Theorem cgraswap 27181

Description: Swap rays in a congruence relation. Theorem 11.9 of [Schwabhauser] p. 96. (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)
cgraid.1	⊢ (𝜑 → 𝐴 ≠ 𝐵)
cgraid.2	⊢ (𝜑 → 𝐵 ≠ 𝐶)

Assertion

Ref	Expression
cgraswap	⊢ (𝜑 → ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐶𝐵𝐴”⟩)

Proof of Theorem cgraswap

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

Step	Hyp	Ref	Expression
1		cgraid.p	. . . . 5 ⊢ 𝑃 = (Base‘𝐺)
2		eqid 2738	. . . . 5 ⊢ (dist‘𝐺) = (dist‘𝐺)
3		eqid 2738	. . . . 5 ⊢ (cgrG‘𝐺) = (cgrG‘𝐺)
4		cgraid.g	. . . . . 6 ⊢ (𝜑 → 𝐺 ∈ TarskiG)
5	4	ad3antrrr 727	. . . . 5 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ ((𝑥(𝐾‘𝐵)𝐶 ∧ (𝐵(dist‘𝐺)𝑥) = (𝐵(dist‘𝐺)𝐴)) ∧ (𝑦(𝐾‘𝐵)𝐴 ∧ (𝐵(dist‘𝐺)𝑦) = (𝐵(dist‘𝐺)𝐶)))) → 𝐺 ∈ TarskiG)
6		cgraid.a	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ 𝑃)
7	6	ad3antrrr 727	. . . . 5 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ ((𝑥(𝐾‘𝐵)𝐶 ∧ (𝐵(dist‘𝐺)𝑥) = (𝐵(dist‘𝐺)𝐴)) ∧ (𝑦(𝐾‘𝐵)𝐴 ∧ (𝐵(dist‘𝐺)𝑦) = (𝐵(dist‘𝐺)𝐶)))) → 𝐴 ∈ 𝑃)
8		cgraid.b	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ 𝑃)
9	8	ad3antrrr 727	. . . . 5 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ ((𝑥(𝐾‘𝐵)𝐶 ∧ (𝐵(dist‘𝐺)𝑥) = (𝐵(dist‘𝐺)𝐴)) ∧ (𝑦(𝐾‘𝐵)𝐴 ∧ (𝐵(dist‘𝐺)𝑦) = (𝐵(dist‘𝐺)𝐶)))) → 𝐵 ∈ 𝑃)
10		cgraid.c	. . . . . 6 ⊢ (𝜑 → 𝐶 ∈ 𝑃)
11	10	ad3antrrr 727	. . . . 5 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ ((𝑥(𝐾‘𝐵)𝐶 ∧ (𝐵(dist‘𝐺)𝑥) = (𝐵(dist‘𝐺)𝐴)) ∧ (𝑦(𝐾‘𝐵)𝐴 ∧ (𝐵(dist‘𝐺)𝑦) = (𝐵(dist‘𝐺)𝐶)))) → 𝐶 ∈ 𝑃)
12		simpllr 773	. . . . 5 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ ((𝑥(𝐾‘𝐵)𝐶 ∧ (𝐵(dist‘𝐺)𝑥) = (𝐵(dist‘𝐺)𝐴)) ∧ (𝑦(𝐾‘𝐵)𝐴 ∧ (𝐵(dist‘𝐺)𝑦) = (𝐵(dist‘𝐺)𝐶)))) → 𝑥 ∈ 𝑃)
