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Theorem trgcopyeulem 29312

Description: Lemma for trgcopyeu 29313. (Contributed by Thierry Arnoux, 8-Aug-2020.)

**Hypotheses**
Ref	Expression
trgcopy.p	⊢ 𝑃 = (Base‘𝐺)
trgcopy.m	⊢ − = (dist‘𝐺)
trgcopy.i	⊢ 𝐼 = (Itv‘𝐺)
trgcopy.l	⊢ 𝐿 = (LineG‘𝐺)
trgcopy.k	⊢ 𝐾 = (hlG‘𝐺)
trgcopy.g	⊢ (𝜑 → 𝐺 ∈ TarskiG)
trgcopy.a	⊢ (𝜑 → 𝐴 ∈ 𝑃)
trgcopy.b	⊢ (𝜑 → 𝐵 ∈ 𝑃)
trgcopy.c	⊢ (𝜑 → 𝐶 ∈ 𝑃)
trgcopy.d	⊢ (𝜑 → 𝐷 ∈ 𝑃)
trgcopy.e	⊢ (𝜑 → 𝐸 ∈ 𝑃)
trgcopy.f	⊢ (𝜑 → 𝐹 ∈ 𝑃)
trgcopy.1	⊢ (𝜑 → ¬ (𝐴 ∈ (𝐵𝐿𝐶) ∨ 𝐵 = 𝐶))
trgcopy.2	⊢ (𝜑 → ¬ (𝐷 ∈ (𝐸𝐿𝐹) ∨ 𝐸 = 𝐹))
trgcopy.3	⊢ (𝜑 → (𝐴 − 𝐵) = (𝐷 − 𝐸))
trgcopyeulem.o	⊢ 𝑂 = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝑃 ∖ (𝐷𝐿𝐸)) ∧ 𝑏 ∈ (𝑃 ∖ (𝐷𝐿𝐸))) ∧ ∃𝑡 ∈ (𝐷𝐿𝐸)𝑡 ∈ (𝑎𝐼𝑏))}
trgcopyeulem.x	⊢ (𝜑 → 𝑋 ∈ 𝑃)
trgcopyeulem.y	⊢ (𝜑 → 𝑌 ∈ 𝑃)
trgcopyeulem.1	⊢ (𝜑 → ⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑋”⟩)
trgcopyeulem.2	⊢ (𝜑 → ⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑌”⟩)
trgcopyeulem.3	⊢ (𝜑 → 𝑋((hpG‘𝐺)‘(𝐷𝐿𝐸))𝐹)
trgcopyeulem.4	⊢ (𝜑 → 𝑌((hpG‘𝐺)‘(𝐷𝐿𝐸))𝐹)

**Assertion**
Ref	Expression
trgcopyeulem	⊢ (𝜑 → 𝑋 = 𝑌)

Distinct variable groups: − ,𝑎,𝑏,𝑡 𝐴,𝑎,𝑏,𝑡 𝐵,𝑎,𝑏,𝑡 𝐶,𝑎,𝑏,𝑡 𝐷,𝑎,𝑏,𝑡 𝐸,𝑎,𝑏,𝑡 𝐹,𝑎,𝑏,𝑡 𝐺,𝑎,𝑏,𝑡 𝐼,𝑎,𝑏,𝑡 𝐿,𝑎,𝑏,𝑡 𝑃,𝑎,𝑏,𝑡 𝜑,𝑎,𝑏,𝑡 𝐾,𝑎 𝑂,𝑎,𝑏,𝑡 𝑋,𝑎,𝑏,𝑡 𝑌,𝑎,𝑏,𝑡 Allowed substitution hints: 𝐾(𝑡, 𝑏)

**Proof of Theorem trgcopyeulem**
Step	Hyp	Ref	Expression
1		trgcopy.p	. 2 ⊢ 𝑃 = (Base‘𝐺)
2		trgcopy.m	. 2 ⊢ − = (dist‘𝐺)
3		trgcopy.i	. 2 ⊢ 𝐼 = (Itv‘𝐺)
4		trgcopy.g	. 2 ⊢ (𝜑 → 𝐺 ∈ TarskiG)
5		trgcopy.l	. . 3 ⊢ 𝐿 = (LineG‘𝐺)
6		trgcopy.b	. . 3 ⊢ (𝜑 → 𝐵 ∈ 𝑃)
7		trgcopy.c	. . 3 ⊢ (𝜑 → 𝐶 ∈ 𝑃)
8		trgcopy.a	. . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑃)
9		trgcopy.1	. . 3 ⊢ (𝜑 → ¬ (𝐴 ∈ (𝐵𝐿𝐶) ∨ 𝐵 = 𝐶))
10	1, 5, 3, 4, 6, 7, 8, 9	ncoltgdim2 29028	. 2 ⊢ (𝜑 → 𝐺Dim_TarskiG≥2)
11		eqid 2761	. 2 ⊢ ((lInvG‘𝐺)‘(𝐷𝐿𝐸)) = ((lInvG‘𝐺)‘(𝐷𝐿𝐸))
12		trgcopy.d	. . 3 ⊢ (𝜑 → 𝐷 ∈ 𝑃)
13		trgcopy.e	. . 3 ⊢ (𝜑 → 𝐸 ∈ 𝑃)
14		trgcopy.f	. . . 4 ⊢ (𝜑 → 𝐹 ∈ 𝑃)
15		trgcopy.2	. . . 4 ⊢ (𝜑 → ¬ (𝐷 ∈ (𝐸𝐿𝐹) ∨ 𝐸 = 𝐹))
16	1, 3, 5, 4, 12, 13, 14, 15	ncolne1 29093	. . 3 ⊢ (𝜑 → 𝐷 ≠ 𝐸)
17	1, 3, 5, 4, 12, 13, 16	tgelrnln 29098	. 2 ⊢ (𝜑 → (𝐷𝐿𝐸) ∈ ran 𝐿)
18		trgcopyeulem.x	. 2 ⊢ (𝜑 → 𝑋 ∈ 𝑃)
19		trgcopyeulem.y	. 2 ⊢ (𝜑 → 𝑌 ∈ 𝑃)
20		eqid 2761	. . . . . . . . 9 ⊢ (pInvG‘𝐺) = (pInvG‘𝐺)
21	4	ad2antrr 739	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝐷𝐿𝐸)) ∧ 𝑡 ∈ (𝑋𝐼(((lInvG‘𝐺)‘(𝐷𝐿𝐸))‘𝑌))) → 𝐺 ∈ TarskiG)
22	17	ad2antrr 739	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝐷𝐿𝐸)) ∧ 𝑡 ∈ (𝑋𝐼(((lInvG‘𝐺)‘(𝐷𝐿𝐸))‘𝑌))) → (𝐷𝐿𝐸) ∈ ran 𝐿)
