Theorem colperpexlem1 26038

Description: Lemma for colperp 26037. 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 25972	. . . 4 ⊢ (𝜑 → (𝑀‘𝑄) ∈ 𝑃)
12		colperpexlem.c	. . . . . 6 ⊢ (𝜑 → 𝐶 ∈ 𝑃)
13	1, 2, 3, 7, 8, 4, 9, 10, 12	mircl 25972	. . . . 5 ⊢ (𝜑 → (𝑀‘𝐶) ∈ 𝑃)
14		eqid 2824	. . . . . 6 ⊢ (𝑆‘𝐵) = (𝑆‘𝐵)
15	1, 2, 3, 7, 8, 4, 6, 14, 12	mircl 25972	. . . . 5 ⊢ (𝜑 → ((𝑆‘𝐵)‘𝐶) ∈ 𝑃)
16	1, 2, 3, 7, 8, 4, 9, 10, 15	mircl 25972	. . . . 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 25972	. . . . . . . 8 ⊢ (𝜑 → (𝑁‘𝐶) ∈ 𝑃)
20	17, 19	eqeltrd 2905	. . . . . . 7 ⊢ (𝜑 → (𝐾‘(𝑀‘𝐶)) ∈ 𝑃)
21		colperpexlem.k	. . . . . . . 8 ⊢ 𝐾 = (𝑆‘𝑄)
22	1, 2, 3, 7, 8, 4, 5, 21, 13	mirbtwn 25969	. . . . . . 7 ⊢ (𝜑 → 𝑄 ∈ ((𝐾‘(𝑀‘𝐶))𝐼(𝑀‘𝐶)))
23	1, 2, 3, 4, 20, 5, 13, 22	tgbtwncom 25799	. . . . . 6 ⊢ (𝜑 → 𝑄 ∈ ((𝑀‘𝐶)𝐼(𝐾‘(𝑀‘𝐶))))
24	18	fveq1i 6433	. . . . . . . 8 ⊢ (𝑁‘𝐶) = ((𝑆‘𝐵)‘𝐶)
25	17, 24	syl6eq 2876	. . . . . . 7 ⊢ (𝜑 → (𝐾‘(𝑀‘𝐶)) = ((𝑆‘𝐵)‘𝐶))
26	25	oveq2d 6920	. . . . . 6 ⊢ (𝜑 → ((𝑀‘𝐶)𝐼(𝐾‘(𝑀‘𝐶))) = ((𝑀‘𝐶)𝐼((𝑆‘𝐵)‘𝐶)))
27	23, 26	eleqtrd 2907	. . . . 5 ⊢ (𝜑 → 𝑄 ∈ ((𝑀‘𝐶)𝐼((𝑆‘𝐵)‘𝐶)))
28	1, 2, 3, 4, 13, 5, 15, 27	tgbtwncom 25799	. . . . . . 7 ⊢ (𝜑 → 𝑄 ∈ (((𝑆‘𝐵)‘𝐶)𝐼(𝑀‘𝐶)))
29	1, 2, 3, 7, 8, 4, 9, 10, 15, 5, 13, 28	mirbtwni 25982	. . . . . 6 ⊢ (𝜑 → (𝑀‘𝑄) ∈ ((𝑀‘((𝑆‘𝐵)‘𝐶))𝐼(𝑀‘(𝑀‘𝐶))))
30	1, 2, 3, 7, 8, 4, 9, 10, 12	mirmir 25973	. . . . . . 7 ⊢ (𝜑 → (𝑀‘(𝑀‘𝐶)) = 𝐶)
31	30	oveq2d 6920	. . . . . 6 ⊢ (𝜑 → ((𝑀‘((𝑆‘𝐵)‘𝐶))𝐼(𝑀‘(𝑀‘𝐶))) = ((𝑀‘((𝑆‘𝐵)‘𝐶))𝐼𝐶))
32	29, 31	eleqtrd 2907	. . . . 5 ⊢ (𝜑 → (𝑀‘𝑄) ∈ ((𝑀‘((𝑆‘𝐵)‘𝐶))𝐼𝐶))
33	1, 2, 3, 4, 13, 15	axtgcgrrflx 25773	. . . . . 6 ⊢ (𝜑 → ((𝑀‘𝐶) − ((𝑆‘𝐵)‘𝐶)) = (((𝑆‘𝐵)‘𝐶) − (𝑀‘𝐶)))
34	1, 2, 3, 7, 8, 4, 9, 10, 15, 13	miriso 25981	. . . . . 6 ⊢ (𝜑 → ((𝑀‘((𝑆‘𝐵)‘𝐶)) − (𝑀‘(𝑀‘𝐶))) = (((𝑆‘𝐵)‘𝐶) − (𝑀‘𝐶)))
35	30	oveq2d 6920	. . . . . 6 ⊢ (𝜑 → ((𝑀‘((𝑆‘𝐵)‘𝐶)) − (𝑀‘(𝑀‘𝐶))) = ((𝑀‘((𝑆‘𝐵)‘𝐶)) − 𝐶))
36	33, 34, 35	3eqtr2d 2866	. . . . 5 ⊢ (𝜑 → ((𝑀‘𝐶) − ((𝑆‘𝐵)‘𝐶)) = ((𝑀‘((𝑆‘𝐵)‘𝐶)) − 𝐶))
37	25	oveq2d 6920	. . . . . . 7 ⊢ (𝜑 → (𝑄 − (𝐾‘(𝑀‘𝐶))) = (𝑄 − ((𝑆‘𝐵)‘𝐶)))
38	1, 2, 3, 7, 8, 4, 5, 21, 13	mircgr 25968	. . . . . . 7 ⊢ (𝜑 → (𝑄 − (𝐾‘(𝑀‘𝐶))) = (𝑄 − (𝑀‘𝐶)))
39	37, 38	eqtr3d 2862	. . . . . 6 ⊢ (𝜑 → (𝑄 − ((𝑆‘𝐵)‘𝐶)) = (𝑄 − (𝑀‘𝐶)))
40	1, 2, 3, 7, 8, 4, 9, 10, 5, 13	miriso 25981	. . . . . 6 ⊢ (𝜑 → ((𝑀‘𝑄) − (𝑀‘(𝑀‘𝐶))) = (𝑄 − (𝑀‘𝐶)))
