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Theorem hypcgrlem2 28315

Description: Lemma for hypcgr 28316, case where triangles share one vertex 𝐵. (Contributed by Thierry Arnoux, 16-Dec-2019.)

**Hypotheses**
Ref	Expression
hypcgr.p	⊢ 𝑃 = (Base‘𝐺)
hypcgr.m	⊢ − = (dist‘𝐺)
hypcgr.i	⊢ 𝐼 = (Itv‘𝐺)
hypcgr.g	⊢ (𝜑 → 𝐺 ∈ TarskiG)
hypcgr.h	⊢ (𝜑 → 𝐺Dim_TarskiG≥2)
hypcgr.a	⊢ (𝜑 → 𝐴 ∈ 𝑃)
hypcgr.b	⊢ (𝜑 → 𝐵 ∈ 𝑃)
hypcgr.c	⊢ (𝜑 → 𝐶 ∈ 𝑃)
hypcgr.d	⊢ (𝜑 → 𝐷 ∈ 𝑃)
hypcgr.e	⊢ (𝜑 → 𝐸 ∈ 𝑃)
hypcgr.f	⊢ (𝜑 → 𝐹 ∈ 𝑃)
hypcgr.1	⊢ (𝜑 → ⟨“𝐴𝐵𝐶”⟩ ∈ (∟G‘𝐺))
hypcgr.2	⊢ (𝜑 → ⟨“𝐷𝐸𝐹”⟩ ∈ (∟G‘𝐺))
hypcgr.3	⊢ (𝜑 → (𝐴 − 𝐵) = (𝐷 − 𝐸))
hypcgr.4	⊢ (𝜑 → (𝐵 − 𝐶) = (𝐸 − 𝐹))
hypcgrlem2.b	⊢ (𝜑 → 𝐵 = 𝐸)
hypcgrlem2.s	⊢ 𝑆 = ((lInvG‘𝐺)‘((𝐶(midG‘𝐺)𝐹)(LineG‘𝐺)𝐵))

**Assertion**
Ref	Expression
hypcgrlem2	⊢ (𝜑 → (𝐴 − 𝐶) = (𝐷 − 𝐹))

**Proof of Theorem hypcgrlem2**
Step	Hyp	Ref	Expression
1		hypcgr.p	. . . 4 ⊢ 𝑃 = (Base‘𝐺)
2		hypcgr.m	. . . 4 ⊢ − = (dist‘𝐺)
3		hypcgr.i	. . . 4 ⊢ 𝐼 = (Itv‘𝐺)
4		hypcgr.g	. . . . 5 ⊢ (𝜑 → 𝐺 ∈ TarskiG)
5	4	adantr 480	. . . 4 ⊢ ((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) = 𝐵) → 𝐺 ∈ TarskiG)
6		hypcgr.h	. . . . 5 ⊢ (𝜑 → 𝐺Dim_TarskiG≥2)
7	6	adantr 480	. . . 4 ⊢ ((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) = 𝐵) → 𝐺Dim_TarskiG≥2)
8		hypcgr.a	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ 𝑃)
9	8	adantr 480	. . . 4 ⊢ ((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) = 𝐵) → 𝐴 ∈ 𝑃)
10		hypcgr.b	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ 𝑃)
11	10	adantr 480	. . . 4 ⊢ ((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) = 𝐵) → 𝐵 ∈ 𝑃)
12		hypcgr.c	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ 𝑃)
13	12	adantr 480	. . . 4 ⊢ ((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) = 𝐵) → 𝐶 ∈ 𝑃)
14		eqid 2731	. . . . 5 ⊢ (LineG‘𝐺) = (LineG‘𝐺)
15		eqid 2731	. . . . 5 ⊢ (pInvG‘𝐺) = (pInvG‘𝐺)
16		eqid 2731	. . . . 5 ⊢ ((pInvG‘𝐺)‘𝐵) = ((pInvG‘𝐺)‘𝐵)
17		hypcgr.d	. . . . . 6 ⊢ (𝜑 → 𝐷 ∈ 𝑃)
18	17	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) = 𝐵) → 𝐷 ∈ 𝑃)
19	1, 2, 3, 14, 15, 5, 11, 16, 18	mircl 28176	. . . 4 ⊢ ((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) = 𝐵) → (((pInvG‘𝐺)‘𝐵)‘𝐷) ∈ 𝑃)
20		hypcgr.e	. . . . 5 ⊢ (𝜑 → 𝐸 ∈ 𝑃)
21	20	adantr 480	. . . 4 ⊢ ((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) = 𝐵) → 𝐸 ∈ 𝑃)
22		hypcgr.1	. . . . 5 ⊢ (𝜑 → ⟨“𝐴𝐵𝐶”⟩ ∈ (∟G‘𝐺))
