opphllem6 - Metamath Proof Explorer

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Theorem opphllem6 29008

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 2761	. . . 4 ⊢ (pInvG‘𝐺) = (pInvG‘𝐺)
6		opphl.g	. . . . 5 ⊢ (𝜑 → 𝐺 ∈ TarskiG)
7	6	adantr 485	. . . 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 485	. . . 4 ⊢ ((𝜑 ∧ 𝑅 = 𝑆) → 𝑀 ∈ 𝑃)
12		opphllem5.a	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ 𝑃)
13	12	adantr 485	. . . 4 ⊢ ((𝜑 ∧ 𝑅 = 𝑆) → 𝐴 ∈ 𝑃)
14		opphllem5.c	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ 𝑃)
15	14	adantr 485	. . . 4 ⊢ ((𝜑 ∧ 𝑅 = 𝑆) → 𝐶 ∈ 𝑃)
16		opphllem5.u	. . . . 5 ⊢ (𝜑 → 𝑈 ∈ 𝑃)
17	16	adantr 485	. . . 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 28794	. . . . . . 7 ⊢ (𝜑 → 𝑅 ∈ 𝑃)
21		opphllem5.p	. . . . . . . 8 ⊢ (𝜑 → 𝐷(⟂G‘𝐺)(𝐴𝐿𝑅))
22	4, 6, 21	perpln2 28966	. . . . . . 7 ⊢ (𝜑 → (𝐴𝐿𝑅) ∈ ran 𝐿)
23	1, 3, 4, 6, 12, 20, 22	tglnne 28877	. . . . . 6 ⊢ (𝜑 → 𝐴 ≠ 𝑅)
24	23	adantr 485	. . . . 5 ⊢ ((𝜑 ∧ 𝑅 = 𝑆) → 𝐴 ≠ 𝑅)
25		opphllem6.v	. . . . . . . 8 ⊢ (𝜑 → (𝑁‘𝑅) = 𝑆)
26	25	adantr 485	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑅 = 𝑆) → (𝑁‘𝑅) = 𝑆)
27		simpr 489	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑅 = 𝑆) → 𝑅 = 𝑆)
28	26, 27	eqtr4d 2799	. . . . . 6 ⊢ ((𝜑 ∧ 𝑅 = 𝑆) → (𝑁‘𝑅) = 𝑅)
29	1, 2, 3, 4, 5, 6, 10, 8, 20	mirinv 28919	. . . . . . 7 ⊢ (𝜑 → ((𝑁‘𝑅) = 𝑅 ↔ 𝑀 = 𝑅))
30	29	adantr 485	. . . . . 6 ⊢ ((𝜑 ∧ 𝑅 = 𝑆) → ((𝑁‘𝑅) = 𝑅 ↔ 𝑀 = 𝑅))
31	28, 30	mpbid 235	. . . . 5 ⊢ ((𝜑 ∧ 𝑅 = 𝑆) → 𝑀 = 𝑅)
32	24, 31	neeqtrrd 3030	. . . 4 ⊢ ((𝜑 ∧ 𝑅 = 𝑆) → 𝐴 ≠ 𝑀)
33		opphllem5.s	. . . . . . . 8 ⊢ (𝜑 → 𝑆 ∈ 𝐷)
34	1, 4, 3, 6, 18, 33	tglnpt 28794	. . . . . . 7 ⊢ (𝜑 → 𝑆 ∈ 𝑃)
35		opphllem5.q	. . . . . . . 8 ⊢ (𝜑 → 𝐷(⟂G‘𝐺)(𝐶𝐿𝑆))
36	4, 6, 35	perpln2 28966	. . . . . . 7 ⊢ (𝜑 → (𝐶𝐿𝑆) ∈ ran 𝐿)
37	1, 3, 4, 6, 14, 34, 36	tglnne 28877	. . . . . 6 ⊢ (𝜑 → 𝐶 ≠ 𝑆)
38	37	adantr 485	. . . . 5 ⊢ ((𝜑 ∧ 𝑅 = 𝑆) → 𝐶 ≠ 𝑆)
39	31, 27	eqtrd 2796	. . . . 5 ⊢ ((𝜑 ∧ 𝑅 = 𝑆) → 𝑀 = 𝑆)
40	38, 39	neeqtrrd 3030	. . . 4 ⊢ ((𝜑 ∧ 𝑅 = 𝑆) → 𝐶 ≠ 𝑀)
41		simpr 489	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) ∧ 𝑅 = 𝑡) → 𝑅 = 𝑡)
42	6	ad4antr 744	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) ∧ 𝑅 ≠ 𝑡) → 𝐺 ∈ TarskiG)
43	14	ad4antr 744	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) ∧ 𝑅 ≠ 𝑡) → 𝐶 ∈ 𝑃)
