Theorem cgratr 28891

Description: Angle congruence is transitive. Theorem 11.8 of [Schwabhauser] p. 97. (Contributed by Thierry Arnoux, 5-Mar-2020.)

Hypotheses

Ref	Expression
cgraid.p	⊢ 𝑃 = (Base‘𝐺)
cgraid.i	⊢ 𝐼 = (Itv‘𝐺)
cgraid.g	⊢ (𝜑 → 𝐺 ∈ TarskiG)
cgraid.k	⊢ 𝐾 = (hlG‘𝐺)
cgraid.a	⊢ (𝜑 → 𝐴 ∈ 𝑃)
cgraid.b	⊢ (𝜑 → 𝐵 ∈ 𝑃)
cgraid.c	⊢ (𝜑 → 𝐶 ∈ 𝑃)
cgracom.d	⊢ (𝜑 → 𝐷 ∈ 𝑃)
cgracom.e	⊢ (𝜑 → 𝐸 ∈ 𝑃)
cgracom.f	⊢ (𝜑 → 𝐹 ∈ 𝑃)
cgracom.1	⊢ (𝜑 → ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩)
cgratr.h	⊢ (𝜑 → 𝐻 ∈ 𝑃)
cgratr.i	⊢ (𝜑 → 𝑈 ∈ 𝑃)
cgratr.j	⊢ (𝜑 → 𝐽 ∈ 𝑃)
cgratr.1	⊢ (𝜑 → ⟨“𝐷𝐸𝐹”⟩(cgrA‘𝐺)⟨“𝐻𝑈𝐽”⟩)

Assertion

Ref	Expression
cgratr	⊢ (𝜑 → ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐻𝑈𝐽”⟩)

Proof of Theorem cgratr

Dummy variables 𝑥 𝑦 𝑢 𝑣 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		cgraid.p	. . . . 5 ⊢ 𝑃 = (Base‘𝐺)
2		eqid 2736	. . . . 5 ⊢ (dist‘𝐺) = (dist‘𝐺)
3		eqid 2736	. . . . 5 ⊢ (cgrG‘𝐺) = (cgrG‘𝐺)
4		cgraid.g	. . . . . 6 ⊢ (𝜑 → 𝐺 ∈ TarskiG)
5	4	ad3antrrr 731	. . . . 5 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ ((𝑥(𝐾‘𝑈)𝐻 ∧ (𝑈(dist‘𝐺)𝑥) = (𝐵(dist‘𝐺)𝐴)) ∧ (𝑦(𝐾‘𝑈)𝐽 ∧ (𝑈(dist‘𝐺)𝑦) = (𝐵(dist‘𝐺)𝐶)))) → 𝐺 ∈ TarskiG)
6		cgraid.a	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ 𝑃)
7	6	ad3antrrr 731	. . . . 5 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ ((𝑥(𝐾‘𝑈)𝐻 ∧ (𝑈(dist‘𝐺)𝑥) = (𝐵(dist‘𝐺)𝐴)) ∧ (𝑦(𝐾‘𝑈)𝐽 ∧ (𝑈(dist‘𝐺)𝑦) = (𝐵(dist‘𝐺)𝐶)))) → 𝐴 ∈ 𝑃)
8		cgraid.b	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ 𝑃)
9	8	ad3antrrr 731	. . . . 5 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ ((𝑥(𝐾‘𝑈)𝐻 ∧ (𝑈(dist‘𝐺)𝑥) = (𝐵(dist‘𝐺)𝐴)) ∧ (𝑦(𝐾‘𝑈)𝐽 ∧ (𝑈(dist‘𝐺)𝑦) = (𝐵(dist‘𝐺)𝐶)))) → 𝐵 ∈ 𝑃)
10		cgraid.c	. . . . . 6 ⊢ (𝜑 → 𝐶 ∈ 𝑃)
11	10	ad3antrrr 731	. . . . 5 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ ((𝑥(𝐾‘𝑈)𝐻 ∧ (𝑈(dist‘𝐺)𝑥) = (𝐵(dist‘𝐺)𝐴)) ∧ (𝑦(𝐾‘𝑈)𝐽 ∧ (𝑈(dist‘𝐺)𝑦) = (𝐵(dist‘𝐺)𝐶)))) → 𝐶 ∈ 𝑃)
12		simpllr 776	. . . . 5 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ ((𝑥(𝐾‘𝑈)𝐻 ∧ (𝑈(dist‘𝐺)𝑥) = (𝐵(dist‘𝐺)𝐴)) ∧ (𝑦(𝐾‘𝑈)𝐽 ∧ (𝑈(dist‘𝐺)𝑦) = (𝐵(dist‘𝐺)𝐶)))) → 𝑥 ∈ 𝑃)
13		cgratr.i	. . . . . 6 ⊢ (𝜑 → 𝑈 ∈ 𝑃)
14	13	ad3antrrr 731	. . . . 5 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ ((𝑥(𝐾‘𝑈)𝐻 ∧ (𝑈(dist‘𝐺)𝑥) = (𝐵(dist‘𝐺)𝐴)) ∧ (𝑦(𝐾‘𝑈)𝐽 ∧ (𝑈(dist‘𝐺)𝑦) = (𝐵(dist‘𝐺)𝐶)))) → 𝑈 ∈ 𝑃)
15		simplr 769	. . . . 5 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ ((𝑥(𝐾‘𝑈)𝐻 ∧ (𝑈(dist‘𝐺)𝑥) = (𝐵(dist‘𝐺)𝐴)) ∧ (𝑦(𝐾‘𝑈)𝐽 ∧ (𝑈(dist‘𝐺)𝑦) = (𝐵(dist‘𝐺)𝐶)))) → 𝑦 ∈ 𝑃)
16		cgraid.i	. . . . . 6 ⊢ 𝐼 = (Itv‘𝐺)
