Theorem colperpexlem1 28756

Description: Lemma for colperp 28755. 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 28687	. . . 4 ⊢ (𝜑 → (𝑀‘𝑄) ∈ 𝑃)
12		colperpexlem.c	. . . . . 6 ⊢ (𝜑 → 𝐶 ∈ 𝑃)
13	1, 2, 3, 7, 8, 4, 9, 10, 12	mircl 28687	. . . . 5 ⊢ (𝜑 → (𝑀‘𝐶) ∈ 𝑃)
14		eqid 2740	. . . . . 6 ⊢ (𝑆‘𝐵) = (𝑆‘𝐵)
15	1, 2, 3, 7, 8, 4, 6, 14, 12	mircl 28687	. . . . 5 ⊢ (𝜑 → ((𝑆‘𝐵)‘𝐶) ∈ 𝑃)
16	1, 2, 3, 7, 8, 4, 9, 10, 15	mircl 28687	. . . . 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 28687	. . . . . . . 8 ⊢ (𝜑 → (𝑁‘𝐶) ∈ 𝑃)
20	17, 19	eqeltrd 2844	. . . . . . 7 ⊢ (𝜑 → (𝐾‘(𝑀‘𝐶)) ∈ 𝑃)
21		colperpexlem.k	. . . . . . . 8 ⊢ 𝐾 = (𝑆‘𝑄)
22	1, 2, 3, 7, 8, 4, 5, 21, 13	mirbtwn 28684	. . . . . . 7 ⊢ (𝜑 → 𝑄 ∈ ((𝐾‘(𝑀‘𝐶))𝐼(𝑀‘𝐶)))
23	1, 2, 3, 4, 20, 5, 13, 22	tgbtwncom 28514	. . . . . 6 ⊢ (𝜑 → 𝑄 ∈ ((𝑀‘𝐶)𝐼(𝐾‘(𝑀‘𝐶))))
24	18	fveq1i 6921	. . . . . . . 8 ⊢ (𝑁‘𝐶) = ((𝑆‘𝐵)‘𝐶)
25	17, 24	eqtrdi 2796	. . . . . . 7 ⊢ (𝜑 → (𝐾‘(𝑀‘𝐶)) = ((𝑆‘𝐵)‘𝐶))
26	25	oveq2d 7464	. . . . . 6 ⊢ (𝜑 → ((𝑀‘𝐶)𝐼(𝐾‘(𝑀‘𝐶))) = ((𝑀‘𝐶)𝐼((𝑆‘𝐵)‘𝐶)))
27	23, 26	eleqtrd 2846	. . . . 5 ⊢ (𝜑 → 𝑄 ∈ ((𝑀‘𝐶)𝐼((𝑆‘𝐵)‘𝐶)))
28	1, 2, 3, 4, 13, 5, 15, 27	tgbtwncom 28514	. . . . . . 7 ⊢ (𝜑 → 𝑄 ∈ (((𝑆‘𝐵)‘𝐶)𝐼(𝑀‘𝐶)))
29	1, 2, 3, 7, 8, 4, 9, 10, 15, 5, 13, 28	mirbtwni 28697	. . . . . 6 ⊢ (𝜑 → (𝑀‘𝑄) ∈ ((𝑀‘((𝑆‘𝐵)‘𝐶))𝐼(𝑀‘(𝑀‘𝐶))))
30	1, 2, 3, 7, 8, 4, 9, 10, 12	mirmir 28688	. . . . . . 7 ⊢ (𝜑 → (𝑀‘(𝑀‘𝐶)) = 𝐶)
31	30	oveq2d 7464	. . . . . 6 ⊢ (𝜑 → ((𝑀‘((𝑆‘𝐵)‘𝐶))𝐼(𝑀‘(𝑀‘𝐶))) = ((𝑀‘((𝑆‘𝐵)‘𝐶))𝐼𝐶))
32	29, 31	eleqtrd 2846	. . . . 5 ⊢ (𝜑 → (𝑀‘𝑄) ∈ ((𝑀‘((𝑆‘𝐵)‘𝐶))𝐼𝐶))
33	1, 2, 3, 4, 13, 15	axtgcgrrflx 28488	. . . . . 6 ⊢ (𝜑 → ((𝑀‘𝐶) − ((𝑆‘𝐵)‘𝐶)) = (((𝑆‘𝐵)‘𝐶) − (𝑀‘𝐶)))
34	1, 2, 3, 7, 8, 4, 9, 10, 15, 13	miriso 28696	. . . . . 6 ⊢ (𝜑 → ((𝑀‘((𝑆‘𝐵)‘𝐶)) − (𝑀‘(𝑀‘𝐶))) = (((𝑆‘𝐵)‘𝐶) − (𝑀‘𝐶)))
35	30	oveq2d 7464	. . . . . 6 ⊢ (𝜑 → ((𝑀‘((𝑆‘𝐵)‘𝐶)) − (𝑀‘(𝑀‘𝐶))) = ((𝑀‘((𝑆‘𝐵)‘𝐶)) − 𝐶))
36	33, 34, 35	3eqtr2d 2786	. . . . 5 ⊢ (𝜑 → ((𝑀‘𝐶) − ((𝑆‘𝐵)‘𝐶)) = ((𝑀‘((𝑆‘𝐵)‘𝐶)) − 𝐶))
37	25	oveq2d 7464	. . . . . . 7 ⊢ (𝜑 → (𝑄 − (𝐾‘(𝑀‘𝐶))) = (𝑄 − ((𝑆‘𝐵)‘𝐶)))
38	1, 2, 3, 7, 8, 4, 5, 21, 13	mircgr 28683	. . . . . . 7 ⊢ (𝜑 → (𝑄 − (𝐾‘(𝑀‘𝐶))) = (𝑄 − (𝑀‘𝐶)))
39	37, 38	eqtr3d 2782	. . . . . 6 ⊢ (𝜑 → (𝑄 − ((𝑆‘𝐵)‘𝐶)) = (𝑄 − (𝑀‘𝐶)))
40	1, 2, 3, 7, 8, 4, 9, 10, 5, 13	miriso 28696	. . . . . 6 ⊢ (𝜑 → ((𝑀‘𝑄) − (𝑀‘(𝑀‘𝐶))) = (𝑄 − (𝑀‘𝐶)))
41	30	oveq2d 7464	. . . . . 6 ⊢ (𝜑 → ((𝑀‘𝑄) − (𝑀‘(𝑀‘𝐶))) = ((𝑀‘𝑄) − 𝐶))
