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Theorem hypcgrlem2 28676

Description: Lemma for hypcgr 28677, 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 479	. . . 4 ⊢ ((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) = 𝐵) → 𝐺 ∈ TarskiG)
6		hypcgr.h	. . . . 5 ⊢ (𝜑 → 𝐺Dim_TarskiG≥2)
7	6	adantr 479	. . . 4 ⊢ ((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) = 𝐵) → 𝐺Dim_TarskiG≥2)
8		hypcgr.a	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ 𝑃)
9	8	adantr 479	. . . 4 ⊢ ((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) = 𝐵) → 𝐴 ∈ 𝑃)
10		hypcgr.b	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ 𝑃)
11	10	adantr 479	. . . 4 ⊢ ((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) = 𝐵) → 𝐵 ∈ 𝑃)
12		hypcgr.c	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ 𝑃)
13	12	adantr 479	. . . 4 ⊢ ((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) = 𝐵) → 𝐶 ∈ 𝑃)
14		eqid 2725	. . . . 5 ⊢ (LineG‘𝐺) = (LineG‘𝐺)
15		eqid 2725	. . . . 5 ⊢ (pInvG‘𝐺) = (pInvG‘𝐺)
16		eqid 2725	. . . . 5 ⊢ ((pInvG‘𝐺)‘𝐵) = ((pInvG‘𝐺)‘𝐵)
17		hypcgr.d	. . . . . 6 ⊢ (𝜑 → 𝐷 ∈ 𝑃)
18	17	adantr 479	. . . . 5 ⊢ ((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) = 𝐵) → 𝐷 ∈ 𝑃)
19	1, 2, 3, 14, 15, 5, 11, 16, 18	mircl 28537	. . . 4 ⊢ ((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) = 𝐵) → (((pInvG‘𝐺)‘𝐵)‘𝐷) ∈ 𝑃)
20		hypcgr.e	. . . . 5 ⊢ (𝜑 → 𝐸 ∈ 𝑃)
21	20	adantr 479	. . . 4 ⊢ ((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) = 𝐵) → 𝐸 ∈ 𝑃)
22		hypcgr.1	. . . . 5 ⊢ (𝜑 → ⟨“𝐴𝐵𝐶”⟩ ∈ (∟G‘𝐺))
23	22	adantr 479	. . . 4 ⊢ ((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) = 𝐵) → ⟨“𝐴𝐵𝐶”⟩ ∈ (∟G‘𝐺))
24		eqidd 2726	. . . . . 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 479	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) = 𝐵) → 𝐵 = 𝐸)
27	1, 2, 3, 14, 15, 5, 11, 16, 21	mirinv 28542	. . . . . . . 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 2731	. . . . . 6 ⊢ ((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) = 𝐵) → 𝐸 = (((pInvG‘𝐺)‘𝐵)‘𝐸))
30		hypcgr.f	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹 ∈ 𝑃)
31	30	adantr 479	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) = 𝐵) → 𝐹 ∈ 𝑃)
32	1, 2, 3, 5, 7, 13, 31	midcom 28658	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) = 𝐵) → (𝐶(midG‘𝐺)𝐹) = (𝐹(midG‘𝐺)𝐶))
33		simpr 483	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) = 𝐵) → (𝐶(midG‘𝐺)𝐹) = 𝐵)
34	32, 33	eqtr3d 2767	. . . . . . 7 ⊢ ((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) = 𝐵) → (𝐹(midG‘𝐺)𝐶) = 𝐵)
35	1, 2, 3, 5, 7, 31, 13, 15, 11	ismidb 28654	. . . . . . 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 14851	. . . . 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 479	. . . . . 6 ⊢ ((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) = 𝐵) → ⟨“𝐷𝐸𝐹”⟩ ∈ (∟G‘𝐺))
