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Theorem opphllem5 28678

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 28476	. . . . . . . . 9 ⊢ (𝜑 → 𝑅 ∈ 𝑃)
13		opphllem5.1	. . . . . . . . . 10 ⊢ (𝜑 → 𝑈(𝐾‘𝑅)𝐴)
14	1, 3, 7, 10, 9, 12, 5, 13	hlne2 28533	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 ≠ 𝑅)
15	1, 3, 4, 5, 9, 12, 14	tglinecom 28562	. . . . . . . 8 ⊢ (𝜑 → (𝐴𝐿𝑅) = (𝑅𝐿𝐴))
16	11, 15	breqtrd 5133	. . . . . . 7 ⊢ (𝜑 → 𝐷(⟂G‘𝐺)(𝑅𝐿𝐴))
17	1, 3, 7, 10, 9, 12, 5, 13	hlcomd 28531	. . . . . . 7 ⊢ (𝜑 → 𝐴(𝐾‘𝑅)𝑈)
18	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 16, 17	hlperpnel 28652	. . . . . 6 ⊢ (𝜑 → ¬ 𝑈 ∈ 𝐷)
19	18	ad3antrrr 730	. . . . 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 28476	. . . . . . . . 9 ⊢ (𝜑 → 𝑆 ∈ 𝑃)
25		opphllem5.2	. . . . . . . . . 10 ⊢ (𝜑 → 𝑉(𝐾‘𝑆)𝐶)
26	1, 3, 7, 22, 21, 24, 5, 25	hlne2 28533	. . . . . . . . 9 ⊢ (𝜑 → 𝐶 ≠ 𝑆)
27	1, 3, 4, 5, 21, 24, 26	tglinecom 28562	. . . . . . . 8 ⊢ (𝜑 → (𝐶𝐿𝑆) = (𝑆𝐿𝐶))
28	23, 27	breqtrd 5133	. . . . . . 7 ⊢ (𝜑 → 𝐷(⟂G‘𝐺)(𝑆𝐿𝐶))
29	1, 3, 7, 22, 21, 24, 5, 25	hlcomd 28531	. . . . . . 7 ⊢ (𝜑 → 𝐶(𝐾‘𝑆)𝑉)
30	1, 2, 3, 4, 5, 6, 7, 20, 21, 22, 28, 29	hlperpnel 28652	. . . . . 6 ⊢ (𝜑 → ¬ 𝑉 ∈ 𝐷)
31	30	ad3antrrr 730	. . . . 5 ⊢ ((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) → ¬ 𝑉 ∈ 𝐷)
32		simplr 768	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) → 𝑡 ∈ 𝐷)
33		simpr 484	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) ∧ 𝑅 = 𝑡) → 𝑅 = 𝑡)
34		eqid 2729	. . . . . . . . 9 ⊢ (pInvG‘𝐺) = (pInvG‘𝐺)
35	5	ad4antr 732	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) ∧ 𝑅 ≠ 𝑡) → 𝐺 ∈ TarskiG)
36	21	ad4antr 732	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) ∧ 𝑅 ≠ 𝑡) → 𝐶 ∈ 𝑃)
37	12	ad4antr 732	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) ∧ 𝑅 ≠ 𝑡) → 𝑅 ∈ 𝑃)
38	5	ad3antrrr 730	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) → 𝐺 ∈ TarskiG)
39	6	ad3antrrr 730	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) → 𝐷 ∈ ran 𝐿)
40	1, 4, 3, 38, 39, 32	tglnpt 28476	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) → 𝑡 ∈ 𝑃)
41	40	adantr 480	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) ∧ 𝑅 ≠ 𝑡) → 𝑡 ∈ 𝑃)
42	9	ad4antr 732	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) ∧ 𝑅 ≠ 𝑡) → 𝐴 ∈ 𝑃)
43	24	ad4antr 732	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) ∧ 𝑅 ≠ 𝑡) → 𝑆 ∈ 𝑃)
