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Theorem angmndaddeu1 29250

Description: There exists a unique point 𝑠 satisfying the conditions of angle addition. General case. (Contributed by Thierry Arnoux, 23-Aug-2026.)

**Hypotheses**
Ref	Expression
angmndadd.p	⊢ 𝑃 = (Base‘𝐺)
angmndadd.a	⊢ 𝐴 = {𝑑 ∈ (𝑃 ↑_m (0..^3)) ∣ ((𝑑‘0) ≠ (𝑑‘1) ∧ (𝑑‘1) ≠ (𝑑‘2))}
angmndadd.i	⊢ 𝐼 = (Itv‘𝐺)
angmndadd.d	⊢ − = (dist‘𝐺)
angmndadd.c	⊢ ∼ = (cgrA‘𝐺)
angmndadd.l	⊢ 𝐿 = (LineG‘𝐺)
angmndadd.g	⊢ (𝜑 → 𝐺 ∈ TarskiG)
angmndaddov.u	⊢ (𝜑 → 𝑈 ∈ 𝑃)
angmndaddov.v	⊢ (𝜑 → 𝑉 ∈ 𝑃)
angmndaddov.w	⊢ (𝜑 → 𝑊 ∈ 𝑃)
angmndaddov.x	⊢ (𝜑 → 𝑋 ∈ 𝑃)
angmndaddov.y	⊢ (𝜑 → 𝑌 ∈ 𝑃)
angmndaddov.z	⊢ (𝜑 → 𝑍 ∈ 𝑃)
angmndaddeu.1	⊢ (𝜑 → 𝑈 ≠ 𝑉)
angmndaddeu.2	⊢ (𝜑 → 𝑉 ≠ 𝑊)
angmndaddeu.3	⊢ (𝜑 → 𝑋 ≠ 𝑌)
angmndaddeu.4	⊢ (𝜑 → 𝑌 ≠ 𝑍)
angmndaddeu1.1	⊢ (𝜑 → ¬ 𝑋 ∈ (𝑌𝐿𝑍))
angmndaddeu1.2	⊢ (𝜑 → ¬ 𝑈 ∈ (𝑉𝐿𝑊))

**Assertion**
Ref	Expression
angmndaddeu1	⊢ (𝜑 → ∃!𝑠 ∈ 𝑃 (⟨“𝑍𝑌𝑠”⟩ ∼ ⟨“𝑈𝑉𝑊”⟩ ∧ (𝑌 − 𝑠) = (𝑉 − 𝑈) ∧ ((𝑌𝐿𝑍) ∩ (𝑠𝐼𝑋)) ≠ ∅))

Distinct variable groups: − ,𝑑,𝑠 𝐺,𝑑,𝑠 𝐼,𝑑,𝑠 𝐿,𝑑,𝑠 𝑃,𝑑,𝑠 𝑈,𝑑,𝑠 𝑉,𝑑,𝑠 𝑊,𝑑,𝑠 𝑋,𝑑,𝑠 𝑌,𝑑,𝑠 𝑍,𝑑,𝑠 𝜑,𝑑,𝑠 Allowed substitution hints: 𝐴(𝑠, 𝑑) ∼ (𝑠, 𝑑)

Proof of Theorem angmndaddeu1 Dummy variables 𝑐 𝑡 𝑤 𝑟 𝑎 𝑏 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		angmndadd.p	. . . . 5 ⊢ 𝑃 = (Base‘𝐺)
2		angmndadd.d	. . . . 5 ⊢ − = (dist‘𝐺)
3		angmndadd.i	. . . . 5 ⊢ 𝐼 = (Itv‘𝐺)
4		angmndadd.l	. . . . 5 ⊢ 𝐿 = (LineG‘𝐺)
5		eqid 2762	. . . . 5 ⊢ (hlG‘𝐺) = (hlG‘𝐺)
6		angmndadd.g	. . . . . 6 ⊢ (𝜑 → 𝐺 ∈ TarskiG)
7	6	ad3antrrr 743	. . . . 5 ⊢ ((((𝜑 ∧ 𝑤 ∈ 𝑃) ∧ 𝑤((hlG‘𝐺)‘𝑉)𝑊) ∧ (𝑉 − 𝑤) = (𝑌 − 𝑍)) → 𝐺 ∈ TarskiG)
8		simpllr 788	. . . . 5 ⊢ ((((𝜑 ∧ 𝑤 ∈ 𝑃) ∧ 𝑤((hlG‘𝐺)‘𝑉)𝑊) ∧ (𝑉 − 𝑤) = (𝑌 − 𝑍)) → 𝑤 ∈ 𝑃)
9		angmndaddov.v	. . . . . 6 ⊢ (𝜑 → 𝑉 ∈ 𝑃)
10	9	ad3antrrr 743	. . . . 5 ⊢ ((((𝜑 ∧ 𝑤 ∈ 𝑃) ∧ 𝑤((hlG‘𝐺)‘𝑉)𝑊) ∧ (𝑉 − 𝑤) = (𝑌 − 𝑍)) → 𝑉 ∈ 𝑃)
11		angmndaddov.u	. . . . . 6 ⊢ (𝜑 → 𝑈 ∈ 𝑃)
12	11	ad3antrrr 743	. . . . 5 ⊢ ((((𝜑 ∧ 𝑤 ∈ 𝑃) ∧ 𝑤((hlG‘𝐺)‘𝑉)𝑊) ∧ (𝑉 − 𝑤) = (𝑌 − 𝑍)) → 𝑈 ∈ 𝑃)
13		angmndaddov.z	. . . . . 6 ⊢ (𝜑 → 𝑍 ∈ 𝑃)
14	13	ad3antrrr 743	. . . . 5 ⊢ ((((𝜑 ∧ 𝑤 ∈ 𝑃) ∧ 𝑤((hlG‘𝐺)‘𝑉)𝑊) ∧ (𝑉 − 𝑤) = (𝑌 − 𝑍)) → 𝑍 ∈ 𝑃)
15		angmndaddov.y	. . . . . 6 ⊢ (𝜑 → 𝑌 ∈ 𝑃)
16	15	ad3antrrr 743	. . . . 5 ⊢ ((((𝜑 ∧ 𝑤 ∈ 𝑃) ∧ 𝑤((hlG‘𝐺)‘𝑉)𝑊) ∧ (𝑉 − 𝑤) = (𝑌 − 𝑍)) → 𝑌 ∈ 𝑃)
17		angmndaddov.w	. . . . . . . 8 ⊢ (𝜑 → 𝑊 ∈ 𝑃)
18		angmndaddeu1.2	. . . . . . . . . 10 ⊢ (𝜑 → ¬ 𝑈 ∈ (𝑉𝐿𝑊))
19		angmndaddeu.2	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑉 ≠ 𝑊)
20	19	neneqd 2962	. . . . . . . . . 10 ⊢ (𝜑 → ¬ 𝑉 = 𝑊)
21		ioran 999	. . . . . . . . . 10 ⊢ (¬ (𝑈 ∈ (𝑉𝐿𝑊) ∨ 𝑉 = 𝑊) ↔ (¬ 𝑈 ∈ (𝑉𝐿𝑊) ∧ ¬ 𝑉 = 𝑊))
22	18, 20, 21	sylanbrc 595	. . . . . . . . 9 ⊢ (𝜑 → ¬ (𝑈 ∈ (𝑉𝐿𝑊) ∨ 𝑉 = 𝑊))
23	1, 4, 3, 6, 9, 17, 11, 22	ncolrot2 28901	. . . . . . . 8 ⊢ (𝜑 → ¬ (𝑊 ∈ (𝑈𝐿𝑉) ∨ 𝑈 = 𝑉))
24	1, 4, 3, 6, 11, 9, 17, 23	ncoltgdim2 28903	. . . . . . 7 ⊢ (𝜑 → 𝐺Dim_TarskiG≥2)
25		eqid 2762	. . . . . . 7 ⊢ ((lInvG‘𝐺)‘(𝑌𝐿𝑍)) = ((lInvG‘𝐺)‘(𝑌𝐿𝑍))
26		angmndaddeu.4	. . . . . . . 8 ⊢ (𝜑 → 𝑌 ≠ 𝑍)
27	1, 3, 4, 6, 15, 13, 26	tgelrnln 28973	. . . . . . 7 ⊢ (𝜑 → (𝑌𝐿𝑍) ∈ ran 𝐿)
28		angmndaddov.x	. . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ 𝑃)
29	1, 2, 3, 6, 24, 25, 4, 27, 28	lmicl 29166	. . . . . 6 ⊢ (𝜑 → (((lInvG‘𝐺)‘(𝑌𝐿𝑍))‘𝑋) ∈ 𝑃)
30	29	ad3antrrr 743	. . . . 5 ⊢ ((((𝜑 ∧ 𝑤 ∈ 𝑃) ∧ 𝑤((hlG‘𝐺)‘𝑉)𝑊) ∧ (𝑉 − 𝑤) = (𝑌 − 𝑍)) → (((lInvG‘𝐺)‘(𝑌𝐿𝑍))‘𝑋) ∈ 𝑃)
31	23	ad3antrrr 743	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑤 ∈ 𝑃) ∧ 𝑤((hlG‘𝐺)‘𝑉)𝑊) ∧ (𝑉 − 𝑤) = (𝑌 − 𝑍)) → ¬ (𝑊 ∈ (𝑈𝐿𝑉) ∨ 𝑈 = 𝑉))
32	7	adantr 486	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑤 ∈ 𝑃) ∧ 𝑤((hlG‘𝐺)‘𝑉)𝑊) ∧ (𝑉 − 𝑤) = (𝑌 − 𝑍)) ∧ (𝑤 ∈ (𝑉𝐿𝑈) ∨ 𝑉 = 𝑈)) → 𝐺 ∈ TarskiG)
33	10	adantr 486	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑤 ∈ 𝑃) ∧ 𝑤((hlG‘𝐺)‘𝑉)𝑊) ∧ (𝑉 − 𝑤) = (𝑌 − 𝑍)) ∧ (𝑤 ∈ (𝑉𝐿𝑈) ∨ 𝑉 = 𝑈)) → 𝑉 ∈ 𝑃)
34	12	adantr 486	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑤 ∈ 𝑃) ∧ 𝑤((hlG‘𝐺)‘𝑉)𝑊) ∧ (𝑉 − 𝑤) = (𝑌 − 𝑍)) ∧ (𝑤 ∈ (𝑉𝐿𝑈) ∨ 𝑉 = 𝑈)) → 𝑈 ∈ 𝑃)
35		angmndaddeu.1	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑈 ≠ 𝑉)
36	35	neneqd 2962	. . . . . . . . . . . 12 ⊢ (𝜑 → ¬ 𝑈 = 𝑉)
37	36	neqcomd 2772	. . . . . . . . . . 11 ⊢ (𝜑 → ¬ 𝑉 = 𝑈)
38	37	neqned 2964	. . . . . . . . . 10 ⊢ (𝜑 → 𝑉 ≠ 𝑈)
39	38	ad4antr 745	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑤 ∈ 𝑃) ∧ 𝑤((hlG‘𝐺)‘𝑉)𝑊) ∧ (𝑉 − 𝑤) = (𝑌 − 𝑍)) ∧ (𝑤 ∈ (𝑉𝐿𝑈) ∨ 𝑉 = 𝑈)) → 𝑉 ≠ 𝑈)
40	8	adantr 486	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑤 ∈ 𝑃) ∧ 𝑤((hlG‘𝐺)‘𝑉)𝑊) ∧ (𝑉 − 𝑤) = (𝑌 − 𝑍)) ∧ (𝑤 ∈ (𝑉𝐿𝑈) ∨ 𝑉 = 𝑈)) → 𝑤 ∈ 𝑃)
