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Theorem angmgmaddeu1 29313

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
angmgmadd.p	⊢ 𝑃 = (Base‘𝐺)
angmgmadd.a	⊢ 𝐴 = {𝑑 ∈ (𝑃 ↑_m (0..^3)) ∣ ((𝑑‘0) ≠ (𝑑‘1) ∧ (𝑑‘1) ≠ (𝑑‘2))}
angmgmadd.i	⊢ 𝐼 = (Itv‘𝐺)
angmgmadd.d	⊢ − = (dist‘𝐺)
angmgmadd.c	⊢ ∼ = (cgrA‘𝐺)
angmgmadd.l	⊢ 𝐿 = (LineG‘𝐺)
angmgmadd.g	⊢ (𝜑 → 𝐺 ∈ TarskiG)
angmgmaddov.u	⊢ (𝜑 → 𝑈 ∈ 𝑃)
angmgmaddov.v	⊢ (𝜑 → 𝑉 ∈ 𝑃)
angmgmaddov.w	⊢ (𝜑 → 𝑊 ∈ 𝑃)
angmgmaddov.x	⊢ (𝜑 → 𝑋 ∈ 𝑃)
angmgmaddov.y	⊢ (𝜑 → 𝑌 ∈ 𝑃)
angmgmaddov.z	⊢ (𝜑 → 𝑍 ∈ 𝑃)
angmgmaddeu.1	⊢ (𝜑 → 𝑈 ≠ 𝑉)
angmgmaddeu.2	⊢ (𝜑 → 𝑉 ≠ 𝑊)
angmgmaddeu.3	⊢ (𝜑 → 𝑋 ≠ 𝑌)
angmgmaddeu.4	⊢ (𝜑 → 𝑌 ≠ 𝑍)
angmgmaddeu1.1	⊢ (𝜑 → ¬ 𝑋 ∈ (𝑌𝐿𝑍))
angmgmaddeu1.2	⊢ (𝜑 → ¬ 𝑈 ∈ (𝑉𝐿𝑊))