23		simplr 781	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝐷𝐿𝐸)) ∧ 𝑡 ∈ (𝑋𝐼(((lInvG‘𝐺)‘(𝐷𝐿𝐸))‘𝑌))) → 𝑡 ∈ (𝐷𝐿𝐸))
24	1, 5, 3, 21, 22, 23	tglnpt 29012	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝐷𝐿𝐸)) ∧ 𝑡 ∈ (𝑋𝐼(((lInvG‘𝐺)‘(𝐷𝐿𝐸))‘𝑌))) → 𝑡 ∈ 𝑃)
25		eqid 2761	. . . . . . . . 9 ⊢ ((pInvG‘𝐺)‘𝑡) = ((pInvG‘𝐺)‘𝑡)
26	1, 2, 3, 4, 10, 11, 5, 17, 19	lmicl 29291	. . . . . . . . . 10 ⊢ (𝜑 → (((lInvG‘𝐺)‘(𝐷𝐿𝐸))‘𝑌) ∈ 𝑃)
27	26	ad2antrr 739	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝐷𝐿𝐸)) ∧ 𝑡 ∈ (𝑋𝐼(((lInvG‘𝐺)‘(𝐷𝐿𝐸))‘𝑌))) → (((lInvG‘𝐺)‘(𝐷𝐿𝐸))‘𝑌) ∈ 𝑃)
28	18	ad2antrr 739	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝐷𝐿𝐸)) ∧ 𝑡 ∈ (𝑋𝐼(((lInvG‘𝐺)‘(𝐷𝐿𝐸))‘𝑌))) → 𝑋 ∈ 𝑃)
29	12	ad2antrr 739	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝐷𝐿𝐸)) ∧ 𝑡 ∈ (𝑋𝐼(((lInvG‘𝐺)‘(𝐷𝐿𝐸))‘𝑌))) → 𝐷 ∈ 𝑃)
30	13	ad2antrr 739	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝐷𝐿𝐸)) ∧ 𝑡 ∈ (𝑋𝐼(((lInvG‘𝐺)‘(𝐷𝐿𝐸))‘𝑌))) → 𝐸 ∈ 𝑃)
31		eqid 2761	. . . . . . . . . . . 12 ⊢ (cgrG‘𝐺) = (cgrG‘𝐺)
32	16	ad2antrr 739	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝐷𝐿𝐸)) ∧ 𝑡 ∈ (𝑋𝐼(((lInvG‘𝐺)‘(𝐷𝐿𝐸))‘𝑌))) → 𝐷 ≠ 𝐸)
33	32	necomd 3011	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝐷𝐿𝐸)) ∧ 𝑡 ∈ (𝑋𝐼(((lInvG‘𝐺)‘(𝐷𝐿𝐸))‘𝑌))) → 𝐸 ≠ 𝐷)
34	1, 3, 5, 21, 30, 29, 24, 33, 23	lncom 29090	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝐷𝐿𝐸)) ∧ 𝑡 ∈ (𝑋𝐼(((lInvG‘𝐺)‘(𝐷𝐿𝐸))‘𝑌))) → 𝑡 ∈ (𝐸𝐿𝐷))
35	34	orcd 887	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝐷𝐿𝐸)) ∧ 𝑡 ∈ (𝑋𝐼(((lInvG‘𝐺)‘(𝐷𝐿𝐸))‘𝑌))) → (𝑡 ∈ (𝐸𝐿𝐷) ∨ 𝐸 = 𝐷))
36	1, 5, 3, 21, 30, 29, 24, 35	colrot1 29022	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝐷𝐿𝐸)) ∧ 𝑡 ∈ (𝑋𝐼(((lInvG‘𝐺)‘(𝐷𝐿𝐸))‘𝑌))) → (𝐸 ∈ (𝐷𝐿𝑡) ∨ 𝐷 = 𝑡))
37		trgcopyeulem.1	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑋”⟩)
38	1, 2, 3, 31, 4, 8, 6, 7, 12, 13, 18, 37	cgr3simp3 28985	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝐶 − 𝐴) = (𝑋 − 𝐷))
39	1, 2, 3, 4, 7, 8, 18, 12, 38	tgcgrcomlr 28942	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐴 − 𝐶) = (𝐷 − 𝑋))
40		trgcopyeulem.2	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑌”⟩)
41	1, 2, 3, 31, 4, 8, 6, 7, 12, 13, 19, 40	cgr3simp3 28985	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝐶 − 𝐴) = (𝑌 − 𝐷))
42	1, 2, 3, 4, 7, 8, 19, 12, 41	tgcgrcomlr 28942	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐴 − 𝐶) = (𝐷 − 𝑌))
43	39, 42	eqtr3d 2798	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐷 − 𝑋) = (𝐷 − 𝑌))
44	1, 2, 3, 4, 10, 11, 5, 17, 12, 19	lmiiso 29302	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((((lInvG‘𝐺)‘(𝐷𝐿𝐸))‘𝐷) − (((lInvG‘𝐺)‘(𝐷𝐿𝐸))‘𝑌)) = (𝐷 − 𝑌))