23	22	adantr 480	. . . 4 ⊢ ((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) = 𝐵) → ⟨“𝐴𝐵𝐶”⟩ ∈ (∟G‘𝐺))
24		eqidd 2732	. . . . . 6 ⊢ ((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) = 𝐵) → (((pInvG‘𝐺)‘𝐵)‘𝐷) = (((pInvG‘𝐺)‘𝐵)‘𝐷))
25		hypcgrlem2.b	. . . . . . . . 9 ⊢ (𝜑 → 𝐵 = 𝐸)
26	25	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) = 𝐵) → 𝐵 = 𝐸)
27	1, 2, 3, 14, 15, 5, 11, 16, 21	mirinv 28181	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) = 𝐵) → ((((pInvG‘𝐺)‘𝐵)‘𝐸) = 𝐸 ↔ 𝐵 = 𝐸))
28	26, 27	mpbird 256	. . . . . . 7 ⊢ ((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) = 𝐵) → (((pInvG‘𝐺)‘𝐵)‘𝐸) = 𝐸)
29	28	eqcomd 2737	. . . . . 6 ⊢ ((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) = 𝐵) → 𝐸 = (((pInvG‘𝐺)‘𝐵)‘𝐸))
30		hypcgr.f	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹 ∈ 𝑃)
31	30	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) = 𝐵) → 𝐹 ∈ 𝑃)
32	1, 2, 3, 5, 7, 13, 31	midcom 28297	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) = 𝐵) → (𝐶(midG‘𝐺)𝐹) = (𝐹(midG‘𝐺)𝐶))
33		simpr 484	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) = 𝐵) → (𝐶(midG‘𝐺)𝐹) = 𝐵)
34	32, 33	eqtr3d 2773	. . . . . . 7 ⊢ ((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) = 𝐵) → (𝐹(midG‘𝐺)𝐶) = 𝐵)
35	1, 2, 3, 5, 7, 31, 13, 15, 11	ismidb 28293	. . . . . . 7 ⊢ ((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) = 𝐵) → (𝐶 = (((pInvG‘𝐺)‘𝐵)‘𝐹) ↔ (𝐹(midG‘𝐺)𝐶) = 𝐵))
36	34, 35	mpbird 256	. . . . . 6 ⊢ ((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) = 𝐵) → 𝐶 = (((pInvG‘𝐺)‘𝐵)‘𝐹))
37	24, 29, 36	s3eqd 14820	. . . . 5 ⊢ ((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) = 𝐵) → ⟨“(((pInvG‘𝐺)‘𝐵)‘𝐷)𝐸𝐶”⟩ = ⟨“(((pInvG‘𝐺)‘𝐵)‘𝐷)(((pInvG‘𝐺)‘𝐵)‘𝐸)(((pInvG‘𝐺)‘𝐵)‘𝐹)”⟩)
38		hypcgr.2	. . . . . . 7 ⊢ (𝜑 → ⟨“𝐷𝐸𝐹”⟩ ∈ (∟G‘𝐺))
39	38	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) = 𝐵) → ⟨“𝐷𝐸𝐹”⟩ ∈ (∟G‘𝐺))
40	1, 2, 3, 14, 15, 5, 18, 21, 31, 39, 16, 11	mirrag 28216	. . . . 5 ⊢ ((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) = 𝐵) → ⟨“(((pInvG‘𝐺)‘𝐵)‘𝐷)(((pInvG‘𝐺)‘𝐵)‘𝐸)(((pInvG‘𝐺)‘𝐵)‘𝐹)”⟩ ∈ (∟G‘𝐺))
41	37, 40	eqeltrd 2832	. . . 4 ⊢ ((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) = 𝐵) → ⟨“(((pInvG‘𝐺)‘𝐵)‘𝐷)𝐸𝐶”⟩ ∈ (∟G‘𝐺))
42		hypcgr.3	. . . . . 6 ⊢ (𝜑 → (𝐴 − 𝐵) = (𝐷 − 𝐸))
43	42	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) = 𝐵) → (𝐴 − 𝐵) = (𝐷 − 𝐸))
