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Theorem flatcgra 26627
Description: Flat angles are congruent. (Contributed by Thierry Arnoux, 13-Feb-2023.)
Hypotheses
Ref Expression
cgracol.p 𝑃 = (Base‘𝐺)
cgracol.i 𝐼 = (Itv‘𝐺)
cgracol.m = (dist‘𝐺)
cgracol.g (𝜑𝐺 ∈ TarskiG)
cgracol.a (𝜑𝐴𝑃)
cgracol.b (𝜑𝐵𝑃)
cgracol.c (𝜑𝐶𝑃)
cgracol.d (𝜑𝐷𝑃)
cgracol.e (𝜑𝐸𝑃)
cgracol.f (𝜑𝐹𝑃)
flatcgra.1 (𝜑𝐵 ∈ (𝐴𝐼𝐶))
flatcgra.2 (𝜑𝐸 ∈ (𝐷𝐼𝐹))
flatcgra.3 (𝜑𝐴𝐵)
flatcgra.4 (𝜑𝐶𝐵)
flatcgra.5 (𝜑𝐷𝐸)
flatcgra.6 (𝜑𝐹𝐸)
Assertion
Ref Expression
flatcgra (𝜑 → ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩)

Proof of Theorem flatcgra
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cgracol.p . . . . 5 𝑃 = (Base‘𝐺)
2 cgracol.m . . . . 5 = (dist‘𝐺)
3 eqid 2824 . . . . 5 (cgrG‘𝐺) = (cgrG‘𝐺)
4 cgracol.g . . . . . 6 (𝜑𝐺 ∈ TarskiG)
54ad3antrrr 729 . . . . 5 ((((𝜑𝑥𝑃) ∧ 𝑦𝑃) ∧ ((𝐸 ∈ (𝐹𝐼𝑥) ∧ (𝐸 𝑥) = (𝐵 𝐴)) ∧ (𝐸 ∈ (𝐷𝐼𝑦) ∧ (𝐸 𝑦) = (𝐵 𝐶)))) → 𝐺 ∈ TarskiG)
6 cgracol.a . . . . . 6 (𝜑𝐴𝑃)
76ad3antrrr 729 . . . . 5 ((((𝜑𝑥𝑃) ∧ 𝑦𝑃) ∧ ((𝐸 ∈ (𝐹𝐼𝑥) ∧ (𝐸 𝑥) = (𝐵 𝐴)) ∧ (𝐸 ∈ (𝐷𝐼𝑦) ∧ (𝐸 𝑦) = (𝐵 𝐶)))) → 𝐴𝑃)
8 cgracol.b . . . . . 6 (𝜑𝐵𝑃)
98ad3antrrr 729 . . . . 5 ((((𝜑𝑥𝑃) ∧ 𝑦𝑃) ∧ ((𝐸 ∈ (𝐹𝐼𝑥) ∧ (𝐸 𝑥) = (𝐵 𝐴)) ∧ (𝐸 ∈ (𝐷𝐼𝑦) ∧ (𝐸 𝑦) = (𝐵 𝐶)))) → 𝐵𝑃)
10 cgracol.c . . . . . 6 (𝜑𝐶𝑃)
1110ad3antrrr 729 . . . . 5 ((((𝜑𝑥𝑃) ∧ 𝑦𝑃) ∧ ((𝐸 ∈ (𝐹𝐼𝑥) ∧ (𝐸 𝑥) = (𝐵 𝐴)) ∧ (𝐸 ∈ (𝐷𝐼𝑦) ∧ (𝐸 𝑦) = (𝐵 𝐶)))) → 𝐶𝑃)
12 simpllr 775 . . . . 5 ((((𝜑𝑥𝑃) ∧ 𝑦𝑃) ∧ ((𝐸 ∈ (𝐹𝐼𝑥) ∧ (𝐸 𝑥) = (𝐵 𝐴)) ∧ (𝐸 ∈ (𝐷𝐼𝑦) ∧ (𝐸 𝑦) = (𝐵 𝐶)))) → 𝑥𝑃)