44		simpllr 775	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) → 𝑅 = 𝑆)
45	1, 3, 4, 5, 21, 24, 26	tglinerflx2 28561	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑆 ∈ (𝐶𝐿𝑆))
46	45	ad3antrrr 730	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) → 𝑆 ∈ (𝐶𝐿𝑆))
47	44, 46	eqeltrd 2828	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) → 𝑅 ∈ (𝐶𝐿𝑆))
48	47	adantr 480	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) ∧ 𝑅 ≠ 𝑡) → 𝑅 ∈ (𝐶𝐿𝑆))
49	4, 5, 23	perpln2 28638	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐶𝐿𝑆) ∈ ran 𝐿)
50	1, 2, 3, 4, 5, 6, 49, 23	perpcom 28640	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐶𝐿𝑆)(⟂G‘𝐺)𝐷)
51	50	ad4antr 732	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) ∧ 𝑅 ≠ 𝑡) → (𝐶𝐿𝑆)(⟂G‘𝐺)𝐷)
52		simpr 484	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) ∧ 𝑅 ≠ 𝑡) → 𝑅 ≠ 𝑡)
53	6	ad4antr 732	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) ∧ 𝑅 ≠ 𝑡) → 𝐷 ∈ ran 𝐿)
54	8	ad4antr 732	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) ∧ 𝑅 ≠ 𝑡) → 𝑅 ∈ 𝐷)
55		simpllr 775	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) ∧ 𝑅 ≠ 𝑡) → 𝑡 ∈ 𝐷)
56	1, 3, 4, 35, 37, 41, 52, 52, 53, 54, 55	tglinethru 28563	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) ∧ 𝑅 ≠ 𝑡) → 𝐷 = (𝑅𝐿𝑡))
57	51, 56	breqtrd 5133	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) ∧ 𝑅 ≠ 𝑡) → (𝐶𝐿𝑆)(⟂G‘𝐺)(𝑅𝐿𝑡))
58	1, 2, 3, 4, 35, 36, 43, 48, 41, 57	perprag 28653	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) ∧ 𝑅 ≠ 𝑡) → ⟨“𝐶𝑅𝑡”⟩ ∈ (∟G‘𝐺))
59	1, 3, 4, 5, 9, 12, 14	tglinerflx2 28561	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑅 ∈ (𝐴𝐿𝑅))
60	59	ad4antr 732	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) ∧ 𝑅 ≠ 𝑡) → 𝑅 ∈ (𝐴𝐿𝑅))
61	4, 5, 11	perpln2 28638	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐴𝐿𝑅) ∈ ran 𝐿)
62	1, 2, 3, 4, 5, 6, 61, 11	perpcom 28640	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐴𝐿𝑅)(⟂G‘𝐺)𝐷)
63	62	ad4antr 732	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) ∧ 𝑅 ≠ 𝑡) → (𝐴𝐿𝑅)(⟂G‘𝐺)𝐷)
64	63, 56	breqtrd 5133	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) ∧ 𝑅 ≠ 𝑡) → (𝐴𝐿𝑅)(⟂G‘𝐺)(𝑅𝐿𝑡))
65	1, 2, 3, 4, 35, 42, 37, 60, 41, 64	perprag 28653	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) ∧ 𝑅 ≠ 𝑡) → ⟨“𝐴𝑅𝑡”⟩ ∈ (∟G‘𝐺))
66		simplr 768	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) ∧ 𝑅 ≠ 𝑡) → 𝑡 ∈ (𝐴𝐼𝐶))
