Theorem cgrg3col4 29125

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 2769	. . . . 5 ⊢ (Itv‘𝐺) = (Itv‘𝐺)
4		isleag.g	. . . . . 6 ⊢ (𝜑 → 𝐺 ∈ TarskiG)
5	4	ad2antrr 738	. . . . 5 ⊢ (((𝜑 ∧ 𝐴 = 𝐶) ∧ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) → 𝐺 ∈ TarskiG)
6		isleag.a	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ 𝑃)
7	6	ad2antrr 738	. . . . 5 ⊢ (((𝜑 ∧ 𝐴 = 𝐶) ∧ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) → 𝐴 ∈ 𝑃)
8		isleag.b	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ 𝑃)
9	8	ad2antrr 738	. . . . 5 ⊢ (((𝜑 ∧ 𝐴 = 𝐶) ∧ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) → 𝐵 ∈ 𝑃)
10		cgrg3col4.x	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ 𝑃)
11	10	ad2antrr 738	. . . . 5 ⊢ (((𝜑 ∧ 𝐴 = 𝐶) ∧ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) → 𝑋 ∈ 𝑃)
12		eqid 2769	. . . . 5 ⊢ (cgrG‘𝐺) = (cgrG‘𝐺)
13		isleag.d	. . . . . 6 ⊢ (𝜑 → 𝐷 ∈ 𝑃)
14	13	ad2antrr 738	. . . . 5 ⊢ (((𝜑 ∧ 𝐴 = 𝐶) ∧ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) → 𝐷 ∈ 𝑃)
15		isleag.e	. . . . . 6 ⊢ (𝜑 → 𝐸 ∈ 𝑃)
16	15	ad2antrr 738	. . . . 5 ⊢ (((𝜑 ∧ 𝐴 = 𝐶) ∧ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) → 𝐸 ∈ 𝑃)
17		eqid 2769	. . . . 5 ⊢ (dist‘𝐺) = (dist‘𝐺)
18		simpr 489	. . . . 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 28755	. . . . . 6 ⊢ (𝜑 → (𝐴(dist‘𝐺)𝐵) = (𝐷(dist‘𝐺)𝐸))
23	22	ad2antrr 738	. . . . 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 28802	. . . 4 ⊢ (((𝜑 ∧ 𝐴 = 𝐶) ∧ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) → ∃𝑦 ∈ 𝑃 ⟨“𝐴𝐵𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩)
25	21	ad4antr 744	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝐴 = 𝐶) ∧ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑦 ∈ 𝑃) ∧ ⟨“𝐴𝐵𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩) → ⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝐹”⟩)
26	5	ad2antrr 738	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝐴 = 𝐶) ∧ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑦 ∈ 𝑃) ∧ ⟨“𝐴𝐵𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩) → 𝐺 ∈ TarskiG)
27	11	ad2antrr 738	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝐴 = 𝐶) ∧ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑦 ∈ 𝑃) ∧ ⟨“𝐴𝐵𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩) → 𝑋 ∈ 𝑃)
28	7	ad2antrr 738	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝐴 = 𝐶) ∧ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑦 ∈ 𝑃) ∧ ⟨“𝐴𝐵𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩) → 𝐴 ∈ 𝑃)
29		simplr 780	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝐴 = 𝐶) ∧ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑦 ∈ 𝑃) ∧ ⟨“𝐴𝐵𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩) → 𝑦 ∈ 𝑃)
30	14	ad2antrr 738	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝐴 = 𝐶) ∧ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑦 ∈ 𝑃) ∧ ⟨“𝐴𝐵𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩) → 𝐷 ∈ 𝑃)
