opphllem6 - Metamath Proof Explorer

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Theorem opphllem6 26538

Description: First part of Lemma 9.4 of [Schwabhauser] p. 68. (Contributed by Thierry Arnoux, 3-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)
opphllem6.v	⊢ (𝜑 → (𝑁‘𝑅) = 𝑆)

**Assertion**
Ref	Expression
opphllem6	⊢ (𝜑 → (𝑈(𝐾‘𝑅)𝐴 ↔ (𝑁‘𝑈)(𝐾‘𝑆)𝐶))

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 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)

**Proof of Theorem opphllem6**
Step	Hyp	Ref	Expression
1		hpg.p	. . . 4 ⊢ 𝑃 = (Base‘𝐺)
2		hpg.d	. . . 4 ⊢ − = (dist‘𝐺)
3		hpg.i	. . . 4 ⊢ 𝐼 = (Itv‘𝐺)
4		opphl.l	. . . 4 ⊢ 𝐿 = (LineG‘𝐺)
5		eqid 2821	. . . 4 ⊢ (pInvG‘𝐺) = (pInvG‘𝐺)
6		opphl.g	. . . . 5 ⊢ (𝜑 → 𝐺 ∈ TarskiG)
7	6	adantr 483	. . . 4 ⊢ ((𝜑 ∧ 𝑅 = 𝑆) → 𝐺 ∈ TarskiG)
8		opphllem5.n	. . . 4 ⊢ 𝑁 = ((pInvG‘𝐺)‘𝑀)
9		opphl.k	. . . 4 ⊢ 𝐾 = (hlG‘𝐺)
10		opphllem5.m	. . . . 5 ⊢ (𝜑 → 𝑀 ∈ 𝑃)
11	10	adantr 483	. . . 4 ⊢ ((𝜑 ∧ 𝑅 = 𝑆) → 𝑀 ∈ 𝑃)
12		opphllem5.a	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ 𝑃)
13	12	adantr 483	. . . 4 ⊢ ((𝜑 ∧ 𝑅 = 𝑆) → 𝐴 ∈ 𝑃)
14		opphllem5.c	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ 𝑃)
15	14	adantr 483	. . . 4 ⊢ ((𝜑 ∧ 𝑅 = 𝑆) → 𝐶 ∈ 𝑃)
16		opphllem5.u	. . . . 5 ⊢ (𝜑 → 𝑈 ∈ 𝑃)
17	16	adantr 483	. . . 4 ⊢ ((𝜑 ∧ 𝑅 = 𝑆) → 𝑈 ∈ 𝑃)
18		opphl.d	. . . . . . . 8 ⊢ (𝜑 → 𝐷 ∈ ran 𝐿)
19		opphllem5.r	. . . . . . . 8 ⊢ (𝜑 → 𝑅 ∈ 𝐷)
20	1, 4, 3, 6, 18, 19	tglnpt 26335	. . . . . . 7 ⊢ (𝜑 → 𝑅 ∈ 𝑃)
21		opphllem5.p	. . . . . . . 8 ⊢ (𝜑 → 𝐷(⟂G‘𝐺)(𝐴𝐿𝑅))
22	4, 6, 21	perpln2 26497	. . . . . . 7 ⊢ (𝜑 → (𝐴𝐿𝑅) ∈ ran 𝐿)
23	1, 3, 4, 6, 12, 20, 22	tglnne 26414	. . . . . 6 ⊢ (𝜑 → 𝐴 ≠ 𝑅)
24	23	adantr 483	. . . . 5 ⊢ ((𝜑 ∧ 𝑅 = 𝑆) → 𝐴 ≠ 𝑅)
25		opphllem6.v	. . . . . . . 8 ⊢ (𝜑 → (𝑁‘𝑅) = 𝑆)
26	25	adantr 483	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑅 = 𝑆) → (𝑁‘𝑅) = 𝑆)
27		simpr 487	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑅 = 𝑆) → 𝑅 = 𝑆)
28	26, 27	eqtr4d 2859	. . . . . 6 ⊢ ((𝜑 ∧ 𝑅 = 𝑆) → (𝑁‘𝑅) = 𝑅)
29	1, 2, 3, 4, 5, 6, 10, 8, 20	mirinv 26452	. . . . . . 7 ⊢ (𝜑 → ((𝑁‘𝑅) = 𝑅 ↔ 𝑀 = 𝑅))
30	29	adantr 483	. . . . . 6 ⊢ ((𝜑 ∧ 𝑅 = 𝑆) → ((𝑁‘𝑅) = 𝑅 ↔ 𝑀 = 𝑅))
31	28, 30	mpbid 234	. . . . 5 ⊢ ((𝜑 ∧ 𝑅 = 𝑆) → 𝑀 = 𝑅)
32	24, 31	neeqtrrd 3090	. . . 4 ⊢ ((𝜑 ∧ 𝑅 = 𝑆) → 𝐴 ≠ 𝑀)
33		opphllem5.s	. . . . . . . 8 ⊢ (𝜑 → 𝑆 ∈ 𝐷)
34	1, 4, 3, 6, 18, 33	tglnpt 26335	. . . . . . 7 ⊢ (𝜑 → 𝑆 ∈ 𝑃)