67	1, 2, 3, 35, 42, 41, 36, 66	tgbtwncom 28415	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) ∧ 𝑅 ≠ 𝑡) → 𝑡 ∈ (𝐶𝐼𝐴))
68	1, 2, 3, 4, 34, 35, 36, 37, 41, 42, 58, 65, 67	ragflat2 28630	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) ∧ 𝑅 ≠ 𝑡) → 𝑅 = 𝑡)
69	33, 68	pm2.61dane 3012	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) → 𝑅 = 𝑡)
70	9	ad3antrrr 730	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) → 𝐴 ∈ 𝑃)
71	10	ad3antrrr 730	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) → 𝑈 ∈ 𝑃)
72	22	ad3antrrr 730	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) → 𝑉 ∈ 𝑃)
73	12	ad3antrrr 730	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) → 𝑅 ∈ 𝑃)
74	17	ad3antrrr 730	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) → 𝐴(𝐾‘𝑅)𝑈)
75	21	ad3antrrr 730	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) → 𝐶 ∈ 𝑃)
76	25	ad3antrrr 730	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) → 𝑉(𝐾‘𝑆)𝐶)
77	44	fveq2d 6862	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) → (𝐾‘𝑅) = (𝐾‘𝑆))
78	77	breqd 5118	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) → (𝑉(𝐾‘𝑅)𝐶 ↔ 𝑉(𝐾‘𝑆)𝐶))
79	76, 78	mpbird 257	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) → 𝑉(𝐾‘𝑅)𝐶)
80	1, 3, 7, 72, 75, 73, 38, 79	hlcomd 28531	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) → 𝐶(𝐾‘𝑅)𝑉)
81		simpr 484	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) → 𝑡 ∈ (𝐴𝐼𝐶))
82	69, 81	eqeltrd 2828	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) → 𝑅 ∈ (𝐴𝐼𝐶))
83	1, 2, 3, 38, 70, 73, 75, 82	tgbtwncom 28415	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) → 𝑅 ∈ (𝐶𝐼𝐴))
84	1, 3, 7, 75, 72, 70, 38, 73, 80, 83	btwnhl 28541	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) → 𝑅 ∈ (𝑉𝐼𝐴))
85	1, 2, 3, 38, 72, 73, 70, 84	tgbtwncom 28415	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) → 𝑅 ∈ (𝐴𝐼𝑉))
86	1, 3, 7, 70, 71, 72, 38, 73, 74, 85	btwnhl 28541	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) → 𝑅 ∈ (𝑈𝐼𝑉))
87	69, 86	eqeltrrd 2829	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) → 𝑡 ∈ (𝑈𝐼𝑉))
88		rspe 3227	. . . . . 6 ⊢ ((𝑡 ∈ 𝐷 ∧ 𝑡 ∈ (𝑈𝐼𝑉)) → ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝑈𝐼𝑉))
89	32, 87, 88	syl2anc 584	. . . . 5 ⊢ ((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) → ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝑈𝐼𝑉))