**Assertion**
Ref	Expression
angmgmaddeu1	⊢ (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 angmgmaddeu1 Dummy variables 𝑐 𝑡 𝑤 𝑟 𝑎 𝑏 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		angmgmadd.p	. . . . 5 ⊢ 𝑃 = (Base‘𝐺)
2		angmgmadd.d	. . . . 5 ⊢ − = (dist‘𝐺)
3		angmgmadd.i	. . . . 5 ⊢ 𝐼 = (Itv‘𝐺)
4		angmgmadd.l	. . . . 5 ⊢ 𝐿 = (LineG‘𝐺)
5		eqid 2760	. . . . 5 ⊢ (hlG‘𝐺) = (hlG‘𝐺)
6		angmgmadd.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		angmgmaddov.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		angmgmaddov.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		angmgmaddov.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		angmgmaddov.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		angmgmaddov.w	. . . . . . . 8 ⊢ (𝜑 → 𝑊 ∈ 𝑃)
18		angmgmaddeu1.2	. . . . . . . . . 10 ⊢ (𝜑 → ¬ 𝑈 ∈ (𝑉𝐿𝑊))
19		angmgmaddeu.2	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑉 ≠ 𝑊)
20	19	neneqd 2960	. . . . . . . . . 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 28960	. . . . . . . 8 ⊢ (𝜑 → ¬ (𝑊 ∈ (𝑈𝐿𝑉) ∨ 𝑈 = 𝑉))
24	1, 4, 3, 6, 11, 9, 17, 23	ncoltgdim2 28962	. . . . . . 7 ⊢ (𝜑 → 𝐺Dim_TarskiG≥2)
25		eqid 2760	. . . . . . 7 ⊢ ((lInvG‘𝐺)‘(𝑌𝐿𝑍)) = ((lInvG‘𝐺)‘(𝑌𝐿𝑍))
26		angmgmaddeu.4	. . . . . . . 8 ⊢ (𝜑 → 𝑌 ≠ 𝑍)
27	1, 3, 4, 6, 15, 13, 26	tgelrnln 29032	. . . . . . 7 ⊢ (𝜑 → (𝑌𝐿𝑍) ∈ ran 𝐿)
28		angmgmaddov.x	. . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ 𝑃)
29	1, 2, 3, 6, 24, 25, 4, 27, 28	lmicl 29225	. . . . . 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		angmgmaddeu.1	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑈 ≠ 𝑉)
36	35	neneqd 2960	. . . . . . . . . . . 12 ⊢ (𝜑 → ¬ 𝑈 = 𝑉)
37	36	neqcomd 2770	. . . . . . . . . . 11 ⊢ (𝜑 → ¬ 𝑉 = 𝑈)
38	37	neqned 2962	. . . . . . . . . 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 2766	. . . . . . . . . . . 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 28879	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑤 ∈ 𝑃) ∧ 𝑤((hlG‘𝐺)‘𝑉)𝑊) ∧ (𝑉 − 𝑤) = (𝑌 − 𝑍)) → 𝑉 ≠ 𝑤)
45	44	necomd 3010	. . . . . . . . . 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 29004	. . . . . . . . . . . 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 29007	. . . . . . . . . . 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 29024	. . . . . . . . . 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 29033	. . . . . . . 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 29037	. . . . . . . . 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 2862	. . . . . . 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 2848	. . . . . . . . . . . . . 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 2848	. . . . . . . . . . . . . 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 7417	. . . . . . . . . . . . . . 15 ⊢ ((𝑎 = 𝑐 ∧ 𝑏 = 𝑑) → (𝑎𝐼𝑏) = (𝑐𝐼𝑑))
69	68	eleq2d 2846	. . . . . . . . . . . . . 14 ⊢ ((𝑎 = 𝑐 ∧ 𝑏 = 𝑑) → (𝑠 ∈ (𝑎𝐼𝑏) ↔ 𝑠 ∈ (𝑐𝐼𝑑)))
70	69	rexbidv 3186	. . . . . . . . . . . . 13 ⊢ ((𝑎 = 𝑐 ∧ 𝑏 = 𝑑) → (∃𝑠 ∈ (𝑌𝐿𝑍)𝑠 ∈ (𝑎𝐼𝑏) ↔ ∃𝑠 ∈ (𝑌𝐿𝑍)𝑠 ∈ (𝑐𝐼𝑑)))
71		eleq1 2848	. . . . . . . . . . . . . 14 ⊢ (𝑠 = 𝑡 → (𝑠 ∈ (𝑐𝐼𝑑) ↔ 𝑡 ∈ (𝑐𝐼𝑑)))
72	71	cbvrexvw 3241	. . . . . . . . . . . . 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 5177	. . . . . . . . . 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		angmgmaddeu1.1	. . . . . . . . . . 11 ⊢ (𝜑 → ¬ 𝑋 ∈ (𝑌𝐿𝑍))
77	1, 2, 3, 4, 6, 24, 27, 75, 25, 28, 76	lmiopp 29242	. . . . . . . . . 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 29152	. . . . . . . . 9 ⊢ (𝜑 → ¬ (((lInvG‘𝐺)‘(𝑌𝐿𝑍))‘𝑋) ∈ (𝑌𝐿𝑍))
79	1, 3, 4, 6, 15, 13, 26	tglinecom 29037	. . . . . . . . 9 ⊢ (𝜑 → (𝑌𝐿𝑍) = (𝑍𝐿𝑌))
80	78, 79	neleqtrd 2882	. . . . . . . 8 ⊢ (𝜑 → ¬ (((lInvG‘𝐺)‘(𝑌𝐿𝑍))‘𝑋) ∈ (𝑍𝐿𝑌))
81	26	necomd 3010	. . . . . . . . 9 ⊢ (𝜑 → 𝑍 ≠ 𝑌)
82	81	neneqd 2960	. . . . . . . 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 28959	. . . . . 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 28876	. . . . 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 29247	. . . 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		angmgmadd.c	. . . . . . . . . . 11 ⊢ ∼ = (cgrA‘𝐺)
90	89	eqcomi 2769	. . . . . . . . . 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 29261	. . . . . . . . . . . 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 29262	. . . . . . . . . . . 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 29264	. . . . . . . . . . 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 29263	. . . . . . . . . 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 29258	. . . . . . . . 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 5117	. . . . . . . 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 2760	. . . . . . . . . 10 ⊢ (cgrG‘𝐺) = (cgrG‘𝐺)
113	1, 2, 3, 112, 92, 98, 97, 96, 93, 94, 95, 104	cgr3simp2 28918	. . . . . . . . 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 2766	. . . . . . . 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 2761	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (hpG‘𝐺) = (hpG‘𝐺))
120	119, 79	fveq12d 6880	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((hpG‘𝐺)‘(𝑌𝐿𝑍)) = ((hpG‘𝐺)‘(𝑍𝐿𝑌)))
121	120	eqcomd 2766	. . . . . . . . . . . . 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 5117	. . . . . . . . . . 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 29179	. . . . . . . . . 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 29154	. . . . . . . . . . . 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 29175	. . . . . . . . . . 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 29165	. . . . . . . 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 2766	. . . . . . . . 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 28879	. . . . . . . . . . . . . 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 3010	. . . . . . . . . . . . 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 29261	. . . . . . . . . . . 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 5117	. . . . . . . . . . . 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 29264	. . . . . . . . . . 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 29263	. . . . . . . . . 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 29009	. . . . . . . . . 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 29259	. . . . . . . . 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 28913	. . . . . . . 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 4149	. . . . . . . . . . . . 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 29271	. . . . . . . . . . . . . . . 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 28959	. . . . . . . . . . . . . . 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 4150	. . . . . . . . . . . . 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 29150	. . . . . . . . . . . 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 4300	. . . . . . . . . . 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 29154	. . . . . . . . . . . 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 29175	. . . . . . . . . . 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 29179	. . . . . . . . 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 5117	. . . . . . . 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 3379	. . . 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 3010	. . 3 ⊢ (𝜑 → 𝑊 ≠ 𝑉)
197	1, 3, 5, 9, 15, 13, 6, 17, 2, 196, 26	hlcgrex 29016	. 2 ⊢ (𝜑 → ∃𝑤 ∈ 𝑃 (𝑤((hlG‘𝐺)‘𝑉)𝑊 ∧ (𝑉 − 𝑤) = (𝑌 − 𝑍)))
198	195, 197	r19.29a 3170	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 2955 ∃wrex 3086 ∃!wreu 3363 {crab 3412 ∖ cdif 3895 ∩ cin 3897 ∅c0 4278 class class class wbr 5102 {copab 5166 ran crn 5648 ‘cfv 6527 (class class class)co 7408 ↑_m cmap 8825 0cc0 11172 1c1 11173 2c2 12367 3c3 12368 ..^cfzo 13757 ⟨“cs3 14961 Basecbs 17349 distcds 17399 TarskiGcstrkg 28823 Itvcitv 28829 LineGclng 28830 cgrGccgrg 28907 hlGchlg 28997 hpGchpg 29169 lInvGclmi 29212 cgrAccgra 29248

