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Theorem hypcgrlem1 26587

Description: Lemma for hypcgr 26589, 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 483	. . 3 ⊢ ((𝜑 ∧ (𝐴(midG‘𝐺)𝐷) = 𝐵) → 𝐺 ∈ TarskiG)
6		hypcgr.c	. . . 4 ⊢ (𝜑 → 𝐶 ∈ 𝑃)
7	6	adantr 483	. . 3 ⊢ ((𝜑 ∧ (𝐴(midG‘𝐺)𝐷) = 𝐵) → 𝐶 ∈ 𝑃)
8		hypcgr.a	. . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑃)
9	8	adantr 483	. . 3 ⊢ ((𝜑 ∧ (𝐴(midG‘𝐺)𝐷) = 𝐵) → 𝐴 ∈ 𝑃)
10		hypcgr.f	. . . 4 ⊢ (𝜑 → 𝐹 ∈ 𝑃)
11	10	adantr 483	. . 3 ⊢ ((𝜑 ∧ (𝐴(midG‘𝐺)𝐷) = 𝐵) → 𝐹 ∈ 𝑃)
12		hypcgr.d	. . . 4 ⊢ (𝜑 → 𝐷 ∈ 𝑃)
13	12	adantr 483	. . 3 ⊢ ((𝜑 ∧ (𝐴(midG‘𝐺)𝐷) = 𝐵) → 𝐷 ∈ 𝑃)
14		eqid 2823	. . . . . . 7 ⊢ (LineG‘𝐺) = (LineG‘𝐺)
15		eqid 2823	. . . . . . 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 26486	. . . . . 6 ⊢ (𝜑 → ⟨“𝐶𝐵𝐴”⟩ ∈ (∟G‘𝐺))
19	1, 2, 3, 14, 15, 4, 6, 16, 8	israg 26485	. . . . . 6 ⊢ (𝜑 → (⟨“𝐶𝐵𝐴”⟩ ∈ (∟G‘𝐺) ↔ (𝐶 − 𝐴) = (𝐶 − (((pInvG‘𝐺)‘𝐵)‘𝐴))))
20	18, 19	mpbid 234	. . . . 5 ⊢ (𝜑 → (𝐶 − 𝐴) = (𝐶 − (((pInvG‘𝐺)‘𝐵)‘𝐴)))
21	20	adantr 483	. . . 4 ⊢ ((𝜑 ∧ (𝐴(midG‘𝐺)𝐷) = 𝐵) → (𝐶 − 𝐴) = (𝐶 − (((pInvG‘𝐺)‘𝐵)‘𝐴)))
22		hypcgrlem1.a	. . . . . . 7 ⊢ (𝜑 → 𝐶 = 𝐹)
23	22	eqcomd 2829	. . . . . 6 ⊢ (𝜑 → 𝐹 = 𝐶)
24	23	adantr 483	. . . . 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 26566	. . . . . 6 ⊢ (𝜑 → (𝐷 = (((pInvG‘𝐺)‘𝐵)‘𝐴) ↔ (𝐴(midG‘𝐺)𝐷) = 𝐵))
27	26	biimpar 480	. . . . 5 ⊢ ((𝜑 ∧ (𝐴(midG‘𝐺)𝐷) = 𝐵) → 𝐷 = (((pInvG‘𝐺)‘𝐵)‘𝐴))
28	24, 27	oveq12d 7176	. . . 4 ⊢ ((𝜑 ∧ (𝐴(midG‘𝐺)𝐷) = 𝐵) → (𝐹 − 𝐷) = (𝐶 − (((pInvG‘𝐺)‘𝐵)‘𝐴)))
29	21, 28	eqtr4d 2861	. . 3 ⊢ ((𝜑 ∧ (𝐴(midG‘𝐺)𝐷) = 𝐵) → (𝐶 − 𝐴) = (𝐹 − 𝐷))
30	1, 2, 3, 5, 7, 9, 11, 13, 29	tgcgrcomlr 26268	. 2 ⊢ ((𝜑 ∧ (𝐴(midG‘𝐺)𝐷) = 𝐵) → (𝐴 − 𝐶) = (𝐷 − 𝐹))
31		simpr 487	. . . 4 ⊢ (((𝜑 ∧ (𝐴(midG‘𝐺)𝐷) ≠ 𝐵) ∧ 𝐴 = 𝐷) → 𝐴 = 𝐷)
32	22	ad2antrr 724	. . . 4 ⊢ (((𝜑 ∧ (𝐴(midG‘𝐺)𝐷) ≠ 𝐵) ∧ 𝐴 = 𝐷) → 𝐶 = 𝐹)
33	31, 32	oveq12d 7176	. . 3 ⊢ (((𝜑 ∧ (𝐴(midG‘𝐺)𝐷) ≠ 𝐵) ∧ 𝐴 = 𝐷) → (𝐴 − 𝐶) = (𝐷 − 𝐹))
34	17	ad2antrr 724	. . . . . 6 ⊢ (((𝜑 ∧ (𝐴(midG‘𝐺)𝐷) ≠ 𝐵) ∧ 𝐴 ≠ 𝐷) → ⟨“𝐴𝐵𝐶”⟩ ∈ (∟G‘𝐺))
35	4	ad2antrr 724	. . . . . . 7 ⊢ (((𝜑 ∧ (𝐴(midG‘𝐺)𝐷) ≠ 𝐵) ∧ 𝐴 ≠ 𝐷) → 𝐺 ∈ TarskiG)
36	8	ad2antrr 724	. . . . . . 7 ⊢ (((𝜑 ∧ (𝐴(midG‘𝐺)𝐷) ≠ 𝐵) ∧ 𝐴 ≠ 𝐷) → 𝐴 ∈ 𝑃)
37	16	ad2antrr 724	. . . . . . 7 ⊢ (((𝜑 ∧ (𝐴(midG‘𝐺)𝐷) ≠ 𝐵) ∧ 𝐴 ≠ 𝐷) → 𝐵 ∈ 𝑃)
