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Theorem trgcopyeulem 28783

Description: Lemma for trgcopyeu 28784. (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 28543	. 2 ⊢ (𝜑 → 𝐺Dim_TarskiG≥2)
11		eqid 2731	. 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 28603	. . 3 ⊢ (𝜑 → 𝐷 ≠ 𝐸)
17	1, 3, 5, 4, 12, 13, 16	tgelrnln 28608	. 2 ⊢ (𝜑 → (𝐷𝐿𝐸) ∈ ran 𝐿)
18		trgcopyeulem.x	. 2 ⊢ (𝜑 → 𝑋 ∈ 𝑃)
19		trgcopyeulem.y	. 2 ⊢ (𝜑 → 𝑌 ∈ 𝑃)
20		eqid 2731	. . . . . . . . 9 ⊢ (pInvG‘𝐺) = (pInvG‘𝐺)
21	4	ad2antrr 726	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝐷𝐿𝐸)) ∧ 𝑡 ∈ (𝑋𝐼(((lInvG‘𝐺)‘(𝐷𝐿𝐸))‘𝑌))) → 𝐺 ∈ TarskiG)
22	17	ad2antrr 726	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝐷𝐿𝐸)) ∧ 𝑡 ∈ (𝑋𝐼(((lInvG‘𝐺)‘(𝐷𝐿𝐸))‘𝑌))) → (𝐷𝐿𝐸) ∈ ran 𝐿)
23		simplr 768	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝐷𝐿𝐸)) ∧ 𝑡 ∈ (𝑋𝐼(((lInvG‘𝐺)‘(𝐷𝐿𝐸))‘𝑌))) → 𝑡 ∈ (𝐷𝐿𝐸))
24	1, 5, 3, 21, 22, 23	tglnpt 28527	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝐷𝐿𝐸)) ∧ 𝑡 ∈ (𝑋𝐼(((lInvG‘𝐺)‘(𝐷𝐿𝐸))‘𝑌))) → 𝑡 ∈ 𝑃)
25		eqid 2731	. . . . . . . . 9 ⊢ ((pInvG‘𝐺)‘𝑡) = ((pInvG‘𝐺)‘𝑡)
26	1, 2, 3, 4, 10, 11, 5, 17, 19	lmicl 28764	. . . . . . . . . 10 ⊢ (𝜑 → (((lInvG‘𝐺)‘(𝐷𝐿𝐸))‘𝑌) ∈ 𝑃)
27	26	ad2antrr 726	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝐷𝐿𝐸)) ∧ 𝑡 ∈ (𝑋𝐼(((lInvG‘𝐺)‘(𝐷𝐿𝐸))‘𝑌))) → (((lInvG‘𝐺)‘(𝐷𝐿𝐸))‘𝑌) ∈ 𝑃)
28	18	ad2antrr 726	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝐷𝐿𝐸)) ∧ 𝑡 ∈ (𝑋𝐼(((lInvG‘𝐺)‘(𝐷𝐿𝐸))‘𝑌))) → 𝑋 ∈ 𝑃)
29	12	ad2antrr 726	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝐷𝐿𝐸)) ∧ 𝑡 ∈ (𝑋𝐼(((lInvG‘𝐺)‘(𝐷𝐿𝐸))‘𝑌))) → 𝐷 ∈ 𝑃)
30	13	ad2antrr 726	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝐷𝐿𝐸)) ∧ 𝑡 ∈ (𝑋𝐼(((lInvG‘𝐺)‘(𝐷𝐿𝐸))‘𝑌))) → 𝐸 ∈ 𝑃)
31		eqid 2731	. . . . . . . . . . . 12 ⊢ (cgrG‘𝐺) = (cgrG‘𝐺)
32	16	ad2antrr 726	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝐷𝐿𝐸)) ∧ 𝑡 ∈ (𝑋𝐼(((lInvG‘𝐺)‘(𝐷𝐿𝐸))‘𝑌))) → 𝐷 ≠ 𝐸)
33	32	necomd 2983	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝐷𝐿𝐸)) ∧ 𝑡 ∈ (𝑋𝐼(((lInvG‘𝐺)‘(𝐷𝐿𝐸))‘𝑌))) → 𝐸 ≠ 𝐷)