44	1, 2, 3, 14, 15, 5, 11, 16, 18, 21	miriso 28185	. . . . 5 ⊢ ((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) = 𝐵) → ((((pInvG‘𝐺)‘𝐵)‘𝐷) − (((pInvG‘𝐺)‘𝐵)‘𝐸)) = (𝐷 − 𝐸))
45	28	oveq2d 7428	. . . . 5 ⊢ ((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) = 𝐵) → ((((pInvG‘𝐺)‘𝐵)‘𝐷) − (((pInvG‘𝐺)‘𝐵)‘𝐸)) = ((((pInvG‘𝐺)‘𝐵)‘𝐷) − 𝐸))
46	43, 44, 45	3eqtr2d 2777	. . . 4 ⊢ ((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) = 𝐵) → (𝐴 − 𝐵) = ((((pInvG‘𝐺)‘𝐵)‘𝐷) − 𝐸))
47	26	oveq1d 7427	. . . 4 ⊢ ((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) = 𝐵) → (𝐵 − 𝐶) = (𝐸 − 𝐶))
48		eqid 2731	. . . 4 ⊢ ((lInvG‘𝐺)‘((𝐴(midG‘𝐺)(((pInvG‘𝐺)‘𝐵)‘𝐷))(LineG‘𝐺)𝐵)) = ((lInvG‘𝐺)‘((𝐴(midG‘𝐺)(((pInvG‘𝐺)‘𝐵)‘𝐷))(LineG‘𝐺)𝐵))
49		eqidd 2732	. . . 4 ⊢ ((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) = 𝐵) → 𝐶 = 𝐶)
50	1, 2, 3, 5, 7, 9, 11, 13, 19, 21, 13, 23, 41, 46, 47, 26, 48, 49	hypcgrlem1 28314	. . 3 ⊢ ((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) = 𝐵) → (𝐴 − 𝐶) = ((((pInvG‘𝐺)‘𝐵)‘𝐷) − 𝐶))
51	36	oveq2d 7428	. . 3 ⊢ ((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) = 𝐵) → ((((pInvG‘𝐺)‘𝐵)‘𝐷) − 𝐶) = ((((pInvG‘𝐺)‘𝐵)‘𝐷) − (((pInvG‘𝐺)‘𝐵)‘𝐹)))
52	1, 2, 3, 14, 15, 5, 11, 16, 18, 31	miriso 28185	. . 3 ⊢ ((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) = 𝐵) → ((((pInvG‘𝐺)‘𝐵)‘𝐷) − (((pInvG‘𝐺)‘𝐵)‘𝐹)) = (𝐷 − 𝐹))
53	50, 51, 52	3eqtrd 2775	. 2 ⊢ ((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) = 𝐵) → (𝐴 − 𝐶) = (𝐷 − 𝐹))
54	4	ad2antrr 723	. . . 4 ⊢ (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶 = 𝐹) → 𝐺 ∈ TarskiG)
55	6	ad2antrr 723	. . . 4 ⊢ (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶 = 𝐹) → 𝐺Dim_TarskiG≥2)
56	8	ad2antrr 723	. . . 4 ⊢ (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶 = 𝐹) → 𝐴 ∈ 𝑃)
57	10	ad2antrr 723	. . . 4 ⊢ (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶 = 𝐹) → 𝐵 ∈ 𝑃)
58	12	ad2antrr 723	. . . 4 ⊢ (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶 = 𝐹) → 𝐶 ∈ 𝑃)
59	17	ad2antrr 723	. . . 4 ⊢ (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶 = 𝐹) → 𝐷 ∈ 𝑃)
60	20	ad2antrr 723	. . . 4 ⊢ (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶 = 𝐹) → 𝐸 ∈ 𝑃)
61	30	ad2antrr 723	. . . 4 ⊢ (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶 = 𝐹) → 𝐹 ∈ 𝑃)
