Step | Hyp | Ref
| Expression |
1 | | df-br 5054 |
. . 3
⊢ (𝐴(⟂G‘𝐺)𝐵 ↔ 〈𝐴, 𝐵〉 ∈ (⟂G‘𝐺)) |
2 | | df-perpg 26787 |
. . . . 5
⊢ ⟂G
= (𝑔 ∈ V ↦
{〈𝑎, 𝑏〉 ∣ ((𝑎 ∈ ran (LineG‘𝑔) ∧ 𝑏 ∈ ran (LineG‘𝑔)) ∧ ∃𝑥 ∈ (𝑎 ∩ 𝑏)∀𝑢 ∈ 𝑎 ∀𝑣 ∈ 𝑏 〈“𝑢𝑥𝑣”〉 ∈ (∟G‘𝑔))}) |
3 | | simpr 488 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑔 = 𝐺) → 𝑔 = 𝐺) |
4 | 3 | fveq2d 6721 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑔 = 𝐺) → (LineG‘𝑔) = (LineG‘𝐺)) |
5 | | isperp.l |
. . . . . . . . . . 11
⊢ 𝐿 = (LineG‘𝐺) |
6 | 4, 5 | eqtr4di 2796 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑔 = 𝐺) → (LineG‘𝑔) = 𝐿) |
7 | 6 | rneqd 5807 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑔 = 𝐺) → ran (LineG‘𝑔) = ran 𝐿) |
8 | 7 | eleq2d 2823 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑔 = 𝐺) → (𝑎 ∈ ran (LineG‘𝑔) ↔ 𝑎 ∈ ran 𝐿)) |
9 | 7 | eleq2d 2823 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑔 = 𝐺) → (𝑏 ∈ ran (LineG‘𝑔) ↔ 𝑏 ∈ ran 𝐿)) |
10 | 8, 9 | anbi12d 634 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑔 = 𝐺) → ((𝑎 ∈ ran (LineG‘𝑔) ∧ 𝑏 ∈ ran (LineG‘𝑔)) ↔ (𝑎 ∈ ran 𝐿 ∧ 𝑏 ∈ ran 𝐿))) |
11 | 3 | fveq2d 6721 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑔 = 𝐺) → (∟G‘𝑔) = (∟G‘𝐺)) |
12 | 11 | eleq2d 2823 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑔 = 𝐺) → (〈“𝑢𝑥𝑣”〉 ∈ (∟G‘𝑔) ↔ 〈“𝑢𝑥𝑣”〉 ∈ (∟G‘𝐺))) |
13 | 12 | ralbidv 3118 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑔 = 𝐺) → (∀𝑣 ∈ 𝑏 〈“𝑢𝑥𝑣”〉 ∈ (∟G‘𝑔) ↔ ∀𝑣 ∈ 𝑏 〈“𝑢𝑥𝑣”〉 ∈ (∟G‘𝐺))) |
