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Theorem hypcgrlem1 29066

Description: Lemma for hypcgr 29068, case where triangles share a cathetus. (Contributed by Thierry Arnoux, 15-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)
hypcgrlem1.s	⊢ 𝑆 = ((lInvG‘𝐺)‘((𝐴(midG‘𝐺)𝐷)(LineG‘𝐺)𝐵))
hypcgrlem1.a	⊢ (𝜑 → 𝐶 = 𝐹)

**Assertion**
Ref	Expression
hypcgrlem1	⊢ (𝜑 → (𝐴 − 𝐶) = (𝐷 − 𝐹))

**Proof of Theorem hypcgrlem1**
Step	Hyp	Ref	Expression
1		hypcgr.p	. . 3 ⊢ 𝑃 = (Base‘𝐺)
2		hypcgr.m	. . 3 ⊢ − = (dist‘𝐺)
3		hypcgr.i	. . 3 ⊢ 𝐼 = (Itv‘𝐺)
4		hypcgr.g	. . . 4 ⊢ (𝜑 → 𝐺 ∈ TarskiG)
5	4	adantr 485	. . 3 ⊢ ((𝜑 ∧ (𝐴(midG‘𝐺)𝐷) = 𝐵) → 𝐺 ∈ TarskiG)
6		hypcgr.c	. . . 4 ⊢ (𝜑 → 𝐶 ∈ 𝑃)
7	6	adantr 485	. . 3 ⊢ ((𝜑 ∧ (𝐴(midG‘𝐺)𝐷) = 𝐵) → 𝐶 ∈ 𝑃)
8		hypcgr.a	. . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑃)
9	8	adantr 485	. . 3 ⊢ ((𝜑 ∧ (𝐴(midG‘𝐺)𝐷) = 𝐵) → 𝐴 ∈ 𝑃)
10		hypcgr.f	. . . 4 ⊢ (𝜑 → 𝐹 ∈ 𝑃)
11	10	adantr 485	. . 3 ⊢ ((𝜑 ∧ (𝐴(midG‘𝐺)𝐷) = 𝐵) → 𝐹 ∈ 𝑃)
12		hypcgr.d	. . . 4 ⊢ (𝜑 → 𝐷 ∈ 𝑃)
13	12	adantr 485	. . 3 ⊢ ((𝜑 ∧ (𝐴(midG‘𝐺)𝐷) = 𝐵) → 𝐷 ∈ 𝑃)
14		eqid 2769	. . . . . . 7 ⊢ (LineG‘𝐺) = (LineG‘𝐺)
15		eqid 2769	. . . . . . 7 ⊢ (pInvG‘𝐺) = (pInvG‘𝐺)
16		hypcgr.b	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ 𝑃)
17		hypcgr.1	. . . . . . 7 ⊢ (𝜑 → ⟨“𝐴𝐵𝐶”⟩ ∈ (∟G‘𝐺))
18	1, 2, 3, 14, 15, 4, 8, 16, 6, 17	ragcom 28937	. . . . . 6 ⊢ (𝜑 → ⟨“𝐶𝐵𝐴”⟩ ∈ (∟G‘𝐺))
19	1, 2, 3, 14, 15, 4, 6, 16, 8	israg 28936	. . . . . 6 ⊢ (𝜑 → (⟨“𝐶𝐵𝐴”⟩ ∈ (∟G‘𝐺) ↔ (𝐶 − 𝐴) = (𝐶 − (((pInvG‘𝐺)‘𝐵)‘𝐴))))
20	18, 19	mpbid 235	. . . . 5 ⊢ (𝜑 → (𝐶 − 𝐴) = (𝐶 − (((pInvG‘𝐺)‘𝐵)‘𝐴)))
21	20	adantr 485	. . . 4 ⊢ ((𝜑 ∧ (𝐴(midG‘𝐺)𝐷) = 𝐵) → (𝐶 − 𝐴) = (𝐶 − (((pInvG‘𝐺)‘𝐵)‘𝐴)))
22		hypcgrlem1.a	. . . . . . 7 ⊢ (𝜑 → 𝐶 = 𝐹)
23	22	eqcomd 2775	. . . . . 6 ⊢ (𝜑 → 𝐹 = 𝐶)
24	23	adantr 485	. . . . 5 ⊢ ((𝜑 ∧ (𝐴(midG‘𝐺)𝐷) = 𝐵) → 𝐹 = 𝐶)
25		hypcgr.h	. . . . . . 7 ⊢ (𝜑 → 𝐺Dim_TarskiG≥2)