62	22	ad2antrr 723	. . . 4 ⊢ (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶 = 𝐹) → ⟨“𝐴𝐵𝐶”⟩ ∈ (∟G‘𝐺))
63	38	ad2antrr 723	. . . 4 ⊢ (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶 = 𝐹) → ⟨“𝐷𝐸𝐹”⟩ ∈ (∟G‘𝐺))
64	42	ad2antrr 723	. . . 4 ⊢ (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶 = 𝐹) → (𝐴 − 𝐵) = (𝐷 − 𝐸))
65		hypcgr.4	. . . . 5 ⊢ (𝜑 → (𝐵 − 𝐶) = (𝐸 − 𝐹))
66	65	ad2antrr 723	. . . 4 ⊢ (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶 = 𝐹) → (𝐵 − 𝐶) = (𝐸 − 𝐹))
67	25	ad2antrr 723	. . . 4 ⊢ (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶 = 𝐹) → 𝐵 = 𝐸)
68		eqid 2731	. . . 4 ⊢ ((lInvG‘𝐺)‘((𝐴(midG‘𝐺)𝐷)(LineG‘𝐺)𝐵)) = ((lInvG‘𝐺)‘((𝐴(midG‘𝐺)𝐷)(LineG‘𝐺)𝐵))
69		simpr 484	. . . 4 ⊢ (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶 = 𝐹) → 𝐶 = 𝐹)
70	1, 2, 3, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 66, 67, 68, 69	hypcgrlem1 28314	. . 3 ⊢ (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶 = 𝐹) → (𝐴 − 𝐶) = (𝐷 − 𝐹))
71	4	ad2antrr 723	. . . . 5 ⊢ (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶 ≠ 𝐹) → 𝐺 ∈ TarskiG)
72	6	ad2antrr 723	. . . . 5 ⊢ (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶 ≠ 𝐹) → 𝐺Dim_TarskiG≥2)
73	8	ad2antrr 723	. . . . 5 ⊢ (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶 ≠ 𝐹) → 𝐴 ∈ 𝑃)
74	10	ad2antrr 723	. . . . 5 ⊢ (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶 ≠ 𝐹) → 𝐵 ∈ 𝑃)
75	12	ad2antrr 723	. . . . 5 ⊢ (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶 ≠ 𝐹) → 𝐶 ∈ 𝑃)
76		hypcgrlem2.s	. . . . . 6 ⊢ 𝑆 = ((lInvG‘𝐺)‘((𝐶(midG‘𝐺)𝐹)(LineG‘𝐺)𝐵))
77	30	ad2antrr 723	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶 ≠ 𝐹) → 𝐹 ∈ 𝑃)
78	1, 2, 3, 71, 72, 75, 77	midcl 28292	. . . . . . 7 ⊢ (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶 ≠ 𝐹) → (𝐶(midG‘𝐺)𝐹) ∈ 𝑃)
79		simplr 766	. . . . . . 7 ⊢ (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶 ≠ 𝐹) → (𝐶(midG‘𝐺)𝐹) ≠ 𝐵)
80	1, 3, 14, 71, 78, 74, 79	tgelrnln 28145	. . . . . 6 ⊢ (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶 ≠ 𝐹) → ((𝐶(midG‘𝐺)𝐹)(LineG‘𝐺)𝐵) ∈ ran (LineG‘𝐺))
81	17	ad2antrr 723	. . . . . 6 ⊢ (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶 ≠ 𝐹) → 𝐷 ∈ 𝑃)
82	1, 2, 3, 71, 72, 76, 14, 80, 81	lmicl 28301	. . . . 5 ⊢ (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶 ≠ 𝐹) → (𝑆‘𝐷) ∈ 𝑃)
83	20	ad2antrr 723	. . . . . 6 ⊢ (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶 ≠ 𝐹) → 𝐸 ∈ 𝑃)