41		simpr 490	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑤 ∈ 𝑃) ∧ 𝑤((hlG‘𝐺)‘𝑉)𝑊) ∧ (𝑉 − 𝑤) = (𝑌 − 𝑍)) → (𝑉 − 𝑤) = (𝑌 − 𝑍))
42	41	eqcomd 2768	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑤 ∈ 𝑃) ∧ 𝑤((hlG‘𝐺)‘𝑉)𝑊) ∧ (𝑉 − 𝑤) = (𝑌 − 𝑍)) → (𝑌 − 𝑍) = (𝑉 − 𝑤))
43	26	ad3antrrr 743	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑤 ∈ 𝑃) ∧ 𝑤((hlG‘𝐺)‘𝑉)𝑊) ∧ (𝑉 − 𝑤) = (𝑌 − 𝑍)) → 𝑌 ≠ 𝑍)
44	1, 2, 3, 7, 16, 14, 10, 8, 42, 43	tgcgrneq 28820	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑤 ∈ 𝑃) ∧ 𝑤((hlG‘𝐺)‘𝑉)𝑊) ∧ (𝑉 − 𝑤) = (𝑌 − 𝑍)) → 𝑉 ≠ 𝑤)
45	44	necomd 3012	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑤 ∈ 𝑃) ∧ 𝑤((hlG‘𝐺)‘𝑉)𝑊) ∧ (𝑉 − 𝑤) = (𝑌 − 𝑍)) → 𝑤 ≠ 𝑉)
46	45	adantr 486	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑤 ∈ 𝑃) ∧ 𝑤((hlG‘𝐺)‘𝑉)𝑊) ∧ (𝑉 − 𝑤) = (𝑌 − 𝑍)) ∧ (𝑤 ∈ (𝑉𝐿𝑈) ∨ 𝑉 = 𝑈)) → 𝑤 ≠ 𝑉)
47		simpr 490	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑤 ∈ 𝑃) ∧ 𝑤((hlG‘𝐺)‘𝑉)𝑊) ∧ (𝑉 − 𝑤) = (𝑌 − 𝑍)) ∧ (𝑤 ∈ (𝑉𝐿𝑈) ∨ 𝑉 = 𝑈)) → (𝑤 ∈ (𝑉𝐿𝑈) ∨ 𝑉 = 𝑈))
48	37	ad4antr 745	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑤 ∈ 𝑃) ∧ 𝑤((hlG‘𝐺)‘𝑉)𝑊) ∧ (𝑉 − 𝑤) = (𝑌 − 𝑍)) ∧ (𝑤 ∈ (𝑉𝐿𝑈) ∨ 𝑉 = 𝑈)) → ¬ 𝑉 = 𝑈)
49	47, 48	olcnd 891	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑤 ∈ 𝑃) ∧ 𝑤((hlG‘𝐺)‘𝑉)𝑊) ∧ (𝑉 − 𝑤) = (𝑌 − 𝑍)) ∧ (𝑤 ∈ (𝑉𝐿𝑈) ∨ 𝑉 = 𝑈)) → 𝑤 ∈ (𝑉𝐿𝑈))
50	17	ad4antr 745	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑤 ∈ 𝑃) ∧ 𝑤((hlG‘𝐺)‘𝑉)𝑊) ∧ (𝑉 − 𝑤) = (𝑌 − 𝑍)) ∧ (𝑤 ∈ (𝑉𝐿𝑈) ∨ 𝑉 = 𝑈)) → 𝑊 ∈ 𝑃)
51	17	ad3antrrr 743	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑤 ∈ 𝑃) ∧ 𝑤((hlG‘𝐺)‘𝑉)𝑊) ∧ (𝑉 − 𝑤) = (𝑌 − 𝑍)) → 𝑊 ∈ 𝑃)
52		simplr 781	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑤 ∈ 𝑃) ∧ 𝑤((hlG‘𝐺)‘𝑉)𝑊) ∧ (𝑉 − 𝑤) = (𝑌 − 𝑍)) → 𝑤((hlG‘𝐺)‘𝑉)𝑊)
53	1, 3, 5, 8, 51, 10, 7, 52	hlcomd 28945	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑤 ∈ 𝑃) ∧ 𝑤((hlG‘𝐺)‘𝑉)𝑊) ∧ (𝑉 − 𝑤) = (𝑌 − 𝑍)) → 𝑊((hlG‘𝐺)‘𝑉)𝑤)
54	1, 3, 5, 51, 8, 10, 7, 4, 53	hlln 28948	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑤 ∈ 𝑃) ∧ 𝑤((hlG‘𝐺)‘𝑉)𝑊) ∧ (𝑉 − 𝑤) = (𝑌 − 𝑍)) → 𝑊 ∈ (𝑤𝐿𝑉))
55	1, 3, 4, 7, 10, 8, 51, 44, 54	lncom 28965	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑤 ∈ 𝑃) ∧ 𝑤((hlG‘𝐺)‘𝑉)𝑊) ∧ (𝑉 − 𝑤) = (𝑌 − 𝑍)) → 𝑊 ∈ (𝑉𝐿𝑤))
56	55	adantr 486	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑤 ∈ 𝑃) ∧ 𝑤((hlG‘𝐺)‘𝑉)𝑊) ∧ (𝑉 − 𝑤) = (𝑌 − 𝑍)) ∧ (𝑤 ∈ (𝑉𝐿𝑈) ∨ 𝑉 = 𝑈)) → 𝑊 ∈ (𝑉𝐿𝑤))
57	1, 3, 4, 32, 33, 34, 39, 40, 46, 49, 50, 56	tglineeltr 28974	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑤 ∈ 𝑃) ∧ 𝑤((hlG‘𝐺)‘𝑉)𝑊) ∧ (𝑉 − 𝑤) = (𝑌 − 𝑍)) ∧ (𝑤 ∈ (𝑉𝐿𝑈) ∨ 𝑉 = 𝑈)) → 𝑊 ∈ (𝑉𝐿𝑈))
58	1, 3, 4, 6, 9, 11, 38	tglinecom 28978	. . . . . . . . 9 ⊢ (𝜑 → (𝑉𝐿𝑈) = (𝑈𝐿𝑉))
59	58	ad4antr 745	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑤 ∈ 𝑃) ∧ 𝑤((hlG‘𝐺)‘𝑉)𝑊) ∧ (𝑉 − 𝑤) = (𝑌 − 𝑍)) ∧ (𝑤 ∈ (𝑉𝐿𝑈) ∨ 𝑉 = 𝑈)) → (𝑉𝐿𝑈) = (𝑈𝐿𝑉))
60	57, 59	eleqtrd 2864	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝑤 ∈ 𝑃) ∧ 𝑤((hlG‘𝐺)‘𝑉)𝑊) ∧ (𝑉 − 𝑤) = (𝑌 − 𝑍)) ∧ (𝑤 ∈ (𝑉𝐿𝑈) ∨ 𝑉 = 𝑈)) → 𝑊 ∈ (𝑈𝐿𝑉))
61	60	orcd 887	. . . . . 6 ⊢ (((((𝜑 ∧ 𝑤 ∈ 𝑃) ∧ 𝑤((hlG‘𝐺)‘𝑉)𝑊) ∧ (𝑉 − 𝑤) = (𝑌 − 𝑍)) ∧ (𝑤 ∈ (𝑉𝐿𝑈) ∨ 𝑉 = 𝑈)) → (𝑊 ∈ (𝑈𝐿𝑉) ∨ 𝑈 = 𝑉))
62	31, 61	mtand 828	. . . . 5 ⊢ ((((𝜑 ∧ 𝑤 ∈ 𝑃) ∧ 𝑤((hlG‘𝐺)‘𝑉)𝑊) ∧ (𝑉 − 𝑤) = (𝑌 − 𝑍)) → ¬ (𝑤 ∈ (𝑉𝐿𝑈) ∨ 𝑉 = 𝑈))
63		eleq1 2850	. . . . . . . . . . . . . 14 ⊢ (𝑎 = 𝑐 → (𝑎 ∈ (𝑃 ∖ (𝑌𝐿𝑍)) ↔ 𝑐 ∈ (𝑃 ∖ (𝑌𝐿𝑍))))
64	63	adantr 486	. . . . . . . . . . . . 13 ⊢ ((𝑎 = 𝑐 ∧ 𝑏 = 𝑑) → (𝑎 ∈ (𝑃 ∖ (𝑌𝐿𝑍)) ↔ 𝑐 ∈ (𝑃 ∖ (𝑌𝐿𝑍))))
65		eleq1 2850	. . . . . . . . . . . . . 14 ⊢ (𝑏 = 𝑑 → (𝑏 ∈ (𝑃 ∖ (𝑌𝐿𝑍)) ↔ 𝑑 ∈ (𝑃 ∖ (𝑌𝐿𝑍))))
66	65	adantl 487	. . . . . . . . . . . . 13 ⊢ ((𝑎 = 𝑐 ∧ 𝑏 = 𝑑) → (𝑏 ∈ (𝑃 ∖ (𝑌𝐿𝑍)) ↔ 𝑑 ∈ (𝑃 ∖ (𝑌𝐿𝑍))))
67	64, 66	anbi12d 644	. . . . . . . . . . . 12 ⊢ ((𝑎 = 𝑐 ∧ 𝑏 = 𝑑) → ((𝑎 ∈ (𝑃 ∖ (𝑌𝐿𝑍)) ∧ 𝑏 ∈ (𝑃 ∖ (𝑌𝐿𝑍))) ↔ (𝑐 ∈ (𝑃 ∖ (𝑌𝐿𝑍)) ∧ 𝑑 ∈ (𝑃 ∖ (𝑌𝐿𝑍)))))
68		oveq12 7425	. . . . . . . . . . . . . . 15 ⊢ ((𝑎 = 𝑐 ∧ 𝑏 = 𝑑) → (𝑎𝐼𝑏) = (𝑐𝐼𝑑))
69	68	eleq2d 2848	. . . . . . . . . . . . . 14 ⊢ ((𝑎 = 𝑐 ∧ 𝑏 = 𝑑) → (𝑠 ∈ (𝑎𝐼𝑏) ↔ 𝑠 ∈ (𝑐𝐼𝑑)))
70	69	rexbidv 3188	. . . . . . . . . . . . 13 ⊢ ((𝑎 = 𝑐 ∧ 𝑏 = 𝑑) → (∃𝑠 ∈ (𝑌𝐿𝑍)𝑠 ∈ (𝑎𝐼𝑏) ↔ ∃𝑠 ∈ (𝑌𝐿𝑍)𝑠 ∈ (𝑐𝐼𝑑)))
71		eleq1 2850	. . . . . . . . . . . . . 14 ⊢ (𝑠 = 𝑡 → (𝑠 ∈ (𝑐𝐼𝑑) ↔ 𝑡 ∈ (𝑐𝐼𝑑)))
72	71	cbvrexvw 3243	. . . . . . . . . . . . 13 ⊢ (∃𝑠 ∈ (𝑌𝐿𝑍)𝑠 ∈ (𝑐𝐼𝑑) ↔ ∃𝑡 ∈ (𝑌𝐿𝑍)𝑡 ∈ (𝑐𝐼𝑑))
73	70, 72	bitrdi 290	. . . . . . . . . . . 12 ⊢ ((𝑎 = 𝑐 ∧ 𝑏 = 𝑑) → (∃𝑠 ∈ (𝑌𝐿𝑍)𝑠 ∈ (𝑎𝐼𝑏) ↔ ∃𝑡 ∈ (𝑌𝐿𝑍)𝑡 ∈ (𝑐𝐼𝑑)))
74	67, 73	anbi12d 644	. . . . . . . . . . 11 ⊢ ((𝑎 = 𝑐 ∧ 𝑏 = 𝑑) → (((𝑎 ∈ (𝑃 ∖ (𝑌𝐿𝑍)) ∧ 𝑏 ∈ (𝑃 ∖ (𝑌𝐿𝑍))) ∧ ∃𝑠 ∈ (𝑌𝐿𝑍)𝑠 ∈ (𝑎𝐼𝑏)) ↔ ((𝑐 ∈ (𝑃 ∖ (𝑌𝐿𝑍)) ∧ 𝑑 ∈ (𝑃 ∖ (𝑌𝐿𝑍))) ∧ ∃𝑡 ∈ (𝑌𝐿𝑍)𝑡 ∈ (𝑐𝐼𝑑))))