38	6	ad2antrr 724	. . . . . . 7 ⊢ (((𝜑 ∧ (𝐴(midG‘𝐺)𝐷) ≠ 𝐵) ∧ 𝐴 ≠ 𝐷) → 𝐶 ∈ 𝑃)
39	1, 2, 3, 14, 15, 35, 36, 37, 38	israg 26485	. . . . . 6 ⊢ (((𝜑 ∧ (𝐴(midG‘𝐺)𝐷) ≠ 𝐵) ∧ 𝐴 ≠ 𝐷) → (⟨“𝐴𝐵𝐶”⟩ ∈ (∟G‘𝐺) ↔ (𝐴 − 𝐶) = (𝐴 − (((pInvG‘𝐺)‘𝐵)‘𝐶))))
40	34, 39	mpbid 234	. . . . 5 ⊢ (((𝜑 ∧ (𝐴(midG‘𝐺)𝐷) ≠ 𝐵) ∧ 𝐴 ≠ 𝐷) → (𝐴 − 𝐶) = (𝐴 − (((pInvG‘𝐺)‘𝐵)‘𝐶)))
41	25	ad2antrr 724	. . . . . . 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 724	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝐴(midG‘𝐺)𝐷) ≠ 𝐵) ∧ 𝐴 ≠ 𝐷) → 𝐷 ∈ 𝑃)
44	1, 2, 3, 35, 41, 36, 43	midcl 26565	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝐴(midG‘𝐺)𝐷) ≠ 𝐵) ∧ 𝐴 ≠ 𝐷) → (𝐴(midG‘𝐺)𝐷) ∈ 𝑃)
45		simplr 767	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝐴(midG‘𝐺)𝐷) ≠ 𝐵) ∧ 𝐴 ≠ 𝐷) → (𝐴(midG‘𝐺)𝐷) ≠ 𝐵)
46	1, 3, 14, 35, 44, 37, 45	tgelrnln 26418	. . . . . . 7 ⊢ (((𝜑 ∧ (𝐴(midG‘𝐺)𝐷) ≠ 𝐵) ∧ 𝐴 ≠ 𝐷) → ((𝐴(midG‘𝐺)𝐷)(LineG‘𝐺)𝐵) ∈ ran (LineG‘𝐺))
47		eqid 2823	. . . . . . 7 ⊢ ((pInvG‘𝐺)‘𝐵) = ((pInvG‘𝐺)‘𝐵)
48		eqid 2823	. . . . . . . . 9 ⊢ (cgrG‘𝐺) = (cgrG‘𝐺)
49	1, 2, 3, 14, 15, 35, 37, 47, 38	mircl 26449	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝐴(midG‘𝐺)𝐷) ≠ 𝐵) ∧ 𝐴 ≠ 𝐷) → (((pInvG‘𝐺)‘𝐵)‘𝐶) ∈ 𝑃)
50		simpr 487	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝐴(midG‘𝐺)𝐷) ≠ 𝐵) ∧ 𝐴 ≠ 𝐷) → 𝐴 ≠ 𝐷)
51	1, 2, 3, 35, 41, 36, 43	midbtwn 26567	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝐴(midG‘𝐺)𝐷) ≠ 𝐵) ∧ 𝐴 ≠ 𝐷) → (𝐴(midG‘𝐺)𝐷) ∈ (𝐴𝐼𝐷))
52	1, 14, 3, 35, 36, 44, 43, 51	btwncolg3 26345	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝐴(midG‘𝐺)𝐷) ≠ 𝐵) ∧ 𝐴 ≠ 𝐷) → (𝐷 ∈ (𝐴(LineG‘𝐺)(𝐴(midG‘𝐺)𝐷)) ∨ 𝐴 = (𝐴(midG‘𝐺)𝐷)))
53		eqidd 2824	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐷 = 𝐷)
54		hypcgrlem2.b	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐵 = 𝐸)
55	53, 54, 22	s3eqd 14228	. . . . . . . . . . . 12 ⊢ (𝜑 → ⟨“𝐷𝐵𝐶”⟩ = ⟨“𝐷𝐸𝐹”⟩)
56	55	ad2antrr 724	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝐴(midG‘𝐺)𝐷) ≠ 𝐵) ∧ 𝐴 ≠ 𝐷) → ⟨“𝐷𝐵𝐶”⟩ = ⟨“𝐷𝐸𝐹”⟩)
57		hypcgr.2	. . . . . . . . . . . 12 ⊢ (𝜑 → ⟨“𝐷𝐸𝐹”⟩ ∈ (∟G‘𝐺))
58	57	ad2antrr 724	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝐴(midG‘𝐺)𝐷) ≠ 𝐵) ∧ 𝐴 ≠ 𝐷) → ⟨“𝐷𝐸𝐹”⟩ ∈ (∟G‘𝐺))