26	1, 2, 3, 4, 25, 8, 12, 15, 16	ismidb 29045	. . . . . 6 ⊢ (𝜑 → (𝐷 = (((pInvG‘𝐺)‘𝐵)‘𝐴) ↔ (𝐴(midG‘𝐺)𝐷) = 𝐵))
27	26	biimpar 482	. . . . 5 ⊢ ((𝜑 ∧ (𝐴(midG‘𝐺)𝐷) = 𝐵) → 𝐷 = (((pInvG‘𝐺)‘𝐵)‘𝐴))
28	24, 27	oveq12d 7429	. . . 4 ⊢ ((𝜑 ∧ (𝐴(midG‘𝐺)𝐷) = 𝐵) → (𝐹 − 𝐷) = (𝐶 − (((pInvG‘𝐺)‘𝐵)‘𝐴)))
29	21, 28	eqtr4d 2807	. . 3 ⊢ ((𝜑 ∧ (𝐴(midG‘𝐺)𝐷) = 𝐵) → (𝐶 − 𝐴) = (𝐹 − 𝐷))
30	1, 2, 3, 5, 7, 9, 11, 13, 29	tgcgrcomlr 28715	. 2 ⊢ ((𝜑 ∧ (𝐴(midG‘𝐺)𝐷) = 𝐵) → (𝐴 − 𝐶) = (𝐷 − 𝐹))
31		simpr 489	. . . 4 ⊢ (((𝜑 ∧ (𝐴(midG‘𝐺)𝐷) ≠ 𝐵) ∧ 𝐴 = 𝐷) → 𝐴 = 𝐷)
32	22	ad2antrr 738	. . . 4 ⊢ (((𝜑 ∧ (𝐴(midG‘𝐺)𝐷) ≠ 𝐵) ∧ 𝐴 = 𝐷) → 𝐶 = 𝐹)
33	31, 32	oveq12d 7429	. . 3 ⊢ (((𝜑 ∧ (𝐴(midG‘𝐺)𝐷) ≠ 𝐵) ∧ 𝐴 = 𝐷) → (𝐴 − 𝐶) = (𝐷 − 𝐹))
34	17	ad2antrr 738	. . . . . 6 ⊢ (((𝜑 ∧ (𝐴(midG‘𝐺)𝐷) ≠ 𝐵) ∧ 𝐴 ≠ 𝐷) → ⟨“𝐴𝐵𝐶”⟩ ∈ (∟G‘𝐺))
35	4	ad2antrr 738	. . . . . . 7 ⊢ (((𝜑 ∧ (𝐴(midG‘𝐺)𝐷) ≠ 𝐵) ∧ 𝐴 ≠ 𝐷) → 𝐺 ∈ TarskiG)
36	8	ad2antrr 738	. . . . . . 7 ⊢ (((𝜑 ∧ (𝐴(midG‘𝐺)𝐷) ≠ 𝐵) ∧ 𝐴 ≠ 𝐷) → 𝐴 ∈ 𝑃)
37	16	ad2antrr 738	. . . . . . 7 ⊢ (((𝜑 ∧ (𝐴(midG‘𝐺)𝐷) ≠ 𝐵) ∧ 𝐴 ≠ 𝐷) → 𝐵 ∈ 𝑃)
38	6	ad2antrr 738	. . . . . . 7 ⊢ (((𝜑 ∧ (𝐴(midG‘𝐺)𝐷) ≠ 𝐵) ∧ 𝐴 ≠ 𝐷) → 𝐶 ∈ 𝑃)
39	1, 2, 3, 14, 15, 35, 36, 37, 38	israg 28936	. . . . . 6 ⊢ (((𝜑 ∧ (𝐴(midG‘𝐺)𝐷) ≠ 𝐵) ∧ 𝐴 ≠ 𝐷) → (⟨“𝐴𝐵𝐶”⟩ ∈ (∟G‘𝐺) ↔ (𝐴 − 𝐶) = (𝐴 − (((pInvG‘𝐺)‘𝐵)‘𝐶))))
40	34, 39	mpbid 235	. . . . 5 ⊢ (((𝜑 ∧ (𝐴(midG‘𝐺)𝐷) ≠ 𝐵) ∧ 𝐴 ≠ 𝐷) → (𝐴 − 𝐶) = (𝐴 − (((pInvG‘𝐺)‘𝐵)‘𝐶)))
41	25	ad2antrr 738	. . . . . . 7 ⊢ (((𝜑 ∧ (𝐴(midG‘𝐺)𝐷) ≠ 𝐵) ∧ 𝐴 ≠ 𝐷) → 𝐺Dim_TarskiG≥2)
42		hypcgrlem1.s	. . . . . . 7 ⊢ 𝑆 = ((lInvG‘𝐺)‘((𝐴(midG‘𝐺)𝐷)(LineG‘𝐺)𝐵))
43	12	ad2antrr 738	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝐴(midG‘𝐺)𝐷) ≠ 𝐵) ∧ 𝐴 ≠ 𝐷) → 𝐷 ∈ 𝑃)
44	1, 2, 3, 35, 41, 36, 43	midcl 29044	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝐴(midG‘𝐺)𝐷) ≠ 𝐵) ∧ 𝐴 ≠ 𝐷) → (𝐴(midG‘𝐺)𝐷) ∈ 𝑃)
45		simplr 780	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝐴(midG‘𝐺)𝐷) ≠ 𝐵) ∧ 𝐴 ≠ 𝐷) → (𝐴(midG‘𝐺)𝐷) ≠ 𝐵)
