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Theorem hypcgrlem1 28702

Description: Lemma for hypcgr 28704, 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 480	. . 3 ⊢ ((𝜑 ∧ (𝐴(midG‘𝐺)𝐷) = 𝐵) → 𝐺 ∈ TarskiG)
6		hypcgr.c	. . . 4 ⊢ (𝜑 → 𝐶 ∈ 𝑃)
7	6	adantr 480	. . 3 ⊢ ((𝜑 ∧ (𝐴(midG‘𝐺)𝐷) = 𝐵) → 𝐶 ∈ 𝑃)
8		hypcgr.a	. . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑃)
9	8	adantr 480	. . 3 ⊢ ((𝜑 ∧ (𝐴(midG‘𝐺)𝐷) = 𝐵) → 𝐴 ∈ 𝑃)
10		hypcgr.f	. . . 4 ⊢ (𝜑 → 𝐹 ∈ 𝑃)
11	10	adantr 480	. . 3 ⊢ ((𝜑 ∧ (𝐴(midG‘𝐺)𝐷) = 𝐵) → 𝐹 ∈ 𝑃)
12		hypcgr.d	. . . 4 ⊢ (𝜑 → 𝐷 ∈ 𝑃)
13	12	adantr 480	. . 3 ⊢ ((𝜑 ∧ (𝐴(midG‘𝐺)𝐷) = 𝐵) → 𝐷 ∈ 𝑃)
14		eqid 2729	. . . . . . 7 ⊢ (LineG‘𝐺) = (LineG‘𝐺)
15		eqid 2729	. . . . . . 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 28601	. . . . . 6 ⊢ (𝜑 → ⟨“𝐶𝐵𝐴”⟩ ∈ (∟G‘𝐺))
19	1, 2, 3, 14, 15, 4, 6, 16, 8	israg 28600	. . . . . 6 ⊢ (𝜑 → (⟨“𝐶𝐵𝐴”⟩ ∈ (∟G‘𝐺) ↔ (𝐶 − 𝐴) = (𝐶 − (((pInvG‘𝐺)‘𝐵)‘𝐴))))
20	18, 19	mpbid 232	. . . . 5 ⊢ (𝜑 → (𝐶 − 𝐴) = (𝐶 − (((pInvG‘𝐺)‘𝐵)‘𝐴)))
21	20	adantr 480	. . . 4 ⊢ ((𝜑 ∧ (𝐴(midG‘𝐺)𝐷) = 𝐵) → (𝐶 − 𝐴) = (𝐶 − (((pInvG‘𝐺)‘𝐵)‘𝐴)))
22		hypcgrlem1.a	. . . . . . 7 ⊢ (𝜑 → 𝐶 = 𝐹)
23	22	eqcomd 2735	. . . . . 6 ⊢ (𝜑 → 𝐹 = 𝐶)
24	23	adantr 480	. . . . 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 28681	. . . . . 6 ⊢ (𝜑 → (𝐷 = (((pInvG‘𝐺)‘𝐵)‘𝐴) ↔ (𝐴(midG‘𝐺)𝐷) = 𝐵))
27	26	biimpar 477	. . . . 5 ⊢ ((𝜑 ∧ (𝐴(midG‘𝐺)𝐷) = 𝐵) → 𝐷 = (((pInvG‘𝐺)‘𝐵)‘𝐴))
28	24, 27	oveq12d 7387	. . . 4 ⊢ ((𝜑 ∧ (𝐴(midG‘𝐺)𝐷) = 𝐵) → (𝐹 − 𝐷) = (𝐶 − (((pInvG‘𝐺)‘𝐵)‘𝐴)))
29	21, 28	eqtr4d 2767	. . 3 ⊢ ((𝜑 ∧ (𝐴(midG‘𝐺)𝐷) = 𝐵) → (𝐶 − 𝐴) = (𝐹 − 𝐷))
30	1, 2, 3, 5, 7, 9, 11, 13, 29	tgcgrcomlr 28383	. 2 ⊢ ((𝜑 ∧ (𝐴(midG‘𝐺)𝐷) = 𝐵) → (𝐴 − 𝐶) = (𝐷 − 𝐹))
31		simpr 484	. . . 4 ⊢ (((𝜑 ∧ (𝐴(midG‘𝐺)𝐷) ≠ 𝐵) ∧ 𝐴 = 𝐷) → 𝐴 = 𝐷)
32	22	ad2antrr 726	. . . 4 ⊢ (((𝜑 ∧ (𝐴(midG‘𝐺)𝐷) ≠ 𝐵) ∧ 𝐴 = 𝐷) → 𝐶 = 𝐹)
33	31, 32	oveq12d 7387	. . 3 ⊢ (((𝜑 ∧ (𝐴(midG‘𝐺)𝐷) ≠ 𝐵) ∧ 𝐴 = 𝐷) → (𝐴 − 𝐶) = (𝐷 − 𝐹))
34	17	ad2antrr 726	. . . . . 6 ⊢ (((𝜑 ∧ (𝐴(midG‘𝐺)𝐷) ≠ 𝐵) ∧ 𝐴 ≠ 𝐷) → ⟨“𝐴𝐵𝐶”⟩ ∈ (∟G‘𝐺))
