opphllem5 - Metamath Proof Explorer

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Theorem opphllem5 25443

Description: Second part of Lemma 9.4 of [Schwabhauser] p. 68. (Contributed by Thierry Arnoux, 2-Mar-2020.)

**Hypotheses**
Ref	Expression
hpg.p	⊢ 𝑃 = (Base‘𝐺)
hpg.d	⊢ − = (dist‘𝐺)
hpg.i	⊢ 𝐼 = (Itv‘𝐺)
hpg.o	⊢ 𝑂 = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝑃 ∖ 𝐷) ∧ 𝑏 ∈ (𝑃 ∖ 𝐷)) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝑎𝐼𝑏))}
opphl.l	⊢ 𝐿 = (LineG‘𝐺)
opphl.d	⊢ (𝜑 → 𝐷 ∈ ran 𝐿)
opphl.g	⊢ (𝜑 → 𝐺 ∈ TarskiG)
opphl.k	⊢ 𝐾 = (hlG‘𝐺)
opphllem5.n	⊢ 𝑁 = ((pInvG‘𝐺)‘𝑀)
opphllem5.a	⊢ (𝜑 → 𝐴 ∈ 𝑃)
opphllem5.c	⊢ (𝜑 → 𝐶 ∈ 𝑃)
opphllem5.r	⊢ (𝜑 → 𝑅 ∈ 𝐷)
opphllem5.s	⊢ (𝜑 → 𝑆 ∈ 𝐷)
opphllem5.m	⊢ (𝜑 → 𝑀 ∈ 𝑃)
opphllem5.o	⊢ (𝜑 → 𝐴𝑂𝐶)
opphllem5.p	⊢ (𝜑 → 𝐷(⟂G‘𝐺)(𝐴𝐿𝑅))
opphllem5.q	⊢ (𝜑 → 𝐷(⟂G‘𝐺)(𝐶𝐿𝑆))
opphllem5.u	⊢ (𝜑 → 𝑈 ∈ 𝑃)
opphllem5.v	⊢ (𝜑 → 𝑉 ∈ 𝑃)
opphllem5.1	⊢ (𝜑 → 𝑈(𝐾‘𝑅)𝐴)
opphllem5.2	⊢ (𝜑 → 𝑉(𝐾‘𝑆)𝐶)

**Assertion**
Ref	Expression
opphllem5	⊢ (𝜑 → 𝑈𝑂𝑉)

Distinct variable groups: 𝐷,𝑎,𝑏 𝐼,𝑎,𝑏 𝑃,𝑎,𝑏 𝑡,𝐴 𝑡,𝐷 𝑡,𝑅 𝑡,𝐶 𝑡,𝐺 𝑡,𝐿 𝑡,𝑈 𝑡,𝐼 𝑡,𝐾 𝑡,𝑀 𝑡,𝑂 𝑡,𝑁 𝑡,𝑃 𝑡,𝑆 𝑡,𝑉 𝜑,𝑡 𝑡, − 𝑡,𝑎,𝑏 Allowed substitution hints: 𝜑(𝑎,𝑏) 𝐴(𝑎,𝑏) 𝐶(𝑎,𝑏) 𝑅(𝑎,𝑏) 𝑆(𝑎,𝑏) 𝑈(𝑎,𝑏) 𝐺(𝑎,𝑏) 𝐾(𝑎,𝑏) 𝐿(𝑎,𝑏) 𝑀(𝑎,𝑏) − (𝑎,𝑏) 𝑁(𝑎,𝑏) 𝑂(𝑎,𝑏) 𝑉(𝑎,𝑏)

Proof of Theorem opphllem5 Dummy variable 𝑚 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		hpg.p	. . . . . . 7 ⊢ 𝑃 = (Base‘𝐺)
2		hpg.d	. . . . . . 7 ⊢ − = (dist‘𝐺)
3		hpg.i	. . . . . . 7 ⊢ 𝐼 = (Itv‘𝐺)
4		opphl.l	. . . . . . 7 ⊢ 𝐿 = (LineG‘𝐺)
5		opphl.g	. . . . . . 7 ⊢ (𝜑 → 𝐺 ∈ TarskiG)
6		opphl.d	. . . . . . 7 ⊢ (𝜑 → 𝐷 ∈ ran 𝐿)
7		opphl.k	. . . . . . 7 ⊢ 𝐾 = (hlG‘𝐺)
8		opphllem5.r	. . . . . . 7 ⊢ (𝜑 → 𝑅 ∈ 𝐷)
9		opphllem5.a	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ 𝑃)
10		opphllem5.u	. . . . . . 7 ⊢ (𝜑 → 𝑈 ∈ 𝑃)
11		opphllem5.p	. . . . . . . 8 ⊢ (𝜑 → 𝐷(⟂G‘𝐺)(𝐴𝐿𝑅))
12	1, 4, 3, 5, 6, 8	tglnpt 25244	. . . . . . . . 9 ⊢ (𝜑 → 𝑅 ∈ 𝑃)
13		opphllem5.1	. . . . . . . . . 10 ⊢ (𝜑 → 𝑈(𝐾‘𝑅)𝐴)
14	1, 3, 7, 10, 9, 12, 5, 13	hlne2 25301	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 ≠ 𝑅)
15	1, 3, 4, 5, 9, 12, 14	tglinecom 25330	. . . . . . . 8 ⊢ (𝜑 → (𝐴𝐿𝑅) = (𝑅𝐿𝐴))
16	11, 15	breqtrd 4609	. . . . . . 7 ⊢ (𝜑 → 𝐷(⟂G‘𝐺)(𝑅𝐿𝐴))
17	1, 3, 7, 10, 9, 12, 5, 13	hlcomd 25299	. . . . . . 7 ⊢ (𝜑 → 𝐴(𝐾‘𝑅)𝑈)
18	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 16, 17	hlperpnel 25417	. . . . . 6 ⊢ (𝜑 → ¬ 𝑈 ∈ 𝐷)
19	18	ad3antrrr 762	. . . . 5 ⊢ ((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) → ¬ 𝑈 ∈ 𝐷)
20		opphllem5.s	. . . . . . 7 ⊢ (𝜑 → 𝑆 ∈ 𝐷)