44	20	ad4antr 744	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) ∧ 𝑅 ≠ 𝑡) → 𝑅 ∈ 𝑃)
45	6	ad3antrrr 742	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) → 𝐺 ∈ TarskiG)
46	18	ad3antrrr 742	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) → 𝐷 ∈ ran 𝐿)
47		simplr 780	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) → 𝑡 ∈ 𝐷)
48	1, 4, 3, 45, 46, 47	tglnpt 28794	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) → 𝑡 ∈ 𝑃)
49	48	adantr 485	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) ∧ 𝑅 ≠ 𝑡) → 𝑡 ∈ 𝑃)
50	12	ad4antr 744	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) ∧ 𝑅 ≠ 𝑡) → 𝐴 ∈ 𝑃)
51	34	ad4antr 744	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) ∧ 𝑅 ≠ 𝑡) → 𝑆 ∈ 𝑃)
52		simpllr 787	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) → 𝑅 = 𝑆)
53	1, 3, 4, 6, 14, 34, 37	tglinerflx2 28883	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑆 ∈ (𝐶𝐿𝑆))
54	53	ad3antrrr 742	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) → 𝑆 ∈ (𝐶𝐿𝑆))
55	52, 54	eqeltrd 2861	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) → 𝑅 ∈ (𝐶𝐿𝑆))
56	55	adantr 485	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) ∧ 𝑅 ≠ 𝑡) → 𝑅 ∈ (𝐶𝐿𝑆))
57	1, 2, 3, 4, 6, 18, 36, 35	perpcom 28968	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐶𝐿𝑆)(⟂G‘𝐺)𝐷)
58	57	ad4antr 744	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) ∧ 𝑅 ≠ 𝑡) → (𝐶𝐿𝑆)(⟂G‘𝐺)𝐷)
59		simpr 489	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) ∧ 𝑅 ≠ 𝑡) → 𝑅 ≠ 𝑡)
60	18	ad4antr 744	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) ∧ 𝑅 ≠ 𝑡) → 𝐷 ∈ ran 𝐿)
61	19	ad4antr 744	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) ∧ 𝑅 ≠ 𝑡) → 𝑅 ∈ 𝐷)
62		simpllr 787	. . . . . . . . . . . 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 28885	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) ∧ 𝑅 ≠ 𝑡) → 𝐷 = (𝑅𝐿𝑡))
64	58, 63	breqtrd 5136	. . . . . . . . . 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 28982	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) ∧ 𝑅 ≠ 𝑡) → ⟨“𝐶𝑅𝑡”⟩ ∈ (∟G‘𝐺))
66	1, 3, 4, 6, 12, 20, 23	tglinerflx2 28883	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑅 ∈ (𝐴𝐿𝑅))
67	66	ad4antr 744	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) ∧ 𝑅 ≠ 𝑡) → 𝑅 ∈ (𝐴𝐿𝑅))
68	1, 2, 3, 4, 6, 18, 22, 21	perpcom 28968	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐴𝐿𝑅)(⟂G‘𝐺)𝐷)