75	74	cbvopabv 5182	. . . . . . . . . 10 ⊢ {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝑃 ∖ (𝑌𝐿𝑍)) ∧ 𝑏 ∈ (𝑃 ∖ (𝑌𝐿𝑍))) ∧ ∃𝑠 ∈ (𝑌𝐿𝑍)𝑠 ∈ (𝑎𝐼𝑏))} = {⟨𝑐, 𝑑⟩ ∣ ((𝑐 ∈ (𝑃 ∖ (𝑌𝐿𝑍)) ∧ 𝑑 ∈ (𝑃 ∖ (𝑌𝐿𝑍))) ∧ ∃𝑡 ∈ (𝑌𝐿𝑍)𝑡 ∈ (𝑐𝐼𝑑))}
76		angmndaddeu1.1	. . . . . . . . . . 11 ⊢ (𝜑 → ¬ 𝑋 ∈ (𝑌𝐿𝑍))
77	1, 2, 3, 4, 6, 24, 27, 75, 25, 28, 76	lmiopp 29183	. . . . . . . . . 10 ⊢ (𝜑 → 𝑋{⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝑃 ∖ (𝑌𝐿𝑍)) ∧ 𝑏 ∈ (𝑃 ∖ (𝑌𝐿𝑍))) ∧ ∃𝑠 ∈ (𝑌𝐿𝑍)𝑠 ∈ (𝑎𝐼𝑏))} (((lInvG‘𝐺)‘(𝑌𝐿𝑍))‘𝑋))
78	1, 2, 3, 75, 4, 27, 6, 28, 29, 77	oppne2 29093	. . . . . . . . 9 ⊢ (𝜑 → ¬ (((lInvG‘𝐺)‘(𝑌𝐿𝑍))‘𝑋) ∈ (𝑌𝐿𝑍))
79	1, 3, 4, 6, 15, 13, 26	tglinecom 28978	. . . . . . . . 9 ⊢ (𝜑 → (𝑌𝐿𝑍) = (𝑍𝐿𝑌))
80	78, 79	neleqtrd 2884	. . . . . . . 8 ⊢ (𝜑 → ¬ (((lInvG‘𝐺)‘(𝑌𝐿𝑍))‘𝑋) ∈ (𝑍𝐿𝑌))
81	26	necomd 3012	. . . . . . . . 9 ⊢ (𝜑 → 𝑍 ≠ 𝑌)
82	81	neneqd 2962	. . . . . . . 8 ⊢ (𝜑 → ¬ 𝑍 = 𝑌)
83		ioran 999	. . . . . . . 8 ⊢ (¬ ((((lInvG‘𝐺)‘(𝑌𝐿𝑍))‘𝑋) ∈ (𝑍𝐿𝑌) ∨ 𝑍 = 𝑌) ↔ (¬ (((lInvG‘𝐺)‘(𝑌𝐿𝑍))‘𝑋) ∈ (𝑍𝐿𝑌) ∧ ¬ 𝑍 = 𝑌))
84	80, 82, 83	sylanbrc 595	. . . . . . 7 ⊢ (𝜑 → ¬ ((((lInvG‘𝐺)‘(𝑌𝐿𝑍))‘𝑋) ∈ (𝑍𝐿𝑌) ∨ 𝑍 = 𝑌))
85	1, 4, 3, 6, 13, 15, 29, 84	ncolrot1 28900	. . . . . 6 ⊢ (𝜑 → ¬ (𝑍 ∈ (𝑌𝐿(((lInvG‘𝐺)‘(𝑌𝐿𝑍))‘𝑋)) ∨ 𝑌 = (((lInvG‘𝐺)‘(𝑌𝐿𝑍))‘𝑋)))
86	85	ad3antrrr 743	. . . . 5 ⊢ ((((𝜑 ∧ 𝑤 ∈ 𝑃) ∧ 𝑤((hlG‘𝐺)‘𝑉)𝑊) ∧ (𝑉 − 𝑤) = (𝑌 − 𝑍)) → ¬ (𝑍 ∈ (𝑌𝐿(((lInvG‘𝐺)‘(𝑌𝐿𝑍))‘𝑋)) ∨ 𝑌 = (((lInvG‘𝐺)‘(𝑌𝐿𝑍))‘𝑋)))
87	1, 2, 3, 7, 10, 8, 16, 14, 41	tgcgrcomlr 28817	. . . . 5 ⊢ ((((𝜑 ∧ 𝑤 ∈ 𝑃) ∧ 𝑤((hlG‘𝐺)‘𝑉)𝑊) ∧ (𝑉 − 𝑤) = (𝑌 − 𝑍)) → (𝑤 − 𝑉) = (𝑍 − 𝑌))
88	1, 2, 3, 4, 5, 7, 8, 10, 12, 14, 16, 30, 62, 86, 87	trgcopyeu 29188	. . . 4 ⊢ ((((𝜑 ∧ 𝑤 ∈ 𝑃) ∧ 𝑤((hlG‘𝐺)‘𝑉)𝑊) ∧ (𝑉 − 𝑤) = (𝑌 − 𝑍)) → ∃!𝑠 ∈ 𝑃 (⟨“𝑤𝑉𝑈”⟩(cgrG‘𝐺)⟨“𝑍𝑌𝑠”⟩ ∧ 𝑠((hpG‘𝐺)‘(𝑍𝐿𝑌))(((lInvG‘𝐺)‘(𝑌𝐿𝑍))‘𝑋)))
89		angmndadd.c	. . . . . . . . . . 11 ⊢ ∼ = (cgrA‘𝐺)
90	89	eqcomi 2771	. . . . . . . . . 10 ⊢ (cgrA‘𝐺) = ∼
91	90	a1i 11	. . . . . . . . 9 ⊢ (((((((𝜑 ∧ 𝑤 ∈ 𝑃) ∧ 𝑤((hlG‘𝐺)‘𝑉)𝑊) ∧ (𝑉 − 𝑤) = (𝑌 − 𝑍)) ∧ 𝑠 ∈ 𝑃) ∧ ⟨“𝑤𝑉𝑈”⟩(cgrG‘𝐺)⟨“𝑍𝑌𝑠”⟩) ∧ 𝑠((hpG‘𝐺)‘(𝑍𝐿𝑌))(((lInvG‘𝐺)‘(𝑌𝐿𝑍))‘𝑋)) → (cgrA‘𝐺) = ∼ )
92	7	ad3antrrr 743	. . . . . . . . . 10 ⊢ (((((((𝜑 ∧ 𝑤 ∈ 𝑃) ∧ 𝑤((hlG‘𝐺)‘𝑉)𝑊) ∧ (𝑉 − 𝑤) = (𝑌 − 𝑍)) ∧ 𝑠 ∈ 𝑃) ∧ ⟨“𝑤𝑉𝑈”⟩(cgrG‘𝐺)⟨“𝑍𝑌𝑠”⟩) ∧ 𝑠((hpG‘𝐺)‘(𝑍𝐿𝑌))(((lInvG‘𝐺)‘(𝑌𝐿𝑍))‘𝑋)) → 𝐺 ∈ TarskiG)
93	14	ad3antrrr 743	. . . . . . . . . 10 ⊢ (((((((𝜑 ∧ 𝑤 ∈ 𝑃) ∧ 𝑤((hlG‘𝐺)‘𝑉)𝑊) ∧ (𝑉 − 𝑤) = (𝑌 − 𝑍)) ∧ 𝑠 ∈ 𝑃) ∧ ⟨“𝑤𝑉𝑈”⟩(cgrG‘𝐺)⟨“𝑍𝑌𝑠”⟩) ∧ 𝑠((hpG‘𝐺)‘(𝑍𝐿𝑌))(((lInvG‘𝐺)‘(𝑌𝐿𝑍))‘𝑋)) → 𝑍 ∈ 𝑃)
94	16	ad3antrrr 743	. . . . . . . . . 10 ⊢ (((((((𝜑 ∧ 𝑤 ∈ 𝑃) ∧ 𝑤((hlG‘𝐺)‘𝑉)𝑊) ∧ (𝑉 − 𝑤) = (𝑌 − 𝑍)) ∧ 𝑠 ∈ 𝑃) ∧ ⟨“𝑤𝑉𝑈”⟩(cgrG‘𝐺)⟨“𝑍𝑌𝑠”⟩) ∧ 𝑠((hpG‘𝐺)‘(𝑍𝐿𝑌))(((lInvG‘𝐺)‘(𝑌𝐿𝑍))‘𝑋)) → 𝑌 ∈ 𝑃)
95		simpllr 788	. . . . . . . . . 10 ⊢ (((((((𝜑 ∧ 𝑤 ∈ 𝑃) ∧ 𝑤((hlG‘𝐺)‘𝑉)𝑊) ∧ (𝑉 − 𝑤) = (𝑌 − 𝑍)) ∧ 𝑠 ∈ 𝑃) ∧ ⟨“𝑤𝑉𝑈”⟩(cgrG‘𝐺)⟨“𝑍𝑌𝑠”⟩) ∧ 𝑠((hpG‘𝐺)‘(𝑍𝐿𝑌))(((lInvG‘𝐺)‘(𝑌𝐿𝑍))‘𝑋)) → 𝑠 ∈ 𝑃)
96	12	ad3antrrr 743	. . . . . . . . . 10 ⊢ (((((((𝜑 ∧ 𝑤 ∈ 𝑃) ∧ 𝑤((hlG‘𝐺)‘𝑉)𝑊) ∧ (𝑉 − 𝑤) = (𝑌 − 𝑍)) ∧ 𝑠 ∈ 𝑃) ∧ ⟨“𝑤𝑉𝑈”⟩(cgrG‘𝐺)⟨“𝑍𝑌𝑠”⟩) ∧ 𝑠((hpG‘𝐺)‘(𝑍𝐿𝑌))(((lInvG‘𝐺)‘(𝑌𝐿𝑍))‘𝑋)) → 𝑈 ∈ 𝑃)
97	10	ad3antrrr 743	. . . . . . . . . 10 ⊢ (((((((𝜑 ∧ 𝑤 ∈ 𝑃) ∧ 𝑤((hlG‘𝐺)‘𝑉)𝑊) ∧ (𝑉 − 𝑤) = (𝑌 − 𝑍)) ∧ 𝑠 ∈ 𝑃) ∧ ⟨“𝑤𝑉𝑈”⟩(cgrG‘𝐺)⟨“𝑍𝑌𝑠”⟩) ∧ 𝑠((hpG‘𝐺)‘(𝑍𝐿𝑌))(((lInvG‘𝐺)‘(𝑌𝐿𝑍))‘𝑋)) → 𝑉 ∈ 𝑃)
98	8	ad3antrrr 743	. . . . . . . . . 10 ⊢ (((((((𝜑 ∧ 𝑤 ∈ 𝑃) ∧ 𝑤((hlG‘𝐺)‘𝑉)𝑊) ∧ (𝑉 − 𝑤) = (𝑌 − 𝑍)) ∧ 𝑠 ∈ 𝑃) ∧ ⟨“𝑤𝑉𝑈”⟩(cgrG‘𝐺)⟨“𝑍𝑌𝑠”⟩) ∧ 𝑠((hpG‘𝐺)‘(𝑍𝐿𝑌))(((lInvG‘𝐺)‘(𝑌𝐿𝑍))‘𝑋)) → 𝑤 ∈ 𝑃)
99	35	ad6antr 749	. . . . . . . . . . . . 13 ⊢ (((((((𝜑 ∧ 𝑤 ∈ 𝑃) ∧ 𝑤((hlG‘𝐺)‘𝑉)𝑊) ∧ (𝑉 − 𝑤) = (𝑌 − 𝑍)) ∧ 𝑠 ∈ 𝑃) ∧ ⟨“𝑤𝑉𝑈”⟩(cgrG‘𝐺)⟨“𝑍𝑌𝑠”⟩) ∧ 𝑠((hpG‘𝐺)‘(𝑍𝐿𝑌))(((lInvG‘𝐺)‘(𝑌𝐿𝑍))‘𝑋)) → 𝑈 ≠ 𝑉)
100	44	ad3antrrr 743	. . . . . . . . . . . . 13 ⊢ (((((((𝜑 ∧ 𝑤 ∈ 𝑃) ∧ 𝑤((hlG‘𝐺)‘𝑉)𝑊) ∧ (𝑉 − 𝑤) = (𝑌 − 𝑍)) ∧ 𝑠 ∈ 𝑃) ∧ ⟨“𝑤𝑉𝑈”⟩(cgrG‘𝐺)⟨“𝑍𝑌𝑠”⟩) ∧ 𝑠((hpG‘𝐺)‘(𝑍𝐿𝑌))(((lInvG‘𝐺)‘(𝑌𝐿𝑍))‘𝑋)) → 𝑉 ≠ 𝑤)
101	1, 3, 92, 5, 96, 97, 98, 99, 100	cgraswap 29202	. . . . . . . . . . . 12 ⊢ (((((((𝜑 ∧ 𝑤 ∈ 𝑃) ∧ 𝑤((hlG‘𝐺)‘𝑉)𝑊) ∧ (𝑉 − 𝑤) = (𝑌 − 𝑍)) ∧ 𝑠 ∈ 𝑃) ∧ ⟨“𝑤𝑉𝑈”⟩(cgrG‘𝐺)⟨“𝑍𝑌𝑠”⟩) ∧ 𝑠((hpG‘𝐺)‘(𝑍𝐿𝑌))(((lInvG‘𝐺)‘(𝑌𝐿𝑍))‘𝑋)) → ⟨“𝑈𝑉𝑤”⟩(cgrA‘𝐺)⟨“𝑤𝑉𝑈”⟩)
102	45	ad3antrrr 743	. . . . . . . . . . . . 13 ⊢ (((((((𝜑 ∧ 𝑤 ∈ 𝑃) ∧ 𝑤((hlG‘𝐺)‘𝑉)𝑊) ∧ (𝑉 − 𝑤) = (𝑌 − 𝑍)) ∧ 𝑠 ∈ 𝑃) ∧ ⟨“𝑤𝑉𝑈”⟩(cgrG‘𝐺)⟨“𝑍𝑌𝑠”⟩) ∧ 𝑠((hpG‘𝐺)‘(𝑍𝐿𝑌))(((lInvG‘𝐺)‘(𝑌𝐿𝑍))‘𝑋)) → 𝑤 ≠ 𝑉)
103	38	ad6antr 749	. . . . . . . . . . . . 13 ⊢ (((((((𝜑 ∧ 𝑤 ∈ 𝑃) ∧ 𝑤((hlG‘𝐺)‘𝑉)𝑊) ∧ (𝑉 − 𝑤) = (𝑌 − 𝑍)) ∧ 𝑠 ∈ 𝑃) ∧ ⟨“𝑤𝑉𝑈”⟩(cgrG‘𝐺)⟨“𝑍𝑌𝑠”⟩) ∧ 𝑠((hpG‘𝐺)‘(𝑍𝐿𝑌))(((lInvG‘𝐺)‘(𝑌𝐿𝑍))‘𝑋)) → 𝑉 ≠ 𝑈)
104		simplr 781	. . . . . . . . . . . . 13 ⊢ (((((((𝜑 ∧ 𝑤 ∈ 𝑃) ∧ 𝑤((hlG‘𝐺)‘𝑉)𝑊) ∧ (𝑉 − 𝑤) = (𝑌 − 𝑍)) ∧ 𝑠 ∈ 𝑃) ∧ ⟨“𝑤𝑉𝑈”⟩(cgrG‘𝐺)⟨“𝑍𝑌𝑠”⟩) ∧ 𝑠((hpG‘𝐺)‘(𝑍𝐿𝑌))(((lInvG‘𝐺)‘(𝑌𝐿𝑍))‘𝑋)) → ⟨“𝑤𝑉𝑈”⟩(cgrG‘𝐺)⟨“𝑍𝑌𝑠”⟩)
