Theorem cgrg3col4 25534

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 2610	. . . . 5 ⊢ (Itv‘𝐺) = (Itv‘𝐺)
4		isleag.g	. . . . . 6 ⊢ (𝜑 → 𝐺 ∈ TarskiG)
5	4	ad2antrr 758	. . . . 5 ⊢ (((𝜑 ∧ 𝐴 = 𝐶) ∧ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) → 𝐺 ∈ TarskiG)
6		isleag.a	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ 𝑃)
7	6	ad2antrr 758	. . . . 5 ⊢ (((𝜑 ∧ 𝐴 = 𝐶) ∧ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) → 𝐴 ∈ 𝑃)
8		isleag.b	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ 𝑃)
9	8	ad2antrr 758	. . . . 5 ⊢ (((𝜑 ∧ 𝐴 = 𝐶) ∧ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) → 𝐵 ∈ 𝑃)
10		cgrg3col4.x	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ 𝑃)
11	10	ad2antrr 758	. . . . 5 ⊢ (((𝜑 ∧ 𝐴 = 𝐶) ∧ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) → 𝑋 ∈ 𝑃)
12		eqid 2610	. . . . 5 ⊢ (cgrG‘𝐺) = (cgrG‘𝐺)
13		isleag.d	. . . . . 6 ⊢ (𝜑 → 𝐷 ∈ 𝑃)
14	13	ad2antrr 758	. . . . 5 ⊢ (((𝜑 ∧ 𝐴 = 𝐶) ∧ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) → 𝐷 ∈ 𝑃)
15		isleag.e	. . . . . 6 ⊢ (𝜑 → 𝐸 ∈ 𝑃)
16	15	ad2antrr 758	. . . . 5 ⊢ (((𝜑 ∧ 𝐴 = 𝐶) ∧ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) → 𝐸 ∈ 𝑃)
17		eqid 2610	. . . . 5 ⊢ (dist‘𝐺) = (dist‘𝐺)
18		simpr 476	. . . . 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 25215	. . . . . 6 ⊢ (𝜑 → (𝐴(dist‘𝐺)𝐵) = (𝐷(dist‘𝐺)𝐸))
23	22	ad2antrr 758	. . . . 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 25262	. . . 4 ⊢ (((𝜑 ∧ 𝐴 = 𝐶) ∧ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) → ∃𝑦 ∈ 𝑃 ⟨“𝐴𝐵𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩)
25	21	ad4antr 764	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝐴 = 𝐶) ∧ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑦 ∈ 𝑃) ∧ ⟨“𝐴𝐵𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩) → ⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝐹”⟩)
26	5	ad2antrr 758	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝐴 = 𝐶) ∧ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑦 ∈ 𝑃) ∧ ⟨“𝐴𝐵𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩) → 𝐺 ∈ TarskiG)
27	11	ad2antrr 758	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝐴 = 𝐶) ∧ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑦 ∈ 𝑃) ∧ ⟨“𝐴𝐵𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩) → 𝑋 ∈ 𝑃)
28	7	ad2antrr 758	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝐴 = 𝐶) ∧ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑦 ∈ 𝑃) ∧ ⟨“𝐴𝐵𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩) → 𝐴 ∈ 𝑃)
29		simplr 788	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝐴 = 𝐶) ∧ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑦 ∈ 𝑃) ∧ ⟨“𝐴𝐵𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩) → 𝑦 ∈ 𝑃)
30	14	ad2antrr 758	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝐴 = 𝐶) ∧ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑦 ∈ 𝑃) ∧ ⟨“𝐴𝐵𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩) → 𝐷 ∈ 𝑃)