17		simprlr 780	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ ((𝑥(𝐾‘𝑈)𝐻 ∧ (𝑈(dist‘𝐺)𝑥) = (𝐵(dist‘𝐺)𝐴)) ∧ (𝑦(𝐾‘𝑈)𝐽 ∧ (𝑈(dist‘𝐺)𝑦) = (𝐵(dist‘𝐺)𝐶)))) → (𝑈(dist‘𝐺)𝑥) = (𝐵(dist‘𝐺)𝐴))
18	17	eqcomd 2742	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ ((𝑥(𝐾‘𝑈)𝐻 ∧ (𝑈(dist‘𝐺)𝑥) = (𝐵(dist‘𝐺)𝐴)) ∧ (𝑦(𝐾‘𝑈)𝐽 ∧ (𝑈(dist‘𝐺)𝑦) = (𝐵(dist‘𝐺)𝐶)))) → (𝐵(dist‘𝐺)𝐴) = (𝑈(dist‘𝐺)𝑥))
19	1, 2, 16, 5, 9, 7, 14, 12, 18	tgcgrcomlr 28548	. . . . 5 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ ((𝑥(𝐾‘𝑈)𝐻 ∧ (𝑈(dist‘𝐺)𝑥) = (𝐵(dist‘𝐺)𝐴)) ∧ (𝑦(𝐾‘𝑈)𝐽 ∧ (𝑈(dist‘𝐺)𝑦) = (𝐵(dist‘𝐺)𝐶)))) → (𝐴(dist‘𝐺)𝐵) = (𝑥(dist‘𝐺)𝑈))
20		simprrr 782	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ ((𝑥(𝐾‘𝑈)𝐻 ∧ (𝑈(dist‘𝐺)𝑥) = (𝐵(dist‘𝐺)𝐴)) ∧ (𝑦(𝐾‘𝑈)𝐽 ∧ (𝑈(dist‘𝐺)𝑦) = (𝐵(dist‘𝐺)𝐶)))) → (𝑈(dist‘𝐺)𝑦) = (𝐵(dist‘𝐺)𝐶))
21	20	eqcomd 2742	. . . . 5 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ ((𝑥(𝐾‘𝑈)𝐻 ∧ (𝑈(dist‘𝐺)𝑥) = (𝐵(dist‘𝐺)𝐴)) ∧ (𝑦(𝐾‘𝑈)𝐽 ∧ (𝑈(dist‘𝐺)𝑦) = (𝐵(dist‘𝐺)𝐶)))) → (𝐵(dist‘𝐺)𝐶) = (𝑈(dist‘𝐺)𝑦))
22	5	ad3antrrr 731	. . . . . . . 8 ⊢ (((((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ ((𝑥(𝐾‘𝑈)𝐻 ∧ (𝑈(dist‘𝐺)𝑥) = (𝐵(dist‘𝐺)𝐴)) ∧ (𝑦(𝐾‘𝑈)𝐽 ∧ (𝑈(dist‘𝐺)𝑦) = (𝐵(dist‘𝐺)𝐶)))) ∧ 𝑢 ∈ 𝑃) ∧ 𝑣 ∈ 𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑢𝐸𝑣”⟩ ∧ 𝑢(𝐾‘𝐸)𝐷 ∧ 𝑣(𝐾‘𝐸)𝐹)) → 𝐺 ∈ TarskiG)
23	7	ad3antrrr 731	. . . . . . . 8 ⊢ (((((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ ((𝑥(𝐾‘𝑈)𝐻 ∧ (𝑈(dist‘𝐺)𝑥) = (𝐵(dist‘𝐺)𝐴)) ∧ (𝑦(𝐾‘𝑈)𝐽 ∧ (𝑈(dist‘𝐺)𝑦) = (𝐵(dist‘𝐺)𝐶)))) ∧ 𝑢 ∈ 𝑃) ∧ 𝑣 ∈ 𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑢𝐸𝑣”⟩ ∧ 𝑢(𝐾‘𝐸)𝐷 ∧ 𝑣(𝐾‘𝐸)𝐹)) → 𝐴 ∈ 𝑃)
24	9	ad3antrrr 731	. . . . . . . 8 ⊢ (((((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ ((𝑥(𝐾‘𝑈)𝐻 ∧ (𝑈(dist‘𝐺)𝑥) = (𝐵(dist‘𝐺)𝐴)) ∧ (𝑦(𝐾‘𝑈)𝐽 ∧ (𝑈(dist‘𝐺)𝑦) = (𝐵(dist‘𝐺)𝐶)))) ∧ 𝑢 ∈ 𝑃) ∧ 𝑣 ∈ 𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑢𝐸𝑣”⟩ ∧ 𝑢(𝐾‘𝐸)𝐷 ∧ 𝑣(𝐾‘𝐸)𝐹)) → 𝐵 ∈ 𝑃)
25	11	ad3antrrr 731	. . . . . . . 8 ⊢ (((((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ ((𝑥(𝐾‘𝑈)𝐻 ∧ (𝑈(dist‘𝐺)𝑥) = (𝐵(dist‘𝐺)𝐴)) ∧ (𝑦(𝐾‘𝑈)𝐽 ∧ (𝑈(dist‘𝐺)𝑦) = (𝐵(dist‘𝐺)𝐶)))) ∧ 𝑢 ∈ 𝑃) ∧ 𝑣 ∈ 𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑢𝐸𝑣”⟩ ∧ 𝑢(𝐾‘𝐸)𝐷 ∧ 𝑣(𝐾‘𝐸)𝐹)) → 𝐶 ∈ 𝑃)
26		simpllr 776	. . . . . . . 8 ⊢ (((((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ ((𝑥(𝐾‘𝑈)𝐻 ∧ (𝑈(dist‘𝐺)𝑥) = (𝐵(dist‘𝐺)𝐴)) ∧ (𝑦(𝐾‘𝑈)𝐽 ∧ (𝑈(dist‘𝐺)𝑦) = (𝐵(dist‘𝐺)𝐶)))) ∧ 𝑢 ∈ 𝑃) ∧ 𝑣 ∈ 𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑢𝐸𝑣”⟩ ∧ 𝑢(𝐾‘𝐸)𝐷 ∧ 𝑣(𝐾‘𝐸)𝐹)) → 𝑢 ∈ 𝑃)
27		cgracom.e	. . . . . . . . 9 ⊢ (𝜑 → 𝐸 ∈ 𝑃)