45	1, 3, 5, 4, 12, 13, 16	tglinerflx1 29101	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐷 ∈ (𝐷𝐿𝐸))
46	1, 2, 3, 4, 10, 11, 5, 17, 12, 45	lmicinv 29298	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (((lInvG‘𝐺)‘(𝐷𝐿𝐸))‘𝐷) = 𝐷)
47	46	oveq1d 7435	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((((lInvG‘𝐺)‘(𝐷𝐿𝐸))‘𝐷) − (((lInvG‘𝐺)‘(𝐷𝐿𝐸))‘𝑌)) = (𝐷 − (((lInvG‘𝐺)‘(𝐷𝐿𝐸))‘𝑌)))
48	43, 44, 47	3eqtr2d 2802	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐷 − 𝑋) = (𝐷 − (((lInvG‘𝐺)‘(𝐷𝐿𝐸))‘𝑌)))
49	48	ad2antrr 739	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝐷𝐿𝐸)) ∧ 𝑡 ∈ (𝑋𝐼(((lInvG‘𝐺)‘(𝐷𝐿𝐸))‘𝑌))) → (𝐷 − 𝑋) = (𝐷 − (((lInvG‘𝐺)‘(𝐷𝐿𝐸))‘𝑌)))
50	1, 2, 3, 31, 4, 8, 6, 7, 12, 13, 18, 37	cgr3simp2 28984	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐵 − 𝐶) = (𝐸 − 𝑋))
51	1, 2, 3, 31, 4, 8, 6, 7, 12, 13, 19, 40	cgr3simp2 28984	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐵 − 𝐶) = (𝐸 − 𝑌))
52	50, 51	eqtr3d 2798	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐸 − 𝑋) = (𝐸 − 𝑌))
53	1, 2, 3, 4, 10, 11, 5, 17, 13, 19	lmiiso 29302	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((((lInvG‘𝐺)‘(𝐷𝐿𝐸))‘𝐸) − (((lInvG‘𝐺)‘(𝐷𝐿𝐸))‘𝑌)) = (𝐸 − 𝑌))
54	1, 3, 5, 4, 12, 13, 16	tglinerflx2 29102	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐸 ∈ (𝐷𝐿𝐸))
55	1, 2, 3, 4, 10, 11, 5, 17, 13, 54	lmicinv 29298	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (((lInvG‘𝐺)‘(𝐷𝐿𝐸))‘𝐸) = 𝐸)
56	55	oveq1d 7435	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((((lInvG‘𝐺)‘(𝐷𝐿𝐸))‘𝐸) − (((lInvG‘𝐺)‘(𝐷𝐿𝐸))‘𝑌)) = (𝐸 − (((lInvG‘𝐺)‘(𝐷𝐿𝐸))‘𝑌)))
57	52, 53, 56	3eqtr2d 2802	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐸 − 𝑋) = (𝐸 − (((lInvG‘𝐺)‘(𝐷𝐿𝐸))‘𝑌)))
58	57	ad2antrr 739	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝐷𝐿𝐸)) ∧ 𝑡 ∈ (𝑋𝐼(((lInvG‘𝐺)‘(𝐷𝐿𝐸))‘𝑌))) → (𝐸 − 𝑋) = (𝐸 − (((lInvG‘𝐺)‘(𝐷𝐿𝐸))‘𝑌)))
59	1, 5, 3, 21, 29, 30, 24, 31, 28, 27, 2, 32, 36, 49, 58	lncgr 29032	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝐷𝐿𝐸)) ∧ 𝑡 ∈ (𝑋𝐼(((lInvG‘𝐺)‘(𝐷𝐿𝐸))‘𝑌))) → (𝑡 − 𝑋) = (𝑡 − (((lInvG‘𝐺)‘(𝐷𝐿𝐸))‘𝑌)))
60		simpr 490	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝐷𝐿𝐸)) ∧ 𝑡 ∈ (𝑋𝐼(((lInvG‘𝐺)‘(𝐷𝐿𝐸))‘𝑌))) → 𝑡 ∈ (𝑋𝐼(((lInvG‘𝐺)‘(𝐷𝐿𝐸))‘𝑌)))
61	1, 2, 3, 5, 20, 21, 24, 25, 27, 28, 59, 60	ismir 29131	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝐷𝐿𝐸)) ∧ 𝑡 ∈ (𝑋𝐼(((lInvG‘𝐺)‘(𝐷𝐿𝐸))‘𝑌))) → 𝑋 = (((pInvG‘𝐺)‘𝑡)‘(((lInvG‘𝐺)‘(𝐷𝐿𝐸))‘𝑌)))
62	61	eqcomd 2767	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝐷𝐿𝐸)) ∧ 𝑡 ∈ (𝑋𝐼(((lInvG‘𝐺)‘(𝐷𝐿𝐸))‘𝑌))) → (((pInvG‘𝐺)‘𝑡)‘(((lInvG‘𝐺)‘(𝐷𝐿𝐸))‘𝑌)) = 𝑋)