46	1, 3, 14, 35, 44, 37, 45	tgelrnln 28865	. . . . . . 7 ⊢ (((𝜑 ∧ (𝐴(midG‘𝐺)𝐷) ≠ 𝐵) ∧ 𝐴 ≠ 𝐷) → ((𝐴(midG‘𝐺)𝐷)(LineG‘𝐺)𝐵) ∈ ran (LineG‘𝐺))
47		eqid 2769	. . . . . . 7 ⊢ ((pInvG‘𝐺)‘𝐵) = ((pInvG‘𝐺)‘𝐵)
48		eqid 2769	. . . . . . . . 9 ⊢ (cgrG‘𝐺) = (cgrG‘𝐺)
49	1, 2, 3, 14, 15, 35, 37, 47, 38	mircl 28900	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝐴(midG‘𝐺)𝐷) ≠ 𝐵) ∧ 𝐴 ≠ 𝐷) → (((pInvG‘𝐺)‘𝐵)‘𝐶) ∈ 𝑃)
50		simpr 489	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝐴(midG‘𝐺)𝐷) ≠ 𝐵) ∧ 𝐴 ≠ 𝐷) → 𝐴 ≠ 𝐷)
51	1, 2, 3, 35, 41, 36, 43	midbtwn 29046	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝐴(midG‘𝐺)𝐷) ≠ 𝐵) ∧ 𝐴 ≠ 𝐷) → (𝐴(midG‘𝐺)𝐷) ∈ (𝐴𝐼𝐷))
52	1, 14, 3, 35, 36, 44, 43, 51	btwncolg3 28792	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝐴(midG‘𝐺)𝐷) ≠ 𝐵) ∧ 𝐴 ≠ 𝐷) → (𝐷 ∈ (𝐴(LineG‘𝐺)(𝐴(midG‘𝐺)𝐷)) ∨ 𝐴 = (𝐴(midG‘𝐺)𝐷)))
53		eqidd 2770	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐷 = 𝐷)
54		hypcgrlem2.b	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐵 = 𝐸)
55	53, 54, 22	s3eqd 14901	. . . . . . . . . . . 12 ⊢ (𝜑 → ⟨“𝐷𝐵𝐶”⟩ = ⟨“𝐷𝐸𝐹”⟩)
56	55	ad2antrr 738	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝐴(midG‘𝐺)𝐷) ≠ 𝐵) ∧ 𝐴 ≠ 𝐷) → ⟨“𝐷𝐵𝐶”⟩ = ⟨“𝐷𝐸𝐹”⟩)
57		hypcgr.2	. . . . . . . . . . . 12 ⊢ (𝜑 → ⟨“𝐷𝐸𝐹”⟩ ∈ (∟G‘𝐺))
58	57	ad2antrr 738	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝐴(midG‘𝐺)𝐷) ≠ 𝐵) ∧ 𝐴 ≠ 𝐷) → ⟨“𝐷𝐸𝐹”⟩ ∈ (∟G‘𝐺))
59	56, 58	eqeltrd 2869	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝐴(midG‘𝐺)𝐷) ≠ 𝐵) ∧ 𝐴 ≠ 𝐷) → ⟨“𝐷𝐵𝐶”⟩ ∈ (∟G‘𝐺))
60	1, 2, 3, 14, 15, 35, 43, 37, 38	israg 28936	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝐴(midG‘𝐺)𝐷) ≠ 𝐵) ∧ 𝐴 ≠ 𝐷) → (⟨“𝐷𝐵𝐶”⟩ ∈ (∟G‘𝐺) ↔ (𝐷 − 𝐶) = (𝐷 − (((pInvG‘𝐺)‘𝐵)‘𝐶))))
61	59, 60	mpbid 235	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝐴(midG‘𝐺)𝐷) ≠ 𝐵) ∧ 𝐴 ≠ 𝐷) → (𝐷 − 𝐶) = (𝐷 − (((pInvG‘𝐺)‘𝐵)‘𝐶)))
62	1, 14, 3, 35, 36, 43, 44, 48, 38, 49, 2, 50, 52, 40, 61	lncgr 28804	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝐴(midG‘𝐺)𝐷) ≠ 𝐵) ∧ 𝐴 ≠ 𝐷) → ((𝐴(midG‘𝐺)𝐷) − 𝐶) = ((𝐴(midG‘𝐺)𝐷) − (((pInvG‘𝐺)‘𝐵)‘𝐶)))