28	27	ad6antr 737	. . . . . . . 8 ⊢ (((((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ ((𝑥(𝐾‘𝑈)𝐻 ∧ (𝑈(dist‘𝐺)𝑥) = (𝐵(dist‘𝐺)𝐴)) ∧ (𝑦(𝐾‘𝑈)𝐽 ∧ (𝑈(dist‘𝐺)𝑦) = (𝐵(dist‘𝐺)𝐶)))) ∧ 𝑢 ∈ 𝑃) ∧ 𝑣 ∈ 𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑢𝐸𝑣”⟩ ∧ 𝑢(𝐾‘𝐸)𝐷 ∧ 𝑣(𝐾‘𝐸)𝐹)) → 𝐸 ∈ 𝑃)
29		simplr 769	. . . . . . . 8 ⊢ (((((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ ((𝑥(𝐾‘𝑈)𝐻 ∧ (𝑈(dist‘𝐺)𝑥) = (𝐵(dist‘𝐺)𝐴)) ∧ (𝑦(𝐾‘𝑈)𝐽 ∧ (𝑈(dist‘𝐺)𝑦) = (𝐵(dist‘𝐺)𝐶)))) ∧ 𝑢 ∈ 𝑃) ∧ 𝑣 ∈ 𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑢𝐸𝑣”⟩ ∧ 𝑢(𝐾‘𝐸)𝐷 ∧ 𝑣(𝐾‘𝐸)𝐹)) → 𝑣 ∈ 𝑃)
30		simpr1 1196	. . . . . . . 8 ⊢ (((((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ ((𝑥(𝐾‘𝑈)𝐻 ∧ (𝑈(dist‘𝐺)𝑥) = (𝐵(dist‘𝐺)𝐴)) ∧ (𝑦(𝐾‘𝑈)𝐽 ∧ (𝑈(dist‘𝐺)𝑦) = (𝐵(dist‘𝐺)𝐶)))) ∧ 𝑢 ∈ 𝑃) ∧ 𝑣 ∈ 𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑢𝐸𝑣”⟩ ∧ 𝑢(𝐾‘𝐸)𝐷 ∧ 𝑣(𝐾‘𝐸)𝐹)) → ⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑢𝐸𝑣”⟩)
31	1, 2, 16, 3, 22, 23, 24, 25, 26, 28, 29, 30	cgr3simp3 28590	. . . . . . 7 ⊢ (((((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ ((𝑥(𝐾‘𝑈)𝐻 ∧ (𝑈(dist‘𝐺)𝑥) = (𝐵(dist‘𝐺)𝐴)) ∧ (𝑦(𝐾‘𝑈)𝐽 ∧ (𝑈(dist‘𝐺)𝑦) = (𝐵(dist‘𝐺)𝐶)))) ∧ 𝑢 ∈ 𝑃) ∧ 𝑣 ∈ 𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑢𝐸𝑣”⟩ ∧ 𝑢(𝐾‘𝐸)𝐷 ∧ 𝑣(𝐾‘𝐸)𝐹)) → (𝐶(dist‘𝐺)𝐴) = (𝑣(dist‘𝐺)𝑢))
32	12	ad3antrrr 731	. . . . . . . 8 ⊢ (((((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ ((𝑥(𝐾‘𝑈)𝐻 ∧ (𝑈(dist‘𝐺)𝑥) = (𝐵(dist‘𝐺)𝐴)) ∧ (𝑦(𝐾‘𝑈)𝐽 ∧ (𝑈(dist‘𝐺)𝑦) = (𝐵(dist‘𝐺)𝐶)))) ∧ 𝑢 ∈ 𝑃) ∧ 𝑣 ∈ 𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑢𝐸𝑣”⟩ ∧ 𝑢(𝐾‘𝐸)𝐷 ∧ 𝑣(𝐾‘𝐸)𝐹)) → 𝑥 ∈ 𝑃)
33	15	ad3antrrr 731	. . . . . . . 8 ⊢ (((((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ ((𝑥(𝐾‘𝑈)𝐻 ∧ (𝑈(dist‘𝐺)𝑥) = (𝐵(dist‘𝐺)𝐴)) ∧ (𝑦(𝐾‘𝑈)𝐽 ∧ (𝑈(dist‘𝐺)𝑦) = (𝐵(dist‘𝐺)𝐶)))) ∧ 𝑢 ∈ 𝑃) ∧ 𝑣 ∈ 𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑢𝐸𝑣”⟩ ∧ 𝑢(𝐾‘𝐸)𝐷 ∧ 𝑣(𝐾‘𝐸)𝐹)) → 𝑦 ∈ 𝑃)
34		cgraid.k	. . . . . . . . 9 ⊢ 𝐾 = (hlG‘𝐺)
35		cgracom.d	. . . . . . . . . 10 ⊢ (𝜑 → 𝐷 ∈ 𝑃)
36	35	ad6antr 737	. . . . . . . . 9 ⊢ (((((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ ((𝑥(𝐾‘𝑈)𝐻 ∧ (𝑈(dist‘𝐺)𝑥) = (𝐵(dist‘𝐺)𝐴)) ∧ (𝑦(𝐾‘𝑈)𝐽 ∧ (𝑈(dist‘𝐺)𝑦) = (𝐵(dist‘𝐺)𝐶)))) ∧ 𝑢 ∈ 𝑃) ∧ 𝑣 ∈ 𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑢𝐸𝑣”⟩ ∧ 𝑢(𝐾‘𝐸)𝐷 ∧ 𝑣(𝐾‘𝐸)𝐹)) → 𝐷 ∈ 𝑃)
37		cgracom.f	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹 ∈ 𝑃)
38	37	ad6antr 737	. . . . . . . . 9 ⊢ (((((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ ((𝑥(𝐾‘𝑈)𝐻 ∧ (𝑈(dist‘𝐺)𝑥) = (𝐵(dist‘𝐺)𝐴)) ∧ (𝑦(𝐾‘𝑈)𝐽 ∧ (𝑈(dist‘𝐺)𝑦) = (𝐵(dist‘𝐺)𝐶)))) ∧ 𝑢 ∈ 𝑃) ∧ 𝑣 ∈ 𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑢𝐸𝑣”⟩ ∧ 𝑢(𝐾‘𝐸)𝐷 ∧ 𝑣(𝐾‘𝐸)𝐹)) → 𝐹 ∈ 𝑃)
