Theorem cgrg3col4 28539

Description: Lemma 11.28 of [Schwabhauser] p. 102. Extend a congruence of three points with a fourth colinear point. (Contributed by Thierry Arnoux, 8-Oct-2020.)

Hypotheses

Ref	Expression
isleag.p	⊢ 𝑃 = (Base‘𝐺)
isleag.g	⊢ (𝜑 → 𝐺 ∈ TarskiG)
isleag.a	⊢ (𝜑 → 𝐴 ∈ 𝑃)
isleag.b	⊢ (𝜑 → 𝐵 ∈ 𝑃)
isleag.c	⊢ (𝜑 → 𝐶 ∈ 𝑃)
isleag.d	⊢ (𝜑 → 𝐷 ∈ 𝑃)
isleag.e	⊢ (𝜑 → 𝐸 ∈ 𝑃)
isleag.f	⊢ (𝜑 → 𝐹 ∈ 𝑃)
cgrg3col4.l	⊢ 𝐿 = (LineG‘𝐺)
cgrg3col4.x	⊢ (𝜑 → 𝑋 ∈ 𝑃)
cgrg3col4.1	⊢ (𝜑 → ⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝐹”⟩)
cgrg3col4.2	⊢ (𝜑 → (𝑋 ∈ (𝐴𝐿𝐶) ∨ 𝐴 = 𝐶))

Assertion

Ref	Expression
cgrg3col4	⊢ (𝜑 → ∃𝑦 ∈ 𝑃 ⟨“𝐴𝐵𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝐹𝑦”⟩)

Distinct variable groups:   𝑦,𝐴   𝑦,𝐵   𝑦,𝐶   𝑦,𝐷   𝑦,𝐸   𝑦,𝐹   𝑦,𝐺   𝑦,𝐿   𝑦,𝑃   𝑦,𝑋   𝜑,𝑦

Proof of Theorem cgrg3col4

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		isleag.p	. . . . 5 ⊢ 𝑃 = (Base‘𝐺)
2		cgrg3col4.l	. . . . 5 ⊢ 𝐿 = (LineG‘𝐺)
3		eqid 2724	. . . . 5 ⊢ (Itv‘𝐺) = (Itv‘𝐺)
4		isleag.g	. . . . . 6 ⊢ (𝜑 → 𝐺 ∈ TarskiG)
5	4	ad2antrr 723	. . . . 5 ⊢ (((𝜑 ∧ 𝐴 = 𝐶) ∧ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) → 𝐺 ∈ TarskiG)
6		isleag.a	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ 𝑃)
7	6	ad2antrr 723	. . . . 5 ⊢ (((𝜑 ∧ 𝐴 = 𝐶) ∧ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) → 𝐴 ∈ 𝑃)
8		isleag.b	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ 𝑃)
9	8	ad2antrr 723	. . . . 5 ⊢ (((𝜑 ∧ 𝐴 = 𝐶) ∧ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) → 𝐵 ∈ 𝑃)
10		cgrg3col4.x	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ 𝑃)
11	10	ad2antrr 723	. . . . 5 ⊢ (((𝜑 ∧ 𝐴 = 𝐶) ∧ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) → 𝑋 ∈ 𝑃)
12		eqid 2724	. . . . 5 ⊢ (cgrG‘𝐺) = (cgrG‘𝐺)
13		isleag.d	. . . . . 6 ⊢ (𝜑 → 𝐷 ∈ 𝑃)
14	13	ad2antrr 723	. . . . 5 ⊢ (((𝜑 ∧ 𝐴 = 𝐶) ∧ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) → 𝐷 ∈ 𝑃)
15		isleag.e	. . . . . 6 ⊢ (𝜑 → 𝐸 ∈ 𝑃)
16	15	ad2antrr 723	. . . . 5 ⊢ (((𝜑 ∧ 𝐴 = 𝐶) ∧ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) → 𝐸 ∈ 𝑃)
17		eqid 2724	. . . . 5 ⊢ (dist‘𝐺) = (dist‘𝐺)
18		simpr 484	. . . . 5 ⊢ (((𝜑 ∧ 𝐴 = 𝐶) ∧ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) → (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋))
19		isleag.c	. . . . . . 7 ⊢ (𝜑 → 𝐶 ∈ 𝑃)
20		isleag.f	. . . . . . 7 ⊢ (𝜑 → 𝐹 ∈ 𝑃)
21		cgrg3col4.1	. . . . . . 7 ⊢ (𝜑 → ⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝐹”⟩)
22	1, 17, 3, 12, 4, 6, 8, 19, 13, 15, 20, 21	cgr3simp1 28206	. . . . . 6 ⊢ (𝜑 → (𝐴(dist‘𝐺)𝐵) = (𝐷(dist‘𝐺)𝐸))
23	22	ad2antrr 723	. . . . 5 ⊢ (((𝜑 ∧ 𝐴 = 𝐶) ∧ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) → (𝐴(dist‘𝐺)𝐵) = (𝐷(dist‘𝐺)𝐸))
24	1, 2, 3, 5, 7, 9, 11, 12, 14, 16, 17, 18, 23	lnext 28253	. . . 4 ⊢ (((𝜑 ∧ 𝐴 = 𝐶) ∧ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) → ∃𝑦 ∈ 𝑃 ⟨“𝐴𝐵𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩)
25	21	ad4antr 729	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝐴 = 𝐶) ∧ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑦 ∈ 𝑃) ∧ ⟨“𝐴𝐵𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩) → ⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝐹”⟩)
26	5	ad2antrr 723	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝐴 = 𝐶) ∧ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑦 ∈ 𝑃) ∧ ⟨“𝐴𝐵𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩) → 𝐺 ∈ TarskiG)
27	11	ad2antrr 723	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝐴 = 𝐶) ∧ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑦 ∈ 𝑃) ∧ ⟨“𝐴𝐵𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩) → 𝑋 ∈ 𝑃)