13 cgracol.e . . . . . 6 (𝜑𝐸𝑃)
1413ad3antrrr 729 . . . . 5 ((((𝜑𝑥𝑃) ∧ 𝑦𝑃) ∧ ((𝐸 ∈ (𝐹𝐼𝑥) ∧ (𝐸 𝑥) = (𝐵 𝐴)) ∧ (𝐸 ∈ (𝐷𝐼𝑦) ∧ (𝐸 𝑦) = (𝐵 𝐶)))) → 𝐸𝑃)
15 simplr 768 . . . . 5 ((((𝜑𝑥𝑃) ∧ 𝑦𝑃) ∧ ((𝐸 ∈ (𝐹𝐼𝑥) ∧ (𝐸 𝑥) = (𝐵 𝐴)) ∧ (𝐸 ∈ (𝐷𝐼𝑦) ∧ (𝐸 𝑦) = (𝐵 𝐶)))) → 𝑦𝑃)
16 cgracol.i . . . . . . 7 𝐼 = (Itv‘𝐺)
17 simprlr 779 . . . . . . 7 ((((𝜑𝑥𝑃) ∧ 𝑦𝑃) ∧ ((𝐸 ∈ (𝐹𝐼𝑥) ∧ (𝐸 𝑥) = (𝐵 𝐴)) ∧ (𝐸 ∈ (𝐷𝐼𝑦) ∧ (𝐸 𝑦) = (𝐵 𝐶)))) → (𝐸 𝑥) = (𝐵 𝐴))
181, 2, 16, 5, 14, 12, 9, 7, 17tgcgrcomlr 26283 . . . . . 6 ((((𝜑𝑥𝑃) ∧ 𝑦𝑃) ∧ ((𝐸 ∈ (𝐹𝐼𝑥) ∧ (𝐸 𝑥) = (𝐵 𝐴)) ∧ (𝐸 ∈ (𝐷𝐼𝑦) ∧ (𝐸 𝑦) = (𝐵 𝐶)))) → (𝑥 𝐸) = (𝐴 𝐵))
1918eqcomd 2830 . . . . 5 ((((𝜑𝑥𝑃) ∧ 𝑦𝑃) ∧ ((𝐸 ∈ (𝐹𝐼𝑥) ∧ (𝐸 𝑥) = (𝐵 𝐴)) ∧ (𝐸 ∈ (𝐷𝐼𝑦) ∧ (𝐸 𝑦) = (𝐵 𝐶)))) → (𝐴 𝐵) = (𝑥 𝐸))
20 simprrr 781 . . . . . 6 ((((𝜑𝑥𝑃) ∧ 𝑦𝑃) ∧ ((𝐸 ∈ (𝐹𝐼𝑥) ∧ (𝐸 𝑥) = (𝐵 𝐴)) ∧ (𝐸 ∈ (𝐷𝐼𝑦) ∧ (𝐸 𝑦) = (𝐵 𝐶)))) → (𝐸 𝑦) = (𝐵 𝐶))
2120eqcomd 2830 . . . . 5 ((((𝜑𝑥𝑃) ∧ 𝑦𝑃) ∧ ((𝐸 ∈ (𝐹𝐼𝑥) ∧ (𝐸 𝑥) = (𝐵 𝐴)) ∧ (𝐸 ∈ (𝐷𝐼𝑦) ∧ (𝐸 𝑦) = (𝐵 𝐶)))) → (𝐵 𝐶) = (𝐸 𝑦))
22 cgracol.f . . . . . . . . . 10 (𝜑𝐹𝑃)
2322ad3antrrr 729 . . . . . . . . 9 ((((𝜑𝑥𝑃) ∧ 𝑦𝑃) ∧ ((𝐸 ∈ (𝐹𝐼𝑥) ∧ (𝐸 𝑥) = (𝐵 𝐴)) ∧ (𝐸 ∈ (𝐷𝐼𝑦) ∧ (𝐸 𝑦) = (𝐵 𝐶)))) → 𝐹𝑃)
24 cgracol.d . . . . . . . . . 10 (𝜑𝐷𝑃)
2524ad3antrrr 729 . . . . . . . . 9 ((((𝜑𝑥𝑃) ∧ 𝑦𝑃) ∧ ((𝐸 ∈ (𝐹𝐼𝑥) ∧ (𝐸 𝑥) = (𝐵 𝐴)) ∧ (𝐸 ∈ (𝐷𝐼𝑦) ∧ (𝐸 𝑦) = (𝐵 𝐶)))) → 𝐷𝑃)
26 flatcgra.6 . . . . . . . . . 10 (𝜑𝐹𝐸)
2726ad3antrrr 729 . . . . . . . . 9 ((((𝜑𝑥𝑃) ∧ 𝑦𝑃) ∧ ((𝐸 ∈ (𝐹𝐼𝑥) ∧ (𝐸 𝑥) = (𝐵 𝐴)) ∧ (𝐸 ∈ (𝐷𝐼𝑦) ∧ (𝐸 𝑦) = (𝐵 𝐶)))) → 𝐹𝐸)
28 flatcgra.5 . . . . . . . . . 10 (𝜑𝐷𝐸)