21		opphllem5.c	. . . . . . 7 ⊢ (𝜑 → 𝐶 ∈ 𝑃)
22		opphllem5.v	. . . . . . 7 ⊢ (𝜑 → 𝑉 ∈ 𝑃)
23		opphllem5.q	. . . . . . . 8 ⊢ (𝜑 → 𝐷(⟂G‘𝐺)(𝐶𝐿𝑆))
24	1, 4, 3, 5, 6, 20	tglnpt 25244	. . . . . . . . 9 ⊢ (𝜑 → 𝑆 ∈ 𝑃)
25		opphllem5.2	. . . . . . . . . 10 ⊢ (𝜑 → 𝑉(𝐾‘𝑆)𝐶)
26	1, 3, 7, 22, 21, 24, 5, 25	hlne2 25301	. . . . . . . . 9 ⊢ (𝜑 → 𝐶 ≠ 𝑆)
27	1, 3, 4, 5, 21, 24, 26	tglinecom 25330	. . . . . . . 8 ⊢ (𝜑 → (𝐶𝐿𝑆) = (𝑆𝐿𝐶))
28	23, 27	breqtrd 4609	. . . . . . 7 ⊢ (𝜑 → 𝐷(⟂G‘𝐺)(𝑆𝐿𝐶))
29	1, 3, 7, 22, 21, 24, 5, 25	hlcomd 25299	. . . . . . 7 ⊢ (𝜑 → 𝐶(𝐾‘𝑆)𝑉)
30	1, 2, 3, 4, 5, 6, 7, 20, 21, 22, 28, 29	hlperpnel 25417	. . . . . 6 ⊢ (𝜑 → ¬ 𝑉 ∈ 𝐷)
31	30	ad3antrrr 762	. . . . 5 ⊢ ((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) → ¬ 𝑉 ∈ 𝐷)
32		simplr 788	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) → 𝑡 ∈ 𝐷)
33		simpr 476	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) ∧ 𝑅 = 𝑡) → 𝑅 = 𝑡)
34		eqid 2610	. . . . . . . . 9 ⊢ (pInvG‘𝐺) = (pInvG‘𝐺)
35	5	ad3antrrr 762	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) → 𝐺 ∈ TarskiG)
36	35	adantr 480	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) ∧ 𝑅 ≠ 𝑡) → 𝐺 ∈ TarskiG)
37	21	ad3antrrr 762	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) → 𝐶 ∈ 𝑃)
38	37	adantr 480	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) ∧ 𝑅 ≠ 𝑡) → 𝐶 ∈ 𝑃)
39	12	ad3antrrr 762	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) → 𝑅 ∈ 𝑃)
40	39	adantr 480	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) ∧ 𝑅 ≠ 𝑡) → 𝑅 ∈ 𝑃)
41	6	ad3antrrr 762	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) → 𝐷 ∈ ran 𝐿)
42	1, 4, 3, 35, 41, 32	tglnpt 25244	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) → 𝑡 ∈ 𝑃)
43	42	adantr 480	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) ∧ 𝑅 ≠ 𝑡) → 𝑡 ∈ 𝑃)
44	9	ad3antrrr 762	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) → 𝐴 ∈ 𝑃)
45	44	adantr 480	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) ∧ 𝑅 ≠ 𝑡) → 𝐴 ∈ 𝑃)
46	24	ad3antrrr 762	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) → 𝑆 ∈ 𝑃)
47	46	adantr 480	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) ∧ 𝑅 ≠ 𝑡) → 𝑆 ∈ 𝑃)
48		simpllr 795	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) → 𝑅 = 𝑆)
49	1, 3, 4, 5, 21, 24, 26	tglinerflx2 25329	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑆 ∈ (𝐶𝐿𝑆))
50	49	ad3antrrr 762	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) → 𝑆 ∈ (𝐶𝐿𝑆))
51	48, 50	eqeltrd 2688	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) → 𝑅 ∈ (𝐶𝐿𝑆))
52	51	adantr 480	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) ∧ 𝑅 ≠ 𝑡) → 𝑅 ∈ (𝐶𝐿𝑆))