69	68	ad4antr 744	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) ∧ 𝑅 ≠ 𝑡) → (𝐴𝐿𝑅)(⟂G‘𝐺)𝐷)
70	69, 63	breqtrd 5136	. . . . . . . . . 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 28982	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) ∧ 𝑅 ≠ 𝑡) → ⟨“𝐴𝑅𝑡”⟩ ∈ (∟G‘𝐺))
72		simplr 780	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) ∧ 𝑅 ≠ 𝑡) → 𝑡 ∈ (𝐴𝐼𝐶))
73	1, 2, 3, 42, 50, 49, 43, 72	tgbtwncom 28733	. . . . . . . . 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 28958	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) ∧ 𝑅 ≠ 𝑡) → 𝑅 = 𝑡)
75	41, 74	pm2.61dane 3043	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) → 𝑅 = 𝑡)
76		simpr 489	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) → 𝑡 ∈ (𝐴𝐼𝐶))
77	75, 76	eqeltrd 2861	. . . . . 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 28995	. . . . . . . . 9 ⊢ (𝜑 → (𝐴𝑂𝐶 ↔ ((¬ 𝐴 ∈ 𝐷 ∧ ¬ 𝐶 ∈ 𝐷) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝐴𝐼𝐶))))
81	78, 80	mpbid 235	. . . . . . . 8 ⊢ (𝜑 → ((¬ 𝐴 ∈ 𝐷 ∧ ¬ 𝐶 ∈ 𝐷) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝐴𝐼𝐶)))
82	81	simprd 500	. . . . . . 7 ⊢ (𝜑 → ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝐴𝐼𝐶))
83	82	adantr 485	. . . . . 6 ⊢ ((𝜑 ∧ 𝑅 = 𝑆) → ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝐴𝐼𝐶))
84	77, 83	r19.29a 3171	. . . . 5 ⊢ ((𝜑 ∧ 𝑅 = 𝑆) → 𝑅 ∈ (𝐴𝐼𝐶))
85	31, 84	eqeltrd 2861	. . . 4 ⊢ ((𝜑 ∧ 𝑅 = 𝑆) → 𝑀 ∈ (𝐴𝐼𝐶))
86	1, 2, 3, 4, 5, 7, 8, 9, 11, 13, 15, 17, 32, 40, 85	mirbtwnhl 28933	. . 3 ⊢ ((𝜑 ∧ 𝑅 = 𝑆) → (𝑈(𝐾‘𝑀)𝐴 ↔ (𝑁‘𝑈)(𝐾‘𝑀)𝐶))
87	31	fveq2d 6885	. . . 4 ⊢ ((𝜑 ∧ 𝑅 = 𝑆) → (𝐾‘𝑀) = (𝐾‘𝑅))
88	87	breqd 5119	. . 3 ⊢ ((𝜑 ∧ 𝑅 = 𝑆) → (𝑈(𝐾‘𝑀)𝐴 ↔ 𝑈(𝐾‘𝑅)𝐴))
89	39	fveq2d 6885	. . . 4 ⊢ ((𝜑 ∧ 𝑅 = 𝑆) → (𝐾‘𝑀) = (𝐾‘𝑆))
90	89	breqd 5119	. . 3 ⊢ ((𝜑 ∧ 𝑅 = 𝑆) → ((𝑁‘𝑈)(𝐾‘𝑀)𝐶 ↔ (𝑁‘𝑈)(𝐾‘𝑆)𝐶))
91	86, 88, 90	3bitr3d 312	. 2 ⊢ ((𝜑 ∧ 𝑅 = 𝑆) → (𝑈(𝐾‘𝑅)𝐴 ↔ (𝑁‘𝑈)(𝐾‘𝑆)𝐶))
92	18	ad2antrr 738	. . . 4 ⊢ (((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ (𝑆 − 𝐶)(≤G‘𝐺)(𝑅 − 𝐴)) → 𝐷 ∈ ran 𝐿)
93	6	ad2antrr 738	. . . 4 ⊢ (((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ (𝑆 − 𝐶)(≤G‘𝐺)(𝑅 − 𝐴)) → 𝐺 ∈ TarskiG)
94	12	ad2antrr 738	. . . 4 ⊢ (((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ (𝑆 − 𝐶)(≤G‘𝐺)(𝑅 − 𝐴)) → 𝐴 ∈ 𝑃)
95	14	ad2antrr 738	. . . 4 ⊢ (((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ (𝑆 − 𝐶)(≤G‘𝐺)(𝑅 − 𝐴)) → 𝐶 ∈ 𝑃)