14 | 13 | rexralbidv 3220 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑔 = 𝐺) → (∃𝑥 ∈ (𝑎 ∩ 𝑏)∀𝑢 ∈ 𝑎 ∀𝑣 ∈ 𝑏 〈“𝑢𝑥𝑣”〉 ∈ (∟G‘𝑔) ↔ ∃𝑥 ∈ (𝑎 ∩ 𝑏)∀𝑢 ∈ 𝑎 ∀𝑣 ∈ 𝑏 〈“𝑢𝑥𝑣”〉 ∈ (∟G‘𝐺))) |
15 | 10, 14 | anbi12d 634 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑔 = 𝐺) → (((𝑎 ∈ ran (LineG‘𝑔) ∧ 𝑏 ∈ ran (LineG‘𝑔)) ∧ ∃𝑥 ∈ (𝑎 ∩ 𝑏)∀𝑢 ∈ 𝑎 ∀𝑣 ∈ 𝑏 〈“𝑢𝑥𝑣”〉 ∈ (∟G‘𝑔)) ↔ ((𝑎 ∈ ran 𝐿 ∧ 𝑏 ∈ ran 𝐿) ∧ ∃𝑥 ∈ (𝑎 ∩ 𝑏)∀𝑢 ∈ 𝑎 ∀𝑣 ∈ 𝑏 〈“𝑢𝑥𝑣”〉 ∈ (∟G‘𝐺)))) |
16 | 15 | opabbidv 5119 |
. . . . 5
⊢ ((𝜑 ∧ 𝑔 = 𝐺) → {〈𝑎, 𝑏〉 ∣ ((𝑎 ∈ ran (LineG‘𝑔) ∧ 𝑏 ∈ ran (LineG‘𝑔)) ∧ ∃𝑥 ∈ (𝑎 ∩ 𝑏)∀𝑢 ∈ 𝑎 ∀𝑣 ∈ 𝑏 〈“𝑢𝑥𝑣”〉 ∈ (∟G‘𝑔))} = {〈𝑎, 𝑏〉 ∣ ((𝑎 ∈ ran 𝐿 ∧ 𝑏 ∈ ran 𝐿) ∧ ∃𝑥 ∈ (𝑎 ∩ 𝑏)∀𝑢 ∈ 𝑎 ∀𝑣 ∈ 𝑏 〈“𝑢𝑥𝑣”〉 ∈ (∟G‘𝐺))}) |
17 | | isperp.g |
. . . . . 6
⊢ (𝜑 → 𝐺 ∈ TarskiG) |
18 | 17 | elexd 3428 |
. . . . 5
⊢ (𝜑 → 𝐺 ∈ V) |
19 | 5 | fvexi 6731 |
. . . . . . . 8
⊢ 𝐿 ∈ V |
20 | | rnexg 7682 |
. . . . . . . 8
⊢ (𝐿 ∈ V → ran 𝐿 ∈ V) |
21 | 19, 20 | mp1i 13 |
. . . . . . 7
⊢ (𝜑 → ran 𝐿 ∈ V) |
22 | 21, 21 | xpexd 7536 |
. . . . . 6
⊢ (𝜑 → (ran 𝐿 × ran 𝐿) ∈ V) |
23 | | opabssxp 5640 |
. . . . . . 7
⊢
{〈𝑎, 𝑏〉 ∣ ((𝑎 ∈ ran 𝐿 ∧ 𝑏 ∈ ran 𝐿) ∧ ∃𝑥 ∈ (𝑎 ∩ 𝑏)∀𝑢 ∈ 𝑎 ∀𝑣 ∈ 𝑏 〈“𝑢𝑥𝑣”〉 ∈ (∟G‘𝐺))} ⊆ (ran 𝐿 × ran 𝐿) |
24 | 23 | a1i 11 |
. . . . . 6
⊢ (𝜑 → {〈𝑎, 𝑏〉 ∣ ((𝑎 ∈ ran 𝐿 ∧ 𝑏 ∈ ran 𝐿) ∧ ∃𝑥 ∈ (𝑎 ∩ 𝑏)∀𝑢 ∈ 𝑎 ∀𝑣 ∈ 𝑏 〈“𝑢𝑥𝑣”〉 ∈ (∟G‘𝐺))} ⊆ (ran 𝐿 × ran 𝐿)) |
25 | 22, 24 | ssexd 5217 |
. . . . 5
⊢ (𝜑 → {〈𝑎, 𝑏〉 ∣ ((𝑎 ∈ ran 𝐿 ∧ 𝑏 ∈ ran 𝐿) ∧ ∃𝑥 ∈ (𝑎 ∩ 𝑏)∀𝑢 ∈ 𝑎 ∀𝑣 ∈ 𝑏 〈“𝑢𝑥𝑣”〉 ∈ (∟G‘𝐺))} ∈ V) |
26 | 2, 16, 18, 25 | fvmptd2 6826 |
. . . 4
⊢ (𝜑 → (⟂G‘𝐺) = {〈𝑎, 𝑏〉 ∣ ((𝑎 ∈ ran 𝐿 ∧ 𝑏 ∈ ran 𝐿) ∧ ∃𝑥 ∈ (𝑎 ∩ 𝑏)∀𝑢 ∈ 𝑎 ∀𝑣 ∈ 𝑏 〈“𝑢𝑥𝑣”〉 ∈ (∟G‘𝐺))}) |
27 | 26 | eleq2d 2823 |
