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Theorem ragcgra 29400

Description: Right angles are congruent with each other. Theorem 11.16 of [Schwabhauser] p. 98. (Contributed by Thierry Arnoux, 5-Jul-2026.)

**Hypotheses**
Ref	Expression
ragcgra.p	⊢ 𝑃 = (Base‘𝐺)
ragcgra.g	⊢ (𝜑 → 𝐺 ∈ TarskiG)
ragcgra.x	⊢ (𝜑 → 𝑋 ∈ 𝑃)
ragcgra.y	⊢ (𝜑 → 𝑌 ∈ 𝑃)
ragcgra.z	⊢ (𝜑 → 𝑍 ∈ 𝑃)
ragcgra.a	⊢ (𝜑 → 𝐴 ∈ 𝑃)
ragcgra.b	⊢ (𝜑 → 𝐵 ∈ 𝑃)
ragcgra.c	⊢ (𝜑 → 𝐶 ∈ 𝑃)
ragcgra.1	⊢ (𝜑 → ⟨“𝑋𝑌𝑍”⟩ ∈ (∟G‘𝐺))
ragcgra.2	⊢ (𝜑 → ⟨“𝐴𝐵𝐶”⟩ ∈ (∟G‘𝐺))
ragcgra.3	⊢ (𝜑 → 𝐴 ≠ 𝐵)
ragcgra.4	⊢ (𝜑 → 𝐵 ≠ 𝐶)
ragcgra.5	⊢ (𝜑 → 𝑋 ≠ 𝑌)
ragcgra.6	⊢ (𝜑 → 𝑌 ≠ 𝑍)

**Assertion**
Ref	Expression
ragcgra	⊢ (𝜑 → ⟨“𝑋𝑌𝑍”⟩(cgrA‘𝐺)⟨“𝐴𝐵𝐶”⟩)

Proof of Theorem ragcgra Dummy variables 𝑎 𝑐 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ragcgra.p	. . . . . 6 ⊢ 𝑃 = (Base‘𝐺)
2		eqid 2763	. . . . . 6 ⊢ (Itv‘𝐺) = (Itv‘𝐺)
3		eqid 2763	. . . . . 6 ⊢ (hlG‘𝐺) = (hlG‘𝐺)
4		ragcgra.g	. . . . . . 7 ⊢ (𝜑 → 𝐺 ∈ TarskiG)
5	4	ad6antr 749	. . . . . 6 ⊢ (((((((𝜑 ∧ 𝑎 ∈ 𝑃) ∧ 𝑎((hlG‘𝐺)‘𝐵)𝐴) ∧ (𝐵(dist‘𝐺)𝑎) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑐 ∈ 𝑃) ∧ 𝑐((hlG‘𝐺)‘𝐵)𝐶) ∧ (𝐵(dist‘𝐺)𝑐) = (𝑌(dist‘𝐺)𝑍)) → 𝐺 ∈ TarskiG)
6		ragcgra.x	. . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ 𝑃)
7	6	ad6antr 749	. . . . . 6 ⊢ (((((((𝜑 ∧ 𝑎 ∈ 𝑃) ∧ 𝑎((hlG‘𝐺)‘𝐵)𝐴) ∧ (𝐵(dist‘𝐺)𝑎) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑐 ∈ 𝑃) ∧ 𝑐((hlG‘𝐺)‘𝐵)𝐶) ∧ (𝐵(dist‘𝐺)𝑐) = (𝑌(dist‘𝐺)𝑍)) → 𝑋 ∈ 𝑃)
8		ragcgra.y	. . . . . . 7 ⊢ (𝜑 → 𝑌 ∈ 𝑃)
9	8	ad6antr 749	. . . . . 6 ⊢ (((((((𝜑 ∧ 𝑎 ∈ 𝑃) ∧ 𝑎((hlG‘𝐺)‘𝐵)𝐴) ∧ (𝐵(dist‘𝐺)𝑎) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑐 ∈ 𝑃) ∧ 𝑐((hlG‘𝐺)‘𝐵)𝐶) ∧ (𝐵(dist‘𝐺)𝑐) = (𝑌(dist‘𝐺)𝑍)) → 𝑌 ∈ 𝑃)
10		ragcgra.z	. . . . . . 7 ⊢ (𝜑 → 𝑍 ∈ 𝑃)
11	10	ad6antr 749	. . . . . 6 ⊢ (((((((𝜑 ∧ 𝑎 ∈ 𝑃) ∧ 𝑎((hlG‘𝐺)‘𝐵)𝐴) ∧ (𝐵(dist‘𝐺)𝑎) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑐 ∈ 𝑃) ∧ 𝑐((hlG‘𝐺)‘𝐵)𝐶) ∧ (𝐵(dist‘𝐺)𝑐) = (𝑌(dist‘𝐺)𝑍)) → 𝑍 ∈ 𝑃)