41	30	oveq2d 6920	. . . . . 6 ⊢ (𝜑 → ((𝑀‘𝑄) − (𝑀‘(𝑀‘𝐶))) = ((𝑀‘𝑄) − 𝐶))
42	39, 40, 41	3eqtr2d 2866	. . . . 5 ⊢ (𝜑 → (𝑄 − ((𝑆‘𝐵)‘𝐶)) = ((𝑀‘𝑄) − 𝐶))
43	1, 2, 3, 7, 8, 4, 9, 10, 6	mirmir 25973	. . . . . . . . . 10 ⊢ (𝜑 → (𝑀‘(𝑀‘𝐵)) = 𝐵)
44		eqidd 2825	. . . . . . . . . 10 ⊢ (𝜑 → (𝑀‘𝐵) = (𝑀‘𝐵))
45		eqidd 2825	. . . . . . . . . 10 ⊢ (𝜑 → (𝑀‘𝐶) = (𝑀‘𝐶))
46	43, 44, 45	s3eqd 13984	. . . . . . . . 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 25972	. . . . . . . . . 10 ⊢ (𝜑 → (𝑀‘𝐵) ∈ 𝑃)
48		simpr 479	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → 𝐴 = 𝐵)
49	48	fveq2d 6436	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → (𝑀‘𝐴) = (𝑀‘𝐵))
50	4	adantr 474	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → 𝐺 ∈ TarskiG)
51	9	adantr 474	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → 𝐴 ∈ 𝑃)
52	1, 2, 3, 7, 8, 50, 51, 10	mircinv 25979	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → (𝑀‘𝐴) = 𝐴)
53	49, 52	eqtr3d 2862	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → (𝑀‘𝐵) = 𝐴)
54		eqidd 2825	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → 𝐵 = 𝐵)
55		eqidd 2825	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → 𝐶 = 𝐶)
56	53, 54, 55	s3eqd 13984	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → ⟨“(𝑀‘𝐵)𝐵𝐶”⟩ = ⟨“𝐴𝐵𝐶”⟩)
57		colperpexlem.1	. . . . . . . . . . . . 13 ⊢ (𝜑 → ⟨“𝐴𝐵𝐶”⟩ ∈ (∟G‘𝐺))
58	57	adantr 474	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → ⟨“𝐴𝐵𝐶”⟩ ∈ (∟G‘𝐺))
59	56, 58	eqeltrd 2905	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → ⟨“(𝑀‘𝐵)𝐵𝐶”⟩ ∈ (∟G‘𝐺))
60	4	adantr 474	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐴 ≠ 𝐵) → 𝐺 ∈ TarskiG)
61	9	adantr 474	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐴 ≠ 𝐵) → 𝐴 ∈ 𝑃)
62	6	adantr 474	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐴 ≠ 𝐵) → 𝐵 ∈ 𝑃)
63	12	adantr 474	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐴 ≠ 𝐵) → 𝐶 ∈ 𝑃)
64	1, 2, 3, 7, 8, 60, 61, 10, 62	mircl 25972	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐴 ≠ 𝐵) → (𝑀‘𝐵) ∈ 𝑃)
65	57	adantr 474	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐴 ≠ 𝐵) → ⟨“𝐴𝐵𝐶”⟩ ∈ (∟G‘𝐺))
66		simpr 479	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐴 ≠ 𝐵) → 𝐴 ≠ 𝐵)