34	1, 3, 5, 21, 30, 29, 24, 33, 23	lncom 28600	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝐷𝐿𝐸)) ∧ 𝑡 ∈ (𝑋𝐼(((lInvG‘𝐺)‘(𝐷𝐿𝐸))‘𝑌))) → 𝑡 ∈ (𝐸𝐿𝐷))
35	34	orcd 873	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝐷𝐿𝐸)) ∧ 𝑡 ∈ (𝑋𝐼(((lInvG‘𝐺)‘(𝐷𝐿𝐸))‘𝑌))) → (𝑡 ∈ (𝐸𝐿𝐷) ∨ 𝐸 = 𝐷))
36	1, 5, 3, 21, 30, 29, 24, 35	colrot1 28537	. . . . . . . . . . . 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 28500	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝐶 − 𝐴) = (𝑋 − 𝐷))
39	1, 2, 3, 4, 7, 8, 18, 12, 38	tgcgrcomlr 28458	. . . . . . . . . . . . . . 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 28500	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝐶 − 𝐴) = (𝑌 − 𝐷))
42	1, 2, 3, 4, 7, 8, 19, 12, 41	tgcgrcomlr 28458	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐴 − 𝐶) = (𝐷 − 𝑌))
43	39, 42	eqtr3d 2768	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐷 − 𝑋) = (𝐷 − 𝑌))
44	1, 2, 3, 4, 10, 11, 5, 17, 12, 19	lmiiso 28775	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((((lInvG‘𝐺)‘(𝐷𝐿𝐸))‘𝐷) − (((lInvG‘𝐺)‘(𝐷𝐿𝐸))‘𝑌)) = (𝐷 − 𝑌))
45	1, 3, 5, 4, 12, 13, 16	tglinerflx1 28611	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐷 ∈ (𝐷𝐿𝐸))
46	1, 2, 3, 4, 10, 11, 5, 17, 12, 45	lmicinv 28771	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (((lInvG‘𝐺)‘(𝐷𝐿𝐸))‘𝐷) = 𝐷)
47	46	oveq1d 7361	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((((lInvG‘𝐺)‘(𝐷𝐿𝐸))‘𝐷) − (((lInvG‘𝐺)‘(𝐷𝐿𝐸))‘𝑌)) = (𝐷 − (((lInvG‘𝐺)‘(𝐷𝐿𝐸))‘𝑌)))
48	43, 44, 47	3eqtr2d 2772	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐷 − 𝑋) = (𝐷 − (((lInvG‘𝐺)‘(𝐷𝐿𝐸))‘𝑌)))
49	48	ad2antrr 726	. . . . . . . . . . . 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 28499	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐵 − 𝐶) = (𝐸 − 𝑋))
51	1, 2, 3, 31, 4, 8, 6, 7, 12, 13, 19, 40	cgr3simp2 28499	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐵 − 𝐶) = (𝐸 − 𝑌))
52	50, 51	eqtr3d 2768	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐸 − 𝑋) = (𝐸 − 𝑌))