63	1, 2, 3, 14, 15, 35, 44, 37, 38	israg 28936	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝐴(midG‘𝐺)𝐷) ≠ 𝐵) ∧ 𝐴 ≠ 𝐷) → (⟨“(𝐴(midG‘𝐺)𝐷)𝐵𝐶”⟩ ∈ (∟G‘𝐺) ↔ ((𝐴(midG‘𝐺)𝐷) − 𝐶) = ((𝐴(midG‘𝐺)𝐷) − (((pInvG‘𝐺)‘𝐵)‘𝐶))))
64	62, 63	mpbird 260	. . . . . . 7 ⊢ (((𝜑 ∧ (𝐴(midG‘𝐺)𝐷) ≠ 𝐵) ∧ 𝐴 ≠ 𝐷) → ⟨“(𝐴(midG‘𝐺)𝐷)𝐵𝐶”⟩ ∈ (∟G‘𝐺))
65	1, 3, 14, 35, 44, 37, 45	tglinerflx1 28868	. . . . . . 7 ⊢ (((𝜑 ∧ (𝐴(midG‘𝐺)𝐷) ≠ 𝐵) ∧ 𝐴 ≠ 𝐷) → (𝐴(midG‘𝐺)𝐷) ∈ ((𝐴(midG‘𝐺)𝐷)(LineG‘𝐺)𝐵))
66	1, 3, 14, 35, 44, 37, 45	tglinerflx2 28869	. . . . . . 7 ⊢ (((𝜑 ∧ (𝐴(midG‘𝐺)𝐷) ≠ 𝐵) ∧ 𝐴 ≠ 𝐷) → 𝐵 ∈ ((𝐴(midG‘𝐺)𝐷)(LineG‘𝐺)𝐵))
67	1, 2, 3, 35, 41, 42, 14, 46, 44, 47, 64, 65, 66, 38, 45	lmimid 29061	. . . . . 6 ⊢ (((𝜑 ∧ (𝐴(midG‘𝐺)𝐷) ≠ 𝐵) ∧ 𝐴 ≠ 𝐷) → (𝑆‘𝐶) = (((pInvG‘𝐺)‘𝐵)‘𝐶))
68	67	oveq2d 7427	. . . . 5 ⊢ (((𝜑 ∧ (𝐴(midG‘𝐺)𝐷) ≠ 𝐵) ∧ 𝐴 ≠ 𝐷) → (𝐴 − (𝑆‘𝐶)) = (𝐴 − (((pInvG‘𝐺)‘𝐵)‘𝐶)))
69	40, 68	eqtr4d 2807	. . . 4 ⊢ (((𝜑 ∧ (𝐴(midG‘𝐺)𝐷) ≠ 𝐵) ∧ 𝐴 ≠ 𝐷) → (𝐴 − 𝐶) = (𝐴 − (𝑆‘𝐶)))
70	1, 2, 3, 35, 41, 43, 36	midcom 29049	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝐴(midG‘𝐺)𝐷) ≠ 𝐵) ∧ 𝐴 ≠ 𝐷) → (𝐷(midG‘𝐺)𝐴) = (𝐴(midG‘𝐺)𝐷))
71	70, 65	eqeltrd 2869	. . . . . . 7 ⊢ (((𝜑 ∧ (𝐴(midG‘𝐺)𝐷) ≠ 𝐵) ∧ 𝐴 ≠ 𝐷) → (𝐷(midG‘𝐺)𝐴) ∈ ((𝐴(midG‘𝐺)𝐷)(LineG‘𝐺)𝐵))
72	50	necomd 3019	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝐴(midG‘𝐺)𝐷) ≠ 𝐵) ∧ 𝐴 ≠ 𝐷) → 𝐷 ≠ 𝐴)
73	1, 3, 14, 35, 43, 36, 72	tgelrnln 28865	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝐴(midG‘𝐺)𝐷) ≠ 𝐵) ∧ 𝐴 ≠ 𝐷) → (𝐷(LineG‘𝐺)𝐴) ∈ ran (LineG‘𝐺))
74	1, 2, 3, 35, 36, 44, 43, 51	tgbtwncom 28723	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝐴(midG‘𝐺)𝐷) ≠ 𝐵) ∧ 𝐴 ≠ 𝐷) → (𝐴(midG‘𝐺)𝐷) ∈ (𝐷𝐼𝐴))
75	1, 3, 14, 35, 43, 36, 44, 72, 74	btwnlng1 28854	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝐴(midG‘𝐺)𝐷) ≠ 𝐵) ∧ 𝐴 ≠ 𝐷) → (𝐴(midG‘𝐺)𝐷) ∈ (𝐷(LineG‘𝐺)𝐴))