59	56, 58	eqeltrd 2915	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝐴(midG‘𝐺)𝐷) ≠ 𝐵) ∧ 𝐴 ≠ 𝐷) → ⟨“𝐷𝐵𝐶”⟩ ∈ (∟G‘𝐺))
60	1, 2, 3, 14, 15, 35, 43, 37, 38	israg 26485	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝐴(midG‘𝐺)𝐷) ≠ 𝐵) ∧ 𝐴 ≠ 𝐷) → (⟨“𝐷𝐵𝐶”⟩ ∈ (∟G‘𝐺) ↔ (𝐷 − 𝐶) = (𝐷 − (((pInvG‘𝐺)‘𝐵)‘𝐶))))
61	59, 60	mpbid 234	. . . . . . . . 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 26357	. . . . . . . 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 26485	. . . . . . . 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 259	. . . . . . 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 26421	. . . . . . 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 26422	. . . . . . 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 26582	. . . . . 6 ⊢ (((𝜑 ∧ (𝐴(midG‘𝐺)𝐷) ≠ 𝐵) ∧ 𝐴 ≠ 𝐷) → (𝑆‘𝐶) = (((pInvG‘𝐺)‘𝐵)‘𝐶))
68	67	oveq2d 7174	. . . . 5 ⊢ (((𝜑 ∧ (𝐴(midG‘𝐺)𝐷) ≠ 𝐵) ∧ 𝐴 ≠ 𝐷) → (𝐴 − (𝑆‘𝐶)) = (𝐴 − (((pInvG‘𝐺)‘𝐵)‘𝐶)))
69	40, 68	eqtr4d 2861	. . . 4 ⊢ (((𝜑 ∧ (𝐴(midG‘𝐺)𝐷) ≠ 𝐵) ∧ 𝐴 ≠ 𝐷) → (𝐴 − 𝐶) = (𝐴 − (𝑆‘𝐶)))
70	1, 2, 3, 35, 41, 43, 36	midcom 26570	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝐴(midG‘𝐺)𝐷) ≠ 𝐵) ∧ 𝐴 ≠ 𝐷) → (𝐷(midG‘𝐺)𝐴) = (𝐴(midG‘𝐺)𝐷))
71	70, 65	eqeltrd 2915	. . . . . . 7 ⊢ (((𝜑 ∧ (𝐴(midG‘𝐺)𝐷) ≠ 𝐵) ∧ 𝐴 ≠ 𝐷) → (𝐷(midG‘𝐺)𝐴) ∈ ((𝐴(midG‘𝐺)𝐷)(LineG‘𝐺)𝐵))
72	50	necomd 3073	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝐴(midG‘𝐺)𝐷) ≠ 𝐵) ∧ 𝐴 ≠ 𝐷) → 𝐷 ≠ 𝐴)
73	1, 3, 14, 35, 43, 36, 72	tgelrnln 26418	. . . . . . . . 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 26276	. . . . . . . . . . 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 26407	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝐴(midG‘𝐺)𝐷) ≠ 𝐵) ∧ 𝐴 ≠ 𝐷) → (𝐴(midG‘𝐺)𝐷) ∈ (𝐷(LineG‘𝐺)𝐴))
76	65, 75	elind 4173	. . . . . . . . 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 26422	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝐴(midG‘𝐺)𝐷) ≠ 𝐵) ∧ 𝐴 ≠ 𝐷) → 𝐴 ∈ (𝐷(LineG‘𝐺)𝐴))
78	45	necomd 3073	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝐴(midG‘𝐺)𝐷) ≠ 𝐵) ∧ 𝐴 ≠ 𝐷) → 𝐵 ≠ (𝐴(midG‘𝐺)𝐷))
79	4	ad2antrr 724	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝐴(midG‘𝐺)𝐷) ≠ 𝐵) ∧ 𝐴 = (𝐴(midG‘𝐺)𝐷)) → 𝐺 ∈ TarskiG)