53	1, 2, 3, 4, 10, 11, 5, 17, 13, 19	lmiiso 28775	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((((lInvG‘𝐺)‘(𝐷𝐿𝐸))‘𝐸) − (((lInvG‘𝐺)‘(𝐷𝐿𝐸))‘𝑌)) = (𝐸 − 𝑌))
54	1, 3, 5, 4, 12, 13, 16	tglinerflx2 28612	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐸 ∈ (𝐷𝐿𝐸))
55	1, 2, 3, 4, 10, 11, 5, 17, 13, 54	lmicinv 28771	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (((lInvG‘𝐺)‘(𝐷𝐿𝐸))‘𝐸) = 𝐸)
56	55	oveq1d 7361	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((((lInvG‘𝐺)‘(𝐷𝐿𝐸))‘𝐸) − (((lInvG‘𝐺)‘(𝐷𝐿𝐸))‘𝑌)) = (𝐸 − (((lInvG‘𝐺)‘(𝐷𝐿𝐸))‘𝑌)))
57	52, 53, 56	3eqtr2d 2772	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐸 − 𝑋) = (𝐸 − (((lInvG‘𝐺)‘(𝐷𝐿𝐸))‘𝑌)))
58	57	ad2antrr 726	. . . . . . . . . . . 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 28547	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝐷𝐿𝐸)) ∧ 𝑡 ∈ (𝑋𝐼(((lInvG‘𝐺)‘(𝐷𝐿𝐸))‘𝑌))) → (𝑡 − 𝑋) = (𝑡 − (((lInvG‘𝐺)‘(𝐷𝐿𝐸))‘𝑌)))
60		simpr 484	. . . . . . . . . . 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 28637	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝐷𝐿𝐸)) ∧ 𝑡 ∈ (𝑋𝐼(((lInvG‘𝐺)‘(𝐷𝐿𝐸))‘𝑌))) → 𝑋 = (((pInvG‘𝐺)‘𝑡)‘(((lInvG‘𝐺)‘(𝐷𝐿𝐸))‘𝑌)))
62	61	eqcomd 2737	. . . . . . . . 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 28641	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝐷𝐿𝐸)) ∧ 𝑡 ∈ (𝑋𝐼(((lInvG‘𝐺)‘(𝐷𝐿𝐸))‘𝑌))) → (((pInvG‘𝐺)‘𝑡)‘𝑋) = (((lInvG‘𝐺)‘(𝐷𝐿𝐸))‘𝑌))
64	63	eqcomd 2737	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝐷𝐿𝐸)) ∧ 𝑡 ∈ (𝑋𝐼(((lInvG‘𝐺)‘(𝐷𝐿𝐸))‘𝑌))) → (((lInvG‘𝐺)‘(𝐷𝐿𝐸))‘𝑌) = (((pInvG‘𝐺)‘𝑡)‘𝑋))
65	10	ad2antrr 726	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝐷𝐿𝐸)) ∧ 𝑡 ∈ (𝑋𝐼(((lInvG‘𝐺)‘(𝐷𝐿𝐸))‘𝑌))) → 𝐺Dim_TarskiG≥2)
66	1, 2, 3, 21, 65, 28, 27, 20, 24	ismidb 28756	. . . . . . 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 232	. . . . . 6 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝐷𝐿𝐸)) ∧ 𝑡 ∈ (𝑋𝐼(((lInvG‘𝐺)‘(𝐷𝐿𝐸))‘𝑌))) → (𝑋(midG‘𝐺)(((lInvG‘𝐺)‘(𝐷𝐿𝐸))‘𝑌)) = 𝑡)
68	67, 23	eqeltrd 2831	. . . . 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 28745	. . . . . . . . 9 ⊢ (𝜑 → 𝐹((hpG‘𝐺)‘(𝐷𝐿𝐸))𝑋)