105	1, 3, 92, 5, 98, 97, 96, 93, 94, 95, 102, 103, 104	cgrcgra 29203	. . . . . . . . . . . 12 ⊢ (((((((𝜑 ∧ 𝑤 ∈ 𝑃) ∧ 𝑤((hlG‘𝐺)‘𝑉)𝑊) ∧ (𝑉 − 𝑤) = (𝑌 − 𝑍)) ∧ 𝑠 ∈ 𝑃) ∧ ⟨“𝑤𝑉𝑈”⟩(cgrG‘𝐺)⟨“𝑍𝑌𝑠”⟩) ∧ 𝑠((hpG‘𝐺)‘(𝑍𝐿𝑌))(((lInvG‘𝐺)‘(𝑌𝐿𝑍))‘𝑋)) → ⟨“𝑤𝑉𝑈”⟩(cgrA‘𝐺)⟨“𝑍𝑌𝑠”⟩)
106	1, 3, 92, 5, 96, 97, 98, 98, 97, 96, 101, 93, 94, 95, 105	cgratr 29205	. . . . . . . . . . 11 ⊢ (((((((𝜑 ∧ 𝑤 ∈ 𝑃) ∧ 𝑤((hlG‘𝐺)‘𝑉)𝑊) ∧ (𝑉 − 𝑤) = (𝑌 − 𝑍)) ∧ 𝑠 ∈ 𝑃) ∧ ⟨“𝑤𝑉𝑈”⟩(cgrG‘𝐺)⟨“𝑍𝑌𝑠”⟩) ∧ 𝑠((hpG‘𝐺)‘(𝑍𝐿𝑌))(((lInvG‘𝐺)‘(𝑌𝐿𝑍))‘𝑋)) → ⟨“𝑈𝑉𝑤”⟩(cgrA‘𝐺)⟨“𝑍𝑌𝑠”⟩)
107	1, 3, 92, 5, 96, 97, 98, 93, 94, 95, 106	cgracom 29204	. . . . . . . . . 10 ⊢ (((((((𝜑 ∧ 𝑤 ∈ 𝑃) ∧ 𝑤((hlG‘𝐺)‘𝑉)𝑊) ∧ (𝑉 − 𝑤) = (𝑌 − 𝑍)) ∧ 𝑠 ∈ 𝑃) ∧ ⟨“𝑤𝑉𝑈”⟩(cgrG‘𝐺)⟨“𝑍𝑌𝑠”⟩) ∧ 𝑠((hpG‘𝐺)‘(𝑍𝐿𝑌))(((lInvG‘𝐺)‘(𝑌𝐿𝑍))‘𝑋)) → ⟨“𝑍𝑌𝑠”⟩(cgrA‘𝐺)⟨“𝑈𝑉𝑤”⟩)
108	51	ad3antrrr 743	. . . . . . . . . 10 ⊢ (((((((𝜑 ∧ 𝑤 ∈ 𝑃) ∧ 𝑤((hlG‘𝐺)‘𝑉)𝑊) ∧ (𝑉 − 𝑤) = (𝑌 − 𝑍)) ∧ 𝑠 ∈ 𝑃) ∧ ⟨“𝑤𝑉𝑈”⟩(cgrG‘𝐺)⟨“𝑍𝑌𝑠”⟩) ∧ 𝑠((hpG‘𝐺)‘(𝑍𝐿𝑌))(((lInvG‘𝐺)‘(𝑌𝐿𝑍))‘𝑋)) → 𝑊 ∈ 𝑃)
109	53	ad3antrrr 743	. . . . . . . . . 10 ⊢ (((((((𝜑 ∧ 𝑤 ∈ 𝑃) ∧ 𝑤((hlG‘𝐺)‘𝑉)𝑊) ∧ (𝑉 − 𝑤) = (𝑌 − 𝑍)) ∧ 𝑠 ∈ 𝑃) ∧ ⟨“𝑤𝑉𝑈”⟩(cgrG‘𝐺)⟨“𝑍𝑌𝑠”⟩) ∧ 𝑠((hpG‘𝐺)‘(𝑍𝐿𝑌))(((lInvG‘𝐺)‘(𝑌𝐿𝑍))‘𝑋)) → 𝑊((hlG‘𝐺)‘𝑉)𝑤)
110	1, 3, 5, 92, 93, 94, 95, 96, 97, 98, 107, 108, 109	cgrahl2 29199	. . . . . . . . 9 ⊢ (((((((𝜑 ∧ 𝑤 ∈ 𝑃) ∧ 𝑤((hlG‘𝐺)‘𝑉)𝑊) ∧ (𝑉 − 𝑤) = (𝑌 − 𝑍)) ∧ 𝑠 ∈ 𝑃) ∧ ⟨“𝑤𝑉𝑈”⟩(cgrG‘𝐺)⟨“𝑍𝑌𝑠”⟩) ∧ 𝑠((hpG‘𝐺)‘(𝑍𝐿𝑌))(((lInvG‘𝐺)‘(𝑌𝐿𝑍))‘𝑋)) → ⟨“𝑍𝑌𝑠”⟩(cgrA‘𝐺)⟨“𝑈𝑉𝑊”⟩)
111	91, 110	breqdi 5122	. . . . . . . 8 ⊢ (((((((𝜑 ∧ 𝑤 ∈ 𝑃) ∧ 𝑤((hlG‘𝐺)‘𝑉)𝑊) ∧ (𝑉 − 𝑤) = (𝑌 − 𝑍)) ∧ 𝑠 ∈ 𝑃) ∧ ⟨“𝑤𝑉𝑈”⟩(cgrG‘𝐺)⟨“𝑍𝑌𝑠”⟩) ∧ 𝑠((hpG‘𝐺)‘(𝑍𝐿𝑌))(((lInvG‘𝐺)‘(𝑌𝐿𝑍))‘𝑋)) → ⟨“𝑍𝑌𝑠”⟩ ∼ ⟨“𝑈𝑉𝑊”⟩)
112		eqid 2762	. . . . . . . . . 10 ⊢ (cgrG‘𝐺) = (cgrG‘𝐺)
113	1, 2, 3, 112, 92, 98, 97, 96, 93, 94, 95, 104	cgr3simp2 28859	. . . . . . . . 9 ⊢ (((((((𝜑 ∧ 𝑤 ∈ 𝑃) ∧ 𝑤((hlG‘𝐺)‘𝑉)𝑊) ∧ (𝑉 − 𝑤) = (𝑌 − 𝑍)) ∧ 𝑠 ∈ 𝑃) ∧ ⟨“𝑤𝑉𝑈”⟩(cgrG‘𝐺)⟨“𝑍𝑌𝑠”⟩) ∧ 𝑠((hpG‘𝐺)‘(𝑍𝐿𝑌))(((lInvG‘𝐺)‘(𝑌𝐿𝑍))‘𝑋)) → (𝑉 − 𝑈) = (𝑌 − 𝑠))
114	113	eqcomd 2768	. . . . . . . 8 ⊢ (((((((𝜑 ∧ 𝑤 ∈ 𝑃) ∧ 𝑤((hlG‘𝐺)‘𝑉)𝑊) ∧ (𝑉 − 𝑤) = (𝑌 − 𝑍)) ∧ 𝑠 ∈ 𝑃) ∧ ⟨“𝑤𝑉𝑈”⟩(cgrG‘𝐺)⟨“𝑍𝑌𝑠”⟩) ∧ 𝑠((hpG‘𝐺)‘(𝑍𝐿𝑌))(((lInvG‘𝐺)‘(𝑌𝐿𝑍))‘𝑋)) → (𝑌 − 𝑠) = (𝑉 − 𝑈))
115	27	ad6antr 749	. . . . . . . . 9 ⊢ (((((((𝜑 ∧ 𝑤 ∈ 𝑃) ∧ 𝑤((hlG‘𝐺)‘𝑉)𝑊) ∧ (𝑉 − 𝑤) = (𝑌 − 𝑍)) ∧ 𝑠 ∈ 𝑃) ∧ ⟨“𝑤𝑉𝑈”⟩(cgrG‘𝐺)⟨“𝑍𝑌𝑠”⟩) ∧ 𝑠((hpG‘𝐺)‘(𝑍𝐿𝑌))(((lInvG‘𝐺)‘(𝑌𝐿𝑍))‘𝑋)) → (𝑌𝐿𝑍) ∈ ran 𝐿)
116	28	ad3antrrr 743	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑤 ∈ 𝑃) ∧ 𝑤((hlG‘𝐺)‘𝑉)𝑊) ∧ (𝑉 − 𝑤) = (𝑌 − 𝑍)) → 𝑋 ∈ 𝑃)
117	116	ad3antrrr 743	. . . . . . . . 9 ⊢ (((((((𝜑 ∧ 𝑤 ∈ 𝑃) ∧ 𝑤((hlG‘𝐺)‘𝑉)𝑊) ∧ (𝑉 − 𝑤) = (𝑌 − 𝑍)) ∧ 𝑠 ∈ 𝑃) ∧ ⟨“𝑤𝑉𝑈”⟩(cgrG‘𝐺)⟨“𝑍𝑌𝑠”⟩) ∧ 𝑠((hpG‘𝐺)‘(𝑍𝐿𝑌))(((lInvG‘𝐺)‘(𝑌𝐿𝑍))‘𝑋)) → 𝑋 ∈ 𝑃)
118	30	ad3antrrr 743	. . . . . . . . . . 11 ⊢ (((((((𝜑 ∧ 𝑤 ∈ 𝑃) ∧ 𝑤((hlG‘𝐺)‘𝑉)𝑊) ∧ (𝑉 − 𝑤) = (𝑌 − 𝑍)) ∧ 𝑠 ∈ 𝑃) ∧ ⟨“𝑤𝑉𝑈”⟩(cgrG‘𝐺)⟨“𝑍𝑌𝑠”⟩) ∧ 𝑠((hpG‘𝐺)‘(𝑍𝐿𝑌))(((lInvG‘𝐺)‘(𝑌𝐿𝑍))‘𝑋)) → (((lInvG‘𝐺)‘(𝑌𝐿𝑍))‘𝑋) ∈ 𝑃)
119		eqidd 2763	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (hpG‘𝐺) = (hpG‘𝐺))
120	119, 79	fveq12d 6889	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((hpG‘𝐺)‘(𝑌𝐿𝑍)) = ((hpG‘𝐺)‘(𝑍𝐿𝑌)))
121	120	eqcomd 2768	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((hpG‘𝐺)‘(𝑍𝐿𝑌)) = ((hpG‘𝐺)‘(𝑌𝐿𝑍)))
122	121	ad6antr 749	. . . . . . . . . . . 12 ⊢ (((((((𝜑 ∧ 𝑤 ∈ 𝑃) ∧ 𝑤((hlG‘𝐺)‘𝑉)𝑊) ∧ (𝑉 − 𝑤) = (𝑌 − 𝑍)) ∧ 𝑠 ∈ 𝑃) ∧ ⟨“𝑤𝑉𝑈”⟩(cgrG‘𝐺)⟨“𝑍𝑌𝑠”⟩) ∧ 𝑠((hpG‘𝐺)‘(𝑍𝐿𝑌))(((lInvG‘𝐺)‘(𝑌𝐿𝑍))‘𝑋)) → ((hpG‘𝐺)‘(𝑍𝐿𝑌)) = ((hpG‘𝐺)‘(𝑌𝐿𝑍)))
123		simpr 490	. . . . . . . . . . . 12 ⊢ (((((((𝜑 ∧ 𝑤 ∈ 𝑃) ∧ 𝑤((hlG‘𝐺)‘𝑉)𝑊) ∧ (𝑉 − 𝑤) = (𝑌 − 𝑍)) ∧ 𝑠 ∈ 𝑃) ∧ ⟨“𝑤𝑉𝑈”⟩(cgrG‘𝐺)⟨“𝑍𝑌𝑠”⟩) ∧ 𝑠((hpG‘𝐺)‘(𝑍𝐿𝑌))(((lInvG‘𝐺)‘(𝑌𝐿𝑍))‘𝑋)) → 𝑠((hpG‘𝐺)‘(𝑍𝐿𝑌))(((lInvG‘𝐺)‘(𝑌𝐿𝑍))‘𝑋))
124	122, 123	breqdi 5122	. . . . . . . . . . 11 ⊢ (((((((𝜑 ∧ 𝑤 ∈ 𝑃) ∧ 𝑤((hlG‘𝐺)‘𝑉)𝑊) ∧ (𝑉 − 𝑤) = (𝑌 − 𝑍)) ∧ 𝑠 ∈ 𝑃) ∧ ⟨“𝑤𝑉𝑈”⟩(cgrG‘𝐺)⟨“𝑍𝑌𝑠”⟩) ∧ 𝑠((hpG‘𝐺)‘(𝑍𝐿𝑌))(((lInvG‘𝐺)‘(𝑌𝐿𝑍))‘𝑋)) → 𝑠((hpG‘𝐺)‘(𝑌𝐿𝑍))(((lInvG‘𝐺)‘(𝑌𝐿𝑍))‘𝑋))
125	1, 3, 4, 92, 115, 95, 75, 118, 124	hpgcom 29120	. . . . . . . . . 10 ⊢ (((((((𝜑 ∧ 𝑤 ∈ 𝑃) ∧ 𝑤((hlG‘𝐺)‘𝑉)𝑊) ∧ (𝑉 − 𝑤) = (𝑌 − 𝑍)) ∧ 𝑠 ∈ 𝑃) ∧ ⟨“𝑤𝑉𝑈”⟩(cgrG‘𝐺)⟨“𝑍𝑌𝑠”⟩) ∧ 𝑠((hpG‘𝐺)‘(𝑍𝐿𝑌))(((lInvG‘𝐺)‘(𝑌𝐿𝑍))‘𝑋)) → (((lInvG‘𝐺)‘(𝑌𝐿𝑍))‘𝑋)((hpG‘𝐺)‘(𝑌𝐿𝑍))𝑠)
126	7	ad2antrr 739	. . . . . . . . . . . 12 ⊢ ((((((𝜑 ∧ 𝑤 ∈ 𝑃) ∧ 𝑤((hlG‘𝐺)‘𝑉)𝑊) ∧ (𝑉 − 𝑤) = (𝑌 − 𝑍)) ∧ 𝑠 ∈ 𝑃) ∧ ⟨“𝑤𝑉𝑈”⟩(cgrG‘𝐺)⟨“𝑍𝑌𝑠”⟩) → 𝐺 ∈ TarskiG)
127	27	ad5antr 747	. . . . . . . . . . . 12 ⊢ ((((((𝜑 ∧ 𝑤 ∈ 𝑃) ∧ 𝑤((hlG‘𝐺)‘𝑉)𝑊) ∧ (𝑉 − 𝑤) = (𝑌 − 𝑍)) ∧ 𝑠 ∈ 𝑃) ∧ ⟨“𝑤𝑉𝑈”⟩(cgrG‘𝐺)⟨“𝑍𝑌𝑠”⟩) → (𝑌𝐿𝑍) ∈ ran 𝐿)
128	30	ad2antrr 739	. . . . . . . . . . . 12 ⊢ ((((((𝜑 ∧ 𝑤 ∈ 𝑃) ∧ 𝑤((hlG‘𝐺)‘𝑉)𝑊) ∧ (𝑉 − 𝑤) = (𝑌 − 𝑍)) ∧ 𝑠 ∈ 𝑃) ∧ ⟨“𝑤𝑉𝑈”⟩(cgrG‘𝐺)⟨“𝑍𝑌𝑠”⟩) → (((lInvG‘𝐺)‘(𝑌𝐿𝑍))‘𝑋) ∈ 𝑃)