40	1, 2, 3, 14, 15, 5, 18, 21, 31, 39, 16, 11	mirrag 28577	. . . . 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 2825	. . . 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 479	. . . . 5 ⊢ ((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) = 𝐵) → (𝐴 − 𝐵) = (𝐷 − 𝐸))
44	1, 2, 3, 14, 15, 5, 11, 16, 18, 21	miriso 28546	. . . . 5 ⊢ ((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) = 𝐵) → ((((pInvG‘𝐺)‘𝐵)‘𝐷) − (((pInvG‘𝐺)‘𝐵)‘𝐸)) = (𝐷 − 𝐸))
45	28	oveq2d 7435	. . . . 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 2771	. . . 4 ⊢ ((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) = 𝐵) → (𝐴 − 𝐵) = ((((pInvG‘𝐺)‘𝐵)‘𝐷) − 𝐸))
47	26	oveq1d 7434	. . . 4 ⊢ ((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) = 𝐵) → (𝐵 − 𝐶) = (𝐸 − 𝐶))
48		eqid 2725	. . . 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 2726	. . . 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 28675	. . 3 ⊢ ((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) = 𝐵) → (𝐴 − 𝐶) = ((((pInvG‘𝐺)‘𝐵)‘𝐷) − 𝐶))
51	36	oveq2d 7435	. . 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 28546	. . 3 ⊢ ((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) = 𝐵) → ((((pInvG‘𝐺)‘𝐵)‘𝐷) − (((pInvG‘𝐺)‘𝐵)‘𝐹)) = (𝐷 − 𝐹))
53	50, 51, 52	3eqtrd 2769	. 2 ⊢ ((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) = 𝐵) → (𝐴 − 𝐶) = (𝐷 − 𝐹))
54	4	ad2antrr 724	. . . 4 ⊢ (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶 = 𝐹) → 𝐺 ∈ TarskiG)
55	6	ad2antrr 724	. . . 4 ⊢ (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶 = 𝐹) → 𝐺Dim_TarskiG≥2)
56	8	ad2antrr 724	. . . 4 ⊢ (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶 = 𝐹) → 𝐴 ∈ 𝑃)
57	10	ad2antrr 724	. . . 4 ⊢ (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶 = 𝐹) → 𝐵 ∈ 𝑃)
58	12	ad2antrr 724	. . . 4 ⊢ (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶 = 𝐹) → 𝐶 ∈ 𝑃)
59	17	ad2antrr 724	. . . 4 ⊢ (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶 = 𝐹) → 𝐷 ∈ 𝑃)
60	20	ad2antrr 724	. . . 4 ⊢ (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶 = 𝐹) → 𝐸 ∈ 𝑃)
61	30	ad2antrr 724	. . . 4 ⊢ (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶 = 𝐹) → 𝐹 ∈ 𝑃)
62	22	ad2antrr 724	. . . 4 ⊢ (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶 = 𝐹) → ⟨“𝐴𝐵𝐶”⟩ ∈ (∟G‘𝐺))
63	38	ad2antrr 724	. . . 4 ⊢ (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶 = 𝐹) → ⟨“𝐷𝐸𝐹”⟩ ∈ (∟G‘𝐺))
64	42	ad2antrr 724	. . . 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 724	. . . 4 ⊢ (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶 = 𝐹) → (𝐵 − 𝐶) = (𝐸 − 𝐹))
67	25	ad2antrr 724	. . . 4 ⊢ (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶 = 𝐹) → 𝐵 = 𝐸)
68		eqid 2725	. . . 4 ⊢ ((lInvG‘𝐺)‘((𝐴(midG‘𝐺)𝐷)(LineG‘𝐺)𝐵)) = ((lInvG‘𝐺)‘((𝐴(midG‘𝐺)𝐷)(LineG‘𝐺)𝐵))
69		simpr 483	. . . 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 28675	. . 3 ⊢ (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶 = 𝐹) → (𝐴 − 𝐶) = (𝐷 − 𝐹))
71	4	ad2antrr 724	. . . . 5 ⊢ (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶 ≠ 𝐹) → 𝐺 ∈ TarskiG)