13		simplr 766	. . . . 5 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ ((𝑥(𝐾‘𝐵)𝐶 ∧ (𝐵(dist‘𝐺)𝑥) = (𝐵(dist‘𝐺)𝐴)) ∧ (𝑦(𝐾‘𝐵)𝐴 ∧ (𝐵(dist‘𝐺)𝑦) = (𝐵(dist‘𝐺)𝐶)))) → 𝑦 ∈ 𝑃)
14		cgraid.i	. . . . . . 7 ⊢ 𝐼 = (Itv‘𝐺)
15		simprlr 777	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ ((𝑥(𝐾‘𝐵)𝐶 ∧ (𝐵(dist‘𝐺)𝑥) = (𝐵(dist‘𝐺)𝐴)) ∧ (𝑦(𝐾‘𝐵)𝐴 ∧ (𝐵(dist‘𝐺)𝑦) = (𝐵(dist‘𝐺)𝐶)))) → (𝐵(dist‘𝐺)𝑥) = (𝐵(dist‘𝐺)𝐴))
16	1, 2, 14, 5, 9, 12, 9, 7, 15	tgcgrcomlr 26841	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ ((𝑥(𝐾‘𝐵)𝐶 ∧ (𝐵(dist‘𝐺)𝑥) = (𝐵(dist‘𝐺)𝐴)) ∧ (𝑦(𝐾‘𝐵)𝐴 ∧ (𝐵(dist‘𝐺)𝑦) = (𝐵(dist‘𝐺)𝐶)))) → (𝑥(dist‘𝐺)𝐵) = (𝐴(dist‘𝐺)𝐵))
17	16	eqcomd 2744	. . . . 5 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ ((𝑥(𝐾‘𝐵)𝐶 ∧ (𝐵(dist‘𝐺)𝑥) = (𝐵(dist‘𝐺)𝐴)) ∧ (𝑦(𝐾‘𝐵)𝐴 ∧ (𝐵(dist‘𝐺)𝑦) = (𝐵(dist‘𝐺)𝐶)))) → (𝐴(dist‘𝐺)𝐵) = (𝑥(dist‘𝐺)𝐵))
18		simprrr 779	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ ((𝑥(𝐾‘𝐵)𝐶 ∧ (𝐵(dist‘𝐺)𝑥) = (𝐵(dist‘𝐺)𝐴)) ∧ (𝑦(𝐾‘𝐵)𝐴 ∧ (𝐵(dist‘𝐺)𝑦) = (𝐵(dist‘𝐺)𝐶)))) → (𝐵(dist‘𝐺)𝑦) = (𝐵(dist‘𝐺)𝐶))
19	18	eqcomd 2744	. . . . 5 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ ((𝑥(𝐾‘𝐵)𝐶 ∧ (𝐵(dist‘𝐺)𝑥) = (𝐵(dist‘𝐺)𝐴)) ∧ (𝑦(𝐾‘𝐵)𝐴 ∧ (𝐵(dist‘𝐺)𝑦) = (𝐵(dist‘𝐺)𝐶)))) → (𝐵(dist‘𝐺)𝐶) = (𝐵(dist‘𝐺)𝑦))
20		eqid 2738	. . . . . . . 8 ⊢ (LineG‘𝐺) = (LineG‘𝐺)
21		cgraid.k	. . . . . . . . . . 11 ⊢ 𝐾 = (hlG‘𝐺)
22		simprll 776	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ ((𝑥(𝐾‘𝐵)𝐶 ∧ (𝐵(dist‘𝐺)𝑥) = (𝐵(dist‘𝐺)𝐴)) ∧ (𝑦(𝐾‘𝐵)𝐴 ∧ (𝐵(dist‘𝐺)𝑦) = (𝐵(dist‘𝐺)𝐶)))) → 𝑥(𝐾‘𝐵)𝐶)
23	1, 14, 21, 12, 11, 9, 5, 20, 22	hlln 26968	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ ((𝑥(𝐾‘𝐵)𝐶 ∧ (𝐵(dist‘𝐺)𝑥) = (𝐵(dist‘𝐺)𝐴)) ∧ (𝑦(𝐾‘𝐵)𝐴 ∧ (𝐵(dist‘𝐺)𝑦) = (𝐵(dist‘𝐺)𝐶)))) → 𝑥 ∈ (𝐶(LineG‘𝐺)𝐵))
24	23	orcd 870	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ ((𝑥(𝐾‘𝐵)𝐶 ∧ (𝐵(dist‘𝐺)𝑥) = (𝐵(dist‘𝐺)𝐴)) ∧ (𝑦(𝐾‘𝐵)𝐴 ∧ (𝐵(dist‘𝐺)𝑦) = (𝐵(dist‘𝐺)𝐶)))) → (𝑥 ∈ (𝐶(LineG‘𝐺)𝐵) ∨ 𝐶 = 𝐵))
