Theorem colperpexlem1 28664

Description: Lemma for colperp 28663. First part of lemma 8.20 of [Schwabhauser] p. 62. (Contributed by Thierry Arnoux, 27-Oct-2019.)

Hypotheses

Ref	Expression
colperpex.p	⊢ 𝑃 = (Base‘𝐺)
colperpex.d	⊢ − = (dist‘𝐺)
colperpex.i	⊢ 𝐼 = (Itv‘𝐺)
colperpex.l	⊢ 𝐿 = (LineG‘𝐺)
colperpex.g	⊢ (𝜑 → 𝐺 ∈ TarskiG)
colperpexlem.s	⊢ 𝑆 = (pInvG‘𝐺)
colperpexlem.m	⊢ 𝑀 = (𝑆‘𝐴)
colperpexlem.n	⊢ 𝑁 = (𝑆‘𝐵)
colperpexlem.k	⊢ 𝐾 = (𝑆‘𝑄)
colperpexlem.a	⊢ (𝜑 → 𝐴 ∈ 𝑃)
colperpexlem.b	⊢ (𝜑 → 𝐵 ∈ 𝑃)
colperpexlem.c	⊢ (𝜑 → 𝐶 ∈ 𝑃)
colperpexlem.q	⊢ (𝜑 → 𝑄 ∈ 𝑃)
colperpexlem.1	⊢ (𝜑 → ⟨“𝐴𝐵𝐶”⟩ ∈ (∟G‘𝐺))
colperpexlem.2	⊢ (𝜑 → (𝐾‘(𝑀‘𝐶)) = (𝑁‘𝐶))

Assertion

Ref	Expression
colperpexlem1	⊢ (𝜑 → ⟨“𝐵𝐴𝑄”⟩ ∈ (∟G‘𝐺))