. . 3
⊢ (𝜑 → (〈𝐴, 𝐵〉 ∈ (⟂G‘𝐺) ↔ 〈𝐴, 𝐵〉 ∈ {〈𝑎, 𝑏〉 ∣ ((𝑎 ∈ ran 𝐿 ∧ 𝑏 ∈ ran 𝐿) ∧ ∃𝑥 ∈ (𝑎 ∩ 𝑏)∀𝑢 ∈ 𝑎 ∀𝑣 ∈ 𝑏 〈“𝑢𝑥𝑣”〉 ∈ (∟G‘𝐺))})) |
28 | 1, 27 | syl5bb 286 |
. 2
⊢ (𝜑 → (𝐴(⟂G‘𝐺)𝐵 ↔ 〈𝐴, 𝐵〉 ∈ {〈𝑎, 𝑏〉 ∣ ((𝑎 ∈ ran 𝐿 ∧ 𝑏 ∈ ran 𝐿) ∧ ∃𝑥 ∈ (𝑎 ∩ 𝑏)∀𝑢 ∈ 𝑎 ∀𝑣 ∈ 𝑏 〈“𝑢𝑥𝑣”〉 ∈ (∟G‘𝐺))})) |
29 | | isperp.a |
. . 3
⊢ (𝜑 → 𝐴 ∈ ran 𝐿) |
30 | | isperp.b |
. . 3
⊢ (𝜑 → 𝐵 ∈ ran 𝐿) |
31 | | ineq12 4122 |
. . . . 5
⊢ ((𝑎 = 𝐴 ∧ 𝑏 = 𝐵) → (𝑎 ∩ 𝑏) = (𝐴 ∩ 𝐵)) |
32 | | simpll 767 |
. . . . . 6
⊢ (((𝑎 = 𝐴 ∧ 𝑏 = 𝐵) ∧ 𝑥 ∈ (𝑎 ∩ 𝑏)) → 𝑎 = 𝐴) |
33 | | simpllr 776 |
. . . . . . 7
⊢ ((((𝑎 = 𝐴 ∧ 𝑏 = 𝐵) ∧ 𝑥 ∈ (𝑎 ∩ 𝑏)) ∧ 𝑢 ∈ 𝑎) → 𝑏 = 𝐵) |
34 | 33 | raleqdv 3325 |
. . . . . 6
⊢ ((((𝑎 = 𝐴 ∧ 𝑏 = 𝐵) ∧ 𝑥 ∈ (𝑎 ∩ 𝑏)) ∧ 𝑢 ∈ 𝑎) → (∀𝑣 ∈ 𝑏 〈“𝑢𝑥𝑣”〉 ∈ (∟G‘𝐺) ↔ ∀𝑣 ∈ 𝐵 〈“𝑢𝑥𝑣”〉 ∈ (∟G‘𝐺))) |
35 | 32, 34 | raleqbidva 3331 |
. . . . 5
⊢ (((𝑎 = 𝐴 ∧ 𝑏 = 𝐵) ∧ 𝑥 ∈ (𝑎 ∩ 𝑏)) → (∀𝑢 ∈ 𝑎 ∀𝑣 ∈ 𝑏 〈“𝑢𝑥𝑣”〉 ∈ (∟G‘𝐺) ↔ ∀𝑢 ∈ 𝐴 ∀𝑣 ∈ 𝐵 〈“𝑢𝑥𝑣”〉 ∈ (∟G‘𝐺))) |
36 | 31, 35 | rexeqbidva 3332 |
. . . 4
⊢ ((𝑎 = 𝐴 ∧ 𝑏 = 𝐵) → (∃𝑥 ∈ (𝑎 ∩ 𝑏)∀𝑢 ∈ 𝑎 ∀𝑣 ∈ 𝑏 〈“𝑢𝑥𝑣”〉 ∈ (∟G‘𝐺) ↔ ∃𝑥 ∈ (𝐴 ∩ 𝐵)∀𝑢 ∈ 𝐴 ∀𝑣 ∈ 𝐵 〈“𝑢𝑥𝑣”〉 ∈ (∟G‘𝐺))) |
37 | 36 | opelopab2a 5416 |
. . 3
⊢ ((𝐴 ∈ ran 𝐿 ∧ 𝐵 ∈ ran 𝐿) → (〈𝐴, 𝐵〉 ∈ {〈𝑎, 𝑏〉 ∣ ((𝑎 ∈ ran 𝐿 ∧ 𝑏 ∈ ran 𝐿) ∧ ∃𝑥 ∈ (𝑎 ∩ 𝑏)∀𝑢 ∈ 𝑎 ∀𝑣 ∈ 𝑏 〈“𝑢𝑥𝑣”〉 ∈ (∟G‘𝐺))} ↔ ∃𝑥 ∈ (𝐴 ∩ 𝐵)∀𝑢 ∈ 𝐴 ∀𝑣 ∈ 𝐵 〈“𝑢𝑥𝑣”〉 ∈ (∟G‘𝐺))) |
38 | 29, 30, 37 | syl2anc 587 |
. 2
⊢ (𝜑 → (〈𝐴, 𝐵〉 ∈ {〈𝑎, 𝑏〉 ∣ ((𝑎 ∈ ran 𝐿 ∧ 𝑏 ∈ ran 𝐿) ∧ ∃𝑥 ∈ (𝑎 ∩ 𝑏)∀𝑢 ∈ 𝑎 ∀𝑣 ∈ 𝑏 〈“𝑢𝑥𝑣”〉 ∈ (∟G‘𝐺))} ↔ ∃𝑥 ∈ (𝐴 ∩ 𝐵)∀𝑢 ∈ 𝐴 ∀𝑣 ∈ 𝐵 〈“𝑢𝑥𝑣”〉 ∈ (∟G‘𝐺))) |
39 | 28, 38 | bitrd 282 |
1
⊢ (𝜑 → (𝐴(⟂G‘𝐺)𝐵 ↔ ∃𝑥 ∈ (𝐴 ∩ 𝐵)∀𝑢 ∈ 𝐴 ∀𝑣 ∈ 𝐵 〈“𝑢𝑥𝑣”〉 ∈ (∟G‘𝐺))) |