12		ragcgra.a	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ 𝑃)
13	12	ad6antr 749	. . . . . 6 ⊢ (((((((𝜑 ∧ 𝑎 ∈ 𝑃) ∧ 𝑎((hlG‘𝐺)‘𝐵)𝐴) ∧ (𝐵(dist‘𝐺)𝑎) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑐 ∈ 𝑃) ∧ 𝑐((hlG‘𝐺)‘𝐵)𝐶) ∧ (𝐵(dist‘𝐺)𝑐) = (𝑌(dist‘𝐺)𝑍)) → 𝐴 ∈ 𝑃)
14		ragcgra.b	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ 𝑃)
15	14	ad6antr 749	. . . . . 6 ⊢ (((((((𝜑 ∧ 𝑎 ∈ 𝑃) ∧ 𝑎((hlG‘𝐺)‘𝐵)𝐴) ∧ (𝐵(dist‘𝐺)𝑎) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑐 ∈ 𝑃) ∧ 𝑐((hlG‘𝐺)‘𝐵)𝐶) ∧ (𝐵(dist‘𝐺)𝑐) = (𝑌(dist‘𝐺)𝑍)) → 𝐵 ∈ 𝑃)
16		ragcgra.c	. . . . . . 7 ⊢ (𝜑 → 𝐶 ∈ 𝑃)
17	16	ad6antr 749	. . . . . 6 ⊢ (((((((𝜑 ∧ 𝑎 ∈ 𝑃) ∧ 𝑎((hlG‘𝐺)‘𝐵)𝐴) ∧ (𝐵(dist‘𝐺)𝑎) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑐 ∈ 𝑃) ∧ 𝑐((hlG‘𝐺)‘𝐵)𝐶) ∧ (𝐵(dist‘𝐺)𝑐) = (𝑌(dist‘𝐺)𝑍)) → 𝐶 ∈ 𝑃)
18		simp-6r 800	. . . . . 6 ⊢ (((((((𝜑 ∧ 𝑎 ∈ 𝑃) ∧ 𝑎((hlG‘𝐺)‘𝐵)𝐴) ∧ (𝐵(dist‘𝐺)𝑎) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑐 ∈ 𝑃) ∧ 𝑐((hlG‘𝐺)‘𝐵)𝐶) ∧ (𝐵(dist‘𝐺)𝑐) = (𝑌(dist‘𝐺)𝑍)) → 𝑎 ∈ 𝑃)
19		simpllr 788	. . . . . 6 ⊢ (((((((𝜑 ∧ 𝑎 ∈ 𝑃) ∧ 𝑎((hlG‘𝐺)‘𝐵)𝐴) ∧ (𝐵(dist‘𝐺)𝑎) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑐 ∈ 𝑃) ∧ 𝑐((hlG‘𝐺)‘𝐵)𝐶) ∧ (𝐵(dist‘𝐺)𝑐) = (𝑌(dist‘𝐺)𝑍)) → 𝑐 ∈ 𝑃)
20		eqid 2763	. . . . . . 7 ⊢ (dist‘𝐺) = (dist‘𝐺)
21		eqid 2763	. . . . . . 7 ⊢ (cgrG‘𝐺) = (cgrG‘𝐺)
22		simp-4r 796	. . . . . . . . 9 ⊢ (((((((𝜑 ∧ 𝑎 ∈ 𝑃) ∧ 𝑎((hlG‘𝐺)‘𝐵)𝐴) ∧ (𝐵(dist‘𝐺)𝑎) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑐 ∈ 𝑃) ∧ 𝑐((hlG‘𝐺)‘𝐵)𝐶) ∧ (𝐵(dist‘𝐺)𝑐) = (𝑌(dist‘𝐺)𝑍)) → (𝐵(dist‘𝐺)𝑎) = (𝑌(dist‘𝐺)𝑋))
23	22	eqcomd 2769	. . . . . . . 8 ⊢ (((((((𝜑 ∧ 𝑎 ∈ 𝑃) ∧ 𝑎((hlG‘𝐺)‘𝐵)𝐴) ∧ (𝐵(dist‘𝐺)𝑎) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑐 ∈ 𝑃) ∧ 𝑐((hlG‘𝐺)‘𝐵)𝐶) ∧ (𝐵(dist‘𝐺)𝑐) = (𝑌(dist‘𝐺)𝑍)) → (𝑌(dist‘𝐺)𝑋) = (𝐵(dist‘𝐺)𝑎))