129		simplr 781	. . . . . . . . . . . 12 ⊢ ((((((𝜑 ∧ 𝑤 ∈ 𝑃) ∧ 𝑤((hlG‘𝐺)‘𝑉)𝑊) ∧ (𝑉 − 𝑤) = (𝑌 − 𝑍)) ∧ 𝑠 ∈ 𝑃) ∧ ⟨“𝑤𝑉𝑈”⟩(cgrG‘𝐺)⟨“𝑍𝑌𝑠”⟩) → 𝑠 ∈ 𝑃)
130	116	ad2antrr 739	. . . . . . . . . . . 12 ⊢ ((((((𝜑 ∧ 𝑤 ∈ 𝑃) ∧ 𝑤((hlG‘𝐺)‘𝑉)𝑊) ∧ (𝑉 − 𝑤) = (𝑌 − 𝑍)) ∧ 𝑠 ∈ 𝑃) ∧ ⟨“𝑤𝑉𝑈”⟩(cgrG‘𝐺)⟨“𝑍𝑌𝑠”⟩) → 𝑋 ∈ 𝑃)
131	77	ad5antr 747	. . . . . . . . . . . . 13 ⊢ ((((((𝜑 ∧ 𝑤 ∈ 𝑃) ∧ 𝑤((hlG‘𝐺)‘𝑉)𝑊) ∧ (𝑉 − 𝑤) = (𝑌 − 𝑍)) ∧ 𝑠 ∈ 𝑃) ∧ ⟨“𝑤𝑉𝑈”⟩(cgrG‘𝐺)⟨“𝑍𝑌𝑠”⟩) → 𝑋{⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝑃 ∖ (𝑌𝐿𝑍)) ∧ 𝑏 ∈ (𝑃 ∖ (𝑌𝐿𝑍))) ∧ ∃𝑠 ∈ (𝑌𝐿𝑍)𝑠 ∈ (𝑎𝐼𝑏))} (((lInvG‘𝐺)‘(𝑌𝐿𝑍))‘𝑋))
132	1, 2, 3, 75, 4, 127, 126, 130, 128, 131	oppcom 29095	. . . . . . . . . . . 12 ⊢ ((((((𝜑 ∧ 𝑤 ∈ 𝑃) ∧ 𝑤((hlG‘𝐺)‘𝑉)𝑊) ∧ (𝑉 − 𝑤) = (𝑌 − 𝑍)) ∧ 𝑠 ∈ 𝑃) ∧ ⟨“𝑤𝑉𝑈”⟩(cgrG‘𝐺)⟨“𝑍𝑌𝑠”⟩) → (((lInvG‘𝐺)‘(𝑌𝐿𝑍))‘𝑋){⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝑃 ∖ (𝑌𝐿𝑍)) ∧ 𝑏 ∈ (𝑃 ∖ (𝑌𝐿𝑍))) ∧ ∃𝑠 ∈ (𝑌𝐿𝑍)𝑠 ∈ (𝑎𝐼𝑏))}𝑋)
133	1, 3, 4, 75, 126, 127, 128, 129, 130, 132	lnopp2hpgb 29116	. . . . . . . . . . 11 ⊢ ((((((𝜑 ∧ 𝑤 ∈ 𝑃) ∧ 𝑤((hlG‘𝐺)‘𝑉)𝑊) ∧ (𝑉 − 𝑤) = (𝑌 − 𝑍)) ∧ 𝑠 ∈ 𝑃) ∧ ⟨“𝑤𝑉𝑈”⟩(cgrG‘𝐺)⟨“𝑍𝑌𝑠”⟩) → (𝑠{⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝑃 ∖ (𝑌𝐿𝑍)) ∧ 𝑏 ∈ (𝑃 ∖ (𝑌𝐿𝑍))) ∧ ∃𝑠 ∈ (𝑌𝐿𝑍)𝑠 ∈ (𝑎𝐼𝑏))}𝑋 ↔ (((lInvG‘𝐺)‘(𝑌𝐿𝑍))‘𝑋)((hpG‘𝐺)‘(𝑌𝐿𝑍))𝑠))
134	133	adantr 486	. . . . . . . . . 10 ⊢ (((((((𝜑 ∧ 𝑤 ∈ 𝑃) ∧ 𝑤((hlG‘𝐺)‘𝑉)𝑊) ∧ (𝑉 − 𝑤) = (𝑌 − 𝑍)) ∧ 𝑠 ∈ 𝑃) ∧ ⟨“𝑤𝑉𝑈”⟩(cgrG‘𝐺)⟨“𝑍𝑌𝑠”⟩) ∧ 𝑠((hpG‘𝐺)‘(𝑍𝐿𝑌))(((lInvG‘𝐺)‘(𝑌𝐿𝑍))‘𝑋)) → (𝑠{⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝑃 ∖ (𝑌𝐿𝑍)) ∧ 𝑏 ∈ (𝑃 ∖ (𝑌𝐿𝑍))) ∧ ∃𝑠 ∈ (𝑌𝐿𝑍)𝑠 ∈ (𝑎𝐼𝑏))}𝑋 ↔ (((lInvG‘𝐺)‘(𝑌𝐿𝑍))‘𝑋)((hpG‘𝐺)‘(𝑌𝐿𝑍))𝑠))
135	125, 134	mpbird 260	. . . . . . . . 9 ⊢ (((((((𝜑 ∧ 𝑤 ∈ 𝑃) ∧ 𝑤((hlG‘𝐺)‘𝑉)𝑊) ∧ (𝑉 − 𝑤) = (𝑌 − 𝑍)) ∧ 𝑠 ∈ 𝑃) ∧ ⟨“𝑤𝑉𝑈”⟩(cgrG‘𝐺)⟨“𝑍𝑌𝑠”⟩) ∧ 𝑠((hpG‘𝐺)‘(𝑍𝐿𝑌))(((lInvG‘𝐺)‘(𝑌𝐿𝑍))‘𝑋)) → 𝑠{⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝑃 ∖ (𝑌𝐿𝑍)) ∧ 𝑏 ∈ (𝑃 ∖ (𝑌𝐿𝑍))) ∧ ∃𝑠 ∈ (𝑌𝐿𝑍)𝑠 ∈ (𝑎𝐼𝑏))}𝑋)
136	1, 3, 4, 75, 92, 115, 95, 117, 135	lnoppinn0 29106	. . . . . . . 8 ⊢ (((((((𝜑 ∧ 𝑤 ∈ 𝑃) ∧ 𝑤((hlG‘𝐺)‘𝑉)𝑊) ∧ (𝑉 − 𝑤) = (𝑌 − 𝑍)) ∧ 𝑠 ∈ 𝑃) ∧ ⟨“𝑤𝑉𝑈”⟩(cgrG‘𝐺)⟨“𝑍𝑌𝑠”⟩) ∧ 𝑠((hpG‘𝐺)‘(𝑍𝐿𝑌))(((lInvG‘𝐺)‘(𝑌𝐿𝑍))‘𝑋)) → ((𝑌𝐿𝑍) ∩ (𝑠𝐼𝑋)) ≠ ∅)
137	111, 114, 136	3jca 1146	. . . . . . 7 ⊢ (((((((𝜑 ∧ 𝑤 ∈ 𝑃) ∧ 𝑤((hlG‘𝐺)‘𝑉)𝑊) ∧ (𝑉 − 𝑤) = (𝑌 − 𝑍)) ∧ 𝑠 ∈ 𝑃) ∧ ⟨“𝑤𝑉𝑈”⟩(cgrG‘𝐺)⟨“𝑍𝑌𝑠”⟩) ∧ 𝑠((hpG‘𝐺)‘(𝑍𝐿𝑌))(((lInvG‘𝐺)‘(𝑌𝐿𝑍))‘𝑋)) → (⟨“𝑍𝑌𝑠”⟩ ∼ ⟨“𝑈𝑉𝑊”⟩ ∧ (𝑌 − 𝑠) = (𝑉 − 𝑈) ∧ ((𝑌𝐿𝑍) ∩ (𝑠𝐼𝑋)) ≠ ∅))
138	137	anasss 472	. . . . . 6 ⊢ ((((((𝜑 ∧ 𝑤 ∈ 𝑃) ∧ 𝑤((hlG‘𝐺)‘𝑉)𝑊) ∧ (𝑉 − 𝑤) = (𝑌 − 𝑍)) ∧ 𝑠 ∈ 𝑃) ∧ (⟨“𝑤𝑉𝑈”⟩(cgrG‘𝐺)⟨“𝑍𝑌𝑠”⟩ ∧ 𝑠((hpG‘𝐺)‘(𝑍𝐿𝑌))(((lInvG‘𝐺)‘(𝑌𝐿𝑍))‘𝑋))) → (⟨“𝑍𝑌𝑠”⟩ ∼ ⟨“𝑈𝑉𝑊”⟩ ∧ (𝑌 − 𝑠) = (𝑉 − 𝑈) ∧ ((𝑌𝐿𝑍) ∩ (𝑠𝐼𝑋)) ≠ ∅))
139	7	ad4antr 745	. . . . . . . . 9 ⊢ ((((((((𝜑 ∧ 𝑤 ∈ 𝑃) ∧ 𝑤((hlG‘𝐺)‘𝑉)𝑊) ∧ (𝑉 − 𝑤) = (𝑌 − 𝑍)) ∧ 𝑠 ∈ 𝑃) ∧ ⟨“𝑍𝑌𝑠”⟩ ∼ ⟨“𝑈𝑉𝑊”⟩) ∧ (𝑌 − 𝑠) = (𝑉 − 𝑈)) ∧ ((𝑌𝐿𝑍) ∩ (𝑠𝐼𝑋)) ≠ ∅) → 𝐺 ∈ TarskiG)
140	8	ad4antr 745	. . . . . . . . 9 ⊢ ((((((((𝜑 ∧ 𝑤 ∈ 𝑃) ∧ 𝑤((hlG‘𝐺)‘𝑉)𝑊) ∧ (𝑉 − 𝑤) = (𝑌 − 𝑍)) ∧ 𝑠 ∈ 𝑃) ∧ ⟨“𝑍𝑌𝑠”⟩ ∼ ⟨“𝑈𝑉𝑊”⟩) ∧ (𝑌 − 𝑠) = (𝑉 − 𝑈)) ∧ ((𝑌𝐿𝑍) ∩ (𝑠𝐼𝑋)) ≠ ∅) → 𝑤 ∈ 𝑃)
141	10	ad4antr 745	. . . . . . . . 9 ⊢ ((((((((𝜑 ∧ 𝑤 ∈ 𝑃) ∧ 𝑤((hlG‘𝐺)‘𝑉)𝑊) ∧ (𝑉 − 𝑤) = (𝑌 − 𝑍)) ∧ 𝑠 ∈ 𝑃) ∧ ⟨“𝑍𝑌𝑠”⟩ ∼ ⟨“𝑈𝑉𝑊”⟩) ∧ (𝑌 − 𝑠) = (𝑉 − 𝑈)) ∧ ((𝑌𝐿𝑍) ∩ (𝑠𝐼𝑋)) ≠ ∅) → 𝑉 ∈ 𝑃)
142	12	ad4antr 745	. . . . . . . . 9 ⊢ ((((((((𝜑 ∧ 𝑤 ∈ 𝑃) ∧ 𝑤((hlG‘𝐺)‘𝑉)𝑊) ∧ (𝑉 − 𝑤) = (𝑌 − 𝑍)) ∧ 𝑠 ∈ 𝑃) ∧ ⟨“𝑍𝑌𝑠”⟩ ∼ ⟨“𝑈𝑉𝑊”⟩) ∧ (𝑌 − 𝑠) = (𝑉 − 𝑈)) ∧ ((𝑌𝐿𝑍) ∩ (𝑠𝐼𝑋)) ≠ ∅) → 𝑈 ∈ 𝑃)
143	14	ad4antr 745	. . . . . . . . 9 ⊢ ((((((((𝜑 ∧ 𝑤 ∈ 𝑃) ∧ 𝑤((hlG‘𝐺)‘𝑉)𝑊) ∧ (𝑉 − 𝑤) = (𝑌 − 𝑍)) ∧ 𝑠 ∈ 𝑃) ∧ ⟨“𝑍𝑌𝑠”⟩ ∼ ⟨“𝑈𝑉𝑊”⟩) ∧ (𝑌 − 𝑠) = (𝑉 − 𝑈)) ∧ ((𝑌𝐿𝑍) ∩ (𝑠𝐼𝑋)) ≠ ∅) → 𝑍 ∈ 𝑃)
144	16	ad4antr 745	. . . . . . . . 9 ⊢ ((((((((𝜑 ∧ 𝑤 ∈ 𝑃) ∧ 𝑤((hlG‘𝐺)‘𝑉)𝑊) ∧ (𝑉 − 𝑤) = (𝑌 − 𝑍)) ∧ 𝑠 ∈ 𝑃) ∧ ⟨“𝑍𝑌𝑠”⟩ ∼ ⟨“𝑈𝑉𝑊”⟩) ∧ (𝑌 − 𝑠) = (𝑉 − 𝑈)) ∧ ((𝑌𝐿𝑍) ∩ (𝑠𝐼𝑋)) ≠ ∅) → 𝑌 ∈ 𝑃)
145		simp-4r 796	. . . . . . . . 9 ⊢ ((((((((𝜑 ∧ 𝑤 ∈ 𝑃) ∧ 𝑤((hlG‘𝐺)‘𝑉)𝑊) ∧ (𝑉 − 𝑤) = (𝑌 − 𝑍)) ∧ 𝑠 ∈ 𝑃) ∧ ⟨“𝑍𝑌𝑠”⟩ ∼ ⟨“𝑈𝑉𝑊”⟩) ∧ (𝑌 − 𝑠) = (𝑉 − 𝑈)) ∧ ((𝑌𝐿𝑍) ∩ (𝑠𝐼𝑋)) ≠ ∅) → 𝑠 ∈ 𝑃)
146	87	ad4antr 745	. . . . . . . . 9 ⊢ ((((((((𝜑 ∧ 𝑤 ∈ 𝑃) ∧ 𝑤((hlG‘𝐺)‘𝑉)𝑊) ∧ (𝑉 − 𝑤) = (𝑌 − 𝑍)) ∧ 𝑠 ∈ 𝑃) ∧ ⟨“𝑍𝑌𝑠”⟩ ∼ ⟨“𝑈𝑉𝑊”⟩) ∧ (𝑌 − 𝑠) = (𝑉 − 𝑈)) ∧ ((𝑌𝐿𝑍) ∩ (𝑠𝐼𝑋)) ≠ ∅) → (𝑤 − 𝑉) = (𝑍 − 𝑌))
147		simplr 781	. . . . . . . . . 10 ⊢ ((((((((𝜑 ∧ 𝑤 ∈ 𝑃) ∧ 𝑤((hlG‘𝐺)‘𝑉)𝑊) ∧ (𝑉 − 𝑤) = (𝑌 − 𝑍)) ∧ 𝑠 ∈ 𝑃) ∧ ⟨“𝑍𝑌𝑠”⟩ ∼ ⟨“𝑈𝑉𝑊”⟩) ∧ (𝑌 − 𝑠) = (𝑉 − 𝑈)) ∧ ((𝑌𝐿𝑍) ∩ (𝑠𝐼𝑋)) ≠ ∅) → (𝑌 − 𝑠) = (𝑉 − 𝑈))
148	147	eqcomd 2768	. . . . . . . . 9 ⊢ ((((((((𝜑 ∧ 𝑤 ∈ 𝑃) ∧ 𝑤((hlG‘𝐺)‘𝑉)𝑊) ∧ (𝑉 − 𝑤) = (𝑌 − 𝑍)) ∧ 𝑠 ∈ 𝑃) ∧ ⟨“𝑍𝑌𝑠”⟩ ∼ ⟨“𝑈𝑉𝑊”⟩) ∧ (𝑌 − 𝑠) = (𝑉 − 𝑈)) ∧ ((𝑌𝐿𝑍) ∩ (𝑠𝐼𝑋)) ≠ ∅) → (𝑉 − 𝑈) = (𝑌 − 𝑠))