2928ad3antrrr 729 . . . . . . . . 9 ((((𝜑𝑥𝑃) ∧ 𝑦𝑃) ∧ ((𝐸 ∈ (𝐹𝐼𝑥) ∧ (𝐸 𝑥) = (𝐵 𝐴)) ∧ (𝐸 ∈ (𝐷𝐼𝑦) ∧ (𝐸 𝑦) = (𝐵 𝐶)))) → 𝐷𝐸)
30 flatcgra.2 . . . . . . . . . . 11 (𝜑𝐸 ∈ (𝐷𝐼𝐹))
311, 2, 16, 4, 24, 13, 22, 30tgbtwncom 26291 . . . . . . . . . 10 (𝜑𝐸 ∈ (𝐹𝐼𝐷))
3231ad3antrrr 729 . . . . . . . . 9 ((((𝜑𝑥𝑃) ∧ 𝑦𝑃) ∧ ((𝐸 ∈ (𝐹𝐼𝑥) ∧ (𝐸 𝑥) = (𝐵 𝐴)) ∧ (𝐸 ∈ (𝐷𝐼𝑦) ∧ (𝐸 𝑦) = (𝐵 𝐶)))) → 𝐸 ∈ (𝐹𝐼𝐷))
33 simprll 778 . . . . . . . . 9 ((((𝜑𝑥𝑃) ∧ 𝑦𝑃) ∧ ((𝐸 ∈ (𝐹𝐼𝑥) ∧ (𝐸 𝑥) = (𝐵 𝐴)) ∧ (𝐸 ∈ (𝐷𝐼𝑦) ∧ (𝐸 𝑦) = (𝐵 𝐶)))) → 𝐸 ∈ (𝐹𝐼𝑥))
34 simprrl 780 . . . . . . . . 9 ((((𝜑𝑥𝑃) ∧ 𝑦𝑃) ∧ ((𝐸 ∈ (𝐹𝐼𝑥) ∧ (𝐸 𝑥) = (𝐵 𝐴)) ∧ (𝐸 ∈ (𝐷𝐼𝑦) ∧ (𝐸 𝑦) = (𝐵 𝐶)))) → 𝐸 ∈ (𝐷𝐼𝑦))
351, 16, 5, 23, 14, 25, 12, 15, 27, 29, 32, 33, 34tgbtwnconn22 26382 . . . . . . . 8 ((((𝜑𝑥𝑃) ∧ 𝑦𝑃) ∧ ((𝐸 ∈ (𝐹𝐼𝑥) ∧ (𝐸 𝑥) = (𝐵 𝐴)) ∧ (𝐸 ∈ (𝐷𝐼𝑦) ∧ (𝐸 𝑦) = (𝐵 𝐶)))) → 𝐸 ∈ (𝑥𝐼𝑦))
36 flatcgra.1 . . . . . . . . 9 (𝜑𝐵 ∈ (𝐴𝐼𝐶))
3736ad3antrrr 729 . . . . . . . 8 ((((𝜑𝑥𝑃) ∧ 𝑦𝑃) ∧ ((𝐸 ∈ (𝐹𝐼𝑥) ∧ (𝐸 𝑥) = (𝐵 𝐴)) ∧ (𝐸 ∈ (𝐷𝐼𝑦) ∧ (𝐸 𝑦) = (𝐵 𝐶)))) → 𝐵 ∈ (𝐴𝐼𝐶))
381, 2, 16, 5, 12, 14, 15, 7, 9, 11, 35, 37, 18, 20tgcgrextend 26288 . . . . . . 7 ((((𝜑𝑥𝑃) ∧ 𝑦𝑃) ∧ ((𝐸 ∈ (𝐹𝐼𝑥) ∧ (𝐸 𝑥) = (𝐵 𝐴)) ∧ (𝐸 ∈ (𝐷𝐼𝑦) ∧ (𝐸 𝑦) = (𝐵 𝐶)))) → (𝑥 𝑦) = (𝐴 𝐶))
3938eqcomd 2830 . . . . . 6 ((((𝜑𝑥𝑃) ∧ 𝑦𝑃) ∧ ((𝐸 ∈ (𝐹𝐼𝑥) ∧ (𝐸 𝑥) = (𝐵 𝐴)) ∧ (𝐸 ∈ (𝐷𝐼𝑦) ∧ (𝐸 𝑦) = (𝐵 𝐶)))) → (𝐴 𝐶) = (𝑥 𝑦))
401, 2, 16, 5, 7, 11, 12, 15, 39tgcgrcomlr 26283 . . . . 5 ((((𝜑𝑥𝑃) ∧ 𝑦𝑃) ∧ ((𝐸 ∈ (𝐹𝐼𝑥) ∧ (𝐸 𝑥) = (𝐵 𝐴)) ∧ (𝐸 ∈ (𝐷𝐼𝑦) ∧ (𝐸 𝑦) = (𝐵 𝐶)))) → (𝐶 𝐴) = (𝑦 𝑥))
