𝑎 ≠ 𝐶 ∧ 𝑏 ≠ 𝐶 ∧ (𝑎 ∈ (𝐶𝐼𝑏) ∨ 𝑏 ∈ (𝐶𝐼𝑎))))} |
12 | 10, 11 | brab2a 5641 |
. . 3
⊢ (𝐴{〈𝑎, 𝑏〉 ∣ ((𝑎 ∈ 𝑃 ∧ 𝑏 ∈ 𝑃) ∧ (𝑎 ≠ 𝐶 ∧ 𝑏 ≠ 𝐶 ∧ (𝑎 ∈ (𝐶𝐼𝑏) ∨ 𝑏 ∈ (𝐶𝐼𝑎))))}𝐵 ↔ ((𝐴 ∈ 𝑃 ∧ 𝐵 ∈ 𝑃) ∧ (𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶 ∧ (𝐴 ∈ (𝐶𝐼𝐵) ∨ 𝐵 ∈ (𝐶𝐼𝐴))))) |
13 | 12 | a1i 11 |
. 2
⊢ (𝜑 → (𝐴{〈𝑎, 𝑏〉 ∣ ((𝑎 ∈ 𝑃 ∧ 𝑏 ∈ 𝑃) ∧ (𝑎 ≠ 𝐶 ∧ 𝑏 ≠ 𝐶 ∧ (𝑎 ∈ (𝐶𝐼𝑏) ∨ 𝑏 ∈ (𝐶𝐼𝑎))))}𝐵 ↔ ((𝐴 ∈ 𝑃 ∧ 𝐵 ∈ 𝑃) ∧ (𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶 ∧ (𝐴 ∈ (𝐶𝐼𝐵) ∨ 𝐵 ∈ (𝐶𝐼𝐴)))))) |
14 | | ishlg.k |
. . . . 5
⊢ 𝐾 = (hlG‘𝐺) |
15 | | ishlg.g |
. . . . . 6
⊢ (𝜑 → 𝐺 ∈ 𝑉) |
16 | | elex 3426 |
. . . . . 6
⊢ (𝐺 ∈ 𝑉 → 𝐺 ∈ V) |
17 | | fveq2 6717 |
. . . . . . . . 9
⊢ (𝑔 = 𝐺 → (Base‘𝑔) = (Base‘𝐺)) |
18 | | ishlg.p |
. . . . . . . . 9
⊢ 𝑃 = (Base‘𝐺) |
19 | 17, 18 | eqtr4di 2796 |
. . . . . . . 8
⊢ (𝑔 = 𝐺 → (Base‘𝑔) = 𝑃) |
20 | 19 | eleq2d 2823 |
. . . . . . . . . . 11
⊢ (𝑔 = 𝐺 → (𝑎 ∈ (Base‘𝑔) ↔ 𝑎 ∈ 𝑃)) |
21 | 19 | eleq2d 2823 |
. . . . . . . . . . 11
⊢ (𝑔 = 𝐺 → (𝑏 ∈ (Base‘𝑔) ↔ 𝑏 ∈ 𝑃)) |
22 | 20, 21 | anbi12d 634 |
. . . . . . . . . 10
⊢ (𝑔 = 𝐺 → ((𝑎 ∈ (Base‘𝑔) ∧ 𝑏 ∈ (Base‘𝑔)) ↔ (𝑎 ∈ 𝑃 ∧ 𝑏 ∈ 𝑃))) |
23 | | fveq2 6717 |
. . . . . . . . . . . . . . 15