84	1, 2, 3, 71, 72, 76, 14, 80, 83	lmicl 28301	. . . . 5 ⊢ (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶 ≠ 𝐹) → (𝑆‘𝐸) ∈ 𝑃)
85	1, 2, 3, 71, 72, 76, 14, 80, 77	lmicl 28301	. . . . 5 ⊢ (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶 ≠ 𝐹) → (𝑆‘𝐹) ∈ 𝑃)
86	22	ad2antrr 723	. . . . 5 ⊢ (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶 ≠ 𝐹) → ⟨“𝐴𝐵𝐶”⟩ ∈ (∟G‘𝐺))
87	1, 2, 3, 71, 72, 76, 14, 80	lmimot 28313	. . . . . 6 ⊢ (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶 ≠ 𝐹) → 𝑆 ∈ (𝐺Ismt𝐺))
88	38	ad2antrr 723	. . . . . 6 ⊢ (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶 ≠ 𝐹) → ⟨“𝐷𝐸𝐹”⟩ ∈ (∟G‘𝐺))
89	1, 2, 3, 14, 15, 71, 81, 83, 77, 87, 88	motrag 28223	. . . . 5 ⊢ (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶 ≠ 𝐹) → ⟨“(𝑆‘𝐷)(𝑆‘𝐸)(𝑆‘𝐹)”⟩ ∈ (∟G‘𝐺))
90	42	ad2antrr 723	. . . . . 6 ⊢ (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶 ≠ 𝐹) → (𝐴 − 𝐵) = (𝐷 − 𝐸))
91	1, 2, 3, 71, 72, 76, 14, 80, 81, 83	lmiiso 28312	. . . . . 6 ⊢ (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶 ≠ 𝐹) → ((𝑆‘𝐷) − (𝑆‘𝐸)) = (𝐷 − 𝐸))
92	90, 91	eqtr4d 2774	. . . . 5 ⊢ (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶 ≠ 𝐹) → (𝐴 − 𝐵) = ((𝑆‘𝐷) − (𝑆‘𝐸)))
93	65	ad2antrr 723	. . . . . 6 ⊢ (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶 ≠ 𝐹) → (𝐵 − 𝐶) = (𝐸 − 𝐹))
94	1, 2, 3, 71, 72, 76, 14, 80, 83, 77	lmiiso 28312	. . . . . 6 ⊢ (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶 ≠ 𝐹) → ((𝑆‘𝐸) − (𝑆‘𝐹)) = (𝐸 − 𝐹))
95	93, 94	eqtr4d 2774	. . . . 5 ⊢ (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶 ≠ 𝐹) → (𝐵 − 𝐶) = ((𝑆‘𝐸) − (𝑆‘𝐹)))
96	1, 3, 14, 71, 78, 74, 79	tglinerflx2 28149	. . . . . . 7 ⊢ (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶 ≠ 𝐹) → 𝐵 ∈ ((𝐶(midG‘𝐺)𝐹)(LineG‘𝐺)𝐵))
97	1, 2, 3, 71, 72, 76, 14, 80, 74, 96	lmicinv 28308	. . . . . 6 ⊢ (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶 ≠ 𝐹) → (𝑆‘𝐵) = 𝐵)
98	25	ad2antrr 723	. . . . . . 7 ⊢ (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶 ≠ 𝐹) → 𝐵 = 𝐸)
99	98	fveq2d 6896	. . . . . 6 ⊢ (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶 ≠ 𝐹) → (𝑆‘𝐵) = (𝑆‘𝐸))
100	97, 99	eqtr3d 2773	. . . . 5 ⊢ (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶 ≠ 𝐹) → 𝐵 = (𝑆‘𝐸))
101		eqid 2731	. . . . 5 ⊢ ((lInvG‘𝐺)‘((𝐴(midG‘𝐺)(𝑆‘𝐷))(LineG‘𝐺)𝐵)) = ((lInvG‘𝐺)‘((𝐴(midG‘𝐺)(𝑆‘𝐷))(LineG‘𝐺)𝐵))
