opphllem5 - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > opphllem5		Structured version Visualization version GIF version

Theorem opphllem5 27693

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 27491	. . . . . . . . 9 ⊢ (𝜑 → 𝑅 ∈ 𝑃)
13		opphllem5.1	. . . . . . . . . 10 ⊢ (𝜑 → 𝑈(𝐾‘𝑅)𝐴)
14	1, 3, 7, 10, 9, 12, 5, 13	hlne2 27548	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 ≠ 𝑅)
15	1, 3, 4, 5, 9, 12, 14	tglinecom 27577	. . . . . . . 8 ⊢ (𝜑 → (𝐴𝐿𝑅) = (𝑅𝐿𝐴))
16	11, 15	breqtrd 5131	. . . . . . 7 ⊢ (𝜑 → 𝐷(⟂G‘𝐺)(𝑅𝐿𝐴))
17	1, 3, 7, 10, 9, 12, 5, 13	hlcomd 27546	. . . . . . 7 ⊢ (𝜑 → 𝐴(𝐾‘𝑅)𝑈)
18	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 16, 17	hlperpnel 27667	. . . . . 6 ⊢ (𝜑 → ¬ 𝑈 ∈ 𝐷)
19	18	ad3antrrr 728	. . . . 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 27491	. . . . . . . . 9 ⊢ (𝜑 → 𝑆 ∈ 𝑃)
25		opphllem5.2	. . . . . . . . . 10 ⊢ (𝜑 → 𝑉(𝐾‘𝑆)𝐶)
26	1, 3, 7, 22, 21, 24, 5, 25	hlne2 27548	. . . . . . . . 9 ⊢ (𝜑 → 𝐶 ≠ 𝑆)
27	1, 3, 4, 5, 21, 24, 26	tglinecom 27577	. . . . . . . 8 ⊢ (𝜑 → (𝐶𝐿𝑆) = (𝑆𝐿𝐶))
28	23, 27	breqtrd 5131	. . . . . . 7 ⊢ (𝜑 → 𝐷(⟂G‘𝐺)(𝑆𝐿𝐶))
29	1, 3, 7, 22, 21, 24, 5, 25	hlcomd 27546	. . . . . . 7 ⊢ (𝜑 → 𝐶(𝐾‘𝑆)𝑉)
30	1, 2, 3, 4, 5, 6, 7, 20, 21, 22, 28, 29	hlperpnel 27667	. . . . . 6 ⊢ (𝜑 → ¬ 𝑉 ∈ 𝐷)
31	30	ad3antrrr 728	. . . . 5 ⊢ ((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) → ¬ 𝑉 ∈ 𝐷)
32		simplr 767	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) → 𝑡 ∈ 𝐷)
33		simpr 485	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) ∧ 𝑅 = 𝑡) → 𝑅 = 𝑡)
34		eqid 2736	. . . . . . . . 9 ⊢ (pInvG‘𝐺) = (pInvG‘𝐺)
35	5	ad4antr 730	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) ∧ 𝑅 ≠ 𝑡) → 𝐺 ∈ TarskiG)
36	21	ad4antr 730	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) ∧ 𝑅 ≠ 𝑡) → 𝐶 ∈ 𝑃)
37	12	ad4antr 730	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) ∧ 𝑅 ≠ 𝑡) → 𝑅 ∈ 𝑃)
38	5	ad3antrrr 728	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) → 𝐺 ∈ TarskiG)
39	6	ad3antrrr 728	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) → 𝐷 ∈ ran 𝐿)
40	1, 4, 3, 38, 39, 32	tglnpt 27491	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) → 𝑡 ∈ 𝑃)
41	40	adantr 481	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) ∧ 𝑅 ≠ 𝑡) → 𝑡 ∈ 𝑃)
42	9	ad4antr 730	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) ∧ 𝑅 ≠ 𝑡) → 𝐴 ∈ 𝑃)