96	19	ad2antrr 738	. . . 4 ⊢ (((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ (𝑆 − 𝐶)(≤G‘𝐺)(𝑅 − 𝐴)) → 𝑅 ∈ 𝐷)
97	33	ad2antrr 738	. . . 4 ⊢ (((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ (𝑆 − 𝐶)(≤G‘𝐺)(𝑅 − 𝐴)) → 𝑆 ∈ 𝐷)
98	10	ad2antrr 738	. . . 4 ⊢ (((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ (𝑆 − 𝐶)(≤G‘𝐺)(𝑅 − 𝐴)) → 𝑀 ∈ 𝑃)
99	78	ad2antrr 738	. . . 4 ⊢ (((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ (𝑆 − 𝐶)(≤G‘𝐺)(𝑅 − 𝐴)) → 𝐴𝑂𝐶)
100	21	ad2antrr 738	. . . 4 ⊢ (((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ (𝑆 − 𝐶)(≤G‘𝐺)(𝑅 − 𝐴)) → 𝐷(⟂G‘𝐺)(𝐴𝐿𝑅))
101	35	ad2antrr 738	. . . 4 ⊢ (((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ (𝑆 − 𝐶)(≤G‘𝐺)(𝑅 − 𝐴)) → 𝐷(⟂G‘𝐺)(𝐶𝐿𝑆))
102		simplr 780	. . . 4 ⊢ (((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ (𝑆 − 𝐶)(≤G‘𝐺)(𝑅 − 𝐴)) → 𝑅 ≠ 𝑆)
103		simpr 489	. . . 4 ⊢ (((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ (𝑆 − 𝐶)(≤G‘𝐺)(𝑅 − 𝐴)) → (𝑆 − 𝐶)(≤G‘𝐺)(𝑅 − 𝐴))
104	16	ad2antrr 738	. . . 4 ⊢ (((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ (𝑆 − 𝐶)(≤G‘𝐺)(𝑅 − 𝐴)) → 𝑈 ∈ 𝑃)
105	25	ad2antrr 738	. . . 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 29005	. . 3 ⊢ (((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ (𝑆 − 𝐶)(≤G‘𝐺)(𝑅 − 𝐴)) → (𝑈(𝐾‘𝑅)𝐴 ↔ (𝑁‘𝑈)(𝐾‘𝑆)𝐶))
107	18	ad2antrr 738	. . . . 5 ⊢ (((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ (𝑅 − 𝐴)(≤G‘𝐺)(𝑆 − 𝐶)) → 𝐷 ∈ ran 𝐿)
108	6	ad2antrr 738	. . . . 5 ⊢ (((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ (𝑅 − 𝐴)(≤G‘𝐺)(𝑆 − 𝐶)) → 𝐺 ∈ TarskiG)
109	14	ad2antrr 738	. . . . 5 ⊢ (((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ (𝑅 − 𝐴)(≤G‘𝐺)(𝑆 − 𝐶)) → 𝐶 ∈ 𝑃)
110	12	ad2antrr 738	. . . . 5 ⊢ (((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ (𝑅 − 𝐴)(≤G‘𝐺)(𝑆 − 𝐶)) → 𝐴 ∈ 𝑃)
111	33	ad2antrr 738	. . . . 5 ⊢ (((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ (𝑅 − 𝐴)(≤G‘𝐺)(𝑆 − 𝐶)) → 𝑆 ∈ 𝐷)
112	19	ad2antrr 738	. . . . 5 ⊢ (((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ (𝑅 − 𝐴)(≤G‘𝐺)(𝑆 − 𝐶)) → 𝑅 ∈ 𝐷)
113	10	ad2antrr 738	. . . . 5 ⊢ (((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ (𝑅 − 𝐴)(≤G‘𝐺)(𝑆 − 𝐶)) → 𝑀 ∈ 𝑃)
114	78	ad2antrr 738	. . . . . 6 ⊢ (((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ (𝑅 − 𝐴)(≤G‘𝐺)(𝑆 − 𝐶)) → 𝐴𝑂𝐶)
