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Theorem ragcgra 29127

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 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 2763	. . . . . . 7 ⊢ (dist‘𝐺) = (dist‘𝐺)
21		eqid 2763	. . . . . . 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 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 28730	. . . . . . 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 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 3013	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐶 ≠ 𝐵)
33	1, 20, 2, 27, 28, 4, 12, 14, 16, 29, 30, 32	ragncol 28970	. . . . . . . . . . 11 ⊢ (𝜑 → ¬ (𝐶 ∈ (𝐴(LineG‘𝐺)𝐵) ∨ 𝐴 = 𝐵))
34	1, 27, 2, 4, 12, 14, 16, 33	ncoltgdim2 28815	. . . . . . . . . 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 28959	. . . . . . . . . . . 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 28733	. . . . . . . . . . . . . 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 28860	. . . . . . . . . . . . . 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 28877	. . . . . . . . . . . . 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 28960	. . . . . . . . . . 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 28959	. . . . . . . . . 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 3013	. . . . . . . . . . . . . 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 28733	. . . . . . . . . . . 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 28860	. . . . . . . . . . . 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 28877	. . . . . . . . . . 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 28960	. . . . . . . . 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 29092	. . . . . . . 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 28730	. . . . . . 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 28766	. . . . . 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 29103	. . . . 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 28869	. . . . 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 3173	. . 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 28869	. 2 ⊢ (𝜑 → ∃𝑎 ∈ 𝑃 (𝑎((hlG‘𝐺)‘𝐵)𝐴 ∧ (𝐵(dist‘𝐺)𝑎) = (𝑌(dist‘𝐺)𝑋)))
70	68, 69	r19.29a 3173	1 ⊢ (𝜑 → ⟨“𝑋𝑌𝑍”⟩(cgrA‘𝐺)⟨“𝐴𝐵𝐶”⟩)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 400 = wceq 1570 ∈ wcel 2143 ≠ wne 2958 ∃wrex 3089 class class class wbr 5110 ‘cfv 6538 (class class class)co 7412 2c2 12296 ⟨“cs3 14881 Basecbs 17270 distcds 17320 TarskiGcstrkg 28677 Dim_TarskiG≥cstrkgld 28681 Itvcitv 28683 LineGclng 28684 cgrGccgrg 28760 hlGchlg 28850 pInvGcmir 28910 ∟Gcrag 28954 cgrAccgra 29099

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1825 ax-4 1839 ax-5 1940 ax-6 1997 ax-7 2038 ax-8 2145 ax-9 2153 ax-10 2176 ax-11 2192 ax-12 2213 ax-ext 2735 ax-rep 5239 ax-sep 5258 ax-nul 5270 ax-pow 5338 ax-pr 5406 ax-un 7734 ax-cnex 11157 ax-resscn 11158 ax-1cn 11159 ax-icn 11160 ax-addcl 11161 ax-addrcl 11162 ax-mulcl 11163 ax-mulrcl 11164 ax-mulcom 11165 ax-addass 11166 ax-mulass 11167 ax-distr 11168 ax-i2m1 11169 ax-1ne0 11170 ax-1rid 11171 ax-rnegex 11172 ax-rrecex 11173 ax-cnre 11174 ax-pre-lttri 11175 ax-pre-lttrn 11176 ax-pre-ltadd 11177 ax-pre-mulgt0 11178

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1810 df-nf 1814 df-sb 2097 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2838 df-nfc 2912 df-ne 2959 df-nel 3065 df-ral 3080 df-rex 3090 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3457 df-sbc 3746 df-csb 3855 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-pss 3926 df-nul 4288 df-if 4489 df-pw 4565 df-sn 4591 df-pr 4593 df-tp 4595 df-op 4597 df-uni 4874 df-int 4914 df-iun 4959 df-br 5111 df-opab 5175 df-mpt 5194 df-tr 5220 df-id 5558 df-eprel 5563 df-po 5571 df-so 5572 df-fr 5616 df-we 5618 df-xp 5669 df-rel 5670 df-cnv 5671 df-co 5672 df-dm 5673 df-rn 5674 df-res 5675 df-ima 5676 df-pred 6304 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6494 df-fun 6540 df-fn 6541 df-f 6542 df-f1 6543 df-fo 6544 df-f1o 6545 df-fv 6546 df-riota 7369 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7864 df-1st 7987 df-2nd 7988 df-frecs 8279 df-wrecs 8310 df-recs 8359 df-rdg 8398 df-1o 8454 df-oadd 8458 df-er 8695 df-map 8827 df-pm 8828 df-en 8945 df-dom 8946 df-sdom 8947 df-fin 8948 df-dju 9888 df-card 9926 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11444 df-neg 11445 df-nn 12235 df-2 12304 df-3 12305 df-n0 12506 df-xnn0 12579 df-z 12593 df-uz 12864 df-fz 13537 df-fzo 13685 df-hash 14369 df-word 14553 df-concat 14610 df-s1 14636 df-s2 14887 df-s3 14888 df-trkgc 28698 df-trkgb 28699 df-trkgcb 28700 df-trkgld 28702 df-trkg 28703 df-cgrg 28761 df-ismt 28783 df-leg 28833 df-hlg 28851 df-mir 28911 df-rag 28955 df-perpg 28957 df-mid 29064 df-lmi 29065 df-cgra 29100

This theorem is referenced by: ragsupplcgra 29129 ragraghl 29130

W3C validator