53	1, 3, 4, 5, 21, 24, 26	tgelrnln 25325	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐶𝐿𝑆) ∈ ran 𝐿)
54	1, 2, 3, 4, 5, 6, 53, 23	perpcom 25408	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐶𝐿𝑆)(⟂G‘𝐺)𝐷)
55	54	ad4antr 764	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) ∧ 𝑅 ≠ 𝑡) → (𝐶𝐿𝑆)(⟂G‘𝐺)𝐷)
56		simpr 476	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) ∧ 𝑅 ≠ 𝑡) → 𝑅 ≠ 𝑡)
57	41	adantr 480	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) ∧ 𝑅 ≠ 𝑡) → 𝐷 ∈ ran 𝐿)
58	8	ad3antrrr 762	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) → 𝑅 ∈ 𝐷)
59	58	adantr 480	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) ∧ 𝑅 ≠ 𝑡) → 𝑅 ∈ 𝐷)
60	32	adantr 480	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) ∧ 𝑅 ≠ 𝑡) → 𝑡 ∈ 𝐷)
61	1, 3, 4, 36, 40, 43, 56, 56, 57, 59, 60	tglinethru 25331	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) ∧ 𝑅 ≠ 𝑡) → 𝐷 = (𝑅𝐿𝑡))
62	55, 61	breqtrd 4609	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) ∧ 𝑅 ≠ 𝑡) → (𝐶𝐿𝑆)(⟂G‘𝐺)(𝑅𝐿𝑡))
63	1, 2, 3, 4, 36, 38, 47, 52, 43, 62	perprag 25418	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) ∧ 𝑅 ≠ 𝑡) → ⟨“𝐶𝑅𝑡”⟩ ∈ (∟G‘𝐺))
64	1, 3, 4, 5, 9, 12, 14	tglinerflx2 25329	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑅 ∈ (𝐴𝐿𝑅))
65	64	ad3antrrr 762	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) → 𝑅 ∈ (𝐴𝐿𝑅))
66	65	adantr 480	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) ∧ 𝑅 ≠ 𝑡) → 𝑅 ∈ (𝐴𝐿𝑅))
67	1, 3, 4, 5, 9, 12, 14	tgelrnln 25325	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐴𝐿𝑅) ∈ ran 𝐿)
68	1, 2, 3, 4, 5, 6, 67, 11	perpcom 25408	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐴𝐿𝑅)(⟂G‘𝐺)𝐷)
69	68	ad4antr 764	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) ∧ 𝑅 ≠ 𝑡) → (𝐴𝐿𝑅)(⟂G‘𝐺)𝐷)
70	69, 61	breqtrd 4609	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) ∧ 𝑅 ≠ 𝑡) → (𝐴𝐿𝑅)(⟂G‘𝐺)(𝑅𝐿𝑡))
71	1, 2, 3, 4, 36, 45, 40, 66, 43, 70	perprag 25418	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) ∧ 𝑅 ≠ 𝑡) → ⟨“𝐴𝑅𝑡”⟩ ∈ (∟G‘𝐺))
72		simplr 788	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) ∧ 𝑅 ≠ 𝑡) → 𝑡 ∈ (𝐴𝐼𝐶))
73	1, 2, 3, 36, 45, 43, 38, 72	tgbtwncom 25183	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) ∧ 𝑅 ≠ 𝑡) → 𝑡 ∈ (𝐶𝐼𝐴))
74	1, 2, 3, 4, 34, 36, 38, 40, 43, 45, 63, 71, 73	ragflat2 25398	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) ∧ 𝑅 ≠ 𝑡) → 𝑅 = 𝑡)
75	33, 74	pm2.61dane 2869	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) → 𝑅 = 𝑡)
76	10	ad3antrrr 762	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) → 𝑈 ∈ 𝑃)
