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Theorem opphllem5 28993

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 28786	. . . . . . . . 9 ⊢ (𝜑 → 𝑅 ∈ 𝑃)
13		opphllem5.1	. . . . . . . . . 10 ⊢ (𝜑 → 𝑈(𝐾‘𝑅)𝐴)
14	1, 3, 7, 10, 9, 12, 5, 13	hlne2 28843	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 ≠ 𝑅)
15	1, 3, 4, 5, 9, 12, 14	tglinecom 28872	. . . . . . . 8 ⊢ (𝜑 → (𝐴𝐿𝑅) = (𝑅𝐿𝐴))
16	11, 15	breqtrd 5141	. . . . . . 7 ⊢ (𝜑 → 𝐷(⟂G‘𝐺)(𝑅𝐿𝐴))
17	1, 3, 7, 10, 9, 12, 5, 13	hlcomd 28841	. . . . . . 7 ⊢ (𝜑 → 𝐴(𝐾‘𝑅)𝑈)
18	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 16, 17	hlperpnel 28967	. . . . . 6 ⊢ (𝜑 → ¬ 𝑈 ∈ 𝐷)
19	18	ad3antrrr 742	. . . . 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 28786	. . . . . . . . 9 ⊢ (𝜑 → 𝑆 ∈ 𝑃)
25		opphllem5.2	. . . . . . . . . 10 ⊢ (𝜑 → 𝑉(𝐾‘𝑆)𝐶)
26	1, 3, 7, 22, 21, 24, 5, 25	hlne2 28843	. . . . . . . . 9 ⊢ (𝜑 → 𝐶 ≠ 𝑆)
27	1, 3, 4, 5, 21, 24, 26	tglinecom 28872	. . . . . . . 8 ⊢ (𝜑 → (𝐶𝐿𝑆) = (𝑆𝐿𝐶))
28	23, 27	breqtrd 5141	. . . . . . 7 ⊢ (𝜑 → 𝐷(⟂G‘𝐺)(𝑆𝐿𝐶))
29	1, 3, 7, 22, 21, 24, 5, 25	hlcomd 28841	. . . . . . 7 ⊢ (𝜑 → 𝐶(𝐾‘𝑆)𝑉)
30	1, 2, 3, 4, 5, 6, 7, 20, 21, 22, 28, 29	hlperpnel 28967	. . . . . 6 ⊢ (𝜑 → ¬ 𝑉 ∈ 𝐷)
31	30	ad3antrrr 742	. . . . 5 ⊢ ((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) → ¬ 𝑉 ∈ 𝐷)
32		simplr 780	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) → 𝑡 ∈ 𝐷)
33		simpr 489	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) ∧ 𝑅 = 𝑡) → 𝑅 = 𝑡)
34		eqid 2769	. . . . . . . . 9 ⊢ (pInvG‘𝐺) = (pInvG‘𝐺)
35	5	ad4antr 744	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) ∧ 𝑅 ≠ 𝑡) → 𝐺 ∈ TarskiG)
36	21	ad4antr 744	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) ∧ 𝑅 ≠ 𝑡) → 𝐶 ∈ 𝑃)
37	12	ad4antr 744	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) ∧ 𝑅 ≠ 𝑡) → 𝑅 ∈ 𝑃)
38	5	ad3antrrr 742	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) → 𝐺 ∈ TarskiG)
39	6	ad3antrrr 742	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) → 𝐷 ∈ ran 𝐿)
40	1, 4, 3, 38, 39, 32	tglnpt 28786	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) → 𝑡 ∈ 𝑃)
41	40	adantr 485	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) ∧ 𝑅 ≠ 𝑡) → 𝑡 ∈ 𝑃)
42	9	ad4antr 744	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) ∧ 𝑅 ≠ 𝑡) → 𝐴 ∈ 𝑃)
43	24	ad4antr 744	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) ∧ 𝑅 ≠ 𝑡) → 𝑆 ∈ 𝑃)
44		simpllr 787	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) → 𝑅 = 𝑆)