43	24	ad4antr 730	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) ∧ 𝑅 ≠ 𝑡) → 𝑆 ∈ 𝑃)
44		simpllr 774	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) → 𝑅 = 𝑆)
45	1, 3, 4, 5, 21, 24, 26	tglinerflx2 27576	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑆 ∈ (𝐶𝐿𝑆))
46	45	ad3antrrr 728	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) → 𝑆 ∈ (𝐶𝐿𝑆))
47	44, 46	eqeltrd 2838	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) → 𝑅 ∈ (𝐶𝐿𝑆))
48	47	adantr 481	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) ∧ 𝑅 ≠ 𝑡) → 𝑅 ∈ (𝐶𝐿𝑆))
49	4, 5, 23	perpln2 27653	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐶𝐿𝑆) ∈ ran 𝐿)
50	1, 2, 3, 4, 5, 6, 49, 23	perpcom 27655	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐶𝐿𝑆)(⟂G‘𝐺)𝐷)
51	50	ad4antr 730	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) ∧ 𝑅 ≠ 𝑡) → (𝐶𝐿𝑆)(⟂G‘𝐺)𝐷)
52		simpr 485	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) ∧ 𝑅 ≠ 𝑡) → 𝑅 ≠ 𝑡)
53	6	ad4antr 730	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) ∧ 𝑅 ≠ 𝑡) → 𝐷 ∈ ran 𝐿)
54	8	ad4antr 730	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) ∧ 𝑅 ≠ 𝑡) → 𝑅 ∈ 𝐷)
55		simpllr 774	. . . . . . . . . . . 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 27578	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) ∧ 𝑅 ≠ 𝑡) → 𝐷 = (𝑅𝐿𝑡))
57	51, 56	breqtrd 5131	. . . . . . . . . 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 27668	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) ∧ 𝑅 ≠ 𝑡) → ⟨“𝐶𝑅𝑡”⟩ ∈ (∟G‘𝐺))
59	1, 3, 4, 5, 9, 12, 14	tglinerflx2 27576	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑅 ∈ (𝐴𝐿𝑅))
60	59	ad4antr 730	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) ∧ 𝑅 ≠ 𝑡) → 𝑅 ∈ (𝐴𝐿𝑅))
61	4, 5, 11	perpln2 27653	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐴𝐿𝑅) ∈ ran 𝐿)
62	1, 2, 3, 4, 5, 6, 61, 11	perpcom 27655	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐴𝐿𝑅)(⟂G‘𝐺)𝐷)
63	62	ad4antr 730	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) ∧ 𝑅 ≠ 𝑡) → (𝐴𝐿𝑅)(⟂G‘𝐺)𝐷)
64	63, 56	breqtrd 5131	. . . . . . . . . 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 27668	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) ∧ 𝑅 ≠ 𝑡) → ⟨“𝐴𝑅𝑡”⟩ ∈ (∟G‘𝐺))
66		simplr 767	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) ∧ 𝑅 ≠ 𝑡) → 𝑡 ∈ (𝐴𝐼𝐶))
67	1, 2, 3, 35, 42, 41, 36, 66	tgbtwncom 27430	. . . . . . . . 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 27645	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) ∧ 𝑅 ≠ 𝑡) → 𝑅 = 𝑡)