39	14	ad3antrrr 731	. . . . . . . . 9 ⊢ (((((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ ((𝑥(𝐾‘𝑈)𝐻 ∧ (𝑈(dist‘𝐺)𝑥) = (𝐵(dist‘𝐺)𝐴)) ∧ (𝑦(𝐾‘𝑈)𝐽 ∧ (𝑈(dist‘𝐺)𝑦) = (𝐵(dist‘𝐺)𝐶)))) ∧ 𝑢 ∈ 𝑃) ∧ 𝑣 ∈ 𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑢𝐸𝑣”⟩ ∧ 𝑢(𝐾‘𝐸)𝐷 ∧ 𝑣(𝐾‘𝐸)𝐹)) → 𝑈 ∈ 𝑃)
40		cgratr.j	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐽 ∈ 𝑃)
41	40	ad6antr 737	. . . . . . . . . 10 ⊢ (((((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ ((𝑥(𝐾‘𝑈)𝐻 ∧ (𝑈(dist‘𝐺)𝑥) = (𝐵(dist‘𝐺)𝐴)) ∧ (𝑦(𝐾‘𝑈)𝐽 ∧ (𝑈(dist‘𝐺)𝑦) = (𝐵(dist‘𝐺)𝐶)))) ∧ 𝑢 ∈ 𝑃) ∧ 𝑣 ∈ 𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑢𝐸𝑣”⟩ ∧ 𝑢(𝐾‘𝐸)𝐷 ∧ 𝑣(𝐾‘𝐸)𝐹)) → 𝐽 ∈ 𝑃)
42		cgratr.h	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐻 ∈ 𝑃)
43	42	ad6antr 737	. . . . . . . . . . 11 ⊢ (((((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ ((𝑥(𝐾‘𝑈)𝐻 ∧ (𝑈(dist‘𝐺)𝑥) = (𝐵(dist‘𝐺)𝐴)) ∧ (𝑦(𝐾‘𝑈)𝐽 ∧ (𝑈(dist‘𝐺)𝑦) = (𝐵(dist‘𝐺)𝐶)))) ∧ 𝑢 ∈ 𝑃) ∧ 𝑣 ∈ 𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑢𝐸𝑣”⟩ ∧ 𝑢(𝐾‘𝐸)𝐷 ∧ 𝑣(𝐾‘𝐸)𝐹)) → 𝐻 ∈ 𝑃)
44		cgratr.1	. . . . . . . . . . . 12 ⊢ (𝜑 → ⟨“𝐷𝐸𝐹”⟩(cgrA‘𝐺)⟨“𝐻𝑈𝐽”⟩)
45	44	ad6antr 737	. . . . . . . . . . 11 ⊢ (((((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ ((𝑥(𝐾‘𝑈)𝐻 ∧ (𝑈(dist‘𝐺)𝑥) = (𝐵(dist‘𝐺)𝐴)) ∧ (𝑦(𝐾‘𝑈)𝐽 ∧ (𝑈(dist‘𝐺)𝑦) = (𝐵(dist‘𝐺)𝐶)))) ∧ 𝑢 ∈ 𝑃) ∧ 𝑣 ∈ 𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑢𝐸𝑣”⟩ ∧ 𝑢(𝐾‘𝐸)𝐷 ∧ 𝑣(𝐾‘𝐸)𝐹)) → ⟨“𝐷𝐸𝐹”⟩(cgrA‘𝐺)⟨“𝐻𝑈𝐽”⟩)
46		simprll 779	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ ((𝑥(𝐾‘𝑈)𝐻 ∧ (𝑈(dist‘𝐺)𝑥) = (𝐵(dist‘𝐺)𝐴)) ∧ (𝑦(𝐾‘𝑈)𝐽 ∧ (𝑈(dist‘𝐺)𝑦) = (𝐵(dist‘𝐺)𝐶)))) → 𝑥(𝐾‘𝑈)𝐻)
47	46	ad3antrrr 731	. . . . . . . . . . 11 ⊢ (((((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ ((𝑥(𝐾‘𝑈)𝐻 ∧ (𝑈(dist‘𝐺)𝑥) = (𝐵(dist‘𝐺)𝐴)) ∧ (𝑦(𝐾‘𝑈)𝐽 ∧ (𝑈(dist‘𝐺)𝑦) = (𝐵(dist‘𝐺)𝐶)))) ∧ 𝑢 ∈ 𝑃) ∧ 𝑣 ∈ 𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑢𝐸𝑣”⟩ ∧ 𝑢(𝐾‘𝐸)𝐷 ∧ 𝑣(𝐾‘𝐸)𝐹)) → 𝑥(𝐾‘𝑈)𝐻)
48	1, 16, 34, 22, 36, 28, 38, 43, 39, 41, 45, 32, 47	cgrahl1 28884	. . . . . . . . . 10 ⊢ (((((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ ((𝑥(𝐾‘𝑈)𝐻 ∧ (𝑈(dist‘𝐺)𝑥) = (𝐵(dist‘𝐺)𝐴)) ∧ (𝑦(𝐾‘𝑈)𝐽 ∧ (𝑈(dist‘𝐺)𝑦) = (𝐵(dist‘𝐺)𝐶)))) ∧ 𝑢 ∈ 𝑃) ∧ 𝑣 ∈ 𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑢𝐸𝑣”⟩ ∧ 𝑢(𝐾‘𝐸)𝐷 ∧ 𝑣(𝐾‘𝐸)𝐹)) → ⟨“𝐷𝐸𝐹”⟩(cgrA‘𝐺)⟨“𝑥𝑈𝐽”⟩)
49		simprrl 781	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ ((𝑥(𝐾‘𝑈)𝐻 ∧ (𝑈(dist‘𝐺)𝑥) = (𝐵(dist‘𝐺)𝐴)) ∧ (𝑦(𝐾‘𝑈)𝐽 ∧ (𝑈(dist‘𝐺)𝑦) = (𝐵(dist‘𝐺)𝐶)))) → 𝑦(𝐾‘𝑈)𝐽)