72	6	ad2antrr 724	. . . . 5 ⊢ (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶 ≠ 𝐹) → 𝐺Dim_TarskiG≥2)
73	8	ad2antrr 724	. . . . 5 ⊢ (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶 ≠ 𝐹) → 𝐴 ∈ 𝑃)
74	10	ad2antrr 724	. . . . 5 ⊢ (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶 ≠ 𝐹) → 𝐵 ∈ 𝑃)
75	12	ad2antrr 724	. . . . 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 724	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶 ≠ 𝐹) → 𝐹 ∈ 𝑃)
78	1, 2, 3, 71, 72, 75, 77	midcl 28653	. . . . . . 7 ⊢ (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶 ≠ 𝐹) → (𝐶(midG‘𝐺)𝐹) ∈ 𝑃)
79		simplr 767	. . . . . . 7 ⊢ (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶 ≠ 𝐹) → (𝐶(midG‘𝐺)𝐹) ≠ 𝐵)
80	1, 3, 14, 71, 78, 74, 79	tgelrnln 28506	. . . . . 6 ⊢ (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶 ≠ 𝐹) → ((𝐶(midG‘𝐺)𝐹)(LineG‘𝐺)𝐵) ∈ ran (LineG‘𝐺))
81	17	ad2antrr 724	. . . . . 6 ⊢ (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶 ≠ 𝐹) → 𝐷 ∈ 𝑃)
82	1, 2, 3, 71, 72, 76, 14, 80, 81	lmicl 28662	. . . . 5 ⊢ (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶 ≠ 𝐹) → (𝑆‘𝐷) ∈ 𝑃)
83	20	ad2antrr 724	. . . . . 6 ⊢ (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶 ≠ 𝐹) → 𝐸 ∈ 𝑃)
84	1, 2, 3, 71, 72, 76, 14, 80, 83	lmicl 28662	. . . . 5 ⊢ (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶 ≠ 𝐹) → (𝑆‘𝐸) ∈ 𝑃)
85	1, 2, 3, 71, 72, 76, 14, 80, 77	lmicl 28662	. . . . 5 ⊢ (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶 ≠ 𝐹) → (𝑆‘𝐹) ∈ 𝑃)
86	22	ad2antrr 724	. . . . 5 ⊢ (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶 ≠ 𝐹) → ⟨“𝐴𝐵𝐶”⟩ ∈ (∟G‘𝐺))
87	1, 2, 3, 71, 72, 76, 14, 80	lmimot 28674	. . . . . 6 ⊢ (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶 ≠ 𝐹) → 𝑆 ∈ (𝐺Ismt𝐺))
88	38	ad2antrr 724	. . . . . 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 28584	. . . . 5 ⊢ (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶 ≠ 𝐹) → ⟨“(𝑆‘𝐷)(𝑆‘𝐸)(𝑆‘𝐹)”⟩ ∈ (∟G‘𝐺))
90	42	ad2antrr 724	. . . . . 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 28673	. . . . . 6 ⊢ (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶 ≠ 𝐹) → ((𝑆‘𝐷) − (𝑆‘𝐸)) = (𝐷 − 𝐸))
92	90, 91	eqtr4d 2768	. . . . 5 ⊢ (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶 ≠ 𝐹) → (𝐴 − 𝐵) = ((𝑆‘𝐷) − (𝑆‘𝐸)))
93	65	ad2antrr 724	. . . . . 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 28673	. . . . . 6 ⊢ (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶 ≠ 𝐹) → ((𝑆‘𝐸) − (𝑆‘𝐹)) = (𝐸 − 𝐹))
95	93, 94	eqtr4d 2768	. . . . 5 ⊢ (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶 ≠ 𝐹) → (𝐵 − 𝐶) = ((𝑆‘𝐸) − (𝑆‘𝐹)))
96	1, 3, 14, 71, 78, 74, 79	tglinerflx2 28510	. . . . . . 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 28669	. . . . . 6 ⊢ (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶 ≠ 𝐹) → (𝑆‘𝐵) = 𝐵)
98	25	ad2antrr 724	. . . . . . 7 ⊢ (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶 ≠ 𝐹) → 𝐵 = 𝐸)
99	98	fveq2d 6900	. . . . . 6 ⊢ (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶 ≠ 𝐹) → (𝑆‘𝐵) = (𝑆‘𝐸))
