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Theorem ragcgra 29099

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 2769	. . . . . 6 ⊢ (Itv‘𝐺) = (Itv‘𝐺)
3		eqid 2769	. . . . . 6 ⊢ (hlG‘𝐺) = (hlG‘𝐺)
4		ragcgra.g	. . . . . . 7 ⊢ (𝜑 → 𝐺 ∈ TarskiG)
5	4	ad6antr 748	. . . . . 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 748	. . . . . 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 748	. . . . . 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 748	. . . . . 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 748	. . . . . 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 748	. . . . . 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 748	. . . . . 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 799	. . . . . 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 787	. . . . . 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 2769	. . . . . . 7 ⊢ (dist‘𝐺) = (dist‘𝐺)
21		eqid 2769	. . . . . . 7 ⊢ (cgrG‘𝐺) = (cgrG‘𝐺)
22		simp-4r 795	. . . . . . . . 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 2775	. . . . . . . 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 28711	. . . . . . 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 489	. . . . . . . 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 2775	. . . . . . 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 2769	. . . . . . . . . . 11 ⊢ (LineG‘𝐺) = (LineG‘𝐺)
28		eqid 2769	. . . . . . . . . . . 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 3019	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐶 ≠ 𝐵)
33	1, 20, 2, 27, 28, 4, 12, 14, 16, 29, 30, 32	ragncol 28944	. . . . . . . . . . 11 ⊢ (𝜑 → ¬ (𝐶 ∈ (𝐴(LineG‘𝐺)𝐵) ∨ 𝐴 = 𝐵))
34	1, 27, 2, 4, 12, 14, 16, 33	ncoltgdim2 28796	. . . . . . . . . 10 ⊢ (𝜑 → 𝐺Dim_TarskiG≥2)
35	34	ad6antr 748	. . . . . . . . 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 748	. . . . . . . . 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 748	. . . . . . . . . . . . 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 28933	. . . . . . . . . . . 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 748	. . . . . . . . . . . 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 748	. . . . . . . . . . . . . . 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 28714	. . . . . . . . . . . . . 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 780	. . . . . . . . . . . . . . 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 28838	. . . . . . . . . . . . . 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 28854	. . . . . . . . . . . . 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 886	. . . . . . . . . . . 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 28934	. . . . . . . . . . 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 28933	. . . . . . . . . 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 748	. . . . . . . . . 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 3019	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑌 ≠ 𝑋)
53	52	ad6antr 748	. . . . . . . . . . . . 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 28714	. . . . . . . . . . . 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 797	. . . . . . . . . . . . 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 28838	. . . . . . . . . . . 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 28854	. . . . . . . . . . 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 886	. . . . . . . . . 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 28934	. . . . . . . . 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 29064	. . . . . . . 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 28711	. . . . . . 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 28747	. . . . . 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 29075	. . . . 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 471	. . . 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 28847	. . . . 5 ⊢ (𝜑 → ∃𝑐 ∈ 𝑃 (𝑐((hlG‘𝐺)‘𝐵)𝐶 ∧ (𝐵(dist‘𝐺)𝑐) = (𝑌(dist‘𝐺)𝑍)))
66	65	ad3antrrr 742	. . . 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 3179	. . 3 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝑃) ∧ 𝑎((hlG‘𝐺)‘𝐵)𝐴) ∧ (𝐵(dist‘𝐺)𝑎) = (𝑌(dist‘𝐺)𝑋)) → ⟨“𝑋𝑌𝑍”⟩(cgrA‘𝐺)⟨“𝐴𝐵𝐶”⟩)
68	67	anasss 471	. 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 28847	. 2 ⊢ (𝜑 → ∃𝑎 ∈ 𝑃 (𝑎((hlG‘𝐺)‘𝐵)𝐴 ∧ (𝐵(dist‘𝐺)𝑎) = (𝑌(dist‘𝐺)𝑋)))
70	68, 69	r19.29a 3179	1 ⊢ (𝜑 → ⟨“𝑋𝑌𝑍”⟩(cgrA‘𝐺)⟨“𝐴𝐵𝐶”⟩)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 400 = wceq 1567 ∈ wcel 2149 ≠ wne 2964 ∃wrex 3095 class class class wbr 5110 ‘cfv 6533 (class class class)co 7408 2c2 12291 ⟨“cs3 14875 Basecbs 17265 distcds 17315 TarskiGcstrkg 28658 Dim_TarskiG≥cstrkgld 28662 Itvcitv 28664 LineGclng 28665 cgrGccgrg 28741 hlGchlg 28831 pInvGcmir 28887 ∟Gcrag 28928 cgrAccgra 29071