31	9	ad2antrr 738	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝐴 = 𝐶) ∧ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑦 ∈ 𝑃) ∧ ⟨“𝐴𝐵𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩) → 𝐵 ∈ 𝑃)
32	16	ad2antrr 738	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝐴 = 𝐶) ∧ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑦 ∈ 𝑃) ∧ ⟨“𝐴𝐵𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩) → 𝐸 ∈ 𝑃)
33		simpr 489	. . . . . . . . . 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 28757	. . . . . . . . 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 28715	. . . . . . . 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 28756	. . . . . . . 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 744	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝐴 = 𝐶) ∧ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑦 ∈ 𝑃) ∧ ⟨“𝐴𝐵𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩) → 𝐶 ∈ 𝑃)
38	20	ad4antr 744	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝐴 = 𝐶) ∧ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑦 ∈ 𝑃) ∧ ⟨“𝐴𝐵𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩) → 𝐹 ∈ 𝑃)
39		simpr 489	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐴 = 𝐶) → 𝐴 = 𝐶)
40	39	ad3antrrr 742	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝐴 = 𝐶) ∧ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑦 ∈ 𝑃) ∧ ⟨“𝐴𝐵𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩) → 𝐴 = 𝐶)
41	40	oveq2d 7427	. . . . . . . . . 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 485	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝐴 = 𝐶) → 𝐺 ∈ TarskiG)
43	6	adantr 485	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝐴 = 𝐶) → 𝐴 ∈ 𝑃)
44	19	adantr 485	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝐴 = 𝐶) → 𝐶 ∈ 𝑃)
45	13	adantr 485	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝐴 = 𝐶) → 𝐷 ∈ 𝑃)
46	20	adantr 485	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝐴 = 𝐶) → 𝐹 ∈ 𝑃)
47	1, 17, 3, 12, 4, 6, 8, 19, 13, 15, 20, 21	cgr3simp3 28757	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐶(dist‘𝐺)𝐴) = (𝐹(dist‘𝐺)𝐷))
48	1, 17, 3, 4, 19, 6, 20, 13, 47	tgcgrcomlr 28715	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐴(dist‘𝐺)𝐶) = (𝐷(dist‘𝐺)𝐹))
49	48	adantr 485	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝐴 = 𝐶) → (𝐴(dist‘𝐺)𝐶) = (𝐷(dist‘𝐺)𝐹))
50	1, 17, 3, 42, 43, 44, 45, 46, 49, 39	tgcgreq 28717	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐴 = 𝐶) → 𝐷 = 𝐹)
51	50	ad3antrrr 742	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝐴 = 𝐶) ∧ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑦 ∈ 𝑃) ∧ ⟨“𝐴𝐵𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩) → 𝐷 = 𝐹)