28	7	ad2antrr 723	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝐴 = 𝐶) ∧ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑦 ∈ 𝑃) ∧ ⟨“𝐴𝐵𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩) → 𝐴 ∈ 𝑃)
29		simplr 766	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝐴 = 𝐶) ∧ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑦 ∈ 𝑃) ∧ ⟨“𝐴𝐵𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩) → 𝑦 ∈ 𝑃)
30	14	ad2antrr 723	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝐴 = 𝐶) ∧ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑦 ∈ 𝑃) ∧ ⟨“𝐴𝐵𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩) → 𝐷 ∈ 𝑃)
31	9	ad2antrr 723	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝐴 = 𝐶) ∧ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑦 ∈ 𝑃) ∧ ⟨“𝐴𝐵𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩) → 𝐵 ∈ 𝑃)
32	16	ad2antrr 723	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝐴 = 𝐶) ∧ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑦 ∈ 𝑃) ∧ ⟨“𝐴𝐵𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩) → 𝐸 ∈ 𝑃)
33		simpr 484	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝐴 = 𝐶) ∧ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑦 ∈ 𝑃) ∧ ⟨“𝐴𝐵𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩) → ⟨“𝐴𝐵𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩)
34	1, 17, 3, 12, 26, 28, 31, 27, 30, 32, 29, 33	cgr3simp3 28208	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝐴 = 𝐶) ∧ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑦 ∈ 𝑃) ∧ ⟨“𝐴𝐵𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩) → (𝑋(dist‘𝐺)𝐴) = (𝑦(dist‘𝐺)𝐷))
35	1, 17, 3, 26, 27, 28, 29, 30, 34	tgcgrcomlr 28166	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝐴 = 𝐶) ∧ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑦 ∈ 𝑃) ∧ ⟨“𝐴𝐵𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩) → (𝐴(dist‘𝐺)𝑋) = (𝐷(dist‘𝐺)𝑦))
36	1, 17, 3, 12, 26, 28, 31, 27, 30, 32, 29, 33	cgr3simp2 28207	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝐴 = 𝐶) ∧ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑦 ∈ 𝑃) ∧ ⟨“𝐴𝐵𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩) → (𝐵(dist‘𝐺)𝑋) = (𝐸(dist‘𝐺)𝑦))
37	19	ad4antr 729	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝐴 = 𝐶) ∧ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑦 ∈ 𝑃) ∧ ⟨“𝐴𝐵𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩) → 𝐶 ∈ 𝑃)
38	20	ad4antr 729	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝐴 = 𝐶) ∧ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑦 ∈ 𝑃) ∧ ⟨“𝐴𝐵𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩) → 𝐹 ∈ 𝑃)
39		simpr 484	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐴 = 𝐶) → 𝐴 = 𝐶)
40	39	ad3antrrr 727	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝐴 = 𝐶) ∧ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑦 ∈ 𝑃) ∧ ⟨“𝐴𝐵𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩) → 𝐴 = 𝐶)
41	40	oveq2d 7417	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝐴 = 𝐶) ∧ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑦 ∈ 𝑃) ∧ ⟨“𝐴𝐵𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩) → (𝑋(dist‘𝐺)𝐴) = (𝑋(dist‘𝐺)𝐶))
42	4	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝐴 = 𝐶) → 𝐺 ∈ TarskiG)
43	6	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝐴 = 𝐶) → 𝐴 ∈ 𝑃)
44	19	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝐴 = 𝐶) → 𝐶 ∈ 𝑃)
45	13	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝐴 = 𝐶) → 𝐷 ∈ 𝑃)
46	20	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝐴 = 𝐶) → 𝐹 ∈ 𝑃)
47	1, 17, 3, 12, 4, 6, 8, 19, 13, 15, 20, 21	cgr3simp3 28208	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐶(dist‘𝐺)𝐴) = (𝐹(dist‘𝐺)𝐷))