⊢ (𝑔 = 𝐺 → (Itv‘𝑔) = (Itv‘𝐺)) |
24 | | ishlg.i |
. . . . . . . . . . . . . . 15
⊢ 𝐼 = (Itv‘𝐺) |
25 | 23, 24 | eqtr4di 2796 |
. . . . . . . . . . . . . 14
⊢ (𝑔 = 𝐺 → (Itv‘𝑔) = 𝐼) |
26 | 25 | oveqd 7230 |
. . . . . . . . . . . . 13
⊢ (𝑔 = 𝐺 → (𝑐(Itv‘𝑔)𝑏) = (𝑐𝐼𝑏)) |
27 | 26 | eleq2d 2823 |
. . . . . . . . . . . 12
⊢ (𝑔 = 𝐺 → (𝑎 ∈ (𝑐(Itv‘𝑔)𝑏) ↔ 𝑎 ∈ (𝑐𝐼𝑏))) |
28 | 25 | oveqd 7230 |
. . . . . . . . . . . . 13
⊢ (𝑔 = 𝐺 → (𝑐(Itv‘𝑔)𝑎) = (𝑐𝐼𝑎)) |
29 | 28 | eleq2d 2823 |
. . . . . . . . . . . 12
⊢ (𝑔 = 𝐺 → (𝑏 ∈ (𝑐(Itv‘𝑔)𝑎) ↔ 𝑏 ∈ (𝑐𝐼𝑎))) |
30 | 27, 29 | orbi12d 919 |
. . . . . . . . . . 11
⊢ (𝑔 = 𝐺 → ((𝑎 ∈ (𝑐(Itv‘𝑔)𝑏) ∨ 𝑏 ∈ (𝑐(Itv‘𝑔)𝑎)) ↔ (𝑎 ∈ (𝑐𝐼𝑏) ∨ 𝑏 ∈ (𝑐𝐼𝑎)))) |
31 | 30 | 3anbi3d 1444 |
. . . . . . . . . 10
⊢ (𝑔 = 𝐺 → ((𝑎 ≠ 𝑐 ∧ 𝑏 ≠ 𝑐 ∧ (𝑎 ∈ (𝑐(Itv‘𝑔)𝑏) ∨ 𝑏 ∈ (𝑐(Itv‘𝑔)𝑎))) ↔ (𝑎 ≠ 𝑐 ∧ 𝑏 ≠ 𝑐 ∧ (𝑎 ∈ (𝑐𝐼𝑏) ∨ 𝑏 ∈ (𝑐𝐼𝑎))))) |
32 | 22, 31 | anbi12d 634 |
. . . . . . . . 9
⊢ (𝑔 = 𝐺 → (((𝑎 ∈ (Base‘𝑔) ∧ 𝑏 ∈ (Base‘𝑔)) ∧ (𝑎 ≠ 𝑐 ∧ 𝑏 ≠ 𝑐 ∧ (𝑎 ∈ (𝑐(Itv‘𝑔)𝑏) ∨ 𝑏 ∈ (𝑐(Itv‘𝑔)𝑎)))) ↔ ((𝑎 ∈ 𝑃 ∧ 𝑏 ∈ 𝑃) ∧ (𝑎 ≠ 𝑐 ∧ 𝑏 ≠ 𝑐 ∧ (𝑎 ∈ (𝑐𝐼𝑏) ∨ 𝑏 ∈ (𝑐𝐼𝑎)))))) |
33 | 32 | opabbidv 5119 |
. . . . . . . 8