35		opphllem5.q	. . . . . . . 8 ⊢ (𝜑 → 𝐷(⟂G‘𝐺)(𝐶𝐿𝑆))
36	4, 6, 35	perpln2 26497	. . . . . . 7 ⊢ (𝜑 → (𝐶𝐿𝑆) ∈ ran 𝐿)
37	1, 3, 4, 6, 14, 34, 36	tglnne 26414	. . . . . 6 ⊢ (𝜑 → 𝐶 ≠ 𝑆)
38	37	adantr 483	. . . . 5 ⊢ ((𝜑 ∧ 𝑅 = 𝑆) → 𝐶 ≠ 𝑆)
39	31, 27	eqtrd 2856	. . . . 5 ⊢ ((𝜑 ∧ 𝑅 = 𝑆) → 𝑀 = 𝑆)
40	38, 39	neeqtrrd 3090	. . . 4 ⊢ ((𝜑 ∧ 𝑅 = 𝑆) → 𝐶 ≠ 𝑀)
41		simpr 487	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) ∧ 𝑅 = 𝑡) → 𝑅 = 𝑡)
42	6	ad4antr 730	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) ∧ 𝑅 ≠ 𝑡) → 𝐺 ∈ TarskiG)
43	14	ad4antr 730	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) ∧ 𝑅 ≠ 𝑡) → 𝐶 ∈ 𝑃)
44	20	ad4antr 730	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) ∧ 𝑅 ≠ 𝑡) → 𝑅 ∈ 𝑃)
45	6	ad3antrrr 728	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) → 𝐺 ∈ TarskiG)
46	18	ad3antrrr 728	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) → 𝐷 ∈ ran 𝐿)
47		simplr 767	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) → 𝑡 ∈ 𝐷)
48	1, 4, 3, 45, 46, 47	tglnpt 26335	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) → 𝑡 ∈ 𝑃)
49	48	adantr 483	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) ∧ 𝑅 ≠ 𝑡) → 𝑡 ∈ 𝑃)
50	12	ad4antr 730	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) ∧ 𝑅 ≠ 𝑡) → 𝐴 ∈ 𝑃)
51	34	ad4antr 730	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) ∧ 𝑅 ≠ 𝑡) → 𝑆 ∈ 𝑃)
52		simpllr 774	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) → 𝑅 = 𝑆)
53	1, 3, 4, 6, 14, 34, 37	tglinerflx2 26420	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑆 ∈ (𝐶𝐿𝑆))
54	53	ad3antrrr 728	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) → 𝑆 ∈ (𝐶𝐿𝑆))
55	52, 54	eqeltrd 2913	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) → 𝑅 ∈ (𝐶𝐿𝑆))
56	55	adantr 483	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) ∧ 𝑅 ≠ 𝑡) → 𝑅 ∈ (𝐶𝐿𝑆))
57	1, 2, 3, 4, 6, 18, 36, 35	perpcom 26499	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐶𝐿𝑆)(⟂G‘𝐺)𝐷)
58	57	ad4antr 730	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) ∧ 𝑅 ≠ 𝑡) → (𝐶𝐿𝑆)(⟂G‘𝐺)𝐷)
59		simpr 487	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) ∧ 𝑅 ≠ 𝑡) → 𝑅 ≠ 𝑡)
60	18	ad4antr 730	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) ∧ 𝑅 ≠ 𝑡) → 𝐷 ∈ ran 𝐿)
61	19	ad4antr 730	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) ∧ 𝑅 ≠ 𝑡) → 𝑅 ∈ 𝐷)
62		simpllr 774	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) ∧ 𝑅 ≠ 𝑡) → 𝑡 ∈ 𝐷)