50	49	ad3antrrr 731	. . . . . . . . . 10 ⊢ (((((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ ((𝑥(𝐾‘𝑈)𝐻 ∧ (𝑈(dist‘𝐺)𝑥) = (𝐵(dist‘𝐺)𝐴)) ∧ (𝑦(𝐾‘𝑈)𝐽 ∧ (𝑈(dist‘𝐺)𝑦) = (𝐵(dist‘𝐺)𝐶)))) ∧ 𝑢 ∈ 𝑃) ∧ 𝑣 ∈ 𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑢𝐸𝑣”⟩ ∧ 𝑢(𝐾‘𝐸)𝐷 ∧ 𝑣(𝐾‘𝐸)𝐹)) → 𝑦(𝐾‘𝑈)𝐽)
51	1, 16, 34, 22, 36, 28, 38, 32, 39, 41, 48, 33, 50	cgrahl2 28885	. . . . . . . . 9 ⊢ (((((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ ((𝑥(𝐾‘𝑈)𝐻 ∧ (𝑈(dist‘𝐺)𝑥) = (𝐵(dist‘𝐺)𝐴)) ∧ (𝑦(𝐾‘𝑈)𝐽 ∧ (𝑈(dist‘𝐺)𝑦) = (𝐵(dist‘𝐺)𝐶)))) ∧ 𝑢 ∈ 𝑃) ∧ 𝑣 ∈ 𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑢𝐸𝑣”⟩ ∧ 𝑢(𝐾‘𝐸)𝐷 ∧ 𝑣(𝐾‘𝐸)𝐹)) → ⟨“𝐷𝐸𝐹”⟩(cgrA‘𝐺)⟨“𝑥𝑈𝑦”⟩)
52		simpr2 1197	. . . . . . . . 9 ⊢ (((((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ ((𝑥(𝐾‘𝑈)𝐻 ∧ (𝑈(dist‘𝐺)𝑥) = (𝐵(dist‘𝐺)𝐴)) ∧ (𝑦(𝐾‘𝑈)𝐽 ∧ (𝑈(dist‘𝐺)𝑦) = (𝐵(dist‘𝐺)𝐶)))) ∧ 𝑢 ∈ 𝑃) ∧ 𝑣 ∈ 𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑢𝐸𝑣”⟩ ∧ 𝑢(𝐾‘𝐸)𝐷 ∧ 𝑣(𝐾‘𝐸)𝐹)) → 𝑢(𝐾‘𝐸)𝐷)
53		simpr3 1198	. . . . . . . . 9 ⊢ (((((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ ((𝑥(𝐾‘𝑈)𝐻 ∧ (𝑈(dist‘𝐺)𝑥) = (𝐵(dist‘𝐺)𝐴)) ∧ (𝑦(𝐾‘𝑈)𝐽 ∧ (𝑈(dist‘𝐺)𝑦) = (𝐵(dist‘𝐺)𝐶)))) ∧ 𝑢 ∈ 𝑃) ∧ 𝑣 ∈ 𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑢𝐸𝑣”⟩ ∧ 𝑢(𝐾‘𝐸)𝐷 ∧ 𝑣(𝐾‘𝐸)𝐹)) → 𝑣(𝐾‘𝐸)𝐹)
54	1, 2, 16, 3, 22, 23, 24, 25, 26, 28, 29, 30	cgr3simp1 28588	. . . . . . . . . . . 12 ⊢ (((((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ ((𝑥(𝐾‘𝑈)𝐻 ∧ (𝑈(dist‘𝐺)𝑥) = (𝐵(dist‘𝐺)𝐴)) ∧ (𝑦(𝐾‘𝑈)𝐽 ∧ (𝑈(dist‘𝐺)𝑦) = (𝐵(dist‘𝐺)𝐶)))) ∧ 𝑢 ∈ 𝑃) ∧ 𝑣 ∈ 𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑢𝐸𝑣”⟩ ∧ 𝑢(𝐾‘𝐸)𝐷 ∧ 𝑣(𝐾‘𝐸)𝐹)) → (𝐴(dist‘𝐺)𝐵) = (𝑢(dist‘𝐺)𝐸))
55	54	eqcomd 2742	. . . . . . . . . . 11 ⊢ (((((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ ((𝑥(𝐾‘𝑈)𝐻 ∧ (𝑈(dist‘𝐺)𝑥) = (𝐵(dist‘𝐺)𝐴)) ∧ (𝑦(𝐾‘𝑈)𝐽 ∧ (𝑈(dist‘𝐺)𝑦) = (𝐵(dist‘𝐺)𝐶)))) ∧ 𝑢 ∈ 𝑃) ∧ 𝑣 ∈ 𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑢𝐸𝑣”⟩ ∧ 𝑢(𝐾‘𝐸)𝐷 ∧ 𝑣(𝐾‘𝐸)𝐹)) → (𝑢(dist‘𝐺)𝐸) = (𝐴(dist‘𝐺)𝐵))
56	1, 2, 16, 22, 26, 28, 23, 24, 55	tgcgrcomlr 28548	. . . . . . . . . 10 ⊢ (((((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ ((𝑥(𝐾‘𝑈)𝐻 ∧ (𝑈(dist‘𝐺)𝑥) = (𝐵(dist‘𝐺)𝐴)) ∧ (𝑦(𝐾‘𝑈)𝐽 ∧ (𝑈(dist‘𝐺)𝑦) = (𝐵(dist‘𝐺)𝐶)))) ∧ 𝑢 ∈ 𝑃) ∧ 𝑣 ∈ 𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑢𝐸𝑣”⟩ ∧ 𝑢(𝐾‘𝐸)𝐷 ∧ 𝑣(𝐾‘𝐸)𝐹)) → (𝐸(dist‘𝐺)𝑢) = (𝐵(dist‘𝐺)𝐴))
57	18	ad3antrrr 731	. . . . . . . . . 10 ⊢ (((((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ ((𝑥(𝐾‘𝑈)𝐻 ∧ (𝑈(dist‘𝐺)𝑥) = (𝐵(dist‘𝐺)𝐴)) ∧ (𝑦(𝐾‘𝑈)𝐽 ∧ (𝑈(dist‘𝐺)𝑦) = (𝐵(dist‘𝐺)𝐶)))) ∧ 𝑢 ∈ 𝑃) ∧ 𝑣 ∈ 𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑢𝐸𝑣”⟩ ∧ 𝑢(𝐾‘𝐸)𝐷 ∧ 𝑣(𝐾‘𝐸)𝐹)) → (𝐵(dist‘𝐺)𝐴) = (𝑈(dist‘𝐺)𝑥))