90	19, 31, 89	jca31 514	. . . 4 ⊢ ((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) → ((¬ 𝑈 ∈ 𝐷 ∧ ¬ 𝑉 ∈ 𝐷) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝑈𝐼𝑉)))
91		hpg.o	. . . . . 6 ⊢ 𝑂 = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝑃 ∖ 𝐷) ∧ 𝑏 ∈ (𝑃 ∖ 𝐷)) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝑎𝐼𝑏))}
92	1, 2, 3, 91, 10, 22	islnopp 28666	. . . . 5 ⊢ (𝜑 → (𝑈𝑂𝑉 ↔ ((¬ 𝑈 ∈ 𝐷 ∧ ¬ 𝑉 ∈ 𝐷) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝑈𝐼𝑉))))
93	92	ad3antrrr 730	. . . 4 ⊢ ((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) → (𝑈𝑂𝑉 ↔ ((¬ 𝑈 ∈ 𝐷 ∧ ¬ 𝑉 ∈ 𝐷) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝑈𝐼𝑉))))
94	90, 93	mpbird 257	. . 3 ⊢ ((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) → 𝑈𝑂𝑉)
95		opphllem5.o	. . . . . 6 ⊢ (𝜑 → 𝐴𝑂𝐶)
96	1, 2, 3, 91, 9, 21	islnopp 28666	. . . . . 6 ⊢ (𝜑 → (𝐴𝑂𝐶 ↔ ((¬ 𝐴 ∈ 𝐷 ∧ ¬ 𝐶 ∈ 𝐷) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝐴𝐼𝐶))))
97	95, 96	mpbid 232	. . . . 5 ⊢ (𝜑 → ((¬ 𝐴 ∈ 𝐷 ∧ ¬ 𝐶 ∈ 𝐷) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝐴𝐼𝐶)))
98	97	simprd 495	. . . 4 ⊢ (𝜑 → ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝐴𝐼𝐶))
99	98	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝑅 = 𝑆) → ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝐴𝐼𝐶))
100	94, 99	r19.29a 3141	. 2 ⊢ ((𝜑 ∧ 𝑅 = 𝑆) → 𝑈𝑂𝑉)
101	6	ad4antr 732	. . . . 5 ⊢ (((((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ 𝑚 ∈ 𝑃) ∧ 𝑆 = (((pInvG‘𝐺)‘𝑚)‘𝑅)) ∧ (𝑆 − 𝐶)(≤G‘𝐺)(𝑅 − 𝐴)) → 𝐷 ∈ ran 𝐿)
102	5	ad4antr 732	. . . . 5 ⊢ (((((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ 𝑚 ∈ 𝑃) ∧ 𝑆 = (((pInvG‘𝐺)‘𝑚)‘𝑅)) ∧ (𝑆 − 𝐶)(≤G‘𝐺)(𝑅 − 𝐴)) → 𝐺 ∈ TarskiG)
103		eqid 2729	. . . . 5 ⊢ ((pInvG‘𝐺)‘𝑚) = ((pInvG‘𝐺)‘𝑚)
104	9	ad4antr 732	. . . . 5 ⊢ (((((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ 𝑚 ∈ 𝑃) ∧ 𝑆 = (((pInvG‘𝐺)‘𝑚)‘𝑅)) ∧ (𝑆 − 𝐶)(≤G‘𝐺)(𝑅 − 𝐴)) → 𝐴 ∈ 𝑃)
105	21	ad4antr 732	. . . . 5 ⊢ (((((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ 𝑚 ∈ 𝑃) ∧ 𝑆 = (((pInvG‘𝐺)‘𝑚)‘𝑅)) ∧ (𝑆 − 𝐶)(≤G‘𝐺)(𝑅 − 𝐴)) → 𝐶 ∈ 𝑃)
106	8	ad4antr 732	. . . . 5 ⊢ (((((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ 𝑚 ∈ 𝑃) ∧ 𝑆 = (((pInvG‘𝐺)‘𝑚)‘𝑅)) ∧ (𝑆 − 𝐶)(≤G‘𝐺)(𝑅 − 𝐴)) → 𝑅 ∈ 𝐷)
107	20	ad4antr 732	. . . . 5 ⊢ (((((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ 𝑚 ∈ 𝑃) ∧ 𝑆 = (((pInvG‘𝐺)‘𝑚)‘𝑅)) ∧ (𝑆 − 𝐶)(≤G‘𝐺)(𝑅 − 𝐴)) → 𝑆 ∈ 𝐷)