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1822 ax-4 1836 ax-5 1937 ax-6 1994 ax-7 2035 ax-8 2151 ax-9 2159 ax-10 2182 ax-11 2198 ax-12 2219 ax-ext 2741 ax-rep 5239 ax-sep 5258 ax-nul 5268 ax-pow 5334 ax-pr 5402 ax-un 7730 ax-cnex 11152 ax-resscn 11153 ax-1cn 11154 ax-icn 11155 ax-addcl 11156 ax-addrcl 11157 ax-mulcl 11158 ax-mulrcl 11159 ax-mulcom 11160 ax-addass 11161 ax-mulass 11162 ax-distr 11163 ax-i2m1 11164 ax-1ne0 11165 ax-1rid 11166 ax-rnegex 11167 ax-rrecex 11168 ax-cnre 11169 ax-pre-lttri 11170 ax-pre-lttrn 11171 ax-pre-ltadd 11172 ax-pre-mulgt0 11173

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1102 df-3an 1103 df-tru 1570 df-fal 1580 df-ex 1807 df-nf 1811 df-sb 2098 df-mo 2573 df-eu 2603 df-clab 2748 df-cleq 2761 df-clel 2844 df-nfc 2918 df-ne 2965 df-nel 3071 df-ral 3086 df-rex 3096 df-rmo 3376 df-reu 3377 df-rab 3424 df-v 3465 df-sbc 3754 df-csb 3862 df-dif 3916 df-un 3918 df-in 3920 df-ss 3930 df-pss 3933 df-nul 4295 df-if 4490 df-pw 4566 df-sn 4592 df-pr 4594 df-tp 4596 df-op 4598 df-uni 4874 df-int 4914 df-iun 4959 df-br 5111 df-opab 5175 df-mpt 5194 df-tr 5220 df-id 5554 df-eprel 5559 df-po 5567 df-so 5568 df-fr 5612 df-we 5614 df-xp 5665 df-rel 5666 df-cnv 5667 df-co 5668 df-dm 5669 df-rn 5670 df-res 5671 df-ima 5672 df-pred 6299 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6489 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-riota 7365 df-ov 7411 df-oprab 7412 df-mpo 7413 df-om 7859 df-1st 7982 df-2nd 7983 df-frecs 8274 df-wrecs 8305 df-recs 8354 df-rdg 8393 df-1o 8449 df-oadd 8453 df-er 8690 df-map 8822 df-pm 8823 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-dju 9883 df-card 9921 df-pnf 11241 df-mnf 11242 df-xr 11243 df-ltxr 11244 df-le 11245 df-sub 11439 df-neg 11440 df-nn 12230 df-2 12299 df-3 12300 df-n0 12501 df-xnn0 12574 df-z 12588 df-uz 12859 df-fz 13532 df-fzo 13679 df-hash 14363 df-word 14547 df-concat 14604 df-s1 14630 df-s2 14881 df-s3 14882 df-trkgc 28679 df-trkgb 28680 df-trkgcb 28681 df-trkgld 28683 df-trkg 28684 df-cgrg 28742 df-ismt 28764 df-leg 28814 df-hlg 28832 df-mir 28888 df-rag 28929 df-perpg 28931 df-mid 29037 df-lmi 29038 df-cgra 29072

This theorem is referenced by: ragraghl 29100

W3C validator