58	56, 57	eqtrd 2771	. . . . . . . . 9 ⊢ (((((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ ((𝑥(𝐾‘𝑈)𝐻 ∧ (𝑈(dist‘𝐺)𝑥) = (𝐵(dist‘𝐺)𝐴)) ∧ (𝑦(𝐾‘𝑈)𝐽 ∧ (𝑈(dist‘𝐺)𝑦) = (𝐵(dist‘𝐺)𝐶)))) ∧ 𝑢 ∈ 𝑃) ∧ 𝑣 ∈ 𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑢𝐸𝑣”⟩ ∧ 𝑢(𝐾‘𝐸)𝐷 ∧ 𝑣(𝐾‘𝐸)𝐹)) → (𝐸(dist‘𝐺)𝑢) = (𝑈(dist‘𝐺)𝑥))
59	1, 2, 16, 3, 22, 23, 24, 25, 26, 28, 29, 30	cgr3simp2 28589	. . . . . . . . . 10 ⊢ (((((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ ((𝑥(𝐾‘𝑈)𝐻 ∧ (𝑈(dist‘𝐺)𝑥) = (𝐵(dist‘𝐺)𝐴)) ∧ (𝑦(𝐾‘𝑈)𝐽 ∧ (𝑈(dist‘𝐺)𝑦) = (𝐵(dist‘𝐺)𝐶)))) ∧ 𝑢 ∈ 𝑃) ∧ 𝑣 ∈ 𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑢𝐸𝑣”⟩ ∧ 𝑢(𝐾‘𝐸)𝐷 ∧ 𝑣(𝐾‘𝐸)𝐹)) → (𝐵(dist‘𝐺)𝐶) = (𝐸(dist‘𝐺)𝑣))
60	21	ad3antrrr 731	. . . . . . . . . 10 ⊢ (((((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ ((𝑥(𝐾‘𝑈)𝐻 ∧ (𝑈(dist‘𝐺)𝑥) = (𝐵(dist‘𝐺)𝐴)) ∧ (𝑦(𝐾‘𝑈)𝐽 ∧ (𝑈(dist‘𝐺)𝑦) = (𝐵(dist‘𝐺)𝐶)))) ∧ 𝑢 ∈ 𝑃) ∧ 𝑣 ∈ 𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑢𝐸𝑣”⟩ ∧ 𝑢(𝐾‘𝐸)𝐷 ∧ 𝑣(𝐾‘𝐸)𝐹)) → (𝐵(dist‘𝐺)𝐶) = (𝑈(dist‘𝐺)𝑦))
61	59, 60	eqtr3d 2773	. . . . . . . . 9 ⊢ (((((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ ((𝑥(𝐾‘𝑈)𝐻 ∧ (𝑈(dist‘𝐺)𝑥) = (𝐵(dist‘𝐺)𝐴)) ∧ (𝑦(𝐾‘𝑈)𝐽 ∧ (𝑈(dist‘𝐺)𝑦) = (𝐵(dist‘𝐺)𝐶)))) ∧ 𝑢 ∈ 𝑃) ∧ 𝑣 ∈ 𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑢𝐸𝑣”⟩ ∧ 𝑢(𝐾‘𝐸)𝐷 ∧ 𝑣(𝐾‘𝐸)𝐹)) → (𝐸(dist‘𝐺)𝑣) = (𝑈(dist‘𝐺)𝑦))
62	1, 16, 34, 22, 36, 28, 38, 32, 39, 33, 51, 26, 2, 29, 52, 53, 58, 61	cgracgr 28886	. . . . . . . 8 ⊢ (((((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ ((𝑥(𝐾‘𝑈)𝐻 ∧ (𝑈(dist‘𝐺)𝑥) = (𝐵(dist‘𝐺)𝐴)) ∧ (𝑦(𝐾‘𝑈)𝐽 ∧ (𝑈(dist‘𝐺)𝑦) = (𝐵(dist‘𝐺)𝐶)))) ∧ 𝑢 ∈ 𝑃) ∧ 𝑣 ∈ 𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑢𝐸𝑣”⟩ ∧ 𝑢(𝐾‘𝐸)𝐷 ∧ 𝑣(𝐾‘𝐸)𝐹)) → (𝑢(dist‘𝐺)𝑣) = (𝑥(dist‘𝐺)𝑦))
63	1, 2, 16, 22, 26, 29, 32, 33, 62	tgcgrcomlr 28548	. . . . . . 7 ⊢ (((((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ ((𝑥(𝐾‘𝑈)𝐻 ∧ (𝑈(dist‘𝐺)𝑥) = (𝐵(dist‘𝐺)𝐴)) ∧ (𝑦(𝐾‘𝑈)𝐽 ∧ (𝑈(dist‘𝐺)𝑦) = (𝐵(dist‘𝐺)𝐶)))) ∧ 𝑢 ∈ 𝑃) ∧ 𝑣 ∈ 𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑢𝐸𝑣”⟩ ∧ 𝑢(𝐾‘𝐸)𝐷 ∧ 𝑣(𝐾‘𝐸)𝐹)) → (𝑣(dist‘𝐺)𝑢) = (𝑦(dist‘𝐺)𝑥))
64	31, 63	eqtrd 2771	. . . . . 6 ⊢ (((((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ ((𝑥(𝐾‘𝑈)𝐻 ∧ (𝑈(dist‘𝐺)𝑥) = (𝐵(dist‘𝐺)𝐴)) ∧ (𝑦(𝐾‘𝑈)𝐽 ∧ (𝑈(dist‘𝐺)𝑦) = (𝐵(dist‘𝐺)𝐶)))) ∧ 𝑢 ∈ 𝑃) ∧ 𝑣 ∈ 𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑢𝐸𝑣”⟩ ∧ 𝑢(𝐾‘𝐸)𝐷 ∧ 𝑣(𝐾‘𝐸)𝐹)) → (𝐶(dist‘𝐺)𝐴) = (𝑦(dist‘𝐺)𝑥))