24	1, 20, 2, 5, 9, 7, 15, 18, 23	tgcgrcomlr 28999	. . . . . . 7 ⊢ (((((((𝜑 ∧ 𝑎 ∈ 𝑃) ∧ 𝑎((hlG‘𝐺)‘𝐵)𝐴) ∧ (𝐵(dist‘𝐺)𝑎) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑐 ∈ 𝑃) ∧ 𝑐((hlG‘𝐺)‘𝐵)𝐶) ∧ (𝐵(dist‘𝐺)𝑐) = (𝑌(dist‘𝐺)𝑍)) → (𝑋(dist‘𝐺)𝑌) = (𝑎(dist‘𝐺)𝐵))
25		simpr 490	. . . . . . . 8 ⊢ (((((((𝜑 ∧ 𝑎 ∈ 𝑃) ∧ 𝑎((hlG‘𝐺)‘𝐵)𝐴) ∧ (𝐵(dist‘𝐺)𝑎) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑐 ∈ 𝑃) ∧ 𝑐((hlG‘𝐺)‘𝐵)𝐶) ∧ (𝐵(dist‘𝐺)𝑐) = (𝑌(dist‘𝐺)𝑍)) → (𝐵(dist‘𝐺)𝑐) = (𝑌(dist‘𝐺)𝑍))
26	25	eqcomd 2769	. . . . . . 7 ⊢ (((((((𝜑 ∧ 𝑎 ∈ 𝑃) ∧ 𝑎((hlG‘𝐺)‘𝐵)𝐴) ∧ (𝐵(dist‘𝐺)𝑎) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑐 ∈ 𝑃) ∧ 𝑐((hlG‘𝐺)‘𝐵)𝐶) ∧ (𝐵(dist‘𝐺)𝑐) = (𝑌(dist‘𝐺)𝑍)) → (𝑌(dist‘𝐺)𝑍) = (𝐵(dist‘𝐺)𝑐))
27		eqid 2763	. . . . . . . . . . 11 ⊢ (LineG‘𝐺) = (LineG‘𝐺)
28		eqid 2763	. . . . . . . . . . . 12 ⊢ (pInvG‘𝐺) = (pInvG‘𝐺)
29		ragcgra.2	. . . . . . . . . . . 12 ⊢ (𝜑 → ⟨“𝐴𝐵𝐶”⟩ ∈ (∟G‘𝐺))
30		ragcgra.3	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐴 ≠ 𝐵)
31		ragcgra.4	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐵 ≠ 𝐶)
32	31	necomd 3014	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐶 ≠ 𝐵)
33	1, 20, 2, 27, 28, 4, 12, 14, 16, 29, 30, 32	ragncol 29241	. . . . . . . . . . 11 ⊢ (𝜑 → ¬ (𝐶 ∈ (𝐴(LineG‘𝐺)𝐵) ∨ 𝐴 = 𝐵))
34	1, 27, 2, 4, 12, 14, 16, 33	ncoltgdim2 29085	. . . . . . . . . 10 ⊢ (𝜑 → 𝐺Dim_TarskiG≥2)
35	34	ad6antr 749	. . . . . . . . 9 ⊢ (((((((𝜑 ∧ 𝑎 ∈ 𝑃) ∧ 𝑎((hlG‘𝐺)‘𝐵)𝐴) ∧ (𝐵(dist‘𝐺)𝑎) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑐 ∈ 𝑃) ∧ 𝑐((hlG‘𝐺)‘𝐵)𝐶) ∧ (𝐵(dist‘𝐺)𝑐) = (𝑌(dist‘𝐺)𝑍)) → 𝐺Dim_TarskiG≥2)
36		ragcgra.1	. . . . . . . . . 10 ⊢ (𝜑 → ⟨“𝑋𝑌𝑍”⟩ ∈ (∟G‘𝐺))
37	36	ad6antr 749	. . . . . . . . 9 ⊢ (((((((𝜑 ∧ 𝑎 ∈ 𝑃) ∧ 𝑎((hlG‘𝐺)‘𝐵)𝐴) ∧ (𝐵(dist‘𝐺)𝑎) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑐 ∈ 𝑃) ∧ 𝑐((hlG‘𝐺)‘𝐵)𝐶) ∧ (𝐵(dist‘𝐺)𝑐) = (𝑌(dist‘𝐺)𝑍)) → ⟨“𝑋𝑌𝑍”⟩ ∈ (∟G‘𝐺))
38	29	ad6antr 749	. . . . . . . . . . . . 13 ⊢ (((((((𝜑 ∧ 𝑎 ∈ 𝑃) ∧ 𝑎((hlG‘𝐺)‘𝐵)𝐴) ∧ (𝐵(dist‘𝐺)𝑎) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑐 ∈ 𝑃) ∧ 𝑐((hlG‘𝐺)‘𝐵)𝐶) ∧ (𝐵(dist‘𝐺)𝑐) = (𝑌(dist‘𝐺)𝑍)) → ⟨“𝐴𝐵𝐶”⟩ ∈ (∟G‘𝐺))