149	51	ad4antr 745	. . . . . . . . . 10 ⊢ ((((((((𝜑 ∧ 𝑤 ∈ 𝑃) ∧ 𝑤((hlG‘𝐺)‘𝑉)𝑊) ∧ (𝑉 − 𝑤) = (𝑌 − 𝑍)) ∧ 𝑠 ∈ 𝑃) ∧ ⟨“𝑍𝑌𝑠”⟩ ∼ ⟨“𝑈𝑉𝑊”⟩) ∧ (𝑌 − 𝑠) = (𝑉 − 𝑈)) ∧ ((𝑌𝐿𝑍) ∩ (𝑠𝐼𝑋)) ≠ ∅) → 𝑊 ∈ 𝑃)
150	38	ad7antr 751	. . . . . . . . . . . . . . 15 ⊢ ((((((((𝜑 ∧ 𝑤 ∈ 𝑃) ∧ 𝑤((hlG‘𝐺)‘𝑉)𝑊) ∧ (𝑉 − 𝑤) = (𝑌 − 𝑍)) ∧ 𝑠 ∈ 𝑃) ∧ ⟨“𝑍𝑌𝑠”⟩ ∼ ⟨“𝑈𝑉𝑊”⟩) ∧ (𝑌 − 𝑠) = (𝑉 − 𝑈)) ∧ ((𝑌𝐿𝑍) ∩ (𝑠𝐼𝑋)) ≠ ∅) → 𝑉 ≠ 𝑈)
151	1, 2, 3, 139, 141, 142, 144, 145, 148, 150	tgcgrneq 28820	. . . . . . . . . . . . . 14 ⊢ ((((((((𝜑 ∧ 𝑤 ∈ 𝑃) ∧ 𝑤((hlG‘𝐺)‘𝑉)𝑊) ∧ (𝑉 − 𝑤) = (𝑌 − 𝑍)) ∧ 𝑠 ∈ 𝑃) ∧ ⟨“𝑍𝑌𝑠”⟩ ∼ ⟨“𝑈𝑉𝑊”⟩) ∧ (𝑌 − 𝑠) = (𝑉 − 𝑈)) ∧ ((𝑌𝐿𝑍) ∩ (𝑠𝐼𝑋)) ≠ ∅) → 𝑌 ≠ 𝑠)
152	151	necomd 3012	. . . . . . . . . . . . 13 ⊢ ((((((((𝜑 ∧ 𝑤 ∈ 𝑃) ∧ 𝑤((hlG‘𝐺)‘𝑉)𝑊) ∧ (𝑉 − 𝑤) = (𝑌 − 𝑍)) ∧ 𝑠 ∈ 𝑃) ∧ ⟨“𝑍𝑌𝑠”⟩ ∼ ⟨“𝑈𝑉𝑊”⟩) ∧ (𝑌 − 𝑠) = (𝑉 − 𝑈)) ∧ ((𝑌𝐿𝑍) ∩ (𝑠𝐼𝑋)) ≠ ∅) → 𝑠 ≠ 𝑌)
153	43	ad4antr 745	. . . . . . . . . . . . 13 ⊢ ((((((((𝜑 ∧ 𝑤 ∈ 𝑃) ∧ 𝑤((hlG‘𝐺)‘𝑉)𝑊) ∧ (𝑉 − 𝑤) = (𝑌 − 𝑍)) ∧ 𝑠 ∈ 𝑃) ∧ ⟨“𝑍𝑌𝑠”⟩ ∼ ⟨“𝑈𝑉𝑊”⟩) ∧ (𝑌 − 𝑠) = (𝑉 − 𝑈)) ∧ ((𝑌𝐿𝑍) ∩ (𝑠𝐼𝑋)) ≠ ∅) → 𝑌 ≠ 𝑍)
154	1, 3, 139, 5, 145, 144, 143, 152, 153	cgraswap 29202	. . . . . . . . . . . 12 ⊢ ((((((((𝜑 ∧ 𝑤 ∈ 𝑃) ∧ 𝑤((hlG‘𝐺)‘𝑉)𝑊) ∧ (𝑉 − 𝑤) = (𝑌 − 𝑍)) ∧ 𝑠 ∈ 𝑃) ∧ ⟨“𝑍𝑌𝑠”⟩ ∼ ⟨“𝑈𝑉𝑊”⟩) ∧ (𝑌 − 𝑠) = (𝑉 − 𝑈)) ∧ ((𝑌𝐿𝑍) ∩ (𝑠𝐼𝑋)) ≠ ∅) → ⟨“𝑠𝑌𝑍”⟩(cgrA‘𝐺)⟨“𝑍𝑌𝑠”⟩)
155	89	a1i 11	. . . . . . . . . . . . 13 ⊢ ((((((((𝜑 ∧ 𝑤 ∈ 𝑃) ∧ 𝑤((hlG‘𝐺)‘𝑉)𝑊) ∧ (𝑉 − 𝑤) = (𝑌 − 𝑍)) ∧ 𝑠 ∈ 𝑃) ∧ ⟨“𝑍𝑌𝑠”⟩ ∼ ⟨“𝑈𝑉𝑊”⟩) ∧ (𝑌 − 𝑠) = (𝑉 − 𝑈)) ∧ ((𝑌𝐿𝑍) ∩ (𝑠𝐼𝑋)) ≠ ∅) → ∼ = (cgrA‘𝐺))
156		simpllr 788	. . . . . . . . . . . . 13 ⊢ ((((((((𝜑 ∧ 𝑤 ∈ 𝑃) ∧ 𝑤((hlG‘𝐺)‘𝑉)𝑊) ∧ (𝑉 − 𝑤) = (𝑌 − 𝑍)) ∧ 𝑠 ∈ 𝑃) ∧ ⟨“𝑍𝑌𝑠”⟩ ∼ ⟨“𝑈𝑉𝑊”⟩) ∧ (𝑌 − 𝑠) = (𝑉 − 𝑈)) ∧ ((𝑌𝐿𝑍) ∩ (𝑠𝐼𝑋)) ≠ ∅) → ⟨“𝑍𝑌𝑠”⟩ ∼ ⟨“𝑈𝑉𝑊”⟩)
157	155, 156	breqdi 5122	. . . . . . . . . . . 12 ⊢ ((((((((𝜑 ∧ 𝑤 ∈ 𝑃) ∧ 𝑤((hlG‘𝐺)‘𝑉)𝑊) ∧ (𝑉 − 𝑤) = (𝑌 − 𝑍)) ∧ 𝑠 ∈ 𝑃) ∧ ⟨“𝑍𝑌𝑠”⟩ ∼ ⟨“𝑈𝑉𝑊”⟩) ∧ (𝑌 − 𝑠) = (𝑉 − 𝑈)) ∧ ((𝑌𝐿𝑍) ∩ (𝑠𝐼𝑋)) ≠ ∅) → ⟨“𝑍𝑌𝑠”⟩(cgrA‘𝐺)⟨“𝑈𝑉𝑊”⟩)
158	1, 3, 139, 5, 145, 144, 143, 143, 144, 145, 154, 142, 141, 149, 157	cgratr 29205	. . . . . . . . . . 11 ⊢ ((((((((𝜑 ∧ 𝑤 ∈ 𝑃) ∧ 𝑤((hlG‘𝐺)‘𝑉)𝑊) ∧ (𝑉 − 𝑤) = (𝑌 − 𝑍)) ∧ 𝑠 ∈ 𝑃) ∧ ⟨“𝑍𝑌𝑠”⟩ ∼ ⟨“𝑈𝑉𝑊”⟩) ∧ (𝑌 − 𝑠) = (𝑉 − 𝑈)) ∧ ((𝑌𝐿𝑍) ∩ (𝑠𝐼𝑋)) ≠ ∅) → ⟨“𝑠𝑌𝑍”⟩(cgrA‘𝐺)⟨“𝑈𝑉𝑊”⟩)
159	1, 3, 139, 5, 145, 144, 143, 142, 141, 149, 158	cgracom 29204	. . . . . . . . . 10 ⊢ ((((((((𝜑 ∧ 𝑤 ∈ 𝑃) ∧ 𝑤((hlG‘𝐺)‘𝑉)𝑊) ∧ (𝑉 − 𝑤) = (𝑌 − 𝑍)) ∧ 𝑠 ∈ 𝑃) ∧ ⟨“𝑍𝑌𝑠”⟩ ∼ ⟨“𝑈𝑉𝑊”⟩) ∧ (𝑌 − 𝑠) = (𝑉 − 𝑈)) ∧ ((𝑌𝐿𝑍) ∩ (𝑠𝐼𝑋)) ≠ ∅) → ⟨“𝑈𝑉𝑊”⟩(cgrA‘𝐺)⟨“𝑠𝑌𝑍”⟩)
160	116	ad4antr 745	. . . . . . . . . . 11 ⊢ ((((((((𝜑 ∧ 𝑤 ∈ 𝑃) ∧ 𝑤((hlG‘𝐺)‘𝑉)𝑊) ∧ (𝑉 − 𝑤) = (𝑌 − 𝑍)) ∧ 𝑠 ∈ 𝑃) ∧ ⟨“𝑍𝑌𝑠”⟩ ∼ ⟨“𝑈𝑉𝑊”⟩) ∧ (𝑌 − 𝑠) = (𝑉 − 𝑈)) ∧ ((𝑌𝐿𝑍) ∩ (𝑠𝐼𝑋)) ≠ ∅) → 𝑋 ∈ 𝑃)
161	35	ad7antr 751	. . . . . . . . . . 11 ⊢ ((((((((𝜑 ∧ 𝑤 ∈ 𝑃) ∧ 𝑤((hlG‘𝐺)‘𝑉)𝑊) ∧ (𝑉 − 𝑤) = (𝑌 − 𝑍)) ∧ 𝑠 ∈ 𝑃) ∧ ⟨“𝑍𝑌𝑠”⟩ ∼ ⟨“𝑈𝑉𝑊”⟩) ∧ (𝑌 − 𝑠) = (𝑉 − 𝑈)) ∧ ((𝑌𝐿𝑍) ∩ (𝑠𝐼𝑋)) ≠ ∅) → 𝑈 ≠ 𝑉)
162	1, 3, 5, 142, 160, 141, 139, 161	hlid 28950	. . . . . . . . . 10 ⊢ ((((((((𝜑 ∧ 𝑤 ∈ 𝑃) ∧ 𝑤((hlG‘𝐺)‘𝑉)𝑊) ∧ (𝑉 − 𝑤) = (𝑌 − 𝑍)) ∧ 𝑠 ∈ 𝑃) ∧ ⟨“𝑍𝑌𝑠”⟩ ∼ ⟨“𝑈𝑉𝑊”⟩) ∧ (𝑌 − 𝑠) = (𝑉 − 𝑈)) ∧ ((𝑌𝐿𝑍) ∩ (𝑠𝐼𝑋)) ≠ ∅) → 𝑈((hlG‘𝐺)‘𝑉)𝑈)
163	52	ad4antr 745	. . . . . . . . . 10 ⊢ ((((((((𝜑 ∧ 𝑤 ∈ 𝑃) ∧ 𝑤((hlG‘𝐺)‘𝑉)𝑊) ∧ (𝑉 − 𝑤) = (𝑌 − 𝑍)) ∧ 𝑠 ∈ 𝑃) ∧ ⟨“𝑍𝑌𝑠”⟩ ∼ ⟨“𝑈𝑉𝑊”⟩) ∧ (𝑌 − 𝑠) = (𝑉 − 𝑈)) ∧ ((𝑌𝐿𝑍) ∩ (𝑠𝐼𝑋)) ≠ ∅) → 𝑤((hlG‘𝐺)‘𝑉)𝑊)
164	41	ad4antr 745	. . . . . . . . . 10 ⊢ ((((((((𝜑 ∧ 𝑤 ∈ 𝑃) ∧ 𝑤((hlG‘𝐺)‘𝑉)𝑊) ∧ (𝑉 − 𝑤) = (𝑌 − 𝑍)) ∧ 𝑠 ∈ 𝑃) ∧ ⟨“𝑍𝑌𝑠”⟩ ∼ ⟨“𝑈𝑉𝑊”⟩) ∧ (𝑌 − 𝑠) = (𝑉 − 𝑈)) ∧ ((𝑌𝐿𝑍) ∩ (𝑠𝐼𝑋)) ≠ ∅) → (𝑉 − 𝑤) = (𝑌 − 𝑍))
165	1, 3, 5, 139, 142, 141, 149, 145, 144, 143, 159, 142, 2, 140, 162, 163, 148, 164	cgracgr 29200	. . . . . . . . 9 ⊢ ((((((((𝜑 ∧ 𝑤 ∈ 𝑃) ∧ 𝑤((hlG‘𝐺)‘𝑉)𝑊) ∧ (𝑉 − 𝑤) = (𝑌 − 𝑍)) ∧ 𝑠 ∈ 𝑃) ∧ ⟨“𝑍𝑌𝑠”⟩ ∼ ⟨“𝑈𝑉𝑊”⟩) ∧ (𝑌 − 𝑠) = (𝑉 − 𝑈)) ∧ ((𝑌𝐿𝑍) ∩ (𝑠𝐼𝑋)) ≠ ∅) → (𝑈 − 𝑤) = (𝑠 − 𝑍))
166	1, 2, 112, 139, 140, 141, 142, 143, 144, 145, 146, 148, 165	trgcgr 28854	. . . . . . . 8 ⊢ ((((((((𝜑 ∧ 𝑤 ∈ 𝑃) ∧ 𝑤((hlG‘𝐺)‘𝑉)𝑊) ∧ (𝑉 − 𝑤) = (𝑌 − 𝑍)) ∧ 𝑠 ∈ 𝑃) ∧ ⟨“𝑍𝑌𝑠”⟩ ∼ ⟨“𝑈𝑉𝑊”⟩) ∧ (𝑌 − 𝑠) = (𝑉 − 𝑈)) ∧ ((𝑌𝐿𝑍) ∩ (𝑠𝐼𝑋)) ≠ ∅) → ⟨“𝑤𝑉𝑈”⟩(cgrG‘𝐺)⟨“𝑍𝑌𝑠”⟩)
167	120	ad7antr 751	. . . . . . . . 9 ⊢ ((((((((𝜑 ∧ 𝑤 ∈ 𝑃) ∧ 𝑤((hlG‘𝐺)‘𝑉)𝑊) ∧ (𝑉 − 𝑤) = (𝑌 − 𝑍)) ∧ 𝑠 ∈ 𝑃) ∧ ⟨“𝑍𝑌𝑠”⟩ ∼ ⟨“𝑈𝑉𝑊”⟩) ∧ (𝑌 − 𝑠) = (𝑉 − 𝑈)) ∧ ((𝑌𝐿𝑍) ∩ (𝑠𝐼𝑋)) ≠ ∅) → ((hpG‘𝐺)‘(𝑌𝐿𝑍)) = ((hpG‘𝐺)‘(𝑍𝐿𝑌)))
168	27	ad7antr 751	. . . . . . . . . 10 ⊢ ((((((((𝜑 ∧ 𝑤 ∈ 𝑃) ∧ 𝑤((hlG‘𝐺)‘𝑉)𝑊) ∧ (𝑉 − 𝑤) = (𝑌 − 𝑍)) ∧ 𝑠 ∈ 𝑃) ∧ ⟨“𝑍𝑌𝑠”⟩ ∼ ⟨“𝑈𝑉𝑊”⟩) ∧ (𝑌 − 𝑠) = (𝑉 − 𝑈)) ∧ ((𝑌𝐿𝑍) ∩ (𝑠𝐼𝑋)) ≠ ∅) → (𝑌𝐿𝑍) ∈ ran 𝐿)
169	30	ad4antr 745	. . . . . . . . . 10 ⊢ ((((((((𝜑 ∧ 𝑤 ∈ 𝑃) ∧ 𝑤((hlG‘𝐺)‘𝑉)𝑊) ∧ (𝑉 − 𝑤) = (𝑌 − 𝑍)) ∧ 𝑠 ∈ 𝑃) ∧ ⟨“𝑍𝑌𝑠”⟩ ∼ ⟨“𝑈𝑉𝑊”⟩) ∧ (𝑌 − 𝑠) = (𝑉 − 𝑈)) ∧ ((𝑌𝐿𝑍) ∩ (𝑠𝐼𝑋)) ≠ ∅) → (((lInvG‘𝐺)‘(𝑌𝐿𝑍))‘𝑋) ∈ 𝑃)