76	65, 75	elind 4161	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝐴(midG‘𝐺)𝐷) ≠ 𝐵) ∧ 𝐴 ≠ 𝐷) → (𝐴(midG‘𝐺)𝐷) ∈ (((𝐴(midG‘𝐺)𝐷)(LineG‘𝐺)𝐵) ∩ (𝐷(LineG‘𝐺)𝐴)))
77	1, 3, 14, 35, 43, 36, 72	tglinerflx2 28869	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝐴(midG‘𝐺)𝐷) ≠ 𝐵) ∧ 𝐴 ≠ 𝐷) → 𝐴 ∈ (𝐷(LineG‘𝐺)𝐴))
78	45	necomd 3019	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝐴(midG‘𝐺)𝐷) ≠ 𝐵) ∧ 𝐴 ≠ 𝐷) → 𝐵 ≠ (𝐴(midG‘𝐺)𝐷))
79	4	ad2antrr 738	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝐴(midG‘𝐺)𝐷) ≠ 𝐵) ∧ 𝐴 = (𝐴(midG‘𝐺)𝐷)) → 𝐺 ∈ TarskiG)
80	8	ad2antrr 738	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝐴(midG‘𝐺)𝐷) ≠ 𝐵) ∧ 𝐴 = (𝐴(midG‘𝐺)𝐷)) → 𝐴 ∈ 𝑃)
81	12	ad2antrr 738	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝐴(midG‘𝐺)𝐷) ≠ 𝐵) ∧ 𝐴 = (𝐴(midG‘𝐺)𝐷)) → 𝐷 ∈ 𝑃)
82	25	ad2antrr 738	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝐴(midG‘𝐺)𝐷) ≠ 𝐵) ∧ 𝐴 = (𝐴(midG‘𝐺)𝐷)) → 𝐺Dim_TarskiG≥2)
83		simpr 489	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝐴(midG‘𝐺)𝐷) ≠ 𝐵) ∧ 𝐴 = (𝐴(midG‘𝐺)𝐷)) → 𝐴 = (𝐴(midG‘𝐺)𝐷))
84	83	eqcomd 2775	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝐴(midG‘𝐺)𝐷) ≠ 𝐵) ∧ 𝐴 = (𝐴(midG‘𝐺)𝐷)) → (𝐴(midG‘𝐺)𝐷) = 𝐴)
85	1, 2, 3, 79, 82, 80, 81, 84	midcgr 29047	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝐴(midG‘𝐺)𝐷) ≠ 𝐵) ∧ 𝐴 = (𝐴(midG‘𝐺)𝐷)) → (𝐴 − 𝐴) = (𝐴 − 𝐷))
86	85	eqcomd 2775	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝐴(midG‘𝐺)𝐷) ≠ 𝐵) ∧ 𝐴 = (𝐴(midG‘𝐺)𝐷)) → (𝐴 − 𝐷) = (𝐴 − 𝐴))
87	1, 2, 3, 79, 80, 81, 80, 86	axtgcgrid 28698	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝐴(midG‘𝐺)𝐷) ≠ 𝐵) ∧ 𝐴 = (𝐴(midG‘𝐺)𝐷)) → 𝐴 = 𝐷)
88	87	ex 417	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝐴(midG‘𝐺)𝐷) ≠ 𝐵) → (𝐴 = (𝐴(midG‘𝐺)𝐷) → 𝐴 = 𝐷))
89	88	necon3d 2985	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝐴(midG‘𝐺)𝐷) ≠ 𝐵) → (𝐴 ≠ 𝐷 → 𝐴 ≠ (𝐴(midG‘𝐺)𝐷)))