39	1, 20, 2, 27, 28, 5, 13, 15, 17, 38	ragcom 29230	. . . . . . . . . . . 12 ⊢ (((((((𝜑 ∧ 𝑎 ∈ 𝑃) ∧ 𝑎((hlG‘𝐺)‘𝐵)𝐴) ∧ (𝐵(dist‘𝐺)𝑎) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑐 ∈ 𝑃) ∧ 𝑐((hlG‘𝐺)‘𝐵)𝐶) ∧ (𝐵(dist‘𝐺)𝑐) = (𝑌(dist‘𝐺)𝑍)) → ⟨“𝐶𝐵𝐴”⟩ ∈ (∟G‘𝐺))
40	32	ad6antr 749	. . . . . . . . . . . 12 ⊢ (((((((𝜑 ∧ 𝑎 ∈ 𝑃) ∧ 𝑎((hlG‘𝐺)‘𝐵)𝐴) ∧ (𝐵(dist‘𝐺)𝑎) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑐 ∈ 𝑃) ∧ 𝑐((hlG‘𝐺)‘𝐵)𝐶) ∧ (𝐵(dist‘𝐺)𝑐) = (𝑌(dist‘𝐺)𝑍)) → 𝐶 ≠ 𝐵)
41		ragcgra.6	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑌 ≠ 𝑍)
42	41	ad6antr 749	. . . . . . . . . . . . . . 15 ⊢ (((((((𝜑 ∧ 𝑎 ∈ 𝑃) ∧ 𝑎((hlG‘𝐺)‘𝐵)𝐴) ∧ (𝐵(dist‘𝐺)𝑎) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑐 ∈ 𝑃) ∧ 𝑐((hlG‘𝐺)‘𝐵)𝐶) ∧ (𝐵(dist‘𝐺)𝑐) = (𝑌(dist‘𝐺)𝑍)) → 𝑌 ≠ 𝑍)
43	1, 20, 2, 5, 9, 11, 15, 19, 26, 42	tgcgrneq 29002	. . . . . . . . . . . . . 14 ⊢ (((((((𝜑 ∧ 𝑎 ∈ 𝑃) ∧ 𝑎((hlG‘𝐺)‘𝐵)𝐴) ∧ (𝐵(dist‘𝐺)𝑎) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑐 ∈ 𝑃) ∧ 𝑐((hlG‘𝐺)‘𝐵)𝐶) ∧ (𝐵(dist‘𝐺)𝑐) = (𝑌(dist‘𝐺)𝑍)) → 𝐵 ≠ 𝑐)
44		simplr 781	. . . . . . . . . . . . . . 15 ⊢ (((((((𝜑 ∧ 𝑎 ∈ 𝑃) ∧ 𝑎((hlG‘𝐺)‘𝐵)𝐴) ∧ (𝐵(dist‘𝐺)𝑎) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑐 ∈ 𝑃) ∧ 𝑐((hlG‘𝐺)‘𝐵)𝐶) ∧ (𝐵(dist‘𝐺)𝑐) = (𝑌(dist‘𝐺)𝑍)) → 𝑐((hlG‘𝐺)‘𝐵)𝐶)
45	1, 2, 3, 19, 17, 15, 5, 27, 44	hlln 29130	. . . . . . . . . . . . . 14 ⊢ (((((((𝜑 ∧ 𝑎 ∈ 𝑃) ∧ 𝑎((hlG‘𝐺)‘𝐵)𝐴) ∧ (𝐵(dist‘𝐺)𝑎) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑐 ∈ 𝑃) ∧ 𝑐((hlG‘𝐺)‘𝐵)𝐶) ∧ (𝐵(dist‘𝐺)𝑐) = (𝑌(dist‘𝐺)𝑍)) → 𝑐 ∈ (𝐶(LineG‘𝐺)𝐵))
46	1, 2, 27, 5, 15, 19, 17, 43, 45, 40	lnrot1 29148	. . . . . . . . . . . . 13 ⊢ (((((((𝜑 ∧ 𝑎 ∈ 𝑃) ∧ 𝑎((hlG‘𝐺)‘𝐵)𝐴) ∧ (𝐵(dist‘𝐺)𝑎) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑐 ∈ 𝑃) ∧ 𝑐((hlG‘𝐺)‘𝐵)𝐶) ∧ (𝐵(dist‘𝐺)𝑐) = (𝑌(dist‘𝐺)𝑍)) → 𝐶 ∈ (𝐵(LineG‘𝐺)𝑐))