Proof of Theorem colperpexlem1

Step	Hyp	Ref	Expression
1		colperpex.p	. . . 4 ⊢ 𝑃 = (Base‘𝐺)
2		colperpex.d	. . . 4 ⊢ − = (dist‘𝐺)
3		colperpex.i	. . . 4 ⊢ 𝐼 = (Itv‘𝐺)
4		colperpex.g	. . . 4 ⊢ (𝜑 → 𝐺 ∈ TarskiG)
5		colperpexlem.q	. . . 4 ⊢ (𝜑 → 𝑄 ∈ 𝑃)
6		colperpexlem.b	. . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑃)
7		colperpex.l	. . . . 5 ⊢ 𝐿 = (LineG‘𝐺)
8		colperpexlem.s	. . . . 5 ⊢ 𝑆 = (pInvG‘𝐺)
9		colperpexlem.a	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ 𝑃)
10		colperpexlem.m	. . . . 5 ⊢ 𝑀 = (𝑆‘𝐴)
11	1, 2, 3, 7, 8, 4, 9, 10, 5	mircl 28595	. . . 4 ⊢ (𝜑 → (𝑀‘𝑄) ∈ 𝑃)
12		colperpexlem.c	. . . . . 6 ⊢ (𝜑 → 𝐶 ∈ 𝑃)
13	1, 2, 3, 7, 8, 4, 9, 10, 12	mircl 28595	. . . . 5 ⊢ (𝜑 → (𝑀‘𝐶) ∈ 𝑃)
14		eqid 2730	. . . . . 6 ⊢ (𝑆‘𝐵) = (𝑆‘𝐵)
15	1, 2, 3, 7, 8, 4, 6, 14, 12	mircl 28595	. . . . 5 ⊢ (𝜑 → ((𝑆‘𝐵)‘𝐶) ∈ 𝑃)
16	1, 2, 3, 7, 8, 4, 9, 10, 15	mircl 28595	. . . . 5 ⊢ (𝜑 → (𝑀‘((𝑆‘𝐵)‘𝐶)) ∈ 𝑃)
17		colperpexlem.2	. . . . . . . 8 ⊢ (𝜑 → (𝐾‘(𝑀‘𝐶)) = (𝑁‘𝐶))
18		colperpexlem.n	. . . . . . . . 9 ⊢ 𝑁 = (𝑆‘𝐵)
19	1, 2, 3, 7, 8, 4, 6, 18, 12	mircl 28595	. . . . . . . 8 ⊢ (𝜑 → (𝑁‘𝐶) ∈ 𝑃)
20	17, 19	eqeltrd 2829	. . . . . . 7 ⊢ (𝜑 → (𝐾‘(𝑀‘𝐶)) ∈ 𝑃)
21		colperpexlem.k	. . . . . . . 8 ⊢ 𝐾 = (𝑆‘𝑄)
22	1, 2, 3, 7, 8, 4, 5, 21, 13	mirbtwn 28592	. . . . . . 7 ⊢ (𝜑 → 𝑄 ∈ ((𝐾‘(𝑀‘𝐶))𝐼(𝑀‘𝐶)))
23	1, 2, 3, 4, 20, 5, 13, 22	tgbtwncom 28422	. . . . . 6 ⊢ (𝜑 → 𝑄 ∈ ((𝑀‘𝐶)𝐼(𝐾‘(𝑀‘𝐶))))
24	18	fveq1i 6862	. . . . . . . 8 ⊢ (𝑁‘𝐶) = ((𝑆‘𝐵)‘𝐶)
25	17, 24	eqtrdi 2781	. . . . . . 7 ⊢ (𝜑 → (𝐾‘(𝑀‘𝐶)) = ((𝑆‘𝐵)‘𝐶))
26	25	oveq2d 7406	. . . . . 6 ⊢ (𝜑 → ((𝑀‘𝐶)𝐼(𝐾‘(𝑀‘𝐶))) = ((𝑀‘𝐶)𝐼((𝑆‘𝐵)‘𝐶)))
27	23, 26	eleqtrd 2831	. . . . 5 ⊢ (𝜑 → 𝑄 ∈ ((𝑀‘𝐶)𝐼((𝑆‘𝐵)‘𝐶)))
28	1, 2, 3, 4, 13, 5, 15, 27	tgbtwncom 28422	. . . . . . 7 ⊢ (𝜑 → 𝑄 ∈ (((𝑆‘𝐵)‘𝐶)𝐼(𝑀‘𝐶)))
29	1, 2, 3, 7, 8, 4, 9, 10, 15, 5, 13, 28	mirbtwni 28605	. . . . . 6 ⊢ (𝜑 → (𝑀‘𝑄) ∈ ((𝑀‘((𝑆‘𝐵)‘𝐶))𝐼(𝑀‘(𝑀‘𝐶))))
30	1, 2, 3, 7, 8, 4, 9, 10, 12	mirmir 28596	. . . . . . 7 ⊢ (𝜑 → (𝑀‘(𝑀‘𝐶)) = 𝐶)
31	30	oveq2d 7406	. . . . . 6 ⊢ (𝜑 → ((𝑀‘((𝑆‘𝐵)‘𝐶))𝐼(𝑀‘(𝑀‘𝐶))) = ((𝑀‘((𝑆‘𝐵)‘𝐶))𝐼𝐶))
32	29, 31	eleqtrd 2831	. . . . 5 ⊢ (𝜑 → (𝑀‘𝑄) ∈ ((𝑀‘((𝑆‘𝐵)‘𝐶))𝐼𝐶))
33	1, 2, 3, 4, 13, 15	axtgcgrrflx 28396	. . . . . 6 ⊢ (𝜑 → ((𝑀‘𝐶) − ((𝑆‘𝐵)‘𝐶)) = (((𝑆‘𝐵)‘𝐶) − (𝑀‘𝐶)))
34	1, 2, 3, 7, 8, 4, 9, 10, 15, 13	miriso 28604	. . . . . 6 ⊢ (𝜑 → ((𝑀‘((𝑆‘𝐵)‘𝐶)) − (𝑀‘(𝑀‘𝐶))) = (((𝑆‘𝐵)‘𝐶) − (𝑀‘𝐶)))
35	30	oveq2d 7406	. . . . . 6 ⊢ (𝜑 → ((𝑀‘((𝑆‘𝐵)‘𝐶)) − (𝑀‘(𝑀‘𝐶))) = ((𝑀‘((𝑆‘𝐵)‘𝐶)) − 𝐶))
36	33, 34, 35	3eqtr2d 2771	. . . . 5 ⊢ (𝜑 → ((𝑀‘𝐶) − ((𝑆‘𝐵)‘𝐶)) = ((𝑀‘((𝑆‘𝐵)‘𝐶)) − 𝐶))
