Theorem cgrg3col4 28921

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 2736	. . . . 5 ⊢ (Itv‘𝐺) = (Itv‘𝐺)
4		isleag.g	. . . . . 6 ⊢ (𝜑 → 𝐺 ∈ TarskiG)
5	4	ad2antrr 727	. . . . 5 ⊢ (((𝜑 ∧ 𝐴 = 𝐶) ∧ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) → 𝐺 ∈ TarskiG)
6		isleag.a	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ 𝑃)
7	6	ad2antrr 727	. . . . 5 ⊢ (((𝜑 ∧ 𝐴 = 𝐶) ∧ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) → 𝐴 ∈ 𝑃)
8		isleag.b	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ 𝑃)
9	8	ad2antrr 727	. . . . 5 ⊢ (((𝜑 ∧ 𝐴 = 𝐶) ∧ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) → 𝐵 ∈ 𝑃)
10		cgrg3col4.x	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ 𝑃)
11	10	ad2antrr 727	. . . . 5 ⊢ (((𝜑 ∧ 𝐴 = 𝐶) ∧ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) → 𝑋 ∈ 𝑃)
12		eqid 2736	. . . . 5 ⊢ (cgrG‘𝐺) = (cgrG‘𝐺)
13		isleag.d	. . . . . 6 ⊢ (𝜑 → 𝐷 ∈ 𝑃)
14	13	ad2antrr 727	. . . . 5 ⊢ (((𝜑 ∧ 𝐴 = 𝐶) ∧ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) → 𝐷 ∈ 𝑃)
15		isleag.e	. . . . . 6 ⊢ (𝜑 → 𝐸 ∈ 𝑃)
16	15	ad2antrr 727	. . . . 5 ⊢ (((𝜑 ∧ 𝐴 = 𝐶) ∧ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) → 𝐸 ∈ 𝑃)
17		eqid 2736	. . . . 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 28588	. . . . . 6 ⊢ (𝜑 → (𝐴(dist‘𝐺)𝐵) = (𝐷(dist‘𝐺)𝐸))
23	22	ad2antrr 727	. . . . 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 28635	. . . 4 ⊢ (((𝜑 ∧ 𝐴 = 𝐶) ∧ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) → ∃𝑦 ∈ 𝑃 ⟨“𝐴𝐵𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩)
25	21	ad4antr 733	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝐴 = 𝐶) ∧ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑦 ∈ 𝑃) ∧ ⟨“𝐴𝐵𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩) → ⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝐹”⟩)
26	5	ad2antrr 727	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝐴 = 𝐶) ∧ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑦 ∈ 𝑃) ∧ ⟨“𝐴𝐵𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩) → 𝐺 ∈ TarskiG)
27	11	ad2antrr 727	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝐴 = 𝐶) ∧ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑦 ∈ 𝑃) ∧ ⟨“𝐴𝐵𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩) → 𝑋 ∈ 𝑃)
28	7	ad2antrr 727	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝐴 = 𝐶) ∧ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑦 ∈ 𝑃) ∧ ⟨“𝐴𝐵𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩) → 𝐴 ∈ 𝑃)
29		simplr 769	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝐴 = 𝐶) ∧ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑦 ∈ 𝑃) ∧ ⟨“𝐴𝐵𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩) → 𝑦 ∈ 𝑃)
30	14	ad2antrr 727	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝐴 = 𝐶) ∧ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑦 ∈ 𝑃) ∧ ⟨“𝐴𝐵𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩) → 𝐷 ∈ 𝑃)
31	9	ad2antrr 727	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝐴 = 𝐶) ∧ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑦 ∈ 𝑃) ∧ ⟨“𝐴𝐵𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩) → 𝐵 ∈ 𝑃)