100	97, 99	eqtr3d 2767	. . . . 5 ⊢ (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶 ≠ 𝐹) → 𝐵 = (𝑆‘𝐸))
101		eqid 2725	. . . . 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 28658	. . . . . . 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 28509	. . . . . . 7 ⊢ (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶 ≠ 𝐹) → (𝐶(midG‘𝐺)𝐹) ∈ ((𝐶(midG‘𝐺)𝐹)(LineG‘𝐺)𝐵))
104	102, 103	eqeltrrd 2826	. . . . . 6 ⊢ (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶 ≠ 𝐹) → (𝐹(midG‘𝐺)𝐶) ∈ ((𝐶(midG‘𝐺)𝐹)(LineG‘𝐺)𝐵))
105		simpr 483	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶 ≠ 𝐹) → 𝐶 ≠ 𝐹)
106	105	necomd 2985	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶 ≠ 𝐹) → 𝐹 ≠ 𝐶)
107	1, 3, 14, 71, 77, 75, 106	tgelrnln 28506	. . . . . . . 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 28655	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶 ≠ 𝐹) → (𝐶(midG‘𝐺)𝐹) ∈ (𝐶𝐼𝐹))
109	1, 2, 3, 71, 75, 78, 77, 108	tgbtwncom 28364	. . . . . . . . . 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 28495	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶 ≠ 𝐹) → (𝐶(midG‘𝐺)𝐹) ∈ (𝐹(LineG‘𝐺)𝐶))
111	103, 110	elind 4192	. . . . . . . 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 28510	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶 ≠ 𝐹) → 𝐶 ∈ (𝐹(LineG‘𝐺)𝐶))
113	79	necomd 2985	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶 ≠ 𝐹) → 𝐵 ≠ (𝐶(midG‘𝐺)𝐹))
114	4	ad2antrr 724	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶 = (𝐶(midG‘𝐺)𝐹)) → 𝐺 ∈ TarskiG)
115	12	ad2antrr 724	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶 = (𝐶(midG‘𝐺)𝐹)) → 𝐶 ∈ 𝑃)
116	30	ad2antrr 724	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶 = (𝐶(midG‘𝐺)𝐹)) → 𝐹 ∈ 𝑃)
117	6	ad2antrr 724	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶 = (𝐶(midG‘𝐺)𝐹)) → 𝐺Dim_TarskiG≥2)
118		simpr 483	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶 = (𝐶(midG‘𝐺)𝐹)) → 𝐶 = (𝐶(midG‘𝐺)𝐹))
119	118	eqcomd 2731	. . . . . . . . . . . . . 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 28656	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶 = (𝐶(midG‘𝐺)𝐹)) → (𝐶 − 𝐶) = (𝐶 − 𝐹))
121	120	eqcomd 2731	. . . . . . . . . . . 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 28339	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶 = (𝐶(midG‘𝐺)𝐹)) → 𝐶 = 𝐹)
123	122	ex 411	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) → (𝐶 = (𝐶(midG‘𝐺)𝐹) → 𝐶 = 𝐹))
124	123	necon3d 2950	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) → (𝐶 ≠ 𝐹 → 𝐶 ≠ (𝐶(midG‘𝐺)𝐹)))
125	124	imp 405	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶 ≠ 𝐹) → 𝐶 ≠ (𝐶(midG‘𝐺)𝐹))
126	98	eqcomd 2731	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶 ≠ 𝐹) → 𝐸 = 𝐵)
127		eqidd 2726	. . . . . . . . . . . 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 28654	. . . . . . . . . . . 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 7437	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶 ≠ 𝐹) → (𝐸 − 𝐹) = (𝐵 − (((pInvG‘𝐺)‘(𝐶(midG‘𝐺)𝐹))‘𝐶)))