63	1, 2, 3, 5, 20, 21, 24, 25, 27, 62	mircom 29135	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝐷𝐿𝐸)) ∧ 𝑡 ∈ (𝑋𝐼(((lInvG‘𝐺)‘(𝐷𝐿𝐸))‘𝑌))) → (((pInvG‘𝐺)‘𝑡)‘𝑋) = (((lInvG‘𝐺)‘(𝐷𝐿𝐸))‘𝑌))
64	63	eqcomd 2767	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝐷𝐿𝐸)) ∧ 𝑡 ∈ (𝑋𝐼(((lInvG‘𝐺)‘(𝐷𝐿𝐸))‘𝑌))) → (((lInvG‘𝐺)‘(𝐷𝐿𝐸))‘𝑌) = (((pInvG‘𝐺)‘𝑡)‘𝑋))
65	10	ad2antrr 739	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝐷𝐿𝐸)) ∧ 𝑡 ∈ (𝑋𝐼(((lInvG‘𝐺)‘(𝐷𝐿𝐸))‘𝑌))) → 𝐺Dim_TarskiG≥2)
66	1, 2, 3, 21, 65, 28, 27, 20, 24	ismidb 29283	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝐷𝐿𝐸)) ∧ 𝑡 ∈ (𝑋𝐼(((lInvG‘𝐺)‘(𝐷𝐿𝐸))‘𝑌))) → ((((lInvG‘𝐺)‘(𝐷𝐿𝐸))‘𝑌) = (((pInvG‘𝐺)‘𝑡)‘𝑋) ↔ (𝑋(midG‘𝐺)(((lInvG‘𝐺)‘(𝐷𝐿𝐸))‘𝑌)) = 𝑡))
67	64, 66	mpbid 235	. . . . . 6 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝐷𝐿𝐸)) ∧ 𝑡 ∈ (𝑋𝐼(((lInvG‘𝐺)‘(𝐷𝐿𝐸))‘𝑌))) → (𝑋(midG‘𝐺)(((lInvG‘𝐺)‘(𝐷𝐿𝐸))‘𝑌)) = 𝑡)
68	67, 23	eqeltrd 2861	. . . . 5 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝐷𝐿𝐸)) ∧ 𝑡 ∈ (𝑋𝐼(((lInvG‘𝐺)‘(𝐷𝐿𝐸))‘𝑌))) → (𝑋(midG‘𝐺)(((lInvG‘𝐺)‘(𝐷𝐿𝐸))‘𝑌)) ∈ (𝐷𝐿𝐸))
69		trgcopyeulem.o	. . . . . . . . 9 ⊢ 𝑂 = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝑃 ∖ (𝐷𝐿𝐸)) ∧ 𝑏 ∈ (𝑃 ∖ (𝐷𝐿𝐸))) ∧ ∃𝑡 ∈ (𝐷𝐿𝐸)𝑡 ∈ (𝑎𝐼𝑏))}
70		trgcopyeulem.4	. . . . . . . . 9 ⊢ (𝜑 → 𝑌((hpG‘𝐺)‘(𝐷𝐿𝐸))𝐹)
71		trgcopyeulem.3	. . . . . . . . . 10 ⊢ (𝜑 → 𝑋((hpG‘𝐺)‘(𝐷𝐿𝐸))𝐹)
72	1, 3, 5, 4, 17, 18, 69, 14, 71	hpgcom 29245	. . . . . . . . 9 ⊢ (𝜑 → 𝐹((hpG‘𝐺)‘(𝐷𝐿𝐸))𝑋)
73	1, 3, 5, 4, 17, 19, 69, 14, 70, 18, 72	hpgtr 29246	. . . . . . . 8 ⊢ (𝜑 → 𝑌((hpG‘𝐺)‘(𝐷𝐿𝐸))𝑋)
74	1, 3, 5, 69, 4, 17, 19, 14, 70	hpgne1 29239	. . . . . . . . . 10 ⊢ (𝜑 → ¬ 𝑌 ∈ (𝐷𝐿𝐸))
75	1, 2, 3, 5, 4, 10, 17, 69, 11, 19, 74	lmiopp 29308	. . . . . . . . 9 ⊢ (𝜑 → 𝑌𝑂(((lInvG‘𝐺)‘(𝐷𝐿𝐸))‘𝑌))
76	1, 3, 5, 69, 4, 17, 19, 18, 26, 75	lnopp2hpgb 29241	. . . . . . . 8 ⊢ (𝜑 → (𝑋𝑂(((lInvG‘𝐺)‘(𝐷𝐿𝐸))‘𝑌) ↔ 𝑌((hpG‘𝐺)‘(𝐷𝐿𝐸))𝑋))
77	73, 76	mpbird 260	. . . . . . 7 ⊢ (𝜑 → 𝑋𝑂(((lInvG‘𝐺)‘(𝐷𝐿𝐸))‘𝑌))
78	1, 2, 3, 69, 18, 26	islnopp 29215	. . . . . . 7 ⊢ (𝜑 → (𝑋𝑂(((lInvG‘𝐺)‘(𝐷𝐿𝐸))‘𝑌) ↔ ((¬ 𝑋 ∈ (𝐷𝐿𝐸) ∧ ¬ (((lInvG‘𝐺)‘(𝐷𝐿𝐸))‘𝑌) ∈ (𝐷𝐿𝐸)) ∧ ∃𝑡 ∈ (𝐷𝐿𝐸)𝑡 ∈ (𝑋𝐼(((lInvG‘𝐺)‘(𝐷𝐿𝐸))‘𝑌)))))
79	77, 78	mpbid 235	. . . . . 6 ⊢ (𝜑 → ((¬ 𝑋 ∈ (𝐷𝐿𝐸) ∧ ¬ (((lInvG‘𝐺)‘(𝐷𝐿𝐸))‘𝑌) ∈ (𝐷𝐿𝐸)) ∧ ∃𝑡 ∈ (𝐷𝐿𝐸)𝑡 ∈ (𝑋𝐼(((lInvG‘𝐺)‘(𝐷𝐿𝐸))‘𝑌))))
80	79	simprd 501	. . . . 5 ⊢ (𝜑 → ∃𝑡 ∈ (𝐷𝐿𝐸)𝑡 ∈ (𝑋𝐼(((lInvG‘𝐺)‘(𝐷𝐿𝐸))‘𝑌)))
81	68, 80	r19.29a 3171	. . . 4 ⊢ (𝜑 → (𝑋(midG‘𝐺)(((lInvG‘𝐺)‘(𝐷𝐿𝐸))‘𝑌)) ∈ (𝐷𝐿𝐸))
82	21	adantr 486	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑡 ∈ (𝐷𝐿𝐸)) ∧ 𝑡 ∈ (𝑋𝐼(((lInvG‘𝐺)‘(𝐷𝐿𝐸))‘𝑌))) ∧ 𝐸 ≠ 𝑡) → 𝐺 ∈ TarskiG)