35	4	ad2antrr 726	. . . . . . 7 ⊢ (((𝜑 ∧ (𝐴(midG‘𝐺)𝐷) ≠ 𝐵) ∧ 𝐴 ≠ 𝐷) → 𝐺 ∈ TarskiG)
36	8	ad2antrr 726	. . . . . . 7 ⊢ (((𝜑 ∧ (𝐴(midG‘𝐺)𝐷) ≠ 𝐵) ∧ 𝐴 ≠ 𝐷) → 𝐴 ∈ 𝑃)
37	16	ad2antrr 726	. . . . . . 7 ⊢ (((𝜑 ∧ (𝐴(midG‘𝐺)𝐷) ≠ 𝐵) ∧ 𝐴 ≠ 𝐷) → 𝐵 ∈ 𝑃)
38	6	ad2antrr 726	. . . . . . 7 ⊢ (((𝜑 ∧ (𝐴(midG‘𝐺)𝐷) ≠ 𝐵) ∧ 𝐴 ≠ 𝐷) → 𝐶 ∈ 𝑃)
39	1, 2, 3, 14, 15, 35, 36, 37, 38	israg 28600	. . . . . 6 ⊢ (((𝜑 ∧ (𝐴(midG‘𝐺)𝐷) ≠ 𝐵) ∧ 𝐴 ≠ 𝐷) → (⟨“𝐴𝐵𝐶”⟩ ∈ (∟G‘𝐺) ↔ (𝐴 − 𝐶) = (𝐴 − (((pInvG‘𝐺)‘𝐵)‘𝐶))))
40	34, 39	mpbid 232	. . . . 5 ⊢ (((𝜑 ∧ (𝐴(midG‘𝐺)𝐷) ≠ 𝐵) ∧ 𝐴 ≠ 𝐷) → (𝐴 − 𝐶) = (𝐴 − (((pInvG‘𝐺)‘𝐵)‘𝐶)))
41	25	ad2antrr 726	. . . . . . 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 726	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝐴(midG‘𝐺)𝐷) ≠ 𝐵) ∧ 𝐴 ≠ 𝐷) → 𝐷 ∈ 𝑃)
44	1, 2, 3, 35, 41, 36, 43	midcl 28680	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝐴(midG‘𝐺)𝐷) ≠ 𝐵) ∧ 𝐴 ≠ 𝐷) → (𝐴(midG‘𝐺)𝐷) ∈ 𝑃)
45		simplr 768	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝐴(midG‘𝐺)𝐷) ≠ 𝐵) ∧ 𝐴 ≠ 𝐷) → (𝐴(midG‘𝐺)𝐷) ≠ 𝐵)
46	1, 3, 14, 35, 44, 37, 45	tgelrnln 28533	. . . . . . 7 ⊢ (((𝜑 ∧ (𝐴(midG‘𝐺)𝐷) ≠ 𝐵) ∧ 𝐴 ≠ 𝐷) → ((𝐴(midG‘𝐺)𝐷)(LineG‘𝐺)𝐵) ∈ ran (LineG‘𝐺))
47		eqid 2729	. . . . . . 7 ⊢ ((pInvG‘𝐺)‘𝐵) = ((pInvG‘𝐺)‘𝐵)
48		eqid 2729	. . . . . . . . 9 ⊢ (cgrG‘𝐺) = (cgrG‘𝐺)
49	1, 2, 3, 14, 15, 35, 37, 47, 38	mircl 28564	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝐴(midG‘𝐺)𝐷) ≠ 𝐵) ∧ 𝐴 ≠ 𝐷) → (((pInvG‘𝐺)‘𝐵)‘𝐶) ∈ 𝑃)
50		simpr 484	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝐴(midG‘𝐺)𝐷) ≠ 𝐵) ∧ 𝐴 ≠ 𝐷) → 𝐴 ≠ 𝐷)
51	1, 2, 3, 35, 41, 36, 43	midbtwn 28682	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝐴(midG‘𝐺)𝐷) ≠ 𝐵) ∧ 𝐴 ≠ 𝐷) → (𝐴(midG‘𝐺)𝐷) ∈ (𝐴𝐼𝐷))
52	1, 14, 3, 35, 36, 44, 43, 51	btwncolg3 28460	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝐴(midG‘𝐺)𝐷) ≠ 𝐵) ∧ 𝐴 ≠ 𝐷) → (𝐷 ∈ (𝐴(LineG‘𝐺)(𝐴(midG‘𝐺)𝐷)) ∨ 𝐴 = (𝐴(midG‘𝐺)𝐷)))
53		eqidd 2730	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐷 = 𝐷)
54		hypcgrlem2.b	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐵 = 𝐸)
