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Theorem hypcgrlem2 27742

Description: Lemma for hypcgr 27743, 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 481	. . . 4 ⊢ ((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) = 𝐵) → 𝐺 ∈ TarskiG)
6		hypcgr.h	. . . . 5 ⊢ (𝜑 → 𝐺Dim_TarskiG≥2)
7	6	adantr 481	. . . 4 ⊢ ((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) = 𝐵) → 𝐺Dim_TarskiG≥2)
8		hypcgr.a	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ 𝑃)
9	8	adantr 481	. . . 4 ⊢ ((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) = 𝐵) → 𝐴 ∈ 𝑃)
10		hypcgr.b	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ 𝑃)
11	10	adantr 481	. . . 4 ⊢ ((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) = 𝐵) → 𝐵 ∈ 𝑃)
12		hypcgr.c	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ 𝑃)
13	12	adantr 481	. . . 4 ⊢ ((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) = 𝐵) → 𝐶 ∈ 𝑃)
14		eqid 2736	. . . . 5 ⊢ (LineG‘𝐺) = (LineG‘𝐺)
15		eqid 2736	. . . . 5 ⊢ (pInvG‘𝐺) = (pInvG‘𝐺)
16		eqid 2736	. . . . 5 ⊢ ((pInvG‘𝐺)‘𝐵) = ((pInvG‘𝐺)‘𝐵)
17		hypcgr.d	. . . . . 6 ⊢ (𝜑 → 𝐷 ∈ 𝑃)
18	17	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) = 𝐵) → 𝐷 ∈ 𝑃)
19	1, 2, 3, 14, 15, 5, 11, 16, 18	mircl 27603	. . . 4 ⊢ ((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) = 𝐵) → (((pInvG‘𝐺)‘𝐵)‘𝐷) ∈ 𝑃)
20		hypcgr.e	. . . . 5 ⊢ (𝜑 → 𝐸 ∈ 𝑃)
21	20	adantr 481	. . . 4 ⊢ ((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) = 𝐵) → 𝐸 ∈ 𝑃)
22		hypcgr.1	. . . . 5 ⊢ (𝜑 → ⟨“𝐴𝐵𝐶”⟩ ∈ (∟G‘𝐺))
23	22	adantr 481	. . . 4 ⊢ ((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) = 𝐵) → ⟨“𝐴𝐵𝐶”⟩ ∈ (∟G‘𝐺))
24		eqidd 2737	. . . . . 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 481	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) = 𝐵) → 𝐵 = 𝐸)
27	1, 2, 3, 14, 15, 5, 11, 16, 21	mirinv 27608	. . . . . . . 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 2742	. . . . . 6 ⊢ ((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) = 𝐵) → 𝐸 = (((pInvG‘𝐺)‘𝐵)‘𝐸))
30		hypcgr.f	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹 ∈ 𝑃)
31	30	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) = 𝐵) → 𝐹 ∈ 𝑃)
32	1, 2, 3, 5, 7, 13, 31	midcom 27724	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) = 𝐵) → (𝐶(midG‘𝐺)𝐹) = (𝐹(midG‘𝐺)𝐶))
33		simpr 485	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) = 𝐵) → (𝐶(midG‘𝐺)𝐹) = 𝐵)
34	32, 33	eqtr3d 2778	. . . . . . 7 ⊢ ((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) = 𝐵) → (𝐹(midG‘𝐺)𝐶) = 𝐵)
35	1, 2, 3, 5, 7, 31, 13, 15, 11	ismidb 27720	. . . . . . 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 14753	. . . . 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 481	. . . . . 6 ⊢ ((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) = 𝐵) → ⟨“𝐷𝐸𝐹”⟩ ∈ (∟G‘𝐺))
40	1, 2, 3, 14, 15, 5, 18, 21, 31, 39, 16, 11	mirrag 27643	. . . . 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 2838	. . . 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 481	. . . . 5 ⊢ ((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) = 𝐵) → (𝐴 − 𝐵) = (𝐷 − 𝐸))
44	1, 2, 3, 14, 15, 5, 11, 16, 18, 21	miriso 27612	. . . . 5 ⊢ ((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) = 𝐵) → ((((pInvG‘𝐺)‘𝐵)‘𝐷) − (((pInvG‘𝐺)‘𝐵)‘𝐸)) = (𝐷 − 𝐸))
45	28	oveq2d 7373	. . . . 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 2782	. . . 4 ⊢ ((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) = 𝐵) → (𝐴 − 𝐵) = ((((pInvG‘𝐺)‘𝐵)‘𝐷) − 𝐸))
47	26	oveq1d 7372	. . . 4 ⊢ ((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) = 𝐵) → (𝐵 − 𝐶) = (𝐸 − 𝐶))
48		eqid 2736	. . . 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 2737	. . . 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 27741	. . 3 ⊢ ((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) = 𝐵) → (𝐴 − 𝐶) = ((((pInvG‘𝐺)‘𝐵)‘𝐷) − 𝐶))
51	36	oveq2d 7373	. . 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 27612	. . 3 ⊢ ((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) = 𝐵) → ((((pInvG‘𝐺)‘𝐵)‘𝐷) − (((pInvG‘𝐺)‘𝐵)‘𝐹)) = (𝐷 − 𝐹))
53	50, 51, 52	3eqtrd 2780	. 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 2736	. . . 4 ⊢ ((lInvG‘𝐺)‘((𝐴(midG‘𝐺)𝐷)(LineG‘𝐺)𝐵)) = ((lInvG‘𝐺)‘((𝐴(midG‘𝐺)𝐷)(LineG‘𝐺)𝐵))
69		simpr 485	. . . 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 27741	. . 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 27719	. . . . . . 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 27572	. . . . . 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 27728	. . . . 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 27728	. . . . 5 ⊢ (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶 ≠ 𝐹) → (𝑆‘𝐸) ∈ 𝑃)