69	33, 68	pm2.61dane 3032	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) → 𝑅 = 𝑡)
70	9	ad3antrrr 728	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) → 𝐴 ∈ 𝑃)
71	10	ad3antrrr 728	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) → 𝑈 ∈ 𝑃)
72	22	ad3antrrr 728	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) → 𝑉 ∈ 𝑃)
73	12	ad3antrrr 728	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) → 𝑅 ∈ 𝑃)
74	17	ad3antrrr 728	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) → 𝐴(𝐾‘𝑅)𝑈)
75	21	ad3antrrr 728	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) → 𝐶 ∈ 𝑃)
76	25	ad3antrrr 728	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) → 𝑉(𝐾‘𝑆)𝐶)
77	44	fveq2d 6846	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) → (𝐾‘𝑅) = (𝐾‘𝑆))
78	77	breqd 5116	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) → (𝑉(𝐾‘𝑅)𝐶 ↔ 𝑉(𝐾‘𝑆)𝐶))
79	76, 78	mpbird 256	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) → 𝑉(𝐾‘𝑅)𝐶)
80	1, 3, 7, 72, 75, 73, 38, 79	hlcomd 27546	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) → 𝐶(𝐾‘𝑅)𝑉)
81		simpr 485	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) → 𝑡 ∈ (𝐴𝐼𝐶))
82	69, 81	eqeltrd 2838	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) → 𝑅 ∈ (𝐴𝐼𝐶))
83	1, 2, 3, 38, 70, 73, 75, 82	tgbtwncom 27430	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) → 𝑅 ∈ (𝐶𝐼𝐴))
84	1, 3, 7, 75, 72, 70, 38, 73, 80, 83	btwnhl 27556	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) → 𝑅 ∈ (𝑉𝐼𝐴))
85	1, 2, 3, 38, 72, 73, 70, 84	tgbtwncom 27430	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) → 𝑅 ∈ (𝐴𝐼𝑉))
86	1, 3, 7, 70, 71, 72, 38, 73, 74, 85	btwnhl 27556	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) → 𝑅 ∈ (𝑈𝐼𝑉))
87	69, 86	eqeltrrd 2839	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) → 𝑡 ∈ (𝑈𝐼𝑉))
88		rspe 3232	. . . . . 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 515	. . . 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 27681	. . . . 5 ⊢ (𝜑 → (𝑈𝑂𝑉 ↔ ((¬ 𝑈 ∈ 𝐷 ∧ ¬ 𝑉 ∈ 𝐷) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝑈𝐼𝑉))))
93	92	ad3antrrr 728	. . . 4 ⊢ ((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) → (𝑈𝑂𝑉 ↔ ((¬ 𝑈 ∈ 𝐷 ∧ ¬ 𝑉 ∈ 𝐷) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝑈𝐼𝑉))))