⊢ (𝑔 = 𝐺 → {〈𝑎, 𝑏〉 ∣ ((𝑎 ∈ (Base‘𝑔) ∧ 𝑏 ∈ (Base‘𝑔)) ∧ (𝑎 ≠ 𝑐 ∧ 𝑏 ≠ 𝑐 ∧ (𝑎 ∈ (𝑐(Itv‘𝑔)𝑏) ∨ 𝑏 ∈ (𝑐(Itv‘𝑔)𝑎))))} = {〈𝑎, 𝑏〉 ∣ ((𝑎 ∈ 𝑃 ∧ 𝑏 ∈ 𝑃) ∧ (𝑎 ≠ 𝑐 ∧ 𝑏 ≠ 𝑐 ∧ (𝑎 ∈ (𝑐𝐼𝑏) ∨ 𝑏 ∈ (𝑐𝐼𝑎))))}) |
34 | 19, 33 | mpteq12dv 5140 |
. . . . . . 7
⊢ (𝑔 = 𝐺 → (𝑐 ∈ (Base‘𝑔) ↦ {〈𝑎, 𝑏〉 ∣ ((𝑎 ∈ (Base‘𝑔) ∧ 𝑏 ∈ (Base‘𝑔)) ∧ (𝑎 ≠ 𝑐 ∧ 𝑏 ≠ 𝑐 ∧ (𝑎 ∈ (𝑐(Itv‘𝑔)𝑏) ∨ 𝑏 ∈ (𝑐(Itv‘𝑔)𝑎))))}) = (𝑐 ∈ 𝑃 ↦ {〈𝑎, 𝑏〉 ∣ ((𝑎 ∈ 𝑃 ∧ 𝑏 ∈ 𝑃) ∧ (𝑎 ≠ 𝑐 ∧ 𝑏 ≠ 𝑐 ∧ (𝑎 ∈ (𝑐𝐼𝑏) ∨ 𝑏 ∈ (𝑐𝐼𝑎))))})) |
35 | | df-hlg 26692 |
. . . . . . 7
⊢ hlG =
(𝑔 ∈ V ↦ (𝑐 ∈ (Base‘𝑔) ↦ {〈𝑎, 𝑏〉 ∣ ((𝑎 ∈ (Base‘𝑔) ∧ 𝑏 ∈ (Base‘𝑔)) ∧ (𝑎 ≠ 𝑐 ∧ 𝑏 ≠ 𝑐 ∧ (𝑎 ∈ (𝑐(Itv‘𝑔)𝑏) ∨ 𝑏 ∈ (𝑐(Itv‘𝑔)𝑎))))})) |
36 | 34, 35, 18 | mptfvmpt 7044 |
. . . . . 6
⊢ (𝐺 ∈ V →
(hlG‘𝐺) = (𝑐 ∈ 𝑃 ↦ {〈𝑎, 𝑏〉 ∣ ((𝑎 ∈ 𝑃 ∧ 𝑏 ∈ 𝑃) ∧ (𝑎 ≠ 𝑐 ∧ 𝑏 ≠ 𝑐 ∧ (𝑎 ∈ (𝑐𝐼𝑏) ∨ 𝑏 ∈ (𝑐𝐼𝑎))))})) |
37 | 15, 16, 36 | 3syl 18 |
. . . . 5
⊢ (𝜑 → (hlG‘𝐺) = (𝑐 ∈ 𝑃 ↦ {〈𝑎, 𝑏〉 ∣ ((𝑎 ∈ 𝑃 ∧ 𝑏 ∈ 𝑃) ∧ (𝑎 ≠ 𝑐 ∧ 𝑏 ≠ 𝑐 ∧ (𝑎 ∈ (𝑐𝐼𝑏) ∨ 𝑏 ∈ (𝑐𝐼𝑎))))})) |
38 | 14, 37 | syl5eq 2790 |
. . . 4
⊢ (𝜑 → 𝐾 = (𝑐 ∈ 𝑃 ↦ {〈𝑎, 𝑏〉 ∣ ((𝑎 ∈ 𝑃 ∧ 𝑏 ∈ 𝑃) ∧ (𝑎 ≠ 𝑐 ∧ 𝑏 ≠ 𝑐 ∧ (𝑎 ∈ (𝑐𝐼𝑏) ∨ 𝑏 ∈ (𝑐𝐼𝑎))))})) |
39 | | neeq2 3004 |
. . . . . . . 8
⊢ (𝑐 = 𝐶 → (𝑎 ≠ 𝑐 ↔ 𝑎 ≠ 𝐶)) |
40 | | neeq2 3004 |
. . . . . . . 8
⊢ (𝑐 = 𝐶 → (𝑏 ≠ 𝑐 ↔ 𝑏 ≠ 𝐶)) |
41 | | oveq1 7220 |
. . . . . . . . . 10
⊢ (𝑐 = 𝐶 → (𝑐𝐼𝑏) = (𝐶𝐼𝑏)) |