48	1, 17, 3, 4, 19, 6, 20, 13, 47	tgcgrcomlr 28166	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐴(dist‘𝐺)𝐶) = (𝐷(dist‘𝐺)𝐹))
49	48	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝐴 = 𝐶) → (𝐴(dist‘𝐺)𝐶) = (𝐷(dist‘𝐺)𝐹))
50	1, 17, 3, 42, 43, 44, 45, 46, 49, 39	tgcgreq 28168	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐴 = 𝐶) → 𝐷 = 𝐹)
51	50	ad3antrrr 727	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝐴 = 𝐶) ∧ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑦 ∈ 𝑃) ∧ ⟨“𝐴𝐵𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩) → 𝐷 = 𝐹)
52	51	oveq2d 7417	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝐴 = 𝐶) ∧ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑦 ∈ 𝑃) ∧ ⟨“𝐴𝐵𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩) → (𝑦(dist‘𝐺)𝐷) = (𝑦(dist‘𝐺)𝐹))
53	34, 41, 52	3eqtr3d 2772	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝐴 = 𝐶) ∧ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑦 ∈ 𝑃) ∧ ⟨“𝐴𝐵𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩) → (𝑋(dist‘𝐺)𝐶) = (𝑦(dist‘𝐺)𝐹))
54	1, 17, 3, 26, 27, 37, 29, 38, 53	tgcgrcomlr 28166	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝐴 = 𝐶) ∧ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑦 ∈ 𝑃) ∧ ⟨“𝐴𝐵𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩) → (𝐶(dist‘𝐺)𝑋) = (𝐹(dist‘𝐺)𝑦))
55	35, 36, 54	3jca 1125	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝐴 = 𝐶) ∧ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑦 ∈ 𝑃) ∧ ⟨“𝐴𝐵𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩) → ((𝐴(dist‘𝐺)𝑋) = (𝐷(dist‘𝐺)𝑦) ∧ (𝐵(dist‘𝐺)𝑋) = (𝐸(dist‘𝐺)𝑦) ∧ (𝐶(dist‘𝐺)𝑋) = (𝐹(dist‘𝐺)𝑦)))
56	1, 17, 3, 12, 26, 28, 31, 37, 27, 30, 32, 38, 29	tgcgr4 28217	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝐴 = 𝐶) ∧ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑦 ∈ 𝑃) ∧ ⟨“𝐴𝐵𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩) → (⟨“𝐴𝐵𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝐹𝑦”⟩ ↔ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝐹”⟩ ∧ ((𝐴(dist‘𝐺)𝑋) = (𝐷(dist‘𝐺)𝑦) ∧ (𝐵(dist‘𝐺)𝑋) = (𝐸(dist‘𝐺)𝑦) ∧ (𝐶(dist‘𝐺)𝑋) = (𝐹(dist‘𝐺)𝑦)))))
57	25, 55, 56	mpbir2and 710	. . . . . 6 ⊢ (((((𝜑 ∧ 𝐴 = 𝐶) ∧ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑦 ∈ 𝑃) ∧ ⟨“𝐴𝐵𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩) → ⟨“𝐴𝐵𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝐹𝑦”⟩)
58	57	ex 412	. . . . 5 ⊢ ((((𝜑 ∧ 𝐴 = 𝐶) ∧ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑦 ∈ 𝑃) → (⟨“𝐴𝐵𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩ → ⟨“𝐴𝐵𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝐹𝑦”⟩))
59	58	reximdva 3160	. . . 4 ⊢ (((𝜑 ∧ 𝐴 = 𝐶) ∧ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) → (∃𝑦 ∈ 𝑃 ⟨“𝐴𝐵𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩ → ∃𝑦 ∈ 𝑃 ⟨“𝐴𝐵𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝐹𝑦”⟩))
60	24, 59	mpd 15	. . 3 ⊢ (((𝜑 ∧ 𝐴 = 𝐶) ∧ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) → ∃𝑦 ∈ 𝑃 ⟨“𝐴𝐵𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝐹𝑦”⟩)
61		eqid 2724	. . . . . 6 ⊢ (hlG‘𝐺) = (hlG‘𝐺)
62	4	ad2antrr 723	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) → 𝐺 ∈ TarskiG)
63	62	ad2antrr 723	. . . . . 6 ⊢ (((((𝜑 ∧ 𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑥 ∈ 𝑃) ∧ ¬ 𝑥 ∈ (𝐷𝐿𝐸)) → 𝐺 ∈ TarskiG)
64	8	ad2antrr 723	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) → 𝐵 ∈ 𝑃)
65	64	ad2antrr 723	. . . . . 6 ⊢ (((((𝜑 ∧ 𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑥 ∈ 𝑃) ∧ ¬ 𝑥 ∈ (𝐷𝐿𝐸)) → 𝐵 ∈ 𝑃)
66	6	ad2antrr 723	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) → 𝐴 ∈ 𝑃)