94	90, 93	mpbird 256	. . 3 ⊢ ((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) → 𝑈𝑂𝑉)
95		opphllem5.o	. . . . . 6 ⊢ (𝜑 → 𝐴𝑂𝐶)
96	1, 2, 3, 91, 9, 21	islnopp 27681	. . . . . 6 ⊢ (𝜑 → (𝐴𝑂𝐶 ↔ ((¬ 𝐴 ∈ 𝐷 ∧ ¬ 𝐶 ∈ 𝐷) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝐴𝐼𝐶))))
97	95, 96	mpbid 231	. . . . 5 ⊢ (𝜑 → ((¬ 𝐴 ∈ 𝐷 ∧ ¬ 𝐶 ∈ 𝐷) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝐴𝐼𝐶)))
98	97	simprd 496	. . . 4 ⊢ (𝜑 → ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝐴𝐼𝐶))
99	98	adantr 481	. . 3 ⊢ ((𝜑 ∧ 𝑅 = 𝑆) → ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝐴𝐼𝐶))
100	94, 99	r19.29a 3159	. 2 ⊢ ((𝜑 ∧ 𝑅 = 𝑆) → 𝑈𝑂𝑉)
101	6	ad4antr 730	. . . . 5 ⊢ (((((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ 𝑚 ∈ 𝑃) ∧ 𝑆 = (((pInvG‘𝐺)‘𝑚)‘𝑅)) ∧ (𝑆 − 𝐶)(≤G‘𝐺)(𝑅 − 𝐴)) → 𝐷 ∈ ran 𝐿)
102	5	ad4antr 730	. . . . 5 ⊢ (((((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ 𝑚 ∈ 𝑃) ∧ 𝑆 = (((pInvG‘𝐺)‘𝑚)‘𝑅)) ∧ (𝑆 − 𝐶)(≤G‘𝐺)(𝑅 − 𝐴)) → 𝐺 ∈ TarskiG)
103		eqid 2736	. . . . 5 ⊢ ((pInvG‘𝐺)‘𝑚) = ((pInvG‘𝐺)‘𝑚)
104	9	ad4antr 730	. . . . 5 ⊢ (((((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ 𝑚 ∈ 𝑃) ∧ 𝑆 = (((pInvG‘𝐺)‘𝑚)‘𝑅)) ∧ (𝑆 − 𝐶)(≤G‘𝐺)(𝑅 − 𝐴)) → 𝐴 ∈ 𝑃)
105	21	ad4antr 730	. . . . 5 ⊢ (((((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ 𝑚 ∈ 𝑃) ∧ 𝑆 = (((pInvG‘𝐺)‘𝑚)‘𝑅)) ∧ (𝑆 − 𝐶)(≤G‘𝐺)(𝑅 − 𝐴)) → 𝐶 ∈ 𝑃)
106	8	ad4antr 730	. . . . 5 ⊢ (((((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ 𝑚 ∈ 𝑃) ∧ 𝑆 = (((pInvG‘𝐺)‘𝑚)‘𝑅)) ∧ (𝑆 − 𝐶)(≤G‘𝐺)(𝑅 − 𝐴)) → 𝑅 ∈ 𝐷)
107	20	ad4antr 730	. . . . 5 ⊢ (((((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ 𝑚 ∈ 𝑃) ∧ 𝑆 = (((pInvG‘𝐺)‘𝑚)‘𝑅)) ∧ (𝑆 − 𝐶)(≤G‘𝐺)(𝑅 − 𝐴)) → 𝑆 ∈ 𝐷)
108		simpllr 774	. . . . 5 ⊢ (((((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ 𝑚 ∈ 𝑃) ∧ 𝑆 = (((pInvG‘𝐺)‘𝑚)‘𝑅)) ∧ (𝑆 − 𝐶)(≤G‘𝐺)(𝑅 − 𝐴)) → 𝑚 ∈ 𝑃)
109	95	ad4antr 730	. . . . 5 ⊢ (((((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ 𝑚 ∈ 𝑃) ∧ 𝑆 = (((pInvG‘𝐺)‘𝑚)‘𝑅)) ∧ (𝑆 − 𝐶)(≤G‘𝐺)(𝑅 − 𝐴)) → 𝐴𝑂𝐶)
110	11	ad4antr 730	. . . . 5 ⊢ (((((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ 𝑚 ∈ 𝑃) ∧ 𝑆 = (((pInvG‘𝐺)‘𝑚)‘𝑅)) ∧ (𝑆 − 𝐶)(≤G‘𝐺)(𝑅 − 𝐴)) → 𝐷(⟂G‘𝐺)(𝐴𝐿𝑅))
111	23	ad4antr 730	. . . . 5 ⊢ (((((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ 𝑚 ∈ 𝑃) ∧ 𝑆 = (((pInvG‘𝐺)‘𝑚)‘𝑅)) ∧ (𝑆 − 𝐶)(≤G‘𝐺)(𝑅 − 𝐴)) → 𝐷(⟂G‘𝐺)(𝐶𝐿𝑆))
112		simpr 485	. . . . . 6 ⊢ ((𝜑 ∧ 𝑅 ≠ 𝑆) → 𝑅 ≠ 𝑆)
