Theorem cgrg3col4 29047

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 2762	. . . . 5 ⊢ (Itv‘𝐺) = (Itv‘𝐺)
4		isleag.g	. . . . . 6 ⊢ (𝜑 → 𝐺 ∈ TarskiG)
5	4	ad2antrr 736	. . . . 5 ⊢ (((𝜑 ∧ 𝐴 = 𝐶) ∧ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) → 𝐺 ∈ TarskiG)
6		isleag.a	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ 𝑃)
7	6	ad2antrr 736	. . . . 5 ⊢ (((𝜑 ∧ 𝐴 = 𝐶) ∧ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) → 𝐴 ∈ 𝑃)
8		isleag.b	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ 𝑃)
9	8	ad2antrr 736	. . . . 5 ⊢ (((𝜑 ∧ 𝐴 = 𝐶) ∧ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) → 𝐵 ∈ 𝑃)
10		cgrg3col4.x	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ 𝑃)
11	10	ad2antrr 736	. . . . 5 ⊢ (((𝜑 ∧ 𝐴 = 𝐶) ∧ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) → 𝑋 ∈ 𝑃)
12		eqid 2762	. . . . 5 ⊢ (cgrG‘𝐺) = (cgrG‘𝐺)
13		isleag.d	. . . . . 6 ⊢ (𝜑 → 𝐷 ∈ 𝑃)
14	13	ad2antrr 736	. . . . 5 ⊢ (((𝜑 ∧ 𝐴 = 𝐶) ∧ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) → 𝐷 ∈ 𝑃)
15		isleag.e	. . . . . 6 ⊢ (𝜑 → 𝐸 ∈ 𝑃)
16	15	ad2antrr 736	. . . . 5 ⊢ (((𝜑 ∧ 𝐴 = 𝐶) ∧ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) → 𝐸 ∈ 𝑃)
17		eqid 2762	. . . . 5 ⊢ (dist‘𝐺) = (dist‘𝐺)
18		simpr 488	. . . . 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 28689	. . . . . 6 ⊢ (𝜑 → (𝐴(dist‘𝐺)𝐵) = (𝐷(dist‘𝐺)𝐸))
23	22	ad2antrr 736	. . . . 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 28736	. . . 4 ⊢ (((𝜑 ∧ 𝐴 = 𝐶) ∧ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) → ∃𝑦 ∈ 𝑃 ⟨“𝐴𝐵𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩)
25	21	ad4antr 742	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝐴 = 𝐶) ∧ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑦 ∈ 𝑃) ∧ ⟨“𝐴𝐵𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩) → ⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝐹”⟩)
26	5	ad2antrr 736	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝐴 = 𝐶) ∧ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑦 ∈ 𝑃) ∧ ⟨“𝐴𝐵𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩) → 𝐺 ∈ TarskiG)
27	11	ad2antrr 736	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝐴 = 𝐶) ∧ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑦 ∈ 𝑃) ∧ ⟨“𝐴𝐵𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩) → 𝑋 ∈ 𝑃)
28	7	ad2antrr 736	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝐴 = 𝐶) ∧ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑦 ∈ 𝑃) ∧ ⟨“𝐴𝐵𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩) → 𝐴 ∈ 𝑃)
29		simplr 778	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝐴 = 𝐶) ∧ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑦 ∈ 𝑃) ∧ ⟨“𝐴𝐵𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩) → 𝑦 ∈ 𝑃)
30	14	ad2antrr 736	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝐴 = 𝐶) ∧ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑦 ∈ 𝑃) ∧ ⟨“𝐴𝐵𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩) → 𝐷 ∈ 𝑃)
31	9	ad2antrr 736	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝐴 = 𝐶) ∧ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑦 ∈ 𝑃) ∧ ⟨“𝐴𝐵𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩) → 𝐵 ∈ 𝑃)
32	16	ad2antrr 736	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝐴 = 𝐶) ∧ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑦 ∈ 𝑃) ∧ ⟨“𝐴𝐵𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩) → 𝐸 ∈ 𝑃)