131	93, 130	eqtrd 2765	. . . . . . . . 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 28573	. . . . . . . . 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 28593	. . . . . . 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 871	. . . . . 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 28663	. . . . . 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 711	. . . . 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 28675	. . . 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 28673	. . . 4 ⊢ (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶 ≠ 𝐹) → ((𝑆‘𝐷) − (𝑆‘𝐹)) = (𝐷 − 𝐹))
140	138, 139	eqtrd 2765	. . 3 ⊢ (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶 ≠ 𝐹) → (𝐴 − 𝐶) = (𝐷 − 𝐹))
141	70, 140	pm2.61dane 3018	. 2 ⊢ ((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) → (𝐴 − 𝐶) = (𝐷 − 𝐹))
142	53, 141	pm2.61dane 3018	1 ⊢ (𝜑 → (𝐴 − 𝐶) = (𝐷 − 𝐹))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 394 ∨ wo 845 = wceq 1533 ∈ wcel 2098 ≠ wne 2929 class class class wbr 5149 ‘cfv 6549 (class class class)co 7419 2c2 12300 ⟨“cs3 14829 Basecbs 17183 distcds 17245 TarskiGcstrkg 28303 Dim_TarskiG≥cstrkgld 28307 Itvcitv 28309 LineGclng 28310 pInvGcmir 28528 ∟Gcrag 28569 ⟂Gcperpg 28571 midGcmid 28648 lInvGclmi 28649

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 ax-cnex 11196 ax-resscn 11197 ax-1cn 11198 ax-icn 11199 ax-addcl 11200 ax-addrcl 11201 ax-mulcl 11202 ax-mulrcl 11203 ax-mulcom 11204 ax-addass 11205 ax-mulass 11206 ax-distr 11207 ax-i2m1 11208 ax-1ne0 11209 ax-1rid 11210 ax-rnegex 11211 ax-rrecex 11212 ax-cnre 11213 ax-pre-lttri 11214 ax-pre-lttrn 11215 ax-pre-ltadd 11216 ax-pre-mulgt0 11217

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-rmo 3363 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3964 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-tp 4635 df-op 4637 df-uni 4910 df-int 4951 df-iun 4999 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6307 df-ord 6374 df-on 6375 df-lim 6376 df-suc 6377 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-riota 7375 df-ov 7422 df-oprab 7423 df-mpo 7424 df-om 7872 df-1st 7994 df-2nd 7995 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-1o 8487 df-oadd 8491 df-er 8725 df-map 8847 df-pm 8848 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-dju 9926 df-card 9964 df-pnf 11282 df-mnf 11283 df-xr 11284 df-ltxr 11285 df-le 11286 df-sub 11478 df-neg 11479 df-nn 12246 df-2 12308 df-3 12309 df-n0 12506 df-xnn0 12578 df-z 12592 df-uz 12856 df-fz 13520 df-fzo 13663 df-hash 14326 df-word 14501 df-concat 14557 df-s1 14582 df-s2 14835 df-s3 14836 df-trkgc 28324 df-trkgb 28325 df-trkgcb 28326 df-trkgld 28328 df-trkg 28329 df-cgrg 28387 df-ismt 28409 df-leg 28459 df-mir 28529 df-rag 28570 df-perpg 28572 df-mid 28650 df-lmi 28651

This theorem is referenced by: hypcgr 28677

W3C validator