115	1, 2, 3, 79, 4, 107, 108, 110, 109, 114	oppcom 29000	. . . . 5 ⊢ (((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ (𝑅 − 𝐴)(≤G‘𝐺)(𝑆 − 𝐶)) → 𝐶𝑂𝐴)
116	35	ad2antrr 738	. . . . 5 ⊢ (((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ (𝑅 − 𝐴)(≤G‘𝐺)(𝑆 − 𝐶)) → 𝐷(⟂G‘𝐺)(𝐶𝐿𝑆))
117	21	ad2antrr 738	. . . . 5 ⊢ (((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ (𝑅 − 𝐴)(≤G‘𝐺)(𝑆 − 𝐶)) → 𝐷(⟂G‘𝐺)(𝐴𝐿𝑅))
118		simpr 489	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑅 ≠ 𝑆) → 𝑅 ≠ 𝑆)
119	118	necomd 3011	. . . . . 6 ⊢ ((𝜑 ∧ 𝑅 ≠ 𝑆) → 𝑆 ≠ 𝑅)
120	119	adantr 485	. . . . 5 ⊢ (((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ (𝑅 − 𝐴)(≤G‘𝐺)(𝑆 − 𝐶)) → 𝑆 ≠ 𝑅)
121		simpr 489	. . . . 5 ⊢ (((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ (𝑅 − 𝐴)(≤G‘𝐺)(𝑆 − 𝐶)) → (𝑅 − 𝐴)(≤G‘𝐺)(𝑆 − 𝐶))
122	16	ad2antrr 738	. . . . . 6 ⊢ (((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ (𝑅 − 𝐴)(≤G‘𝐺)(𝑆 − 𝐶)) → 𝑈 ∈ 𝑃)
123	1, 2, 3, 4, 5, 108, 113, 8, 122	mircl 28914	. . . . 5 ⊢ (((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ (𝑅 − 𝐴)(≤G‘𝐺)(𝑆 − 𝐶)) → (𝑁‘𝑈) ∈ 𝑃)
124	20	ad2antrr 738	. . . . . 6 ⊢ (((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ (𝑅 − 𝐴)(≤G‘𝐺)(𝑆 − 𝐶)) → 𝑅 ∈ 𝑃)
125	25	ad2antrr 738	. . . . . 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 28916	. . . . 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 29005	. . . 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 28915	. . . . 5 ⊢ (((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ (𝑅 − 𝐴)(≤G‘𝐺)(𝑆 − 𝐶)) → (𝑁‘(𝑁‘𝑈)) = 𝑈)
129	128	breq1d 5118	. . . 4 ⊢ (((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ (𝑅 − 𝐴)(≤G‘𝐺)(𝑆 − 𝐶)) → ((𝑁‘(𝑁‘𝑈))(𝐾‘𝑅)𝐴 ↔ 𝑈(𝐾‘𝑅)𝐴))
130	127, 129	bitr2d 283	. . 3 ⊢ (((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ (𝑅 − 𝐴)(≤G‘𝐺)(𝑆 − 𝐶)) → (𝑈(𝐾‘𝑅)𝐴 ↔ (𝑁‘𝑈)(𝐾‘𝑆)𝐶))
131		eqid 2761	. . . . 5 ⊢ (≤G‘𝐺) = (≤G‘𝐺)
132	1, 2, 3, 131, 6, 34, 14, 20, 12	legtrid 28836	. . . 4 ⊢ (𝜑 → ((𝑆 − 𝐶)(≤G‘𝐺)(𝑅 − 𝐴) ∨ (𝑅 − 𝐴)(≤G‘𝐺)(𝑆 − 𝐶)))
133	132	adantr 485	. . 3 ⊢ ((𝜑 ∧ 𝑅 ≠ 𝑆) → ((𝑆 − 𝐶)(≤G‘𝐺)(𝑅 − 𝐴) ∨ (𝑅 − 𝐴)(≤G‘𝐺)(𝑆 − 𝐶)))
134	106, 130, 133	mpjaodan 973	. 2 ⊢ ((𝜑 ∧ 𝑅 ≠ 𝑆) → (𝑈(𝐾‘𝑅)𝐴 ↔ (𝑁‘𝑈)(𝐾‘𝑆)𝐶))
135	91, 134	pm2.61dane 3043	1 ⊢ (𝜑 → (𝑈(𝐾‘𝑅)𝐴 ↔ (𝑁‘𝑈)(𝐾‘𝑆)𝐶))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 400 ∨ wo 860 = wceq 1568 ∈ wcel 2141 ≠ wne 2956 ∃wrex 3087 ∖ cdif 3901 class class class wbr 5108 {copab 5172 ran crn 5662 ‘cfv 6536 (class class class)co 7410 Basecbs 17268 distcds 17318 TarskiGcstrkg 28672 Itvcitv 28678 LineGclng 28679 ≤Gcleg 28827 hlGchlg 28845 pInvGcmir 28905 ⟂Gcperpg 28950