32	16	ad2antrr 727	. . . . . . . . . 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 28590	. . . . . . . . 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 28548	. . . . . . . 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 28589	. . . . . . . 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 733	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝐴 = 𝐶) ∧ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑦 ∈ 𝑃) ∧ ⟨“𝐴𝐵𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩) → 𝐶 ∈ 𝑃)
38	20	ad4antr 733	. . . . . . . . 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 731	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝐴 = 𝐶) ∧ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑦 ∈ 𝑃) ∧ ⟨“𝐴𝐵𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩) → 𝐴 = 𝐶)
41	40	oveq2d 7383	. . . . . . . . . 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 28590	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐶(dist‘𝐺)𝐴) = (𝐹(dist‘𝐺)𝐷))
48	1, 17, 3, 4, 19, 6, 20, 13, 47	tgcgrcomlr 28548	. . . . . . . . . . . . . 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 28550	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐴 = 𝐶) → 𝐷 = 𝐹)
51	50	ad3antrrr 731	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝐴 = 𝐶) ∧ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑦 ∈ 𝑃) ∧ ⟨“𝐴𝐵𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩) → 𝐷 = 𝐹)
52	51	oveq2d 7383	. . . . . . . . . 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 2779	. . . . . . . . 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 28548	. . . . . . . 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 1129	. . . . . . 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 28599	. . . . . . 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 714	. . . . . 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 3150	. . . 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 2736	. . . . . 6 ⊢ (hlG‘𝐺) = (hlG‘𝐺)
62	4	ad2antrr 727	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) → 𝐺 ∈ TarskiG)
63	62	ad2antrr 727	. . . . . 6 ⊢ (((((𝜑 ∧ 𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑥 ∈ 𝑃) ∧ ¬ 𝑥 ∈ (𝐷𝐿𝐸)) → 𝐺 ∈ TarskiG)
64	8	ad2antrr 727	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) → 𝐵 ∈ 𝑃)
65	64	ad2antrr 727	. . . . . 6 ⊢ (((((𝜑 ∧ 𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑥 ∈ 𝑃) ∧ ¬ 𝑥 ∈ (𝐷𝐿𝐸)) → 𝐵 ∈ 𝑃)
66	6	ad2antrr 727	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) → 𝐴 ∈ 𝑃)
67	66	ad2antrr 727	. . . . . 6 ⊢ (((((𝜑 ∧ 𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑥 ∈ 𝑃) ∧ ¬ 𝑥 ∈ (𝐷𝐿𝐸)) → 𝐴 ∈ 𝑃)
68	10	ad2antrr 727	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) → 𝑋 ∈ 𝑃)
69	68	ad2antrr 727	. . . . . 6 ⊢ (((((𝜑 ∧ 𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑥 ∈ 𝑃) ∧ ¬ 𝑥 ∈ (𝐷𝐿𝐸)) → 𝑋 ∈ 𝑃)
70	15	ad2antrr 727	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) → 𝐸 ∈ 𝑃)
71	70	ad2antrr 727	. . . . . 6 ⊢ (((((𝜑 ∧ 𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑥 ∈ 𝑃) ∧ ¬ 𝑥 ∈ (𝐷𝐿𝐸)) → 𝐸 ∈ 𝑃)
72	13	ad2antrr 727	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) → 𝐷 ∈ 𝑃)
73	72	ad2antrr 727	. . . . . 6 ⊢ (((((𝜑 ∧ 𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑥 ∈ 𝑃) ∧ ¬ 𝑥 ∈ (𝐷𝐿𝐸)) → 𝐷 ∈ 𝑃)
74		simplr 769	. . . . . 6 ⊢ (((((𝜑 ∧ 𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑥 ∈ 𝑃) ∧ ¬ 𝑥 ∈ (𝐷𝐿𝐸)) → 𝑥 ∈ 𝑃)
75		simpllr 776	. . . . . 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 727	. . . . . . . . . . . 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 28693	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) → 𝐵 ≠ 𝐴)