47	46	orcd 887	. . . . . . . . . . . 12 ⊢ (((((((𝜑 ∧ 𝑎 ∈ 𝑃) ∧ 𝑎((hlG‘𝐺)‘𝐵)𝐴) ∧ (𝐵(dist‘𝐺)𝑎) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑐 ∈ 𝑃) ∧ 𝑐((hlG‘𝐺)‘𝐵)𝐶) ∧ (𝐵(dist‘𝐺)𝑐) = (𝑌(dist‘𝐺)𝑍)) → (𝐶 ∈ (𝐵(LineG‘𝐺)𝑐) ∨ 𝐵 = 𝑐))
48	1, 20, 2, 27, 28, 5, 17, 15, 13, 19, 39, 40, 47	ragcol 29231	. . . . . . . . . . 11 ⊢ (((((((𝜑 ∧ 𝑎 ∈ 𝑃) ∧ 𝑎((hlG‘𝐺)‘𝐵)𝐴) ∧ (𝐵(dist‘𝐺)𝑎) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑐 ∈ 𝑃) ∧ 𝑐((hlG‘𝐺)‘𝐵)𝐶) ∧ (𝐵(dist‘𝐺)𝑐) = (𝑌(dist‘𝐺)𝑍)) → ⟨“𝑐𝐵𝐴”⟩ ∈ (∟G‘𝐺))
49	1, 20, 2, 27, 28, 5, 19, 15, 13, 48	ragcom 29230	. . . . . . . . . 10 ⊢ (((((((𝜑 ∧ 𝑎 ∈ 𝑃) ∧ 𝑎((hlG‘𝐺)‘𝐵)𝐴) ∧ (𝐵(dist‘𝐺)𝑎) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑐 ∈ 𝑃) ∧ 𝑐((hlG‘𝐺)‘𝐵)𝐶) ∧ (𝐵(dist‘𝐺)𝑐) = (𝑌(dist‘𝐺)𝑍)) → ⟨“𝐴𝐵𝑐”⟩ ∈ (∟G‘𝐺))
50	30	ad6antr 749	. . . . . . . . . 10 ⊢ (((((((𝜑 ∧ 𝑎 ∈ 𝑃) ∧ 𝑎((hlG‘𝐺)‘𝐵)𝐴) ∧ (𝐵(dist‘𝐺)𝑎) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑐 ∈ 𝑃) ∧ 𝑐((hlG‘𝐺)‘𝐵)𝐶) ∧ (𝐵(dist‘𝐺)𝑐) = (𝑌(dist‘𝐺)𝑍)) → 𝐴 ≠ 𝐵)
51		ragcgra.5	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑋 ≠ 𝑌)
52	51	necomd 3014	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑌 ≠ 𝑋)
53	52	ad6antr 749	. . . . . . . . . . . . 13 ⊢ (((((((𝜑 ∧ 𝑎 ∈ 𝑃) ∧ 𝑎((hlG‘𝐺)‘𝐵)𝐴) ∧ (𝐵(dist‘𝐺)𝑎) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑐 ∈ 𝑃) ∧ 𝑐((hlG‘𝐺)‘𝐵)𝐶) ∧ (𝐵(dist‘𝐺)𝑐) = (𝑌(dist‘𝐺)𝑍)) → 𝑌 ≠ 𝑋)
54	1, 20, 2, 5, 9, 7, 15, 18, 23, 53	tgcgrneq 29002	. . . . . . . . . . . 12 ⊢ (((((((𝜑 ∧ 𝑎 ∈ 𝑃) ∧ 𝑎((hlG‘𝐺)‘𝐵)𝐴) ∧ (𝐵(dist‘𝐺)𝑎) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑐 ∈ 𝑃) ∧ 𝑐((hlG‘𝐺)‘𝐵)𝐶) ∧ (𝐵(dist‘𝐺)𝑐) = (𝑌(dist‘𝐺)𝑍)) → 𝐵 ≠ 𝑎)
55		simp-5r 798	. . . . . . . . . . . . 13 ⊢ (((((((𝜑 ∧ 𝑎 ∈ 𝑃) ∧ 𝑎((hlG‘𝐺)‘𝐵)𝐴) ∧ (𝐵(dist‘𝐺)𝑎) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑐 ∈ 𝑃) ∧ 𝑐((hlG‘𝐺)‘𝐵)𝐶) ∧ (𝐵(dist‘𝐺)𝑐) = (𝑌(dist‘𝐺)𝑍)) → 𝑎((hlG‘𝐺)‘𝐵)𝐴)