31	9	ad2antrr 758	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝐴 = 𝐶) ∧ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑦 ∈ 𝑃) ∧ ⟨“𝐴𝐵𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩) → 𝐵 ∈ 𝑃)
32	16	ad2antrr 758	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝐴 = 𝐶) ∧ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑦 ∈ 𝑃) ∧ ⟨“𝐴𝐵𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩) → 𝐸 ∈ 𝑃)
33		simpr 476	. . . . . . . . . . 11 ⊢ (((((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 25217	. . . . . . . . . 10 ⊢ (((((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 25175	. . . . . . . . 9 ⊢ (((((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 25216	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝐴 = 𝐶) ∧ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑦 ∈ 𝑃) ∧ ⟨“𝐴𝐵𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩) → (𝐵(dist‘𝐺)𝑋) = (𝐸(dist‘𝐺)𝑦))
37	19	ad4antr 764	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝐴 = 𝐶) ∧ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑦 ∈ 𝑃) ∧ ⟨“𝐴𝐵𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩) → 𝐶 ∈ 𝑃)
38	20	ad4antr 764	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝐴 = 𝐶) ∧ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑦 ∈ 𝑃) ∧ ⟨“𝐴𝐵𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩) → 𝐹 ∈ 𝑃)
39		simpr 476	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝐴 = 𝐶) → 𝐴 = 𝐶)
40	39	ad3antrrr 762	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝐴 = 𝐶) ∧ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑦 ∈ 𝑃) ∧ ⟨“𝐴𝐵𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩) → 𝐴 = 𝐶)
41	40	oveq2d 6565	. . . . . . . . . . 11 ⊢ (((((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	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝐴 = 𝐶) → 𝐺 ∈ TarskiG)
43	6	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝐴 = 𝐶) → 𝐴 ∈ 𝑃)
44	19	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝐴 = 𝐶) → 𝐶 ∈ 𝑃)
45	13	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝐴 = 𝐶) → 𝐷 ∈ 𝑃)
46	20	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝐴 = 𝐶) → 𝐹 ∈ 𝑃)
47	1, 17, 3, 12, 4, 6, 8, 19, 13, 15, 20, 21	cgr3simp3 25217	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝐶(dist‘𝐺)𝐴) = (𝐹(dist‘𝐺)𝐷))
48	1, 17, 3, 4, 19, 6, 20, 13, 47	tgcgrcomlr 25175	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐴(dist‘𝐺)𝐶) = (𝐷(dist‘𝐺)𝐹))
49	48	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝐴 = 𝐶) → (𝐴(dist‘𝐺)𝐶) = (𝐷(dist‘𝐺)𝐹))
50	1, 17, 3, 42, 43, 44, 45, 46, 49, 39	tgcgreq 25177	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝐴 = 𝐶) → 𝐷 = 𝐹)
51	50	ad3antrrr 762	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝐴 = 𝐶) ∧ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑦 ∈ 𝑃) ∧ ⟨“𝐴𝐵𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩) → 𝐷 = 𝐹)