73	1, 3, 5, 4, 17, 19, 69, 14, 70, 18, 72	hpgtr 28746	. . . . . . . 8 ⊢ (𝜑 → 𝑌((hpG‘𝐺)‘(𝐷𝐿𝐸))𝑋)
74	1, 3, 5, 69, 4, 17, 19, 14, 70	hpgne1 28739	. . . . . . . . . 10 ⊢ (𝜑 → ¬ 𝑌 ∈ (𝐷𝐿𝐸))
75	1, 2, 3, 5, 4, 10, 17, 69, 11, 19, 74	lmiopp 28780	. . . . . . . . 9 ⊢ (𝜑 → 𝑌𝑂(((lInvG‘𝐺)‘(𝐷𝐿𝐸))‘𝑌))
76	1, 3, 5, 69, 4, 17, 19, 18, 26, 75	lnopp2hpgb 28741	. . . . . . . 8 ⊢ (𝜑 → (𝑋𝑂(((lInvG‘𝐺)‘(𝐷𝐿𝐸))‘𝑌) ↔ 𝑌((hpG‘𝐺)‘(𝐷𝐿𝐸))𝑋))
77	73, 76	mpbird 257	. . . . . . 7 ⊢ (𝜑 → 𝑋𝑂(((lInvG‘𝐺)‘(𝐷𝐿𝐸))‘𝑌))
78	1, 2, 3, 69, 18, 26	islnopp 28717	. . . . . . 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 232	. . . . . 6 ⊢ (𝜑 → ((¬ 𝑋 ∈ (𝐷𝐿𝐸) ∧ ¬ (((lInvG‘𝐺)‘(𝐷𝐿𝐸))‘𝑌) ∈ (𝐷𝐿𝐸)) ∧ ∃𝑡 ∈ (𝐷𝐿𝐸)𝑡 ∈ (𝑋𝐼(((lInvG‘𝐺)‘(𝐷𝐿𝐸))‘𝑌))))
80	79	simprd 495	. . . . 5 ⊢ (𝜑 → ∃𝑡 ∈ (𝐷𝐿𝐸)𝑡 ∈ (𝑋𝐼(((lInvG‘𝐺)‘(𝐷𝐿𝐸))‘𝑌)))
81	68, 80	r19.29a 3140	. . . 4 ⊢ (𝜑 → (𝑋(midG‘𝐺)(((lInvG‘𝐺)‘(𝐷𝐿𝐸))‘𝑌)) ∈ (𝐷𝐿𝐸))
82	21	adantr 480	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑡 ∈ (𝐷𝐿𝐸)) ∧ 𝑡 ∈ (𝑋𝐼(((lInvG‘𝐺)‘(𝐷𝐿𝐸))‘𝑌))) ∧ 𝐸 ≠ 𝑡) → 𝐺 ∈ TarskiG)
83	22	adantr 480	. . . . . . . 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 28721	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑋 ≠ (((lInvG‘𝐺)‘(𝐷𝐿𝐸))‘𝑌))
85	1, 3, 5, 4, 18, 26, 84	tgelrnln 28608	. . . . . . . . . 10 ⊢ (𝜑 → (𝑋𝐿(((lInvG‘𝐺)‘(𝐷𝐿𝐸))‘𝑌)) ∈ ran 𝐿)
86	85	ad2antrr 726	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝐷𝐿𝐸)) ∧ 𝑡 ∈ (𝑋𝐼(((lInvG‘𝐺)‘(𝐷𝐿𝐸))‘𝑌))) → (𝑋𝐿(((lInvG‘𝐺)‘(𝐷𝐿𝐸))‘𝑌)) ∈ ran 𝐿)
87	86	adantr 480	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑡 ∈ (𝐷𝐿𝐸)) ∧ 𝑡 ∈ (𝑋𝐼(((lInvG‘𝐺)‘(𝐷𝐿𝐸))‘𝑌))) ∧ 𝐸 ≠ 𝑡) → (𝑋𝐿(((lInvG‘𝐺)‘(𝐷𝐿𝐸))‘𝑌)) ∈ ran 𝐿)
88	84	ad2antrr 726	. . . . . . . . . . 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 28597	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝐷𝐿𝐸)) ∧ 𝑡 ∈ (𝑋𝐼(((lInvG‘𝐺)‘(𝐷𝐿𝐸))‘𝑌))) → 𝑡 ∈ (𝑋𝐿(((lInvG‘𝐺)‘(𝐷𝐿𝐸))‘𝑌)))
90	23, 89	elind 4147	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝐷𝐿𝐸)) ∧ 𝑡 ∈ (𝑋𝐼(((lInvG‘𝐺)‘(𝐷𝐿𝐸))‘𝑌))) → 𝑡 ∈ ((𝐷𝐿𝐸) ∩ (𝑋𝐿(((lInvG‘𝐺)‘(𝐷𝐿𝐸))‘𝑌))))
91	90	adantr 480	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑡 ∈ (𝐷𝐿𝐸)) ∧ 𝑡 ∈ (𝑋𝐼(((lInvG‘𝐺)‘(𝐷𝐿𝐸))‘𝑌))) ∧ 𝐸 ≠ 𝑡) → 𝑡 ∈ ((𝐷𝐿𝐸) ∩ (𝑋𝐿(((lInvG‘𝐺)‘(𝐷𝐿𝐸))‘𝑌))))