83	22	adantr 486	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑡 ∈ (𝐷𝐿𝐸)) ∧ 𝑡 ∈ (𝑋𝐼(((lInvG‘𝐺)‘(𝐷𝐿𝐸))‘𝑌))) ∧ 𝐸 ≠ 𝑡) → (𝐷𝐿𝐸) ∈ ran 𝐿)
84	1, 2, 3, 69, 5, 17, 4, 18, 26, 77	oppne3 29219	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑋 ≠ (((lInvG‘𝐺)‘(𝐷𝐿𝐸))‘𝑌))
85	1, 3, 5, 4, 18, 26, 84	tgelrnln 29098	. . . . . . . . . 10 ⊢ (𝜑 → (𝑋𝐿(((lInvG‘𝐺)‘(𝐷𝐿𝐸))‘𝑌)) ∈ ran 𝐿)
86	85	ad2antrr 739	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝐷𝐿𝐸)) ∧ 𝑡 ∈ (𝑋𝐼(((lInvG‘𝐺)‘(𝐷𝐿𝐸))‘𝑌))) → (𝑋𝐿(((lInvG‘𝐺)‘(𝐷𝐿𝐸))‘𝑌)) ∈ ran 𝐿)
87	86	adantr 486	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑡 ∈ (𝐷𝐿𝐸)) ∧ 𝑡 ∈ (𝑋𝐼(((lInvG‘𝐺)‘(𝐷𝐿𝐸))‘𝑌))) ∧ 𝐸 ≠ 𝑡) → (𝑋𝐿(((lInvG‘𝐺)‘(𝐷𝐿𝐸))‘𝑌)) ∈ ran 𝐿)
88	84	ad2antrr 739	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝐷𝐿𝐸)) ∧ 𝑡 ∈ (𝑋𝐼(((lInvG‘𝐺)‘(𝐷𝐿𝐸))‘𝑌))) → 𝑋 ≠ (((lInvG‘𝐺)‘(𝐷𝐿𝐸))‘𝑌))
89	1, 3, 5, 21, 28, 27, 24, 88, 60	btwnlng1 29087	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝐷𝐿𝐸)) ∧ 𝑡 ∈ (𝑋𝐼(((lInvG‘𝐺)‘(𝐷𝐿𝐸))‘𝑌))) → 𝑡 ∈ (𝑋𝐿(((lInvG‘𝐺)‘(𝐷𝐿𝐸))‘𝑌)))
90	23, 89	elind 4146	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝐷𝐿𝐸)) ∧ 𝑡 ∈ (𝑋𝐼(((lInvG‘𝐺)‘(𝐷𝐿𝐸))‘𝑌))) → 𝑡 ∈ ((𝐷𝐿𝐸) ∩ (𝑋𝐿(((lInvG‘𝐺)‘(𝐷𝐿𝐸))‘𝑌))))
91	90	adantr 486	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑡 ∈ (𝐷𝐿𝐸)) ∧ 𝑡 ∈ (𝑋𝐼(((lInvG‘𝐺)‘(𝐷𝐿𝐸))‘𝑌))) ∧ 𝐸 ≠ 𝑡) → 𝑡 ∈ ((𝐷𝐿𝐸) ∩ (𝑋𝐿(((lInvG‘𝐺)‘(𝐷𝐿𝐸))‘𝑌))))
92	54	ad3antrrr 743	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑡 ∈ (𝐷𝐿𝐸)) ∧ 𝑡 ∈ (𝑋𝐼(((lInvG‘𝐺)‘(𝐷𝐿𝐸))‘𝑌))) ∧ 𝐸 ≠ 𝑡) → 𝐸 ∈ (𝐷𝐿𝐸))
93	1, 3, 5, 4, 18, 26, 84	tglinerflx1 29101	. . . . . . . . 9 ⊢ (𝜑 → 𝑋 ∈ (𝑋𝐿(((lInvG‘𝐺)‘(𝐷𝐿𝐸))‘𝑌)))
94	93	ad3antrrr 743	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑡 ∈ (𝐷𝐿𝐸)) ∧ 𝑡 ∈ (𝑋𝐼(((lInvG‘𝐺)‘(𝐷𝐿𝐸))‘𝑌))) ∧ 𝐸 ≠ 𝑡) → 𝑋 ∈ (𝑋𝐿(((lInvG‘𝐺)‘(𝐷𝐿𝐸))‘𝑌)))
95		simpr 490	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑡 ∈ (𝐷𝐿𝐸)) ∧ 𝑡 ∈ (𝑋𝐼(((lInvG‘𝐺)‘(𝐷𝐿𝐸))‘𝑌))) ∧ 𝐸 ≠ 𝑡) → 𝐸 ≠ 𝑡)
96	79	simplld 780	. . . . . . . . . . . 12 ⊢ (𝜑 → ¬ 𝑋 ∈ (𝐷𝐿𝐸))
97	96	ad2antrr 739	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝐷𝐿𝐸)) ∧ 𝑡 ∈ (𝑋𝐼(((lInvG‘𝐺)‘(𝐷𝐿𝐸))‘𝑌))) → ¬ 𝑋 ∈ (𝐷𝐿𝐸))
98		nelne2 3054	. . . . . . . . . . 11 ⊢ ((𝑡 ∈ (𝐷𝐿𝐸) ∧ ¬ 𝑋 ∈ (𝐷𝐿𝐸)) → 𝑡 ≠ 𝑋)
99	23, 97, 98	syl2anc 596	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝐷𝐿𝐸)) ∧ 𝑡 ∈ (𝑋𝐼(((lInvG‘𝐺)‘(𝐷𝐿𝐸))‘𝑌))) → 𝑡 ≠ 𝑋)
100	99	necomd 3011	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝐷𝐿𝐸)) ∧ 𝑡 ∈ (𝑋𝐼(((lInvG‘𝐺)‘(𝐷𝐿𝐸))‘𝑌))) → 𝑋 ≠ 𝑡)
101	100	adantr 486	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑡 ∈ (𝐷𝐿𝐸)) ∧ 𝑡 ∈ (𝑋𝐼(((lInvG‘𝐺)‘(𝐷𝐿𝐸))‘𝑌))) ∧ 𝐸 ≠ 𝑡) → 𝑋 ≠ 𝑡)