67	1, 2, 3, 7, 8, 60, 61, 10, 62	mirbtwn 25969	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝐴 ≠ 𝐵) → 𝐴 ∈ ((𝑀‘𝐵)𝐼𝐵))
68	1, 7, 3, 60, 64, 62, 61, 67	btwncolg1 25866	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝐴 ≠ 𝐵) → (𝐴 ∈ ((𝑀‘𝐵)𝐿𝐵) ∨ (𝑀‘𝐵) = 𝐵))
69	1, 7, 3, 60, 64, 62, 61, 68	colcom 25869	. . . . . . . . . . . 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 26010	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐴 ≠ 𝐵) → ⟨“(𝑀‘𝐵)𝐵𝐶”⟩ ∈ (∟G‘𝐺))
71	59, 70	pm2.61dane 3085	. . . . . . . . . 10 ⊢ (𝜑 → ⟨“(𝑀‘𝐵)𝐵𝐶”⟩ ∈ (∟G‘𝐺))
72	1, 2, 3, 7, 8, 4, 47, 6, 12, 71, 10, 9	mirrag 26012	. . . . . . . . 9 ⊢ (𝜑 → ⟨“(𝑀‘(𝑀‘𝐵))(𝑀‘𝐵)(𝑀‘𝐶)”⟩ ∈ (∟G‘𝐺))
73	46, 72	eqeltrrd 2906	. . . . . . . 8 ⊢ (𝜑 → ⟨“𝐵(𝑀‘𝐵)(𝑀‘𝐶)”⟩ ∈ (∟G‘𝐺))
74	1, 2, 3, 7, 8, 4, 6, 47, 13	israg 26008	. . . . . . . 8 ⊢ (𝜑 → (⟨“𝐵(𝑀‘𝐵)(𝑀‘𝐶)”⟩ ∈ (∟G‘𝐺) ↔ (𝐵 − (𝑀‘𝐶)) = (𝐵 − ((𝑆‘(𝑀‘𝐵))‘(𝑀‘𝐶)))))
75	73, 74	mpbid 224	. . . . . . 7 ⊢ (𝜑 → (𝐵 − (𝑀‘𝐶)) = (𝐵 − ((𝑆‘(𝑀‘𝐵))‘(𝑀‘𝐶))))
76	1, 2, 3, 7, 8, 4, 9, 10, 12, 6	mirmir2 25985	. . . . . . . 8 ⊢ (𝜑 → (𝑀‘((𝑆‘𝐵)‘𝐶)) = ((𝑆‘(𝑀‘𝐵))‘(𝑀‘𝐶)))
77	76	oveq2d 6920	. . . . . . 7 ⊢ (𝜑 → (𝐵 − (𝑀‘((𝑆‘𝐵)‘𝐶))) = (𝐵 − ((𝑆‘(𝑀‘𝐵))‘(𝑀‘𝐶))))
78	75, 77	eqtr4d 2863	. . . . . 6 ⊢ (𝜑 → (𝐵 − (𝑀‘𝐶)) = (𝐵 − (𝑀‘((𝑆‘𝐵)‘𝐶))))
79	1, 2, 3, 4, 6, 13, 6, 16, 78	tgcgrcomlr 25791	. . . . 5 ⊢ (𝜑 → ((𝑀‘𝐶) − 𝐵) = ((𝑀‘((𝑆‘𝐵)‘𝐶)) − 𝐵))
80	1, 2, 3, 7, 8, 4, 6, 14, 12	mircgr 25968	. . . . . 6 ⊢ (𝜑 → (𝐵 − ((𝑆‘𝐵)‘𝐶)) = (𝐵 − 𝐶))
81	1, 2, 3, 4, 6, 15, 6, 12, 80	tgcgrcomlr 25791	. . . . 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 25819	. . . 4 ⊢ (𝜑 → (𝑄 − 𝐵) = ((𝑀‘𝑄) − 𝐵))
83	1, 2, 3, 4, 5, 6, 11, 6, 82	tgcgrcomlr 25791	. . 3 ⊢ (𝜑 → (𝐵 − 𝑄) = (𝐵 − (𝑀‘𝑄)))
84	10	fveq1i 6433	. . . 4 ⊢ (𝑀‘𝑄) = ((𝑆‘𝐴)‘𝑄)
85	84	oveq2i 6915	. . 3 ⊢ (𝐵 − (𝑀‘𝑄)) = (𝐵 − ((𝑆‘𝐴)‘𝑄))
86	83, 85	syl6eq 2876	. 2 ⊢ (𝜑 → (𝐵 − 𝑄) = (𝐵 − ((𝑆‘𝐴)‘𝑄)))
87	1, 2, 3, 7, 8, 4, 6, 9, 5	israg 26008	. 2 ⊢ (𝜑 → (⟨“𝐵𝐴𝑄”⟩ ∈ (∟G‘𝐺) ↔ (𝐵 − 𝑄) = (𝐵 − ((𝑆‘𝐴)‘𝑄))))
88	86, 87	mpbird 249	1 ⊢ (𝜑 → ⟨“𝐵𝐴𝑄”⟩ ∈ (∟G‘𝐺))

Syntax hints:   → wi 4   ∧ wa 386   = wceq 1658   ∈ wcel 2166   ≠ wne 2998  ‘cfv 6122  (class class class)co 6904  ⟨“cs3 13962  Basecbs 16221  distcds 16313  TarskiGcstrkg 25741  Itvcitv 25747  LineGclng 25748  pInvGcmir 25963  ∟Gcrag 26004