42	39, 40, 41	3eqtr2d 2786	. . . . 5 ⊢ (𝜑 → (𝑄 − ((𝑆‘𝐵)‘𝐶)) = ((𝑀‘𝑄) − 𝐶))
43	1, 2, 3, 7, 8, 4, 9, 10, 6	mirmir 28688	. . . . . . . . . 10 ⊢ (𝜑 → (𝑀‘(𝑀‘𝐵)) = 𝐵)
44		eqidd 2741	. . . . . . . . . 10 ⊢ (𝜑 → (𝑀‘𝐵) = (𝑀‘𝐵))
45		eqidd 2741	. . . . . . . . . 10 ⊢ (𝜑 → (𝑀‘𝐶) = (𝑀‘𝐶))
46	43, 44, 45	s3eqd 14913	. . . . . . . . 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 28687	. . . . . . . . . 10 ⊢ (𝜑 → (𝑀‘𝐵) ∈ 𝑃)
48		simpr 484	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → 𝐴 = 𝐵)
49	48	fveq2d 6924	. . . . . . . . . . . . . 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 28694	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → (𝑀‘𝐴) = 𝐴)
53	49, 52	eqtr3d 2782	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → (𝑀‘𝐵) = 𝐴)
54		eqidd 2741	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → 𝐵 = 𝐵)
55		eqidd 2741	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → 𝐶 = 𝐶)
56	53, 54, 55	s3eqd 14913	. . . . . . . . . . . 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 2844	. . . . . . . . . . 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 28687	. . . . . . . . . . . 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 28684	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝐴 ≠ 𝐵) → 𝐴 ∈ ((𝑀‘𝐵)𝐼𝐵))
68	1, 7, 3, 60, 64, 62, 61, 67	btwncolg1 28581	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝐴 ≠ 𝐵) → (𝐴 ∈ ((𝑀‘𝐵)𝐿𝐵) ∨ (𝑀‘𝐵) = 𝐵))
69	1, 7, 3, 60, 64, 62, 61, 68	colcom 28584	. . . . . . . . . . . 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 28725	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐴 ≠ 𝐵) → ⟨“(𝑀‘𝐵)𝐵𝐶”⟩ ∈ (∟G‘𝐺))
71	59, 70	pm2.61dane 3035	. . . . . . . . . 10 ⊢ (𝜑 → ⟨“(𝑀‘𝐵)𝐵𝐶”⟩ ∈ (∟G‘𝐺))
72	1, 2, 3, 7, 8, 4, 47, 6, 12, 71, 10, 9	mirrag 28727	. . . . . . . . 9 ⊢ (𝜑 → ⟨“(𝑀‘(𝑀‘𝐵))(𝑀‘𝐵)(𝑀‘𝐶)”⟩ ∈ (∟G‘𝐺))
73	46, 72	eqeltrrd 2845	. . . . . . . 8 ⊢ (𝜑 → ⟨“𝐵(𝑀‘𝐵)(𝑀‘𝐶)”⟩ ∈ (∟G‘𝐺))
74	1, 2, 3, 7, 8, 4, 6, 47, 13	israg 28723	. . . . . . . 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 28700	. . . . . . . 8 ⊢ (𝜑 → (𝑀‘((𝑆‘𝐵)‘𝐶)) = ((𝑆‘(𝑀‘𝐵))‘(𝑀‘𝐶)))
77	76	oveq2d 7464	. . . . . . 7 ⊢ (𝜑 → (𝐵 − (𝑀‘((𝑆‘𝐵)‘𝐶))) = (𝐵 − ((𝑆‘(𝑀‘𝐵))‘(𝑀‘𝐶))))
78	75, 77	eqtr4d 2783	. . . . . 6 ⊢ (𝜑 → (𝐵 − (𝑀‘𝐶)) = (𝐵 − (𝑀‘((𝑆‘𝐵)‘𝐶))))