80	79	necomd 2987	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) → 𝐴 ≠ 𝐵)
81	1, 17, 3, 62, 66, 64, 72, 70, 77, 80	tgcgrneq 28551	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) → 𝐷 ≠ 𝐸)
82	81	ad2antrr 727	. . . . . . . . . 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 986	. . . . . . . . 9 ⊢ (¬ (𝑥 ∈ (𝐷𝐿𝐸) ∨ 𝐷 = 𝐸) ↔ (¬ 𝑥 ∈ (𝐷𝐿𝐸) ∧ ¬ 𝐷 = 𝐸))
85	76, 83, 84	sylanbrc 584	. . . . . . . 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 28629	. . . . . . 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 28630	. . . . . 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 28548	. . . . . . 7 ⊢ (𝜑 → (𝐵(dist‘𝐺)𝐴) = (𝐸(dist‘𝐺)𝐷))
89	88	ad4antr 733	. . . . . 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 28872	. . . . 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 737	. . . . . . . . 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 727	. . . . . . . . . . 11 ⊢ (((((((𝜑 ∧ 𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑥 ∈ 𝑃) ∧ ¬ 𝑥 ∈ (𝐷𝐿𝐸)) ∧ 𝑦 ∈ 𝑃) ∧ ⟨“𝐵𝐴𝑋”⟩(cgrG‘𝐺)⟨“𝐸𝐷𝑦”⟩) → 𝐺 ∈ TarskiG)
93	65	ad2antrr 727	. . . . . . . . . . 11 ⊢ (((((((𝜑 ∧ 𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑥 ∈ 𝑃) ∧ ¬ 𝑥 ∈ (𝐷𝐿𝐸)) ∧ 𝑦 ∈ 𝑃) ∧ ⟨“𝐵𝐴𝑋”⟩(cgrG‘𝐺)⟨“𝐸𝐷𝑦”⟩) → 𝐵 ∈ 𝑃)
94	67	ad2antrr 727	. . . . . . . . . . 11 ⊢ (((((((𝜑 ∧ 𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑥 ∈ 𝑃) ∧ ¬ 𝑥 ∈ (𝐷𝐿𝐸)) ∧ 𝑦 ∈ 𝑃) ∧ ⟨“𝐵𝐴𝑋”⟩(cgrG‘𝐺)⟨“𝐸𝐷𝑦”⟩) → 𝐴 ∈ 𝑃)
95	69	ad2antrr 727	. . . . . . . . . . 11 ⊢ (((((((𝜑 ∧ 𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑥 ∈ 𝑃) ∧ ¬ 𝑥 ∈ (𝐷𝐿𝐸)) ∧ 𝑦 ∈ 𝑃) ∧ ⟨“𝐵𝐴𝑋”⟩(cgrG‘𝐺)⟨“𝐸𝐷𝑦”⟩) → 𝑋 ∈ 𝑃)
96	71	ad2antrr 727	. . . . . . . . . . 11 ⊢ (((((((𝜑 ∧ 𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑥 ∈ 𝑃) ∧ ¬ 𝑥 ∈ (𝐷𝐿𝐸)) ∧ 𝑦 ∈ 𝑃) ∧ ⟨“𝐵𝐴𝑋”⟩(cgrG‘𝐺)⟨“𝐸𝐷𝑦”⟩) → 𝐸 ∈ 𝑃)
97	73	ad2antrr 727	. . . . . . . . . . 11 ⊢ (((((((𝜑 ∧ 𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑥 ∈ 𝑃) ∧ ¬ 𝑥 ∈ (𝐷𝐿𝐸)) ∧ 𝑦 ∈ 𝑃) ∧ ⟨“𝐵𝐴𝑋”⟩(cgrG‘𝐺)⟨“𝐸𝐷𝑦”⟩) → 𝐷 ∈ 𝑃)
98		simplr 769	. . . . . . . . . . 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 28589	. . . . . . . . . 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 28590	. . . . . . . . . . 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 28548	. . . . . . . . . 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 735	. . . . . . . . . . 11 ⊢ (((((((𝜑 ∧ 𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑥 ∈ 𝑃) ∧ ¬ 𝑥 ∈ (𝐷𝐿𝐸)) ∧ 𝑦 ∈ 𝑃) ∧ ⟨“𝐵𝐴𝑋”⟩(cgrG‘𝐺)⟨“𝐸𝐷𝑦”⟩) → 𝐶 ∈ 𝑃)
104	46	ad5antr 735	. . . . . . . . . . 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 28548	. . . . . . . . . . . 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 788	. . . . . . . . . . . . 13 ⊢ (((((((𝜑 ∧ 𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑥 ∈ 𝑃) ∧ ¬ 𝑥 ∈ (𝐷𝐿𝐸)) ∧ 𝑦 ∈ 𝑃) ∧ ⟨“𝐵𝐴𝑋”⟩(cgrG‘𝐺)⟨“𝐸𝐷𝑦”⟩) → 𝐴 = 𝐶)
107	106	oveq2d 7383	. . . . . . . . . . . 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 735	. . . . . . . . . . . . 13 ⊢ (((((((𝜑 ∧ 𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑥 ∈ 𝑃) ∧ ¬ 𝑥 ∈ (𝐷𝐿𝐸)) ∧ 𝑦 ∈ 𝑃) ∧ ⟨“𝐵𝐴𝑋”⟩(cgrG‘𝐺)⟨“𝐸𝐷𝑦”⟩) → 𝐷 = 𝐹)