411, 2, 3, 5, 7, 9, 11, 12, 14, 15, 19, 21, 40trgcgr 26319 . . . 4 ((((𝜑𝑥𝑃) ∧ 𝑦𝑃) ∧ ((𝐸 ∈ (𝐹𝐼𝑥) ∧ (𝐸 𝑥) = (𝐵 𝐴)) ∧ (𝐸 ∈ (𝐷𝐼𝑦) ∧ (𝐸 𝑦) = (𝐵 𝐶)))) → ⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑥𝐸𝑦”⟩)
4217eqcomd 2830 . . . . . . . 8 ((((𝜑𝑥𝑃) ∧ 𝑦𝑃) ∧ ((𝐸 ∈ (𝐹𝐼𝑥) ∧ (𝐸 𝑥) = (𝐵 𝐴)) ∧ (𝐸 ∈ (𝐷𝐼𝑦) ∧ (𝐸 𝑦) = (𝐵 𝐶)))) → (𝐵 𝐴) = (𝐸 𝑥))
43 flatcgra.3 . . . . . . . . . 10 (𝜑𝐴𝐵)
4443necomd 3069 . . . . . . . . 9 (𝜑𝐵𝐴)
4544ad3antrrr 729 . . . . . . . 8 ((((𝜑𝑥𝑃) ∧ 𝑦𝑃) ∧ ((𝐸 ∈ (𝐹𝐼𝑥) ∧ (𝐸 𝑥) = (𝐵 𝐴)) ∧ (𝐸 ∈ (𝐷𝐼𝑦) ∧ (𝐸 𝑦) = (𝐵 𝐶)))) → 𝐵𝐴)
461, 2, 16, 5, 9, 7, 14, 12, 42, 45tgcgrneq 26286 . . . . . . 7 ((((𝜑𝑥𝑃) ∧ 𝑦𝑃) ∧ ((𝐸 ∈ (𝐹𝐼𝑥) ∧ (𝐸 𝑥) = (𝐵 𝐴)) ∧ (𝐸 ∈ (𝐷𝐼𝑦) ∧ (𝐸 𝑦) = (𝐵 𝐶)))) → 𝐸𝑥)
4746necomd 3069 . . . . . 6 ((((𝜑𝑥𝑃) ∧ 𝑦𝑃) ∧ ((𝐸 ∈ (𝐹𝐼𝑥) ∧ (𝐸 𝑥) = (𝐵 𝐴)) ∧ (𝐸 ∈ (𝐷𝐼𝑦) ∧ (𝐸 𝑦) = (𝐵 𝐶)))) → 𝑥𝐸)
481, 16, 5, 23, 14, 12, 25, 27, 33, 32tgbtwnconn2 26379 . . . . . 6 ((((𝜑𝑥𝑃) ∧ 𝑦𝑃) ∧ ((𝐸 ∈ (𝐹𝐼𝑥) ∧ (𝐸 𝑥) = (𝐵 𝐴)) ∧ (𝐸 ∈ (𝐷𝐼𝑦) ∧ (𝐸 𝑦) = (𝐵 𝐶)))) → (𝑥 ∈ (𝐸𝐼𝐷) ∨ 𝐷 ∈ (𝐸𝐼𝑥)))
4947, 29, 483jca 1125 . . . . 5 ((((𝜑𝑥𝑃) ∧ 𝑦𝑃) ∧ ((𝐸 ∈ (𝐹𝐼𝑥) ∧ (𝐸 𝑥) = (𝐵 𝐴)) ∧ (𝐸 ∈ (𝐷𝐼𝑦) ∧ (𝐸 𝑦) = (𝐵 𝐶)))) → (𝑥𝐸𝐷𝐸 ∧ (𝑥 ∈ (𝐸𝐼𝐷) ∨ 𝐷 ∈ (𝐸𝐼𝑥))))
50 eqid 2824 . . . . . 6 (hlG‘𝐺) = (hlG‘𝐺)
511, 16, 50, 12, 25, 14, 5ishlg 26405 . . . . 5 ((((𝜑𝑥𝑃) ∧ 𝑦𝑃) ∧ ((𝐸 ∈ (𝐹𝐼𝑥) ∧ (𝐸 𝑥) = (𝐵 𝐴)) ∧ (𝐸 ∈ (𝐷𝐼𝑦) ∧ (𝐸 𝑦) = (𝐵 𝐶)))) → (𝑥((hlG‘𝐺)‘𝐸)𝐷 ↔ (𝑥𝐸𝐷𝐸 ∧ (𝑥 ∈ (𝐸𝐼𝐷) ∨ 𝐷 ∈ (𝐸𝐼𝑥)))))
5249, 51mpbird 260 . . . 4 ((((𝜑𝑥𝑃) ∧ 𝑦𝑃) ∧ ((𝐸 ∈ (𝐹𝐼𝑥) ∧ (𝐸 𝑥) = (𝐵 𝐴)) ∧ (𝐸 ∈ (𝐷𝐼𝑦) ∧ (𝐸 𝑦) = (𝐵 𝐶)))) → 𝑥((hlG‘𝐺)‘𝐸)𝐷)
53 flatcgra.4 . . . . . . . . . 10 (𝜑𝐶𝐵)
5453necomd 3069 . . . . . . . . 9 (𝜑𝐵𝐶)