65		cgracom.1	. . . . . . . 8 ⊢ (𝜑 → ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩)
66	1, 16, 34, 4, 6, 8, 10, 35, 27, 37	iscgra 28877	. . . . . . . 8 ⊢ (𝜑 → (⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩ ↔ ∃𝑢 ∈ 𝑃 ∃𝑣 ∈ 𝑃 (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑢𝐸𝑣”⟩ ∧ 𝑢(𝐾‘𝐸)𝐷 ∧ 𝑣(𝐾‘𝐸)𝐹)))
67	65, 66	mpbid 232	. . . . . . 7 ⊢ (𝜑 → ∃𝑢 ∈ 𝑃 ∃𝑣 ∈ 𝑃 (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑢𝐸𝑣”⟩ ∧ 𝑢(𝐾‘𝐸)𝐷 ∧ 𝑣(𝐾‘𝐸)𝐹))
68	67	ad3antrrr 731	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ ((𝑥(𝐾‘𝑈)𝐻 ∧ (𝑈(dist‘𝐺)𝑥) = (𝐵(dist‘𝐺)𝐴)) ∧ (𝑦(𝐾‘𝑈)𝐽 ∧ (𝑈(dist‘𝐺)𝑦) = (𝐵(dist‘𝐺)𝐶)))) → ∃𝑢 ∈ 𝑃 ∃𝑣 ∈ 𝑃 (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑢𝐸𝑣”⟩ ∧ 𝑢(𝐾‘𝐸)𝐷 ∧ 𝑣(𝐾‘𝐸)𝐹))
69	64, 68	r19.29vva 3197	. . . . 5 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ ((𝑥(𝐾‘𝑈)𝐻 ∧ (𝑈(dist‘𝐺)𝑥) = (𝐵(dist‘𝐺)𝐴)) ∧ (𝑦(𝐾‘𝑈)𝐽 ∧ (𝑈(dist‘𝐺)𝑦) = (𝐵(dist‘𝐺)𝐶)))) → (𝐶(dist‘𝐺)𝐴) = (𝑦(dist‘𝐺)𝑥))
70	1, 2, 3, 5, 7, 9, 11, 12, 14, 15, 19, 21, 69	trgcgr 28584	. . . 4 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ ((𝑥(𝐾‘𝑈)𝐻 ∧ (𝑈(dist‘𝐺)𝑥) = (𝐵(dist‘𝐺)𝐴)) ∧ (𝑦(𝐾‘𝑈)𝐽 ∧ (𝑈(dist‘𝐺)𝑦) = (𝐵(dist‘𝐺)𝐶)))) → ⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑥𝑈𝑦”⟩)
71	70, 46, 49	3jca 1129	. . 3 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ ((𝑥(𝐾‘𝑈)𝐻 ∧ (𝑈(dist‘𝐺)𝑥) = (𝐵(dist‘𝐺)𝐴)) ∧ (𝑦(𝐾‘𝑈)𝐽 ∧ (𝑈(dist‘𝐺)𝑦) = (𝐵(dist‘𝐺)𝐶)))) → (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑥𝑈𝑦”⟩ ∧ 𝑥(𝐾‘𝑈)𝐻 ∧ 𝑦(𝐾‘𝑈)𝐽))
72	1, 16, 34, 4, 35, 27, 37, 42, 13, 40, 44	cgrane3 28882	. . . . . 6 ⊢ (𝜑 → 𝑈 ≠ 𝐻)
73	72	necomd 2987	. . . . 5 ⊢ (𝜑 → 𝐻 ≠ 𝑈)
74	1, 16, 34, 4, 6, 8, 10, 35, 27, 37, 65	cgrane1 28880	. . . . . 6 ⊢ (𝜑 → 𝐴 ≠ 𝐵)
75	74	necomd 2987	. . . . 5 ⊢ (𝜑 → 𝐵 ≠ 𝐴)
76	1, 16, 34, 13, 8, 6, 4, 42, 2, 73, 75	hlcgrex 28684	. . . 4 ⊢ (𝜑 → ∃𝑥 ∈ 𝑃 (𝑥(𝐾‘𝑈)𝐻 ∧ (𝑈(dist‘𝐺)𝑥) = (𝐵(dist‘𝐺)𝐴)))
77	1, 16, 34, 4, 35, 27, 37, 42, 13, 40, 44	cgrane4 28883	. . . . . 6 ⊢ (𝜑 → 𝑈 ≠ 𝐽)
78	77	necomd 2987	. . . . 5 ⊢ (𝜑 → 𝐽 ≠ 𝑈)
79	1, 16, 34, 4, 6, 8, 10, 35, 27, 37, 65	cgrane2 28881	. . . . 5 ⊢ (𝜑 → 𝐵 ≠ 𝐶)
80	1, 16, 34, 13, 8, 10, 4, 40, 2, 78, 79	hlcgrex 28684	. . . 4 ⊢ (𝜑 → ∃𝑦 ∈ 𝑃 (𝑦(𝐾‘𝑈)𝐽 ∧ (𝑈(dist‘𝐺)𝑦) = (𝐵(dist‘𝐺)𝐶)))
81		reeanv 3209	. . . 4 ⊢ (∃𝑥 ∈ 𝑃 ∃𝑦 ∈ 𝑃 ((𝑥(𝐾‘𝑈)𝐻 ∧ (𝑈(dist‘𝐺)𝑥) = (𝐵(dist‘𝐺)𝐴)) ∧ (𝑦(𝐾‘𝑈)𝐽 ∧ (𝑈(dist‘𝐺)𝑦) = (𝐵(dist‘𝐺)𝐶))) ↔ (∃𝑥 ∈ 𝑃 (𝑥(𝐾‘𝑈)𝐻 ∧ (𝑈(dist‘𝐺)𝑥) = (𝐵(dist‘𝐺)𝐴)) ∧ ∃𝑦 ∈ 𝑃 (𝑦(𝐾‘𝑈)𝐽 ∧ (𝑈(dist‘𝐺)𝑦) = (𝐵(dist‘𝐺)𝐶))))
82	76, 80, 81	sylanbrc 584	. . 3 ⊢ (𝜑 → ∃𝑥 ∈ 𝑃 ∃𝑦 ∈ 𝑃 ((𝑥(𝐾‘𝑈)𝐻 ∧ (𝑈(dist‘𝐺)𝑥) = (𝐵(dist‘𝐺)𝐴)) ∧ (𝑦(𝐾‘𝑈)𝐽 ∧ (𝑈(dist‘𝐺)𝑦) = (𝐵(dist‘𝐺)𝐶))))