52	51	oveq2d 6565	. . . . . . . . . . 11 ⊢ (((((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 2652	. . . . . . . . . 10 ⊢ (((((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 25175	. . . . . . . . 9 ⊢ (((((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 1235	. . . . . . . 8 ⊢ (((((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	25, 55	jca 553	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝐴 = 𝐶) ∧ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑦 ∈ 𝑃) ∧ ⟨“𝐴𝐵𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩) → (⟨“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	1, 17, 3, 12, 26, 28, 31, 37, 27, 30, 32, 38, 29	tgcgr4 25226	. . . . . . 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)))))
58	56, 57	mpbird 246	. . . . . 6 ⊢ (((((𝜑 ∧ 𝐴 = 𝐶) ∧ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑦 ∈ 𝑃) ∧ ⟨“𝐴𝐵𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩) → ⟨“𝐴𝐵𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝐹𝑦”⟩)
59	58	ex 449	. . . . 5 ⊢ ((((𝜑 ∧ 𝐴 = 𝐶) ∧ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑦 ∈ 𝑃) → (⟨“𝐴𝐵𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩ → ⟨“𝐴𝐵𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝐹𝑦”⟩))
60	59	reximdva 3000	. . . 4 ⊢ (((𝜑 ∧ 𝐴 = 𝐶) ∧ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) → (∃𝑦 ∈ 𝑃 ⟨“𝐴𝐵𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩ → ∃𝑦 ∈ 𝑃 ⟨“𝐴𝐵𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝐹𝑦”⟩))
61	24, 60	mpd 15	. . 3 ⊢ (((𝜑 ∧ 𝐴 = 𝐶) ∧ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) → ∃𝑦 ∈ 𝑃 ⟨“𝐴𝐵𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝐹𝑦”⟩)
62		eqid 2610	. . . . . 6 ⊢ (hlG‘𝐺) = (hlG‘𝐺)
63	42	adantr 480	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) → 𝐺 ∈ TarskiG)
64	63	ad2antrr 758	. . . . . 6 ⊢ (((((𝜑 ∧ 𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑥 ∈ 𝑃) ∧ ¬ 𝑥 ∈ (𝐷𝐿𝐸)) → 𝐺 ∈ TarskiG)
65	8	ad2antrr 758	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) → 𝐵 ∈ 𝑃)
66	65	ad2antrr 758	. . . . . 6 ⊢ (((((𝜑 ∧ 𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑥 ∈ 𝑃) ∧ ¬ 𝑥 ∈ (𝐷𝐿𝐸)) → 𝐵 ∈ 𝑃)
67	43	adantr 480	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) → 𝐴 ∈ 𝑃)
68	67	ad2antrr 758	. . . . . 6 ⊢ (((((𝜑 ∧ 𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑥 ∈ 𝑃) ∧ ¬ 𝑥 ∈ (𝐷𝐿𝐸)) → 𝐴 ∈ 𝑃)
69	10	ad2antrr 758	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) → 𝑋 ∈ 𝑃)
70	69	ad2antrr 758	. . . . . 6 ⊢ (((((𝜑 ∧ 𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑥 ∈ 𝑃) ∧ ¬ 𝑥 ∈ (𝐷𝐿𝐸)) → 𝑋 ∈ 𝑃)
71	15	ad2antrr 758	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) → 𝐸 ∈ 𝑃)
72	71	ad2antrr 758	. . . . . 6 ⊢ (((((𝜑 ∧ 𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑥 ∈ 𝑃) ∧ ¬ 𝑥 ∈ (𝐷𝐿𝐸)) → 𝐸 ∈ 𝑃)
73	45	adantr 480	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) → 𝐷 ∈ 𝑃)
74	73	ad2antrr 758	. . . . . 6 ⊢ (((((𝜑 ∧ 𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑥 ∈ 𝑃) ∧ ¬ 𝑥 ∈ (𝐷𝐿𝐸)) → 𝐷 ∈ 𝑃)