42 | 41 | eleq2d 2823 |
. . . . . . . . 9
⊢ (𝑐 = 𝐶 → (𝑎 ∈ (𝑐𝐼𝑏) ↔ 𝑎 ∈ (𝐶𝐼𝑏))) |
43 | | oveq1 7220 |
. . . . . . . . . 10
⊢ (𝑐 = 𝐶 → (𝑐𝐼𝑎) = (𝐶𝐼𝑎)) |
44 | 43 | eleq2d 2823 |
. . . . . . . . 9
⊢ (𝑐 = 𝐶 → (𝑏 ∈ (𝑐𝐼𝑎) ↔ 𝑏 ∈ (𝐶𝐼𝑎))) |
45 | 42, 44 | orbi12d 919 |
. . . . . . . 8
⊢ (𝑐 = 𝐶 → ((𝑎 ∈ (𝑐𝐼𝑏) ∨ 𝑏 ∈ (𝑐𝐼𝑎)) ↔ (𝑎 ∈ (𝐶𝐼𝑏) ∨ 𝑏 ∈ (𝐶𝐼𝑎)))) |
46 | 39, 40, 45 | 3anbi123d 1438 |
. . . . . . 7
⊢ (𝑐 = 𝐶 → ((𝑎 ≠ 𝑐 ∧ 𝑏 ≠ 𝑐 ∧ (𝑎 ∈ (𝑐𝐼𝑏) ∨ 𝑏 ∈ (𝑐𝐼𝑎))) ↔ (𝑎 ≠ 𝐶 ∧ 𝑏 ≠ 𝐶 ∧ (𝑎 ∈ (𝐶𝐼𝑏) ∨ 𝑏 ∈ (𝐶𝐼𝑎))))) |
47 | 46 | anbi2d 632 |
. . . . . 6
⊢ (𝑐 = 𝐶 → (((𝑎 ∈ 𝑃 ∧ 𝑏 ∈ 𝑃) ∧ (𝑎 ≠ 𝑐 ∧ 𝑏 ≠ 𝑐 ∧ (𝑎 ∈ (𝑐𝐼𝑏) ∨ 𝑏 ∈ (𝑐𝐼𝑎)))) ↔ ((𝑎 ∈ 𝑃 ∧ 𝑏 ∈ 𝑃) ∧ (𝑎 ≠ 𝐶 ∧ 𝑏 ≠ 𝐶 ∧ (𝑎 ∈ (𝐶𝐼𝑏) ∨ 𝑏 ∈ (𝐶𝐼𝑎)))))) |
48 | 47 | opabbidv 5119 |
. . . . 5
⊢ (𝑐 = 𝐶 → {〈𝑎, 𝑏〉 ∣ ((𝑎 ∈ 𝑃 ∧ 𝑏 ∈ 𝑃) ∧ (𝑎 ≠ 𝑐 ∧ 𝑏 ≠ 𝑐 ∧ (𝑎 ∈ (𝑐𝐼𝑏) ∨ 𝑏 ∈ (𝑐𝐼𝑎))))} = {〈𝑎, 𝑏〉 ∣ ((𝑎 ∈ 𝑃 ∧ 𝑏 ∈ 𝑃) ∧ (𝑎 ≠ 𝐶 ∧ 𝑏 ≠ 𝐶 ∧ (𝑎 ∈ (𝐶𝐼𝑏) ∨ 𝑏 ∈ (𝐶𝐼𝑎))))}) |
49 | 48 | adantl 485 |
. . . 4
⊢ ((𝜑 ∧ 𝑐 = 𝐶) → {〈𝑎, 𝑏〉 ∣ ((𝑎 ∈ 𝑃 ∧ 𝑏 ∈ 𝑃) ∧ (𝑎 ≠ 𝑐 ∧ 𝑏 ≠ 𝑐 ∧ (𝑎 ∈ (𝑐𝐼𝑏) ∨ 𝑏 ∈ (𝑐𝐼𝑎))))} = {〈𝑎, 𝑏〉 ∣ ((𝑎 ∈ 𝑃 ∧ 𝑏 ∈ 𝑃) ∧ (𝑎 ≠ 𝐶 ∧ 𝑏 ≠ 𝐶 ∧ (𝑎 ∈ (𝐶𝐼𝑏) ∨ 𝑏 ∈ (𝐶𝐼𝑎))))}) |
50 | | ishlg.c |
. . . 4
⊢ (𝜑 → 𝐶 ∈ 𝑃) |
51 | 18 | fvexi 6731 |
. . . . . . 7
⊢ 𝑃 ∈ V |
52 | 51, 51 | xpex 7538 |