102	64	oveq2d 7436	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝐷𝐿𝐸)) ∧ 𝑡 ∈ (𝑋𝐼(((lInvG‘𝐺)‘(𝐷𝐿𝐸))‘𝑌))) → (𝐸 − (((lInvG‘𝐺)‘(𝐷𝐿𝐸))‘𝑌)) = (𝐸 − (((pInvG‘𝐺)‘𝑡)‘𝑋)))
103	58, 102	eqtrd 2796	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝐷𝐿𝐸)) ∧ 𝑡 ∈ (𝑋𝐼(((lInvG‘𝐺)‘(𝐷𝐿𝐸))‘𝑌))) → (𝐸 − 𝑋) = (𝐸 − (((pInvG‘𝐺)‘𝑡)‘𝑋)))
104	103	adantr 486	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑡 ∈ (𝐷𝐿𝐸)) ∧ 𝑡 ∈ (𝑋𝐼(((lInvG‘𝐺)‘(𝐷𝐿𝐸))‘𝑌))) ∧ 𝐸 ≠ 𝑡) → (𝐸 − 𝑋) = (𝐸 − (((pInvG‘𝐺)‘𝑡)‘𝑋)))
105	30	adantr 486	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑡 ∈ (𝐷𝐿𝐸)) ∧ 𝑡 ∈ (𝑋𝐼(((lInvG‘𝐺)‘(𝐷𝐿𝐸))‘𝑌))) ∧ 𝐸 ≠ 𝑡) → 𝐸 ∈ 𝑃)
106	24	adantr 486	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑡 ∈ (𝐷𝐿𝐸)) ∧ 𝑡 ∈ (𝑋𝐼(((lInvG‘𝐺)‘(𝐷𝐿𝐸))‘𝑌))) ∧ 𝐸 ≠ 𝑡) → 𝑡 ∈ 𝑃)
107	28	adantr 486	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑡 ∈ (𝐷𝐿𝐸)) ∧ 𝑡 ∈ (𝑋𝐼(((lInvG‘𝐺)‘(𝐷𝐿𝐸))‘𝑌))) ∧ 𝐸 ≠ 𝑡) → 𝑋 ∈ 𝑃)
108	1, 2, 3, 5, 20, 82, 105, 106, 107	israg 29172	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑡 ∈ (𝐷𝐿𝐸)) ∧ 𝑡 ∈ (𝑋𝐼(((lInvG‘𝐺)‘(𝐷𝐿𝐸))‘𝑌))) ∧ 𝐸 ≠ 𝑡) → (⟨“𝐸𝑡𝑋”⟩ ∈ (∟G‘𝐺) ↔ (𝐸 − 𝑋) = (𝐸 − (((pInvG‘𝐺)‘𝑡)‘𝑋))))
109	104, 108	mpbird 260	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑡 ∈ (𝐷𝐿𝐸)) ∧ 𝑡 ∈ (𝑋𝐼(((lInvG‘𝐺)‘(𝐷𝐿𝐸))‘𝑌))) ∧ 𝐸 ≠ 𝑡) → ⟨“𝐸𝑡𝑋”⟩ ∈ (∟G‘𝐺))
110	1, 2, 3, 5, 82, 83, 87, 91, 92, 94, 95, 101, 109	ragperp 29192	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑡 ∈ (𝐷𝐿𝐸)) ∧ 𝑡 ∈ (𝑋𝐼(((lInvG‘𝐺)‘(𝐷𝐿𝐸))‘𝑌))) ∧ 𝐸 ≠ 𝑡) → (𝐷𝐿𝐸)(⟂G‘𝐺)(𝑋𝐿(((lInvG‘𝐺)‘(𝐷𝐿𝐸))‘𝑌)))
111	21	adantr 486	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑡 ∈ (𝐷𝐿𝐸)) ∧ 𝑡 ∈ (𝑋𝐼(((lInvG‘𝐺)‘(𝐷𝐿𝐸))‘𝑌))) ∧ 𝐷 ≠ 𝑡) → 𝐺 ∈ TarskiG)
112	22	adantr 486	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑡 ∈ (𝐷𝐿𝐸)) ∧ 𝑡 ∈ (𝑋𝐼(((lInvG‘𝐺)‘(𝐷𝐿𝐸))‘𝑌))) ∧ 𝐷 ≠ 𝑡) → (𝐷𝐿𝐸) ∈ ran 𝐿)
113	86	adantr 486	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑡 ∈ (𝐷𝐿𝐸)) ∧ 𝑡 ∈ (𝑋𝐼(((lInvG‘𝐺)‘(𝐷𝐿𝐸))‘𝑌))) ∧ 𝐷 ≠ 𝑡) → (𝑋𝐿(((lInvG‘𝐺)‘(𝐷𝐿𝐸))‘𝑌)) ∈ ran 𝐿)
114	90	adantr 486	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑡 ∈ (𝐷𝐿𝐸)) ∧ 𝑡 ∈ (𝑋𝐼(((lInvG‘𝐺)‘(𝐷𝐿𝐸))‘𝑌))) ∧ 𝐷 ≠ 𝑡) → 𝑡 ∈ ((𝐷𝐿𝐸) ∩ (𝑋𝐿(((lInvG‘𝐺)‘(𝐷𝐿𝐸))‘𝑌))))
115	45	ad3antrrr 743	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑡 ∈ (𝐷𝐿𝐸)) ∧ 𝑡 ∈ (𝑋𝐼(((lInvG‘𝐺)‘(𝐷𝐿𝐸))‘𝑌))) ∧ 𝐷 ≠ 𝑡) → 𝐷 ∈ (𝐷𝐿𝐸))
116	93	ad3antrrr 743	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑡 ∈ (𝐷𝐿𝐸)) ∧ 𝑡 ∈ (𝑋𝐼(((lInvG‘𝐺)‘(𝐷𝐿𝐸))‘𝑌))) ∧ 𝐷 ≠ 𝑡) → 𝑋 ∈ (𝑋𝐿(((lInvG‘𝐺)‘(𝐷𝐿𝐸))‘𝑌)))
117		simpr 490	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑡 ∈ (𝐷𝐿𝐸)) ∧ 𝑡 ∈ (𝑋𝐼(((lInvG‘𝐺)‘(𝐷𝐿𝐸))‘𝑌))) ∧ 𝐷 ≠ 𝑡) → 𝐷 ≠ 𝑡)