52	51	oveq2d 7427	. . . . . . . . . 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 2812	. . . . . . . . 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 28715	. . . . . . . 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 1144	. . . . . . 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 28766	. . . . . . 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 725	. . . . . 6 ⊢ (((((𝜑 ∧ 𝐴 = 𝐶) ∧ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑦 ∈ 𝑃) ∧ ⟨“𝐴𝐵𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩) → ⟨“𝐴𝐵𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝐹𝑦”⟩)
58	57	ex 417	. . . . 5 ⊢ ((((𝜑 ∧ 𝐴 = 𝐶) ∧ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑦 ∈ 𝑃) → (⟨“𝐴𝐵𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩ → ⟨“𝐴𝐵𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝐹𝑦”⟩))
59	58	reximdva 3184	. . . 4 ⊢ (((𝜑 ∧ 𝐴 = 𝐶) ∧ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) → (∃𝑦 ∈ 𝑃 ⟨“𝐴𝐵𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩ → ∃𝑦 ∈ 𝑃 ⟨“𝐴𝐵𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝐹𝑦”⟩))
60	24, 59	mpd 16	. . 3 ⊢ (((𝜑 ∧ 𝐴 = 𝐶) ∧ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) → ∃𝑦 ∈ 𝑃 ⟨“𝐴𝐵𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝐹𝑦”⟩)
61		eqid 2769	. . . . . 6 ⊢ (hlG‘𝐺) = (hlG‘𝐺)
62	4	ad2antrr 738	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) → 𝐺 ∈ TarskiG)
63	62	ad2antrr 738	. . . . . 6 ⊢ (((((𝜑 ∧ 𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑥 ∈ 𝑃) ∧ ¬ 𝑥 ∈ (𝐷𝐿𝐸)) → 𝐺 ∈ TarskiG)
64	8	ad2antrr 738	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) → 𝐵 ∈ 𝑃)
65	64	ad2antrr 738	. . . . . 6 ⊢ (((((𝜑 ∧ 𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑥 ∈ 𝑃) ∧ ¬ 𝑥 ∈ (𝐷𝐿𝐸)) → 𝐵 ∈ 𝑃)
66	6	ad2antrr 738	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) → 𝐴 ∈ 𝑃)
67	66	ad2antrr 738	. . . . . 6 ⊢ (((((𝜑 ∧ 𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑥 ∈ 𝑃) ∧ ¬ 𝑥 ∈ (𝐷𝐿𝐸)) → 𝐴 ∈ 𝑃)
68	10	ad2antrr 738	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) → 𝑋 ∈ 𝑃)
69	68	ad2antrr 738	. . . . . 6 ⊢ (((((𝜑 ∧ 𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑥 ∈ 𝑃) ∧ ¬ 𝑥 ∈ (𝐷𝐿𝐸)) → 𝑋 ∈ 𝑃)
70	15	ad2antrr 738	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) → 𝐸 ∈ 𝑃)
71	70	ad2antrr 738	. . . . . 6 ⊢ (((((𝜑 ∧ 𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑥 ∈ 𝑃) ∧ ¬ 𝑥 ∈ (𝐷𝐿𝐸)) → 𝐸 ∈ 𝑃)
72	13	ad2antrr 738	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) → 𝐷 ∈ 𝑃)
73	72	ad2antrr 738	. . . . . 6 ⊢ (((((𝜑 ∧ 𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑥 ∈ 𝑃) ∧ ¬ 𝑥 ∈ (𝐷𝐿𝐸)) → 𝐷 ∈ 𝑃)
74		simplr 780	. . . . . 6 ⊢ (((((𝜑 ∧ 𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑥 ∈ 𝑃) ∧ ¬ 𝑥 ∈ (𝐷𝐿𝐸)) → 𝑥 ∈ 𝑃)
75		simpllr 787	. . . . . 6 ⊢ (((((𝜑 ∧ 𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑥 ∈ 𝑃) ∧ ¬ 𝑥 ∈ (𝐷𝐿𝐸)) → ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋))
76		simpr 489	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑥 ∈ 𝑃) ∧ ¬ 𝑥 ∈ (𝐷𝐿𝐸)) → ¬ 𝑥 ∈ (𝐷𝐿𝐸))
77	22	ad2antrr 738	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) → (𝐴(dist‘𝐺)𝐵) = (𝐷(dist‘𝐺)𝐸))