. . . . . 6
⊢ (𝑃 × 𝑃) ∈ V |
53 | | opabssxp 5640 |
. . . . . 6
⊢
{〈𝑎, 𝑏〉 ∣ ((𝑎 ∈ 𝑃 ∧ 𝑏 ∈ 𝑃) ∧ (𝑎 ≠ 𝐶 ∧ 𝑏 ≠ 𝐶 ∧ (𝑎 ∈ (𝐶𝐼𝑏) ∨ 𝑏 ∈ (𝐶𝐼𝑎))))} ⊆ (𝑃 × 𝑃) |
54 | 52, 53 | ssexi 5215 |
. . . . 5
⊢
{〈𝑎, 𝑏〉 ∣ ((𝑎 ∈ 𝑃 ∧ 𝑏 ∈ 𝑃) ∧ (𝑎 ≠ 𝐶 ∧ 𝑏 ≠ 𝐶 ∧ (𝑎 ∈ (𝐶𝐼𝑏) ∨ 𝑏 ∈ (𝐶𝐼𝑎))))} ∈ V |
55 | 54 | a1i 11 |
. . . 4
⊢ (𝜑 → {〈𝑎, 𝑏〉 ∣ ((𝑎 ∈ 𝑃 ∧ 𝑏 ∈ 𝑃) ∧ (𝑎 ≠ 𝐶 ∧ 𝑏 ≠ 𝐶 ∧ (𝑎 ∈ (𝐶𝐼𝑏) ∨ 𝑏 ∈ (𝐶𝐼𝑎))))} ∈ V) |
56 | 38, 49, 50, 55 | fvmptd 6825 |
. . 3
⊢ (𝜑 → (𝐾‘𝐶) = {〈𝑎, 𝑏〉 ∣ ((𝑎 ∈ 𝑃 ∧ 𝑏 ∈ 𝑃) ∧ (𝑎 ≠ 𝐶 ∧ 𝑏 ≠ 𝐶 ∧ (𝑎 ∈ (𝐶𝐼𝑏) ∨ 𝑏 ∈ (𝐶𝐼𝑎))))}) |
57 | 56 | breqd 5064 |
. 2
⊢ (𝜑 → (𝐴(𝐾‘𝐶)𝐵 ↔ 𝐴{〈𝑎, 𝑏〉 ∣ ((𝑎 ∈ 𝑃 ∧ 𝑏 ∈ 𝑃) ∧ (𝑎 ≠ 𝐶 ∧ 𝑏 ≠ 𝐶 ∧ (𝑎 ∈ (𝐶𝐼𝑏) ∨ 𝑏 ∈ (𝐶𝐼𝑎))))}𝐵)) |
58 | | ishlg.a |
. . . 4
⊢ (𝜑 → 𝐴 ∈ 𝑃) |
59 | | ishlg.b |
. . . 4
⊢ (𝜑 → 𝐵 ∈ 𝑃) |
60 | 58, 59 | jca 515 |
. . 3
⊢ (𝜑 → (𝐴 ∈ 𝑃 ∧ 𝐵 ∈ 𝑃)) |
61 | 60 | biantrurd 536 |
. 2
⊢ (𝜑 → ((𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶 ∧ (𝐴 ∈ (𝐶𝐼𝐵) ∨ 𝐵 ∈ (𝐶𝐼𝐴))) ↔ ((𝐴 ∈ 𝑃 ∧ 𝐵 ∈ 𝑃) ∧ (𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶 ∧ (𝐴 ∈ (𝐶𝐼𝐵) ∨ 𝐵 ∈ (𝐶𝐼𝐴)))))) |
62 | 13, 57, 61 | 3bitr4d 314 |
1
⊢ (𝜑 → (𝐴(𝐾‘𝐶)𝐵 ↔ (𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶 ∧ (𝐴 ∈ (𝐶𝐼𝐵) ∨ 𝐵 ∈ (𝐶𝐼𝐴))))) |