113	112	ad3antrrr 728	. . . . 5 ⊢ (((((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ 𝑚 ∈ 𝑃) ∧ 𝑆 = (((pInvG‘𝐺)‘𝑚)‘𝑅)) ∧ (𝑆 − 𝐶)(≤G‘𝐺)(𝑅 − 𝐴)) → 𝑅 ≠ 𝑆)
114		simpr 485	. . . . 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 730	. . . . 5 ⊢ (((((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ 𝑚 ∈ 𝑃) ∧ 𝑆 = (((pInvG‘𝐺)‘𝑚)‘𝑅)) ∧ (𝑆 − 𝐶)(≤G‘𝐺)(𝑅 − 𝐴)) → 𝑈 ∈ 𝑃)
116		simplr 767	. . . . . 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 2742	. . . . 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 730	. . . . 5 ⊢ (((((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ 𝑚 ∈ 𝑃) ∧ 𝑆 = (((pInvG‘𝐺)‘𝑚)‘𝑅)) ∧ (𝑆 − 𝐶)(≤G‘𝐺)(𝑅 − 𝐴)) → 𝑉 ∈ 𝑃)
119	13	ad4antr 730	. . . . 5 ⊢ (((((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ 𝑚 ∈ 𝑃) ∧ 𝑆 = (((pInvG‘𝐺)‘𝑚)‘𝑅)) ∧ (𝑆 − 𝐶)(≤G‘𝐺)(𝑅 − 𝐴)) → 𝑈(𝐾‘𝑅)𝐴)
120	25	ad4antr 730	. . . . 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 27692	. . . 4 ⊢ (((((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ 𝑚 ∈ 𝑃) ∧ 𝑆 = (((pInvG‘𝐺)‘𝑚)‘𝑅)) ∧ (𝑆 − 𝐶)(≤G‘𝐺)(𝑅 − 𝐴)) → 𝑈𝑂𝑉)
122	6	ad4antr 730	. . . . 5 ⊢ (((((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ 𝑚 ∈ 𝑃) ∧ 𝑆 = (((pInvG‘𝐺)‘𝑚)‘𝑅)) ∧ (𝑅 − 𝐴)(≤G‘𝐺)(𝑆 − 𝐶)) → 𝐷 ∈ ran 𝐿)
123	5	ad4antr 730	. . . . 5 ⊢ (((((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ 𝑚 ∈ 𝑃) ∧ 𝑆 = (((pInvG‘𝐺)‘𝑚)‘𝑅)) ∧ (𝑅 − 𝐴)(≤G‘𝐺)(𝑆 − 𝐶)) → 𝐺 ∈ TarskiG)
124	22	ad4antr 730	. . . . 5 ⊢ (((((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ 𝑚 ∈ 𝑃) ∧ 𝑆 = (((pInvG‘𝐺)‘𝑚)‘𝑅)) ∧ (𝑅 − 𝐴)(≤G‘𝐺)(𝑆 − 𝐶)) → 𝑉 ∈ 𝑃)
125	10	ad4antr 730	. . . . 5 ⊢ (((((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ 𝑚 ∈ 𝑃) ∧ 𝑆 = (((pInvG‘𝐺)‘𝑚)‘𝑅)) ∧ (𝑅 − 𝐴)(≤G‘𝐺)(𝑆 − 𝐶)) → 𝑈 ∈ 𝑃)
126	21	ad4antr 730	. . . . . 6 ⊢ (((((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ 𝑚 ∈ 𝑃) ∧ 𝑆 = (((pInvG‘𝐺)‘𝑚)‘𝑅)) ∧ (𝑅 − 𝐴)(≤G‘𝐺)(𝑆 − 𝐶)) → 𝐶 ∈ 𝑃)
127	9	ad4antr 730	. . . . . 6 ⊢ (((((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ 𝑚 ∈ 𝑃) ∧ 𝑆 = (((pInvG‘𝐺)‘𝑚)‘𝑅)) ∧ (𝑅 − 𝐴)(≤G‘𝐺)(𝑆 − 𝐶)) → 𝐴 ∈ 𝑃)
128	20	ad4antr 730	. . . . . 6 ⊢ (((((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ 𝑚 ∈ 𝑃) ∧ 𝑆 = (((pInvG‘𝐺)‘𝑚)‘𝑅)) ∧ (𝑅 − 𝐴)(≤G‘𝐺)(𝑆 − 𝐶)) → 𝑆 ∈ 𝐷)
129	8	ad4antr 730	. . . . . 6 ⊢ (((((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ 𝑚 ∈ 𝑃) ∧ 𝑆 = (((pInvG‘𝐺)‘𝑚)‘𝑅)) ∧ (𝑅 − 𝐴)(≤G‘𝐺)(𝑆 − 𝐶)) → 𝑅 ∈ 𝐷)
130		simpllr 774	. . . . . 6 ⊢ (((((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ 𝑚 ∈ 𝑃) ∧ 𝑆 = (((pInvG‘𝐺)‘𝑚)‘𝑅)) ∧ (𝑅 − 𝐴)(≤G‘𝐺)(𝑆 − 𝐶)) → 𝑚 ∈ 𝑃)