109	108	oveq2d 7383	. . . . . . . . . . . 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 2779	. . . . . . . . . . 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 28548	. . . . . . . . . 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 1129	. . . . . . . . 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 28599	. . . . . . . . 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 714	. . . . . . . 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 3150	. . . . 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 28633	. . . . 5 ⊢ (((𝜑 ∧ 𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) → 𝐺DimTarskiG≥2)
120	1, 3, 2, 62, 119, 72, 70, 81	tglowdim2ln 28719	. . . 4 ⊢ (((𝜑 ∧ 𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) → ∃𝑥 ∈ 𝑃 ¬ 𝑥 ∈ (𝐷𝐿𝐸))
121	118, 120	r19.29a 3145	. . 3 ⊢ (((𝜑 ∧ 𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) → ∃𝑦 ∈ 𝑃 ⟨“𝐴𝐵𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝐹𝑦”⟩)
122	60, 121	pm2.61dan 813	. 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 28626	. . . . . 6 ⊢ (𝜑 → (𝑋 ∈ (𝐶𝐿𝐴) ∨ 𝐶 = 𝐴))
125	1, 2, 3, 4, 19, 6, 10, 124	colrot1 28627	. . . . 5 ⊢ (𝜑 → (𝐶 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋))
126	1, 2, 3, 4, 6, 19, 10, 12, 13, 20, 17, 125, 48	lnext 28635	. . . 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 731	. . . . . 6 ⊢ ((((𝜑 ∧ 𝐴 ≠ 𝐶) ∧ 𝑦 ∈ 𝑃) ∧ ⟨“𝐴𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐹𝑦”⟩) → ⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝐹”⟩)
129	4	ad3antrrr 731	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝐴 ≠ 𝐶) ∧ 𝑦 ∈ 𝑃) ∧ ⟨“𝐴𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐹𝑦”⟩) → 𝐺 ∈ TarskiG)
130	10	ad3antrrr 731	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝐴 ≠ 𝐶) ∧ 𝑦 ∈ 𝑃) ∧ ⟨“𝐴𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐹𝑦”⟩) → 𝑋 ∈ 𝑃)
131	6	ad3antrrr 731	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝐴 ≠ 𝐶) ∧ 𝑦 ∈ 𝑃) ∧ ⟨“𝐴𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐹𝑦”⟩) → 𝐴 ∈ 𝑃)
132		simplr 769	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝐴 ≠ 𝐶) ∧ 𝑦 ∈ 𝑃) ∧ ⟨“𝐴𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐹𝑦”⟩) → 𝑦 ∈ 𝑃)
133	13	ad3antrrr 731	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝐴 ≠ 𝐶) ∧ 𝑦 ∈ 𝑃) ∧ ⟨“𝐴𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐹𝑦”⟩) → 𝐷 ∈ 𝑃)
134	19	ad3antrrr 731	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝐴 ≠ 𝐶) ∧ 𝑦 ∈ 𝑃) ∧ ⟨“𝐴𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐹𝑦”⟩) → 𝐶 ∈ 𝑃)
135	20	ad3antrrr 731	. . . . . . . . 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 28590	. . . . . . . 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 28548	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝐴 ≠ 𝐶) ∧ 𝑦 ∈ 𝑃) ∧ ⟨“𝐴𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐹𝑦”⟩) → (𝐴(dist‘𝐺)𝑋) = (𝐷(dist‘𝐺)𝑦))
139	8	ad3antrrr 731	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝐴 ≠ 𝐶) ∧ 𝑦 ∈ 𝑃) ∧ ⟨“𝐴𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐹𝑦”⟩) → 𝐵 ∈ 𝑃)
140	15	ad3antrrr 731	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝐴 ≠ 𝐶) ∧ 𝑦 ∈ 𝑃) ∧ ⟨“𝐴𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐹𝑦”⟩) → 𝐸 ∈ 𝑃)
141	125	ad3antrrr 731	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝐴 ≠ 𝐶) ∧ 𝑦 ∈ 𝑃) ∧ ⟨“𝐴𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐹𝑦”⟩) → (𝐶 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋))
142	22	ad3antrrr 731	. . . . . . . . 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 28589	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐵(dist‘𝐺)𝐶) = (𝐸(dist‘𝐺)𝐹))