90	89	imp 411	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝐴(midG‘𝐺)𝐷) ≠ 𝐵) ∧ 𝐴 ≠ 𝐷) → 𝐴 ≠ (𝐴(midG‘𝐺)𝐷))
91		hypcgr.e	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐸 ∈ 𝑃)
92		hypcgr.3	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐴 − 𝐵) = (𝐷 − 𝐸))
93	1, 2, 3, 4, 8, 16, 12, 91, 92	tgcgrcomlr 28715	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐵 − 𝐴) = (𝐸 − 𝐷))
94	54	oveq1d 7426	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐵 − 𝐷) = (𝐸 − 𝐷))
95	93, 94	eqtr4d 2807	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐵 − 𝐴) = (𝐵 − 𝐷))
96	95	ad2antrr 738	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝐴(midG‘𝐺)𝐷) ≠ 𝐵) ∧ 𝐴 ≠ 𝐷) → (𝐵 − 𝐴) = (𝐵 − 𝐷))
97		eqidd 2770	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝐴(midG‘𝐺)𝐷) ≠ 𝐵) ∧ 𝐴 ≠ 𝐷) → (𝐴(midG‘𝐺)𝐷) = (𝐴(midG‘𝐺)𝐷))
98	1, 2, 3, 35, 41, 36, 43, 15, 44	ismidb 29045	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝐴(midG‘𝐺)𝐷) ≠ 𝐵) ∧ 𝐴 ≠ 𝐷) → (𝐷 = (((pInvG‘𝐺)‘(𝐴(midG‘𝐺)𝐷))‘𝐴) ↔ (𝐴(midG‘𝐺)𝐷) = (𝐴(midG‘𝐺)𝐷)))
99	97, 98	mpbird 260	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝐴(midG‘𝐺)𝐷) ≠ 𝐵) ∧ 𝐴 ≠ 𝐷) → 𝐷 = (((pInvG‘𝐺)‘(𝐴(midG‘𝐺)𝐷))‘𝐴))
100	99	oveq2d 7427	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝐴(midG‘𝐺)𝐷) ≠ 𝐵) ∧ 𝐴 ≠ 𝐷) → (𝐵 − 𝐷) = (𝐵 − (((pInvG‘𝐺)‘(𝐴(midG‘𝐺)𝐷))‘𝐴)))
101	96, 100	eqtrd 2804	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝐴(midG‘𝐺)𝐷) ≠ 𝐵) ∧ 𝐴 ≠ 𝐷) → (𝐵 − 𝐴) = (𝐵 − (((pInvG‘𝐺)‘(𝐴(midG‘𝐺)𝐷))‘𝐴)))
102	1, 2, 3, 14, 15, 35, 37, 44, 36	israg 28936	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝐴(midG‘𝐺)𝐷) ≠ 𝐵) ∧ 𝐴 ≠ 𝐷) → (⟨“𝐵(𝐴(midG‘𝐺)𝐷)𝐴”⟩ ∈ (∟G‘𝐺) ↔ (𝐵 − 𝐴) = (𝐵 − (((pInvG‘𝐺)‘(𝐴(midG‘𝐺)𝐷))‘𝐴))))
103	101, 102	mpbird 260	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝐴(midG‘𝐺)𝐷) ≠ 𝐵) ∧ 𝐴 ≠ 𝐷) → ⟨“𝐵(𝐴(midG‘𝐺)𝐷)𝐴”⟩ ∈ (∟G‘𝐺))
104	1, 2, 3, 14, 35, 46, 73, 76, 66, 77, 78, 90, 103	ragperp 28956	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝐴(midG‘𝐺)𝐷) ≠ 𝐵) ∧ 𝐴 ≠ 𝐷) → ((𝐴(midG‘𝐺)𝐷)(LineG‘𝐺)𝐵)(⟂G‘𝐺)(𝐷(LineG‘𝐺)𝐴))
