Theorem colperpexlem1 29263

Description: Lemma for colperp 29262. 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 29190	. . . 4 ⊢ (𝜑 → (𝑀‘𝑄) ∈ 𝑃)
12		colperpexlem.c	. . . . . 6 ⊢ (𝜑 → 𝐶 ∈ 𝑃)
13	1, 2, 3, 7, 8, 4, 9, 10, 12	mircl 29190	. . . . 5 ⊢ (𝜑 → (𝑀‘𝐶) ∈ 𝑃)
14		eqid 2763	. . . . . 6 ⊢ (𝑆‘𝐵) = (𝑆‘𝐵)
15	1, 2, 3, 7, 8, 4, 6, 14, 12	mircl 29190	. . . . 5 ⊢ (𝜑 → ((𝑆‘𝐵)‘𝐶) ∈ 𝑃)
16	1, 2, 3, 7, 8, 4, 9, 10, 15	mircl 29190	. . . . 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 29190	. . . . . . . 8 ⊢ (𝜑 → (𝑁‘𝐶) ∈ 𝑃)
20	17, 19	eqeltrd 2864	. . . . . . 7 ⊢ (𝜑 → (𝐾‘(𝑀‘𝐶)) ∈ 𝑃)
21		colperpexlem.k	. . . . . . . 8 ⊢ 𝐾 = (𝑆‘𝑄)
22	1, 2, 3, 7, 8, 4, 5, 21, 13	mirbtwn 29187	. . . . . . 7 ⊢ (𝜑 → 𝑄 ∈ ((𝐾‘(𝑀‘𝐶))𝐼(𝑀‘𝐶)))
23	1, 2, 3, 4, 20, 5, 13, 22	tgbtwncom 29008	. . . . . 6 ⊢ (𝜑 → 𝑄 ∈ ((𝑀‘𝐶)𝐼(𝐾‘(𝑀‘𝐶))))
24	18	fveq1i 6917	. . . . . . . 8 ⊢ (𝑁‘𝐶) = ((𝑆‘𝐵)‘𝐶)
25	17, 24	eqtrdi 2814	. . . . . . 7 ⊢ (𝜑 → (𝐾‘(𝑀‘𝐶)) = ((𝑆‘𝐵)‘𝐶))
26	25	oveq2d 7467	. . . . . 6 ⊢ (𝜑 → ((𝑀‘𝐶)𝐼(𝐾‘(𝑀‘𝐶))) = ((𝑀‘𝐶)𝐼((𝑆‘𝐵)‘𝐶)))
27	23, 26	eleqtrd 2866	. . . . 5 ⊢ (𝜑 → 𝑄 ∈ ((𝑀‘𝐶)𝐼((𝑆‘𝐵)‘𝐶)))
28	1, 2, 3, 4, 13, 5, 15, 27	tgbtwncom 29008	. . . . . . 7 ⊢ (𝜑 → 𝑄 ∈ (((𝑆‘𝐵)‘𝐶)𝐼(𝑀‘𝐶)))
29	1, 2, 3, 7, 8, 4, 9, 10, 15, 5, 13, 28	mirbtwni 29200	. . . . . 6 ⊢ (𝜑 → (𝑀‘𝑄) ∈ ((𝑀‘((𝑆‘𝐵)‘𝐶))𝐼(𝑀‘(𝑀‘𝐶))))
30	1, 2, 3, 7, 8, 4, 9, 10, 12	mirmir 29191	. . . . . . 7 ⊢ (𝜑 → (𝑀‘(𝑀‘𝐶)) = 𝐶)
31	30	oveq2d 7467	. . . . . 6 ⊢ (𝜑 → ((𝑀‘((𝑆‘𝐵)‘𝐶))𝐼(𝑀‘(𝑀‘𝐶))) = ((𝑀‘((𝑆‘𝐵)‘𝐶))𝐼𝐶))
32	29, 31	eleqtrd 2866	. . . . 5 ⊢ (𝜑 → (𝑀‘𝑄) ∈ ((𝑀‘((𝑆‘𝐵)‘𝐶))𝐼𝐶))
33	1, 2, 3, 4, 13, 15	axtgcgrrflx 28981	. . . . . 6 ⊢ (𝜑 → ((𝑀‘𝐶) − ((𝑆‘𝐵)‘𝐶)) = (((𝑆‘𝐵)‘𝐶) − (𝑀‘𝐶)))
34	1, 2, 3, 7, 8, 4, 9, 10, 15, 13	miriso 29199	. . . . . 6 ⊢ (𝜑 → ((𝑀‘((𝑆‘𝐵)‘𝐶)) − (𝑀‘(𝑀‘𝐶))) = (((𝑆‘𝐵)‘𝐶) − (𝑀‘𝐶)))
35	30	oveq2d 7467	. . . . . 6 ⊢ (𝜑 → ((𝑀‘((𝑆‘𝐵)‘𝐶)) − (𝑀‘(𝑀‘𝐶))) = ((𝑀‘((𝑆‘𝐵)‘𝐶)) − 𝐶))
36	33, 34, 35	3eqtr2d 2804	. . . . 5 ⊢ (𝜑 → ((𝑀‘𝐶) − ((𝑆‘𝐵)‘𝐶)) = ((𝑀‘((𝑆‘𝐵)‘𝐶)) − 𝐶))
37	25	oveq2d 7467	. . . . . . 7 ⊢ (𝜑 → (𝑄 − (𝐾‘(𝑀‘𝐶))) = (𝑄 − ((𝑆‘𝐵)‘𝐶)))
38	1, 2, 3, 7, 8, 4, 5, 21, 13	mircgr 29186	. . . . . . 7 ⊢ (𝜑 → (𝑄 − (𝐾‘(𝑀‘𝐶))) = (𝑄 − (𝑀‘𝐶)))
