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Theorem cgraswap 28071

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 2733	. . . . 5 ⊢ (dist‘𝐺) = (dist‘𝐺)
3		eqid 2733	. . . . 5 ⊢ (cgrG‘𝐺) = (cgrG‘𝐺)
4		cgraid.g	. . . . . 6 ⊢ (𝜑 → 𝐺 ∈ TarskiG)
5	4	ad3antrrr 729	. . . . 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 729	. . . . 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 729	. . . . 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 729	. . . . 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 775	. . . . 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 768	. . . . 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 779	. . . . . . 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 27731	. . . . . 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 2739	. . . . 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 781	. . . . . 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 2739	. . . . 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 2733	. . . . . . . 8 ⊢ (LineG‘𝐺) = (LineG‘𝐺)
21		cgraid.k	. . . . . . . . . . 11 ⊢ 𝐾 = (hlG‘𝐺)
22		simprll 778	. . . . . . . . . . 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 27858	. . . . . . . . . 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 872	. . . . . . . . 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 27810	. . . . . . . 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 2733	. . . . . . . . . 10 ⊢ (≤G‘𝐺) = (≤G‘𝐺)
27	1, 14, 21, 12, 11, 9, 5	ishlg 27853	. . . . . . . . . . . . 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 1145	. . . . . . . . . . 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 870	. . . . . . . . . 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 780	. . . . . . . . . . . 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 27853	. . . . . . . . . . . 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 1145	. . . . . . . . . 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 27846	. . . . . . . . 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 27767	. . . . . . . 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 27713	. . . . . . . 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 729	. . . . . . . 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 27819	. . . . . . 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 27731	. . . . . 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 2739	. . . . 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 27767	. . . 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 1129	. . 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 2997	. . . . 5 ⊢ (𝜑 → 𝐶 ≠ 𝐵)
46		cgraid.1	. . . . . 6 ⊢ (𝜑 → 𝐴 ≠ 𝐵)
47	46	necomd 2997	. . . . 5 ⊢ (𝜑 → 𝐵 ≠ 𝐴)
48	1, 14, 21, 8, 8, 6, 4, 10, 2, 45, 47	hlcgrex 27867	. . . 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 27867	. . . 4 ⊢ (𝜑 → ∃𝑦 ∈ 𝑃 (𝑦(𝐾‘𝐵)𝐴 ∧ (𝐵(dist‘𝐺)𝑦) = (𝐵(dist‘𝐺)𝐶)))
50		reeanv 3227	. . . 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 584	. . 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 3213	. 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 28060	. 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 257	1 ⊢ (𝜑 → ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐶𝐵𝐴”⟩)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 397 ∨ wo 846 ∧ w3a 1088 = wceq 1542 ∈ wcel 2107 ≠ wne 2941 ∃wrex 3071 class class class wbr 5149 ‘cfv 6544 (class class class)co 7409 ⟨“cs3 14793 Basecbs 17144 distcds 17206 TarskiGcstrkg 27678 Itvcitv 27684 LineGclng 27685 cgrGccgrg 27761 ≤Gcleg 27833 hlGchlg 27851 cgrAccgra 28058

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 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-cnex 11166 ax-resscn 11167 ax-1cn 11168 ax-icn 11169 ax-addcl 11170 ax-addrcl 11171 ax-mulcl 11172 ax-mulrcl 11173 ax-mulcom 11174 ax-addass 11175 ax-mulass 11176 ax-distr 11177 ax-i2m1 11178 ax-1ne0 11179 ax-1rid 11180 ax-rnegex 11181 ax-rrecex 11182 ax-cnre 11183 ax-pre-lttri 11184 ax-pre-lttrn 11185 ax-pre-ltadd 11186 ax-pre-mulgt0 11187

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-tp 4634 df-op 4636 df-uni 4910 df-int 4952 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-riota 7365 df-ov 7412 df-oprab 7413 df-mpo 7414 df-om 7856 df-1st 7975 df-2nd 7976 df-frecs 8266 df-wrecs 8297 df-recs 8371 df-rdg 8410 df-1o 8466 df-oadd 8470 df-er 8703 df-map 8822 df-pm 8823 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-dju 9896 df-card 9934 df-pnf 11250 df-mnf 11251 df-xr 11252 df-ltxr 11253 df-le 11254 df-sub 11446 df-neg 11447 df-nn 12213 df-2 12275 df-3 12276 df-n0 12473 df-xnn0 12545 df-z 12559 df-uz 12823 df-fz 13485 df-fzo 13628 df-hash 14291 df-word 14465 df-concat 14521 df-s1 14546 df-s2 14799 df-s3 14800 df-trkgc 27699 df-trkgb 27700 df-trkgcb 27701 df-trkg 27704 df-cgrg 27762 df-leg 27834 df-hlg 27852 df-cgra 28059

This theorem is referenced by: cgraswaplr 28076 oacgr 28083 tgasa1 28109

W3C validator