77	22	ad3antrrr 762	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) → 𝑉 ∈ 𝑃)
78	17	ad3antrrr 762	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) → 𝐴(𝐾‘𝑅)𝑈)
79	25	ad3antrrr 762	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) → 𝑉(𝐾‘𝑆)𝐶)
80	48	fveq2d 6107	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) → (𝐾‘𝑅) = (𝐾‘𝑆))
81	80	breqd 4594	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) → (𝑉(𝐾‘𝑅)𝐶 ↔ 𝑉(𝐾‘𝑆)𝐶))
82	79, 81	mpbird 246	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) → 𝑉(𝐾‘𝑅)𝐶)
83	1, 3, 7, 77, 37, 39, 35, 82	hlcomd 25299	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) → 𝐶(𝐾‘𝑅)𝑉)
84		simpr 476	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) → 𝑡 ∈ (𝐴𝐼𝐶))
85	75, 84	eqeltrd 2688	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) → 𝑅 ∈ (𝐴𝐼𝐶))
86	1, 2, 3, 35, 44, 39, 37, 85	tgbtwncom 25183	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) → 𝑅 ∈ (𝐶𝐼𝐴))
87	1, 3, 7, 37, 77, 44, 35, 39, 83, 86	btwnhl 25309	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) → 𝑅 ∈ (𝑉𝐼𝐴))
88	1, 2, 3, 35, 77, 39, 44, 87	tgbtwncom 25183	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) → 𝑅 ∈ (𝐴𝐼𝑉))
89	1, 3, 7, 44, 76, 77, 35, 39, 78, 88	btwnhl 25309	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) → 𝑅 ∈ (𝑈𝐼𝑉))
90	75, 89	eqeltrrd 2689	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) → 𝑡 ∈ (𝑈𝐼𝑉))
91		rspe 2986	. . . . . 6 ⊢ ((𝑡 ∈ 𝐷 ∧ 𝑡 ∈ (𝑈𝐼𝑉)) → ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝑈𝐼𝑉))
92	32, 90, 91	syl2anc 691	. . . . 5 ⊢ ((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) → ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝑈𝐼𝑉))
93	19, 31, 92	jca31 555	. . . 4 ⊢ ((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) → ((¬ 𝑈 ∈ 𝐷 ∧ ¬ 𝑉 ∈ 𝐷) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝑈𝐼𝑉)))
94		hpg.o	. . . . . 6 ⊢ 𝑂 = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝑃 ∖ 𝐷) ∧ 𝑏 ∈ (𝑃 ∖ 𝐷)) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝑎𝐼𝑏))}
95	1, 2, 3, 94, 10, 22	islnopp 25431	. . . . 5 ⊢ (𝜑 → (𝑈𝑂𝑉 ↔ ((¬ 𝑈 ∈ 𝐷 ∧ ¬ 𝑉 ∈ 𝐷) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝑈𝐼𝑉))))
96	95	ad3antrrr 762	. . . 4 ⊢ ((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) → (𝑈𝑂𝑉 ↔ ((¬ 𝑈 ∈ 𝐷 ∧ ¬ 𝑉 ∈ 𝐷) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝑈𝐼𝑉))))
97	93, 96	mpbird 246	. . 3 ⊢ ((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) → 𝑈𝑂𝑉)
98		opphllem5.o	. . . . . 6 ⊢ (𝜑 → 𝐴𝑂𝐶)