25	1, 20, 14, 5, 11, 9, 12, 24	colrot1 26920	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ ((𝑥(𝐾‘𝐵)𝐶 ∧ (𝐵(dist‘𝐺)𝑥) = (𝐵(dist‘𝐺)𝐴)) ∧ (𝑦(𝐾‘𝐵)𝐴 ∧ (𝐵(dist‘𝐺)𝑦) = (𝐵(dist‘𝐺)𝐶)))) → (𝐶 ∈ (𝐵(LineG‘𝐺)𝑥) ∨ 𝐵 = 𝑥))
26		eqid 2738	. . . . . . . . . 10 ⊢ (≤G‘𝐺) = (≤G‘𝐺)
27	1, 14, 21, 12, 11, 9, 5	ishlg 26963	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ ((𝑥(𝐾‘𝐵)𝐶 ∧ (𝐵(dist‘𝐺)𝑥) = (𝐵(dist‘𝐺)𝐴)) ∧ (𝑦(𝐾‘𝐵)𝐴 ∧ (𝐵(dist‘𝐺)𝑦) = (𝐵(dist‘𝐺)𝐶)))) → (𝑥(𝐾‘𝐵)𝐶 ↔ (𝑥 ≠ 𝐵 ∧ 𝐶 ≠ 𝐵 ∧ (𝑥 ∈ (𝐵𝐼𝐶) ∨ 𝐶 ∈ (𝐵𝐼𝑥)))))
28	22, 27	mpbid 231	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ ((𝑥(𝐾‘𝐵)𝐶 ∧ (𝐵(dist‘𝐺)𝑥) = (𝐵(dist‘𝐺)𝐴)) ∧ (𝑦(𝐾‘𝐵)𝐴 ∧ (𝐵(dist‘𝐺)𝑦) = (𝐵(dist‘𝐺)𝐶)))) → (𝑥 ≠ 𝐵 ∧ 𝐶 ≠ 𝐵 ∧ (𝑥 ∈ (𝐵𝐼𝐶) ∨ 𝐶 ∈ (𝐵𝐼𝑥))))
29	28	simp3d 1143	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ ((𝑥(𝐾‘𝐵)𝐶 ∧ (𝐵(dist‘𝐺)𝑥) = (𝐵(dist‘𝐺)𝐴)) ∧ (𝑦(𝐾‘𝐵)𝐴 ∧ (𝐵(dist‘𝐺)𝑦) = (𝐵(dist‘𝐺)𝐶)))) → (𝑥 ∈ (𝐵𝐼𝐶) ∨ 𝐶 ∈ (𝐵𝐼𝑥)))
30	29	orcomd 868	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ ((𝑥(𝐾‘𝐵)𝐶 ∧ (𝐵(dist‘𝐺)𝑥) = (𝐵(dist‘𝐺)𝐴)) ∧ (𝑦(𝐾‘𝐵)𝐴 ∧ (𝐵(dist‘𝐺)𝑦) = (𝐵(dist‘𝐺)𝐶)))) → (𝐶 ∈ (𝐵𝐼𝑥) ∨ 𝑥 ∈ (𝐵𝐼𝐶)))
31		simprrl 778	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ ((𝑥(𝐾‘𝐵)𝐶 ∧ (𝐵(dist‘𝐺)𝑥) = (𝐵(dist‘𝐺)𝐴)) ∧ (𝑦(𝐾‘𝐵)𝐴 ∧ (𝐵(dist‘𝐺)𝑦) = (𝐵(dist‘𝐺)𝐶)))) → 𝑦(𝐾‘𝐵)𝐴)
32	1, 14, 21, 13, 7, 9, 5	ishlg 26963	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ ((𝑥(𝐾‘𝐵)𝐶 ∧ (𝐵(dist‘𝐺)𝑥) = (𝐵(dist‘𝐺)𝐴)) ∧ (𝑦(𝐾‘𝐵)𝐴 ∧ (𝐵(dist‘𝐺)𝑦) = (𝐵(dist‘𝐺)𝐶)))) → (𝑦(𝐾‘𝐵)𝐴 ↔ (𝑦 ≠ 𝐵 ∧ 𝐴 ≠ 𝐵 ∧ (𝑦 ∈ (𝐵𝐼𝐴) ∨ 𝐴 ∈ (𝐵𝐼𝑦)))))
33	31, 32	mpbid 231	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ ((𝑥(𝐾‘𝐵)𝐶 ∧ (𝐵(dist‘𝐺)𝑥) = (𝐵(dist‘𝐺)𝐴)) ∧ (𝑦(𝐾‘𝐵)𝐴 ∧ (𝐵(dist‘𝐺)𝑦) = (𝐵(dist‘𝐺)𝐶)))) → (𝑦 ≠ 𝐵 ∧ 𝐴 ≠ 𝐵 ∧ (𝑦 ∈ (𝐵𝐼𝐴) ∨ 𝐴 ∈ (𝐵𝐼𝑦))))