108		simpllr 775	. . . . 5 ⊢ (((((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ 𝑚 ∈ 𝑃) ∧ 𝑆 = (((pInvG‘𝐺)‘𝑚)‘𝑅)) ∧ (𝑆 − 𝐶)(≤G‘𝐺)(𝑅 − 𝐴)) → 𝑚 ∈ 𝑃)
109	95	ad4antr 732	. . . . 5 ⊢ (((((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ 𝑚 ∈ 𝑃) ∧ 𝑆 = (((pInvG‘𝐺)‘𝑚)‘𝑅)) ∧ (𝑆 − 𝐶)(≤G‘𝐺)(𝑅 − 𝐴)) → 𝐴𝑂𝐶)
110	11	ad4antr 732	. . . . 5 ⊢ (((((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ 𝑚 ∈ 𝑃) ∧ 𝑆 = (((pInvG‘𝐺)‘𝑚)‘𝑅)) ∧ (𝑆 − 𝐶)(≤G‘𝐺)(𝑅 − 𝐴)) → 𝐷(⟂G‘𝐺)(𝐴𝐿𝑅))
111	23	ad4antr 732	. . . . 5 ⊢ (((((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ 𝑚 ∈ 𝑃) ∧ 𝑆 = (((pInvG‘𝐺)‘𝑚)‘𝑅)) ∧ (𝑆 − 𝐶)(≤G‘𝐺)(𝑅 − 𝐴)) → 𝐷(⟂G‘𝐺)(𝐶𝐿𝑆))
112		simpr 484	. . . . . 6 ⊢ ((𝜑 ∧ 𝑅 ≠ 𝑆) → 𝑅 ≠ 𝑆)
113	112	ad3antrrr 730	. . . . 5 ⊢ (((((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ 𝑚 ∈ 𝑃) ∧ 𝑆 = (((pInvG‘𝐺)‘𝑚)‘𝑅)) ∧ (𝑆 − 𝐶)(≤G‘𝐺)(𝑅 − 𝐴)) → 𝑅 ≠ 𝑆)
114		simpr 484	. . . . 5 ⊢ (((((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ 𝑚 ∈ 𝑃) ∧ 𝑆 = (((pInvG‘𝐺)‘𝑚)‘𝑅)) ∧ (𝑆 − 𝐶)(≤G‘𝐺)(𝑅 − 𝐴)) → (𝑆 − 𝐶)(≤G‘𝐺)(𝑅 − 𝐴))
115	10	ad4antr 732	. . . . 5 ⊢ (((((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ 𝑚 ∈ 𝑃) ∧ 𝑆 = (((pInvG‘𝐺)‘𝑚)‘𝑅)) ∧ (𝑆 − 𝐶)(≤G‘𝐺)(𝑅 − 𝐴)) → 𝑈 ∈ 𝑃)
116		simplr 768	. . . . . 6 ⊢ (((((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ 𝑚 ∈ 𝑃) ∧ 𝑆 = (((pInvG‘𝐺)‘𝑚)‘𝑅)) ∧ (𝑆 − 𝐶)(≤G‘𝐺)(𝑅 − 𝐴)) → 𝑆 = (((pInvG‘𝐺)‘𝑚)‘𝑅))
117	116	eqcomd 2735	. . . . 5 ⊢ (((((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ 𝑚 ∈ 𝑃) ∧ 𝑆 = (((pInvG‘𝐺)‘𝑚)‘𝑅)) ∧ (𝑆 − 𝐶)(≤G‘𝐺)(𝑅 − 𝐴)) → (((pInvG‘𝐺)‘𝑚)‘𝑅) = 𝑆)
118	22	ad4antr 732	. . . . 5 ⊢ (((((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ 𝑚 ∈ 𝑃) ∧ 𝑆 = (((pInvG‘𝐺)‘𝑚)‘𝑅)) ∧ (𝑆 − 𝐶)(≤G‘𝐺)(𝑅 − 𝐴)) → 𝑉 ∈ 𝑃)
119	13	ad4antr 732	. . . . 5 ⊢ (((((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ 𝑚 ∈ 𝑃) ∧ 𝑆 = (((pInvG‘𝐺)‘𝑚)‘𝑅)) ∧ (𝑆 − 𝐶)(≤G‘𝐺)(𝑅 − 𝐴)) → 𝑈(𝐾‘𝑅)𝐴)
120	25	ad4antr 732	. . . . 5 ⊢ (((((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ 𝑚 ∈ 𝑃) ∧ 𝑆 = (((pInvG‘𝐺)‘𝑚)‘𝑅)) ∧ (𝑆 − 𝐶)(≤G‘𝐺)(𝑅 − 𝐴)) → 𝑉(𝐾‘𝑆)𝐶)
121	1, 2, 3, 91, 4, 101, 102, 7, 103, 104, 105, 106, 107, 108, 109, 110, 111, 113, 114, 115, 117, 118, 119, 120	opphllem4 28677	. . . 4 ⊢ (((((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ 𝑚 ∈ 𝑃) ∧ 𝑆 = (((pInvG‘𝐺)‘𝑚)‘𝑅)) ∧ (𝑆 − 𝐶)(≤G‘𝐺)(𝑅 − 𝐴)) → 𝑈𝑂𝑉)
122	6	ad4antr 732	. . . . 5 ⊢ (((((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ 𝑚 ∈ 𝑃) ∧ 𝑆 = (((pInvG‘𝐺)‘𝑚)‘𝑅)) ∧ (𝑅 − 𝐴)(≤G‘𝐺)(𝑆 − 𝐶)) → 𝐷 ∈ ran 𝐿)