63	1, 3, 4, 42, 44, 49, 59, 59, 60, 61, 62	tglinethru 26422	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) ∧ 𝑅 ≠ 𝑡) → 𝐷 = (𝑅𝐿𝑡))
64	58, 63	breqtrd 5092	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) ∧ 𝑅 ≠ 𝑡) → (𝐶𝐿𝑆)(⟂G‘𝐺)(𝑅𝐿𝑡))
65	1, 2, 3, 4, 42, 43, 51, 56, 49, 64	perprag 26512	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) ∧ 𝑅 ≠ 𝑡) → ⟨“𝐶𝑅𝑡”⟩ ∈ (∟G‘𝐺))
66	1, 3, 4, 6, 12, 20, 23	tglinerflx2 26420	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑅 ∈ (𝐴𝐿𝑅))
67	66	ad4antr 730	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) ∧ 𝑅 ≠ 𝑡) → 𝑅 ∈ (𝐴𝐿𝑅))
68	1, 2, 3, 4, 6, 18, 22, 21	perpcom 26499	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐴𝐿𝑅)(⟂G‘𝐺)𝐷)
69	68	ad4antr 730	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) ∧ 𝑅 ≠ 𝑡) → (𝐴𝐿𝑅)(⟂G‘𝐺)𝐷)
70	69, 63	breqtrd 5092	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) ∧ 𝑅 ≠ 𝑡) → (𝐴𝐿𝑅)(⟂G‘𝐺)(𝑅𝐿𝑡))
71	1, 2, 3, 4, 42, 50, 44, 67, 49, 70	perprag 26512	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) ∧ 𝑅 ≠ 𝑡) → ⟨“𝐴𝑅𝑡”⟩ ∈ (∟G‘𝐺))
72		simplr 767	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) ∧ 𝑅 ≠ 𝑡) → 𝑡 ∈ (𝐴𝐼𝐶))
73	1, 2, 3, 42, 50, 49, 43, 72	tgbtwncom 26274	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) ∧ 𝑅 ≠ 𝑡) → 𝑡 ∈ (𝐶𝐼𝐴))
74	1, 2, 3, 4, 5, 42, 43, 44, 49, 50, 65, 71, 73	ragflat2 26489	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) ∧ 𝑅 ≠ 𝑡) → 𝑅 = 𝑡)
75	41, 74	pm2.61dane 3104	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) → 𝑅 = 𝑡)
76		simpr 487	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) → 𝑡 ∈ (𝐴𝐼𝐶))
77	75, 76	eqeltrd 2913	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) → 𝑅 ∈ (𝐴𝐼𝐶))
78		opphllem5.o	. . . . . . . . 9 ⊢ (𝜑 → 𝐴𝑂𝐶)
79		hpg.o	. . . . . . . . . 10 ⊢ 𝑂 = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝑃 ∖ 𝐷) ∧ 𝑏 ∈ (𝑃 ∖ 𝐷)) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝑎𝐼𝑏))}
80	1, 2, 3, 79, 12, 14	islnopp 26525	. . . . . . . . 9 ⊢ (𝜑 → (𝐴𝑂𝐶 ↔ ((¬ 𝐴 ∈ 𝐷 ∧ ¬ 𝐶 ∈ 𝐷) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝐴𝐼𝐶))))
81	78, 80	mpbid 234	. . . . . . . 8 ⊢ (𝜑 → ((¬ 𝐴 ∈ 𝐷 ∧ ¬ 𝐶 ∈ 𝐷) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝐴𝐼𝐶)))
82	81	simprd 498	. . . . . . 7 ⊢ (𝜑 → ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝐴𝐼𝐶))
83	82	adantr 483	. . . . . 6 ⊢ ((𝜑 ∧ 𝑅 = 𝑆) → ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝐴𝐼𝐶))