45	1, 3, 4, 5, 21, 24, 26	tglinerflx2 28871	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑆 ∈ (𝐶𝐿𝑆))
46	45	ad3antrrr 742	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) → 𝑆 ∈ (𝐶𝐿𝑆))
47	44, 46	eqeltrd 2869	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) → 𝑅 ∈ (𝐶𝐿𝑆))
48	47	adantr 485	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) ∧ 𝑅 ≠ 𝑡) → 𝑅 ∈ (𝐶𝐿𝑆))
49	4, 5, 23	perpln2 28952	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐶𝐿𝑆) ∈ ran 𝐿)
50	1, 2, 3, 4, 5, 6, 49, 23	perpcom 28954	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐶𝐿𝑆)(⟂G‘𝐺)𝐷)
51	50	ad4antr 744	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) ∧ 𝑅 ≠ 𝑡) → (𝐶𝐿𝑆)(⟂G‘𝐺)𝐷)
52		simpr 489	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) ∧ 𝑅 ≠ 𝑡) → 𝑅 ≠ 𝑡)
53	6	ad4antr 744	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) ∧ 𝑅 ≠ 𝑡) → 𝐷 ∈ ran 𝐿)
54	8	ad4antr 744	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) ∧ 𝑅 ≠ 𝑡) → 𝑅 ∈ 𝐷)
55		simpllr 787	. . . . . . . . . . . 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 28873	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) ∧ 𝑅 ≠ 𝑡) → 𝐷 = (𝑅𝐿𝑡))
57	51, 56	breqtrd 5141	. . . . . . . . . 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 28968	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) ∧ 𝑅 ≠ 𝑡) → ⟨“𝐶𝑅𝑡”⟩ ∈ (∟G‘𝐺))
59	1, 3, 4, 5, 9, 12, 14	tglinerflx2 28871	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑅 ∈ (𝐴𝐿𝑅))
60	59	ad4antr 744	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) ∧ 𝑅 ≠ 𝑡) → 𝑅 ∈ (𝐴𝐿𝑅))
61	4, 5, 11	perpln2 28952	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐴𝐿𝑅) ∈ ran 𝐿)
62	1, 2, 3, 4, 5, 6, 61, 11	perpcom 28954	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐴𝐿𝑅)(⟂G‘𝐺)𝐷)
63	62	ad4antr 744	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) ∧ 𝑅 ≠ 𝑡) → (𝐴𝐿𝑅)(⟂G‘𝐺)𝐷)
64	63, 56	breqtrd 5141	. . . . . . . . . 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 28968	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) ∧ 𝑅 ≠ 𝑡) → ⟨“𝐴𝑅𝑡”⟩ ∈ (∟G‘𝐺))
66		simplr 780	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) ∧ 𝑅 ≠ 𝑡) → 𝑡 ∈ (𝐴𝐼𝐶))
67	1, 2, 3, 35, 42, 41, 36, 66	tgbtwncom 28725	. . . . . . . . 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 28944	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) ∧ 𝑅 ≠ 𝑡) → 𝑅 = 𝑡)
69	33, 68	pm2.61dane 3051	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) → 𝑅 = 𝑡)
70	9	ad3antrrr 742	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) → 𝐴 ∈ 𝑃)