92	54	ad3antrrr 730	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑡 ∈ (𝐷𝐿𝐸)) ∧ 𝑡 ∈ (𝑋𝐼(((lInvG‘𝐺)‘(𝐷𝐿𝐸))‘𝑌))) ∧ 𝐸 ≠ 𝑡) → 𝐸 ∈ (𝐷𝐿𝐸))
93	1, 3, 5, 4, 18, 26, 84	tglinerflx1 28611	. . . . . . . . 9 ⊢ (𝜑 → 𝑋 ∈ (𝑋𝐿(((lInvG‘𝐺)‘(𝐷𝐿𝐸))‘𝑌)))
94	93	ad3antrrr 730	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑡 ∈ (𝐷𝐿𝐸)) ∧ 𝑡 ∈ (𝑋𝐼(((lInvG‘𝐺)‘(𝐷𝐿𝐸))‘𝑌))) ∧ 𝐸 ≠ 𝑡) → 𝑋 ∈ (𝑋𝐿(((lInvG‘𝐺)‘(𝐷𝐿𝐸))‘𝑌)))
95		simpr 484	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑡 ∈ (𝐷𝐿𝐸)) ∧ 𝑡 ∈ (𝑋𝐼(((lInvG‘𝐺)‘(𝐷𝐿𝐸))‘𝑌))) ∧ 𝐸 ≠ 𝑡) → 𝐸 ≠ 𝑡)
96	79	simplld 767	. . . . . . . . . . . 12 ⊢ (𝜑 → ¬ 𝑋 ∈ (𝐷𝐿𝐸))
97	96	ad2antrr 726	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝐷𝐿𝐸)) ∧ 𝑡 ∈ (𝑋𝐼(((lInvG‘𝐺)‘(𝐷𝐿𝐸))‘𝑌))) → ¬ 𝑋 ∈ (𝐷𝐿𝐸))
98		nelne2 3026	. . . . . . . . . . 11 ⊢ ((𝑡 ∈ (𝐷𝐿𝐸) ∧ ¬ 𝑋 ∈ (𝐷𝐿𝐸)) → 𝑡 ≠ 𝑋)
99	23, 97, 98	syl2anc 584	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝐷𝐿𝐸)) ∧ 𝑡 ∈ (𝑋𝐼(((lInvG‘𝐺)‘(𝐷𝐿𝐸))‘𝑌))) → 𝑡 ≠ 𝑋)
100	99	necomd 2983	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝐷𝐿𝐸)) ∧ 𝑡 ∈ (𝑋𝐼(((lInvG‘𝐺)‘(𝐷𝐿𝐸))‘𝑌))) → 𝑋 ≠ 𝑡)
101	100	adantr 480	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑡 ∈ (𝐷𝐿𝐸)) ∧ 𝑡 ∈ (𝑋𝐼(((lInvG‘𝐺)‘(𝐷𝐿𝐸))‘𝑌))) ∧ 𝐸 ≠ 𝑡) → 𝑋 ≠ 𝑡)
102	64	oveq2d 7362	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝐷𝐿𝐸)) ∧ 𝑡 ∈ (𝑋𝐼(((lInvG‘𝐺)‘(𝐷𝐿𝐸))‘𝑌))) → (𝐸 − (((lInvG‘𝐺)‘(𝐷𝐿𝐸))‘𝑌)) = (𝐸 − (((pInvG‘𝐺)‘𝑡)‘𝑋)))
103	58, 102	eqtrd 2766	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝐷𝐿𝐸)) ∧ 𝑡 ∈ (𝑋𝐼(((lInvG‘𝐺)‘(𝐷𝐿𝐸))‘𝑌))) → (𝐸 − 𝑋) = (𝐸 − (((pInvG‘𝐺)‘𝑡)‘𝑋)))
104	103	adantr 480	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑡 ∈ (𝐷𝐿𝐸)) ∧ 𝑡 ∈ (𝑋𝐼(((lInvG‘𝐺)‘(𝐷𝐿𝐸))‘𝑌))) ∧ 𝐸 ≠ 𝑡) → (𝐸 − 𝑋) = (𝐸 − (((pInvG‘𝐺)‘𝑡)‘𝑋)))
105	30	adantr 480	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑡 ∈ (𝐷𝐿𝐸)) ∧ 𝑡 ∈ (𝑋𝐼(((lInvG‘𝐺)‘(𝐷𝐿𝐸))‘𝑌))) ∧ 𝐸 ≠ 𝑡) → 𝐸 ∈ 𝑃)
106	24	adantr 480	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑡 ∈ (𝐷𝐿𝐸)) ∧ 𝑡 ∈ (𝑋𝐼(((lInvG‘𝐺)‘(𝐷𝐿𝐸))‘𝑌))) ∧ 𝐸 ≠ 𝑡) → 𝑡 ∈ 𝑃)