99	1, 2, 3, 94, 9, 21	islnopp 25431	. . . . . 6 ⊢ (𝜑 → (𝐴𝑂𝐶 ↔ ((¬ 𝐴 ∈ 𝐷 ∧ ¬ 𝐶 ∈ 𝐷) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝐴𝐼𝐶))))
100	98, 99	mpbid 221	. . . . 5 ⊢ (𝜑 → ((¬ 𝐴 ∈ 𝐷 ∧ ¬ 𝐶 ∈ 𝐷) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝐴𝐼𝐶)))
101	100	simprd 478	. . . 4 ⊢ (𝜑 → ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝐴𝐼𝐶))
102	101	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝑅 = 𝑆) → ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝐴𝐼𝐶))
103	97, 102	r19.29a 3060	. 2 ⊢ ((𝜑 ∧ 𝑅 = 𝑆) → 𝑈𝑂𝑉)
104	6	ad4antr 764	. . . . 5 ⊢ (((((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ 𝑚 ∈ 𝑃) ∧ 𝑆 = (((pInvG‘𝐺)‘𝑚)‘𝑅)) ∧ (𝑆 − 𝐶)(≤G‘𝐺)(𝑅 − 𝐴)) → 𝐷 ∈ ran 𝐿)
105	5	ad4antr 764	. . . . 5 ⊢ (((((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ 𝑚 ∈ 𝑃) ∧ 𝑆 = (((pInvG‘𝐺)‘𝑚)‘𝑅)) ∧ (𝑆 − 𝐶)(≤G‘𝐺)(𝑅 − 𝐴)) → 𝐺 ∈ TarskiG)
106		eqid 2610	. . . . 5 ⊢ ((pInvG‘𝐺)‘𝑚) = ((pInvG‘𝐺)‘𝑚)
107	9	ad4antr 764	. . . . 5 ⊢ (((((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ 𝑚 ∈ 𝑃) ∧ 𝑆 = (((pInvG‘𝐺)‘𝑚)‘𝑅)) ∧ (𝑆 − 𝐶)(≤G‘𝐺)(𝑅 − 𝐴)) → 𝐴 ∈ 𝑃)
108	21	ad4antr 764	. . . . 5 ⊢ (((((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ 𝑚 ∈ 𝑃) ∧ 𝑆 = (((pInvG‘𝐺)‘𝑚)‘𝑅)) ∧ (𝑆 − 𝐶)(≤G‘𝐺)(𝑅 − 𝐴)) → 𝐶 ∈ 𝑃)
109	8	ad4antr 764	. . . . 5 ⊢ (((((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ 𝑚 ∈ 𝑃) ∧ 𝑆 = (((pInvG‘𝐺)‘𝑚)‘𝑅)) ∧ (𝑆 − 𝐶)(≤G‘𝐺)(𝑅 − 𝐴)) → 𝑅 ∈ 𝐷)
110	20	ad4antr 764	. . . . 5 ⊢ (((((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ 𝑚 ∈ 𝑃) ∧ 𝑆 = (((pInvG‘𝐺)‘𝑚)‘𝑅)) ∧ (𝑆 − 𝐶)(≤G‘𝐺)(𝑅 − 𝐴)) → 𝑆 ∈ 𝐷)
111		simpllr 795	. . . . 5 ⊢ (((((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ 𝑚 ∈ 𝑃) ∧ 𝑆 = (((pInvG‘𝐺)‘𝑚)‘𝑅)) ∧ (𝑆 − 𝐶)(≤G‘𝐺)(𝑅 − 𝐴)) → 𝑚 ∈ 𝑃)
112	98	ad4antr 764	. . . . 5 ⊢ (((((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ 𝑚 ∈ 𝑃) ∧ 𝑆 = (((pInvG‘𝐺)‘𝑚)‘𝑅)) ∧ (𝑆 − 𝐶)(≤G‘𝐺)(𝑅 − 𝐴)) → 𝐴𝑂𝐶)
113	11	ad4antr 764	. . . . 5 ⊢ (((((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ 𝑚 ∈ 𝑃) ∧ 𝑆 = (((pInvG‘𝐺)‘𝑚)‘𝑅)) ∧ (𝑆 − 𝐶)(≤G‘𝐺)(𝑅 − 𝐴)) → 𝐷(⟂G‘𝐺)(𝐴𝐿𝑅))
114	23	ad4antr 764	. . . . 5 ⊢ (((((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ 𝑚 ∈ 𝑃) ∧ 𝑆 = (((pInvG‘𝐺)‘𝑚)‘𝑅)) ∧ (𝑆 − 𝐶)(≤G‘𝐺)(𝑅 − 𝐴)) → 𝐷(⟂G‘𝐺)(𝐶𝐿𝑆))
115		simpr 476	. . . . . 6 ⊢ ((𝜑 ∧ 𝑅 ≠ 𝑆) → 𝑅 ≠ 𝑆)
116	115	ad3antrrr 762	. . . . 5 ⊢ (((((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ 𝑚 ∈ 𝑃) ∧ 𝑆 = (((pInvG‘𝐺)‘𝑚)‘𝑅)) ∧ (𝑆 − 𝐶)(≤G‘𝐺)(𝑅 − 𝐴)) → 𝑅 ≠ 𝑆)
117		simpr 476	. . . . 5 ⊢ (((((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ 𝑚 ∈ 𝑃) ∧ 𝑆 = (((pInvG‘𝐺)‘𝑚)‘𝑅)) ∧ (𝑆 − 𝐶)(≤G‘𝐺)(𝑅 − 𝐴)) → (𝑆 − 𝐶)(≤G‘𝐺)(𝑅 − 𝐴))