5554ad3antrrr 729 . . . . . . . 8 ((((𝜑𝑥𝑃) ∧ 𝑦𝑃) ∧ ((𝐸 ∈ (𝐹𝐼𝑥) ∧ (𝐸 𝑥) = (𝐵 𝐴)) ∧ (𝐸 ∈ (𝐷𝐼𝑦) ∧ (𝐸 𝑦) = (𝐵 𝐶)))) → 𝐵𝐶)
561, 2, 16, 5, 9, 11, 14, 15, 21, 55tgcgrneq 26286 . . . . . . 7 ((((𝜑𝑥𝑃) ∧ 𝑦𝑃) ∧ ((𝐸 ∈ (𝐹𝐼𝑥) ∧ (𝐸 𝑥) = (𝐵 𝐴)) ∧ (𝐸 ∈ (𝐷𝐼𝑦) ∧ (𝐸 𝑦) = (𝐵 𝐶)))) → 𝐸𝑦)
5756necomd 3069 . . . . . 6 ((((𝜑𝑥𝑃) ∧ 𝑦𝑃) ∧ ((𝐸 ∈ (𝐹𝐼𝑥) ∧ (𝐸 𝑥) = (𝐵 𝐴)) ∧ (𝐸 ∈ (𝐷𝐼𝑦) ∧ (𝐸 𝑦) = (𝐵 𝐶)))) → 𝑦𝐸)
5830ad3antrrr 729 . . . . . . 7 ((((𝜑𝑥𝑃) ∧ 𝑦𝑃) ∧ ((𝐸 ∈ (𝐹𝐼𝑥) ∧ (𝐸 𝑥) = (𝐵 𝐴)) ∧ (𝐸 ∈ (𝐷𝐼𝑦) ∧ (𝐸 𝑦) = (𝐵 𝐶)))) → 𝐸 ∈ (𝐷𝐼𝐹))
591, 16, 5, 25, 14, 15, 23, 29, 34, 58tgbtwnconn2 26379 . . . . . 6 ((((𝜑𝑥𝑃) ∧ 𝑦𝑃) ∧ ((𝐸 ∈ (𝐹𝐼𝑥) ∧ (𝐸 𝑥) = (𝐵 𝐴)) ∧ (𝐸 ∈ (𝐷𝐼𝑦) ∧ (𝐸 𝑦) = (𝐵 𝐶)))) → (𝑦 ∈ (𝐸𝐼𝐹) ∨ 𝐹 ∈ (𝐸𝐼𝑦)))
6057, 27, 593jca 1125 . . . . 5 ((((𝜑𝑥𝑃) ∧ 𝑦𝑃) ∧ ((𝐸 ∈ (𝐹𝐼𝑥) ∧ (𝐸 𝑥) = (𝐵 𝐴)) ∧ (𝐸 ∈ (𝐷𝐼𝑦) ∧ (𝐸 𝑦) = (𝐵 𝐶)))) → (𝑦𝐸𝐹𝐸 ∧ (𝑦 ∈ (𝐸𝐼𝐹) ∨ 𝐹 ∈ (𝐸𝐼𝑦))))
611, 16, 50, 15, 23, 14, 5ishlg 26405 . . . . 5 ((((𝜑𝑥𝑃) ∧ 𝑦𝑃) ∧ ((𝐸 ∈ (𝐹𝐼𝑥) ∧ (𝐸 𝑥) = (𝐵 𝐴)) ∧ (𝐸 ∈ (𝐷𝐼𝑦) ∧ (𝐸 𝑦) = (𝐵 𝐶)))) → (𝑦((hlG‘𝐺)‘𝐸)𝐹 ↔ (𝑦𝐸𝐹𝐸 ∧ (𝑦 ∈ (𝐸𝐼𝐹) ∨ 𝐹 ∈ (𝐸𝐼𝑦)))))
6260, 61mpbird 260 . . . 4 ((((𝜑𝑥𝑃) ∧ 𝑦𝑃) ∧ ((𝐸 ∈ (𝐹𝐼𝑥) ∧ (𝐸 𝑥) = (𝐵 𝐴)) ∧ (𝐸 ∈ (𝐷𝐼𝑦) ∧ (𝐸 𝑦) = (𝐵 𝐶)))) → 𝑦((hlG‘𝐺)‘𝐸)𝐹)
6341, 52, 623jca 1125 . . 3 ((((𝜑𝑥𝑃) ∧ 𝑦𝑃) ∧ ((𝐸 ∈ (𝐹𝐼𝑥) ∧ (𝐸 𝑥) = (𝐵 𝐴)) ∧ (𝐸 ∈ (𝐷𝐼𝑦) ∧ (𝐸 𝑦) = (𝐵 𝐶)))) → (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑥𝐸𝑦”⟩ ∧ 𝑥((hlG‘𝐺)‘𝐸)𝐷𝑦((hlG‘𝐺)‘𝐸)𝐹))
641, 2, 16, 4, 22, 13, 8, 6axtgsegcon 26267 . . . 4 (𝜑 → ∃𝑥𝑃 (𝐸 ∈ (𝐹𝐼𝑥) ∧ (𝐸 𝑥) = (𝐵 𝐴)))
651, 2, 16, 4, 24, 13, 8, 10axtgsegcon 26267 . . . 4 (𝜑 → ∃𝑦𝑃 (𝐸 ∈ (𝐷𝐼𝑦) ∧ (𝐸 𝑦) = (𝐵 𝐶)))