105	104	orcd 886	. . . . . . 7 ⊢ (((𝜑 ∧ (𝐴(midG‘𝐺)𝐷) ≠ 𝐵) ∧ 𝐴 ≠ 𝐷) → (((𝐴(midG‘𝐺)𝐷)(LineG‘𝐺)𝐵)(⟂G‘𝐺)(𝐷(LineG‘𝐺)𝐴) ∨ 𝐷 = 𝐴))
106	1, 2, 3, 35, 41, 42, 14, 46, 43, 36	islmib 29054	. . . . . . 7 ⊢ (((𝜑 ∧ (𝐴(midG‘𝐺)𝐷) ≠ 𝐵) ∧ 𝐴 ≠ 𝐷) → (𝐴 = (𝑆‘𝐷) ↔ ((𝐷(midG‘𝐺)𝐴) ∈ ((𝐴(midG‘𝐺)𝐷)(LineG‘𝐺)𝐵) ∧ (((𝐴(midG‘𝐺)𝐷)(LineG‘𝐺)𝐵)(⟂G‘𝐺)(𝐷(LineG‘𝐺)𝐴) ∨ 𝐷 = 𝐴))))
107	71, 105, 106	mpbir2and 725	. . . . . 6 ⊢ (((𝜑 ∧ (𝐴(midG‘𝐺)𝐷) ≠ 𝐵) ∧ 𝐴 ≠ 𝐷) → 𝐴 = (𝑆‘𝐷))
108	107	oveq1d 7426	. . . . 5 ⊢ (((𝜑 ∧ (𝐴(midG‘𝐺)𝐷) ≠ 𝐵) ∧ 𝐴 ≠ 𝐷) → (𝐴 − (𝑆‘𝐶)) = ((𝑆‘𝐷) − (𝑆‘𝐶)))
109	1, 2, 3, 35, 41, 42, 14, 46, 43, 38	lmiiso 29064	. . . . 5 ⊢ (((𝜑 ∧ (𝐴(midG‘𝐺)𝐷) ≠ 𝐵) ∧ 𝐴 ≠ 𝐷) → ((𝑆‘𝐷) − (𝑆‘𝐶)) = (𝐷 − 𝐶))
110	22	oveq2d 7427	. . . . . 6 ⊢ (𝜑 → (𝐷 − 𝐶) = (𝐷 − 𝐹))
111	110	ad2antrr 738	. . . . 5 ⊢ (((𝜑 ∧ (𝐴(midG‘𝐺)𝐷) ≠ 𝐵) ∧ 𝐴 ≠ 𝐷) → (𝐷 − 𝐶) = (𝐷 − 𝐹))
112	108, 109, 111	3eqtrd 2808	. . . 4 ⊢ (((𝜑 ∧ (𝐴(midG‘𝐺)𝐷) ≠ 𝐵) ∧ 𝐴 ≠ 𝐷) → (𝐴 − (𝑆‘𝐶)) = (𝐷 − 𝐹))
113	69, 112	eqtrd 2804	. . 3 ⊢ (((𝜑 ∧ (𝐴(midG‘𝐺)𝐷) ≠ 𝐵) ∧ 𝐴 ≠ 𝐷) → (𝐴 − 𝐶) = (𝐷 − 𝐹))
114	33, 113	pm2.61dane 3051	. 2 ⊢ ((𝜑 ∧ (𝐴(midG‘𝐺)𝐷) ≠ 𝐵) → (𝐴 − 𝐶) = (𝐷 − 𝐹))
115	30, 114	pm2.61dane 3051	1 ⊢ (𝜑 → (𝐴 − 𝐶) = (𝐷 − 𝐹))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 400 ∨ wo 860 = wceq 1567 ∈ wcel 2149 ≠ wne 2964 class class class wbr 5113 ‘cfv 6537 (class class class)co 7411 2c2 12295 ⟨“cs3 14879 Basecbs 17269 distcds 17319 TarskiGcstrkg 28662 Dim_TarskiG≥cstrkgld 28666 Itvcitv 28668 LineGclng 28669 cgrGccgrg 28745 pInvGcmir 28891 ∟Gcrag 28932 ⟂Gcperpg 28934 midGcmid 29039 lInvGclmi 29040

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1822 ax-4 1836 ax-5 1937 ax-6 1994 ax-7 2035 ax-8 2151 ax-9 2159 ax-10 2182 ax-11 2198 ax-12 2219 ax-ext 2741 ax-rep 5242 ax-sep 5261 ax-nul 5271 ax-pow 5337 ax-pr 5405 ax-un 7733 ax-cnex 11156 ax-resscn 11157 ax-1cn 11158 ax-icn 11159 