118	10	ad4antr 764	. . . . 5 ⊢ (((((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ 𝑚 ∈ 𝑃) ∧ 𝑆 = (((pInvG‘𝐺)‘𝑚)‘𝑅)) ∧ (𝑆 − 𝐶)(≤G‘𝐺)(𝑅 − 𝐴)) → 𝑈 ∈ 𝑃)
119		simplr 788	. . . . . 6 ⊢ (((((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ 𝑚 ∈ 𝑃) ∧ 𝑆 = (((pInvG‘𝐺)‘𝑚)‘𝑅)) ∧ (𝑆 − 𝐶)(≤G‘𝐺)(𝑅 − 𝐴)) → 𝑆 = (((pInvG‘𝐺)‘𝑚)‘𝑅))
120	119	eqcomd 2616	. . . . 5 ⊢ (((((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ 𝑚 ∈ 𝑃) ∧ 𝑆 = (((pInvG‘𝐺)‘𝑚)‘𝑅)) ∧ (𝑆 − 𝐶)(≤G‘𝐺)(𝑅 − 𝐴)) → (((pInvG‘𝐺)‘𝑚)‘𝑅) = 𝑆)
121	22	ad4antr 764	. . . . 5 ⊢ (((((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ 𝑚 ∈ 𝑃) ∧ 𝑆 = (((pInvG‘𝐺)‘𝑚)‘𝑅)) ∧ (𝑆 − 𝐶)(≤G‘𝐺)(𝑅 − 𝐴)) → 𝑉 ∈ 𝑃)
122	13	ad4antr 764	. . . . 5 ⊢ (((((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ 𝑚 ∈ 𝑃) ∧ 𝑆 = (((pInvG‘𝐺)‘𝑚)‘𝑅)) ∧ (𝑆 − 𝐶)(≤G‘𝐺)(𝑅 − 𝐴)) → 𝑈(𝐾‘𝑅)𝐴)
123	25	ad4antr 764	. . . . 5 ⊢ (((((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ 𝑚 ∈ 𝑃) ∧ 𝑆 = (((pInvG‘𝐺)‘𝑚)‘𝑅)) ∧ (𝑆 − 𝐶)(≤G‘𝐺)(𝑅 − 𝐴)) → 𝑉(𝐾‘𝑆)𝐶)
124	1, 2, 3, 94, 4, 104, 105, 7, 106, 107, 108, 109, 110, 111, 112, 113, 114, 116, 117, 118, 120, 121, 122, 123	opphllem4 25442	. . . 4 ⊢ (((((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ 𝑚 ∈ 𝑃) ∧ 𝑆 = (((pInvG‘𝐺)‘𝑚)‘𝑅)) ∧ (𝑆 − 𝐶)(≤G‘𝐺)(𝑅 − 𝐴)) → 𝑈𝑂𝑉)
125	6	ad4antr 764	. . . . 5 ⊢ (((((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ 𝑚 ∈ 𝑃) ∧ 𝑆 = (((pInvG‘𝐺)‘𝑚)‘𝑅)) ∧ (𝑅 − 𝐴)(≤G‘𝐺)(𝑆 − 𝐶)) → 𝐷 ∈ ran 𝐿)
126	5	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑅 ≠ 𝑆) → 𝐺 ∈ TarskiG)
127	126	ad3antrrr 762	. . . . 5 ⊢ (((((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ 𝑚 ∈ 𝑃) ∧ 𝑆 = (((pInvG‘𝐺)‘𝑚)‘𝑅)) ∧ (𝑅 − 𝐴)(≤G‘𝐺)(𝑆 − 𝐶)) → 𝐺 ∈ TarskiG)
128	22	ad4antr 764	. . . . 5 ⊢ (((((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ 𝑚 ∈ 𝑃) ∧ 𝑆 = (((pInvG‘𝐺)‘𝑚)‘𝑅)) ∧ (𝑅 − 𝐴)(≤G‘𝐺)(𝑆 − 𝐶)) → 𝑉 ∈ 𝑃)
129	10	ad4antr 764	. . . . 5 ⊢ (((((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ 𝑚 ∈ 𝑃) ∧ 𝑆 = (((pInvG‘𝐺)‘𝑚)‘𝑅)) ∧ (𝑅 − 𝐴)(≤G‘𝐺)(𝑆 − 𝐶)) → 𝑈 ∈ 𝑃)
130	21	ad4antr 764	. . . . . 6 ⊢ (((((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ 𝑚 ∈ 𝑃) ∧ 𝑆 = (((pInvG‘𝐺)‘𝑚)‘𝑅)) ∧ (𝑅 − 𝐴)(≤G‘𝐺)(𝑆 − 𝐶)) → 𝐶 ∈ 𝑃)
131	9	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑅 ≠ 𝑆) → 𝐴 ∈ 𝑃)
132	131	ad3antrrr 762	. . . . . 6 ⊢ (((((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ 𝑚 ∈ 𝑃) ∧ 𝑆 = (((pInvG‘𝐺)‘𝑚)‘𝑅)) ∧ (𝑅 − 𝐴)(≤G‘𝐺)(𝑆 − 𝐶)) → 𝐴 ∈ 𝑃)
133	20	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑅 ≠ 𝑆) → 𝑆 ∈ 𝐷)
134	133	ad3antrrr 762	. . . . . 6 ⊢ (((((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ 𝑚 ∈ 𝑃) ∧ 𝑆 = (((pInvG‘𝐺)‘𝑚)‘𝑅)) ∧ (𝑅 − 𝐴)(≤G‘𝐺)(𝑆 − 𝐶)) → 𝑆 ∈ 𝐷)