34	33	simp3d 1143	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ ((𝑥(𝐾‘𝐵)𝐶 ∧ (𝐵(dist‘𝐺)𝑥) = (𝐵(dist‘𝐺)𝐴)) ∧ (𝑦(𝐾‘𝐵)𝐴 ∧ (𝐵(dist‘𝐺)𝑦) = (𝐵(dist‘𝐺)𝐶)))) → (𝑦 ∈ (𝐵𝐼𝐴) ∨ 𝐴 ∈ (𝐵𝐼𝑦)))
35	1, 2, 14, 26, 5, 9, 11, 12, 9, 9, 13, 7, 30, 34, 19, 15	tgcgrsub2 26956	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ ((𝑥(𝐾‘𝐵)𝐶 ∧ (𝐵(dist‘𝐺)𝑥) = (𝐵(dist‘𝐺)𝐴)) ∧ (𝑦(𝐾‘𝐵)𝐴 ∧ (𝐵(dist‘𝐺)𝑦) = (𝐵(dist‘𝐺)𝐶)))) → (𝐶(dist‘𝐺)𝑥) = (𝑦(dist‘𝐺)𝐴))
36	1, 2, 3, 5, 9, 11, 12, 9, 13, 7, 19, 35, 16	trgcgr 26877	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ ((𝑥(𝐾‘𝐵)𝐶 ∧ (𝐵(dist‘𝐺)𝑥) = (𝐵(dist‘𝐺)𝐴)) ∧ (𝑦(𝐾‘𝐵)𝐴 ∧ (𝐵(dist‘𝐺)𝑦) = (𝐵(dist‘𝐺)𝐶)))) → ⟨“𝐵𝐶𝑥”⟩(cgrG‘𝐺)⟨“𝐵𝑦𝐴”⟩)
37	1, 2, 14, 5, 11, 13	axtgcgrrflx 26823	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ ((𝑥(𝐾‘𝐵)𝐶 ∧ (𝐵(dist‘𝐺)𝑥) = (𝐵(dist‘𝐺)𝐴)) ∧ (𝑦(𝐾‘𝐵)𝐴 ∧ (𝐵(dist‘𝐺)𝑦) = (𝐵(dist‘𝐺)𝐶)))) → (𝐶(dist‘𝐺)𝑦) = (𝑦(dist‘𝐺)𝐶))
38		cgraid.2	. . . . . . . . 9 ⊢ (𝜑 → 𝐵 ≠ 𝐶)
39	38	ad3antrrr 727	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ ((𝑥(𝐾‘𝐵)𝐶 ∧ (𝐵(dist‘𝐺)𝑥) = (𝐵(dist‘𝐺)𝐴)) ∧ (𝑦(𝐾‘𝐵)𝐴 ∧ (𝐵(dist‘𝐺)𝑦) = (𝐵(dist‘𝐺)𝐶)))) → 𝐵 ≠ 𝐶)
40	1, 20, 14, 5, 9, 11, 12, 3, 9, 13, 2, 13, 7, 11, 25, 36, 18, 37, 39	tgfscgr 26929	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ ((𝑥(𝐾‘𝐵)𝐶 ∧ (𝐵(dist‘𝐺)𝑥) = (𝐵(dist‘𝐺)𝐴)) ∧ (𝑦(𝐾‘𝐵)𝐴 ∧ (𝐵(dist‘𝐺)𝑦) = (𝐵(dist‘𝐺)𝐶)))) → (𝑥(dist‘𝐺)𝑦) = (𝐴(dist‘𝐺)𝐶))
41	1, 2, 14, 5, 12, 13, 7, 11, 40	tgcgrcomlr 26841	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ ((𝑥(𝐾‘𝐵)𝐶 ∧ (𝐵(dist‘𝐺)𝑥) = (𝐵(dist‘𝐺)𝐴)) ∧ (𝑦(𝐾‘𝐵)𝐴 ∧ (𝐵(dist‘𝐺)𝑦) = (𝐵(dist‘𝐺)𝐶)))) → (𝑦(dist‘𝐺)𝑥) = (𝐶(dist‘𝐺)𝐴))
42	41	eqcomd 2744	. . . . 5 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ ((𝑥(𝐾‘𝐵)𝐶 ∧ (𝐵(dist‘𝐺)𝑥) = (𝐵(dist‘𝐺)𝐴)) ∧ (𝑦(𝐾‘𝐵)𝐴 ∧ (𝐵(dist‘𝐺)𝑦) = (𝐵(dist‘𝐺)𝐶)))) → (𝐶(dist‘𝐺)𝐴) = (𝑦(dist‘𝐺)𝑥))