33		simpr 488	. . . . . . . . . 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 28691	. . . . . . . . 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 28649	. . . . . . . 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 28690	. . . . . . . 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 742	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝐴 = 𝐶) ∧ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑦 ∈ 𝑃) ∧ ⟨“𝐴𝐵𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩) → 𝐶 ∈ 𝑃)
38	20	ad4antr 742	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝐴 = 𝐶) ∧ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑦 ∈ 𝑃) ∧ ⟨“𝐴𝐵𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩) → 𝐹 ∈ 𝑃)
39		simpr 488	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐴 = 𝐶) → 𝐴 = 𝐶)
40	39	ad3antrrr 740	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝐴 = 𝐶) ∧ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑦 ∈ 𝑃) ∧ ⟨“𝐴𝐵𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩) → 𝐴 = 𝐶)
41	40	oveq2d 7412	. . . . . . . . . 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 484	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝐴 = 𝐶) → 𝐺 ∈ TarskiG)
43	6	adantr 484	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝐴 = 𝐶) → 𝐴 ∈ 𝑃)
44	19	adantr 484	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝐴 = 𝐶) → 𝐶 ∈ 𝑃)
45	13	adantr 484	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝐴 = 𝐶) → 𝐷 ∈ 𝑃)
46	20	adantr 484	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝐴 = 𝐶) → 𝐹 ∈ 𝑃)
47	1, 17, 3, 12, 4, 6, 8, 19, 13, 15, 20, 21	cgr3simp3 28691	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐶(dist‘𝐺)𝐴) = (𝐹(dist‘𝐺)𝐷))
48	1, 17, 3, 4, 19, 6, 20, 13, 47	tgcgrcomlr 28649	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐴(dist‘𝐺)𝐶) = (𝐷(dist‘𝐺)𝐹))
49	48	adantr 484	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝐴 = 𝐶) → (𝐴(dist‘𝐺)𝐶) = (𝐷(dist‘𝐺)𝐹))
50	1, 17, 3, 42, 43, 44, 45, 46, 49, 39	tgcgreq 28651	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐴 = 𝐶) → 𝐷 = 𝐹)
51	50	ad3antrrr 740	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝐴 = 𝐶) ∧ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑦 ∈ 𝑃) ∧ ⟨“𝐴𝐵𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩) → 𝐷 = 𝐹)
52	51	oveq2d 7412	. . . . . . . . . 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 2805	. . . . . . . . 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 28649	. . . . . . . 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 1141	. . . . . . 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 28700	. . . . . . 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 723	. . . . . 6 ⊢ (((((𝜑 ∧ 𝐴 = 𝐶) ∧ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑦 ∈ 𝑃) ∧ ⟨“𝐴𝐵𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩) → ⟨“𝐴𝐵𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝐹𝑦”⟩)
58	57	ex 416	. . . . 5 ⊢ ((((𝜑 ∧ 𝐴 = 𝐶) ∧ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑦 ∈ 𝑃) → (⟨“𝐴𝐵𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩ → ⟨“𝐴𝐵𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝐹𝑦”⟩))
59	58	reximdva 3175	. . . 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 2762	. . . . . 6 ⊢ (hlG‘𝐺) = (hlG‘𝐺)