79	1, 2, 3, 4, 6, 13, 6, 16, 78	tgcgrcomlr 28506	. . . . 5 ⊢ (𝜑 → ((𝑀‘𝐶) − 𝐵) = ((𝑀‘((𝑆‘𝐵)‘𝐶)) − 𝐵))
80	1, 2, 3, 7, 8, 4, 6, 14, 12	mircgr 28683	. . . . . 6 ⊢ (𝜑 → (𝐵 − ((𝑆‘𝐵)‘𝐶)) = (𝐵 − 𝐶))
81	1, 2, 3, 4, 6, 15, 6, 12, 80	tgcgrcomlr 28506	. . . . 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 28534	. . . 4 ⊢ (𝜑 → (𝑄 − 𝐵) = ((𝑀‘𝑄) − 𝐵))
83	1, 2, 3, 4, 5, 6, 11, 6, 82	tgcgrcomlr 28506	. . 3 ⊢ (𝜑 → (𝐵 − 𝑄) = (𝐵 − (𝑀‘𝑄)))
84	10	fveq1i 6921	. . . 4 ⊢ (𝑀‘𝑄) = ((𝑆‘𝐴)‘𝑄)
85	84	oveq2i 7459	. . 3 ⊢ (𝐵 − (𝑀‘𝑄)) = (𝐵 − ((𝑆‘𝐴)‘𝑄))
86	83, 85	eqtrdi 2796	. 2 ⊢ (𝜑 → (𝐵 − 𝑄) = (𝐵 − ((𝑆‘𝐴)‘𝑄)))
87	1, 2, 3, 7, 8, 4, 6, 9, 5	israg 28723	. 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 1537   ∈ wcel 2108   ≠ wne 2946  ‘cfv 6573  (class class class)co 7448  ⟨“cs3 14891  Basecbs 17258  distcds 17320  TarskiGcstrkg 28453  Itvcitv 28459  LineGclng 28460  pInvGcmir 28678  ∟Gcrag 28719

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770  ax-cnex 11240  ax-resscn 11241  ax-1cn 11242  ax-icn 11243  ax-addcl 11244  ax-addrcl 11245  ax-mulcl 11246  ax-mulrcl 11247  ax-mulcom 11248  ax-addass 11249  ax-mulass 11250  ax-distr 11251  ax-i2m1 11252  ax-1ne0 11253  ax-1rid 11254  ax-rnegex 11255  ax-rrecex 11256  ax-cnre 11257  ax-pre-lttri 11258  ax-pre-lttrn 11259  ax-pre-ltadd 11260  ax-pre-mulgt0 11261

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-nel 3053  df-ral 3068  df-rex 3077  df-rmo 3388  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-tp 4653  df-op 4655  df-uni 4932  df-int 4971  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-riota 7404  df-ov 7451  df-oprab 7452  df-mpo 7453  df-om 7904  df-1st 8030  df-2nd 8031  df-frecs 8322  df-wrecs 8353  df-recs 8427  df-rdg 8466  df-1o 8522  df-oadd 8526  df-er 8763  df-map 8886  df-pm 8887  df-en 9004  df-dom 9005  df-sdom 9006  df-fin 9007  df-dju 9970  df-card 10008  df-pnf 11326  df-mnf 11327  df-xr 11328  df-ltxr 11329  df-le 11330  df-sub 11522  df-neg 11523  df-nn 12294  df-2 12356  df-3 12357  df-n0 12554  df-xnn0 12626  df-z 12640  df-uz 12904  df-fz 13568  df-fzo 13712  df-hash 14380  df-word 14563  df-concat 14619  df-s1 14644  df-s2 14897  df-s3 14898  df-trkgc 28474  df-trkgb 28475  df-trkgcb 28476  df-trkg 28479  df-cgrg 28537  df-mir 28679  df-rag 28720

This theorem is referenced by:  colperpexlem3  28758