56	1, 2, 3, 18, 13, 15, 5, 27, 55	hlln 29130	. . . . . . . . . . . 12 ⊢ (((((((𝜑 ∧ 𝑎 ∈ 𝑃) ∧ 𝑎((hlG‘𝐺)‘𝐵)𝐴) ∧ (𝐵(dist‘𝐺)𝑎) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑐 ∈ 𝑃) ∧ 𝑐((hlG‘𝐺)‘𝐵)𝐶) ∧ (𝐵(dist‘𝐺)𝑐) = (𝑌(dist‘𝐺)𝑍)) → 𝑎 ∈ (𝐴(LineG‘𝐺)𝐵))
57	1, 2, 27, 5, 15, 18, 13, 54, 56, 50	lnrot1 29148	. . . . . . . . . . 11 ⊢ (((((((𝜑 ∧ 𝑎 ∈ 𝑃) ∧ 𝑎((hlG‘𝐺)‘𝐵)𝐴) ∧ (𝐵(dist‘𝐺)𝑎) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑐 ∈ 𝑃) ∧ 𝑐((hlG‘𝐺)‘𝐵)𝐶) ∧ (𝐵(dist‘𝐺)𝑐) = (𝑌(dist‘𝐺)𝑍)) → 𝐴 ∈ (𝐵(LineG‘𝐺)𝑎))
58	57	orcd 887	. . . . . . . . . 10 ⊢ (((((((𝜑 ∧ 𝑎 ∈ 𝑃) ∧ 𝑎((hlG‘𝐺)‘𝐵)𝐴) ∧ (𝐵(dist‘𝐺)𝑎) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑐 ∈ 𝑃) ∧ 𝑐((hlG‘𝐺)‘𝐵)𝐶) ∧ (𝐵(dist‘𝐺)𝑐) = (𝑌(dist‘𝐺)𝑍)) → (𝐴 ∈ (𝐵(LineG‘𝐺)𝑎) ∨ 𝐵 = 𝑎))
59	1, 20, 2, 27, 28, 5, 13, 15, 19, 18, 49, 50, 58	ragcol 29231	. . . . . . . . 9 ⊢ (((((((𝜑 ∧ 𝑎 ∈ 𝑃) ∧ 𝑎((hlG‘𝐺)‘𝐵)𝐴) ∧ (𝐵(dist‘𝐺)𝑎) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑐 ∈ 𝑃) ∧ 𝑐((hlG‘𝐺)‘𝐵)𝐶) ∧ (𝐵(dist‘𝐺)𝑐) = (𝑌(dist‘𝐺)𝑍)) → ⟨“𝑎𝐵𝑐”⟩ ∈ (∟G‘𝐺))
60	1, 20, 2, 5, 35, 7, 9, 11, 18, 15, 19, 37, 59, 24, 26	hypcgr 29364	. . . . . . . 8 ⊢ (((((((𝜑 ∧ 𝑎 ∈ 𝑃) ∧ 𝑎((hlG‘𝐺)‘𝐵)𝐴) ∧ (𝐵(dist‘𝐺)𝑎) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑐 ∈ 𝑃) ∧ 𝑐((hlG‘𝐺)‘𝐵)𝐶) ∧ (𝐵(dist‘𝐺)𝑐) = (𝑌(dist‘𝐺)𝑍)) → (𝑋(dist‘𝐺)𝑍) = (𝑎(dist‘𝐺)𝑐))
61	1, 20, 2, 5, 7, 11, 18, 19, 60	tgcgrcomlr 28999	. . . . . . 7 ⊢ (((((((𝜑 ∧ 𝑎 ∈ 𝑃) ∧ 𝑎((hlG‘𝐺)‘𝐵)𝐴) ∧ (𝐵(dist‘𝐺)𝑎) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑐 ∈ 𝑃) ∧ 𝑐((hlG‘𝐺)‘𝐵)𝐶) ∧ (𝐵(dist‘𝐺)𝑐) = (𝑌(dist‘𝐺)𝑍)) → (𝑍(dist‘𝐺)𝑋) = (𝑐(dist‘𝐺)𝑎))
62	1, 20, 21, 5, 7, 9, 11, 18, 15, 19, 24, 26, 61	trgcgr 29036	. . . . . 6 ⊢ (((((((𝜑 ∧ 𝑎 ∈ 𝑃) ∧ 𝑎((hlG‘𝐺)‘𝐵)𝐴) ∧ (𝐵(dist‘𝐺)𝑎) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑐 ∈ 𝑃) ∧ 𝑐((hlG‘𝐺)‘𝐵)𝐶) ∧ (𝐵(dist‘𝐺)𝑐) = (𝑌(dist‘𝐺)𝑍)) → ⟨“𝑋𝑌𝑍”⟩(cgrG‘𝐺)⟨“𝑎𝐵𝑐”⟩)