66 reeanv 3358 . . . 4 (∃𝑥𝑃𝑦𝑃 ((𝐸 ∈ (𝐹𝐼𝑥) ∧ (𝐸 𝑥) = (𝐵 𝐴)) ∧ (𝐸 ∈ (𝐷𝐼𝑦) ∧ (𝐸 𝑦) = (𝐵 𝐶))) ↔ (∃𝑥𝑃 (𝐸 ∈ (𝐹𝐼𝑥) ∧ (𝐸 𝑥) = (𝐵 𝐴)) ∧ ∃𝑦𝑃 (𝐸 ∈ (𝐷𝐼𝑦) ∧ (𝐸 𝑦) = (𝐵 𝐶))))
6764, 65, 66sylanbrc 586 . . 3 (𝜑 → ∃𝑥𝑃𝑦𝑃 ((𝐸 ∈ (𝐹𝐼𝑥) ∧ (𝐸 𝑥) = (𝐵 𝐴)) ∧ (𝐸 ∈ (𝐷𝐼𝑦) ∧ (𝐸 𝑦) = (𝐵 𝐶))))
6863, 67reximddv2 3270 . 2 (𝜑 → ∃𝑥𝑃𝑦𝑃 (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑥𝐸𝑦”⟩ ∧ 𝑥((hlG‘𝐺)‘𝐸)𝐷𝑦((hlG‘𝐺)‘𝐸)𝐹))
691, 16, 50, 4, 6, 8, 10, 24, 13, 22iscgra 26612 . 2 (𝜑 → (⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩ ↔ ∃𝑥𝑃𝑦𝑃 (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑥𝐸𝑦”⟩ ∧ 𝑥((hlG‘𝐺)‘𝐸)𝐷𝑦((hlG‘𝐺)‘𝐸)𝐹)))
7068, 69mpbird 260 1 (𝜑 → ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  wo 844  w3a 1084   = wceq 1538  wcel 2115  wne 3014  wrex 3134   class class class wbr 5053  cfv 6345  (class class class)co 7151  ⟨“cs3 14206  Basecbs 16485  distcds 16576  TarskiGcstrkg 26233  Itvcitv 26239  cgrGccgrg 26313  hlGchlg 26403  cgrAccgra 26610
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5177  ax-sep 5190  ax-nul 5197  ax-pow 5254  ax-pr 5318  ax-un 7457  ax-cnex 10593  ax-resscn 10594  ax-1cn 10595  ax-icn 10596  ax-addcl 10597  ax-addrcl 10598  ax-mulcl 10599  ax-mulrcl 10600  ax-mulcom 10601  ax-addass 10602  ax-mulass 10603  ax-distr 10604  ax-i2m1 10605  ax-1ne0 10606  ax-1rid 10607  ax-rnegex 10608  ax-rrecex 10609  ax-cnre 10610  ax-pre-lttri 10611  ax-pre-lttrn 10612  ax-pre-ltadd 10613  ax-pre-mulgt0 10614