131	95	ad4antr 730	. . . . . . 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 27686	. . . . . 6 ⊢ (((((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ 𝑚 ∈ 𝑃) ∧ 𝑆 = (((pInvG‘𝐺)‘𝑚)‘𝑅)) ∧ (𝑅 − 𝐴)(≤G‘𝐺)(𝑆 − 𝐶)) → 𝐶𝑂𝐴)
133	23	ad4antr 730	. . . . . 6 ⊢ (((((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ 𝑚 ∈ 𝑃) ∧ 𝑆 = (((pInvG‘𝐺)‘𝑚)‘𝑅)) ∧ (𝑅 − 𝐴)(≤G‘𝐺)(𝑆 − 𝐶)) → 𝐷(⟂G‘𝐺)(𝐶𝐿𝑆))
134	11	ad4antr 730	. . . . . 6 ⊢ (((((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ 𝑚 ∈ 𝑃) ∧ 𝑆 = (((pInvG‘𝐺)‘𝑚)‘𝑅)) ∧ (𝑅 − 𝐴)(≤G‘𝐺)(𝑆 − 𝐶)) → 𝐷(⟂G‘𝐺)(𝐴𝐿𝑅))
135	112	necomd 2999	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑅 ≠ 𝑆) → 𝑆 ≠ 𝑅)
136	135	ad3antrrr 728	. . . . . 6 ⊢ (((((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ 𝑚 ∈ 𝑃) ∧ 𝑆 = (((pInvG‘𝐺)‘𝑚)‘𝑅)) ∧ (𝑅 − 𝐴)(≤G‘𝐺)(𝑆 − 𝐶)) → 𝑆 ≠ 𝑅)
137		simpr 485	. . . . . 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 730	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ 𝑚 ∈ 𝑃) ∧ 𝑆 = (((pInvG‘𝐺)‘𝑚)‘𝑅)) ∧ (𝑅 − 𝐴)(≤G‘𝐺)(𝑆 − 𝐶)) → 𝑅 ∈ 𝑃)
139		simplr 767	. . . . . . . 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 2742	. . . . . . 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 27605	. . . . . 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 730	. . . . . 6 ⊢ (((((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ 𝑚 ∈ 𝑃) ∧ 𝑆 = (((pInvG‘𝐺)‘𝑚)‘𝑅)) ∧ (𝑅 − 𝐴)(≤G‘𝐺)(𝑆 − 𝐶)) → 𝑉(𝐾‘𝑆)𝐶)
143	13	ad4antr 730	. . . . . 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 27692	. . . . 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 27686	. . . 4 ⊢ (((((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ 𝑚 ∈ 𝑃) ∧ 𝑆 = (((pInvG‘𝐺)‘𝑚)‘𝑅)) ∧ (𝑅 − 𝐴)(≤G‘𝐺)(𝑆 − 𝐶)) → 𝑈𝑂𝑉)
146		eqid 2736	. . . . . 6 ⊢ (≤G‘𝐺) = (≤G‘𝐺)
147	1, 2, 3, 146, 5, 24, 21, 12, 9	legtrid 27533	. . . . 5 ⊢ (𝜑 → ((𝑆 − 𝐶)(≤G‘𝐺)(𝑅 − 𝐴) ∨ (𝑅 − 𝐴)(≤G‘𝐺)(𝑆 − 𝐶)))
148	147	ad3antrrr 728	. . . 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 957	. . 3 ⊢ ((((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ 𝑚 ∈ 𝑃) ∧ 𝑆 = (((pInvG‘𝐺)‘𝑚)‘𝑅)) → 𝑈𝑂𝑉)
150	5	adantr 481	. . . 4 ⊢ ((𝜑 ∧ 𝑅 ≠ 𝑆) → 𝐺 ∈ TarskiG)
151	12	adantr 481	. . . 4 ⊢ ((𝜑 ∧ 𝑅 ≠ 𝑆) → 𝑅 ∈ 𝑃)
152	24	adantr 481	. . . 4 ⊢ ((𝜑 ∧ 𝑅 ≠ 𝑆) → 𝑆 ∈ 𝑃)
153	1, 2, 3, 91, 4, 6, 5, 9, 21, 95	opptgdim2 27687	. . . . 5 ⊢ (𝜑 → 𝐺Dim_TarskiG≥2)