84	77, 83	r19.29a 3289	. . . . 5 ⊢ ((𝜑 ∧ 𝑅 = 𝑆) → 𝑅 ∈ (𝐴𝐼𝐶))
85	31, 84	eqeltrd 2913	. . . 4 ⊢ ((𝜑 ∧ 𝑅 = 𝑆) → 𝑀 ∈ (𝐴𝐼𝐶))
86	1, 2, 3, 4, 5, 7, 8, 9, 11, 13, 15, 17, 32, 40, 85	mirbtwnhl 26466	. . 3 ⊢ ((𝜑 ∧ 𝑅 = 𝑆) → (𝑈(𝐾‘𝑀)𝐴 ↔ (𝑁‘𝑈)(𝐾‘𝑀)𝐶))
87	31	fveq2d 6674	. . . 4 ⊢ ((𝜑 ∧ 𝑅 = 𝑆) → (𝐾‘𝑀) = (𝐾‘𝑅))
88	87	breqd 5077	. . 3 ⊢ ((𝜑 ∧ 𝑅 = 𝑆) → (𝑈(𝐾‘𝑀)𝐴 ↔ 𝑈(𝐾‘𝑅)𝐴))
89	39	fveq2d 6674	. . . 4 ⊢ ((𝜑 ∧ 𝑅 = 𝑆) → (𝐾‘𝑀) = (𝐾‘𝑆))
90	89	breqd 5077	. . 3 ⊢ ((𝜑 ∧ 𝑅 = 𝑆) → ((𝑁‘𝑈)(𝐾‘𝑀)𝐶 ↔ (𝑁‘𝑈)(𝐾‘𝑆)𝐶))
91	86, 88, 90	3bitr3d 311	. 2 ⊢ ((𝜑 ∧ 𝑅 = 𝑆) → (𝑈(𝐾‘𝑅)𝐴 ↔ (𝑁‘𝑈)(𝐾‘𝑆)𝐶))
92	18	ad2antrr 724	. . . 4 ⊢ (((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ (𝑆 − 𝐶)(≤G‘𝐺)(𝑅 − 𝐴)) → 𝐷 ∈ ran 𝐿)
93	6	ad2antrr 724	. . . 4 ⊢ (((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ (𝑆 − 𝐶)(≤G‘𝐺)(𝑅 − 𝐴)) → 𝐺 ∈ TarskiG)
94	12	ad2antrr 724	. . . 4 ⊢ (((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ (𝑆 − 𝐶)(≤G‘𝐺)(𝑅 − 𝐴)) → 𝐴 ∈ 𝑃)
95	14	ad2antrr 724	. . . 4 ⊢ (((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ (𝑆 − 𝐶)(≤G‘𝐺)(𝑅 − 𝐴)) → 𝐶 ∈ 𝑃)
96	19	ad2antrr 724	. . . 4 ⊢ (((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ (𝑆 − 𝐶)(≤G‘𝐺)(𝑅 − 𝐴)) → 𝑅 ∈ 𝐷)
97	33	ad2antrr 724	. . . 4 ⊢ (((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ (𝑆 − 𝐶)(≤G‘𝐺)(𝑅 − 𝐴)) → 𝑆 ∈ 𝐷)
98	10	ad2antrr 724	. . . 4 ⊢ (((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ (𝑆 − 𝐶)(≤G‘𝐺)(𝑅 − 𝐴)) → 𝑀 ∈ 𝑃)
99	78	ad2antrr 724	. . . 4 ⊢ (((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ (𝑆 − 𝐶)(≤G‘𝐺)(𝑅 − 𝐴)) → 𝐴𝑂𝐶)
100	21	ad2antrr 724	. . . 4 ⊢ (((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ (𝑆 − 𝐶)(≤G‘𝐺)(𝑅 − 𝐴)) → 𝐷(⟂G‘𝐺)(𝐴𝐿𝑅))
101	35	ad2antrr 724	. . . 4 ⊢ (((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ (𝑆 − 𝐶)(≤G‘𝐺)(𝑅 − 𝐴)) → 𝐷(⟂G‘𝐺)(𝐶𝐿𝑆))
102		simplr 767	. . . 4 ⊢ (((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ (𝑆 − 𝐶)(≤G‘𝐺)(𝑅 − 𝐴)) → 𝑅 ≠ 𝑆)
103		simpr 487	. . . 4 ⊢ (((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ (𝑆 − 𝐶)(≤G‘𝐺)(𝑅 − 𝐴)) → (𝑆 − 𝐶)(≤G‘𝐺)(𝑅 − 𝐴))
104	16	ad2antrr 724	. . . 4 ⊢ (((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ (𝑆 − 𝐶)(≤G‘𝐺)(𝑅 − 𝐴)) → 𝑈 ∈ 𝑃)
105	25	ad2antrr 724	. . . 4 ⊢ (((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ (𝑆 − 𝐶)(≤G‘𝐺)(𝑅 − 𝐴)) → (𝑁‘𝑅) = 𝑆)