107	28	adantr 480	. . . . . . . . . 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 28675	. . . . . . . . 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 257	. . . . . . . 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 28695	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑡 ∈ (𝐷𝐿𝐸)) ∧ 𝑡 ∈ (𝑋𝐼(((lInvG‘𝐺)‘(𝐷𝐿𝐸))‘𝑌))) ∧ 𝐸 ≠ 𝑡) → (𝐷𝐿𝐸)(⟂G‘𝐺)(𝑋𝐿(((lInvG‘𝐺)‘(𝐷𝐿𝐸))‘𝑌)))
111	21	adantr 480	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑡 ∈ (𝐷𝐿𝐸)) ∧ 𝑡 ∈ (𝑋𝐼(((lInvG‘𝐺)‘(𝐷𝐿𝐸))‘𝑌))) ∧ 𝐷 ≠ 𝑡) → 𝐺 ∈ TarskiG)
112	22	adantr 480	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑡 ∈ (𝐷𝐿𝐸)) ∧ 𝑡 ∈ (𝑋𝐼(((lInvG‘𝐺)‘(𝐷𝐿𝐸))‘𝑌))) ∧ 𝐷 ≠ 𝑡) → (𝐷𝐿𝐸) ∈ ran 𝐿)
113	86	adantr 480	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑡 ∈ (𝐷𝐿𝐸)) ∧ 𝑡 ∈ (𝑋𝐼(((lInvG‘𝐺)‘(𝐷𝐿𝐸))‘𝑌))) ∧ 𝐷 ≠ 𝑡) → (𝑋𝐿(((lInvG‘𝐺)‘(𝐷𝐿𝐸))‘𝑌)) ∈ ran 𝐿)
114	90	adantr 480	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑡 ∈ (𝐷𝐿𝐸)) ∧ 𝑡 ∈ (𝑋𝐼(((lInvG‘𝐺)‘(𝐷𝐿𝐸))‘𝑌))) ∧ 𝐷 ≠ 𝑡) → 𝑡 ∈ ((𝐷𝐿𝐸) ∩ (𝑋𝐿(((lInvG‘𝐺)‘(𝐷𝐿𝐸))‘𝑌))))
115	45	ad3antrrr 730	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑡 ∈ (𝐷𝐿𝐸)) ∧ 𝑡 ∈ (𝑋𝐼(((lInvG‘𝐺)‘(𝐷𝐿𝐸))‘𝑌))) ∧ 𝐷 ≠ 𝑡) → 𝐷 ∈ (𝐷𝐿𝐸))
116	93	ad3antrrr 730	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑡 ∈ (𝐷𝐿𝐸)) ∧ 𝑡 ∈ (𝑋𝐼(((lInvG‘𝐺)‘(𝐷𝐿𝐸))‘𝑌))) ∧ 𝐷 ≠ 𝑡) → 𝑋 ∈ (𝑋𝐿(((lInvG‘𝐺)‘(𝐷𝐿𝐸))‘𝑌)))
117		simpr 484	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑡 ∈ (𝐷𝐿𝐸)) ∧ 𝑡 ∈ (𝑋𝐼(((lInvG‘𝐺)‘(𝐷𝐿𝐸))‘𝑌))) ∧ 𝐷 ≠ 𝑡) → 𝐷 ≠ 𝑡)
118	100	adantr 480	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑡 ∈ (𝐷𝐿𝐸)) ∧ 𝑡 ∈ (𝑋𝐼(((lInvG‘𝐺)‘(𝐷𝐿𝐸))‘𝑌))) ∧ 𝐷 ≠ 𝑡) → 𝑋 ≠ 𝑡)
119	64	oveq2d 7362	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝐷𝐿𝐸)) ∧ 𝑡 ∈ (𝑋𝐼(((lInvG‘𝐺)‘(𝐷𝐿𝐸))‘𝑌))) → (𝐷 − (((lInvG‘𝐺)‘(𝐷𝐿𝐸))‘𝑌)) = (𝐷 − (((pInvG‘𝐺)‘𝑡)‘𝑋)))
120	49, 119	eqtrd 2766	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝐷𝐿𝐸)) ∧ 𝑡 ∈ (𝑋𝐼(((lInvG‘𝐺)‘(𝐷𝐿𝐸))‘𝑌))) → (𝐷 − 𝑋) = (𝐷 − (((pInvG‘𝐺)‘𝑡)‘𝑋)))
121	120	adantr 480	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑡 ∈ (𝐷𝐿𝐸)) ∧ 𝑡 ∈ (𝑋𝐼(((lInvG‘𝐺)‘(𝐷𝐿𝐸))‘𝑌))) ∧ 𝐷 ≠ 𝑡) → (𝐷 − 𝑋) = (𝐷 − (((pInvG‘𝐺)‘𝑡)‘𝑋)))