85	1, 2, 3, 71, 72, 76, 14, 80, 77	lmicl 27728	. . . . 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 27740	. . . . . 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 27650	. . . . 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 27739	. . . . . 6 ⊢ (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶 ≠ 𝐹) → ((𝑆‘𝐷) − (𝑆‘𝐸)) = (𝐷 − 𝐸))
92	90, 91	eqtr4d 2779	. . . . 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 27739	. . . . . 6 ⊢ (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶 ≠ 𝐹) → ((𝑆‘𝐸) − (𝑆‘𝐹)) = (𝐸 − 𝐹))
95	93, 94	eqtr4d 2779	. . . . 5 ⊢ (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶 ≠ 𝐹) → (𝐵 − 𝐶) = ((𝑆‘𝐸) − (𝑆‘𝐹)))
96	1, 3, 14, 71, 78, 74, 79	tglinerflx2 27576	. . . . . . 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 27735	. . . . . 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 6846	. . . . . 6 ⊢ (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶 ≠ 𝐹) → (𝑆‘𝐵) = (𝑆‘𝐸))
100	97, 99	eqtr3d 2778	. . . . 5 ⊢ (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶 ≠ 𝐹) → 𝐵 = (𝑆‘𝐸))
101		eqid 2736	. . . . 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 27724	. . . . . . 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 27575	. . . . . . 7 ⊢ (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶 ≠ 𝐹) → (𝐶(midG‘𝐺)𝐹) ∈ ((𝐶(midG‘𝐺)𝐹)(LineG‘𝐺)𝐵))
104	102, 103	eqeltrrd 2839	. . . . . 6 ⊢ (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶 ≠ 𝐹) → (𝐹(midG‘𝐺)𝐶) ∈ ((𝐶(midG‘𝐺)𝐹)(LineG‘𝐺)𝐵))
105		simpr 485	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶 ≠ 𝐹) → 𝐶 ≠ 𝐹)
106	105	necomd 2999	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶 ≠ 𝐹) → 𝐹 ≠ 𝐶)
107	1, 3, 14, 71, 77, 75, 106	tgelrnln 27572	. . . . . . . 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 27721	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶 ≠ 𝐹) → (𝐶(midG‘𝐺)𝐹) ∈ (𝐶𝐼𝐹))
109	1, 2, 3, 71, 75, 78, 77, 108	tgbtwncom 27430	. . . . . . . . . 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 27561	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶 ≠ 𝐹) → (𝐶(midG‘𝐺)𝐹) ∈ (𝐹(LineG‘𝐺)𝐶))
111	103, 110	elind 4154	. . . . . . . 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 27576	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶 ≠ 𝐹) → 𝐶 ∈ (𝐹(LineG‘𝐺)𝐶))
113	79	necomd 2999	. . . . . . . 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 485	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶 = (𝐶(midG‘𝐺)𝐹)) → 𝐶 = (𝐶(midG‘𝐺)𝐹))
119	118	eqcomd 2742	. . . . . . . . . . . . . 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 27722	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶 = (𝐶(midG‘𝐺)𝐹)) → (𝐶 − 𝐶) = (𝐶 − 𝐹))
121	120	eqcomd 2742	. . . . . . . . . . . 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 27405	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶 = (𝐶(midG‘𝐺)𝐹)) → 𝐶 = 𝐹)
123	122	ex 413	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) → (𝐶 = (𝐶(midG‘𝐺)𝐹) → 𝐶 = 𝐹))
124	123	necon3d 2964	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) → (𝐶 ≠ 𝐹 → 𝐶 ≠ (𝐶(midG‘𝐺)𝐹)))
125	124	imp 407	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶 ≠ 𝐹) → 𝐶 ≠ (𝐶(midG‘𝐺)𝐹))
126	98	eqcomd 2742	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶 ≠ 𝐹) → 𝐸 = 𝐵)
127		eqidd 2737	. . . . . . . . . . . 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 27720	. . . . . . . . . . . 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 7375	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶 ≠ 𝐹) → (𝐸 − 𝐹) = (𝐵 − (((pInvG‘𝐺)‘(𝐶(midG‘𝐺)𝐹))‘𝐶)))
131	93, 130	eqtrd 2776	. . . . . . . . 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 27639	. . . . . . . . 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 27659	. . . . . . 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 27729	. . . . . 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 27741	. . . 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 27739	. . . 4 ⊢ (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶 ≠ 𝐹) → ((𝑆‘𝐷) − (𝑆‘𝐹)) = (𝐷 − 𝐹))
140	138, 139	eqtrd 2776	. . 3 ⊢ (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶 ≠ 𝐹) → (𝐴 − 𝐶) = (𝐷 − 𝐹))
141	70, 140	pm2.61dane 3032	. 2 ⊢ ((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) → (𝐴 − 𝐶) = (𝐷 − 𝐹))
142	53, 141	pm2.61dane 3032	1 ⊢ (𝜑 → (𝐴 − 𝐶) = (𝐷 − 𝐹))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 ∨ wo 845 = wceq 1541 ∈ wcel 2106 ≠ wne 2943 class class class wbr 5105 ‘cfv 6496 (class class class)co 7357 2c2 12208 ⟨“cs3 14731 Basecbs 17083 distcds 17142 TarskiGcstrkg 27369 Dim_TarskiG≥cstrkgld 27373 Itvcitv 27375 LineGclng 27376 pInvGcmir 27594 ∟Gcrag 27635 ⟂Gcperpg 27637 midGcmid 27714 lInvGclmi 27715