78		simpr 489	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) → ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋))
79	1, 3, 2, 62, 64, 66, 68, 78	ncolne1 28860	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) → 𝐵 ≠ 𝐴)
80	79	necomd 3019	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) → 𝐴 ≠ 𝐵)
81	1, 17, 3, 62, 66, 64, 72, 70, 77, 80	tgcgrneq 28718	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) → 𝐷 ≠ 𝐸)
82	81	ad2antrr 738	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑥 ∈ 𝑃) ∧ ¬ 𝑥 ∈ (𝐷𝐿𝐸)) → 𝐷 ≠ 𝐸)
83	82	neneqd 2969	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑥 ∈ 𝑃) ∧ ¬ 𝑥 ∈ (𝐷𝐿𝐸)) → ¬ 𝐷 = 𝐸)
84		ioran 999	. . . . . . . . 9 ⊢ (¬ (𝑥 ∈ (𝐷𝐿𝐸) ∨ 𝐷 = 𝐸) ↔ (¬ 𝑥 ∈ (𝐷𝐿𝐸) ∧ ¬ 𝐷 = 𝐸))
85	76, 83, 84	sylanbrc 594	. . . . . . . 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 28796	. . . . . . 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 28797	. . . . . 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 28715	. . . . . . 7 ⊢ (𝜑 → (𝐵(dist‘𝐺)𝐴) = (𝐸(dist‘𝐺)𝐷))
89	88	ad4antr 744	. . . . . 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 29072	. . . . 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 748	. . . . . . . . 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 738	. . . . . . . . . . 11 ⊢ (((((((𝜑 ∧ 𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑥 ∈ 𝑃) ∧ ¬ 𝑥 ∈ (𝐷𝐿𝐸)) ∧ 𝑦 ∈ 𝑃) ∧ ⟨“𝐵𝐴𝑋”⟩(cgrG‘𝐺)⟨“𝐸𝐷𝑦”⟩) → 𝐺 ∈ TarskiG)
93	65	ad2antrr 738	. . . . . . . . . . 11 ⊢ (((((((𝜑 ∧ 𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑥 ∈ 𝑃) ∧ ¬ 𝑥 ∈ (𝐷𝐿𝐸)) ∧ 𝑦 ∈ 𝑃) ∧ ⟨“𝐵𝐴𝑋”⟩(cgrG‘𝐺)⟨“𝐸𝐷𝑦”⟩) → 𝐵 ∈ 𝑃)
94	67	ad2antrr 738	. . . . . . . . . . 11 ⊢ (((((((𝜑 ∧ 𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑥 ∈ 𝑃) ∧ ¬ 𝑥 ∈ (𝐷𝐿𝐸)) ∧ 𝑦 ∈ 𝑃) ∧ ⟨“𝐵𝐴𝑋”⟩(cgrG‘𝐺)⟨“𝐸𝐷𝑦”⟩) → 𝐴 ∈ 𝑃)
95	69	ad2antrr 738	. . . . . . . . . . 11 ⊢ (((((((𝜑 ∧ 𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑥 ∈ 𝑃) ∧ ¬ 𝑥 ∈ (𝐷𝐿𝐸)) ∧ 𝑦 ∈ 𝑃) ∧ ⟨“𝐵𝐴𝑋”⟩(cgrG‘𝐺)⟨“𝐸𝐷𝑦”⟩) → 𝑋 ∈ 𝑃)
96	71	ad2antrr 738	. . . . . . . . . . 11 ⊢ (((((((𝜑 ∧ 𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑥 ∈ 𝑃) ∧ ¬ 𝑥 ∈ (𝐷𝐿𝐸)) ∧ 𝑦 ∈ 𝑃) ∧ ⟨“𝐵𝐴𝑋”⟩(cgrG‘𝐺)⟨“𝐸𝐷𝑦”⟩) → 𝐸 ∈ 𝑃)
97	73	ad2antrr 738	. . . . . . . . . . 11 ⊢ (((((((𝜑 ∧ 𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑥 ∈ 𝑃) ∧ ¬ 𝑥 ∈ (𝐷𝐿𝐸)) ∧ 𝑦 ∈ 𝑃) ∧ ⟨“𝐵𝐴𝑋”⟩(cgrG‘𝐺)⟨“𝐸𝐷𝑦”⟩) → 𝐷 ∈ 𝑃)
98		simplr 780	. . . . . . . . . . 11 ⊢ (((((((𝜑 ∧ 𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑥 ∈ 𝑃) ∧ ¬ 𝑥 ∈ (𝐷𝐿𝐸)) ∧ 𝑦 ∈ 𝑃) ∧ ⟨“𝐵𝐴𝑋”⟩(cgrG‘𝐺)⟨“𝐸𝐷𝑦”⟩) → 𝑦 ∈ 𝑃)
99		simpr 489	. . . . . . . . . . 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 28756	. . . . . . . . . 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 28757	. . . . . . . . . . 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 28715	. . . . . . . . . 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 746	. . . . . . . . . . 11 ⊢ (((((((𝜑 ∧ 𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑥 ∈ 𝑃) ∧ ¬ 𝑥 ∈ (𝐷𝐿𝐸)) ∧ 𝑦 ∈ 𝑃) ∧ ⟨“𝐵𝐴𝑋”⟩(cgrG‘𝐺)⟨“𝐸𝐷𝑦”⟩) → 𝐶 ∈ 𝑃)