39	37, 38	eqtr3d 2800	. . . . . 6 ⊢ (𝜑 → (𝑄 − ((𝑆‘𝐵)‘𝐶)) = (𝑄 − (𝑀‘𝐶)))
40	1, 2, 3, 7, 8, 4, 9, 10, 5, 13	miriso 29199	. . . . . 6 ⊢ (𝜑 → ((𝑀‘𝑄) − (𝑀‘(𝑀‘𝐶))) = (𝑄 − (𝑀‘𝐶)))
41	30	oveq2d 7467	. . . . . 6 ⊢ (𝜑 → ((𝑀‘𝑄) − (𝑀‘(𝑀‘𝐶))) = ((𝑀‘𝑄) − 𝐶))
42	39, 40, 41	3eqtr2d 2804	. . . . 5 ⊢ (𝜑 → (𝑄 − ((𝑆‘𝐵)‘𝐶)) = ((𝑀‘𝑄) − 𝐶))
43	1, 2, 3, 7, 8, 4, 9, 10, 6	mirmir 29191	. . . . . . . . . 10 ⊢ (𝜑 → (𝑀‘(𝑀‘𝐵)) = 𝐵)
44		eqidd 2764	. . . . . . . . . 10 ⊢ (𝜑 → (𝑀‘𝐵) = (𝑀‘𝐵))
45		eqidd 2764	. . . . . . . . . 10 ⊢ (𝜑 → (𝑀‘𝐶) = (𝑀‘𝐶))
46	43, 44, 45	s3eqd 15057	. . . . . . . . 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 29190	. . . . . . . . . 10 ⊢ (𝜑 → (𝑀‘𝐵) ∈ 𝑃)
48		simpr 490	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → 𝐴 = 𝐵)
49	48	fveq2d 6920	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → (𝑀‘𝐴) = (𝑀‘𝐵))
50	4	adantr 486	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → 𝐺 ∈ TarskiG)
51	9	adantr 486	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → 𝐴 ∈ 𝑃)
52	1, 2, 3, 7, 8, 50, 51, 10	mircinv 29197	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → (𝑀‘𝐴) = 𝐴)
53	49, 52	eqtr3d 2800	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → (𝑀‘𝐵) = 𝐴)
54		eqidd 2764	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → 𝐵 = 𝐵)
55		eqidd 2764	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → 𝐶 = 𝐶)
56	53, 54, 55	s3eqd 15057	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → ⟨“(𝑀‘𝐵)𝐵𝐶”⟩ = ⟨“𝐴𝐵𝐶”⟩)
57		colperpexlem.1	. . . . . . . . . . . . 13 ⊢ (𝜑 → ⟨“𝐴𝐵𝐶”⟩ ∈ (∟G‘𝐺))
58	57	adantr 486	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → ⟨“𝐴𝐵𝐶”⟩ ∈ (∟G‘𝐺))
59	56, 58	eqeltrd 2864	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → ⟨“(𝑀‘𝐵)𝐵𝐶”⟩ ∈ (∟G‘𝐺))
60	4	adantr 486	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐴 ≠ 𝐵) → 𝐺 ∈ TarskiG)
61	9	adantr 486	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐴 ≠ 𝐵) → 𝐴 ∈ 𝑃)
62	6	adantr 486	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐴 ≠ 𝐵) → 𝐵 ∈ 𝑃)
63	12	adantr 486	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐴 ≠ 𝐵) → 𝐶 ∈ 𝑃)