67	66	ad2antrr 723	. . . . . 6 ⊢ (((((𝜑 ∧ 𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑥 ∈ 𝑃) ∧ ¬ 𝑥 ∈ (𝐷𝐿𝐸)) → 𝐴 ∈ 𝑃)
68	10	ad2antrr 723	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) → 𝑋 ∈ 𝑃)
69	68	ad2antrr 723	. . . . . 6 ⊢ (((((𝜑 ∧ 𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑥 ∈ 𝑃) ∧ ¬ 𝑥 ∈ (𝐷𝐿𝐸)) → 𝑋 ∈ 𝑃)
70	15	ad2antrr 723	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) → 𝐸 ∈ 𝑃)
71	70	ad2antrr 723	. . . . . 6 ⊢ (((((𝜑 ∧ 𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑥 ∈ 𝑃) ∧ ¬ 𝑥 ∈ (𝐷𝐿𝐸)) → 𝐸 ∈ 𝑃)
72	13	ad2antrr 723	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) → 𝐷 ∈ 𝑃)
73	72	ad2antrr 723	. . . . . 6 ⊢ (((((𝜑 ∧ 𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑥 ∈ 𝑃) ∧ ¬ 𝑥 ∈ (𝐷𝐿𝐸)) → 𝐷 ∈ 𝑃)
74		simplr 766	. . . . . 6 ⊢ (((((𝜑 ∧ 𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑥 ∈ 𝑃) ∧ ¬ 𝑥 ∈ (𝐷𝐿𝐸)) → 𝑥 ∈ 𝑃)
75		simpllr 773	. . . . . 6 ⊢ (((((𝜑 ∧ 𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑥 ∈ 𝑃) ∧ ¬ 𝑥 ∈ (𝐷𝐿𝐸)) → ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋))
76		simpr 484	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑥 ∈ 𝑃) ∧ ¬ 𝑥 ∈ (𝐷𝐿𝐸)) → ¬ 𝑥 ∈ (𝐷𝐿𝐸))
77	22	ad2antrr 723	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) → (𝐴(dist‘𝐺)𝐵) = (𝐷(dist‘𝐺)𝐸))
78		simpr 484	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) → ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋))
79	1, 3, 2, 62, 64, 66, 68, 78	ncolne1 28311	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) → 𝐵 ≠ 𝐴)
80	79	necomd 2988	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) → 𝐴 ≠ 𝐵)
81	1, 17, 3, 62, 66, 64, 72, 70, 77, 80	tgcgrneq 28169	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) → 𝐷 ≠ 𝐸)
82	81	ad2antrr 723	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑥 ∈ 𝑃) ∧ ¬ 𝑥 ∈ (𝐷𝐿𝐸)) → 𝐷 ≠ 𝐸)
83	82	neneqd 2937	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑥 ∈ 𝑃) ∧ ¬ 𝑥 ∈ (𝐷𝐿𝐸)) → ¬ 𝐷 = 𝐸)
84		ioran 980	. . . . . . . . 9 ⊢ (¬ (𝑥 ∈ (𝐷𝐿𝐸) ∨ 𝐷 = 𝐸) ↔ (¬ 𝑥 ∈ (𝐷𝐿𝐸) ∧ ¬ 𝐷 = 𝐸))
85	76, 83, 84	sylanbrc 582	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑥 ∈ 𝑃) ∧ ¬ 𝑥 ∈ (𝐷𝐿𝐸)) → ¬ (𝑥 ∈ (𝐷𝐿𝐸) ∨ 𝐷 = 𝐸))
86	1, 2, 3, 63, 73, 71, 74, 85	ncolcom 28247	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑥 ∈ 𝑃) ∧ ¬ 𝑥 ∈ (𝐷𝐿𝐸)) → ¬ (𝑥 ∈ (𝐸𝐿𝐷) ∨ 𝐸 = 𝐷))
87	1, 2, 3, 63, 71, 73, 74, 86	ncolrot1 28248	. . . . . 6 ⊢ (((((𝜑 ∧ 𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑥 ∈ 𝑃) ∧ ¬ 𝑥 ∈ (𝐷𝐿𝐸)) → ¬ (𝐸 ∈ (𝐷𝐿𝑥) ∨ 𝐷 = 𝑥))
88	1, 17, 3, 4, 6, 8, 13, 15, 22	tgcgrcomlr 28166	. . . . . . 7 ⊢ (𝜑 → (𝐵(dist‘𝐺)𝐴) = (𝐸(dist‘𝐺)𝐷))
89	88	ad4antr 729	. . . . . 6 ⊢ (((((𝜑 ∧ 𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑥 ∈ 𝑃) ∧ ¬ 𝑥 ∈ (𝐷𝐿𝐸)) → (𝐵(dist‘𝐺)𝐴) = (𝐸(dist‘𝐺)𝐷))
90	1, 17, 3, 2, 61, 63, 65, 67, 69, 71, 73, 74, 75, 87, 89	trgcopy 28490	. . . . 5 ⊢ (((((𝜑 ∧ 𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑥 ∈ 𝑃) ∧ ¬ 𝑥 ∈ (𝐷𝐿𝐸)) → ∃𝑦 ∈ 𝑃 (⟨“𝐵𝐴𝑋”⟩(cgrG‘𝐺)⟨“𝐸𝐷𝑦”⟩ ∧ 𝑦((hpG‘𝐺)‘(𝐸𝐿𝐷))𝑥))
91	21	ad6antr 733	. . . . . . . . 9 ⊢ (((((((𝜑 ∧ 𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑥 ∈ 𝑃) ∧ ¬ 𝑥 ∈ (𝐷𝐿𝐸)) ∧ 𝑦 ∈ 𝑃) ∧ ⟨“𝐵𝐴𝑋”⟩(cgrG‘𝐺)⟨“𝐸𝐷𝑦”⟩) → ⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝐹”⟩)
92	63	ad2antrr 723	. . . . . . . . . . 11 ⊢ (((((((𝜑 ∧ 𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑥 ∈ 𝑃) ∧ ¬ 𝑥 ∈ (𝐷𝐿𝐸)) ∧ 𝑦 ∈ 𝑃) ∧ ⟨“𝐵𝐴𝑋”⟩(cgrG‘𝐺)⟨“𝐸𝐷𝑦”⟩) → 𝐺 ∈ TarskiG)
93	65	ad2antrr 723	. . . . . . . . . . 11 ⊢ (((((((𝜑 ∧ 𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑥 ∈ 𝑃) ∧ ¬ 𝑥 ∈ (𝐷𝐿𝐸)) ∧ 𝑦 ∈ 𝑃) ∧ ⟨“𝐵𝐴𝑋”⟩(cgrG‘𝐺)⟨“𝐸𝐷𝑦”⟩) → 𝐵 ∈ 𝑃)
94	67	ad2antrr 723	. . . . . . . . . . 11 ⊢ (((((((𝜑 ∧ 𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑥 ∈ 𝑃) ∧ ¬ 𝑥 ∈ (𝐷𝐿𝐸)) ∧ 𝑦 ∈ 𝑃) ∧ ⟨“𝐵𝐴𝑋”⟩(cgrG‘𝐺)⟨“𝐸𝐷𝑦”⟩) → 𝐴 ∈ 𝑃)
