Theorem cgratr 28895

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 730	. . . . 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 730	. . . . 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 730	. . . . 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 730	. . . . 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 775	. . . . 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 730	. . . . 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 768	. . . . 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 779	. . . . . . 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 28552	. . . . 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 781	. . . . . 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 730	. . . . . . . 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 730	. . . . . . . 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 730	. . . . . . . 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 730	. . . . . . . 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 775	. . . . . . . 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 736	. . . . . . . 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 768	. . . . . . . 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 1195	. . . . . . . 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 28594	. . . . . . 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 730	. . . . . . . 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 730	. . . . . . . 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 736	. . . . . . . . 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 736	. . . . . . . . 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 730	. . . . . . . . 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 736	. . . . . . . . . 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 736	. . . . . . . . . . 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 736	. . . . . . . . . . 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 778	. . . . . . . . . . . 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 730	. . . . . . . . . . 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 28888	. . . . . . . . . 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 780	. . . . . . . . . . 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 730	. . . . . . . . . 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 28889	. . . . . . . . 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 1196	. . . . . . . . 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 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)) → 𝑣(𝐾‘𝐸)𝐹)
54	1, 2, 16, 3, 22, 23, 24, 25, 26, 28, 29, 30	cgr3simp1 28592	. . . . . . . . . . . 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 28552	. . . . . . . . . 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 730	. . . . . . . . . 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 28593	. . . . . . . . . 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 730	. . . . . . . . . 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 28890	. . . . . . . 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 28552	. . . . . . 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 28881	. . . . . . . 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 730	. . . . . 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 3196	. . . . 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 28588	. . . 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 1128	. . 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 28886	. . . . . 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 28884	. . . . . 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 28688	. . . 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 28887	. . . . . 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 28885	. . . . 5 ⊢ (𝜑 → 𝐵 ≠ 𝐶)
80	1, 16, 34, 13, 8, 10, 4, 40, 2, 78, 79	hlcgrex 28688	. . . 4 ⊢ (𝜑 → ∃𝑦 ∈ 𝑃 (𝑦(𝐾‘𝑈)𝐽 ∧ (𝑈(dist‘𝐺)𝑦) = (𝐵(dist‘𝐺)𝐶)))
81		reeanv 3208	. . . 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 583	. . 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 3195	. 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 28881	. 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 1086   = wceq 1541   ∈ wcel 2113  ∃wrex 3060   class class class wbr 5098  ‘cfv 6492  (class class class)co 7358  ⟨“cs3 14765  Basecbs 17136  distcds 17186  TarskiGcstrkg 28499  Itvcitv 28505  cgrGccgrg 28582  hlGchlg 28672  cgrAccgra 28879

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-rep 5224  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680  ax-cnex 11082  ax-resscn 11083  ax-1cn 11084  ax-icn 11085  ax-addcl 11086  ax-addrcl 11087  ax-mulcl 11088  ax-mulrcl 11089  ax-mulcom 11090  ax-addass 11091  ax-mulass 11092  ax-distr 11093  ax-i2m1 11094  ax-1ne0 11095  ax-1rid 11096  ax-rnegex 11097  ax-rrecex 11098  ax-cnre 11099  ax-pre-lttri 11100  ax-pre-lttrn 11101  ax-pre-ltadd 11102  ax-pre-mulgt0 11103

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  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 3061  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-pss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-tp 4585  df-op 4587  df-uni 4864  df-int 4903  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-tr 5206  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-riota 7315  df-ov 7361  df-oprab 7362  df-mpo 7363  df-om 7809  df-1st 7933  df-2nd 7934  df-frecs 8223  df-wrecs 8254  df-recs 8303  df-rdg 8341  df-1o 8397  df-oadd 8401  df-er 8635  df-map 8765  df-pm 8766  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-dju 9813  df-card 9851  df-pnf 11168  df-mnf 11169  df-xr 11170  df-ltxr 11171  df-le 11172  df-sub 11366  df-neg 11367  df-nn 12146  df-2 12208  df-3 12209  df-n0 12402  df-xnn0 12475  df-z 12489  df-uz 12752  df-fz 13424  df-fzo 13571  df-hash 14254  df-word 14437  df-concat 14494  df-s1 14520  df-s2 14771  df-s3 14772  df-trkgc 28520  df-trkgb 28521  df-trkgcb 28522  df-trkg 28525  df-cgrg 28583  df-leg 28655  df-hlg 28673  df-cgra 28880

This theorem is referenced by:  cgraswaplr  28897  sacgr  28903  oacgr  28904  tgasa1  28930