123	5	ad4antr 732	. . . . 5 ⊢ (((((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ 𝑚 ∈ 𝑃) ∧ 𝑆 = (((pInvG‘𝐺)‘𝑚)‘𝑅)) ∧ (𝑅 − 𝐴)(≤G‘𝐺)(𝑆 − 𝐶)) → 𝐺 ∈ TarskiG)
124	22	ad4antr 732	. . . . 5 ⊢ (((((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ 𝑚 ∈ 𝑃) ∧ 𝑆 = (((pInvG‘𝐺)‘𝑚)‘𝑅)) ∧ (𝑅 − 𝐴)(≤G‘𝐺)(𝑆 − 𝐶)) → 𝑉 ∈ 𝑃)
125	10	ad4antr 732	. . . . 5 ⊢ (((((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ 𝑚 ∈ 𝑃) ∧ 𝑆 = (((pInvG‘𝐺)‘𝑚)‘𝑅)) ∧ (𝑅 − 𝐴)(≤G‘𝐺)(𝑆 − 𝐶)) → 𝑈 ∈ 𝑃)
126	21	ad4antr 732	. . . . . 6 ⊢ (((((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ 𝑚 ∈ 𝑃) ∧ 𝑆 = (((pInvG‘𝐺)‘𝑚)‘𝑅)) ∧ (𝑅 − 𝐴)(≤G‘𝐺)(𝑆 − 𝐶)) → 𝐶 ∈ 𝑃)
127	9	ad4antr 732	. . . . . 6 ⊢ (((((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ 𝑚 ∈ 𝑃) ∧ 𝑆 = (((pInvG‘𝐺)‘𝑚)‘𝑅)) ∧ (𝑅 − 𝐴)(≤G‘𝐺)(𝑆 − 𝐶)) → 𝐴 ∈ 𝑃)
128	20	ad4antr 732	. . . . . 6 ⊢ (((((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ 𝑚 ∈ 𝑃) ∧ 𝑆 = (((pInvG‘𝐺)‘𝑚)‘𝑅)) ∧ (𝑅 − 𝐴)(≤G‘𝐺)(𝑆 − 𝐶)) → 𝑆 ∈ 𝐷)
129	8	ad4antr 732	. . . . . 6 ⊢ (((((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ 𝑚 ∈ 𝑃) ∧ 𝑆 = (((pInvG‘𝐺)‘𝑚)‘𝑅)) ∧ (𝑅 − 𝐴)(≤G‘𝐺)(𝑆 − 𝐶)) → 𝑅 ∈ 𝐷)
130		simpllr 775	. . . . . 6 ⊢ (((((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ 𝑚 ∈ 𝑃) ∧ 𝑆 = (((pInvG‘𝐺)‘𝑚)‘𝑅)) ∧ (𝑅 − 𝐴)(≤G‘𝐺)(𝑆 − 𝐶)) → 𝑚 ∈ 𝑃)
131	95	ad4antr 732	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ 𝑚 ∈ 𝑃) ∧ 𝑆 = (((pInvG‘𝐺)‘𝑚)‘𝑅)) ∧ (𝑅 − 𝐴)(≤G‘𝐺)(𝑆 − 𝐶)) → 𝐴𝑂𝐶)
132	1, 2, 3, 91, 4, 122, 123, 127, 126, 131	oppcom 28671	. . . . . 6 ⊢ (((((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ 𝑚 ∈ 𝑃) ∧ 𝑆 = (((pInvG‘𝐺)‘𝑚)‘𝑅)) ∧ (𝑅 − 𝐴)(≤G‘𝐺)(𝑆 − 𝐶)) → 𝐶𝑂𝐴)
133	23	ad4antr 732	. . . . . 6 ⊢ (((((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ 𝑚 ∈ 𝑃) ∧ 𝑆 = (((pInvG‘𝐺)‘𝑚)‘𝑅)) ∧ (𝑅 − 𝐴)(≤G‘𝐺)(𝑆 − 𝐶)) → 𝐷(⟂G‘𝐺)(𝐶𝐿𝑆))
134	11	ad4antr 732	. . . . . 6 ⊢ (((((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ 𝑚 ∈ 𝑃) ∧ 𝑆 = (((pInvG‘𝐺)‘𝑚)‘𝑅)) ∧ (𝑅 − 𝐴)(≤G‘𝐺)(𝑆 − 𝐶)) → 𝐷(⟂G‘𝐺)(𝐴𝐿𝑅))
135	112	necomd 2980	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑅 ≠ 𝑆) → 𝑆 ≠ 𝑅)
136	135	ad3antrrr 730	. . . . . 6 ⊢ (((((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ 𝑚 ∈ 𝑃) ∧ 𝑆 = (((pInvG‘𝐺)‘𝑚)‘𝑅)) ∧ (𝑅 − 𝐴)(≤G‘𝐺)(𝑆 − 𝐶)) → 𝑆 ≠ 𝑅)
137		simpr 484	. . . . . 6 ⊢ (((((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ 𝑚 ∈ 𝑃) ∧ 𝑆 = (((pInvG‘𝐺)‘𝑚)‘𝑅)) ∧ (𝑅 − 𝐴)(≤G‘𝐺)(𝑆 − 𝐶)) → (𝑅 − 𝐴)(≤G‘𝐺)(𝑆 − 𝐶))