95	69	ad2antrr 723	. . . . . . . . . . 11 ⊢ (((((((𝜑 ∧ 𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑥 ∈ 𝑃) ∧ ¬ 𝑥 ∈ (𝐷𝐿𝐸)) ∧ 𝑦 ∈ 𝑃) ∧ ⟨“𝐵𝐴𝑋”⟩(cgrG‘𝐺)⟨“𝐸𝐷𝑦”⟩) → 𝑋 ∈ 𝑃)
96	71	ad2antrr 723	. . . . . . . . . . 11 ⊢ (((((((𝜑 ∧ 𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑥 ∈ 𝑃) ∧ ¬ 𝑥 ∈ (𝐷𝐿𝐸)) ∧ 𝑦 ∈ 𝑃) ∧ ⟨“𝐵𝐴𝑋”⟩(cgrG‘𝐺)⟨“𝐸𝐷𝑦”⟩) → 𝐸 ∈ 𝑃)
97	73	ad2antrr 723	. . . . . . . . . . 11 ⊢ (((((((𝜑 ∧ 𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑥 ∈ 𝑃) ∧ ¬ 𝑥 ∈ (𝐷𝐿𝐸)) ∧ 𝑦 ∈ 𝑃) ∧ ⟨“𝐵𝐴𝑋”⟩(cgrG‘𝐺)⟨“𝐸𝐷𝑦”⟩) → 𝐷 ∈ 𝑃)
98		simplr 766	. . . . . . . . . . 11 ⊢ (((((((𝜑 ∧ 𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑥 ∈ 𝑃) ∧ ¬ 𝑥 ∈ (𝐷𝐿𝐸)) ∧ 𝑦 ∈ 𝑃) ∧ ⟨“𝐵𝐴𝑋”⟩(cgrG‘𝐺)⟨“𝐸𝐷𝑦”⟩) → 𝑦 ∈ 𝑃)
99		simpr 484	. . . . . . . . . . 11 ⊢ (((((((𝜑 ∧ 𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑥 ∈ 𝑃) ∧ ¬ 𝑥 ∈ (𝐷𝐿𝐸)) ∧ 𝑦 ∈ 𝑃) ∧ ⟨“𝐵𝐴𝑋”⟩(cgrG‘𝐺)⟨“𝐸𝐷𝑦”⟩) → ⟨“𝐵𝐴𝑋”⟩(cgrG‘𝐺)⟨“𝐸𝐷𝑦”⟩)
100	1, 17, 3, 12, 92, 93, 94, 95, 96, 97, 98, 99	cgr3simp2 28207	. . . . . . . . . 10 ⊢ (((((((𝜑 ∧ 𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑥 ∈ 𝑃) ∧ ¬ 𝑥 ∈ (𝐷𝐿𝐸)) ∧ 𝑦 ∈ 𝑃) ∧ ⟨“𝐵𝐴𝑋”⟩(cgrG‘𝐺)⟨“𝐸𝐷𝑦”⟩) → (𝐴(dist‘𝐺)𝑋) = (𝐷(dist‘𝐺)𝑦))
101	1, 17, 3, 12, 92, 93, 94, 95, 96, 97, 98, 99	cgr3simp3 28208	. . . . . . . . . . 11 ⊢ (((((((𝜑 ∧ 𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑥 ∈ 𝑃) ∧ ¬ 𝑥 ∈ (𝐷𝐿𝐸)) ∧ 𝑦 ∈ 𝑃) ∧ ⟨“𝐵𝐴𝑋”⟩(cgrG‘𝐺)⟨“𝐸𝐷𝑦”⟩) → (𝑋(dist‘𝐺)𝐵) = (𝑦(dist‘𝐺)𝐸))
102	1, 17, 3, 92, 95, 93, 98, 96, 101	tgcgrcomlr 28166	. . . . . . . . . 10 ⊢ (((((((𝜑 ∧ 𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑥 ∈ 𝑃) ∧ ¬ 𝑥 ∈ (𝐷𝐿𝐸)) ∧ 𝑦 ∈ 𝑃) ∧ ⟨“𝐵𝐴𝑋”⟩(cgrG‘𝐺)⟨“𝐸𝐷𝑦”⟩) → (𝐵(dist‘𝐺)𝑋) = (𝐸(dist‘𝐺)𝑦))
103	44	ad5antr 731	. . . . . . . . . . 11 ⊢ (((((((𝜑 ∧ 𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑥 ∈ 𝑃) ∧ ¬ 𝑥 ∈ (𝐷𝐿𝐸)) ∧ 𝑦 ∈ 𝑃) ∧ ⟨“𝐵𝐴𝑋”⟩(cgrG‘𝐺)⟨“𝐸𝐷𝑦”⟩) → 𝐶 ∈ 𝑃)
104	46	ad5antr 731	. . . . . . . . . . 11 ⊢ (((((((𝜑 ∧ 𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑥 ∈ 𝑃) ∧ ¬ 𝑥 ∈ (𝐷𝐿𝐸)) ∧ 𝑦 ∈ 𝑃) ∧ ⟨“𝐵𝐴𝑋”⟩(cgrG‘𝐺)⟨“𝐸𝐷𝑦”⟩) → 𝐹 ∈ 𝑃)
105	1, 17, 3, 92, 94, 95, 97, 98, 100	tgcgrcomlr 28166	. . . . . . . . . . . 12 ⊢ (((((((𝜑 ∧ 𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑥 ∈ 𝑃) ∧ ¬ 𝑥 ∈ (𝐷𝐿𝐸)) ∧ 𝑦 ∈ 𝑃) ∧ ⟨“𝐵𝐴𝑋”⟩(cgrG‘𝐺)⟨“𝐸𝐷𝑦”⟩) → (𝑋(dist‘𝐺)𝐴) = (𝑦(dist‘𝐺)𝐷))
106		simp-6r 785	. . . . . . . . . . . . 13 ⊢ (((((((𝜑 ∧ 𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑥 ∈ 𝑃) ∧ ¬ 𝑥 ∈ (𝐷𝐿𝐸)) ∧ 𝑦 ∈ 𝑃) ∧ ⟨“𝐵𝐴𝑋”⟩(cgrG‘𝐺)⟨“𝐸𝐷𝑦”⟩) → 𝐴 = 𝐶)
107	106	oveq2d 7417	. . . . . . . . . . . 12 ⊢ (((((((𝜑 ∧ 𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑥 ∈ 𝑃) ∧ ¬ 𝑥 ∈ (𝐷𝐿𝐸)) ∧ 𝑦 ∈ 𝑃) ∧ ⟨“𝐵𝐴𝑋”⟩(cgrG‘𝐺)⟨“𝐸𝐷𝑦”⟩) → (𝑋(dist‘𝐺)𝐴) = (𝑋(dist‘𝐺)𝐶))
108	50	ad5antr 731	. . . . . . . . . . . . 13 ⊢ (((((((𝜑 ∧ 𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑥 ∈ 𝑃) ∧ ¬ 𝑥 ∈ (𝐷𝐿𝐸)) ∧ 𝑦 ∈ 𝑃) ∧ ⟨“𝐵𝐴𝑋”⟩(cgrG‘𝐺)⟨“𝐸𝐷𝑦”⟩) → 𝐷 = 𝐹)
109	108	oveq2d 7417	. . . . . . . . . . . 12 ⊢ (((((((𝜑 ∧ 𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑥 ∈ 𝑃) ∧ ¬ 𝑥 ∈ (𝐷𝐿𝐸)) ∧ 𝑦 ∈ 𝑃) ∧ ⟨“𝐵𝐴𝑋”⟩(cgrG‘𝐺)⟨“𝐸𝐷𝑦”⟩) → (𝑦(dist‘𝐺)𝐷) = (𝑦(dist‘𝐺)𝐹))
110	105, 107, 109	3eqtr3d 2772	. . . . . . . . . . 11 ⊢ (((((((𝜑 ∧ 𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑥 ∈ 𝑃) ∧ ¬ 𝑥 ∈ (𝐷𝐿𝐸)) ∧ 𝑦 ∈ 𝑃) ∧ ⟨“𝐵𝐴𝑋”⟩(cgrG‘𝐺)⟨“𝐸𝐷𝑦”⟩) → (𝑋(dist‘𝐺)𝐶) = (𝑦(dist‘𝐺)𝐹))
111	1, 17, 3, 92, 95, 103, 98, 104, 110	tgcgrcomlr 28166	. . . . . . . . . 10 ⊢ (((((((𝜑 ∧ 𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑥 ∈ 𝑃) ∧ ¬ 𝑥 ∈ (𝐷𝐿𝐸)) ∧ 𝑦 ∈ 𝑃) ∧ ⟨“𝐵𝐴𝑋”⟩(cgrG‘𝐺)⟨“𝐸𝐷𝑦”⟩) → (𝐶(dist‘𝐺)𝑋) = (𝐹(dist‘𝐺)𝑦))
112	100, 102, 111	3jca 1125	. . . . . . . . 9 ⊢ (((((((𝜑 ∧ 𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑥 ∈ 𝑃) ∧ ¬ 𝑥 ∈ (𝐷𝐿𝐸)) ∧ 𝑦 ∈ 𝑃) ∧ ⟨“𝐵𝐴𝑋”⟩(cgrG‘𝐺)⟨“𝐸𝐷𝑦”⟩) → ((𝐴(dist‘𝐺)𝑋) = (𝐷(dist‘𝐺)𝑦) ∧ (𝐵(dist‘𝐺)𝑋) = (𝐸(dist‘𝐺)𝑦) ∧ (𝐶(dist‘𝐺)𝑋) = (𝐹(dist‘𝐺)𝑦)))