64	1, 2, 3, 7, 8, 60, 61, 10, 62	mircl 29190	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐴 ≠ 𝐵) → (𝑀‘𝐵) ∈ 𝑃)
65	57	adantr 486	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐴 ≠ 𝐵) → ⟨“𝐴𝐵𝐶”⟩ ∈ (∟G‘𝐺))
66		simpr 490	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐴 ≠ 𝐵) → 𝐴 ≠ 𝐵)
67	1, 2, 3, 7, 8, 60, 61, 10, 62	mirbtwn 29187	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝐴 ≠ 𝐵) → 𝐴 ∈ ((𝑀‘𝐵)𝐼𝐵))
68	1, 7, 3, 60, 64, 62, 61, 67	btwncolg1 29075	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝐴 ≠ 𝐵) → (𝐴 ∈ ((𝑀‘𝐵)𝐿𝐵) ∨ (𝑀‘𝐵) = 𝐵))
69	1, 7, 3, 60, 64, 62, 61, 68	colcom 29078	. . . . . . . . . . . 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 29231	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐴 ≠ 𝐵) → ⟨“(𝑀‘𝐵)𝐵𝐶”⟩ ∈ (∟G‘𝐺))
71	59, 70	pm2.61dane 3046	. . . . . . . . . 10 ⊢ (𝜑 → ⟨“(𝑀‘𝐵)𝐵𝐶”⟩ ∈ (∟G‘𝐺))
72	1, 2, 3, 7, 8, 4, 47, 6, 12, 71, 10, 9	mirrag 29233	. . . . . . . . 9 ⊢ (𝜑 → ⟨“(𝑀‘(𝑀‘𝐵))(𝑀‘𝐵)(𝑀‘𝐶)”⟩ ∈ (∟G‘𝐺))
73	46, 72	eqeltrrd 2865	. . . . . . . 8 ⊢ (𝜑 → ⟨“𝐵(𝑀‘𝐵)(𝑀‘𝐶)”⟩ ∈ (∟G‘𝐺))
74	1, 2, 3, 7, 8, 4, 6, 47, 13	israg 29229	. . . . . . . 8 ⊢ (𝜑 → (⟨“𝐵(𝑀‘𝐵)(𝑀‘𝐶)”⟩ ∈ (∟G‘𝐺) ↔ (𝐵 − (𝑀‘𝐶)) = (𝐵 − ((𝑆‘(𝑀‘𝐵))‘(𝑀‘𝐶)))))
75	73, 74	mpbid 235	. . . . . . 7 ⊢ (𝜑 → (𝐵 − (𝑀‘𝐶)) = (𝐵 − ((𝑆‘(𝑀‘𝐵))‘(𝑀‘𝐶))))
76	1, 2, 3, 7, 8, 4, 9, 10, 12, 6	mirmir2 29203	. . . . . . . 8 ⊢ (𝜑 → (𝑀‘((𝑆‘𝐵)‘𝐶)) = ((𝑆‘(𝑀‘𝐵))‘(𝑀‘𝐶)))
77	76	oveq2d 7467	. . . . . . 7 ⊢ (𝜑 → (𝐵 − (𝑀‘((𝑆‘𝐵)‘𝐶))) = (𝐵 − ((𝑆‘(𝑀‘𝐵))‘(𝑀‘𝐶))))
78	75, 77	eqtr4d 2801	. . . . . 6 ⊢ (𝜑 → (𝐵 − (𝑀‘𝐶)) = (𝐵 − (𝑀‘((𝑆‘𝐵)‘𝐶))))
79	1, 2, 3, 4, 6, 13, 6, 16, 78	tgcgrcomlr 28999	. . . . 5 ⊢ (𝜑 → ((𝑀‘𝐶) − 𝐵) = ((𝑀‘((𝑆‘𝐵)‘𝐶)) − 𝐵))
80	1, 2, 3, 7, 8, 4, 6, 14, 12	mircgr 29186	. . . . . 6 ⊢ (𝜑 → (𝐵 − ((𝑆‘𝐵)‘𝐶)) = (𝐵 − 𝐶))
81	1, 2, 3, 4, 6, 15, 6, 12, 80	tgcgrcomlr 28999	. . . . 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 29028	. . . 4 ⊢ (𝜑 → (𝑄 − 𝐵) = ((𝑀‘𝑄) − 𝐵))
83	1, 2, 3, 4, 5, 6, 11, 6, 82	tgcgrcomlr 28999	. . 3 ⊢ (𝜑 → (𝐵 − 𝑄) = (𝐵 − (𝑀‘𝑄)))
84	10	fveq1i 6917	. . . 4 ⊢ (𝑀‘𝑄) = ((𝑆‘𝐴)‘𝑄)
85	84	oveq2i 7462	. . 3 ⊢ (𝐵 − (𝑀‘𝑄)) = (𝐵 − ((𝑆‘𝐴)‘𝑄))
86	83, 85	eqtrdi 2814	. 2 ⊢ (𝜑 → (𝐵 − 𝑄) = (𝐵 − ((𝑆‘𝐴)‘𝑄)))