135	8	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑅 ≠ 𝑆) → 𝑅 ∈ 𝐷)
136	135	ad3antrrr 762	. . . . . 6 ⊢ (((((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ 𝑚 ∈ 𝑃) ∧ 𝑆 = (((pInvG‘𝐺)‘𝑚)‘𝑅)) ∧ (𝑅 − 𝐴)(≤G‘𝐺)(𝑆 − 𝐶)) → 𝑅 ∈ 𝐷)
137		simpllr 795	. . . . . 6 ⊢ (((((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ 𝑚 ∈ 𝑃) ∧ 𝑆 = (((pInvG‘𝐺)‘𝑚)‘𝑅)) ∧ (𝑅 − 𝐴)(≤G‘𝐺)(𝑆 − 𝐶)) → 𝑚 ∈ 𝑃)
138	98	ad4antr 764	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ 𝑚 ∈ 𝑃) ∧ 𝑆 = (((pInvG‘𝐺)‘𝑚)‘𝑅)) ∧ (𝑅 − 𝐴)(≤G‘𝐺)(𝑆 − 𝐶)) → 𝐴𝑂𝐶)
139	1, 2, 3, 94, 4, 125, 127, 132, 130, 138	oppcom 25436	. . . . . 6 ⊢ (((((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ 𝑚 ∈ 𝑃) ∧ 𝑆 = (((pInvG‘𝐺)‘𝑚)‘𝑅)) ∧ (𝑅 − 𝐴)(≤G‘𝐺)(𝑆 − 𝐶)) → 𝐶𝑂𝐴)
140	23	ad4antr 764	. . . . . 6 ⊢ (((((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ 𝑚 ∈ 𝑃) ∧ 𝑆 = (((pInvG‘𝐺)‘𝑚)‘𝑅)) ∧ (𝑅 − 𝐴)(≤G‘𝐺)(𝑆 − 𝐶)) → 𝐷(⟂G‘𝐺)(𝐶𝐿𝑆))
141	11	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑅 ≠ 𝑆) → 𝐷(⟂G‘𝐺)(𝐴𝐿𝑅))
142	141	ad3antrrr 762	. . . . . 6 ⊢ (((((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ 𝑚 ∈ 𝑃) ∧ 𝑆 = (((pInvG‘𝐺)‘𝑚)‘𝑅)) ∧ (𝑅 − 𝐴)(≤G‘𝐺)(𝑆 − 𝐶)) → 𝐷(⟂G‘𝐺)(𝐴𝐿𝑅))
143	115	necomd 2837	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑅 ≠ 𝑆) → 𝑆 ≠ 𝑅)
144	143	ad3antrrr 762	. . . . . 6 ⊢ (((((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ 𝑚 ∈ 𝑃) ∧ 𝑆 = (((pInvG‘𝐺)‘𝑚)‘𝑅)) ∧ (𝑅 − 𝐴)(≤G‘𝐺)(𝑆 − 𝐶)) → 𝑆 ≠ 𝑅)
145		simpr 476	. . . . . 6 ⊢ (((((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ 𝑚 ∈ 𝑃) ∧ 𝑆 = (((pInvG‘𝐺)‘𝑚)‘𝑅)) ∧ (𝑅 − 𝐴)(≤G‘𝐺)(𝑆 − 𝐶)) → (𝑅 − 𝐴)(≤G‘𝐺)(𝑆 − 𝐶))
146	12	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑅 ≠ 𝑆) → 𝑅 ∈ 𝑃)
147	146	ad3antrrr 762	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ 𝑚 ∈ 𝑃) ∧ 𝑆 = (((pInvG‘𝐺)‘𝑚)‘𝑅)) ∧ (𝑅 − 𝐴)(≤G‘𝐺)(𝑆 − 𝐶)) → 𝑅 ∈ 𝑃)
148		simplr 788	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ 𝑚 ∈ 𝑃) ∧ 𝑆 = (((pInvG‘𝐺)‘𝑚)‘𝑅)) ∧ (𝑅 − 𝐴)(≤G‘𝐺)(𝑆 − 𝐶)) → 𝑆 = (((pInvG‘𝐺)‘𝑚)‘𝑅))
149	148	eqcomd 2616	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ 𝑚 ∈ 𝑃) ∧ 𝑆 = (((pInvG‘𝐺)‘𝑚)‘𝑅)) ∧ (𝑅 − 𝐴)(≤G‘𝐺)(𝑆 − 𝐶)) → (((pInvG‘𝐺)‘𝑚)‘𝑅) = 𝑆)
150	1, 2, 3, 4, 34, 127, 137, 106, 147, 149	mircom 25358	. . . . . 6 ⊢ (((((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ 𝑚 ∈ 𝑃) ∧ 𝑆 = (((pInvG‘𝐺)‘𝑚)‘𝑅)) ∧ (𝑅 − 𝐴)(≤G‘𝐺)(𝑆 − 𝐶)) → (((pInvG‘𝐺)‘𝑚)‘𝑆) = 𝑅)
151	25	ad4antr 764	. . . . . 6 ⊢ (((((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ 𝑚 ∈ 𝑃) ∧ 𝑆 = (((pInvG‘𝐺)‘𝑚)‘𝑅)) ∧ (𝑅 − 𝐴)(≤G‘𝐺)(𝑆 − 𝐶)) → 𝑉(𝐾‘𝑆)𝐶)