62	4	ad2antrr 736	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) → 𝐺 ∈ TarskiG)
63	62	ad2antrr 736	. . . . . 6 ⊢ (((((𝜑 ∧ 𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑥 ∈ 𝑃) ∧ ¬ 𝑥 ∈ (𝐷𝐿𝐸)) → 𝐺 ∈ TarskiG)
64	8	ad2antrr 736	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) → 𝐵 ∈ 𝑃)
65	64	ad2antrr 736	. . . . . 6 ⊢ (((((𝜑 ∧ 𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑥 ∈ 𝑃) ∧ ¬ 𝑥 ∈ (𝐷𝐿𝐸)) → 𝐵 ∈ 𝑃)
66	6	ad2antrr 736	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) → 𝐴 ∈ 𝑃)
67	66	ad2antrr 736	. . . . . 6 ⊢ (((((𝜑 ∧ 𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑥 ∈ 𝑃) ∧ ¬ 𝑥 ∈ (𝐷𝐿𝐸)) → 𝐴 ∈ 𝑃)
68	10	ad2antrr 736	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) → 𝑋 ∈ 𝑃)
69	68	ad2antrr 736	. . . . . 6 ⊢ (((((𝜑 ∧ 𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑥 ∈ 𝑃) ∧ ¬ 𝑥 ∈ (𝐷𝐿𝐸)) → 𝑋 ∈ 𝑃)
70	15	ad2antrr 736	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) → 𝐸 ∈ 𝑃)
71	70	ad2antrr 736	. . . . . 6 ⊢ (((((𝜑 ∧ 𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑥 ∈ 𝑃) ∧ ¬ 𝑥 ∈ (𝐷𝐿𝐸)) → 𝐸 ∈ 𝑃)
72	13	ad2antrr 736	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) → 𝐷 ∈ 𝑃)
73	72	ad2antrr 736	. . . . . 6 ⊢ (((((𝜑 ∧ 𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑥 ∈ 𝑃) ∧ ¬ 𝑥 ∈ (𝐷𝐿𝐸)) → 𝐷 ∈ 𝑃)
74		simplr 778	. . . . . 6 ⊢ (((((𝜑 ∧ 𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑥 ∈ 𝑃) ∧ ¬ 𝑥 ∈ (𝐷𝐿𝐸)) → 𝑥 ∈ 𝑃)
75		simpllr 785	. . . . . 6 ⊢ (((((𝜑 ∧ 𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑥 ∈ 𝑃) ∧ ¬ 𝑥 ∈ (𝐷𝐿𝐸)) → ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋))
76		simpr 488	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑥 ∈ 𝑃) ∧ ¬ 𝑥 ∈ (𝐷𝐿𝐸)) → ¬ 𝑥 ∈ (𝐷𝐿𝐸))
77	22	ad2antrr 736	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) → (𝐴(dist‘𝐺)𝐵) = (𝐷(dist‘𝐺)𝐸))
78		simpr 488	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) → ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋))
79	1, 3, 2, 62, 64, 66, 68, 78	ncolne1 28794	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) → 𝐵 ≠ 𝐴)
80	79	necomd 3012	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) → 𝐴 ≠ 𝐵)
81	1, 17, 3, 62, 66, 64, 72, 70, 77, 80	tgcgrneq 28652	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) → 𝐷 ≠ 𝐸)
82	81	ad2antrr 736	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑥 ∈ 𝑃) ∧ ¬ 𝑥 ∈ (𝐷𝐿𝐸)) → 𝐷 ≠ 𝐸)
83	82	neneqd 2962	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑥 ∈ 𝑃) ∧ ¬ 𝑥 ∈ (𝐷𝐿𝐸)) → ¬ 𝐷 = 𝐸)
84		ioran 997	. . . . . . . . 9 ⊢ (¬ (𝑥 ∈ (𝐷𝐿𝐸) ∨ 𝐷 = 𝐸) ↔ (¬ 𝑥 ∈ (𝐷𝐿𝐸) ∧ ¬ 𝐷 = 𝐸))
85	76, 83, 84	sylanbrc 592	. . . . . . . 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 28730	. . . . . . 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 28731	. . . . . 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 28649	. . . . . . 7 ⊢ (𝜑 → (𝐵(dist‘𝐺)𝐴) = (𝐸(dist‘𝐺)𝐷))
89	88	ad4antr 742	. . . . . 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 28977	. . . . 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 746	. . . . . . . . 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 736	. . . . . . . . . . 11 ⊢ (((((((𝜑 ∧ 𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑥 ∈ 𝑃) ∧ ¬ 𝑥 ∈ (𝐷𝐿𝐸)) ∧ 𝑦 ∈ 𝑃) ∧ ⟨“𝐵𝐴𝑋”⟩(cgrG‘𝐺)⟨“𝐸𝐷𝑦”⟩) → 𝐺 ∈ TarskiG)
93	65	ad2antrr 736	. . . . . . . . . . 11 ⊢ (((((((𝜑 ∧ 𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑥 ∈ 𝑃) ∧ ¬ 𝑥 ∈ (𝐷𝐿𝐸)) ∧ 𝑦 ∈ 𝑃) ∧ ⟨“𝐵𝐴𝑋”⟩(cgrG‘𝐺)⟨“𝐸𝐷𝑦”⟩) → 𝐵 ∈ 𝑃)