71	10	ad3antrrr 742	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) → 𝑈 ∈ 𝑃)
72	22	ad3antrrr 742	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) → 𝑉 ∈ 𝑃)
73	12	ad3antrrr 742	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) → 𝑅 ∈ 𝑃)
74	17	ad3antrrr 742	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) → 𝐴(𝐾‘𝑅)𝑈)
75	21	ad3antrrr 742	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) → 𝐶 ∈ 𝑃)
76	25	ad3antrrr 742	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) → 𝑉(𝐾‘𝑆)𝐶)
77	44	fveq2d 6888	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) → (𝐾‘𝑅) = (𝐾‘𝑆))
78	77	breqd 5124	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) → (𝑉(𝐾‘𝑅)𝐶 ↔ 𝑉(𝐾‘𝑆)𝐶))
79	76, 78	mpbird 260	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) → 𝑉(𝐾‘𝑅)𝐶)
80	1, 3, 7, 72, 75, 73, 38, 79	hlcomd 28841	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) → 𝐶(𝐾‘𝑅)𝑉)
81		simpr 489	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) → 𝑡 ∈ (𝐴𝐼𝐶))
82	69, 81	eqeltrd 2869	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) → 𝑅 ∈ (𝐴𝐼𝐶))
83	1, 2, 3, 38, 70, 73, 75, 82	tgbtwncom 28725	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) → 𝑅 ∈ (𝐶𝐼𝐴))
84	1, 3, 7, 75, 72, 70, 38, 73, 80, 83	btwnhl 28851	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) → 𝑅 ∈ (𝑉𝐼𝐴))
85	1, 2, 3, 38, 72, 73, 70, 84	tgbtwncom 28725	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) → 𝑅 ∈ (𝐴𝐼𝑉))
86	1, 3, 7, 70, 71, 72, 38, 73, 74, 85	btwnhl 28851	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) → 𝑅 ∈ (𝑈𝐼𝑉))
87	69, 86	eqeltrrd 2870	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) → 𝑡 ∈ (𝑈𝐼𝑉))
88		rspe 3261	. . . . . 6 ⊢ ((𝑡 ∈ 𝐷 ∧ 𝑡 ∈ (𝑈𝐼𝑉)) → ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝑈𝐼𝑉))
89	32, 87, 88	syl2anc 595	. . . . 5 ⊢ ((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) → ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝑈𝐼𝑉))
90	19, 31, 89	jca31 523	. . . 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 28981	. . . . 5 ⊢ (𝜑 → (𝑈𝑂𝑉 ↔ ((¬ 𝑈 ∈ 𝐷 ∧ ¬ 𝑉 ∈ 𝐷) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝑈𝐼𝑉))))
93	92	ad3antrrr 742	. . . 4 ⊢ ((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) → (𝑈𝑂𝑉 ↔ ((¬ 𝑈 ∈ 𝐷 ∧ ¬ 𝑉 ∈ 𝐷) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝑈𝐼𝑉))))
94	90, 93	mpbird 260	. . 3 ⊢ ((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) → 𝑈𝑂𝑉)
95		opphllem5.o	. . . . . 6 ⊢ (𝜑 → 𝐴𝑂𝐶)