144	1, 17, 3, 4, 8, 19, 15, 20, 143	tgcgrcomlr 28548	. . . . . . . . . 10 ⊢ (𝜑 → (𝐶(dist‘𝐺)𝐵) = (𝐹(dist‘𝐺)𝐸))
145	144	ad3antrrr 731	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝐴 ≠ 𝐶) ∧ 𝑦 ∈ 𝑃) ∧ ⟨“𝐴𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐹𝑦”⟩) → (𝐶(dist‘𝐺)𝐵) = (𝐹(dist‘𝐺)𝐸))
146		simpllr 776	. . . . . . . . 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 28636	. . . . . . . 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 28548	. . . . . . 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 28589	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝐴 ≠ 𝐶) ∧ 𝑦 ∈ 𝑃) ∧ ⟨“𝐴𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐹𝑦”⟩) → (𝐶(dist‘𝐺)𝑋) = (𝐹(dist‘𝐺)𝑦))
150	138, 148, 149	3jca 1129	. . . . . 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 28599	. . . . . 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 714	. . . . 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 3150	. . 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 3019	1 ⊢ (𝜑 → ∃𝑦 ∈ 𝑃 ⟨“𝐴𝐵𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝐹𝑦”⟩)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   ∨ wo 848   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114   ≠ wne 2932  ∃wrex 3061   class class class wbr 5085  ‘cfv 6498  (class class class)co 7367  ⟨“cs3 14804  ⟨“cs4 14805  Basecbs 17179  distcds 17229  TarskiGcstrkg 28495  Itvcitv 28501  LineGclng 28502  cgrGccgrg 28578  hlGchlg 28668  hpGchpg 28825

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689  ax-cnex 11094  ax-resscn 11095  ax-1cn 11096  ax-icn 11097  ax-addcl 11098  ax-addrcl 11099  ax-mulcl 11100  ax-mulrcl 11101  ax-mulcom 11102  ax-addass 11103  ax-mulass 11104  ax-distr 11105  ax-i2m1 11106  ax-1ne0 11107  ax-1rid 11108  ax-rnegex 11109  ax-rrecex 11110  ax-cnre 11111  ax-pre-lttri 11112  ax-pre-lttrn 11113  ax-pre-ltadd 11114  ax-pre-mulgt0 11115

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3062  df-rmo 3342  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-pss 3909  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-tp 4572  df-op 4574  df-uni 4851  df-int 4890  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-tr 5193  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6265  df-ord 6326  df-on 6327  df-lim 6328  df-suc 6329  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-riota 7324  df-ov 7370  df-oprab 7371  df-mpo 7372  df-om 7818  df-1st 7942  df-2nd 7943  df-frecs 8231  df-wrecs 8262  df-recs 8311  df-rdg 8349  df-1o 8405  df-oadd 8409  df-er 8643  df-map 8775  df-pm 8776  df-en 8894  df-dom 8895  df-sdom 8896  df-fin 8897  df-dju 9825  df-card 9863  df-pnf 11181  df-mnf 11182  df-xr 11183  df-ltxr 11184  df-le 11185  df-sub 11379  df-neg 11380  df-nn 12175  df-2 12244  df-3 12245  df-4 12246  df-n0 12438  df-xnn0 12511  df-z 12525  df-uz 12789  df-fz 13462  df-fzo 13609  df-hash 14293  df-word 14476  df-concat 14533  df-s1 14559  df-s2 14810  df-s3 14811  df-s4 14812  df-trkgc 28516  df-trkgb 28517  df-trkgcb 28518  df-trkgld 28520  df-trkg 28521  df-cgrg 28579  df-ismt 28601  df-leg 28651  df-hlg 28669  df-mir 28721  df-rag 28762  df-perpg 28764  df-hpg 28826  df-mid 28842  df-lmi 28843

This theorem is referenced by: (None)