118	100	adantr 486	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑡 ∈ (𝐷𝐿𝐸)) ∧ 𝑡 ∈ (𝑋𝐼(((lInvG‘𝐺)‘(𝐷𝐿𝐸))‘𝑌))) ∧ 𝐷 ≠ 𝑡) → 𝑋 ≠ 𝑡)
119	64	oveq2d 7436	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝐷𝐿𝐸)) ∧ 𝑡 ∈ (𝑋𝐼(((lInvG‘𝐺)‘(𝐷𝐿𝐸))‘𝑌))) → (𝐷 − (((lInvG‘𝐺)‘(𝐷𝐿𝐸))‘𝑌)) = (𝐷 − (((pInvG‘𝐺)‘𝑡)‘𝑋)))
120	49, 119	eqtrd 2796	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝐷𝐿𝐸)) ∧ 𝑡 ∈ (𝑋𝐼(((lInvG‘𝐺)‘(𝐷𝐿𝐸))‘𝑌))) → (𝐷 − 𝑋) = (𝐷 − (((pInvG‘𝐺)‘𝑡)‘𝑋)))
121	120	adantr 486	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑡 ∈ (𝐷𝐿𝐸)) ∧ 𝑡 ∈ (𝑋𝐼(((lInvG‘𝐺)‘(𝐷𝐿𝐸))‘𝑌))) ∧ 𝐷 ≠ 𝑡) → (𝐷 − 𝑋) = (𝐷 − (((pInvG‘𝐺)‘𝑡)‘𝑋)))
122	29	adantr 486	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑡 ∈ (𝐷𝐿𝐸)) ∧ 𝑡 ∈ (𝑋𝐼(((lInvG‘𝐺)‘(𝐷𝐿𝐸))‘𝑌))) ∧ 𝐷 ≠ 𝑡) → 𝐷 ∈ 𝑃)
123	24	adantr 486	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑡 ∈ (𝐷𝐿𝐸)) ∧ 𝑡 ∈ (𝑋𝐼(((lInvG‘𝐺)‘(𝐷𝐿𝐸))‘𝑌))) ∧ 𝐷 ≠ 𝑡) → 𝑡 ∈ 𝑃)
124	28	adantr 486	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑡 ∈ (𝐷𝐿𝐸)) ∧ 𝑡 ∈ (𝑋𝐼(((lInvG‘𝐺)‘(𝐷𝐿𝐸))‘𝑌))) ∧ 𝐷 ≠ 𝑡) → 𝑋 ∈ 𝑃)
125	1, 2, 3, 5, 20, 111, 122, 123, 124	israg 29172	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑡 ∈ (𝐷𝐿𝐸)) ∧ 𝑡 ∈ (𝑋𝐼(((lInvG‘𝐺)‘(𝐷𝐿𝐸))‘𝑌))) ∧ 𝐷 ≠ 𝑡) → (⟨“𝐷𝑡𝑋”⟩ ∈ (∟G‘𝐺) ↔ (𝐷 − 𝑋) = (𝐷 − (((pInvG‘𝐺)‘𝑡)‘𝑋))))
126	121, 125	mpbird 260	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑡 ∈ (𝐷𝐿𝐸)) ∧ 𝑡 ∈ (𝑋𝐼(((lInvG‘𝐺)‘(𝐷𝐿𝐸))‘𝑌))) ∧ 𝐷 ≠ 𝑡) → ⟨“𝐷𝑡𝑋”⟩ ∈ (∟G‘𝐺))
127	1, 2, 3, 5, 111, 112, 113, 114, 115, 116, 117, 118, 126	ragperp 29192	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑡 ∈ (𝐷𝐿𝐸)) ∧ 𝑡 ∈ (𝑋𝐼(((lInvG‘𝐺)‘(𝐷𝐿𝐸))‘𝑌))) ∧ 𝐷 ≠ 𝑡) → (𝐷𝐿𝐸)(⟂G‘𝐺)(𝑋𝐿(((lInvG‘𝐺)‘(𝐷𝐿𝐸))‘𝑌)))
128		neneor 3058	. . . . . . . 8 ⊢ (𝐸 ≠ 𝐷 → (𝐸 ≠ 𝑡 ∨ 𝐷 ≠ 𝑡))
129	33, 128	syl 18	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝐷𝐿𝐸)) ∧ 𝑡 ∈ (𝑋𝐼(((lInvG‘𝐺)‘(𝐷𝐿𝐸))‘𝑌))) → (𝐸 ≠ 𝑡 ∨ 𝐷 ≠ 𝑡))
130	110, 127, 129	mpjaodan 973	. . . . . 6 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝐷𝐿𝐸)) ∧ 𝑡 ∈ (𝑋𝐼(((lInvG‘𝐺)‘(𝐷𝐿𝐸))‘𝑌))) → (𝐷𝐿𝐸)(⟂G‘𝐺)(𝑋𝐿(((lInvG‘𝐺)‘(𝐷𝐿𝐸))‘𝑌)))
131	130	orcd 887	. . . . 5 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝐷𝐿𝐸)) ∧ 𝑡 ∈ (𝑋𝐼(((lInvG‘𝐺)‘(𝐷𝐿𝐸))‘𝑌))) → ((𝐷𝐿𝐸)(⟂G‘𝐺)(𝑋𝐿(((lInvG‘𝐺)‘(𝐷𝐿𝐸))‘𝑌)) ∨ 𝑋 = (((lInvG‘𝐺)‘(𝐷𝐿𝐸))‘𝑌)))
132	131, 80	r19.29a 3171	. . . 4 ⊢ (𝜑 → ((𝐷𝐿𝐸)(⟂G‘𝐺)(𝑋𝐿(((lInvG‘𝐺)‘(𝐷𝐿𝐸))‘𝑌)) ∨ 𝑋 = (((lInvG‘𝐺)‘(𝐷𝐿𝐸))‘𝑌)))
133	1, 2, 3, 4, 10, 11, 5, 17, 18, 26	islmib 29292	. . . 4 ⊢ (𝜑 → ((((lInvG‘𝐺)‘(𝐷𝐿𝐸))‘𝑌) = (((lInvG‘𝐺)‘(𝐷𝐿𝐸))‘𝑋) ↔ ((𝑋(midG‘𝐺)(((lInvG‘𝐺)‘(𝐷𝐿𝐸))‘𝑌)) ∈ (𝐷𝐿𝐸) ∧ ((𝐷𝐿𝐸)(⟂G‘𝐺)(𝑋𝐿(((lInvG‘𝐺)‘(𝐷𝐿𝐸))‘𝑌)) ∨ 𝑋 = (((lInvG‘𝐺)‘(𝐷𝐿𝐸))‘𝑌)))))
134	81, 132, 133	mpbir2and 726	. . 3 ⊢ (𝜑 → (((lInvG‘𝐺)‘(𝐷𝐿𝐸))‘𝑌) = (((lInvG‘𝐺)‘(𝐷𝐿𝐸))‘𝑋))