138	12	ad4antr 732	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ 𝑚 ∈ 𝑃) ∧ 𝑆 = (((pInvG‘𝐺)‘𝑚)‘𝑅)) ∧ (𝑅 − 𝐴)(≤G‘𝐺)(𝑆 − 𝐶)) → 𝑅 ∈ 𝑃)
139		simplr 768	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ 𝑚 ∈ 𝑃) ∧ 𝑆 = (((pInvG‘𝐺)‘𝑚)‘𝑅)) ∧ (𝑅 − 𝐴)(≤G‘𝐺)(𝑆 − 𝐶)) → 𝑆 = (((pInvG‘𝐺)‘𝑚)‘𝑅))
140	139	eqcomd 2735	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ 𝑚 ∈ 𝑃) ∧ 𝑆 = (((pInvG‘𝐺)‘𝑚)‘𝑅)) ∧ (𝑅 − 𝐴)(≤G‘𝐺)(𝑆 − 𝐶)) → (((pInvG‘𝐺)‘𝑚)‘𝑅) = 𝑆)
141	1, 2, 3, 4, 34, 123, 130, 103, 138, 140	mircom 28590	. . . . . 6 ⊢ (((((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ 𝑚 ∈ 𝑃) ∧ 𝑆 = (((pInvG‘𝐺)‘𝑚)‘𝑅)) ∧ (𝑅 − 𝐴)(≤G‘𝐺)(𝑆 − 𝐶)) → (((pInvG‘𝐺)‘𝑚)‘𝑆) = 𝑅)
142	25	ad4antr 732	. . . . . 6 ⊢ (((((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ 𝑚 ∈ 𝑃) ∧ 𝑆 = (((pInvG‘𝐺)‘𝑚)‘𝑅)) ∧ (𝑅 − 𝐴)(≤G‘𝐺)(𝑆 − 𝐶)) → 𝑉(𝐾‘𝑆)𝐶)
143	13	ad4antr 732	. . . . . 6 ⊢ (((((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ 𝑚 ∈ 𝑃) ∧ 𝑆 = (((pInvG‘𝐺)‘𝑚)‘𝑅)) ∧ (𝑅 − 𝐴)(≤G‘𝐺)(𝑆 − 𝐶)) → 𝑈(𝐾‘𝑅)𝐴)
144	1, 2, 3, 91, 4, 122, 123, 7, 103, 126, 127, 128, 129, 130, 132, 133, 134, 136, 137, 124, 141, 125, 142, 143	opphllem4 28677	. . . . 5 ⊢ (((((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ 𝑚 ∈ 𝑃) ∧ 𝑆 = (((pInvG‘𝐺)‘𝑚)‘𝑅)) ∧ (𝑅 − 𝐴)(≤G‘𝐺)(𝑆 − 𝐶)) → 𝑉𝑂𝑈)
145	1, 2, 3, 91, 4, 122, 123, 124, 125, 144	oppcom 28671	. . . 4 ⊢ (((((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ 𝑚 ∈ 𝑃) ∧ 𝑆 = (((pInvG‘𝐺)‘𝑚)‘𝑅)) ∧ (𝑅 − 𝐴)(≤G‘𝐺)(𝑆 − 𝐶)) → 𝑈𝑂𝑉)
146		eqid 2729	. . . . . 6 ⊢ (≤G‘𝐺) = (≤G‘𝐺)
147	1, 2, 3, 146, 5, 24, 21, 12, 9	legtrid 28518	. . . . 5 ⊢ (𝜑 → ((𝑆 − 𝐶)(≤G‘𝐺)(𝑅 − 𝐴) ∨ (𝑅 − 𝐴)(≤G‘𝐺)(𝑆 − 𝐶)))
148	147	ad3antrrr 730	. . . 4 ⊢ ((((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ 𝑚 ∈ 𝑃) ∧ 𝑆 = (((pInvG‘𝐺)‘𝑚)‘𝑅)) → ((𝑆 − 𝐶)(≤G‘𝐺)(𝑅 − 𝐴) ∨ (𝑅 − 𝐴)(≤G‘𝐺)(𝑆 − 𝐶)))
149	121, 145, 148	mpjaodan 960	. . 3 ⊢ ((((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ 𝑚 ∈ 𝑃) ∧ 𝑆 = (((pInvG‘𝐺)‘𝑚)‘𝑅)) → 𝑈𝑂𝑉)
150	5	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑅 ≠ 𝑆) → 𝐺 ∈ TarskiG)
151	12	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑅 ≠ 𝑆) → 𝑅 ∈ 𝑃)
152	24	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑅 ≠ 𝑆) → 𝑆 ∈ 𝑃)
153	1, 2, 3, 91, 4, 6, 5, 9, 21, 95	opptgdim2 28672	. . . . 5 ⊢ (𝜑 → 𝐺Dim_TarskiG≥2)
154	153	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑅 ≠ 𝑆) → 𝐺Dim_TarskiG≥2)
155	1, 2, 3, 4, 150, 34, 151, 152, 154	midex 28664	. . 3 ⊢ ((𝜑 ∧ 𝑅 ≠ 𝑆) → ∃𝑚 ∈ 𝑃 𝑆 = (((pInvG‘𝐺)‘𝑚)‘𝑅))