94	67	ad2antrr 736	. . . . . . . . . . 11 ⊢ (((((((𝜑 ∧ 𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑥 ∈ 𝑃) ∧ ¬ 𝑥 ∈ (𝐷𝐿𝐸)) ∧ 𝑦 ∈ 𝑃) ∧ ⟨“𝐵𝐴𝑋”⟩(cgrG‘𝐺)⟨“𝐸𝐷𝑦”⟩) → 𝐴 ∈ 𝑃)
95	69	ad2antrr 736	. . . . . . . . . . 11 ⊢ (((((((𝜑 ∧ 𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑥 ∈ 𝑃) ∧ ¬ 𝑥 ∈ (𝐷𝐿𝐸)) ∧ 𝑦 ∈ 𝑃) ∧ ⟨“𝐵𝐴𝑋”⟩(cgrG‘𝐺)⟨“𝐸𝐷𝑦”⟩) → 𝑋 ∈ 𝑃)
96	71	ad2antrr 736	. . . . . . . . . . 11 ⊢ (((((((𝜑 ∧ 𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑥 ∈ 𝑃) ∧ ¬ 𝑥 ∈ (𝐷𝐿𝐸)) ∧ 𝑦 ∈ 𝑃) ∧ ⟨“𝐵𝐴𝑋”⟩(cgrG‘𝐺)⟨“𝐸𝐷𝑦”⟩) → 𝐸 ∈ 𝑃)
97	73	ad2antrr 736	. . . . . . . . . . 11 ⊢ (((((((𝜑 ∧ 𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑥 ∈ 𝑃) ∧ ¬ 𝑥 ∈ (𝐷𝐿𝐸)) ∧ 𝑦 ∈ 𝑃) ∧ ⟨“𝐵𝐴𝑋”⟩(cgrG‘𝐺)⟨“𝐸𝐷𝑦”⟩) → 𝐷 ∈ 𝑃)
98		simplr 778	. . . . . . . . . . 11 ⊢ (((((((𝜑 ∧ 𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑥 ∈ 𝑃) ∧ ¬ 𝑥 ∈ (𝐷𝐿𝐸)) ∧ 𝑦 ∈ 𝑃) ∧ ⟨“𝐵𝐴𝑋”⟩(cgrG‘𝐺)⟨“𝐸𝐷𝑦”⟩) → 𝑦 ∈ 𝑃)
99		simpr 488	. . . . . . . . . . 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 28690	. . . . . . . . . 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 28691	. . . . . . . . . . 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 28649	. . . . . . . . . 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 744	. . . . . . . . . . 11 ⊢ (((((((𝜑 ∧ 𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑥 ∈ 𝑃) ∧ ¬ 𝑥 ∈ (𝐷𝐿𝐸)) ∧ 𝑦 ∈ 𝑃) ∧ ⟨“𝐵𝐴𝑋”⟩(cgrG‘𝐺)⟨“𝐸𝐷𝑦”⟩) → 𝐶 ∈ 𝑃)
104	46	ad5antr 744	. . . . . . . . . . 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 28649	. . . . . . . . . . . 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 797	. . . . . . . . . . . . 13 ⊢ (((((((𝜑 ∧ 𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑥 ∈ 𝑃) ∧ ¬ 𝑥 ∈ (𝐷𝐿𝐸)) ∧ 𝑦 ∈ 𝑃) ∧ ⟨“𝐵𝐴𝑋”⟩(cgrG‘𝐺)⟨“𝐸𝐷𝑦”⟩) → 𝐴 = 𝐶)
107	106	oveq2d 7412	. . . . . . . . . . . 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 744	. . . . . . . . . . . . 13 ⊢ (((((((𝜑 ∧ 𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑥 ∈ 𝑃) ∧ ¬ 𝑥 ∈ (𝐷𝐿𝐸)) ∧ 𝑦 ∈ 𝑃) ∧ ⟨“𝐵𝐴𝑋”⟩(cgrG‘𝐺)⟨“𝐸𝐷𝑦”⟩) → 𝐷 = 𝐹)
109	108	oveq2d 7412	. . . . . . . . . . . 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 2805	. . . . . . . . . . 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 28649	. . . . . . . . . 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 1141	. . . . . . . . 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 28700	. . . . . . . . 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 723	. . . . . . . 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 416	. . . . . . 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 495	. . . . . 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 3175	. . . . 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 28734	. . . . 5 ⊢ (((𝜑 ∧ 𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) → 𝐺DimTarskiG≥2)
120	1, 3, 2, 62, 119, 72, 70, 81	tglowdim2ln 28821	. . . 4 ⊢ (((𝜑 ∧ 𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) → ∃𝑥 ∈ 𝑃 ¬ 𝑥 ∈ (𝐷𝐿𝐸))
121	118, 120	r19.29a 3170	. . 3 ⊢ (((𝜑 ∧ 𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) → ∃𝑦 ∈ 𝑃 ⟨“𝐴𝐵𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝐹𝑦”⟩)
122	60, 121	pm2.61dan 822	. 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 28727	. . . . . 6 ⊢ (𝜑 → (𝑋 ∈ (𝐶𝐿𝐴) ∨ 𝐶 = 𝐴))
125	1, 2, 3, 4, 19, 6, 10, 124	colrot1 28728	. . . . 5 ⊢ (𝜑 → (𝐶 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋))