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-8 2168  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2390  ax-ext 2802  ax-rep 4993  ax-sep 5004  ax-nul 5012  ax-pow 5064  ax-pr 5126  ax-un 7208  ax-cnex 10307  ax-resscn 10308  ax-1cn 10309  ax-icn 10310  ax-addcl 10311  ax-addrcl 10312  ax-mulcl 10313  ax-mulrcl 10314  ax-mulcom 10315  ax-addass 10316  ax-mulass 10317  ax-distr 10318  ax-i2m1 10319  ax-1ne0 10320  ax-1rid 10321  ax-rnegex 10322  ax-rrecex 10323  ax-cnre 10324  ax-pre-lttri 10325  ax-pre-lttrn 10326  ax-pre-ltadd 10327  ax-pre-mulgt0 10328

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3or 1114  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2604  df-eu 2639  df-clab 2811  df-cleq 2817  df-clel 2820  df-nfc 2957  df-ne 2999  df-nel 3102  df-ral 3121  df-rex 3122  df-reu 3123  df-rmo 3124  df-rab 3125  df-v 3415  df-sbc 3662  df-csb 3757  df-dif 3800  df-un 3802  df-in 3804  df-ss 3811  df-pss 3813  df-nul 4144  df-if 4306  df-pw 4379  df-sn 4397  df-pr 4399  df-tp 4401  df-op 4403  df-uni 4658  df-int 4697  df-iun 4741  df-br 4873  df-opab 4935  df-mpt 4952  df-tr 4975  df-id 5249  df-eprel 5254  df-po 5262  df-so 5263  df-fr 5300  df-we 5302  df-xp 5347  df-rel 5348  df-cnv 5349  df-co 5350  df-dm 5351  df-rn 5352  df-res 5353  df-ima 5354  df-pred 5919  df-ord 5965  df-on 5966  df-lim 5967  df-suc 5968  df-iota 6085  df-fun 6124  df-fn 6125  df-f 6126  df-f1 6127  df-fo 6128  df-f1o 6129  df-fv 6130  df-riota 6865  df-ov 6907  df-oprab 6908  df-mpt2 6909  df-om 7326  df-1st 7427  df-2nd 7428  df-wrecs 7671  df-recs 7733  df-rdg 7771  df-1o 7825  df-oadd 7829  df-er 8008  df-map 8123  df-pm 8124  df-en 8222  df-dom 8223  df-sdom 8224  df-fin 8225  df-card 9077  df-cda 9304  df-pnf 10392  df-mnf 10393  df-xr 10394  df-ltxr 10395  df-le 10396  df-sub 10586  df-neg 10587  df-nn 11350  df-2 11413  df-3 11414  df-n0 11618  df-xnn0 11690  df-z 11704  df-uz 11968  df-fz 12619  df-fzo 12760  df-hash 13410  df-word 13574  df-concat 13630  df-s1 13655  df-s2 13968  df-s3 13969  df-trkgc 25759  df-trkgb 25760  df-trkgcb 25761  df-trkg 25764  df-cgrg 25822  df-mir 25964  df-rag 26005

This theorem is referenced by:  colperpexlem3  26040