106	1, 2, 3, 79, 4, 92, 93, 9, 8, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104, 105	opphllem3 26535	. . 3 ⊢ (((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ (𝑆 − 𝐶)(≤G‘𝐺)(𝑅 − 𝐴)) → (𝑈(𝐾‘𝑅)𝐴 ↔ (𝑁‘𝑈)(𝐾‘𝑆)𝐶))
107	18	ad2antrr 724	. . . . 5 ⊢ (((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ (𝑅 − 𝐴)(≤G‘𝐺)(𝑆 − 𝐶)) → 𝐷 ∈ ran 𝐿)
108	6	ad2antrr 724	. . . . 5 ⊢ (((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ (𝑅 − 𝐴)(≤G‘𝐺)(𝑆 − 𝐶)) → 𝐺 ∈ TarskiG)
109	14	ad2antrr 724	. . . . 5 ⊢ (((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ (𝑅 − 𝐴)(≤G‘𝐺)(𝑆 − 𝐶)) → 𝐶 ∈ 𝑃)
110	12	ad2antrr 724	. . . . 5 ⊢ (((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ (𝑅 − 𝐴)(≤G‘𝐺)(𝑆 − 𝐶)) → 𝐴 ∈ 𝑃)
111	33	ad2antrr 724	. . . . 5 ⊢ (((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ (𝑅 − 𝐴)(≤G‘𝐺)(𝑆 − 𝐶)) → 𝑆 ∈ 𝐷)
112	19	ad2antrr 724	. . . . 5 ⊢ (((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ (𝑅 − 𝐴)(≤G‘𝐺)(𝑆 − 𝐶)) → 𝑅 ∈ 𝐷)
113	10	ad2antrr 724	. . . . 5 ⊢ (((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ (𝑅 − 𝐴)(≤G‘𝐺)(𝑆 − 𝐶)) → 𝑀 ∈ 𝑃)
114	78	ad2antrr 724	. . . . . 6 ⊢ (((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ (𝑅 − 𝐴)(≤G‘𝐺)(𝑆 − 𝐶)) → 𝐴𝑂𝐶)
115	1, 2, 3, 79, 4, 107, 108, 110, 109, 114	oppcom 26530	. . . . 5 ⊢ (((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ (𝑅 − 𝐴)(≤G‘𝐺)(𝑆 − 𝐶)) → 𝐶𝑂𝐴)
116	35	ad2antrr 724	. . . . 5 ⊢ (((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ (𝑅 − 𝐴)(≤G‘𝐺)(𝑆 − 𝐶)) → 𝐷(⟂G‘𝐺)(𝐶𝐿𝑆))
117	21	ad2antrr 724	. . . . 5 ⊢ (((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ (𝑅 − 𝐴)(≤G‘𝐺)(𝑆 − 𝐶)) → 𝐷(⟂G‘𝐺)(𝐴𝐿𝑅))
118		simpr 487	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑅 ≠ 𝑆) → 𝑅 ≠ 𝑆)
119	118	necomd 3071	. . . . . 6 ⊢ ((𝜑 ∧ 𝑅 ≠ 𝑆) → 𝑆 ≠ 𝑅)
120	119	adantr 483	. . . . 5 ⊢ (((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ (𝑅 − 𝐴)(≤G‘𝐺)(𝑆 − 𝐶)) → 𝑆 ≠ 𝑅)
121		simpr 487	. . . . 5 ⊢ (((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ (𝑅 − 𝐴)(≤G‘𝐺)(𝑆 − 𝐶)) → (𝑅 − 𝐴)(≤G‘𝐺)(𝑆 − 𝐶))
122	16	ad2antrr 724	. . . . . 6 ⊢ (((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ (𝑅 − 𝐴)(≤G‘𝐺)(𝑆 − 𝐶)) → 𝑈 ∈ 𝑃)
123	1, 2, 3, 4, 5, 108, 113, 8, 122	mircl 26447	. . . . 5 ⊢ (((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ (𝑅 − 𝐴)(≤G‘𝐺)(𝑆 − 𝐶)) → (𝑁‘𝑈) ∈ 𝑃)
124	20	ad2antrr 724	. . . . . 6 ⊢ (((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ (𝑅 − 𝐴)(≤G‘𝐺)(𝑆 − 𝐶)) → 𝑅 ∈ 𝑃)