75		simplr 788	. . . . . 6 ⊢ (((((𝜑 ∧ 𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑥 ∈ 𝑃) ∧ ¬ 𝑥 ∈ (𝐷𝐿𝐸)) → 𝑥 ∈ 𝑃)
76		simpr 476	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) → ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋))
77	76	ad2antrr 758	. . . . . 6 ⊢ (((((𝜑 ∧ 𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑥 ∈ 𝑃) ∧ ¬ 𝑥 ∈ (𝐷𝐿𝐸)) → ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋))
78		simpr 476	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑥 ∈ 𝑃) ∧ ¬ 𝑥 ∈ (𝐷𝐿𝐸)) → ¬ 𝑥 ∈ (𝐷𝐿𝐸))
79	22	ad2antrr 758	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) → (𝐴(dist‘𝐺)𝐵) = (𝐷(dist‘𝐺)𝐸))
80	1, 3, 2, 63, 65, 67, 69, 76	ncolne1 25320	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) → 𝐵 ≠ 𝐴)
81	80	necomd 2837	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) → 𝐴 ≠ 𝐵)
82	1, 17, 3, 63, 67, 65, 73, 71, 79, 81	tgcgrneq 25178	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) → 𝐷 ≠ 𝐸)
83	82	ad2antrr 758	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑥 ∈ 𝑃) ∧ ¬ 𝑥 ∈ (𝐷𝐿𝐸)) → 𝐷 ≠ 𝐸)
84	83	neneqd 2787	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑥 ∈ 𝑃) ∧ ¬ 𝑥 ∈ (𝐷𝐿𝐸)) → ¬ 𝐷 = 𝐸)
85	78, 84	jca 553	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑥 ∈ 𝑃) ∧ ¬ 𝑥 ∈ (𝐷𝐿𝐸)) → (¬ 𝑥 ∈ (𝐷𝐿𝐸) ∧ ¬ 𝐷 = 𝐸))
86		ioran 510	. . . . . . . . 9 ⊢ (¬ (𝑥 ∈ (𝐷𝐿𝐸) ∨ 𝐷 = 𝐸) ↔ (¬ 𝑥 ∈ (𝐷𝐿𝐸) ∧ ¬ 𝐷 = 𝐸))
87	85, 86	sylibr 223	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑥 ∈ 𝑃) ∧ ¬ 𝑥 ∈ (𝐷𝐿𝐸)) → ¬ (𝑥 ∈ (𝐷𝐿𝐸) ∨ 𝐷 = 𝐸))
88	1, 2, 3, 64, 74, 72, 75, 87	ncolcom 25256	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑥 ∈ 𝑃) ∧ ¬ 𝑥 ∈ (𝐷𝐿𝐸)) → ¬ (𝑥 ∈ (𝐸𝐿𝐷) ∨ 𝐸 = 𝐷))
89	1, 2, 3, 64, 72, 74, 75, 88	ncolrot1 25257	. . . . . 6 ⊢ (((((𝜑 ∧ 𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑥 ∈ 𝑃) ∧ ¬ 𝑥 ∈ (𝐷𝐿𝐸)) → ¬ (𝐸 ∈ (𝐷𝐿𝑥) ∨ 𝐷 = 𝑥))
90	1, 17, 3, 4, 6, 8, 13, 15, 22	tgcgrcomlr 25175	. . . . . . 7 ⊢ (𝜑 → (𝐵(dist‘𝐺)𝐴) = (𝐸(dist‘𝐺)𝐷))
91	90	ad4antr 764	. . . . . 6 ⊢ (((((𝜑 ∧ 𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑥 ∈ 𝑃) ∧ ¬ 𝑥 ∈ (𝐷𝐿𝐸)) → (𝐵(dist‘𝐺)𝐴) = (𝐸(dist‘𝐺)𝐷))
92	1, 17, 3, 2, 62, 64, 66, 68, 70, 72, 74, 75, 77, 89, 91	trgcopy 25496	. . . . 5 ⊢ (((((𝜑 ∧ 𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑥 ∈ 𝑃) ∧ ¬ 𝑥 ∈ (𝐷𝐿𝐸)) → ∃𝑦 ∈ 𝑃 (⟨“𝐵𝐴𝑋”⟩(cgrG‘𝐺)⟨“𝐸𝐷𝑦”⟩ ∧ 𝑦((hpG‘𝐺)‘(𝐸𝐿𝐷))𝑥))
93	21	ad6antr 768	. . . . . . . . . 10 ⊢ (((((((𝜑 ∧ 𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑥 ∈ 𝑃) ∧ ¬ 𝑥 ∈ (𝐷𝐿𝐸)) ∧ 𝑦 ∈ 𝑃) ∧ ⟨“𝐵𝐴𝑋”⟩(cgrG‘𝐺)⟨“𝐸𝐷𝑦”⟩) → ⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝐹”⟩)
94	64	ad2antrr 758	. . . . . . . . . . . 12 ⊢ (((((((𝜑 ∧ 𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑥 ∈ 𝑃) ∧ ¬ 𝑥 ∈ (𝐷𝐿𝐸)) ∧ 𝑦 ∈ 𝑃) ∧ ⟨“𝐵𝐴𝑋”⟩(cgrG‘𝐺)⟨“𝐸𝐷𝑦”⟩) → 𝐺 ∈ TarskiG)
95	66	ad2antrr 758	. . . . . . . . . . . 12 ⊢ (((((((𝜑 ∧ 𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑥 ∈ 𝑃) ∧ ¬ 𝑥 ∈ (𝐷𝐿𝐸)) ∧ 𝑦 ∈ 𝑃) ∧ ⟨“𝐵𝐴𝑋”⟩(cgrG‘𝐺)⟨“𝐸𝐷𝑦”⟩) → 𝐵 ∈ 𝑃)
96	68	ad2antrr 758	. . . . . . . . . . . 12 ⊢ (((((((𝜑 ∧ 𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑥 ∈ 𝑃) ∧ ¬ 𝑥 ∈ (𝐷𝐿𝐸)) ∧ 𝑦 ∈ 𝑃) ∧ ⟨“𝐵𝐴𝑋”⟩(cgrG‘𝐺)⟨“𝐸𝐷𝑦”⟩) → 𝐴 ∈ 𝑃)