96	1, 2, 3, 91, 9, 21	islnopp 28981	. . . . . 6 ⊢ (𝜑 → (𝐴𝑂𝐶 ↔ ((¬ 𝐴 ∈ 𝐷 ∧ ¬ 𝐶 ∈ 𝐷) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝐴𝐼𝐶))))
97	95, 96	mpbid 235	. . . . 5 ⊢ (𝜑 → ((¬ 𝐴 ∈ 𝐷 ∧ ¬ 𝐶 ∈ 𝐷) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝐴𝐼𝐶)))
98	97	simprd 500	. . . 4 ⊢ (𝜑 → ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝐴𝐼𝐶))
99	98	adantr 485	. . 3 ⊢ ((𝜑 ∧ 𝑅 = 𝑆) → ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝐴𝐼𝐶))
100	94, 99	r19.29a 3179	. 2 ⊢ ((𝜑 ∧ 𝑅 = 𝑆) → 𝑈𝑂𝑉)
101	6	ad4antr 744	. . . . 5 ⊢ (((((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ 𝑚 ∈ 𝑃) ∧ 𝑆 = (((pInvG‘𝐺)‘𝑚)‘𝑅)) ∧ (𝑆 − 𝐶)(≤G‘𝐺)(𝑅 − 𝐴)) → 𝐷 ∈ ran 𝐿)
102	5	ad4antr 744	. . . . 5 ⊢ (((((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ 𝑚 ∈ 𝑃) ∧ 𝑆 = (((pInvG‘𝐺)‘𝑚)‘𝑅)) ∧ (𝑆 − 𝐶)(≤G‘𝐺)(𝑅 − 𝐴)) → 𝐺 ∈ TarskiG)
103		eqid 2769	. . . . 5 ⊢ ((pInvG‘𝐺)‘𝑚) = ((pInvG‘𝐺)‘𝑚)
104	9	ad4antr 744	. . . . 5 ⊢ (((((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ 𝑚 ∈ 𝑃) ∧ 𝑆 = (((pInvG‘𝐺)‘𝑚)‘𝑅)) ∧ (𝑆 − 𝐶)(≤G‘𝐺)(𝑅 − 𝐴)) → 𝐴 ∈ 𝑃)
105	21	ad4antr 744	. . . . 5 ⊢ (((((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ 𝑚 ∈ 𝑃) ∧ 𝑆 = (((pInvG‘𝐺)‘𝑚)‘𝑅)) ∧ (𝑆 − 𝐶)(≤G‘𝐺)(𝑅 − 𝐴)) → 𝐶 ∈ 𝑃)
106	8	ad4antr 744	. . . . 5 ⊢ (((((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ 𝑚 ∈ 𝑃) ∧ 𝑆 = (((pInvG‘𝐺)‘𝑚)‘𝑅)) ∧ (𝑆 − 𝐶)(≤G‘𝐺)(𝑅 − 𝐴)) → 𝑅 ∈ 𝐷)
107	20	ad4antr 744	. . . . 5 ⊢ (((((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ 𝑚 ∈ 𝑃) ∧ 𝑆 = (((pInvG‘𝐺)‘𝑚)‘𝑅)) ∧ (𝑆 − 𝐶)(≤G‘𝐺)(𝑅 − 𝐴)) → 𝑆 ∈ 𝐷)
108		simpllr 787	. . . . 5 ⊢ (((((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ 𝑚 ∈ 𝑃) ∧ 𝑆 = (((pInvG‘𝐺)‘𝑚)‘𝑅)) ∧ (𝑆 − 𝐶)(≤G‘𝐺)(𝑅 − 𝐴)) → 𝑚 ∈ 𝑃)
109	95	ad4antr 744	. . . . 5 ⊢ (((((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ 𝑚 ∈ 𝑃) ∧ 𝑆 = (((pInvG‘𝐺)‘𝑚)‘𝑅)) ∧ (𝑆 − 𝐶)(≤G‘𝐺)(𝑅 − 𝐴)) → 𝐴𝑂𝐶)
110	11	ad4antr 744	. . . . 5 ⊢ (((((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ 𝑚 ∈ 𝑃) ∧ 𝑆 = (((pInvG‘𝐺)‘𝑚)‘𝑅)) ∧ (𝑆 − 𝐶)(≤G‘𝐺)(𝑅 − 𝐴)) → 𝐷(⟂G‘𝐺)(𝐴𝐿𝑅))
111	23	ad4antr 744	. . . . 5 ⊢ (((((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ 𝑚 ∈ 𝑃) ∧ 𝑆 = (((pInvG‘𝐺)‘𝑚)‘𝑅)) ∧ (𝑆 − 𝐶)(≤G‘𝐺)(𝑅 − 𝐴)) → 𝐷(⟂G‘𝐺)(𝐶𝐿𝑆))
112		simpr 489	. . . . . 6 ⊢ ((𝜑 ∧ 𝑅 ≠ 𝑆) → 𝑅 ≠ 𝑆)
113	112	ad3antrrr 742	. . . . 5 ⊢ (((((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ 𝑚 ∈ 𝑃) ∧ 𝑆 = (((pInvG‘𝐺)‘𝑚)‘𝑅)) ∧ (𝑆 − 𝐶)(≤G‘𝐺)(𝑅 − 𝐴)) → 𝑅 ≠ 𝑆)