55	53, 54, 22	s3eqd 14806	. . . . . . . . . . . 12 ⊢ (𝜑 → ⟨“𝐷𝐵𝐶”⟩ = ⟨“𝐷𝐸𝐹”⟩)
56	55	ad2antrr 726	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝐴(midG‘𝐺)𝐷) ≠ 𝐵) ∧ 𝐴 ≠ 𝐷) → ⟨“𝐷𝐵𝐶”⟩ = ⟨“𝐷𝐸𝐹”⟩)
57		hypcgr.2	. . . . . . . . . . . 12 ⊢ (𝜑 → ⟨“𝐷𝐸𝐹”⟩ ∈ (∟G‘𝐺))
58	57	ad2antrr 726	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝐴(midG‘𝐺)𝐷) ≠ 𝐵) ∧ 𝐴 ≠ 𝐷) → ⟨“𝐷𝐸𝐹”⟩ ∈ (∟G‘𝐺))
59	56, 58	eqeltrd 2828	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝐴(midG‘𝐺)𝐷) ≠ 𝐵) ∧ 𝐴 ≠ 𝐷) → ⟨“𝐷𝐵𝐶”⟩ ∈ (∟G‘𝐺))
60	1, 2, 3, 14, 15, 35, 43, 37, 38	israg 28600	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝐴(midG‘𝐺)𝐷) ≠ 𝐵) ∧ 𝐴 ≠ 𝐷) → (⟨“𝐷𝐵𝐶”⟩ ∈ (∟G‘𝐺) ↔ (𝐷 − 𝐶) = (𝐷 − (((pInvG‘𝐺)‘𝐵)‘𝐶))))
61	59, 60	mpbid 232	. . . . . . . . 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 28472	. . . . . . . 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 28600	. . . . . . . 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 257	. . . . . . 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 28536	. . . . . . 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 28537	. . . . . . 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 28697	. . . . . 6 ⊢ (((𝜑 ∧ (𝐴(midG‘𝐺)𝐷) ≠ 𝐵) ∧ 𝐴 ≠ 𝐷) → (𝑆‘𝐶) = (((pInvG‘𝐺)‘𝐵)‘𝐶))
68	67	oveq2d 7385	. . . . 5 ⊢ (((𝜑 ∧ (𝐴(midG‘𝐺)𝐷) ≠ 𝐵) ∧ 𝐴 ≠ 𝐷) → (𝐴 − (𝑆‘𝐶)) = (𝐴 − (((pInvG‘𝐺)‘𝐵)‘𝐶)))
69	40, 68	eqtr4d 2767	. . . 4 ⊢ (((𝜑 ∧ (𝐴(midG‘𝐺)𝐷) ≠ 𝐵) ∧ 𝐴 ≠ 𝐷) → (𝐴 − 𝐶) = (𝐴 − (𝑆‘𝐶)))
70	1, 2, 3, 35, 41, 43, 36	midcom 28685	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝐴(midG‘𝐺)𝐷) ≠ 𝐵) ∧ 𝐴 ≠ 𝐷) → (𝐷(midG‘𝐺)𝐴) = (𝐴(midG‘𝐺)𝐷))
71	70, 65	eqeltrd 2828	. . . . . . 7 ⊢ (((𝜑 ∧ (𝐴(midG‘𝐺)𝐷) ≠ 𝐵) ∧ 𝐴 ≠ 𝐷) → (𝐷(midG‘𝐺)𝐴) ∈ ((𝐴(midG‘𝐺)𝐷)(LineG‘𝐺)𝐵))
72	50	necomd 2980	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝐴(midG‘𝐺)𝐷) ≠ 𝐵) ∧ 𝐴 ≠ 𝐷) → 𝐷 ≠ 𝐴)
73	1, 3, 14, 35, 43, 36, 72	tgelrnln 28533	. . . . . . . . 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 28391	. . . . . . . . . . 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 28522	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝐴(midG‘𝐺)𝐷) ≠ 𝐵) ∧ 𝐴 ≠ 𝐷) → (𝐴(midG‘𝐺)𝐷) ∈ (𝐷(LineG‘𝐺)𝐴))