97	70	ad2antrr 758	. . . . . . . . . . . 12 ⊢ (((((((𝜑 ∧ 𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑥 ∈ 𝑃) ∧ ¬ 𝑥 ∈ (𝐷𝐿𝐸)) ∧ 𝑦 ∈ 𝑃) ∧ ⟨“𝐵𝐴𝑋”⟩(cgrG‘𝐺)⟨“𝐸𝐷𝑦”⟩) → 𝑋 ∈ 𝑃)
98	72	ad2antrr 758	. . . . . . . . . . . 12 ⊢ (((((((𝜑 ∧ 𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑥 ∈ 𝑃) ∧ ¬ 𝑥 ∈ (𝐷𝐿𝐸)) ∧ 𝑦 ∈ 𝑃) ∧ ⟨“𝐵𝐴𝑋”⟩(cgrG‘𝐺)⟨“𝐸𝐷𝑦”⟩) → 𝐸 ∈ 𝑃)
99	74	ad2antrr 758	. . . . . . . . . . . 12 ⊢ (((((((𝜑 ∧ 𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑥 ∈ 𝑃) ∧ ¬ 𝑥 ∈ (𝐷𝐿𝐸)) ∧ 𝑦 ∈ 𝑃) ∧ ⟨“𝐵𝐴𝑋”⟩(cgrG‘𝐺)⟨“𝐸𝐷𝑦”⟩) → 𝐷 ∈ 𝑃)
100		simplr 788	. . . . . . . . . . . 12 ⊢ (((((((𝜑 ∧ 𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑥 ∈ 𝑃) ∧ ¬ 𝑥 ∈ (𝐷𝐿𝐸)) ∧ 𝑦 ∈ 𝑃) ∧ ⟨“𝐵𝐴𝑋”⟩(cgrG‘𝐺)⟨“𝐸𝐷𝑦”⟩) → 𝑦 ∈ 𝑃)
101		simpr 476	. . . . . . . . . . . 12 ⊢ (((((((𝜑 ∧ 𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑥 ∈ 𝑃) ∧ ¬ 𝑥 ∈ (𝐷𝐿𝐸)) ∧ 𝑦 ∈ 𝑃) ∧ ⟨“𝐵𝐴𝑋”⟩(cgrG‘𝐺)⟨“𝐸𝐷𝑦”⟩) → ⟨“𝐵𝐴𝑋”⟩(cgrG‘𝐺)⟨“𝐸𝐷𝑦”⟩)
102	1, 17, 3, 12, 94, 95, 96, 97, 98, 99, 100, 101	cgr3simp2 25216	. . . . . . . . . . 11 ⊢ (((((((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))
103	1, 17, 3, 12, 94, 95, 96, 97, 98, 99, 100, 101	cgr3simp3 25217	. . . . . . . . . . . 12 ⊢ (((((((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))
104	1, 17, 3, 94, 97, 95, 100, 98, 103	tgcgrcomlr 25175	. . . . . . . . . . 11 ⊢ (((((((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))
105	44	ad5antr 766	. . . . . . . . . . . 12 ⊢ (((((((𝜑 ∧ 𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑥 ∈ 𝑃) ∧ ¬ 𝑥 ∈ (𝐷𝐿𝐸)) ∧ 𝑦 ∈ 𝑃) ∧ ⟨“𝐵𝐴𝑋”⟩(cgrG‘𝐺)⟨“𝐸𝐷𝑦”⟩) → 𝐶 ∈ 𝑃)
106	46	ad5antr 766	. . . . . . . . . . . 12 ⊢ (((((((𝜑 ∧ 𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑥 ∈ 𝑃) ∧ ¬ 𝑥 ∈ (𝐷𝐿𝐸)) ∧ 𝑦 ∈ 𝑃) ∧ ⟨“𝐵𝐴𝑋”⟩(cgrG‘𝐺)⟨“𝐸𝐷𝑦”⟩) → 𝐹 ∈ 𝑃)
107	1, 17, 3, 94, 96, 97, 99, 100, 102	tgcgrcomlr 25175	. . . . . . . . . . . . 13 ⊢ (((((((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))
108		simp-6r 807	. . . . . . . . . . . . . 14 ⊢ (((((((𝜑 ∧ 𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑥 ∈ 𝑃) ∧ ¬ 𝑥 ∈ (𝐷𝐿𝐸)) ∧ 𝑦 ∈ 𝑃) ∧ ⟨“𝐵𝐴𝑋”⟩(cgrG‘𝐺)⟨“𝐸𝐷𝑦”⟩) → 𝐴 = 𝐶)
109	108	oveq2d 6565	. . . . . . . . . . . . 13 ⊢ (((((((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))
110	50	ad5antr 766	. . . . . . . . . . . . . 14 ⊢ (((((((𝜑 ∧ 𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑥 ∈ 𝑃) ∧ ¬ 𝑥 ∈ (𝐷𝐿𝐸)) ∧ 𝑦 ∈ 𝑃) ∧ ⟨“𝐵𝐴𝑋”⟩(cgrG‘𝐺)⟨“𝐸𝐷𝑦”⟩) → 𝐷 = 𝐹)
111	110	oveq2d 6565	. . . . . . . . . . . . 13 ⊢ (((((((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))
112	107, 109, 111	3eqtr3d 2652	. . . . . . . . . . . 12 ⊢ (((((((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))
113	1, 17, 3, 94, 97, 105, 100, 106, 112	tgcgrcomlr 25175	. . . . . . . . . . 11 ⊢ (((((((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))
114	102, 104, 113	3jca 1235	. . . . . . . . . 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) ∧ (𝐵(dist‘𝐺)𝑋) = (𝐸(dist‘𝐺)𝑦) ∧ (𝐶(dist‘𝐺)𝑋) = (𝐹(dist‘𝐺)𝑦)))