113	1, 17, 3, 12, 92, 94, 93, 103, 95, 97, 96, 104, 98	tgcgr4 28217	. . . . . . . . 9 ⊢ (((((((𝜑 ∧ 𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑥 ∈ 𝑃) ∧ ¬ 𝑥 ∈ (𝐷𝐿𝐸)) ∧ 𝑦 ∈ 𝑃) ∧ ⟨“𝐵𝐴𝑋”⟩(cgrG‘𝐺)⟨“𝐸𝐷𝑦”⟩) → (⟨“𝐴𝐵𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝐹𝑦”⟩ ↔ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝐹”⟩ ∧ ((𝐴(dist‘𝐺)𝑋) = (𝐷(dist‘𝐺)𝑦) ∧ (𝐵(dist‘𝐺)𝑋) = (𝐸(dist‘𝐺)𝑦) ∧ (𝐶(dist‘𝐺)𝑋) = (𝐹(dist‘𝐺)𝑦)))))
114	91, 112, 113	mpbir2and 710	. . . . . . . 8 ⊢ (((((((𝜑 ∧ 𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑥 ∈ 𝑃) ∧ ¬ 𝑥 ∈ (𝐷𝐿𝐸)) ∧ 𝑦 ∈ 𝑃) ∧ ⟨“𝐵𝐴𝑋”⟩(cgrG‘𝐺)⟨“𝐸𝐷𝑦”⟩) → ⟨“𝐴𝐵𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝐹𝑦”⟩)
115	114	ex 412	. . . . . . 7 ⊢ ((((((𝜑 ∧ 𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑥 ∈ 𝑃) ∧ ¬ 𝑥 ∈ (𝐷𝐿𝐸)) ∧ 𝑦 ∈ 𝑃) → (⟨“𝐵𝐴𝑋”⟩(cgrG‘𝐺)⟨“𝐸𝐷𝑦”⟩ → ⟨“𝐴𝐵𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝐹𝑦”⟩))
116	115	adantrd 491	. . . . . 6 ⊢ ((((((𝜑 ∧ 𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑥 ∈ 𝑃) ∧ ¬ 𝑥 ∈ (𝐷𝐿𝐸)) ∧ 𝑦 ∈ 𝑃) → ((⟨“𝐵𝐴𝑋”⟩(cgrG‘𝐺)⟨“𝐸𝐷𝑦”⟩ ∧ 𝑦((hpG‘𝐺)‘(𝐸𝐿𝐷))𝑥) → ⟨“𝐴𝐵𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝐹𝑦”⟩))
117	116	reximdva 3160	. . . . 5 ⊢ (((((𝜑 ∧ 𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑥 ∈ 𝑃) ∧ ¬ 𝑥 ∈ (𝐷𝐿𝐸)) → (∃𝑦 ∈ 𝑃 (⟨“𝐵𝐴𝑋”⟩(cgrG‘𝐺)⟨“𝐸𝐷𝑦”⟩ ∧ 𝑦((hpG‘𝐺)‘(𝐸𝐿𝐷))𝑥) → ∃𝑦 ∈ 𝑃 ⟨“𝐴𝐵𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝐹𝑦”⟩))
118	90, 117	mpd 15	. . . 4 ⊢ (((((𝜑 ∧ 𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑥 ∈ 𝑃) ∧ ¬ 𝑥 ∈ (𝐷𝐿𝐸)) → ∃𝑦 ∈ 𝑃 ⟨“𝐴𝐵𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝐹𝑦”⟩)
119	1, 2, 3, 62, 66, 68, 64, 78	ncoltgdim2 28251	. . . . 5 ⊢ (((𝜑 ∧ 𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) → 𝐺DimTarskiG≥2)
120	1, 3, 2, 62, 119, 72, 70, 81	tglowdim2ln 28337	. . . 4 ⊢ (((𝜑 ∧ 𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) → ∃𝑥 ∈ 𝑃 ¬ 𝑥 ∈ (𝐷𝐿𝐸))
121	118, 120	r19.29a 3154	. . 3 ⊢ (((𝜑 ∧ 𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) → ∃𝑦 ∈ 𝑃 ⟨“𝐴𝐵𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝐹𝑦”⟩)
122	60, 121	pm2.61dan 810	. 2 ⊢ ((𝜑 ∧ 𝐴 = 𝐶) → ∃𝑦 ∈ 𝑃 ⟨“𝐴𝐵𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝐹𝑦”⟩)
123		cgrg3col4.2	. . . . . . 7 ⊢ (𝜑 → (𝑋 ∈ (𝐴𝐿𝐶) ∨ 𝐴 = 𝐶))
124	1, 2, 3, 4, 6, 19, 10, 123	colcom 28244	. . . . . 6 ⊢ (𝜑 → (𝑋 ∈ (𝐶𝐿𝐴) ∨ 𝐶 = 𝐴))
125	1, 2, 3, 4, 19, 6, 10, 124	colrot1 28245	. . . . 5 ⊢ (𝜑 → (𝐶 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋))
126	1, 2, 3, 4, 6, 19, 10, 12, 13, 20, 17, 125, 48	lnext 28253	. . . 4 ⊢ (𝜑 → ∃𝑦 ∈ 𝑃 ⟨“𝐴𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐹𝑦”⟩)
127	126	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝐴 ≠ 𝐶) → ∃𝑦 ∈ 𝑃 ⟨“𝐴𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐹𝑦”⟩)
128	21	ad3antrrr 727	. . . . . 6 ⊢ ((((𝜑 ∧ 𝐴 ≠ 𝐶) ∧ 𝑦 ∈ 𝑃) ∧ ⟨“𝐴𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐹𝑦”⟩) → ⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝐹”⟩)
129	4	ad3antrrr 727	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝐴 ≠ 𝐶) ∧ 𝑦 ∈ 𝑃) ∧ ⟨“𝐴𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐹𝑦”⟩) → 𝐺 ∈ TarskiG)
130	10	ad3antrrr 727	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝐴 ≠ 𝐶) ∧ 𝑦 ∈ 𝑃) ∧ ⟨“𝐴𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐹𝑦”⟩) → 𝑋 ∈ 𝑃)
131	6	ad3antrrr 727	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝐴 ≠ 𝐶) ∧ 𝑦 ∈ 𝑃) ∧ ⟨“𝐴𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐹𝑦”⟩) → 𝐴 ∈ 𝑃)