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-rep 5242 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7672 ax-cnex 11107 ax-resscn 11108 ax-1cn 11109 ax-icn 11110 ax-addcl 11111 ax-addrcl 11112 ax-mulcl 11113 ax-mulrcl 11114 ax-mulcom 11115 ax-addass 11116 ax-mulass 11117 ax-distr 11118 ax-i2m1 11119 ax-1ne0 11120 ax-1rid 11121 ax-rnegex 11122 ax-rrecex 11123 ax-cnre 11124 ax-pre-lttri 11125 ax-pre-lttrn 11126 ax-pre-ltadd 11127 ax-pre-mulgt0 11128

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3065 df-rex 3074 df-rmo 3353 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-pss 3929 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-tp 4591 df-op 4593 df-uni 4866 df-int 4908 df-iun 4956 df-br 5106 df-opab 5168 df-mpt 5189 df-tr 5223 df-id 5531 df-eprel 5537 df-po 5545 df-so 5546 df-fr 5588 df-we 5590 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-pred 6253 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-riota 7313 df-ov 7360 df-oprab 7361 df-mpo 7362 df-om 7803 df-1st 7921 df-2nd 7922 df-frecs 8212 df-wrecs 8243 df-recs 8317 df-rdg 8356 df-1o 8412 df-oadd 8416 df-er 8648 df-map 8767 df-pm 8768 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-dju 9837 df-card 9875 df-pnf 11191 df-mnf 11192 df-xr 11193 df-ltxr 11194 df-le 11195 df-sub 11387 df-neg 11388 df-nn 12154 df-2 12216 df-3 12217 df-n0 12414 df-xnn0 12486 df-z 12500 df-uz 12764 df-fz 13425 df-fzo 13568 df-hash 14231 df-word 14403 df-concat 14459 df-s1 14484 df-s2 14737 df-s3 14738 df-trkgc 27390 df-trkgb 27391 df-trkgcb 27392 df-trkgld 27394 df-trkg 27395 df-cgrg 27453 df-ismt 27475 df-leg 27525 df-mir 27595 df-rag 27636 df-perpg 27638 df-mid 27716 df-lmi 27717

This theorem is referenced by: hypcgr 27743

W3C validator