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1823 ax-4 1837 ax-5 1938 ax-6 1995 ax-7 2036 ax-8 2143 ax-9 2151 ax-10 2174 ax-11 2190 ax-12 2211 ax-ext 2733 ax-rep 5237 ax-sep 5256 ax-nul 5268 ax-pow 5336 ax-pr 5404 ax-un 7732 ax-cnex 11155 ax-resscn 11156 ax-1cn 11157 ax-icn 11158 ax-addcl 11159 ax-addrcl 11160 ax-mulcl 11161 ax-mulrcl 11162 ax-mulcom 11163 ax-addass 11164 ax-mulass 11165 ax-distr 11166 ax-i2m1 11167 ax-1ne0 11168 ax-1rid 11169 ax-rnegex 11170 ax-rrecex 11171 ax-cnre 11172 ax-pre-lttri 11173 ax-pre-lttrn 11174 ax-pre-ltadd 11175 ax-pre-mulgt0 11176

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1102 df-3an 1103 df-tru 1571 df-fal 1581 df-ex 1808 df-nf 1812 df-sb 2095 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3063 df-ral 3078 df-rex 3088 df-rmo 3367 df-reu 3368 df-rab 3415 df-v 3455 df-sbc 3744 df-csb 3853 df-dif 3907 df-un 3909 df-in 3911 df-ss 3921 df-pss 3924 df-nul 4286 df-if 4487 df-pw 4563 df-sn 4589 df-pr 4591 df-tp 4593 df-op 4595 df-uni 4872 df-int 4912 df-iun 4957 df-br 5109 df-opab 5173 df-mpt 5192 df-tr 5218 df-id 5556 df-eprel 5561 df-po 5569 df-so 5570 df-fr 5614 df-we 5616 df-xp 5667 df-rel 5668 df-cnv 5669 df-co 5670 df-dm 5671 df-rn 5672 df-res 5673 df-ima 5674 df-pred 6302 df-ord 6363 df-on 6364 df-lim 6365 df-suc 6366 df-iota 6492 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7367 df-ov 7413 df-oprab 7414 df-mpo 7415 df-om 7862 df-1st 7985 df-2nd 7986 df-frecs 8277 df-wrecs 8308 df-recs 8357 df-rdg 8396 df-1o 8452 df-oadd 8456 df-er 8693 df-map 8825 df-pm 8826 df-en 8943 df-dom 8944 df-sdom 8945 df-fin 8946 df-dju 9886 df-card 9924 df-pnf 11244 df-mnf 11245 df-xr 11246 df-ltxr 11247 df-le 11248 df-sub 11442 df-neg 11443 df-nn 12233 df-2 12302 df-3 12303 df-n0 12504 df-xnn0 12577 df-z 12591 df-uz 12862 df-fz 13535 df-fzo 13682 df-hash 14366 df-word 14550 df-concat 14607 df-s1 14633 df-s2 14884 df-s3 14885 df-trkgc 28693 df-trkgb 28694 df-trkgcb 28695 df-trkg 28698 df-cgrg 28756 df-leg 28828 df-hlg 28846 df-mir 28906 df-rag 28949 df-perpg 28951

This theorem is referenced by: opphl 29010

W3C validator