122	29	adantr 480	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑡 ∈ (𝐷𝐿𝐸)) ∧ 𝑡 ∈ (𝑋𝐼(((lInvG‘𝐺)‘(𝐷𝐿𝐸))‘𝑌))) ∧ 𝐷 ≠ 𝑡) → 𝐷 ∈ 𝑃)
123	24	adantr 480	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑡 ∈ (𝐷𝐿𝐸)) ∧ 𝑡 ∈ (𝑋𝐼(((lInvG‘𝐺)‘(𝐷𝐿𝐸))‘𝑌))) ∧ 𝐷 ≠ 𝑡) → 𝑡 ∈ 𝑃)
124	28	adantr 480	. . . . . . . . . 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 28675	. . . . . . . . 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 257	. . . . . . . 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 28695	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑡 ∈ (𝐷𝐿𝐸)) ∧ 𝑡 ∈ (𝑋𝐼(((lInvG‘𝐺)‘(𝐷𝐿𝐸))‘𝑌))) ∧ 𝐷 ≠ 𝑡) → (𝐷𝐿𝐸)(⟂G‘𝐺)(𝑋𝐿(((lInvG‘𝐺)‘(𝐷𝐿𝐸))‘𝑌)))
128		neneor 3028	. . . . . . . 8 ⊢ (𝐸 ≠ 𝐷 → (𝐸 ≠ 𝑡 ∨ 𝐷 ≠ 𝑡))
129	33, 128	syl 17	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝐷𝐿𝐸)) ∧ 𝑡 ∈ (𝑋𝐼(((lInvG‘𝐺)‘(𝐷𝐿𝐸))‘𝑌))) → (𝐸 ≠ 𝑡 ∨ 𝐷 ≠ 𝑡))
130	110, 127, 129	mpjaodan 960	. . . . . 6 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝐷𝐿𝐸)) ∧ 𝑡 ∈ (𝑋𝐼(((lInvG‘𝐺)‘(𝐷𝐿𝐸))‘𝑌))) → (𝐷𝐿𝐸)(⟂G‘𝐺)(𝑋𝐿(((lInvG‘𝐺)‘(𝐷𝐿𝐸))‘𝑌)))
131	130	orcd 873	. . . . 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 3140	. . . 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 28765	. . . 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 713	. . 3 ⊢ (𝜑 → (((lInvG‘𝐺)‘(𝐷𝐿𝐸))‘𝑌) = (((lInvG‘𝐺)‘(𝐷𝐿𝐸))‘𝑋))
135	134	eqcomd 2737	. 2 ⊢ (𝜑 → (((lInvG‘𝐺)‘(𝐷𝐿𝐸))‘𝑋) = (((lInvG‘𝐺)‘(𝐷𝐿𝐸))‘𝑌))
136	1, 2, 3, 4, 10, 11, 5, 17, 18, 19, 135	lmieq 28769	1 ⊢ (𝜑 → 𝑋 = 𝑌)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 ∨ wo 847 = wceq 1541 ∈ wcel 2111 ≠ wne 2928 ∃wrex 3056 ∖ cdif 3894 ∩ cin 3896 class class class wbr 5089 {copab 5151 ran crn 5615 ‘cfv 6481 (class class class)co 7346 2c2 12180 ⟨“cs3 14749 Basecbs 17120 distcds 17170 TarskiGcstrkg 28405 Dim_TarskiG≥cstrkgld 28409 Itvcitv 28411 LineGclng 28412 cgrGccgrg 28488 hlGchlg 28578 pInvGcmir 28630 ∟Gcrag 28671 ⟂Gcperpg 28673 hpGchpg 28735 midGcmid 28750 lInvGclmi 28751