37	25	oveq2d 7406	. . . . . . 7 ⊢ (𝜑 → (𝑄 − (𝐾‘(𝑀‘𝐶))) = (𝑄 − ((𝑆‘𝐵)‘𝐶)))
38	1, 2, 3, 7, 8, 4, 5, 21, 13	mircgr 28591	. . . . . . 7 ⊢ (𝜑 → (𝑄 − (𝐾‘(𝑀‘𝐶))) = (𝑄 − (𝑀‘𝐶)))
39	37, 38	eqtr3d 2767	. . . . . 6 ⊢ (𝜑 → (𝑄 − ((𝑆‘𝐵)‘𝐶)) = (𝑄 − (𝑀‘𝐶)))
40	1, 2, 3, 7, 8, 4, 9, 10, 5, 13	miriso 28604	. . . . . 6 ⊢ (𝜑 → ((𝑀‘𝑄) − (𝑀‘(𝑀‘𝐶))) = (𝑄 − (𝑀‘𝐶)))
41	30	oveq2d 7406	. . . . . 6 ⊢ (𝜑 → ((𝑀‘𝑄) − (𝑀‘(𝑀‘𝐶))) = ((𝑀‘𝑄) − 𝐶))
42	39, 40, 41	3eqtr2d 2771	. . . . 5 ⊢ (𝜑 → (𝑄 − ((𝑆‘𝐵)‘𝐶)) = ((𝑀‘𝑄) − 𝐶))
43	1, 2, 3, 7, 8, 4, 9, 10, 6	mirmir 28596	. . . . . . . . . 10 ⊢ (𝜑 → (𝑀‘(𝑀‘𝐵)) = 𝐵)
44		eqidd 2731	. . . . . . . . . 10 ⊢ (𝜑 → (𝑀‘𝐵) = (𝑀‘𝐵))
45		eqidd 2731	. . . . . . . . . 10 ⊢ (𝜑 → (𝑀‘𝐶) = (𝑀‘𝐶))
46	43, 44, 45	s3eqd 14837	. . . . . . . . 9 ⊢ (𝜑 → ⟨“(𝑀‘(𝑀‘𝐵))(𝑀‘𝐵)(𝑀‘𝐶)”⟩ = ⟨“𝐵(𝑀‘𝐵)(𝑀‘𝐶)”⟩)
47	1, 2, 3, 7, 8, 4, 9, 10, 6	mircl 28595	. . . . . . . . . 10 ⊢ (𝜑 → (𝑀‘𝐵) ∈ 𝑃)
48		simpr 484	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → 𝐴 = 𝐵)
49	48	fveq2d 6865	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → (𝑀‘𝐴) = (𝑀‘𝐵))
50	4	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → 𝐺 ∈ TarskiG)
51	9	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → 𝐴 ∈ 𝑃)
52	1, 2, 3, 7, 8, 50, 51, 10	mircinv 28602	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → (𝑀‘𝐴) = 𝐴)
53	49, 52	eqtr3d 2767	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → (𝑀‘𝐵) = 𝐴)
54		eqidd 2731	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → 𝐵 = 𝐵)
55		eqidd 2731	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → 𝐶 = 𝐶)
56	53, 54, 55	s3eqd 14837	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → ⟨“(𝑀‘𝐵)𝐵𝐶”⟩ = ⟨“𝐴𝐵𝐶”⟩)
57		colperpexlem.1	. . . . . . . . . . . . 13 ⊢ (𝜑 → ⟨“𝐴𝐵𝐶”⟩ ∈ (∟G‘𝐺))
58	57	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → ⟨“𝐴𝐵𝐶”⟩ ∈ (∟G‘𝐺))
59	56, 58	eqeltrd 2829	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → ⟨“(𝑀‘𝐵)𝐵𝐶”⟩ ∈ (∟G‘𝐺))
60	4	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐴 ≠ 𝐵) → 𝐺 ∈ TarskiG)
61	9	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐴 ≠ 𝐵) → 𝐴 ∈ 𝑃)
62	6	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐴 ≠ 𝐵) → 𝐵 ∈ 𝑃)
63	12	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐴 ≠ 𝐵) → 𝐶 ∈ 𝑃)
64	1, 2, 3, 7, 8, 60, 61, 10, 62	mircl 28595	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐴 ≠ 𝐵) → (𝑀‘𝐵) ∈ 𝑃)
65	57	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐴 ≠ 𝐵) → ⟨“𝐴𝐵𝐶”⟩ ∈ (∟G‘𝐺))