87	1, 2, 3, 7, 8, 4, 6, 9, 5	israg 29229	. 2 ⊢ (𝜑 → (⟨“𝐵𝐴𝑄”⟩ ∈ (∟G‘𝐺) ↔ (𝐵 − 𝑄) = (𝐵 − ((𝑆‘𝐴)‘𝑄))))
88	86, 87	mpbird 260	1 ⊢ (𝜑 → ⟨“𝐵𝐴𝑄”⟩ ∈ (∟G‘𝐺))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2147   ≠ wne 2959  ‘cfv 6566  (class class class)co 7451  ⟨“cs3 15035  Basecbs 17431  distcds 17481  TarskiGcstrkg 28946  Itvcitv 28952  LineGclng 28953  pInvGcmir 29181  ∟Gcrag 29225

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5238  ax-sep 5255  ax-nul 5266  ax-pow 5334  ax-pr 5398  ax-un 7783  ax-cnex 11295  ax-resscn 11296  ax-1cn 11297  ax-icn 11298  ax-addcl 11299  ax-addrcl 11300  ax-mulcl 11301  ax-mulrcl 11302  ax-mulcom 11303  ax-addass 11304  ax-mulass 11305  ax-distr 11306  ax-i2m1 11307  ax-1ne0 11308  ax-1rid 11309  ax-rnegex 11310  ax-rrecex 11311  ax-cnre 11312  ax-pre-lttri 11313  ax-pre-lttrn 11314  ax-pre-ltadd 11315  ax-pre-mulgt0 11316

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-nel 3066  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-tp 4593  df-op 4841  df-uni 4873  df-int 4913  df-iun 4958  df-br 5109  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-we 5613  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-pred 6330  df-ord 6391  df-on 6392  df-lim 6393  df-suc 6394  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-f1 6571  df-fo 6572  df-f1o 6573  df-fv 6574  df-riota 7408  df-ov 7454  df-oprab 7455  df-mpo 7456  df-om 7912  df-1st 8035  df-2nd 8036  df-frecs 8337  df-wrecs 8368  df-recs 8417  df-rdg 8456  df-1o 8514  df-oadd 8518  df-er 8755  df-map 8887  df-pm 8888  df-en 9012  df-dom 9013  df-sdom 9014  df-fin 9015  df-dju 10019  df-card 10057  df-pnf 11384  df-mnf 11385  df-xr 11386  df-ltxr 11387  df-le 11388  df-sub 11582  df-neg 11583  df-nn 12376  df-2 12445  df-3 12446  df-n0 12647  df-xnn0 12720  df-z 12734  df-uz 13006  df-fz 13680  df-fzo 13829  df-hash 14516  df-word 14700  df-concat 14757  df-s1 14784  df-s2 15041  df-s3 15042  df-trkgc 28967  df-trkgb 28968  df-trkgcb 28969  df-trkg 28972  df-cgrg 29031  df-mir 29182  df-rag 29226

This theorem is used by:  colperpexlem3  29265