102	1, 2, 3, 71, 72, 75, 77	midcom 28297	. . . . . . 7 ⊢ (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶 ≠ 𝐹) → (𝐶(midG‘𝐺)𝐹) = (𝐹(midG‘𝐺)𝐶))
103	1, 3, 14, 71, 78, 74, 79	tglinerflx1 28148	. . . . . . 7 ⊢ (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶 ≠ 𝐹) → (𝐶(midG‘𝐺)𝐹) ∈ ((𝐶(midG‘𝐺)𝐹)(LineG‘𝐺)𝐵))
104	102, 103	eqeltrrd 2833	. . . . . 6 ⊢ (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶 ≠ 𝐹) → (𝐹(midG‘𝐺)𝐶) ∈ ((𝐶(midG‘𝐺)𝐹)(LineG‘𝐺)𝐵))
105		simpr 484	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶 ≠ 𝐹) → 𝐶 ≠ 𝐹)
106	105	necomd 2995	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶 ≠ 𝐹) → 𝐹 ≠ 𝐶)
107	1, 3, 14, 71, 77, 75, 106	tgelrnln 28145	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶 ≠ 𝐹) → (𝐹(LineG‘𝐺)𝐶) ∈ ran (LineG‘𝐺))
108	1, 2, 3, 71, 72, 75, 77	midbtwn 28294	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶 ≠ 𝐹) → (𝐶(midG‘𝐺)𝐹) ∈ (𝐶𝐼𝐹))
109	1, 2, 3, 71, 75, 78, 77, 108	tgbtwncom 28003	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶 ≠ 𝐹) → (𝐶(midG‘𝐺)𝐹) ∈ (𝐹𝐼𝐶))
110	1, 3, 14, 71, 77, 75, 78, 106, 109	btwnlng1 28134	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶 ≠ 𝐹) → (𝐶(midG‘𝐺)𝐹) ∈ (𝐹(LineG‘𝐺)𝐶))
111	103, 110	elind 4195	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶 ≠ 𝐹) → (𝐶(midG‘𝐺)𝐹) ∈ (((𝐶(midG‘𝐺)𝐹)(LineG‘𝐺)𝐵) ∩ (𝐹(LineG‘𝐺)𝐶)))
112	1, 3, 14, 71, 77, 75, 106	tglinerflx2 28149	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶 ≠ 𝐹) → 𝐶 ∈ (𝐹(LineG‘𝐺)𝐶))
113	79	necomd 2995	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶 ≠ 𝐹) → 𝐵 ≠ (𝐶(midG‘𝐺)𝐹))
114	4	ad2antrr 723	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶 = (𝐶(midG‘𝐺)𝐹)) → 𝐺 ∈ TarskiG)
115	12	ad2antrr 723	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶 = (𝐶(midG‘𝐺)𝐹)) → 𝐶 ∈ 𝑃)
116	30	ad2antrr 723	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶 = (𝐶(midG‘𝐺)𝐹)) → 𝐹 ∈ 𝑃)
117	6	ad2antrr 723	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶 = (𝐶(midG‘𝐺)𝐹)) → 𝐺Dim_TarskiG≥2)
118		simpr 484	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶 = (𝐶(midG‘𝐺)𝐹)) → 𝐶 = (𝐶(midG‘𝐺)𝐹))
119	118	eqcomd 2737	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶 = (𝐶(midG‘𝐺)𝐹)) → (𝐶(midG‘𝐺)𝐹) = 𝐶)
120	1, 2, 3, 114, 117, 115, 116, 119	midcgr 28295	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶 = (𝐶(midG‘𝐺)𝐹)) → (𝐶 − 𝐶) = (𝐶 − 𝐹))