66		simpr 484	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐴 ≠ 𝐵) → 𝐴 ≠ 𝐵)
67	1, 2, 3, 7, 8, 60, 61, 10, 62	mirbtwn 28592	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝐴 ≠ 𝐵) → 𝐴 ∈ ((𝑀‘𝐵)𝐼𝐵))
68	1, 7, 3, 60, 64, 62, 61, 67	btwncolg1 28489	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝐴 ≠ 𝐵) → (𝐴 ∈ ((𝑀‘𝐵)𝐿𝐵) ∨ (𝑀‘𝐵) = 𝐵))
69	1, 7, 3, 60, 64, 62, 61, 68	colcom 28492	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐴 ≠ 𝐵) → (𝐴 ∈ (𝐵𝐿(𝑀‘𝐵)) ∨ 𝐵 = (𝑀‘𝐵)))
70	1, 2, 3, 7, 8, 60, 61, 62, 63, 64, 65, 66, 69	ragcol 28633	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐴 ≠ 𝐵) → ⟨“(𝑀‘𝐵)𝐵𝐶”⟩ ∈ (∟G‘𝐺))
71	59, 70	pm2.61dane 3013	. . . . . . . . . 10 ⊢ (𝜑 → ⟨“(𝑀‘𝐵)𝐵𝐶”⟩ ∈ (∟G‘𝐺))
72	1, 2, 3, 7, 8, 4, 47, 6, 12, 71, 10, 9	mirrag 28635	. . . . . . . . 9 ⊢ (𝜑 → ⟨“(𝑀‘(𝑀‘𝐵))(𝑀‘𝐵)(𝑀‘𝐶)”⟩ ∈ (∟G‘𝐺))
73	46, 72	eqeltrrd 2830	. . . . . . . 8 ⊢ (𝜑 → ⟨“𝐵(𝑀‘𝐵)(𝑀‘𝐶)”⟩ ∈ (∟G‘𝐺))
74	1, 2, 3, 7, 8, 4, 6, 47, 13	israg 28631	. . . . . . . 8 ⊢ (𝜑 → (⟨“𝐵(𝑀‘𝐵)(𝑀‘𝐶)”⟩ ∈ (∟G‘𝐺) ↔ (𝐵 − (𝑀‘𝐶)) = (𝐵 − ((𝑆‘(𝑀‘𝐵))‘(𝑀‘𝐶)))))
75	73, 74	mpbid 232	. . . . . . 7 ⊢ (𝜑 → (𝐵 − (𝑀‘𝐶)) = (𝐵 − ((𝑆‘(𝑀‘𝐵))‘(𝑀‘𝐶))))
76	1, 2, 3, 7, 8, 4, 9, 10, 12, 6	mirmir2 28608	. . . . . . . 8 ⊢ (𝜑 → (𝑀‘((𝑆‘𝐵)‘𝐶)) = ((𝑆‘(𝑀‘𝐵))‘(𝑀‘𝐶)))
77	76	oveq2d 7406	. . . . . . 7 ⊢ (𝜑 → (𝐵 − (𝑀‘((𝑆‘𝐵)‘𝐶))) = (𝐵 − ((𝑆‘(𝑀‘𝐵))‘(𝑀‘𝐶))))
78	75, 77	eqtr4d 2768	. . . . . 6 ⊢ (𝜑 → (𝐵 − (𝑀‘𝐶)) = (𝐵 − (𝑀‘((𝑆‘𝐵)‘𝐶))))
79	1, 2, 3, 4, 6, 13, 6, 16, 78	tgcgrcomlr 28414	. . . . 5 ⊢ (𝜑 → ((𝑀‘𝐶) − 𝐵) = ((𝑀‘((𝑆‘𝐵)‘𝐶)) − 𝐵))
80	1, 2, 3, 7, 8, 4, 6, 14, 12	mircgr 28591	. . . . . 6 ⊢ (𝜑 → (𝐵 − ((𝑆‘𝐵)‘𝐶)) = (𝐵 − 𝐶))
81	1, 2, 3, 4, 6, 15, 6, 12, 80	tgcgrcomlr 28414	. . . . 5 ⊢ (𝜑 → (((𝑆‘𝐵)‘𝐶) − 𝐵) = (𝐶 − 𝐵))
82	1, 2, 3, 4, 13, 5, 15, 6, 16, 11, 12, 6, 27, 32, 36, 42, 79, 81	tgifscgr 28442	. . . 4 ⊢ (𝜑 → (𝑄 − 𝐵) = ((𝑀‘𝑄) − 𝐵))
83	1, 2, 3, 4, 5, 6, 11, 6, 82	tgcgrcomlr 28414	. . 3 ⊢ (𝜑 → (𝐵 − 𝑄) = (𝐵 − (𝑀‘𝑄)))
84	10	fveq1i 6862	. . . 4 ⊢ (𝑀‘𝑄) = ((𝑆‘𝐴)‘𝑄)
85	84	oveq2i 7401	. . 3 ⊢ (𝐵 − (𝑀‘𝑄)) = (𝐵 − ((𝑆‘𝐴)‘𝑄))
86	83, 85	eqtrdi 2781	. 2 ⊢ (𝜑 → (𝐵 − 𝑄) = (𝐵 − ((𝑆‘𝐴)‘𝑄)))
87	1, 2, 3, 7, 8, 4, 6, 9, 5	israg 28631	. 2 ⊢ (𝜑 → (⟨“𝐵𝐴𝑄”⟩ ∈ (∟G‘𝐺) ↔ (𝐵 − 𝑄) = (𝐵 − ((𝑆‘𝐴)‘𝑄))))
88	86, 87	mpbird 257	1 ⊢ (𝜑 → ⟨“𝐵𝐴𝑄”⟩ ∈ (∟G‘𝐺))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109   ≠ wne 2926  ‘cfv 6514  (class class class)co 7390  ⟨“cs3 14815  Basecbs 17186  distcds 17236  TarskiGcstrkg 28361  Itvcitv 28367  LineGclng 28368  pInvGcmir 28586  ∟Gcrag 28627