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 2732 ax-rep 5231 ax-sep 5248 ax-nul 5259 ax-pow 5326 ax-pr 5390 ax-un 7734 ax-cnex 11228 ax-resscn 11229 ax-1cn 11230 ax-icn 11231 ax-addcl 11232 ax-addrcl 11233 ax-mulcl 11234 ax-mulrcl 11235 ax-mulcom 11236 ax-addass 11237 ax-mulass 11238 ax-distr 11239 ax-i2m1 11240 ax-1ne0 11241 ax-1rid 11242 ax-rnegex 11243 ax-rrecex 11244 ax-cnre 11245 ax-pre-lttri 11246 ax-pre-lttrn 11247 ax-pre-ltadd 11248 ax-pre-mulgt0 11249

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 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3739 df-csb 3847 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-pss 3918 df-nul 4279 df-if 4482 df-pw 4558 df-sn 4584 df-pr 4586 df-tp 4588 df-op 4590 df-uni 4867 df-int 4907 df-iun 4952 df-br 5103 df-opab 5167 df-mpt 5186 df-tr 5212 df-id 5542 df-eprel 5547 df-po 5555 df-so 5556 df-fr 5600 df-we 5602 df-xp 5653 df-rel 5654 df-cnv 5655 df-co 5656 df-dm 5657 df-rn 5658 df-res 5659 df-ima 5660 df-pred 6293 df-ord 6354 df-on 6355 df-lim 6356 df-suc 6357 df-iota 6483 df-fun 6529 df-fn 6530 df-f 6531 df-f1 6532 df-fo 6533 df-f1o 6534 df-fv 6535 df-riota 7365 df-ov 7411 df-oprab 7412 df-mpo 7413 df-om 7861 df-1st 7984 df-2nd 7985 df-frecs 8277 df-wrecs 8308 df-recs 8357 df-rdg 8396 df-1o 8454 df-oadd 8458 df-er 8695 df-map 8827 df-pm 8828 df-en 8952 df-dom 8953 df-sdom 8954 df-fin 8955 df-dju 9954 df-card 9992 df-pnf 11317 df-mnf 11318 df-xr 11319 df-ltxr 11320 df-le 11321 df-sub 11515 df-neg 11516 df-nn 12306 df-2 12375 df-3 12376 df-n0 12577 df-xnn0 12650 df-z 12664 df-uz 12936 df-fz 13610 df-fzo 13758 df-hash 14443 df-word 14627 df-concat 14684 df-s1 14711 df-s2 14967 df-s3 14968 df-trkgc 28844 df-trkgb 28845 df-trkgcb 28846 df-trkgld 28848 df-trkg 28849 df-cgrg 28908 df-ismt 28930 df-leg 28980 df-hlg 28998 df-mir 29059 df-rag 29103 df-perpg 29105 df-hpg 29170 df-mid 29213 df-lmi 29214 df-cgra 29249

This theorem is used by: angmgmaddov1lem 29320

W3C validator