115	93, 114	jca 553	. . . . . . . . 9 ⊢ (((((((𝜑 ∧ 𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑥 ∈ 𝑃) ∧ ¬ 𝑥 ∈ (𝐷𝐿𝐸)) ∧ 𝑦 ∈ 𝑃) ∧ ⟨“𝐵𝐴𝑋”⟩(cgrG‘𝐺)⟨“𝐸𝐷𝑦”⟩) → (⟨“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))))
116	1, 17, 3, 12, 94, 96, 95, 105, 97, 99, 98, 106, 100	tgcgr4 25226	. . . . . . . . 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)))))
117	115, 116	mpbird 246	. . . . . . . 8 ⊢ (((((((𝜑 ∧ 𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑥 ∈ 𝑃) ∧ ¬ 𝑥 ∈ (𝐷𝐿𝐸)) ∧ 𝑦 ∈ 𝑃) ∧ ⟨“𝐵𝐴𝑋”⟩(cgrG‘𝐺)⟨“𝐸𝐷𝑦”⟩) → ⟨“𝐴𝐵𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝐹𝑦”⟩)
118	117	ex 449	. . . . . . 7 ⊢ ((((((𝜑 ∧ 𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑥 ∈ 𝑃) ∧ ¬ 𝑥 ∈ (𝐷𝐿𝐸)) ∧ 𝑦 ∈ 𝑃) → (⟨“𝐵𝐴𝑋”⟩(cgrG‘𝐺)⟨“𝐸𝐷𝑦”⟩ → ⟨“𝐴𝐵𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝐹𝑦”⟩))
119	118	adantrd 483	. . . . . 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”⟩))
120	119	reximdva 3000	. . . . 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”⟩))
121	92, 120	mpd 15	. . . 4 ⊢ (((((𝜑 ∧ 𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑥 ∈ 𝑃) ∧ ¬ 𝑥 ∈ (𝐷𝐿𝐸)) → ∃𝑦 ∈ 𝑃 ⟨“𝐴𝐵𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝐹𝑦”⟩)
122	1, 2, 3, 63, 67, 69, 65, 76	ncoltgdim2 25260	. . . . 5 ⊢ (((𝜑 ∧ 𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) → 𝐺DimTarskiG≥2)
123	1, 3, 2, 63, 122, 73, 71, 82	tglowdim2ln 25346	. . . 4 ⊢ (((𝜑 ∧ 𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) → ∃𝑥 ∈ 𝑃 ¬ 𝑥 ∈ (𝐷𝐿𝐸))
124	121, 123	r19.29a 3060	. . 3 ⊢ (((𝜑 ∧ 𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) → ∃𝑦 ∈ 𝑃 ⟨“𝐴𝐵𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝐹𝑦”⟩)
125	61, 124	pm2.61dan 828	. 2 ⊢ ((𝜑 ∧ 𝐴 = 𝐶) → ∃𝑦 ∈ 𝑃 ⟨“𝐴𝐵𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝐹𝑦”⟩)
126		cgrg3col4.2	. . . . . . 7 ⊢ (𝜑 → (𝑋 ∈ (𝐴𝐿𝐶) ∨ 𝐴 = 𝐶))
127	1, 2, 3, 4, 6, 19, 10, 126	colcom 25253	. . . . . 6 ⊢ (𝜑 → (𝑋 ∈ (𝐶𝐿𝐴) ∨ 𝐶 = 𝐴))
128	1, 2, 3, 4, 19, 6, 10, 127	colrot1 25254	. . . . 5 ⊢ (𝜑 → (𝐶 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋))
129	1, 2, 3, 4, 6, 19, 10, 12, 13, 20, 17, 128, 48	lnext 25262	. . . 4 ⊢ (𝜑 → ∃𝑦 ∈ 𝑃 ⟨“𝐴𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐹𝑦”⟩)
130	129	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝐴 ≠ 𝐶) → ∃𝑦 ∈ 𝑃 ⟨“𝐴𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐹𝑦”⟩)
131	21	ad3antrrr 762	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝐴 ≠ 𝐶) ∧ 𝑦 ∈ 𝑃) ∧ ⟨“𝐴𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐹𝑦”⟩) → ⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝐹”⟩)
132	4	ad3antrrr 762	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝐴 ≠ 𝐶) ∧ 𝑦 ∈ 𝑃) ∧ ⟨“𝐴𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐹𝑦”⟩) → 𝐺 ∈ TarskiG)
133	10	ad3antrrr 762	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝐴 ≠ 𝐶) ∧ 𝑦 ∈ 𝑃) ∧ ⟨“𝐴𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐹𝑦”⟩) → 𝑋 ∈ 𝑃)