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-rep 5237  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714  ax-cnex 11131  ax-resscn 11132  ax-1cn 11133  ax-icn 11134  ax-addcl 11135  ax-addrcl 11136  ax-mulcl 11137  ax-mulrcl 11138  ax-mulcom 11139  ax-addass 11140  ax-mulass 11141  ax-distr 11142  ax-i2m1 11143  ax-1ne0 11144  ax-1rid 11145  ax-rnegex 11146  ax-rrecex 11147  ax-cnre 11148  ax-pre-lttri 11149  ax-pre-lttrn 11150  ax-pre-ltadd 11151  ax-pre-mulgt0 11152

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-nel 3031  df-ral 3046  df-rex 3055  df-rmo 3356  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-pss 3937  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-tp 4597  df-op 4599  df-uni 4875  df-int 4914  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5536  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5594  df-we 5596  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-pred 6277  df-ord 6338  df-on 6339  df-lim 6340  df-suc 6341  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-riota 7347  df-ov 7393  df-oprab 7394  df-mpo 7395  df-om 7846  df-1st 7971  df-2nd 7972  df-frecs 8263  df-wrecs 8294  df-recs 8343  df-rdg 8381  df-1o 8437  df-oadd 8441  df-er 8674  df-map 8804  df-pm 8805  df-en 8922  df-dom 8923  df-sdom 8924  df-fin 8925  df-dju 9861  df-card 9899  df-pnf 11217  df-mnf 11218  df-xr 11219  df-ltxr 11220  df-le 11221  df-sub 11414  df-neg 11415  df-nn 12194  df-2 12256  df-3 12257  df-n0 12450  df-xnn0 12523  df-z 12537  df-uz 12801  df-fz 13476  df-fzo 13623  df-hash 14303  df-word 14486  df-concat 14543  df-s1 14568  df-s2 14821  df-s3 14822  df-trkgc 28382  df-trkgb 28383  df-trkgcb 28384  df-trkg 28387  df-cgrg 28445  df-mir 28587  df-rag 28628

This theorem is referenced by:  colperpexlem3  28666