156	149, 155	r19.29a 3141	. 2 ⊢ ((𝜑 ∧ 𝑅 ≠ 𝑆) → 𝑈𝑂𝑉)
157	100, 156	pm2.61dane 3012	1 ⊢ (𝜑 → 𝑈𝑂𝑉)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∨ wo 847 = wceq 1540 ∈ wcel 2109 ≠ wne 2925 ∃wrex 3053 ∖ cdif 3911 class class class wbr 5107 {copab 5169 ran crn 5639 ‘cfv 6511 (class class class)co 7387 2c2 12241 Basecbs 17179 distcds 17229 TarskiGcstrkg 28354 Dim_TarskiG≥cstrkgld 28358 Itvcitv 28360 LineGclng 28361 ≤Gcleg 28509 hlGchlg 28527 pInvGcmir 28579 ⟂Gcperpg 28622

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5234 ax-sep 5251 ax-nul 5261 ax-pow 5320 ax-pr 5387 ax-un 7711 ax-cnex 11124 ax-resscn 11125 ax-1cn 11126 ax-icn 11127 ax-addcl 11128 ax-addrcl 11129 ax-mulcl 11130 ax-mulrcl 11131 ax-mulcom 11132 ax-addass 11133 ax-mulass 11134 ax-distr 11135 ax-i2m1 11136 ax-1ne0 11137 ax-1rid 11138 ax-rnegex 11139 ax-rrecex 11140 ax-cnre 11141 ax-pre-lttri 11142 ax-pre-lttrn 11143 ax-pre-ltadd 11144 ax-pre-mulgt0 11145

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3354 df-reu 3355 df-rab 3406 df-v 3449 df-sbc 3754 df-csb 3863 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3934 df-nul 4297 df-if 4489 df-pw 4565 df-sn 4590 df-pr 4592 df-tp 4594 df-op 4596 df-uni 4872 df-int 4911 df-iun 4957 df-br 5108 df-opab 5170 df-mpt 5189 df-tr 5215 df-id 5533 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5591 df-we 5593 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6274 df-ord 6335 df-on 6336 df-lim 6337 df-suc 6338 df-iota 6464 df-fun 6513 df-fn 6514 df-f 6515 df-f1 6516 df-fo 6517 df-f1o 6518 df-fv 6519 df-riota 7344 df-ov 7390 df-oprab 7391 df-mpo 7392 df-om 7843 df-1st 7968 df-2nd 7969 df-frecs 8260 df-wrecs 8291 df-recs 8340 df-rdg 8378 df-1o 8434 df-oadd 8438 df-er 8671 df-map 8801 df-pm 8802 df-en 8919 df-dom 8920 df-sdom 8921 df-fin 8922 df-dju 9854 df-card 9892 df-pnf 11210 df-mnf 11211 df-xr 11212 df-ltxr 11213 df-le 11214 df-sub 11407 df-neg 11408 df-nn 12187 df-2 12249 df-3 12250 df-n0 12443 df-xnn0 12516 df-z 12530 df-uz 12794 df-fz 13469 df-fzo 13616 df-hash 14296 df-word 14479 df-concat 14536 df-s1 14561 df-s2 14814 df-s3 14815 df-trkgc 28375 df-trkgb 28376 df-trkgcb 28377 df-trkgld 28379 df-trkg 28380 df-cgrg 28438 df-leg 28510 df-hlg 28528 df-mir 28580 df-rag 28621 df-perpg 28623

This theorem is referenced by: opphl 28681

W3C validator