121	120	eqcomd 2737	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶 = (𝐶(midG‘𝐺)𝐹)) → (𝐶 − 𝐹) = (𝐶 − 𝐶))
122	1, 2, 3, 114, 115, 116, 115, 121	axtgcgrid 27978	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶 = (𝐶(midG‘𝐺)𝐹)) → 𝐶 = 𝐹)
123	122	ex 412	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) → (𝐶 = (𝐶(midG‘𝐺)𝐹) → 𝐶 = 𝐹))
124	123	necon3d 2960	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) → (𝐶 ≠ 𝐹 → 𝐶 ≠ (𝐶(midG‘𝐺)𝐹)))
125	124	imp 406	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶 ≠ 𝐹) → 𝐶 ≠ (𝐶(midG‘𝐺)𝐹))
126	98	eqcomd 2737	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶 ≠ 𝐹) → 𝐸 = 𝐵)
127		eqidd 2732	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶 ≠ 𝐹) → (𝐶(midG‘𝐺)𝐹) = (𝐶(midG‘𝐺)𝐹))
128	1, 2, 3, 71, 72, 75, 77, 15, 78	ismidb 28293	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶 ≠ 𝐹) → (𝐹 = (((pInvG‘𝐺)‘(𝐶(midG‘𝐺)𝐹))‘𝐶) ↔ (𝐶(midG‘𝐺)𝐹) = (𝐶(midG‘𝐺)𝐹)))
129	127, 128	mpbird 256	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶 ≠ 𝐹) → 𝐹 = (((pInvG‘𝐺)‘(𝐶(midG‘𝐺)𝐹))‘𝐶))
130	126, 129	oveq12d 7430	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶 ≠ 𝐹) → (𝐸 − 𝐹) = (𝐵 − (((pInvG‘𝐺)‘(𝐶(midG‘𝐺)𝐹))‘𝐶)))
131	93, 130	eqtrd 2771	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶 ≠ 𝐹) → (𝐵 − 𝐶) = (𝐵 − (((pInvG‘𝐺)‘(𝐶(midG‘𝐺)𝐹))‘𝐶)))
132	1, 2, 3, 14, 15, 71, 74, 78, 75	israg 28212	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶 ≠ 𝐹) → (⟨“𝐵(𝐶(midG‘𝐺)𝐹)𝐶”⟩ ∈ (∟G‘𝐺) ↔ (𝐵 − 𝐶) = (𝐵 − (((pInvG‘𝐺)‘(𝐶(midG‘𝐺)𝐹))‘𝐶))))
133	131, 132	mpbird 256	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶 ≠ 𝐹) → ⟨“𝐵(𝐶(midG‘𝐺)𝐹)𝐶”⟩ ∈ (∟G‘𝐺))
134	1, 2, 3, 14, 71, 80, 107, 111, 96, 112, 113, 125, 133	ragperp 28232	. . . . . . 7 ⊢ (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶 ≠ 𝐹) → ((𝐶(midG‘𝐺)𝐹)(LineG‘𝐺)𝐵)(⟂G‘𝐺)(𝐹(LineG‘𝐺)𝐶))
135	134	orcd 870	. . . . . 6 ⊢ (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶 ≠ 𝐹) → (((𝐶(midG‘𝐺)𝐹)(LineG‘𝐺)𝐵)(⟂G‘𝐺)(𝐹(LineG‘𝐺)𝐶) ∨ 𝐹 = 𝐶))
136	1, 2, 3, 71, 72, 76, 14, 80, 77, 75	islmib 28302	. . . . . 6 ⊢ (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶 ≠ 𝐹) → (𝐶 = (𝑆‘𝐹) ↔ ((𝐹(midG‘𝐺)𝐶) ∈ ((𝐶(midG‘𝐺)𝐹)(LineG‘𝐺)𝐵) ∧ (((𝐶(midG‘𝐺)𝐹)(LineG‘𝐺)𝐵)(⟂G‘𝐺)(𝐹(LineG‘𝐺)𝐶) ∨ 𝐹 = 𝐶))))
137	104, 135, 136	mpbir2and 710	. . . . 5 ⊢ (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶 ≠ 𝐹) → 𝐶 = (𝑆‘𝐹))