134	6	ad3antrrr 762	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝐴 ≠ 𝐶) ∧ 𝑦 ∈ 𝑃) ∧ ⟨“𝐴𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐹𝑦”⟩) → 𝐴 ∈ 𝑃)
135		simplr 788	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝐴 ≠ 𝐶) ∧ 𝑦 ∈ 𝑃) ∧ ⟨“𝐴𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐹𝑦”⟩) → 𝑦 ∈ 𝑃)
136	13	ad3antrrr 762	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝐴 ≠ 𝐶) ∧ 𝑦 ∈ 𝑃) ∧ ⟨“𝐴𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐹𝑦”⟩) → 𝐷 ∈ 𝑃)
137	19	ad3antrrr 762	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝐴 ≠ 𝐶) ∧ 𝑦 ∈ 𝑃) ∧ ⟨“𝐴𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐹𝑦”⟩) → 𝐶 ∈ 𝑃)
138	20	ad3antrrr 762	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝐴 ≠ 𝐶) ∧ 𝑦 ∈ 𝑃) ∧ ⟨“𝐴𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐹𝑦”⟩) → 𝐹 ∈ 𝑃)
139		simpr 476	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝐴 ≠ 𝐶) ∧ 𝑦 ∈ 𝑃) ∧ ⟨“𝐴𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐹𝑦”⟩) → ⟨“𝐴𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐹𝑦”⟩)
140	1, 17, 3, 12, 132, 134, 137, 133, 136, 138, 135, 139	cgr3simp3 25217	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝐴 ≠ 𝐶) ∧ 𝑦 ∈ 𝑃) ∧ ⟨“𝐴𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐹𝑦”⟩) → (𝑋(dist‘𝐺)𝐴) = (𝑦(dist‘𝐺)𝐷))
141	1, 17, 3, 132, 133, 134, 135, 136, 140	tgcgrcomlr 25175	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝐴 ≠ 𝐶) ∧ 𝑦 ∈ 𝑃) ∧ ⟨“𝐴𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐹𝑦”⟩) → (𝐴(dist‘𝐺)𝑋) = (𝐷(dist‘𝐺)𝑦))
142	8	ad3antrrr 762	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝐴 ≠ 𝐶) ∧ 𝑦 ∈ 𝑃) ∧ ⟨“𝐴𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐹𝑦”⟩) → 𝐵 ∈ 𝑃)
143	15	ad3antrrr 762	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝐴 ≠ 𝐶) ∧ 𝑦 ∈ 𝑃) ∧ ⟨“𝐴𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐹𝑦”⟩) → 𝐸 ∈ 𝑃)
144	128	ad3antrrr 762	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝐴 ≠ 𝐶) ∧ 𝑦 ∈ 𝑃) ∧ ⟨“𝐴𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐹𝑦”⟩) → (𝐶 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋))
145	22	ad3antrrr 762	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝐴 ≠ 𝐶) ∧ 𝑦 ∈ 𝑃) ∧ ⟨“𝐴𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐹𝑦”⟩) → (𝐴(dist‘𝐺)𝐵) = (𝐷(dist‘𝐺)𝐸))
146	1, 17, 3, 12, 4, 6, 8, 19, 13, 15, 20, 21	cgr3simp2 25216	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐵(dist‘𝐺)𝐶) = (𝐸(dist‘𝐺)𝐹))
147	1, 17, 3, 4, 8, 19, 15, 20, 146	tgcgrcomlr 25175	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐶(dist‘𝐺)𝐵) = (𝐹(dist‘𝐺)𝐸))
148	147	ad3antrrr 762	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝐴 ≠ 𝐶) ∧ 𝑦 ∈ 𝑃) ∧ ⟨“𝐴𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐹𝑦”⟩) → (𝐶(dist‘𝐺)𝐵) = (𝐹(dist‘𝐺)𝐸))
149		simpllr 795	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝐴 ≠ 𝐶) ∧ 𝑦 ∈ 𝑃) ∧ ⟨“𝐴𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐹𝑦”⟩) → 𝐴 ≠ 𝐶)