ax-addcl 11160 ax-addrcl 11161 ax-mulcl 11162 ax-mulrcl 11163 ax-mulcom 11164 ax-addass 11165 ax-mulass 11166 ax-distr 11167 ax-i2m1 11168 ax-1ne0 11169 ax-1rid 11170 ax-rnegex 11171 ax-rrecex 11172 ax-cnre 11173 ax-pre-lttri 11174 ax-pre-lttrn 11175 ax-pre-ltadd 11176 ax-pre-mulgt0 11177

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1102 df-3an 1103 df-tru 1570 df-fal 1580 df-ex 1807 df-nf 1811 df-sb 2098 df-mo 2573 df-eu 2603 df-clab 2748 df-cleq 2761 df-clel 2844 df-nfc 2918 df-ne 2965 df-nel 3071 df-ral 3086 df-rex 3096 df-rmo 3376 df-reu 3377 df-rab 3424 df-v 3465 df-sbc 3754 df-csb 3862 df-dif 3916 df-un 3918 df-in 3920 df-ss 3930 df-pss 3933 df-nul 4295 df-if 4493 df-pw 4569 df-sn 4595 df-pr 4597 df-tp 4599 df-op 4601 df-uni 4877 df-int 4917 df-iun 4962 df-br 5114 df-opab 5178 df-mpt 5197 df-tr 5223 df-id 5557 df-eprel 5562 df-po 5570 df-so 5571 df-fr 5615 df-we 5617 df-xp 5668 df-rel 5669 df-cnv 5670 df-co 5671 df-dm 5672 df-rn 5673 df-res 5674 df-ima 5675 df-pred 6303 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6493 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7368 df-ov 7414 df-oprab 7415 df-mpo 7416 df-om 7863 df-1st 7986 df-2nd 7987 df-frecs 8278 df-wrecs 8309 df-recs 8358 df-rdg 8397 df-1o 8453 df-oadd 8457 df-er 8694 df-map 8826 df-pm 8827 df-en 8944 df-dom 8945 df-sdom 8946 df-fin 8947 df-dju 9887 df-card 9925 df-pnf 11245 df-mnf 11246 df-xr 11247 df-ltxr 11248 df-le 11249 df-sub 11443 df-neg 11444 df-nn 12234 df-2 12303 df-3 12304 df-n0 12505 df-xnn0 12578 df-z 12592 df-uz 12863 df-fz 13536 df-fzo 13683 df-hash 14367 df-word 14551 df-concat 14608 df-s1 14634 df-s2 14885 df-s3 14886 df-trkgc 28683 df-trkgb 28684 df-trkgcb 28685 df-trkgld 28687 df-trkg 28688 df-cgrg 28746 df-leg 28818 df-mir 28892 df-rag 28933 df-perpg 28935 df-mid 29041 df-lmi 29042

This theorem is referenced by: hypcgrlem2 29067

W3C validator