126	1, 2, 3, 4, 6, 19, 10, 12, 13, 20, 17, 125, 48	lnext 28736	. . . 4 ⊢ (𝜑 → ∃𝑦 ∈ 𝑃 ⟨“𝐴𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐹𝑦”⟩)
127	126	adantr 484	. . 3 ⊢ ((𝜑 ∧ 𝐴 ≠ 𝐶) → ∃𝑦 ∈ 𝑃 ⟨“𝐴𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐹𝑦”⟩)
128	21	ad3antrrr 740	. . . . . 6 ⊢ ((((𝜑 ∧ 𝐴 ≠ 𝐶) ∧ 𝑦 ∈ 𝑃) ∧ ⟨“𝐴𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐹𝑦”⟩) → ⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝐹”⟩)
129	4	ad3antrrr 740	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝐴 ≠ 𝐶) ∧ 𝑦 ∈ 𝑃) ∧ ⟨“𝐴𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐹𝑦”⟩) → 𝐺 ∈ TarskiG)
130	10	ad3antrrr 740	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝐴 ≠ 𝐶) ∧ 𝑦 ∈ 𝑃) ∧ ⟨“𝐴𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐹𝑦”⟩) → 𝑋 ∈ 𝑃)
131	6	ad3antrrr 740	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝐴 ≠ 𝐶) ∧ 𝑦 ∈ 𝑃) ∧ ⟨“𝐴𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐹𝑦”⟩) → 𝐴 ∈ 𝑃)
132		simplr 778	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝐴 ≠ 𝐶) ∧ 𝑦 ∈ 𝑃) ∧ ⟨“𝐴𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐹𝑦”⟩) → 𝑦 ∈ 𝑃)
133	13	ad3antrrr 740	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝐴 ≠ 𝐶) ∧ 𝑦 ∈ 𝑃) ∧ ⟨“𝐴𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐹𝑦”⟩) → 𝐷 ∈ 𝑃)
134	19	ad3antrrr 740	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝐴 ≠ 𝐶) ∧ 𝑦 ∈ 𝑃) ∧ ⟨“𝐴𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐹𝑦”⟩) → 𝐶 ∈ 𝑃)
135	20	ad3antrrr 740	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝐴 ≠ 𝐶) ∧ 𝑦 ∈ 𝑃) ∧ ⟨“𝐴𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐹𝑦”⟩) → 𝐹 ∈ 𝑃)
136		simpr 488	. . . . . . . . 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 28691	. . . . . . . 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 28649	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝐴 ≠ 𝐶) ∧ 𝑦 ∈ 𝑃) ∧ ⟨“𝐴𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐹𝑦”⟩) → (𝐴(dist‘𝐺)𝑋) = (𝐷(dist‘𝐺)𝑦))
139	8	ad3antrrr 740	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝐴 ≠ 𝐶) ∧ 𝑦 ∈ 𝑃) ∧ ⟨“𝐴𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐹𝑦”⟩) → 𝐵 ∈ 𝑃)
140	15	ad3antrrr 740	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝐴 ≠ 𝐶) ∧ 𝑦 ∈ 𝑃) ∧ ⟨“𝐴𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐹𝑦”⟩) → 𝐸 ∈ 𝑃)
141	125	ad3antrrr 740	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝐴 ≠ 𝐶) ∧ 𝑦 ∈ 𝑃) ∧ ⟨“𝐴𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐹𝑦”⟩) → (𝐶 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋))
142	22	ad3antrrr 740	. . . . . . . . 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 28690	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐵(dist‘𝐺)𝐶) = (𝐸(dist‘𝐺)𝐹))
144	1, 17, 3, 4, 8, 19, 15, 20, 143	tgcgrcomlr 28649	. . . . . . . . . 10 ⊢ (𝜑 → (𝐶(dist‘𝐺)𝐵) = (𝐹(dist‘𝐺)𝐸))
145	144	ad3antrrr 740	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝐴 ≠ 𝐶) ∧ 𝑦 ∈ 𝑃) ∧ ⟨“𝐴𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐹𝑦”⟩) → (𝐶(dist‘𝐺)𝐵) = (𝐹(dist‘𝐺)𝐸))
146		simpllr 785	. . . . . . . . 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 28737	. . . . . . . 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 28649	. . . . . . 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 28690	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝐴 ≠ 𝐶) ∧ 𝑦 ∈ 𝑃) ∧ ⟨“𝐴𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐹𝑦”⟩) → (𝐶(dist‘𝐺)𝑋) = (𝐹(dist‘𝐺)𝑦))