132		simplr 766	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝐴 ≠ 𝐶) ∧ 𝑦 ∈ 𝑃) ∧ ⟨“𝐴𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐹𝑦”⟩) → 𝑦 ∈ 𝑃)
133	13	ad3antrrr 727	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝐴 ≠ 𝐶) ∧ 𝑦 ∈ 𝑃) ∧ ⟨“𝐴𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐹𝑦”⟩) → 𝐷 ∈ 𝑃)
134	19	ad3antrrr 727	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝐴 ≠ 𝐶) ∧ 𝑦 ∈ 𝑃) ∧ ⟨“𝐴𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐹𝑦”⟩) → 𝐶 ∈ 𝑃)
135	20	ad3antrrr 727	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝐴 ≠ 𝐶) ∧ 𝑦 ∈ 𝑃) ∧ ⟨“𝐴𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐹𝑦”⟩) → 𝐹 ∈ 𝑃)
136		simpr 484	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝐴 ≠ 𝐶) ∧ 𝑦 ∈ 𝑃) ∧ ⟨“𝐴𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐹𝑦”⟩) → ⟨“𝐴𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐹𝑦”⟩)
137	1, 17, 3, 12, 129, 131, 134, 130, 133, 135, 132, 136	cgr3simp3 28208	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝐴 ≠ 𝐶) ∧ 𝑦 ∈ 𝑃) ∧ ⟨“𝐴𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐹𝑦”⟩) → (𝑋(dist‘𝐺)𝐴) = (𝑦(dist‘𝐺)𝐷))
138	1, 17, 3, 129, 130, 131, 132, 133, 137	tgcgrcomlr 28166	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝐴 ≠ 𝐶) ∧ 𝑦 ∈ 𝑃) ∧ ⟨“𝐴𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐹𝑦”⟩) → (𝐴(dist‘𝐺)𝑋) = (𝐷(dist‘𝐺)𝑦))
139	8	ad3antrrr 727	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝐴 ≠ 𝐶) ∧ 𝑦 ∈ 𝑃) ∧ ⟨“𝐴𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐹𝑦”⟩) → 𝐵 ∈ 𝑃)
140	15	ad3antrrr 727	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝐴 ≠ 𝐶) ∧ 𝑦 ∈ 𝑃) ∧ ⟨“𝐴𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐹𝑦”⟩) → 𝐸 ∈ 𝑃)
141	125	ad3antrrr 727	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝐴 ≠ 𝐶) ∧ 𝑦 ∈ 𝑃) ∧ ⟨“𝐴𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐹𝑦”⟩) → (𝐶 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋))
142	22	ad3antrrr 727	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝐴 ≠ 𝐶) ∧ 𝑦 ∈ 𝑃) ∧ ⟨“𝐴𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐹𝑦”⟩) → (𝐴(dist‘𝐺)𝐵) = (𝐷(dist‘𝐺)𝐸))
143	1, 17, 3, 12, 4, 6, 8, 19, 13, 15, 20, 21	cgr3simp2 28207	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐵(dist‘𝐺)𝐶) = (𝐸(dist‘𝐺)𝐹))
144	1, 17, 3, 4, 8, 19, 15, 20, 143	tgcgrcomlr 28166	. . . . . . . . . 10 ⊢ (𝜑 → (𝐶(dist‘𝐺)𝐵) = (𝐹(dist‘𝐺)𝐸))
145	144	ad3antrrr 727	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝐴 ≠ 𝐶) ∧ 𝑦 ∈ 𝑃) ∧ ⟨“𝐴𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐹𝑦”⟩) → (𝐶(dist‘𝐺)𝐵) = (𝐹(dist‘𝐺)𝐸))
146		simpllr 773	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝐴 ≠ 𝐶) ∧ 𝑦 ∈ 𝑃) ∧ ⟨“𝐴𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐹𝑦”⟩) → 𝐴 ≠ 𝐶)
147	1, 2, 3, 129, 131, 134, 130, 12, 133, 135, 17, 139, 132, 140, 141, 136, 142, 145, 146	tgfscgr 28254	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝐴 ≠ 𝐶) ∧ 𝑦 ∈ 𝑃) ∧ ⟨“𝐴𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐹𝑦”⟩) → (𝑋(dist‘𝐺)𝐵) = (𝑦(dist‘𝐺)𝐸))
148	1, 17, 3, 129, 130, 139, 132, 140, 147	tgcgrcomlr 28166	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝐴 ≠ 𝐶) ∧ 𝑦 ∈ 𝑃) ∧ ⟨“𝐴𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐹𝑦”⟩) → (𝐵(dist‘𝐺)𝑋) = (𝐸(dist‘𝐺)𝑦))
149	1, 17, 3, 12, 129, 131, 134, 130, 133, 135, 132, 136	cgr3simp2 28207	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝐴 ≠ 𝐶) ∧ 𝑦 ∈ 𝑃) ∧ ⟨“𝐴𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐹𝑦”⟩) → (𝐶(dist‘𝐺)𝑋) = (𝐹(dist‘𝐺)𝑦))
150	138, 148, 149	3jca 1125	. . . . . 6 ⊢ ((((𝜑 ∧ 𝐴 ≠ 𝐶) ∧ 𝑦 ∈ 𝑃) ∧ ⟨“𝐴𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐹𝑦”⟩) → ((𝐴(dist‘𝐺)𝑋) = (𝐷(dist‘𝐺)𝑦) ∧ (𝐵(dist‘𝐺)𝑋) = (𝐸(dist‘𝐺)𝑦) ∧ (𝐶(dist‘𝐺)𝑋) = (𝐹(dist‘𝐺)𝑦)))