83	71, 82	reximddv2 3196	. 2 ⊢ (𝜑 → ∃𝑥 ∈ 𝑃 ∃𝑦 ∈ 𝑃 (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑥𝑈𝑦”⟩ ∧ 𝑥(𝐾‘𝑈)𝐻 ∧ 𝑦(𝐾‘𝑈)𝐽))
84	1, 16, 34, 4, 6, 8, 10, 42, 13, 40	iscgra 28877	. 2 ⊢ (𝜑 → (⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐻𝑈𝐽”⟩ ↔ ∃𝑥 ∈ 𝑃 ∃𝑦 ∈ 𝑃 (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑥𝑈𝑦”⟩ ∧ 𝑥(𝐾‘𝑈)𝐻 ∧ 𝑦(𝐾‘𝑈)𝐽)))
85	83, 84	mpbird 257	1 ⊢ (𝜑 → ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐻𝑈𝐽”⟩)

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114  ∃wrex 3061   class class class wbr 5085  ‘cfv 6498  (class class class)co 7367  ⟨“cs3 14804  Basecbs 17179  distcds 17229  TarskiGcstrkg 28495  Itvcitv 28501  cgrGccgrg 28578  hlGchlg 28668  cgrAccgra 28875

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 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689  ax-cnex 11094  ax-resscn 11095  ax-1cn 11096  ax-icn 11097  ax-addcl 11098  ax-addrcl 11099  ax-mulcl 11100  ax-mulrcl 11101  ax-mulcom 11102  ax-addass 11103  ax-mulass 11104  ax-distr 11105  ax-i2m1 11106  ax-1ne0 11107  ax-1rid 11108  ax-rnegex 11109  ax-rrecex 11110  ax-cnre 11111  ax-pre-lttri 11112  ax-pre-lttrn 11113  ax-pre-ltadd 11114  ax-pre-mulgt0 11115

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3062  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-pss 3909  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-tp 4572  df-op 4574  df-uni 4851  df-int 4890  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-tr 5193  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6265  df-ord 6326  df-on 6327  df-lim 6328  df-suc 6329  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-riota 7324  df-ov 7370  df-oprab 7371  df-mpo 7372  df-om 7818  df-1st 7942  df-2nd 7943  df-frecs 8231  df-wrecs 8262  df-recs 8311  df-rdg 8349  df-1o 8405  df-oadd 8409  df-er 8643  df-map 8775  df-pm 8776  df-en 8894  df-dom 8895  df-sdom 8896  df-fin 8897  df-dju 9825  df-card 9863  df-pnf 11181  df-mnf 11182  df-xr 11183  df-ltxr 11184  df-le 11185  df-sub 11379  df-neg 11380  df-nn 12175  df-2 12244  df-3 12245  df-n0 12438  df-xnn0 12511  df-z 12525  df-uz 12789  df-fz 13462  df-fzo 13609  df-hash 14293  df-word 14476  df-concat 14533  df-s1 14559  df-s2 14810  df-s3 14811  df-trkgc 28516  df-trkgb 28517  df-trkgcb 28518  df-trkg 28521  df-cgrg 28579  df-leg 28651  df-hlg 28669  df-cgra 28876

This theorem is referenced by:  cgraswaplr  28893  sacgr  28899  oacgr  28900  tgasa1  28926