104	46	ad5antr 746	. . . . . . . . . . 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 28715	. . . . . . . . . . . 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 799	. . . . . . . . . . . . 13 ⊢ (((((((𝜑 ∧ 𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑥 ∈ 𝑃) ∧ ¬ 𝑥 ∈ (𝐷𝐿𝐸)) ∧ 𝑦 ∈ 𝑃) ∧ ⟨“𝐵𝐴𝑋”⟩(cgrG‘𝐺)⟨“𝐸𝐷𝑦”⟩) → 𝐴 = 𝐶)
107	106	oveq2d 7427	. . . . . . . . . . . 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 746	. . . . . . . . . . . . 13 ⊢ (((((((𝜑 ∧ 𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑥 ∈ 𝑃) ∧ ¬ 𝑥 ∈ (𝐷𝐿𝐸)) ∧ 𝑦 ∈ 𝑃) ∧ ⟨“𝐵𝐴𝑋”⟩(cgrG‘𝐺)⟨“𝐸𝐷𝑦”⟩) → 𝐷 = 𝐹)
109	108	oveq2d 7427	. . . . . . . . . . . 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 2812	. . . . . . . . . . 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 28715	. . . . . . . . . 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 1144	. . . . . . . . 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 28766	. . . . . . . . 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 725	. . . . . . . 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 417	. . . . . . 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 496	. . . . . 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 3184	. . . . 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 16	. . . 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 28800	. . . . 5 ⊢ (((𝜑 ∧ 𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) → 𝐺DimTarskiG≥2)
120	1, 3, 2, 62, 119, 72, 70, 81	tglowdim2ln 28887	. . . 4 ⊢ (((𝜑 ∧ 𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) → ∃𝑥 ∈ 𝑃 ¬ 𝑥 ∈ (𝐷𝐿𝐸))
121	118, 120	r19.29a 3179	. . 3 ⊢ (((𝜑 ∧ 𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) → ∃𝑦 ∈ 𝑃 ⟨“𝐴𝐵𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝐹𝑦”⟩)
122	60, 121	pm2.61dan 824	. 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 28793	. . . . . 6 ⊢ (𝜑 → (𝑋 ∈ (𝐶𝐿𝐴) ∨ 𝐶 = 𝐴))
125	1, 2, 3, 4, 19, 6, 10, 124	colrot1 28794	. . . . 5 ⊢ (𝜑 → (𝐶 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋))
126	1, 2, 3, 4, 6, 19, 10, 12, 13, 20, 17, 125, 48	lnext 28802	. . . 4 ⊢ (𝜑 → ∃𝑦 ∈ 𝑃 ⟨“𝐴𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐹𝑦”⟩)
127	126	adantr 485	. . 3 ⊢ ((𝜑 ∧ 𝐴 ≠ 𝐶) → ∃𝑦 ∈ 𝑃 ⟨“𝐴𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐹𝑦”⟩)
128	21	ad3antrrr 742	. . . . . 6 ⊢ ((((𝜑 ∧ 𝐴 ≠ 𝐶) ∧ 𝑦 ∈ 𝑃) ∧ ⟨“𝐴𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐹𝑦”⟩) → ⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝐹”⟩)
129	4	ad3antrrr 742	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝐴 ≠ 𝐶) ∧ 𝑦 ∈ 𝑃) ∧ ⟨“𝐴𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐹𝑦”⟩) → 𝐺 ∈ TarskiG)
130	10	ad3antrrr 742	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝐴 ≠ 𝐶) ∧ 𝑦 ∈ 𝑃) ∧ ⟨“𝐴𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐹𝑦”⟩) → 𝑋 ∈ 𝑃)