135	134	eqcomd 2767	. 2 ⊢ (𝜑 → (((lInvG‘𝐺)‘(𝐷𝐿𝐸))‘𝑋) = (((lInvG‘𝐺)‘(𝐷𝐿𝐸))‘𝑌))
136	1, 2, 3, 4, 10, 11, 5, 17, 18, 19, 135	lmieq 29296	1 ⊢ (𝜑 → 𝑋 = 𝑌)

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ∧ wa 401 ∨ wo 861 = wceq 1570 ∈ wcel 2145 ≠ wne 2956 ∃wrex 3087 ∖ cdif 3896 ∩ cin 3898 class class class wbr 5103 {copab 5167 ran crn 5652 ‘cfv 6538 (class class class)co 7420 2c2 12397 ⟨“cs3 14993 Basecbs 17387 distcds 17437 TarskiGcstrkg 28889 Dim_TarskiG≥cstrkgld 28893 Itvcitv 28895 LineGclng 28896 cgrGccgrg 28973 hlGchlg 29063 pInvGcmir 29124 ∟Gcrag 29168 ⟂Gcperpg 29170 hpGchpg 29235 midGcmid 29277 lInvGclmi 29278

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2733 ax-rep 5232 ax-sep 5249 ax-nul 5260 ax-pow 5327 ax-pr 5391 ax-un 7751 ax-cnex 11256 ax-resscn 11257 ax-1cn 11258 ax-icn 11259 ax-addcl 11260 ax-addrcl 11261 ax-mulcl 11262 ax-mulrcl 11263 ax-mulcom 11264 ax-addass 11265 ax-mulass 11266 ax-distr 11267 ax-i2m1 11268 ax-1ne0 11269 ax-1rid 11270 ax-rnegex 11271 ax-rrecex 11272 ax-cnre 11273 ax-pre-lttri 11274 ax-pre-lttrn 11275 ax-pre-ltadd 11276 ax-pre-mulgt0 11277

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3063 df-ral 3078 df-rex 3088 df-rmo 3366 df-reu 3367 df-rab 3414 df-v 3453 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4591 df-uni 4868 df-int 4908 df-iun 4953 df-br 5104 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5546 df-eprel 5551 df-po 5559 df-so 5560 df-fr 5604 df-we 5606 df-xp 5657 df-rel 5658 df-cnv 5659 df-co 5660 df-dm 5661 df-rn 5662 df-res 5663 df-ima 5664 df-pred 6304 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6494 df-fun 6540 df-fn 6541 df-f 6542 df-f1 6543 df-fo 6544 df-f1o 6545 df-fv 6546 df-riota 7377 df-ov 7423 df-oprab 7424 df-mpo 7425 df-om 7878 df-1st 8001 df-2nd 8002 df-frecs 8299 df-wrecs 8330 df-recs 8379 df-rdg 8418 df-1o 8476 df-oadd 8480 df-er 8717 df-map 8849 df-pm 8850 df-en 8974 df-dom 8975 df-sdom 8976 df-fin 8977 df-dju 9982 df-card 10020 df-pnf 11345 df-mnf 11346 df-xr 11347 df-ltxr 11348 df-le 11349 df-sub 11543 df-neg 11544 df-nn 12336 df-2 12405 df-3 12406 df-n0 12607 df-xnn0 12680 df-z 12694 df-uz 12966 df-fz 13640 df-fzo 13789 df-hash 14475 df-word 14659 df-concat 14716 df-s1 14743 df-s2 14999 df-s3 15000 df-trkgc 28910 df-trkgb 28911 df-trkgcb 28912 df-trkgld 28914 df-trkg 28915 df-cgrg 28974 df-leg 29046 df-hlg 29064 df-mir 29125 df-rag 29169 df-perpg 29171 df-hpg 29236 df-mid 29279 df-lmi 29280

This theorem is used by: trgcopyeu 29313 acopyeu 29342

W3C validator