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-nel 3119  df-ral 3138  df-rex 3139  df-reu 3140  df-rmo 3141  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-pss 3938  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-tp 4555  df-op 4557  df-uni 4825  df-int 4863  df-iun 4907  df-br 5054  df-opab 5116  df-mpt 5134  df-tr 5160  df-id 5448  df-eprel 5453  df-po 5462  df-so 5463  df-fr 5502  df-we 5504  df-xp 5549  df-rel 5550  df-cnv 5551  df-co 5552  df-dm 5553  df-rn 5554  df-res 5555  df-ima 5556  df-pred 6137  df-ord 6183  df-on 6184  df-lim 6185  df-suc 6186  df-iota 6304  df-fun 6347  df-fn 6348  df-f 6349  df-f1 6350  df-fo 6351  df-f1o 6352  df-fv 6353  df-riota 7109  df-ov 7154  df-oprab 7155  df-mpo 7156  df-om 7577  df-1st 7686  df-2nd 7687  df-wrecs 7945  df-recs 8006  df-rdg 8044  df-1o 8100  df-oadd 8104  df-er 8287  df-map 8406  df-pm 8407  df-en 8508  df-dom 8509  df-sdom 8510  df-fin 8511  df-dju 9329  df-card 9367  df-pnf 10677  df-mnf 10678  df-xr 10679  df-ltxr 10680  df-le 10681  df-sub 10872  df-neg 10873  df-nn 11637  df-2 11699  df-3 11700  df-n0 11897  df-xnn0 11967  df-z 11981  df-uz 12243  df-fz 12897  df-fzo 13040  df-hash 13698  df-word 13869  df-concat 13925  df-s1 13952  df-s2 14212  df-s3 14213  df-trkgc 26251  df-trkgb 26252  df-trkgcb 26253  df-trkg 26256  df-cgrg 26314  df-hlg 26404  df-cgra 26611
This theorem is referenced by: (None)
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