131	6	ad3antrrr 742	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝐴 ≠ 𝐶) ∧ 𝑦 ∈ 𝑃) ∧ ⟨“𝐴𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐹𝑦”⟩) → 𝐴 ∈ 𝑃)
132		simplr 780	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝐴 ≠ 𝐶) ∧ 𝑦 ∈ 𝑃) ∧ ⟨“𝐴𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐹𝑦”⟩) → 𝑦 ∈ 𝑃)
133	13	ad3antrrr 742	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝐴 ≠ 𝐶) ∧ 𝑦 ∈ 𝑃) ∧ ⟨“𝐴𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐹𝑦”⟩) → 𝐷 ∈ 𝑃)
134	19	ad3antrrr 742	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝐴 ≠ 𝐶) ∧ 𝑦 ∈ 𝑃) ∧ ⟨“𝐴𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐹𝑦”⟩) → 𝐶 ∈ 𝑃)
135	20	ad3antrrr 742	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝐴 ≠ 𝐶) ∧ 𝑦 ∈ 𝑃) ∧ ⟨“𝐴𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐹𝑦”⟩) → 𝐹 ∈ 𝑃)
136		simpr 489	. . . . . . . . 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 28757	. . . . . . . 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 28715	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝐴 ≠ 𝐶) ∧ 𝑦 ∈ 𝑃) ∧ ⟨“𝐴𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐹𝑦”⟩) → (𝐴(dist‘𝐺)𝑋) = (𝐷(dist‘𝐺)𝑦))
139	8	ad3antrrr 742	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝐴 ≠ 𝐶) ∧ 𝑦 ∈ 𝑃) ∧ ⟨“𝐴𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐹𝑦”⟩) → 𝐵 ∈ 𝑃)
140	15	ad3antrrr 742	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝐴 ≠ 𝐶) ∧ 𝑦 ∈ 𝑃) ∧ ⟨“𝐴𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐹𝑦”⟩) → 𝐸 ∈ 𝑃)
141	125	ad3antrrr 742	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝐴 ≠ 𝐶) ∧ 𝑦 ∈ 𝑃) ∧ ⟨“𝐴𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐹𝑦”⟩) → (𝐶 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋))
142	22	ad3antrrr 742	. . . . . . . . 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 28756	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐵(dist‘𝐺)𝐶) = (𝐸(dist‘𝐺)𝐹))
144	1, 17, 3, 4, 8, 19, 15, 20, 143	tgcgrcomlr 28715	. . . . . . . . . 10 ⊢ (𝜑 → (𝐶(dist‘𝐺)𝐵) = (𝐹(dist‘𝐺)𝐸))
145	144	ad3antrrr 742	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝐴 ≠ 𝐶) ∧ 𝑦 ∈ 𝑃) ∧ ⟨“𝐴𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐹𝑦”⟩) → (𝐶(dist‘𝐺)𝐵) = (𝐹(dist‘𝐺)𝐸))
146		simpllr 787	. . . . . . . . 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 28803	. . . . . . . 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 28715	. . . . . . 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 28756	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝐴 ≠ 𝐶) ∧ 𝑦 ∈ 𝑃) ∧ ⟨“𝐴𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐹𝑦”⟩) → (𝐶(dist‘𝐺)𝑋) = (𝐹(dist‘𝐺)𝑦))
150	138, 148, 149	3jca 1144	. . . . . 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 28766	. . . . . 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 725	. . . . 5 ⊢ ((((𝜑 ∧ 𝐴 ≠ 𝐶) ∧ 𝑦 ∈ 𝑃) ∧ ⟨“𝐴𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐹𝑦”⟩) → ⟨“𝐴𝐵𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝐹𝑦”⟩)