80	8	ad2antrr 724	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝐴(midG‘𝐺)𝐷) ≠ 𝐵) ∧ 𝐴 = (𝐴(midG‘𝐺)𝐷)) → 𝐴 ∈ 𝑃)
81	12	ad2antrr 724	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝐴(midG‘𝐺)𝐷) ≠ 𝐵) ∧ 𝐴 = (𝐴(midG‘𝐺)𝐷)) → 𝐷 ∈ 𝑃)
82	25	ad2antrr 724	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝐴(midG‘𝐺)𝐷) ≠ 𝐵) ∧ 𝐴 = (𝐴(midG‘𝐺)𝐷)) → 𝐺Dim_TarskiG≥2)
83		simpr 487	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝐴(midG‘𝐺)𝐷) ≠ 𝐵) ∧ 𝐴 = (𝐴(midG‘𝐺)𝐷)) → 𝐴 = (𝐴(midG‘𝐺)𝐷))
84	83	eqcomd 2829	. . . . . . . . . . . . . . 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 26568	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝐴(midG‘𝐺)𝐷) ≠ 𝐵) ∧ 𝐴 = (𝐴(midG‘𝐺)𝐷)) → (𝐴 − 𝐴) = (𝐴 − 𝐷))
86	85	eqcomd 2829	. . . . . . . . . . . . 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 26251	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝐴(midG‘𝐺)𝐷) ≠ 𝐵) ∧ 𝐴 = (𝐴(midG‘𝐺)𝐷)) → 𝐴 = 𝐷)
88	87	ex 415	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝐴(midG‘𝐺)𝐷) ≠ 𝐵) → (𝐴 = (𝐴(midG‘𝐺)𝐷) → 𝐴 = 𝐷))
89	88	necon3d 3039	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝐴(midG‘𝐺)𝐷) ≠ 𝐵) → (𝐴 ≠ 𝐷 → 𝐴 ≠ (𝐴(midG‘𝐺)𝐷)))
90	89	imp 409	. . . . . . . . 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 26268	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐵 − 𝐴) = (𝐸 − 𝐷))
94	54	oveq1d 7173	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐵 − 𝐷) = (𝐸 − 𝐷))
95	93, 94	eqtr4d 2861	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐵 − 𝐴) = (𝐵 − 𝐷))
96	95	ad2antrr 724	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝐴(midG‘𝐺)𝐷) ≠ 𝐵) ∧ 𝐴 ≠ 𝐷) → (𝐵 − 𝐴) = (𝐵 − 𝐷))
97		eqidd 2824	. . . . . . . . . . . . 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 26566	. . . . . . . . . . . . 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 259	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝐴(midG‘𝐺)𝐷) ≠ 𝐵) ∧ 𝐴 ≠ 𝐷) → 𝐷 = (((pInvG‘𝐺)‘(𝐴(midG‘𝐺)𝐷))‘𝐴))
100	99	oveq2d 7174	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝐴(midG‘𝐺)𝐷) ≠ 𝐵) ∧ 𝐴 ≠ 𝐷) → (𝐵 − 𝐷) = (𝐵 − (((pInvG‘𝐺)‘(𝐴(midG‘𝐺)𝐷))‘𝐴)))