76	65, 75	elind 4159	. . . . . . . . 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 28537	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝐴(midG‘𝐺)𝐷) ≠ 𝐵) ∧ 𝐴 ≠ 𝐷) → 𝐴 ∈ (𝐷(LineG‘𝐺)𝐴))
78	45	necomd 2980	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝐴(midG‘𝐺)𝐷) ≠ 𝐵) ∧ 𝐴 ≠ 𝐷) → 𝐵 ≠ (𝐴(midG‘𝐺)𝐷))
79	4	ad2antrr 726	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝐴(midG‘𝐺)𝐷) ≠ 𝐵) ∧ 𝐴 = (𝐴(midG‘𝐺)𝐷)) → 𝐺 ∈ TarskiG)
80	8	ad2antrr 726	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝐴(midG‘𝐺)𝐷) ≠ 𝐵) ∧ 𝐴 = (𝐴(midG‘𝐺)𝐷)) → 𝐴 ∈ 𝑃)
81	12	ad2antrr 726	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝐴(midG‘𝐺)𝐷) ≠ 𝐵) ∧ 𝐴 = (𝐴(midG‘𝐺)𝐷)) → 𝐷 ∈ 𝑃)
82	25	ad2antrr 726	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝐴(midG‘𝐺)𝐷) ≠ 𝐵) ∧ 𝐴 = (𝐴(midG‘𝐺)𝐷)) → 𝐺Dim_TarskiG≥2)
83		simpr 484	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝐴(midG‘𝐺)𝐷) ≠ 𝐵) ∧ 𝐴 = (𝐴(midG‘𝐺)𝐷)) → 𝐴 = (𝐴(midG‘𝐺)𝐷))
84	83	eqcomd 2735	. . . . . . . . . . . . . . 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 28683	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝐴(midG‘𝐺)𝐷) ≠ 𝐵) ∧ 𝐴 = (𝐴(midG‘𝐺)𝐷)) → (𝐴 − 𝐴) = (𝐴 − 𝐷))
86	85	eqcomd 2735	. . . . . . . . . . . . 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 28366	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝐴(midG‘𝐺)𝐷) ≠ 𝐵) ∧ 𝐴 = (𝐴(midG‘𝐺)𝐷)) → 𝐴 = 𝐷)
88	87	ex 412	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝐴(midG‘𝐺)𝐷) ≠ 𝐵) → (𝐴 = (𝐴(midG‘𝐺)𝐷) → 𝐴 = 𝐷))
89	88	necon3d 2946	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝐴(midG‘𝐺)𝐷) ≠ 𝐵) → (𝐴 ≠ 𝐷 → 𝐴 ≠ (𝐴(midG‘𝐺)𝐷)))
90	89	imp 406	. . . . . . . . 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 28383	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐵 − 𝐴) = (𝐸 − 𝐷))
94	54	oveq1d 7384	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐵 − 𝐷) = (𝐸 − 𝐷))
95	93, 94	eqtr4d 2767	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐵 − 𝐴) = (𝐵 − 𝐷))
