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Theorem ragcgra 29336

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 2761	. . . . . 6 ⊢ (Itv‘𝐺) = (Itv‘𝐺)
3		eqid 2761	. . . . . 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 2761	. . . . . . 7 ⊢ (dist‘𝐺) = (dist‘𝐺)
21		eqid 2761	. . . . . . 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 2767	. . . . . . . 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 28935	. . . . . . 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 2767	. . . . . . 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 2761	. . . . . . . . . . 11 ⊢ (LineG‘𝐺) = (LineG‘𝐺)
28		eqid 2761	. . . . . . . . . . . 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 3011	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐶 ≠ 𝐵)
33	1, 20, 2, 27, 28, 4, 12, 14, 16, 29, 30, 32	ragncol 29177	. . . . . . . . . . 11 ⊢ (𝜑 → ¬ (𝐶 ∈ (𝐴(LineG‘𝐺)𝐵) ∨ 𝐴 = 𝐵))
34	1, 27, 2, 4, 12, 14, 16, 33	ncoltgdim2 29021	. . . . . . . . . 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 29166	. . . . . . . . . . . 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 28938	. . . . . . . . . . . . . 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 29066	. . . . . . . . . . . . . 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 29084	. . . . . . . . . . . . 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 29167	. . . . . . . . . . 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 29166	. . . . . . . . . 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 3011	. . . . . . . . . . . . . 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 28938	. . . . . . . . . . . 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 29066	. . . . . . . . . . . 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 29084	. . . . . . . . . . 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 29167	. . . . . . . . 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 29300	. . . . . . . 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 28935	. . . . . . 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 28972	. . . . . 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 29311	. . . . 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 29075	. . . . 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 3171	. . 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 29075	. 2 ⊢ (𝜑 → ∃𝑎 ∈ 𝑃 (𝑎((hlG‘𝐺)‘𝐵)𝐴 ∧ (𝐵(dist‘𝐺)𝑎) = (𝑌(dist‘𝐺)𝑋)))
70	68, 69	r19.29a 3171	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 2145 ≠ wne 2956 ∃wrex 3087 class class class wbr 5103 ‘cfv 6537 (class class class)co 7418 2c2 12390 ⟨“cs3 14986 Basecbs 17380 distcds 17430 TarskiGcstrkg 28882 Dim_TarskiG≥cstrkgld 28886 Itvcitv 28888 LineGclng 28889 cgrGccgrg 28966 hlGchlg 29056 pInvGcmir 29117 ∟Gcrag 29161 cgrAccgra 29307

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 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2733 ax-rep 5232 ax-sep 5249 ax-nul 5260 ax-pow 5327 ax-pr 5391 ax-un 7749 ax-cnex 11249 ax-resscn 11250 ax-1cn 11251 ax-icn 11252 ax-addcl 11253 ax-addrcl 11254 ax-mulcl 11255 ax-mulrcl 11256 ax-mulcom 11257 ax-addass 11258 ax-mulass 11259 ax-distr 11260 ax-i2m1 11261 ax-1ne0 11262 ax-1rid 11263 ax-rnegex 11264 ax-rrecex 11265 ax-cnre 11266 ax-pre-lttri 11267 ax-pre-lttrn 11268 ax-pre-ltadd 11269 ax-pre-mulgt0 11270

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 2100 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3063 df-ral 3078 df-rex 3088 df-rmo 3366 df-reu 3367 df-rab 3414 df-v 3453 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4591 df-uni 4868 df-int 4908 df-iun 4953 df-br 5104 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5546 df-eprel 5551 df-po 5559 df-so 5560 df-fr 5604 df-we 5606 df-xp 5657 df-rel 5658 df-cnv 5659 df-co 5660 df-dm 5661 df-rn 5662 df-res 5663 df-ima 5664 df-pred 6303 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6493 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7375 df-ov 7421 df-oprab 7422 df-mpo 7423 df-om 7876 df-1st 7999 df-2nd 8000 df-frecs 8292 df-wrecs 8323 df-recs 8372 df-rdg 8411 df-1o 8469 df-oadd 8473 df-er 8710 df-map 8842 df-pm 8843 df-en 8967 df-dom 8968 df-sdom 8969 df-fin 8970 df-dju 9975 df-card 10013 df-pnf 11338 df-mnf 11339 df-xr 11340 df-ltxr 11341 df-le 11342 df-sub 11536 df-neg 11537 df-nn 12329 df-2 12398 df-3 12399 df-n0 12600 df-xnn0 12673 df-z 12687 df-uz 12959 df-fz 13633 df-fzo 13782 df-hash 14468 df-word 14652 df-concat 14709 df-s1 14736 df-s2 14992 df-s3 14993 df-trkgc 28903 df-trkgb 28904 df-trkgcb 28905 df-trkgld 28907 df-trkg 28908 df-cgrg 28967 df-ismt 28989 df-leg 29039 df-hlg 29057 df-mir 29118 df-rag 29162 df-perpg 29164 df-mid 29272 df-lmi 29273 df-cgra 29308

This theorem is used by: ragsupplcgra 29338 ragraghl 29339

W3C validator