151	1, 17, 3, 12, 129, 131, 139, 134, 130, 133, 140, 135, 132	tgcgr4 28217	. . . . . 6 ⊢ ((((𝜑 ∧ 𝐴 ≠ 𝐶) ∧ 𝑦 ∈ 𝑃) ∧ ⟨“𝐴𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐹𝑦”⟩) → (⟨“𝐴𝐵𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝐹𝑦”⟩ ↔ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝐹”⟩ ∧ ((𝐴(dist‘𝐺)𝑋) = (𝐷(dist‘𝐺)𝑦) ∧ (𝐵(dist‘𝐺)𝑋) = (𝐸(dist‘𝐺)𝑦) ∧ (𝐶(dist‘𝐺)𝑋) = (𝐹(dist‘𝐺)𝑦)))))
152	128, 150, 151	mpbir2and 710	. . . . 5 ⊢ ((((𝜑 ∧ 𝐴 ≠ 𝐶) ∧ 𝑦 ∈ 𝑃) ∧ ⟨“𝐴𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐹𝑦”⟩) → ⟨“𝐴𝐵𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝐹𝑦”⟩)
153	152	ex 412	. . . 4 ⊢ (((𝜑 ∧ 𝐴 ≠ 𝐶) ∧ 𝑦 ∈ 𝑃) → (⟨“𝐴𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐹𝑦”⟩ → ⟨“𝐴𝐵𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝐹𝑦”⟩))
154	153	reximdva 3160	. . 3 ⊢ ((𝜑 ∧ 𝐴 ≠ 𝐶) → (∃𝑦 ∈ 𝑃 ⟨“𝐴𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐹𝑦”⟩ → ∃𝑦 ∈ 𝑃 ⟨“𝐴𝐵𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝐹𝑦”⟩))
155	127, 154	mpd 15	. 2 ⊢ ((𝜑 ∧ 𝐴 ≠ 𝐶) → ∃𝑦 ∈ 𝑃 ⟨“𝐴𝐵𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝐹𝑦”⟩)
156	122, 155	pm2.61dane 3021	1 ⊢ (𝜑 → ∃𝑦 ∈ 𝑃 ⟨“𝐴𝐵𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝐹𝑦”⟩)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   ∨ wo 844   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098   ≠ wne 2932  ∃wrex 3062   class class class wbr 5138  ‘cfv 6533  (class class class)co 7401  ⟨“cs3 14789  ⟨“cs4 14790  Basecbs 17142  distcds 17204  TarskiGcstrkg 28113  Itvcitv 28119  LineGclng 28120  cgrGccgrg 28196  hlGchlg 28286  hpGchpg 28443

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-rep 5275  ax-sep 5289  ax-nul 5296  ax-pow 5353  ax-pr 5417  ax-un 7718  ax-cnex 11161  ax-resscn 11162  ax-1cn 11163  ax-icn 11164  ax-addcl 11165  ax-addrcl 11166  ax-mulcl 11167  ax-mulrcl 11168  ax-mulcom 11169  ax-addass 11170  ax-mulass 11171  ax-distr 11172  ax-i2m1 11173  ax-1ne0 11174  ax-1rid 11175  ax-rnegex 11176  ax-rrecex 11177  ax-cnre 11178  ax-pre-lttri 11179  ax-pre-lttrn 11180  ax-pre-ltadd 11181  ax-pre-mulgt0 11182

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-nel 3039  df-ral 3054  df-rex 3063  df-rmo 3368  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3770  df-csb 3886  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-pss 3959  df-nul 4315  df-if 4521  df-pw 4596  df-sn 4621  df-pr 4623  df-tp 4625  df-op 4627  df-uni 4900  df-int 4941  df-iun 4989  df-br 5139  df-opab 5201  df-mpt 5222  df-tr 5256  df-id 5564  df-eprel 5570  df-po 5578  df-so 5579  df-fr 5621  df-we 5623  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-pred 6290  df-ord 6357  df-on 6358  df-lim 6359  df-suc 6360  df-iota 6485  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-riota 7357  df-ov 7404  df-oprab 7405  df-mpo 7406  df-om 7849  df-1st 7968  df-2nd 7969  df-frecs 8261  df-wrecs 8292  df-recs 8366  df-rdg 8405  df-1o 8461  df-oadd 8465  df-er 8698  df-map 8817  df-pm 8818  df-en 8935  df-dom 8936  df-sdom 8937  df-fin 8938  df-dju 9891  df-card 9929  df-pnf 11246  df-mnf 11247  df-xr 11248  df-ltxr 11249  df-le 11250  df-sub 11442  df-neg 11443  df-nn 12209  df-2 12271  df-3 12272  df-4 12273  df-n0 12469  df-xnn0 12541  df-z 12555  df-uz 12819  df-fz 13481  df-fzo 13624  df-hash 14287  df-word 14461  df-concat 14517  df-s1 14542  df-s2 14795  df-s3 14796  df-s4 14797  df-trkgc 28134  df-trkgb 28135  df-trkgcb 28136  df-trkgld 28138  df-trkg 28139  df-cgrg 28197  df-ismt 28219  df-leg 28269  df-hlg 28287  df-mir 28339  df-rag 28380  df-perpg 28382  df-hpg 28444  df-mid 28460  df-lmi 28461

This theorem is referenced by: (None)