150	138, 148, 149	3jca 1141	. . . . . 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 28700	. . . . . 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 723	. . . . 5 ⊢ ((((𝜑 ∧ 𝐴 ≠ 𝐶) ∧ 𝑦 ∈ 𝑃) ∧ ⟨“𝐴𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐹𝑦”⟩) → ⟨“𝐴𝐵𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝐹𝑦”⟩)
153	152	ex 416	. . . 4 ⊢ (((𝜑 ∧ 𝐴 ≠ 𝐶) ∧ 𝑦 ∈ 𝑃) → (⟨“𝐴𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐹𝑦”⟩ → ⟨“𝐴𝐵𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝐹𝑦”⟩))
154	153	reximdva 3175	. . 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 3044	1 ⊢ (𝜑 → ∃𝑦 ∈ 𝑃 ⟨“𝐴𝐵𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝐹𝑦”⟩)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399   ∨ wo 858   ∧ w3a 1098   = wceq 1560   ∈ wcel 2142   ≠ wne 2957  ∃wrex 3086   class class class wbr 5100  ‘cfv 6521  (class class class)co 7396  ⟨“cs3 14855  ⟨“cs4 14856  Basecbs 17245  distcds 17295  TarskiGcstrkg 28596  Itvcitv 28602  LineGclng 28603  cgrGccgrg 28679  hlGchlg 28769  hpGchpg 28930

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-rep 5227  ax-sep 5246  ax-nul 5256  ax-pow 5322  ax-pr 5390  ax-un 7718  ax-cnex 11129  ax-resscn 11130  ax-1cn 11131  ax-icn 11132  ax-addcl 11133  ax-addrcl 11134  ax-mulcl 11135  ax-mulrcl 11136  ax-mulcom 11137  ax-addass 11138  ax-mulass 11139  ax-distr 11140  ax-i2m1 11141  ax-1ne0 11142  ax-1rid 11143  ax-rnegex 11144  ax-rrecex 11145  ax-cnre 11146  ax-pre-lttri 11147  ax-pre-lttrn 11148  ax-pre-ltadd 11149  ax-pre-mulgt0 11150

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1099  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-nf 1804  df-sb 2091  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-nel 3062  df-ral 3077  df-rex 3087  df-rmo 3367  df-reu 3368  df-rab 3415  df-v 3456  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4481  df-pw 4557  df-sn 4583  df-pr 4585  df-tp 4587  df-op 4589  df-uni 4866  df-int 4906  df-iun 4951  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5542  df-eprel 5547  df-po 5555  df-so 5556  df-fr 5600  df-we 5602  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-pred 6288  df-ord 6349  df-on 6350  df-lim 6351  df-suc 6352  df-iota 6477  df-fun 6523  df-fn 6524  df-f 6525  df-f1 6526  df-fo 6527  df-f1o 6528  df-fv 6529  df-riota 7353  df-ov 7399  df-oprab 7400  df-mpo 7401  df-om 7847  df-1st 7970  df-2nd 7971  df-frecs 8262  df-wrecs 8293  df-recs 8342  df-rdg 8381  df-1o 8437  df-oadd 8441  df-er 8678  df-map 8810  df-pm 8811  df-en 8928  df-dom 8929  df-sdom 8930  df-fin 8931  df-dju 9859  df-card 9897  df-pnf 11218  df-mnf 11219  df-xr 11220  df-ltxr 11221  df-le 11222  df-sub 11416  df-neg 11417  df-nn 12211  df-2 12280  df-3 12281  df-4 12282  df-n0 12482  df-xnn0 12555  df-z 12569  df-uz 12840  df-fz 13513  df-fzo 13660  df-hash 14344  df-word 14527  df-concat 14584  df-s1 14610  df-s2 14861  df-s3 14862  df-s4 14863  df-trkgc 28617  df-trkgb 28618  df-trkgcb 28619  df-trkgld 28621  df-trkg 28622  df-cgrg 28680  df-ismt 28702  df-leg 28752  df-hlg 28770  df-mir 28826  df-rag 28867  df-perpg 28869  df-hpg 28931  df-mid 28947  df-lmi 28948

This theorem is referenced by: (None)