43	1, 2, 3, 5, 7, 9, 11, 12, 9, 13, 17, 19, 42	trgcgr 26877	. . . 4 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ ((𝑥(𝐾‘𝐵)𝐶 ∧ (𝐵(dist‘𝐺)𝑥) = (𝐵(dist‘𝐺)𝐴)) ∧ (𝑦(𝐾‘𝐵)𝐴 ∧ (𝐵(dist‘𝐺)𝑦) = (𝐵(dist‘𝐺)𝐶)))) → ⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑥𝐵𝑦”⟩)
44	43, 22, 31	3jca 1127	. . 3 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ ((𝑥(𝐾‘𝐵)𝐶 ∧ (𝐵(dist‘𝐺)𝑥) = (𝐵(dist‘𝐺)𝐴)) ∧ (𝑦(𝐾‘𝐵)𝐴 ∧ (𝐵(dist‘𝐺)𝑦) = (𝐵(dist‘𝐺)𝐶)))) → (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑥𝐵𝑦”⟩ ∧ 𝑥(𝐾‘𝐵)𝐶 ∧ 𝑦(𝐾‘𝐵)𝐴))
45	38	necomd 2999	. . . . 5 ⊢ (𝜑 → 𝐶 ≠ 𝐵)
46		cgraid.1	. . . . . 6 ⊢ (𝜑 → 𝐴 ≠ 𝐵)
47	46	necomd 2999	. . . . 5 ⊢ (𝜑 → 𝐵 ≠ 𝐴)
48	1, 14, 21, 8, 8, 6, 4, 10, 2, 45, 47	hlcgrex 26977	. . . 4 ⊢ (𝜑 → ∃𝑥 ∈ 𝑃 (𝑥(𝐾‘𝐵)𝐶 ∧ (𝐵(dist‘𝐺)𝑥) = (𝐵(dist‘𝐺)𝐴)))
49	1, 14, 21, 8, 8, 10, 4, 6, 2, 46, 38	hlcgrex 26977	. . . 4 ⊢ (𝜑 → ∃𝑦 ∈ 𝑃 (𝑦(𝐾‘𝐵)𝐴 ∧ (𝐵(dist‘𝐺)𝑦) = (𝐵(dist‘𝐺)𝐶)))
50		reeanv 3294	. . . 4 ⊢ (∃𝑥 ∈ 𝑃 ∃𝑦 ∈ 𝑃 ((𝑥(𝐾‘𝐵)𝐶 ∧ (𝐵(dist‘𝐺)𝑥) = (𝐵(dist‘𝐺)𝐴)) ∧ (𝑦(𝐾‘𝐵)𝐴 ∧ (𝐵(dist‘𝐺)𝑦) = (𝐵(dist‘𝐺)𝐶))) ↔ (∃𝑥 ∈ 𝑃 (𝑥(𝐾‘𝐵)𝐶 ∧ (𝐵(dist‘𝐺)𝑥) = (𝐵(dist‘𝐺)𝐴)) ∧ ∃𝑦 ∈ 𝑃 (𝑦(𝐾‘𝐵)𝐴 ∧ (𝐵(dist‘𝐺)𝑦) = (𝐵(dist‘𝐺)𝐶))))
51	48, 49, 50	sylanbrc 583	. . 3 ⊢ (𝜑 → ∃𝑥 ∈ 𝑃 ∃𝑦 ∈ 𝑃 ((𝑥(𝐾‘𝐵)𝐶 ∧ (𝐵(dist‘𝐺)𝑥) = (𝐵(dist‘𝐺)𝐴)) ∧ (𝑦(𝐾‘𝐵)𝐴 ∧ (𝐵(dist‘𝐺)𝑦) = (𝐵(dist‘𝐺)𝐶))))
52	44, 51	reximddv2 3207	. 2 ⊢ (𝜑 → ∃𝑥 ∈ 𝑃 ∃𝑦 ∈ 𝑃 (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑥𝐵𝑦”⟩ ∧ 𝑥(𝐾‘𝐵)𝐶 ∧ 𝑦(𝐾‘𝐵)𝐴))
53	1, 14, 21, 4, 6, 8, 10, 10, 8, 6	iscgra 27170	. 2 ⊢ (𝜑 → (⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐶𝐵𝐴”⟩ ↔ ∃𝑥 ∈ 𝑃 ∃𝑦 ∈ 𝑃 (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑥𝐵𝑦”⟩ ∧ 𝑥(𝐾‘𝐵)𝐶 ∧ 𝑦(𝐾‘𝐵)𝐴)))
54	52, 53	mpbird 256	1 ⊢ (𝜑 → ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐶𝐵𝐴”⟩)