125	25	ad2antrr 724	. . . . . 6 ⊢ (((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ (𝑅 − 𝐴)(≤G‘𝐺)(𝑆 − 𝐶)) → (𝑁‘𝑅) = 𝑆)
126	1, 2, 3, 4, 5, 108, 113, 8, 124, 125	mircom 26449	. . . . 5 ⊢ (((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ (𝑅 − 𝐴)(≤G‘𝐺)(𝑆 − 𝐶)) → (𝑁‘𝑆) = 𝑅)
127	1, 2, 3, 79, 4, 107, 108, 9, 8, 109, 110, 111, 112, 113, 115, 116, 117, 120, 121, 123, 126	opphllem3 26535	. . . 4 ⊢ (((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ (𝑅 − 𝐴)(≤G‘𝐺)(𝑆 − 𝐶)) → ((𝑁‘𝑈)(𝐾‘𝑆)𝐶 ↔ (𝑁‘(𝑁‘𝑈))(𝐾‘𝑅)𝐴))
128	1, 2, 3, 4, 5, 108, 113, 8, 122	mirmir 26448	. . . . 5 ⊢ (((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ (𝑅 − 𝐴)(≤G‘𝐺)(𝑆 − 𝐶)) → (𝑁‘(𝑁‘𝑈)) = 𝑈)
129	128	breq1d 5076	. . . 4 ⊢ (((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ (𝑅 − 𝐴)(≤G‘𝐺)(𝑆 − 𝐶)) → ((𝑁‘(𝑁‘𝑈))(𝐾‘𝑅)𝐴 ↔ 𝑈(𝐾‘𝑅)𝐴))
130	127, 129	bitr2d 282	. . 3 ⊢ (((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ (𝑅 − 𝐴)(≤G‘𝐺)(𝑆 − 𝐶)) → (𝑈(𝐾‘𝑅)𝐴 ↔ (𝑁‘𝑈)(𝐾‘𝑆)𝐶))
131		eqid 2821	. . . . 5 ⊢ (≤G‘𝐺) = (≤G‘𝐺)
132	1, 2, 3, 131, 6, 34, 14, 20, 12	legtrid 26377	. . . 4 ⊢ (𝜑 → ((𝑆 − 𝐶)(≤G‘𝐺)(𝑅 − 𝐴) ∨ (𝑅 − 𝐴)(≤G‘𝐺)(𝑆 − 𝐶)))
133	132	adantr 483	. . 3 ⊢ ((𝜑 ∧ 𝑅 ≠ 𝑆) → ((𝑆 − 𝐶)(≤G‘𝐺)(𝑅 − 𝐴) ∨ (𝑅 − 𝐴)(≤G‘𝐺)(𝑆 − 𝐶)))
134	106, 130, 133	mpjaodan 955	. 2 ⊢ ((𝜑 ∧ 𝑅 ≠ 𝑆) → (𝑈(𝐾‘𝑅)𝐴 ↔ (𝑁‘𝑈)(𝐾‘𝑆)𝐶))
135	91, 134	pm2.61dane 3104	1 ⊢ (𝜑 → (𝑈(𝐾‘𝑅)𝐴 ↔ (𝑁‘𝑈)(𝐾‘𝑆)𝐶))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∧ wa 398 ∨ wo 843 = wceq 1537 ∈ wcel 2114 ≠ wne 3016 ∃wrex 3139 ∖ cdif 3933 class class class wbr 5066 {copab 5128 ran crn 5556 ‘cfv 6355 (class class class)co 7156 Basecbs 16483 distcds 16574 TarskiGcstrkg 26216 Itvcitv 26222 LineGclng 26223 ≤Gcleg 26368 hlGchlg 26386 pInvGcmir 26438 ⟂Gcperpg 26481

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-1st 7689 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-oadd 8106 df-er 8289 df-map 8408 df-pm 8409 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-dju 9330 df-card 9368 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-nn 11639 df-2 11701 df-3 11702 df-n0 11899 df-xnn0 11969 df-z 11983 df-uz 12245 df-fz 12894 df-fzo 13035 df-hash 13692 df-word 13863 df-concat 13923 df-s1 13950 df-s2 14210 df-s3 14211 df-trkgc 26234 df-trkgb 26235 df-trkgcb 26236 df-trkg 26239 df-cgrg 26297 df-leg 26369 df-hlg 26387 df-mir 26439 df-rag 26480 df-perpg 26482

This theorem is referenced by: opphl 26540

W3C validator