63	1, 2, 3, 5, 7, 9, 11, 13, 15, 17, 18, 19, 62, 55, 44	iscgrad 29375	. . . . 5 ⊢ (((((((𝜑 ∧ 𝑎 ∈ 𝑃) ∧ 𝑎((hlG‘𝐺)‘𝐵)𝐴) ∧ (𝐵(dist‘𝐺)𝑎) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑐 ∈ 𝑃) ∧ 𝑐((hlG‘𝐺)‘𝐵)𝐶) ∧ (𝐵(dist‘𝐺)𝑐) = (𝑌(dist‘𝐺)𝑍)) → ⟨“𝑋𝑌𝑍”⟩(cgrA‘𝐺)⟨“𝐴𝐵𝐶”⟩)
64	63	anasss 472	. . . 4 ⊢ ((((((𝜑 ∧ 𝑎 ∈ 𝑃) ∧ 𝑎((hlG‘𝐺)‘𝐵)𝐴) ∧ (𝐵(dist‘𝐺)𝑎) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑐 ∈ 𝑃) ∧ (𝑐((hlG‘𝐺)‘𝐵)𝐶 ∧ (𝐵(dist‘𝐺)𝑐) = (𝑌(dist‘𝐺)𝑍))) → ⟨“𝑋𝑌𝑍”⟩(cgrA‘𝐺)⟨“𝐴𝐵𝐶”⟩)
65	1, 2, 3, 14, 8, 10, 4, 16, 20, 32, 41	hlcgrex 29139	. . . . 5 ⊢ (𝜑 → ∃𝑐 ∈ 𝑃 (𝑐((hlG‘𝐺)‘𝐵)𝐶 ∧ (𝐵(dist‘𝐺)𝑐) = (𝑌(dist‘𝐺)𝑍)))
66	65	ad3antrrr 743	. . . 4 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝑃) ∧ 𝑎((hlG‘𝐺)‘𝐵)𝐴) ∧ (𝐵(dist‘𝐺)𝑎) = (𝑌(dist‘𝐺)𝑋)) → ∃𝑐 ∈ 𝑃 (𝑐((hlG‘𝐺)‘𝐵)𝐶 ∧ (𝐵(dist‘𝐺)𝑐) = (𝑌(dist‘𝐺)𝑍)))
67	64, 66	r19.29a 3174	. . 3 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝑃) ∧ 𝑎((hlG‘𝐺)‘𝐵)𝐴) ∧ (𝐵(dist‘𝐺)𝑎) = (𝑌(dist‘𝐺)𝑋)) → ⟨“𝑋𝑌𝑍”⟩(cgrA‘𝐺)⟨“𝐴𝐵𝐶”⟩)
68	67	anasss 472	. 2 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝑃) ∧ (𝑎((hlG‘𝐺)‘𝐵)𝐴 ∧ (𝐵(dist‘𝐺)𝑎) = (𝑌(dist‘𝐺)𝑋))) → ⟨“𝑋𝑌𝑍”⟩(cgrA‘𝐺)⟨“𝐴𝐵𝐶”⟩)
69	1, 2, 3, 14, 8, 6, 4, 12, 20, 30, 52	hlcgrex 29139	. 2 ⊢ (𝜑 → ∃𝑎 ∈ 𝑃 (𝑎((hlG‘𝐺)‘𝐵)𝐴 ∧ (𝐵(dist‘𝐺)𝑎) = (𝑌(dist‘𝐺)𝑋)))
70	68, 69	r19.29a 3174	1 ⊢ (𝜑 → ⟨“𝑋𝑌𝑍”⟩(cgrA‘𝐺)⟨“𝐴𝐵𝐶”⟩)

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 401 = wceq 1570 ∈ wcel 2147 ≠ wne 2959 ∃wrex 3090 class class class wbr 5108 ‘cfv 6566 (class class class)co 7451 2c2 12437 ⟨“cs3 15035 Basecbs 17431 distcds 17481 TarskiGcstrkg 28946 Dim_TarskiG≥cstrkgld 28950 Itvcitv 28952 LineGclng 28953 cgrGccgrg 29030 hlGchlg 29120 pInvGcmir 29181 ∟Gcrag 29225 cgrAccgra 29371

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1945 ax-6 2002 ax-7 2043 ax-8 2149 ax-9 2157 ax-10 2180 ax-11 2197 ax-12 2216 ax-ext 2735 ax-rep 5238 ax-sep 5255 ax-nul 5266 ax-pow 5334 ax-pr 5398 ax-un 7783 ax-cnex 11295 ax-resscn 11296 ax-1cn 11297 ax-icn 11298 ax-addcl 11299 ax-addrcl 11300 ax-mulcl 11301 ax-mulrcl 11302 ax-mulcom 11303 ax-addass 11304 ax-mulass 11305 ax-distr 11306 ax-i2m1 11307 ax-1ne0 11308 ax-1rid 11309 ax-rnegex 11310 ax-rrecex 11311 ax-cnre 11312 ax-pre-lttri 11313 ax-pre-lttrn 11314 ax-pre-ltadd 11315 ax-pre-mulgt0 11316