96	95	ad2antrr 726	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝐴(midG‘𝐺)𝐷) ≠ 𝐵) ∧ 𝐴 ≠ 𝐷) → (𝐵 − 𝐴) = (𝐵 − 𝐷))
97		eqidd 2730	. . . . . . . . . . . . 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 28681	. . . . . . . . . . . . 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 257	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝐴(midG‘𝐺)𝐷) ≠ 𝐵) ∧ 𝐴 ≠ 𝐷) → 𝐷 = (((pInvG‘𝐺)‘(𝐴(midG‘𝐺)𝐷))‘𝐴))
100	99	oveq2d 7385	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝐴(midG‘𝐺)𝐷) ≠ 𝐵) ∧ 𝐴 ≠ 𝐷) → (𝐵 − 𝐷) = (𝐵 − (((pInvG‘𝐺)‘(𝐴(midG‘𝐺)𝐷))‘𝐴)))
101	96, 100	eqtrd 2764	. . . . . . . . . 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 28600	. . . . . . . . . 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 257	. . . . . . . . 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 28620	. . . . . . . 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 873	. . . . . . 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 28690	. . . . . . 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 713	. . . . . 6 ⊢ (((𝜑 ∧ (𝐴(midG‘𝐺)𝐷) ≠ 𝐵) ∧ 𝐴 ≠ 𝐷) → 𝐴 = (𝑆‘𝐷))
108	107	oveq1d 7384	. . . . 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 28700	. . . . 5 ⊢ (((𝜑 ∧ (𝐴(midG‘𝐺)𝐷) ≠ 𝐵) ∧ 𝐴 ≠ 𝐷) → ((𝑆‘𝐷) − (𝑆‘𝐶)) = (𝐷 − 𝐶))
110	22	oveq2d 7385	. . . . . 6 ⊢ (𝜑 → (𝐷 − 𝐶) = (𝐷 − 𝐹))
111	110	ad2antrr 726	. . . . 5 ⊢ (((𝜑 ∧ (𝐴(midG‘𝐺)𝐷) ≠ 𝐵) ∧ 𝐴 ≠ 𝐷) → (𝐷 − 𝐶) = (𝐷 − 𝐹))
112	108, 109, 111	3eqtrd 2768	. . . 4 ⊢ (((𝜑 ∧ (𝐴(midG‘𝐺)𝐷) ≠ 𝐵) ∧ 𝐴 ≠ 𝐷) → (𝐴 − (𝑆‘𝐶)) = (𝐷 − 𝐹))
113	69, 112	eqtrd 2764	. . 3 ⊢ (((𝜑 ∧ (𝐴(midG‘𝐺)𝐷) ≠ 𝐵) ∧ 𝐴 ≠ 𝐷) → (𝐴 − 𝐶) = (𝐷 − 𝐹))
114	33, 113	pm2.61dane 3012	. 2 ⊢ ((𝜑 ∧ (𝐴(midG‘𝐺)𝐷) ≠ 𝐵) → (𝐴 − 𝐶) = (𝐷 − 𝐹))
115	30, 114	pm2.61dane 3012	1 ⊢ (𝜑 → (𝐴 − 𝐶) = (𝐷 − 𝐹))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∨ wo 847 = wceq 1540 ∈ wcel 2109 ≠ wne 2925 class class class wbr 5102 ‘cfv 6499 (class class class)co 7369 2c2 12217 ⟨“cs3 14784 Basecbs 17155 distcds 17205 TarskiGcstrkg 28330 Dim_TarskiG≥cstrkgld 28334 Itvcitv 28336 LineGclng 28337 cgrGccgrg 28413 pInvGcmir 28555 ∟Gcrag 28596 ⟂Gcperpg 28598 midGcmid 28675 lInvGclmi 28676