114		simpr 489	. . . . 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 744	. . . . 5 ⊢ (((((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ 𝑚 ∈ 𝑃) ∧ 𝑆 = (((pInvG‘𝐺)‘𝑚)‘𝑅)) ∧ (𝑆 − 𝐶)(≤G‘𝐺)(𝑅 − 𝐴)) → 𝑈 ∈ 𝑃)
116		simplr 780	. . . . . 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 2775	. . . . 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 744	. . . . 5 ⊢ (((((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ 𝑚 ∈ 𝑃) ∧ 𝑆 = (((pInvG‘𝐺)‘𝑚)‘𝑅)) ∧ (𝑆 − 𝐶)(≤G‘𝐺)(𝑅 − 𝐴)) → 𝑉 ∈ 𝑃)
119	13	ad4antr 744	. . . . 5 ⊢ (((((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ 𝑚 ∈ 𝑃) ∧ 𝑆 = (((pInvG‘𝐺)‘𝑚)‘𝑅)) ∧ (𝑆 − 𝐶)(≤G‘𝐺)(𝑅 − 𝐴)) → 𝑈(𝐾‘𝑅)𝐴)
120	25	ad4antr 744	. . . . 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 28992	. . . 4 ⊢ (((((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ 𝑚 ∈ 𝑃) ∧ 𝑆 = (((pInvG‘𝐺)‘𝑚)‘𝑅)) ∧ (𝑆 − 𝐶)(≤G‘𝐺)(𝑅 − 𝐴)) → 𝑈𝑂𝑉)
122	6	ad4antr 744	. . . . 5 ⊢ (((((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ 𝑚 ∈ 𝑃) ∧ 𝑆 = (((pInvG‘𝐺)‘𝑚)‘𝑅)) ∧ (𝑅 − 𝐴)(≤G‘𝐺)(𝑆 − 𝐶)) → 𝐷 ∈ ran 𝐿)
123	5	ad4antr 744	. . . . 5 ⊢ (((((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ 𝑚 ∈ 𝑃) ∧ 𝑆 = (((pInvG‘𝐺)‘𝑚)‘𝑅)) ∧ (𝑅 − 𝐴)(≤G‘𝐺)(𝑆 − 𝐶)) → 𝐺 ∈ TarskiG)
124	22	ad4antr 744	. . . . 5 ⊢ (((((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ 𝑚 ∈ 𝑃) ∧ 𝑆 = (((pInvG‘𝐺)‘𝑚)‘𝑅)) ∧ (𝑅 − 𝐴)(≤G‘𝐺)(𝑆 − 𝐶)) → 𝑉 ∈ 𝑃)
125	10	ad4antr 744	. . . . 5 ⊢ (((((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ 𝑚 ∈ 𝑃) ∧ 𝑆 = (((pInvG‘𝐺)‘𝑚)‘𝑅)) ∧ (𝑅 − 𝐴)(≤G‘𝐺)(𝑆 − 𝐶)) → 𝑈 ∈ 𝑃)
126	21	ad4antr 744	. . . . . 6 ⊢ (((((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ 𝑚 ∈ 𝑃) ∧ 𝑆 = (((pInvG‘𝐺)‘𝑚)‘𝑅)) ∧ (𝑅 − 𝐴)(≤G‘𝐺)(𝑆 − 𝐶)) → 𝐶 ∈ 𝑃)
127	9	ad4antr 744	. . . . . 6 ⊢ (((((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ 𝑚 ∈ 𝑃) ∧ 𝑆 = (((pInvG‘𝐺)‘𝑚)‘𝑅)) ∧ (𝑅 − 𝐴)(≤G‘𝐺)(𝑆 − 𝐶)) → 𝐴 ∈ 𝑃)
128	20	ad4antr 744	. . . . . 6 ⊢ (((((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ 𝑚 ∈ 𝑃) ∧ 𝑆 = (((pInvG‘𝐺)‘𝑚)‘𝑅)) ∧ (𝑅 − 𝐴)(≤G‘𝐺)(𝑆 − 𝐶)) → 𝑆 ∈ 𝐷)
129	8	ad4antr 744	. . . . . 6 ⊢ (((((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ 𝑚 ∈ 𝑃) ∧ 𝑆 = (((pInvG‘𝐺)‘𝑚)‘𝑅)) ∧ (𝑅 − 𝐴)(≤G‘𝐺)(𝑆 − 𝐶)) → 𝑅 ∈ 𝐷)
130		simpllr 787	. . . . . 6 ⊢ (((((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ 𝑚 ∈ 𝑃) ∧ 𝑆 = (((pInvG‘𝐺)‘𝑚)‘𝑅)) ∧ (𝑅 − 𝐴)(≤G‘𝐺)(𝑆 − 𝐶)) → 𝑚 ∈ 𝑃)
131	95	ad4antr 744	. . . . . . 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 28986	. . . . . 6 ⊢ (((((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ 𝑚 ∈ 𝑃) ∧ 𝑆 = (((pInvG‘𝐺)‘𝑚)‘𝑅)) ∧ (𝑅 − 𝐴)(≤G‘𝐺)(𝑆 − 𝐶)) → 𝐶𝑂𝐴)