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2102 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2839 df-nfc 2913 df-ne 2960 df-nel 3066 df-ral 3081 df-rex 3091 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3456 df-sbc 3743 df-csb 3852 df-ss 3901 df-pss 4017 df-dif 4050 df-in 4094 df-un 4154 df-nul 4282 df-if 4486 df-pw 4562 df-sn 4589 df-pr 4591 df-tp 4593 df-op 4841 df-uni 4873 df-int 4913 df-iun 4958 df-br 5109 df-opab 5174 df-mpt 5193 df-tr 5219 df-id 5553 df-eprel 5558 df-po 5566 df-so 5567 df-fr 5611 df-we 5613 df-xp 5664 df-rel 5665 df-cnv 5666 df-co 5667 df-dm 5668 df-rn 5669 df-res 5670 df-ima 5671 df-pred 6330 df-ord 6391 df-on 6392 df-lim 6393 df-suc 6394 df-iota 6520 df-fun 6568 df-fn 6569 df-f 6570 df-f1 6571 df-fo 6572 df-f1o 6573 df-fv 6574 df-riota 7408 df-ov 7454 df-oprab 7455 df-mpo 7456 df-om 7912 df-1st 8035 df-2nd 8036 df-frecs 8337 df-wrecs 8368 df-recs 8417 df-rdg 8456 df-1o 8514 df-oadd 8518 df-er 8755 df-map 8887 df-pm 8888 df-en 9012 df-dom 9013 df-sdom 9014 df-fin 9015 df-dju 10019 df-card 10057 df-pnf 11384 df-mnf 11385 df-xr 11386 df-ltxr 11387 df-le 11388 df-sub 11582 df-neg 11583 df-nn 12376 df-2 12445 df-3 12446 df-n0 12647 df-xnn0 12720 df-z 12734 df-uz 13006 df-fz 13680 df-fzo 13829 df-hash 14516 df-word 14700 df-concat 14757 df-s1 14784 df-s2 15041 df-s3 15042 df-trkgc 28967 df-trkgb 28968 df-trkgcb 28969 df-trkgld 28971 df-trkg 28972 df-cgrg 29031 df-ismt 29053 df-leg 29103 df-hlg 29121 df-mir 29182 df-rag 29226 df-perpg 29228 df-mid 29336 df-lmi 29337 df-cgra 29372

This theorem is used by: ragsupplcgra 29402 ragraghl 29403

W3C validator