Syntax hints:   → wi 4   ∧ wa 396   ∨ wo 844   ∧ w3a 1086   = wceq 1539   ∈ wcel 2106   ≠ wne 2943  ∃wrex 3065   class class class wbr 5074  ‘cfv 6433  (class class class)co 7275  ⟨“cs3 14555  Basecbs 16912  distcds 16971  TarskiGcstrkg 26788  Itvcitv 26794  LineGclng 26795  cgrGccgrg 26871  ≤Gcleg 26943  hlGchlg 26961  cgrAccgra 27168

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588  ax-cnex 10927  ax-resscn 10928  ax-1cn 10929  ax-icn 10930  ax-addcl 10931  ax-addrcl 10932  ax-mulcl 10933  ax-mulrcl 10934  ax-mulcom 10935  ax-addass 10936  ax-mulass 10937  ax-distr 10938  ax-i2m1 10939  ax-1ne0 10940  ax-1rid 10941  ax-rnegex 10942  ax-rrecex 10943  ax-cnre 10944  ax-pre-lttri 10945  ax-pre-lttrn 10946  ax-pre-ltadd 10947  ax-pre-mulgt0 10948

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-tp 4566  df-op 4568  df-uni 4840  df-int 4880  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-riota 7232  df-ov 7278  df-oprab 7279  df-mpo 7280  df-om 7713  df-1st 7831  df-2nd 7832  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-rdg 8241  df-1o 8297  df-oadd 8301  df-er 8498  df-map 8617  df-pm 8618  df-en 8734  df-dom 8735  df-sdom 8736  df-fin 8737  df-dju 9659  df-card 9697  df-pnf 11011  df-mnf 11012  df-xr 11013  df-ltxr 11014  df-le 11015  df-sub 11207  df-neg 11208  df-nn 11974  df-2 12036  df-3 12037  df-n0 12234  df-xnn0 12306  df-z 12320  df-uz 12583  df-fz 13240  df-fzo 13383  df-hash 14045  df-word 14218  df-concat 14274  df-s1 14301  df-s2 14561  df-s3 14562  df-trkgc 26809  df-trkgb 26810  df-trkgcb 26811  df-trkg 26814  df-cgrg 26872  df-leg 26944  df-hlg 26962  df-cgra 27169

This theorem is referenced by:  cgraswaplr  27186  oacgr  27193  tgasa1  27219