153	152	ex 417	. . . 4 ⊢ (((𝜑 ∧ 𝐴 ≠ 𝐶) ∧ 𝑦 ∈ 𝑃) → (⟨“𝐴𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐹𝑦”⟩ → ⟨“𝐴𝐵𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝐹𝑦”⟩))
154	153	reximdva 3184	. . 3 ⊢ ((𝜑 ∧ 𝐴 ≠ 𝐶) → (∃𝑦 ∈ 𝑃 ⟨“𝐴𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐹𝑦”⟩ → ∃𝑦 ∈ 𝑃 ⟨“𝐴𝐵𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝐹𝑦”⟩))
155	127, 154	mpd 16	. 2 ⊢ ((𝜑 ∧ 𝐴 ≠ 𝐶) → ∃𝑦 ∈ 𝑃 ⟨“𝐴𝐵𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝐹𝑦”⟩)
156	122, 155	pm2.61dane 3051	1 ⊢ (𝜑 → ∃𝑦 ∈ 𝑃 ⟨“𝐴𝐵𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝐹𝑦”⟩)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 400   ∨ wo 860   ∧ w3a 1101   = wceq 1567   ∈ wcel 2149   ≠ wne 2964  ∃wrex 3095   class class class wbr 5113  ‘cfv 6537  (class class class)co 7411  ⟨“cs3 14879  ⟨“cs4 14880  Basecbs 17269  distcds 17319  TarskiGcstrkg 28662  Itvcitv 28668  LineGclng 28669  cgrGccgrg 28745  hlGchlg 28835  hpGchpg 28998

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5242  ax-sep 5261  ax-nul 5271  ax-pow 5337  ax-pr 5405  ax-un 7733  ax-cnex 11156  ax-resscn 11157  ax-1cn 11158  ax-icn 11159  ax-addcl 11160  ax-addrcl 11161  ax-mulcl 11162  ax-mulrcl 11163  ax-mulcom 11164  ax-addass 11165  ax-mulass 11166  ax-distr 11167  ax-i2m1 11168  ax-1ne0 11169  ax-1rid 11170  ax-rnegex 11171  ax-rrecex 11172  ax-cnre 11173  ax-pre-lttri 11174  ax-pre-lttrn 11175  ax-pre-ltadd 11176  ax-pre-mulgt0 11177

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-nel 3071  df-ral 3086  df-rex 3096  df-rmo 3376  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3933  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-tp 4599  df-op 4601  df-uni 4877  df-int 4917  df-iun 4962  df-br 5114  df-opab 5178  df-mpt 5197  df-tr 5223  df-id 5557  df-eprel 5562  df-po 5570  df-so 5571  df-fr 5615  df-we 5617  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  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 7368  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7863  df-1st 7986  df-2nd 7987  df-frecs 8278  df-wrecs 8309  df-recs 8358  df-rdg 8397  df-1o 8453  df-oadd 8457  df-er 8694  df-map 8826  df-pm 8827  df-en 8944  df-dom 8945  df-sdom 8946  df-fin 8947  df-dju 9887  df-card 9925  df-pnf 11245  df-mnf 11246  df-xr 11247  df-ltxr 11248  df-le 11249  df-sub 11443  df-neg 11444  df-nn 12234  df-2 12303  df-3 12304  df-4 12305  df-n0 12505  df-xnn0 12578  df-z 12592  df-uz 12863  df-fz 13536  df-fzo 13683  df-hash 14367  df-word 14551  df-concat 14608  df-s1 14634  df-s2 14885  df-s3 14886  df-s4 14887  df-trkgc 28683  df-trkgb 28684  df-trkgcb 28685  df-trkgld 28687  df-trkg 28688  df-cgrg 28746  df-ismt 28768  df-leg 28818  df-hlg 28836  df-mir 28892  df-rag 28933  df-perpg 28935  df-hpg 28999  df-mid 29041  df-lmi 29042

This theorem is referenced by: (None)