170		simpr 490	. . . . . . . . . . . 12 ⊢ ((((((((𝜑 ∧ 𝑤 ∈ 𝑃) ∧ 𝑤((hlG‘𝐺)‘𝑉)𝑊) ∧ (𝑉 − 𝑤) = (𝑌 − 𝑍)) ∧ 𝑠 ∈ 𝑃) ∧ ⟨“𝑍𝑌𝑠”⟩ ∼ ⟨“𝑈𝑉𝑊”⟩) ∧ (𝑌 − 𝑠) = (𝑉 − 𝑈)) ∧ ((𝑌𝐿𝑍) ∩ (𝑠𝐼𝑋)) ≠ ∅) → ((𝑌𝐿𝑍) ∩ (𝑠𝐼𝑋)) ≠ ∅)
171	145	adantr 486	. . . . . . . . . . . . 13 ⊢ (((((((((𝜑 ∧ 𝑤 ∈ 𝑃) ∧ 𝑤((hlG‘𝐺)‘𝑉)𝑊) ∧ (𝑉 − 𝑤) = (𝑌 − 𝑍)) ∧ 𝑠 ∈ 𝑃) ∧ ⟨“𝑍𝑌𝑠”⟩ ∼ ⟨“𝑈𝑉𝑊”⟩) ∧ (𝑌 − 𝑠) = (𝑉 − 𝑈)) ∧ ((𝑌𝐿𝑍) ∩ (𝑠𝐼𝑋)) ≠ ∅) ∧ 𝑟 ∈ ((𝑌𝐿𝑍) ∩ (𝑠𝐼𝑋))) → 𝑠 ∈ 𝑃)
172	160	adantr 486	. . . . . . . . . . . . 13 ⊢ (((((((((𝜑 ∧ 𝑤 ∈ 𝑃) ∧ 𝑤((hlG‘𝐺)‘𝑉)𝑊) ∧ (𝑉 − 𝑤) = (𝑌 − 𝑍)) ∧ 𝑠 ∈ 𝑃) ∧ ⟨“𝑍𝑌𝑠”⟩ ∼ ⟨“𝑈𝑉𝑊”⟩) ∧ (𝑌 − 𝑠) = (𝑉 − 𝑈)) ∧ ((𝑌𝐿𝑍) ∩ (𝑠𝐼𝑋)) ≠ ∅) ∧ 𝑟 ∈ ((𝑌𝐿𝑍) ∩ (𝑠𝐼𝑋))) → 𝑋 ∈ 𝑃)
173		simpr 490	. . . . . . . . . . . . . 14 ⊢ (((((((((𝜑 ∧ 𝑤 ∈ 𝑃) ∧ 𝑤((hlG‘𝐺)‘𝑉)𝑊) ∧ (𝑉 − 𝑤) = (𝑌 − 𝑍)) ∧ 𝑠 ∈ 𝑃) ∧ ⟨“𝑍𝑌𝑠”⟩ ∼ ⟨“𝑈𝑉𝑊”⟩) ∧ (𝑌 − 𝑠) = (𝑉 − 𝑈)) ∧ ((𝑌𝐿𝑍) ∩ (𝑠𝐼𝑋)) ≠ ∅) ∧ 𝑟 ∈ ((𝑌𝐿𝑍) ∩ (𝑠𝐼𝑋))) → 𝑟 ∈ ((𝑌𝐿𝑍) ∩ (𝑠𝐼𝑋)))
174	173	elin1d 4153	. . . . . . . . . . . . 13 ⊢ (((((((((𝜑 ∧ 𝑤 ∈ 𝑃) ∧ 𝑤((hlG‘𝐺)‘𝑉)𝑊) ∧ (𝑉 − 𝑤) = (𝑌 − 𝑍)) ∧ 𝑠 ∈ 𝑃) ∧ ⟨“𝑍𝑌𝑠”⟩ ∼ ⟨“𝑈𝑉𝑊”⟩) ∧ (𝑌 − 𝑠) = (𝑉 − 𝑈)) ∧ ((𝑌𝐿𝑍) ∩ (𝑠𝐼𝑋)) ≠ ∅) ∧ 𝑟 ∈ ((𝑌𝐿𝑍) ∩ (𝑠𝐼𝑋))) → 𝑟 ∈ (𝑌𝐿𝑍))
175	31	ad4antr 745	. . . . . . . . . . . . . . . . 17 ⊢ ((((((((𝜑 ∧ 𝑤 ∈ 𝑃) ∧ 𝑤((hlG‘𝐺)‘𝑉)𝑊) ∧ (𝑉 − 𝑤) = (𝑌 − 𝑍)) ∧ 𝑠 ∈ 𝑃) ∧ ⟨“𝑍𝑌𝑠”⟩ ∼ ⟨“𝑈𝑉𝑊”⟩) ∧ (𝑌 − 𝑠) = (𝑉 − 𝑈)) ∧ ((𝑌𝐿𝑍) ∩ (𝑠𝐼𝑋)) ≠ ∅) → ¬ (𝑊 ∈ (𝑈𝐿𝑉) ∨ 𝑈 = 𝑉))
176	1, 3, 2, 139, 142, 141, 149, 145, 144, 143, 159, 4, 175	cgrancol 29212	. . . . . . . . . . . . . . . 16 ⊢ ((((((((𝜑 ∧ 𝑤 ∈ 𝑃) ∧ 𝑤((hlG‘𝐺)‘𝑉)𝑊) ∧ (𝑉 − 𝑤) = (𝑌 − 𝑍)) ∧ 𝑠 ∈ 𝑃) ∧ ⟨“𝑍𝑌𝑠”⟩ ∼ ⟨“𝑈𝑉𝑊”⟩) ∧ (𝑌 − 𝑠) = (𝑉 − 𝑈)) ∧ ((𝑌𝐿𝑍) ∩ (𝑠𝐼𝑋)) ≠ ∅) → ¬ (𝑍 ∈ (𝑠𝐿𝑌) ∨ 𝑠 = 𝑌))
177	1, 4, 3, 139, 145, 144, 143, 176	ncolrot1 28900	. . . . . . . . . . . . . . 15 ⊢ ((((((((𝜑 ∧ 𝑤 ∈ 𝑃) ∧ 𝑤((hlG‘𝐺)‘𝑉)𝑊) ∧ (𝑉 − 𝑤) = (𝑌 − 𝑍)) ∧ 𝑠 ∈ 𝑃) ∧ ⟨“𝑍𝑌𝑠”⟩ ∼ ⟨“𝑈𝑉𝑊”⟩) ∧ (𝑌 − 𝑠) = (𝑉 − 𝑈)) ∧ ((𝑌𝐿𝑍) ∩ (𝑠𝐼𝑋)) ≠ ∅) → ¬ (𝑠 ∈ (𝑌𝐿𝑍) ∨ 𝑌 = 𝑍))
178	177	orsild 1019	. . . . . . . . . . . . . 14 ⊢ ((((((((𝜑 ∧ 𝑤 ∈ 𝑃) ∧ 𝑤((hlG‘𝐺)‘𝑉)𝑊) ∧ (𝑉 − 𝑤) = (𝑌 − 𝑍)) ∧ 𝑠 ∈ 𝑃) ∧ ⟨“𝑍𝑌𝑠”⟩ ∼ ⟨“𝑈𝑉𝑊”⟩) ∧ (𝑌 − 𝑠) = (𝑉 − 𝑈)) ∧ ((𝑌𝐿𝑍) ∩ (𝑠𝐼𝑋)) ≠ ∅) → ¬ 𝑠 ∈ (𝑌𝐿𝑍))
179	178	adantr 486	. . . . . . . . . . . . 13 ⊢ (((((((((𝜑 ∧ 𝑤 ∈ 𝑃) ∧ 𝑤((hlG‘𝐺)‘𝑉)𝑊) ∧ (𝑉 − 𝑤) = (𝑌 − 𝑍)) ∧ 𝑠 ∈ 𝑃) ∧ ⟨“𝑍𝑌𝑠”⟩ ∼ ⟨“𝑈𝑉𝑊”⟩) ∧ (𝑌 − 𝑠) = (𝑉 − 𝑈)) ∧ ((𝑌𝐿𝑍) ∩ (𝑠𝐼𝑋)) ≠ ∅) ∧ 𝑟 ∈ ((𝑌𝐿𝑍) ∩ (𝑠𝐼𝑋))) → ¬ 𝑠 ∈ (𝑌𝐿𝑍))
180	76	ad8antr 753	. . . . . . . . . . . . 13 ⊢ (((((((((𝜑 ∧ 𝑤 ∈ 𝑃) ∧ 𝑤((hlG‘𝐺)‘𝑉)𝑊) ∧ (𝑉 − 𝑤) = (𝑌 − 𝑍)) ∧ 𝑠 ∈ 𝑃) ∧ ⟨“𝑍𝑌𝑠”⟩ ∼ ⟨“𝑈𝑉𝑊”⟩) ∧ (𝑌 − 𝑠) = (𝑉 − 𝑈)) ∧ ((𝑌𝐿𝑍) ∩ (𝑠𝐼𝑋)) ≠ ∅) ∧ 𝑟 ∈ ((𝑌𝐿𝑍) ∩ (𝑠𝐼𝑋))) → ¬ 𝑋 ∈ (𝑌𝐿𝑍))
181	173	elin2d 4154	. . . . . . . . . . . . 13 ⊢ (((((((((𝜑 ∧ 𝑤 ∈ 𝑃) ∧ 𝑤((hlG‘𝐺)‘𝑉)𝑊) ∧ (𝑉 − 𝑤) = (𝑌 − 𝑍)) ∧ 𝑠 ∈ 𝑃) ∧ ⟨“𝑍𝑌𝑠”⟩ ∼ ⟨“𝑈𝑉𝑊”⟩) ∧ (𝑌 − 𝑠) = (𝑉 − 𝑈)) ∧ ((𝑌𝐿𝑍) ∩ (𝑠𝐼𝑋)) ≠ ∅) ∧ 𝑟 ∈ ((𝑌𝐿𝑍) ∩ (𝑠𝐼𝑋))) → 𝑟 ∈ (𝑠𝐼𝑋))
182	1, 2, 3, 75, 171, 172, 174, 179, 180, 181	islnoppd 29091	. . . . . . . . . . . 12 ⊢ (((((((((𝜑 ∧ 𝑤 ∈ 𝑃) ∧ 𝑤((hlG‘𝐺)‘𝑉)𝑊) ∧ (𝑉 − 𝑤) = (𝑌 − 𝑍)) ∧ 𝑠 ∈ 𝑃) ∧ ⟨“𝑍𝑌𝑠”⟩ ∼ ⟨“𝑈𝑉𝑊”⟩) ∧ (𝑌 − 𝑠) = (𝑉 − 𝑈)) ∧ ((𝑌𝐿𝑍) ∩ (𝑠𝐼𝑋)) ≠ ∅) ∧ 𝑟 ∈ ((𝑌𝐿𝑍) ∩ (𝑠𝐼𝑋))) → 𝑠{⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝑃 ∖ (𝑌𝐿𝑍)) ∧ 𝑏 ∈ (𝑃 ∖ (𝑌𝐿𝑍))) ∧ ∃𝑠 ∈ (𝑌𝐿𝑍)𝑠 ∈ (𝑎𝐼𝑏))}𝑋)
183	170, 182	n0limd 4304	. . . . . . . . . . 11 ⊢ ((((((((𝜑 ∧ 𝑤 ∈ 𝑃) ∧ 𝑤((hlG‘𝐺)‘𝑉)𝑊) ∧ (𝑉 − 𝑤) = (𝑌 − 𝑍)) ∧ 𝑠 ∈ 𝑃) ∧ ⟨“𝑍𝑌𝑠”⟩ ∼ ⟨“𝑈𝑉𝑊”⟩) ∧ (𝑌 − 𝑠) = (𝑉 − 𝑈)) ∧ ((𝑌𝐿𝑍) ∩ (𝑠𝐼𝑋)) ≠ ∅) → 𝑠{⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝑃 ∖ (𝑌𝐿𝑍)) ∧ 𝑏 ∈ (𝑃 ∖ (𝑌𝐿𝑍))) ∧ ∃𝑠 ∈ (𝑌𝐿𝑍)𝑠 ∈ (𝑎𝐼𝑏))}𝑋)
184	77	ad7antr 751	. . . . . . . . . . . . 13 ⊢ ((((((((𝜑 ∧ 𝑤 ∈ 𝑃) ∧ 𝑤((hlG‘𝐺)‘𝑉)𝑊) ∧ (𝑉 − 𝑤) = (𝑌 − 𝑍)) ∧ 𝑠 ∈ 𝑃) ∧ ⟨“𝑍𝑌𝑠”⟩ ∼ ⟨“𝑈𝑉𝑊”⟩) ∧ (𝑌 − 𝑠) = (𝑉 − 𝑈)) ∧ ((𝑌𝐿𝑍) ∩ (𝑠𝐼𝑋)) ≠ ∅) → 𝑋{⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝑃 ∖ (𝑌𝐿𝑍)) ∧ 𝑏 ∈ (𝑃 ∖ (𝑌𝐿𝑍))) ∧ ∃𝑠 ∈ (𝑌𝐿𝑍)𝑠 ∈ (𝑎𝐼𝑏))} (((lInvG‘𝐺)‘(𝑌𝐿𝑍))‘𝑋))
185	1, 2, 3, 75, 4, 168, 139, 160, 169, 184	oppcom 29095	. . . . . . . . . . . 12 ⊢ ((((((((𝜑 ∧ 𝑤 ∈ 𝑃) ∧ 𝑤((hlG‘𝐺)‘𝑉)𝑊) ∧ (𝑉 − 𝑤) = (𝑌 − 𝑍)) ∧ 𝑠 ∈ 𝑃) ∧ ⟨“𝑍𝑌𝑠”⟩ ∼ ⟨“𝑈𝑉𝑊”⟩) ∧ (𝑌 − 𝑠) = (𝑉 − 𝑈)) ∧ ((𝑌𝐿𝑍) ∩ (𝑠𝐼𝑋)) ≠ ∅) → (((lInvG‘𝐺)‘(𝑌𝐿𝑍))‘𝑋){⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝑃 ∖ (𝑌𝐿𝑍)) ∧ 𝑏 ∈ (𝑃 ∖ (𝑌𝐿𝑍))) ∧ ∃𝑠 ∈ (𝑌𝐿𝑍)𝑠 ∈ (𝑎𝐼𝑏))}𝑋)
186	1, 3, 4, 75, 139, 168, 169, 145, 160, 185	lnopp2hpgb 29116	. . . . . . . . . . 11 ⊢ ((((((((𝜑 ∧ 𝑤 ∈ 𝑃) ∧ 𝑤((hlG‘𝐺)‘𝑉)𝑊) ∧ (𝑉 − 𝑤) = (𝑌 − 𝑍)) ∧ 𝑠 ∈ 𝑃) ∧ ⟨“𝑍𝑌𝑠”⟩ ∼ ⟨“𝑈𝑉𝑊”⟩) ∧ (𝑌 − 𝑠) = (𝑉 − 𝑈)) ∧ ((𝑌𝐿𝑍) ∩ (𝑠𝐼𝑋)) ≠ ∅) → (𝑠{⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝑃 ∖ (𝑌𝐿𝑍)) ∧ 𝑏 ∈ (𝑃 ∖ (𝑌𝐿𝑍))) ∧ ∃𝑠 ∈ (𝑌𝐿𝑍)𝑠 ∈ (𝑎𝐼𝑏))}𝑋 ↔ (((lInvG‘𝐺)‘(𝑌𝐿𝑍))‘𝑋)((hpG‘𝐺)‘(𝑌𝐿𝑍))𝑠))
187	183, 186	mpbid 235	. . . . . . . . . 10 ⊢ ((((((((𝜑 ∧ 𝑤 ∈ 𝑃) ∧ 𝑤((hlG‘𝐺)‘𝑉)𝑊) ∧ (𝑉 − 𝑤) = (𝑌 − 𝑍)) ∧ 𝑠 ∈ 𝑃) ∧ ⟨“𝑍𝑌𝑠”⟩ ∼ ⟨“𝑈𝑉𝑊”⟩) ∧ (𝑌 − 𝑠) = (𝑉 − 𝑈)) ∧ ((𝑌𝐿𝑍) ∩ (𝑠𝐼𝑋)) ≠ ∅) → (((lInvG‘𝐺)‘(𝑌𝐿𝑍))‘𝑋)((hpG‘𝐺)‘(𝑌𝐿𝑍))𝑠)