101	96, 100	eqtrd 2858	. . . . . . . . . 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 26485	. . . . . . . . . 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 259	. . . . . . . . 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 26505	. . . . . . . 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 869	. . . . . . 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 26575	. . . . . . 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 711	. . . . . 6 ⊢ (((𝜑 ∧ (𝐴(midG‘𝐺)𝐷) ≠ 𝐵) ∧ 𝐴 ≠ 𝐷) → 𝐴 = (𝑆‘𝐷))
108	107	oveq1d 7173	. . . . 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 26585	. . . . 5 ⊢ (((𝜑 ∧ (𝐴(midG‘𝐺)𝐷) ≠ 𝐵) ∧ 𝐴 ≠ 𝐷) → ((𝑆‘𝐷) − (𝑆‘𝐶)) = (𝐷 − 𝐶))
110	22	oveq2d 7174	. . . . . 6 ⊢ (𝜑 → (𝐷 − 𝐶) = (𝐷 − 𝐹))
111	110	ad2antrr 724	. . . . 5 ⊢ (((𝜑 ∧ (𝐴(midG‘𝐺)𝐷) ≠ 𝐵) ∧ 𝐴 ≠ 𝐷) → (𝐷 − 𝐶) = (𝐷 − 𝐹))
112	108, 109, 111	3eqtrd 2862	. . . 4 ⊢ (((𝜑 ∧ (𝐴(midG‘𝐺)𝐷) ≠ 𝐵) ∧ 𝐴 ≠ 𝐷) → (𝐴 − (𝑆‘𝐶)) = (𝐷 − 𝐹))
113	69, 112	eqtrd 2858	. . 3 ⊢ (((𝜑 ∧ (𝐴(midG‘𝐺)𝐷) ≠ 𝐵) ∧ 𝐴 ≠ 𝐷) → (𝐴 − 𝐶) = (𝐷 − 𝐹))
114	33, 113	pm2.61dane 3106	. 2 ⊢ ((𝜑 ∧ (𝐴(midG‘𝐺)𝐷) ≠ 𝐵) → (𝐴 − 𝐶) = (𝐷 − 𝐹))
115	30, 114	pm2.61dane 3106	1 ⊢ (𝜑 → (𝐴 − 𝐶) = (𝐷 − 𝐹))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 398 ∨ wo 843 = wceq 1537 ∈ wcel 2114 ≠ wne 3018 class class class wbr 5068 ‘cfv 6357 (class class class)co 7158 2c2 11695 ⟨“cs3 14206 Basecbs 16485 distcds 16576 TarskiGcstrkg 26218 Dim_TarskiG≥cstrkgld 26222 Itvcitv 26224 LineGclng 26225 cgrGccgrg 26298 pInvGcmir 26440 ∟Gcrag 26481 ⟂Gcperpg 26483 midGcmid 26560 lInvGclmi 26561

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 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-1st 7691 df-2nd 7692 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-oadd 8108 df-er 8291 df-map 8410 df-pm 8411 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-dju 9332 df-card 9370 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-nn 11641 df-2 11703 df-3 11704 df-n0 11901 df-xnn0 11971 df-z 11985 df-uz 12247 df-fz 12896 df-fzo 13037 df-hash 13694 df-word 13865 df-concat 13925 df-s1 13952 df-s2 14212 df-s3 14213 df-trkgc 26236 df-trkgb 26237 df-trkgcb 26238 df-trkgld 26240 df-trkg 26241 df-cgrg 26299 df-leg 26371 df-mir 26441 df-rag 26482 df-perpg 26484 df-mid 26562 df-lmi 26563

This theorem is referenced by: hypcgrlem2 26588

W3C validator