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 1968 ax-7 2009 ax-8 2113 ax-9 2121 ax-10 2144 ax-11 2160 ax-12 2180 ax-ext 2703 ax-rep 5215 ax-sep 5232 ax-nul 5242 ax-pow 5301 ax-pr 5368 ax-un 7668 ax-cnex 11062 ax-resscn 11063 ax-1cn 11064 ax-icn 11065 ax-addcl 11066 ax-addrcl 11067 ax-mulcl 11068 ax-mulrcl 11069 ax-mulcom 11070 ax-addass 11071 ax-mulass 11072 ax-distr 11073 ax-i2m1 11074 ax-1ne0 11075 ax-1rid 11076 ax-rnegex 11077 ax-rrecex 11078 ax-cnre 11079 ax-pre-lttri 11080 ax-pre-lttrn 11081 ax-pre-ltadd 11082 ax-pre-mulgt0 11083

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2535 df-eu 2564 df-clab 2710 df-cleq 2723 df-clel 2806 df-nfc 2881 df-ne 2929 df-nel 3033 df-ral 3048 df-rex 3057 df-rmo 3346 df-reu 3347 df-rab 3396 df-v 3438 df-sbc 3737 df-csb 3846 df-dif 3900 df-un 3902 df-in 3904 df-ss 3914 df-pss 3917 df-nul 4281 df-if 4473 df-pw 4549 df-sn 4574 df-pr 4576 df-tp 4578 df-op 4580 df-uni 4857 df-int 4896 df-iun 4941 df-br 5090 df-opab 5152 df-mpt 5171 df-tr 5197 df-id 5509 df-eprel 5514 df-po 5522 df-so 5523 df-fr 5567 df-we 5569 df-xp 5620 df-rel 5621 df-cnv 5622 df-co 5623 df-dm 5624 df-rn 5625 df-res 5626 df-ima 5627 df-pred 6248 df-ord 6309 df-on 6310 df-lim 6311 df-suc 6312 df-iota 6437 df-fun 6483 df-fn 6484 df-f 6485 df-f1 6486 df-fo 6487 df-f1o 6488 df-fv 6489 df-riota 7303 df-ov 7349 df-oprab 7350 df-mpo 7351 df-om 7797 df-1st 7921 df-2nd 7922 df-frecs 8211 df-wrecs 8242 df-recs 8291 df-rdg 8329 df-1o 8385 df-oadd 8389 df-er 8622 df-map 8752 df-pm 8753 df-en 8870 df-dom 8871 df-sdom 8872 df-fin 8873 df-dju 9794 df-card 9832 df-pnf 11148 df-mnf 11149 df-xr 11150 df-ltxr 11151 df-le 11152 df-sub 11346 df-neg 11347 df-nn 12126 df-2 12188 df-3 12189 df-n0 12382 df-xnn0 12455 df-z 12469 df-uz 12733 df-fz 13408 df-fzo 13555 df-hash 14238 df-word 14421 df-concat 14478 df-s1 14504 df-s2 14755 df-s3 14756 df-trkgc 28426 df-trkgb 28427 df-trkgcb 28428 df-trkgld 28430 df-trkg 28431 df-cgrg 28489 df-leg 28561 df-hlg 28579 df-mir 28631 df-rag 28672 df-perpg 28674 df-hpg 28736 df-mid 28752 df-lmi 28753

This theorem is referenced by: trgcopyeu 28784 acopyeu 28812

W3C validator