150	1, 2, 3, 132, 134, 137, 133, 12, 136, 138, 17, 142, 135, 143, 144, 139, 145, 148, 149	tgfscgr 25263	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝐴 ≠ 𝐶) ∧ 𝑦 ∈ 𝑃) ∧ ⟨“𝐴𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐹𝑦”⟩) → (𝑋(dist‘𝐺)𝐵) = (𝑦(dist‘𝐺)𝐸))
151	1, 17, 3, 132, 133, 142, 135, 143, 150	tgcgrcomlr 25175	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝐴 ≠ 𝐶) ∧ 𝑦 ∈ 𝑃) ∧ ⟨“𝐴𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐹𝑦”⟩) → (𝐵(dist‘𝐺)𝑋) = (𝐸(dist‘𝐺)𝑦))
152	1, 17, 3, 12, 132, 134, 137, 133, 136, 138, 135, 139	cgr3simp2 25216	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝐴 ≠ 𝐶) ∧ 𝑦 ∈ 𝑃) ∧ ⟨“𝐴𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐹𝑦”⟩) → (𝐶(dist‘𝐺)𝑋) = (𝐹(dist‘𝐺)𝑦))
153	141, 151, 152	3jca 1235	. . . . . . 7 ⊢ ((((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)))
154	131, 153	jca 553	. . . . . 6 ⊢ ((((𝜑 ∧ 𝐴 ≠ 𝐶) ∧ 𝑦 ∈ 𝑃) ∧ ⟨“𝐴𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐹𝑦”⟩) → (⟨“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))))
155	1, 17, 3, 12, 132, 134, 142, 137, 133, 136, 143, 138, 135	tgcgr4 25226	. . . . . 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)))))
156	154, 155	mpbird 246	. . . . 5 ⊢ ((((𝜑 ∧ 𝐴 ≠ 𝐶) ∧ 𝑦 ∈ 𝑃) ∧ ⟨“𝐴𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐹𝑦”⟩) → ⟨“𝐴𝐵𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝐹𝑦”⟩)
157	156	ex 449	. . . 4 ⊢ (((𝜑 ∧ 𝐴 ≠ 𝐶) ∧ 𝑦 ∈ 𝑃) → (⟨“𝐴𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐹𝑦”⟩ → ⟨“𝐴𝐵𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝐹𝑦”⟩))
158	157	reximdva 3000	. . 3 ⊢ ((𝜑 ∧ 𝐴 ≠ 𝐶) → (∃𝑦 ∈ 𝑃 ⟨“𝐴𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐹𝑦”⟩ → ∃𝑦 ∈ 𝑃 ⟨“𝐴𝐵𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝐹𝑦”⟩))
159	130, 158	mpd 15	. 2 ⊢ ((𝜑 ∧ 𝐴 ≠ 𝐶) → ∃𝑦 ∈ 𝑃 ⟨“𝐴𝐵𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝐹𝑦”⟩)
160	125, 159	pm2.61dane 2869	1 ⊢ (𝜑 → ∃𝑦 ∈ 𝑃 ⟨“𝐴𝐵𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝐹𝑦”⟩)

Syntax hints:  ¬ wn 3   → wi 4   ∨ wo 382   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∃wrex 2897   class class class wbr 4583  ‘cfv 5804  (class class class)co 6549  ⟨“cs3 13438  ⟨“cs4 13439  Basecbs 15695  distcds 15777  TarskiGcstrkg 25129  Itvcitv 25135  LineGclng 25136  cgrGccgrg 25205  hlGchlg 25295  hpGchpg 25449

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-map 7746  df-pm 7747  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-card 8648  df-cda 8873  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-3 10957  df-4 10958  df-n0 11170  df-xnn0 11241  df-z 11255  df-uz 11564  df-fz 12198  df-fzo 12335  df-hash 12980  df-word 13154  df-concat 13156  df-s1 13157  df-s2 13444  df-s3 13445  df-s4 13446  df-trkgc 25147  df-trkgb 25148  df-trkgcb 25149  df-trkgld 25151  df-trkg 25152  df-cgrg 25206  df-ismt 25228  df-leg 25278  df-hlg 25296  df-mir 25348  df-rag 25389  df-perpg 25391  df-hpg 25450  df-mid 25466  df-lmi 25467

This theorem is referenced by: (None)