152	13	ad4antr 764	. . . . . 6 ⊢ (((((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ 𝑚 ∈ 𝑃) ∧ 𝑆 = (((pInvG‘𝐺)‘𝑚)‘𝑅)) ∧ (𝑅 − 𝐴)(≤G‘𝐺)(𝑆 − 𝐶)) → 𝑈(𝐾‘𝑅)𝐴)
153	1, 2, 3, 94, 4, 125, 127, 7, 106, 130, 132, 134, 136, 137, 139, 140, 142, 144, 145, 128, 150, 129, 151, 152	opphllem4 25442	. . . . 5 ⊢ (((((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ 𝑚 ∈ 𝑃) ∧ 𝑆 = (((pInvG‘𝐺)‘𝑚)‘𝑅)) ∧ (𝑅 − 𝐴)(≤G‘𝐺)(𝑆 − 𝐶)) → 𝑉𝑂𝑈)
154	1, 2, 3, 94, 4, 125, 127, 128, 129, 153	oppcom 25436	. . . 4 ⊢ (((((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ 𝑚 ∈ 𝑃) ∧ 𝑆 = (((pInvG‘𝐺)‘𝑚)‘𝑅)) ∧ (𝑅 − 𝐴)(≤G‘𝐺)(𝑆 − 𝐶)) → 𝑈𝑂𝑉)
155		eqid 2610	. . . . . 6 ⊢ (≤G‘𝐺) = (≤G‘𝐺)
156	1, 2, 3, 155, 5, 24, 21, 12, 9	legtrid 25286	. . . . 5 ⊢ (𝜑 → ((𝑆 − 𝐶)(≤G‘𝐺)(𝑅 − 𝐴) ∨ (𝑅 − 𝐴)(≤G‘𝐺)(𝑆 − 𝐶)))
157	156	ad3antrrr 762	. . . 4 ⊢ ((((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ 𝑚 ∈ 𝑃) ∧ 𝑆 = (((pInvG‘𝐺)‘𝑚)‘𝑅)) → ((𝑆 − 𝐶)(≤G‘𝐺)(𝑅 − 𝐴) ∨ (𝑅 − 𝐴)(≤G‘𝐺)(𝑆 − 𝐶)))
158	124, 154, 157	mpjaodan 823	. . 3 ⊢ ((((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ 𝑚 ∈ 𝑃) ∧ 𝑆 = (((pInvG‘𝐺)‘𝑚)‘𝑅)) → 𝑈𝑂𝑉)
159	24	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑅 ≠ 𝑆) → 𝑆 ∈ 𝑃)
160	1, 2, 3, 94, 4, 6, 5, 9, 21, 98	opptgdim2 25437	. . . . 5 ⊢ (𝜑 → 𝐺Dim_TarskiG≥2)
161	160	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑅 ≠ 𝑆) → 𝐺Dim_TarskiG≥2)
162	1, 2, 3, 4, 126, 34, 146, 159, 161	midex 25429	. . 3 ⊢ ((𝜑 ∧ 𝑅 ≠ 𝑆) → ∃𝑚 ∈ 𝑃 𝑆 = (((pInvG‘𝐺)‘𝑚)‘𝑅))
163	158, 162	r19.29a 3060	. 2 ⊢ ((𝜑 ∧ 𝑅 ≠ 𝑆) → 𝑈𝑂𝑉)
164	103, 163	pm2.61dane 2869	1 ⊢ (𝜑 → 𝑈𝑂𝑉)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 195 ∨ wo 382 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 ∃wrex 2897 ∖ cdif 3537 class class class wbr 4583 {copab 4642 ran crn 5039 ‘cfv 5804 (class class class)co 6549 2c2 10947 Basecbs 15695 distcds 15777 TarskiGcstrkg 25129 Dim_TarskiG≥cstrkgld 25133 Itvcitv 25135 LineGclng 25136 ≤Gcleg 25277 hlGchlg 25295 pInvGcmir 25347 ⟂Gcperpg 25390

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-er 7629 df-map 7746 df-pm 7747 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-card 8648 df-cda 8873 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-2 10956 df-3 10957 df-n0 11170 df-xnn0 11241 df-z 11255 df-uz 11564 df-fz 12198 df-fzo 12335 df-hash 12980 df-word 13154 df-concat 13156 df-s1 13157 df-s2 13444 df-s3 13445 df-trkgc 25147 df-trkgb 25148 df-trkgcb 25149 df-trkgld 25151 df-trkg 25152 df-cgrg 25206 df-leg 25278 df-hlg 25296 df-mir 25348 df-rag 25389 df-perpg 25391

This theorem is referenced by: opphl 25446

W3C validator