154	153	adantr 481	. . . 4 ⊢ ((𝜑 ∧ 𝑅 ≠ 𝑆) → 𝐺Dim_TarskiG≥2)
155	1, 2, 3, 4, 150, 34, 151, 152, 154	midex 27679	. . 3 ⊢ ((𝜑 ∧ 𝑅 ≠ 𝑆) → ∃𝑚 ∈ 𝑃 𝑆 = (((pInvG‘𝐺)‘𝑚)‘𝑅))
156	149, 155	r19.29a 3159	. 2 ⊢ ((𝜑 ∧ 𝑅 ≠ 𝑆) → 𝑈𝑂𝑉)
157	100, 156	pm2.61dane 3032	1 ⊢ (𝜑 → 𝑈𝑂𝑉)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 396 ∨ wo 845 = wceq 1541 ∈ wcel 2106 ≠ wne 2943 ∃wrex 3073 ∖ cdif 3907 class class class wbr 5105 {copab 5167 ran crn 5634 ‘cfv 6496 (class class class)co 7357 2c2 12208 Basecbs 17083 distcds 17142 TarskiGcstrkg 27369 Dim_TarskiG≥cstrkgld 27373 Itvcitv 27375 LineGclng 27376 ≤Gcleg 27524 hlGchlg 27542 pInvGcmir 27594 ⟂Gcperpg 27637

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-rep 5242 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7672 ax-cnex 11107 ax-resscn 11108 ax-1cn 11109 ax-icn 11110 ax-addcl 11111 ax-addrcl 11112 ax-mulcl 11113 ax-mulrcl 11114 ax-mulcom 11115 ax-addass 11116 ax-mulass 11117 ax-distr 11118 ax-i2m1 11119 ax-1ne0 11120 ax-1rid 11121 ax-rnegex 11122 ax-rrecex 11123 ax-cnre 11124 ax-pre-lttri 11125 ax-pre-lttrn 11126 ax-pre-ltadd 11127 ax-pre-mulgt0 11128

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3065 df-rex 3074 df-rmo 3353 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-pss 3929 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-tp 4591 df-op 4593 df-uni 4866 df-int 4908 df-iun 4956 df-br 5106 df-opab 5168 df-mpt 5189 df-tr 5223 df-id 5531 df-eprel 5537 df-po 5545 df-so 5546 df-fr 5588 df-we 5590 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-pred 6253 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-riota 7313 df-ov 7360 df-oprab 7361 df-mpo 7362 df-om 7803 df-1st 7921 df-2nd 7922 df-frecs 8212 df-wrecs 8243 df-recs 8317 df-rdg 8356 df-1o 8412 df-oadd 8416 df-er 8648 df-map 8767 df-pm 8768 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-dju 9837 df-card 9875 df-pnf 11191 df-mnf 11192 df-xr 11193 df-ltxr 11194 df-le 11195 df-sub 11387 df-neg 11388 df-nn 12154 df-2 12216 df-3 12217 df-n0 12414 df-xnn0 12486 df-z 12500 df-uz 12764 df-fz 13425 df-fzo 13568 df-hash 14231 df-word 14403 df-concat 14459 df-s1 14484 df-s2 14737 df-s3 14738 df-trkgc 27390 df-trkgb 27391 df-trkgcb 27392 df-trkgld 27394 df-trkg 27395 df-cgrg 27453 df-leg 27525 df-hlg 27543 df-mir 27595 df-rag 27636 df-perpg 27638

This theorem is referenced by: opphl 27696

W3C validator