188	1, 3, 4, 139, 168, 169, 75, 145, 187	hpgcom 29120	. . . . . . . . 9 ⊢ ((((((((𝜑 ∧ 𝑤 ∈ 𝑃) ∧ 𝑤((hlG‘𝐺)‘𝑉)𝑊) ∧ (𝑉 − 𝑤) = (𝑌 − 𝑍)) ∧ 𝑠 ∈ 𝑃) ∧ ⟨“𝑍𝑌𝑠”⟩ ∼ ⟨“𝑈𝑉𝑊”⟩) ∧ (𝑌 − 𝑠) = (𝑉 − 𝑈)) ∧ ((𝑌𝐿𝑍) ∩ (𝑠𝐼𝑋)) ≠ ∅) → 𝑠((hpG‘𝐺)‘(𝑌𝐿𝑍))(((lInvG‘𝐺)‘(𝑌𝐿𝑍))‘𝑋))
189	167, 188	breqdi 5122	. . . . . . . 8 ⊢ ((((((((𝜑 ∧ 𝑤 ∈ 𝑃) ∧ 𝑤((hlG‘𝐺)‘𝑉)𝑊) ∧ (𝑉 − 𝑤) = (𝑌 − 𝑍)) ∧ 𝑠 ∈ 𝑃) ∧ ⟨“𝑍𝑌𝑠”⟩ ∼ ⟨“𝑈𝑉𝑊”⟩) ∧ (𝑌 − 𝑠) = (𝑉 − 𝑈)) ∧ ((𝑌𝐿𝑍) ∩ (𝑠𝐼𝑋)) ≠ ∅) → 𝑠((hpG‘𝐺)‘(𝑍𝐿𝑌))(((lInvG‘𝐺)‘(𝑌𝐿𝑍))‘𝑋))
190	166, 189	jca 521	. . . . . . 7 ⊢ ((((((((𝜑 ∧ 𝑤 ∈ 𝑃) ∧ 𝑤((hlG‘𝐺)‘𝑉)𝑊) ∧ (𝑉 − 𝑤) = (𝑌 − 𝑍)) ∧ 𝑠 ∈ 𝑃) ∧ ⟨“𝑍𝑌𝑠”⟩ ∼ ⟨“𝑈𝑉𝑊”⟩) ∧ (𝑌 − 𝑠) = (𝑉 − 𝑈)) ∧ ((𝑌𝐿𝑍) ∩ (𝑠𝐼𝑋)) ≠ ∅) → (⟨“𝑤𝑉𝑈”⟩(cgrG‘𝐺)⟨“𝑍𝑌𝑠”⟩ ∧ 𝑠((hpG‘𝐺)‘(𝑍𝐿𝑌))(((lInvG‘𝐺)‘(𝑌𝐿𝑍))‘𝑋)))
191	190	3anasss 1380	. . . . . 6 ⊢ ((((((𝜑 ∧ 𝑤 ∈ 𝑃) ∧ 𝑤((hlG‘𝐺)‘𝑉)𝑊) ∧ (𝑉 − 𝑤) = (𝑌 − 𝑍)) ∧ 𝑠 ∈ 𝑃) ∧ (⟨“𝑍𝑌𝑠”⟩ ∼ ⟨“𝑈𝑉𝑊”⟩ ∧ (𝑌 − 𝑠) = (𝑉 − 𝑈) ∧ ((𝑌𝐿𝑍) ∩ (𝑠𝐼𝑋)) ≠ ∅)) → (⟨“𝑤𝑉𝑈”⟩(cgrG‘𝐺)⟨“𝑍𝑌𝑠”⟩ ∧ 𝑠((hpG‘𝐺)‘(𝑍𝐿𝑌))(((lInvG‘𝐺)‘(𝑌𝐿𝑍))‘𝑋)))
192	138, 191	impbida 813	. . . . 5 ⊢ (((((𝜑 ∧ 𝑤 ∈ 𝑃) ∧ 𝑤((hlG‘𝐺)‘𝑉)𝑊) ∧ (𝑉 − 𝑤) = (𝑌 − 𝑍)) ∧ 𝑠 ∈ 𝑃) → ((⟨“𝑤𝑉𝑈”⟩(cgrG‘𝐺)⟨“𝑍𝑌𝑠”⟩ ∧ 𝑠((hpG‘𝐺)‘(𝑍𝐿𝑌))(((lInvG‘𝐺)‘(𝑌𝐿𝑍))‘𝑋)) ↔ (⟨“𝑍𝑌𝑠”⟩ ∼ ⟨“𝑈𝑉𝑊”⟩ ∧ (𝑌 − 𝑠) = (𝑉 − 𝑈) ∧ ((𝑌𝐿𝑍) ∩ (𝑠𝐼𝑋)) ≠ ∅)))
193	192	reubidva 3381	. . . 4 ⊢ ((((𝜑 ∧ 𝑤 ∈ 𝑃) ∧ 𝑤((hlG‘𝐺)‘𝑉)𝑊) ∧ (𝑉 − 𝑤) = (𝑌 − 𝑍)) → (∃!𝑠 ∈ 𝑃 (⟨“𝑤𝑉𝑈”⟩(cgrG‘𝐺)⟨“𝑍𝑌𝑠”⟩ ∧ 𝑠((hpG‘𝐺)‘(𝑍𝐿𝑌))(((lInvG‘𝐺)‘(𝑌𝐿𝑍))‘𝑋)) ↔ ∃!𝑠 ∈ 𝑃 (⟨“𝑍𝑌𝑠”⟩ ∼ ⟨“𝑈𝑉𝑊”⟩ ∧ (𝑌 − 𝑠) = (𝑉 − 𝑈) ∧ ((𝑌𝐿𝑍) ∩ (𝑠𝐼𝑋)) ≠ ∅)))
194	88, 193	mpbid 235	. . 3 ⊢ ((((𝜑 ∧ 𝑤 ∈ 𝑃) ∧ 𝑤((hlG‘𝐺)‘𝑉)𝑊) ∧ (𝑉 − 𝑤) = (𝑌 − 𝑍)) → ∃!𝑠 ∈ 𝑃 (⟨“𝑍𝑌𝑠”⟩ ∼ ⟨“𝑈𝑉𝑊”⟩ ∧ (𝑌 − 𝑠) = (𝑉 − 𝑈) ∧ ((𝑌𝐿𝑍) ∩ (𝑠𝐼𝑋)) ≠ ∅))
195	194	anasss 472	. 2 ⊢ (((𝜑 ∧ 𝑤 ∈ 𝑃) ∧ (𝑤((hlG‘𝐺)‘𝑉)𝑊 ∧ (𝑉 − 𝑤) = (𝑌 − 𝑍))) → ∃!𝑠 ∈ 𝑃 (⟨“𝑍𝑌𝑠”⟩ ∼ ⟨“𝑈𝑉𝑊”⟩ ∧ (𝑌 − 𝑠) = (𝑉 − 𝑈) ∧ ((𝑌𝐿𝑍) ∩ (𝑠𝐼𝑋)) ≠ ∅))
196	19	necomd 3012	. . 3 ⊢ (𝜑 → 𝑊 ≠ 𝑉)
197	1, 3, 5, 9, 15, 13, 6, 17, 2, 196, 26	hlcgrex 28957	. 2 ⊢ (𝜑 → ∃𝑤 ∈ 𝑃 (𝑤((hlG‘𝐺)‘𝑉)𝑊 ∧ (𝑉 − 𝑤) = (𝑌 − 𝑍)))
198	195, 197	r19.29a 3172	1 ⊢ (𝜑 → ∃!𝑠 ∈ 𝑃 (⟨“𝑍𝑌𝑠”⟩ ∼ ⟨“𝑈𝑉𝑊”⟩ ∧ (𝑌 − 𝑠) = (𝑉 − 𝑈) ∧ ((𝑌𝐿𝑍) ∩ (𝑠𝐼𝑋)) ≠ ∅))

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 401 ∨ wo 861 ∧ w3a 1103 = wceq 1570 ∈ wcel 2145 ≠ wne 2957 ∃wrex 3088 ∃!wreu 3365 {crab 3414 ∖ cdif 3899 ∩ cin 3901 ∅c0 4282 class class class wbr 5107 {copab 5171 ran crn 5660 ‘cfv 6537 (class class class)co 7416 ↑_m cmap 8829 0cc0 11125 1c1 11126 2c2 12320 3c3 12321 ..^cfzo 13709 ⟨“cs3 14913 Basecbs 17303 distcds 17353 TarskiGcstrkg 28764 Itvcitv 28770 LineGclng 28771 cgrGccgrg 28848 hlGchlg 28938 hpGchpg 29110 lInvGclmi 29153 cgrAccgra 29189

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 2215 ax-ext 2734 ax-rep 5236 ax-sep 5255 ax-nul 5267 ax-pow 5334 ax-pr 5402 ax-un 7739 ax-cnex 11181 ax-resscn 11182 ax-1cn 11183 ax-icn 11184 ax-addcl 11185 ax-addrcl 11186 ax-mulcl 11187 ax-mulrcl 11188 ax-mulcom 11189 ax-addass 11190 ax-mulass 11191 ax-distr 11192 ax-i2m1 11193 ax-1ne0 11194 ax-1rid 11195 ax-rnegex 11196 ax-rrecex 11197 ax-cnre 11198 ax-pre-lttri 11199 ax-pre-lttrn 11200 ax-pre-ltadd 11201 ax-pre-mulgt0 11202

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 2566 df-eu 2596 df-clab 2741 df-cleq 2754 df-clel 2837 df-nfc 2911 df-ne 2958 df-nel 3064 df-ral 3079 df-rex 3089 df-rmo 3367 df-reu 3368 df-rab 3415 df-v 3455 df-sbc 3743 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-nul 4283 df-if 4486 df-pw 4562 df-sn 4588 df-pr 4590 df-tp 4592 df-op 4594 df-uni 4871 df-int 4911 df-iun 4956 df-br 5108 df-opab 5172 df-mpt 5191 df-tr 5217 df-id 5554 df-eprel 5559 df-po 5567 df-so 5568 df-fr 5612 df-we 5614 df-xp 5665 df-rel 5666 df-cnv 5667 df-co 5668 df-dm 5669 df-rn 5670 df-res 5671 df-ima 5672 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 7373 df-ov 7419 df-oprab 7420 df-mpo 7421 df-om 7866 df-1st 7989 df-2nd 7990 df-frecs 8283 df-wrecs 8314 df-recs 8363 df-rdg 8402 df-1o 8458 df-oadd 8462 df-er 8699 df-map 8831 df-pm 8832 df-en 8956 df-dom 8957 df-sdom 8958 df-fin 8959 df-dju 9909 df-card 9947 df-pnf 11270 df-mnf 11271 df-xr 11272 df-ltxr 11273 df-le 11274 df-sub 11468 df-neg 11469 df-nn 12259 df-2 12328 df-3 12329 df-n0 12530 df-xnn0 12603 df-z 12617 df-uz 12889 df-fz 13562 df-fzo 13710 df-hash 14395 df-word 14579 df-concat 14636 df-s1 14663 df-s2 14919 df-s3 14920 df-trkgc 28785 df-trkgb 28786 df-trkgcb 28787 df-trkgld 28789 df-trkg 28790 df-cgrg 28849 df-ismt 28871 df-leg 28921 df-hlg 28939 df-mir 29000 df-rag 29044 df-perpg 29046 df-hpg 29111 df-mid 29154 df-lmi 29155 df-cgra 29190

This theorem is used by: angmndaddov1lem 29257

W3C validator