138	1, 2, 3, 71, 72, 73, 74, 75, 82, 84, 85, 86, 89, 92, 95, 100, 101, 137	hypcgrlem1 28314	. . . 4 ⊢ (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶 ≠ 𝐹) → (𝐴 − 𝐶) = ((𝑆‘𝐷) − (𝑆‘𝐹)))
139	1, 2, 3, 71, 72, 76, 14, 80, 81, 77	lmiiso 28312	. . . 4 ⊢ (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶 ≠ 𝐹) → ((𝑆‘𝐷) − (𝑆‘𝐹)) = (𝐷 − 𝐹))
140	138, 139	eqtrd 2771	. . 3 ⊢ (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶 ≠ 𝐹) → (𝐴 − 𝐶) = (𝐷 − 𝐹))
141	70, 140	pm2.61dane 3028	. 2 ⊢ ((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) → (𝐴 − 𝐶) = (𝐷 − 𝐹))
142	53, 141	pm2.61dane 3028	1 ⊢ (𝜑 → (𝐴 − 𝐶) = (𝐷 − 𝐹))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∨ wo 844 = wceq 1540 ∈ wcel 2105 ≠ wne 2939 class class class wbr 5149 ‘cfv 6544 (class class class)co 7412 2c2 12272 ⟨“cs3 14798 Basecbs 17149 distcds 17211 TarskiGcstrkg 27942 Dim_TarskiG≥cstrkgld 27946 Itvcitv 27948 LineGclng 27949 pInvGcmir 28167 ∟Gcrag 28208 ⟂Gcperpg 28210 midGcmid 28287 lInvGclmi 28288

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7728 ax-cnex 11169 ax-resscn 11170 ax-1cn 11171 ax-icn 11172 ax-addcl 11173 ax-addrcl 11174 ax-mulcl 11175 ax-mulrcl 11176 ax-mulcom 11177 ax-addass 11178 ax-mulass 11179 ax-distr 11180 ax-i2m1 11181 ax-1ne0 11182 ax-1rid 11183 ax-rnegex 11184 ax-rrecex 11185 ax-cnre 11186 ax-pre-lttri 11187 ax-pre-lttrn 11188 ax-pre-ltadd 11189 ax-pre-mulgt0 11190

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-tp 4634 df-op 4636 df-uni 4910 df-int 4952 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7859 df-1st 7978 df-2nd 7979 df-frecs 8269 df-wrecs 8300 df-recs 8374 df-rdg 8413 df-1o 8469 df-oadd 8473 df-er 8706 df-map 8825 df-pm 8826 df-en 8943 df-dom 8944 df-sdom 8945 df-fin 8946 df-dju 9899 df-card 9937 df-pnf 11255 df-mnf 11256 df-xr 11257 df-ltxr 11258 df-le 11259 df-sub 11451 df-neg 11452 df-nn 12218 df-2 12280 df-3 12281 df-n0 12478 df-xnn0 12550 df-z 12564 df-uz 12828 df-fz 13490 df-fzo 13633 df-hash 14296 df-word 14470 df-concat 14526 df-s1 14551 df-s2 14804 df-s3 14805 df-trkgc 27963 df-trkgb 27964 df-trkgcb 27965 df-trkgld 27967 df-trkg 27968 df-cgrg 28026 df-ismt 28048 df-leg 28098 df-mir 28168 df-rag 28209 df-perpg 28211 df-mid 28289 df-lmi 28290

This theorem is referenced by: hypcgr 28316

W3C validator