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5229 ax-sep 5246 ax-nul 5256 ax-pow 5315 ax-pr 5382 ax-un 7691 ax-cnex 11100 ax-resscn 11101 ax-1cn 11102 ax-icn 11103 ax-addcl 11104 ax-addrcl 11105 ax-mulcl 11106 ax-mulrcl 11107 ax-mulcom 11108 ax-addass 11109 ax-mulass 11110 ax-distr 11111 ax-i2m1 11112 ax-1ne0 11113 ax-1rid 11114 ax-rnegex 11115 ax-rrecex 11116 ax-cnre 11117 ax-pre-lttri 11118 ax-pre-lttrn 11119 ax-pre-ltadd 11120 ax-pre-mulgt0 11121

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3351 df-reu 3352 df-rab 3403 df-v 3446 df-sbc 3751 df-csb 3860 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3931 df-nul 4293 df-if 4485 df-pw 4561 df-sn 4586 df-pr 4588 df-tp 4590 df-op 4592 df-uni 4868 df-int 4907 df-iun 4953 df-br 5103 df-opab 5165 df-mpt 5184 df-tr 5210 df-id 5526 df-eprel 5531 df-po 5539 df-so 5540 df-fr 5584 df-we 5586 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6262 df-ord 6323 df-on 6324 df-lim 6325 df-suc 6326 df-iota 6452 df-fun 6501 df-fn 6502 df-f 6503 df-f1 6504 df-fo 6505 df-f1o 6506 df-fv 6507 df-riota 7326 df-ov 7372 df-oprab 7373 df-mpo 7374 df-om 7823 df-1st 7947 df-2nd 7948 df-frecs 8237 df-wrecs 8268 df-recs 8317 df-rdg 8355 df-1o 8411 df-oadd 8415 df-er 8648 df-map 8778 df-pm 8779 df-en 8896 df-dom 8897 df-sdom 8898 df-fin 8899 df-dju 9830 df-card 9868 df-pnf 11186 df-mnf 11187 df-xr 11188 df-ltxr 11189 df-le 11190 df-sub 11383 df-neg 11384 df-nn 12163 df-2 12225 df-3 12226 df-n0 12419 df-xnn0 12492 df-z 12506 df-uz 12770 df-fz 13445 df-fzo 13592 df-hash 14272 df-word 14455 df-concat 14512 df-s1 14537 df-s2 14790 df-s3 14791 df-trkgc 28351 df-trkgb 28352 df-trkgcb 28353 df-trkgld 28355 df-trkg 28356 df-cgrg 28414 df-leg 28486 df-mir 28556 df-rag 28597 df-perpg 28599 df-mid 28677 df-lmi 28678

This theorem is referenced by: hypcgrlem2 28703

W3C validator