133	23	ad4antr 744	. . . . . 6 ⊢ (((((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ 𝑚 ∈ 𝑃) ∧ 𝑆 = (((pInvG‘𝐺)‘𝑚)‘𝑅)) ∧ (𝑅 − 𝐴)(≤G‘𝐺)(𝑆 − 𝐶)) → 𝐷(⟂G‘𝐺)(𝐶𝐿𝑆))
134	11	ad4antr 744	. . . . . 6 ⊢ (((((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ 𝑚 ∈ 𝑃) ∧ 𝑆 = (((pInvG‘𝐺)‘𝑚)‘𝑅)) ∧ (𝑅 − 𝐴)(≤G‘𝐺)(𝑆 − 𝐶)) → 𝐷(⟂G‘𝐺)(𝐴𝐿𝑅))
135	112	necomd 3019	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑅 ≠ 𝑆) → 𝑆 ≠ 𝑅)
136	135	ad3antrrr 742	. . . . . 6 ⊢ (((((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ 𝑚 ∈ 𝑃) ∧ 𝑆 = (((pInvG‘𝐺)‘𝑚)‘𝑅)) ∧ (𝑅 − 𝐴)(≤G‘𝐺)(𝑆 − 𝐶)) → 𝑆 ≠ 𝑅)
137		simpr 489	. . . . . 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 744	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ 𝑚 ∈ 𝑃) ∧ 𝑆 = (((pInvG‘𝐺)‘𝑚)‘𝑅)) ∧ (𝑅 − 𝐴)(≤G‘𝐺)(𝑆 − 𝐶)) → 𝑅 ∈ 𝑃)
139		simplr 780	. . . . . . . 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 2775	. . . . . . 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 28904	. . . . . 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 744	. . . . . 6 ⊢ (((((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ 𝑚 ∈ 𝑃) ∧ 𝑆 = (((pInvG‘𝐺)‘𝑚)‘𝑅)) ∧ (𝑅 − 𝐴)(≤G‘𝐺)(𝑆 − 𝐶)) → 𝑉(𝐾‘𝑆)𝐶)
143	13	ad4antr 744	. . . . . 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 28992	. . . . 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 28986	. . . 4 ⊢ (((((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ 𝑚 ∈ 𝑃) ∧ 𝑆 = (((pInvG‘𝐺)‘𝑚)‘𝑅)) ∧ (𝑅 − 𝐴)(≤G‘𝐺)(𝑆 − 𝐶)) → 𝑈𝑂𝑉)
146		eqid 2769	. . . . . 6 ⊢ (≤G‘𝐺) = (≤G‘𝐺)
147	1, 2, 3, 146, 5, 24, 21, 12, 9	legtrid 28828	. . . . 5 ⊢ (𝜑 → ((𝑆 − 𝐶)(≤G‘𝐺)(𝑅 − 𝐴) ∨ (𝑅 − 𝐴)(≤G‘𝐺)(𝑆 − 𝐶)))
148	147	ad3antrrr 742	. . . 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 973	. . 3 ⊢ ((((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ 𝑚 ∈ 𝑃) ∧ 𝑆 = (((pInvG‘𝐺)‘𝑚)‘𝑅)) → 𝑈𝑂𝑉)
150	5	adantr 485	. . . 4 ⊢ ((𝜑 ∧ 𝑅 ≠ 𝑆) → 𝐺 ∈ TarskiG)
151	12	adantr 485	. . . 4 ⊢ ((𝜑 ∧ 𝑅 ≠ 𝑆) → 𝑅 ∈ 𝑃)
152	24	adantr 485	. . . 4 ⊢ ((𝜑 ∧ 𝑅 ≠ 𝑆) → 𝑆 ∈ 𝑃)
153	1, 2, 3, 91, 4, 6, 5, 9, 21, 95	opptgdim2 28987	. . . . 5 ⊢ (𝜑 → 𝐺Dim_TarskiG≥2)
154	153	adantr 485	. . . 4 ⊢ ((𝜑 ∧ 𝑅 ≠ 𝑆) → 𝐺Dim_TarskiG≥2)
155	1, 2, 3, 4, 150, 34, 151, 152, 154	midex 28979	. . 3 ⊢ ((𝜑 ∧ 𝑅 ≠ 𝑆) → ∃𝑚 ∈ 𝑃 𝑆 = (((pInvG‘𝐺)‘𝑚)‘𝑅))
156	149, 155	r19.29a 3179	. 2 ⊢ ((𝜑 ∧ 𝑅 ≠ 𝑆) → 𝑈𝑂𝑉)
157	100, 156	pm2.61dane 3051	1 ⊢ (𝜑 → 𝑈𝑂𝑉)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 400 ∨ wo 860 = wceq 1567 ∈ wcel 2149 ≠ wne 2964 ∃wrex 3095 ∖ cdif 3910 class class class wbr 5113 {copab 5177 ran crn 5665 ‘cfv 6539 (class class class)co 7413 2c2 12297 Basecbs 17271 distcds 17321 TarskiGcstrkg 28664 Dim_TarskiG≥cstrkgld 28668 Itvcitv 28670 LineGclng 28671 ≤Gcleg 28819 hlGchlg 28837 pInvGcmir 28893 ⟂Gcperpg 28936

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1822 ax-4 1836 ax-5 1937 ax-6 1994 ax-7 2035 ax-8 2151 ax-9 2159 ax-10 2182 ax-11 2198 ax-12 2219 ax-ext 2741 ax-rep 5242 ax-sep 5261 ax-nul 5273 ax-pow 5339 ax-pr 5407 ax-un 7735 ax-cnex 11158 ax-resscn 11159 ax-1cn 11160 ax-icn 11161 ax-addcl 11162 ax-addrcl 11163 ax-mulcl 11164 ax-mulrcl 11165 ax-mulcom 11166 ax-addass 11167 ax-mulass 11168 ax-distr 11169 ax-i2m1 11170 ax-1ne0 11171 ax-1rid 11172 ax-rnegex 11173 ax-rrecex 11174 ax-cnre 11175 ax-pre-lttri 11176 ax-pre-lttrn 11177 ax-pre-ltadd 11178 ax-pre-mulgt0 11179

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1102 df-3an 1103 df-tru 1570 df-fal 1580 df-ex 1807 df-nf 1811 df-sb 2098 df-mo 2573 df-eu 2603 df-clab 2748 df-cleq 2761 df-clel 2844 df-nfc 2918 df-ne 2965 df-nel 3071 df-ral 3086 df-rex 3096 df-rmo 3376 df-reu 3377 df-rab 3424 df-v 3465 df-sbc 3754 df-csb 3862 df-dif 3916 df-un 3918 df-in 3920 df-ss 3930 df-pss 3933 df-nul 4295 df-if 4493 df-pw 4569 df-sn 4595 df-pr 4597 df-tp 4599 df-op 4601 df-uni 4877 df-int 4917 df-iun 4962 df-br 5114 df-opab 5178 df-mpt 5197 df-tr 5223 df-id 5559 df-eprel 5564 df-po 5572 df-so 5573 df-fr 5617 df-we 5619 df-xp 5670 df-rel 5671 df-cnv 5672 df-co 5673 df-dm 5674 df-rn 5675 df-res 5676 df-ima 5677 df-pred 6305 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6495 df-fun 6541 df-fn 6542 df-f 6543 df-f1 6544 df-fo 6545 df-f1o 6546 df-fv 6547 df-riota 7370 df-ov 7416 df-oprab 7417 df-mpo 7418 df-om 7865 df-1st 7988 df-2nd 7989 df-frecs 8280 df-wrecs 8311 df-recs 8360 df-rdg 8399 df-1o 8455 df-oadd 8459 df-er 8696 df-map 8828 df-pm 8829 df-en 8946 df-dom 8947 df-sdom 8948 df-fin 8949 df-dju 9889 df-card 9927 df-pnf 11247 df-mnf 11248 df-xr 11249 df-ltxr 11250 df-le 11251 df-sub 11445 df-neg 11446 df-nn 12236 df-2 12305 df-3 12306 df-n0 12507 df-xnn0 12580 df-z 12594 df-uz 12865 df-fz 13538 df-fzo 13685 df-hash 14369 df-word 14553 df-concat 14610 df-s1 14636 df-s2 14887 df-s3 14888 df-trkgc 28685 df-trkgb 28686 df-trkgcb 28687 df-trkgld 28689 df-trkg 28690 df-cgrg 28748 df-leg 28820 df-hlg 28838 df-mir 28894 df-rag 28935 df-perpg 28937

This theorem is referenced by: opphl 28996

W3C validator