opphllem6 - Metamath Proof Explorer

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Theorem opphllem6 28679

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 2729	. . . 4 ⊢ (pInvG‘𝐺) = (pInvG‘𝐺)
6		opphl.g	. . . . 5 ⊢ (𝜑 → 𝐺 ∈ TarskiG)
7	6	adantr 480	. . . 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 480	. . . 4 ⊢ ((𝜑 ∧ 𝑅 = 𝑆) → 𝑀 ∈ 𝑃)
12		opphllem5.a	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ 𝑃)
13	12	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑅 = 𝑆) → 𝐴 ∈ 𝑃)
14		opphllem5.c	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ 𝑃)
15	14	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑅 = 𝑆) → 𝐶 ∈ 𝑃)
16		opphllem5.u	. . . . 5 ⊢ (𝜑 → 𝑈 ∈ 𝑃)
17	16	adantr 480	. . . 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 28476	. . . . . . 7 ⊢ (𝜑 → 𝑅 ∈ 𝑃)
21		opphllem5.p	. . . . . . . 8 ⊢ (𝜑 → 𝐷(⟂G‘𝐺)(𝐴𝐿𝑅))
22	4, 6, 21	perpln2 28638	. . . . . . 7 ⊢ (𝜑 → (𝐴𝐿𝑅) ∈ ran 𝐿)
23	1, 3, 4, 6, 12, 20, 22	tglnne 28555	. . . . . 6 ⊢ (𝜑 → 𝐴 ≠ 𝑅)
24	23	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑅 = 𝑆) → 𝐴 ≠ 𝑅)
25		opphllem6.v	. . . . . . . 8 ⊢ (𝜑 → (𝑁‘𝑅) = 𝑆)
26	25	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑅 = 𝑆) → (𝑁‘𝑅) = 𝑆)
27		simpr 484	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑅 = 𝑆) → 𝑅 = 𝑆)
28	26, 27	eqtr4d 2767	. . . . . 6 ⊢ ((𝜑 ∧ 𝑅 = 𝑆) → (𝑁‘𝑅) = 𝑅)
29	1, 2, 3, 4, 5, 6, 10, 8, 20	mirinv 28593	. . . . . . 7 ⊢ (𝜑 → ((𝑁‘𝑅) = 𝑅 ↔ 𝑀 = 𝑅))
30	29	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑅 = 𝑆) → ((𝑁‘𝑅) = 𝑅 ↔ 𝑀 = 𝑅))
31	28, 30	mpbid 232	. . . . 5 ⊢ ((𝜑 ∧ 𝑅 = 𝑆) → 𝑀 = 𝑅)
32	24, 31	neeqtrrd 2999	. . . 4 ⊢ ((𝜑 ∧ 𝑅 = 𝑆) → 𝐴 ≠ 𝑀)
33		opphllem5.s	. . . . . . . 8 ⊢ (𝜑 → 𝑆 ∈ 𝐷)
34	1, 4, 3, 6, 18, 33	tglnpt 28476	. . . . . . 7 ⊢ (𝜑 → 𝑆 ∈ 𝑃)
35		opphllem5.q	. . . . . . . 8 ⊢ (𝜑 → 𝐷(⟂G‘𝐺)(𝐶𝐿𝑆))
36	4, 6, 35	perpln2 28638	. . . . . . 7 ⊢ (𝜑 → (𝐶𝐿𝑆) ∈ ran 𝐿)
37	1, 3, 4, 6, 14, 34, 36	tglnne 28555	. . . . . 6 ⊢ (𝜑 → 𝐶 ≠ 𝑆)
38	37	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑅 = 𝑆) → 𝐶 ≠ 𝑆)
39	31, 27	eqtrd 2764	. . . . 5 ⊢ ((𝜑 ∧ 𝑅 = 𝑆) → 𝑀 = 𝑆)
40	38, 39	neeqtrrd 2999	. . . 4 ⊢ ((𝜑 ∧ 𝑅 = 𝑆) → 𝐶 ≠ 𝑀)
41		simpr 484	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) ∧ 𝑅 = 𝑡) → 𝑅 = 𝑡)
42	6	ad4antr 732	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) ∧ 𝑅 ≠ 𝑡) → 𝐺 ∈ TarskiG)
43	14	ad4antr 732	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) ∧ 𝑅 ≠ 𝑡) → 𝐶 ∈ 𝑃)
44	20	ad4antr 732	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) ∧ 𝑅 ≠ 𝑡) → 𝑅 ∈ 𝑃)
45	6	ad3antrrr 730	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) → 𝐺 ∈ TarskiG)
46	18	ad3antrrr 730	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) → 𝐷 ∈ ran 𝐿)
47		simplr 768	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) → 𝑡 ∈ 𝐷)
48	1, 4, 3, 45, 46, 47	tglnpt 28476	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) → 𝑡 ∈ 𝑃)
49	48	adantr 480	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) ∧ 𝑅 ≠ 𝑡) → 𝑡 ∈ 𝑃)
50	12	ad4antr 732	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) ∧ 𝑅 ≠ 𝑡) → 𝐴 ∈ 𝑃)
51	34	ad4antr 732	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) ∧ 𝑅 ≠ 𝑡) → 𝑆 ∈ 𝑃)
52		simpllr 775	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) → 𝑅 = 𝑆)
53	1, 3, 4, 6, 14, 34, 37	tglinerflx2 28561	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑆 ∈ (𝐶𝐿𝑆))
54	53	ad3antrrr 730	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) → 𝑆 ∈ (𝐶𝐿𝑆))
55	52, 54	eqeltrd 2828	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) → 𝑅 ∈ (𝐶𝐿𝑆))
56	55	adantr 480	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) ∧ 𝑅 ≠ 𝑡) → 𝑅 ∈ (𝐶𝐿𝑆))
57	1, 2, 3, 4, 6, 18, 36, 35	perpcom 28640	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐶𝐿𝑆)(⟂G‘𝐺)𝐷)
58	57	ad4antr 732	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) ∧ 𝑅 ≠ 𝑡) → (𝐶𝐿𝑆)(⟂G‘𝐺)𝐷)
59		simpr 484	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) ∧ 𝑅 ≠ 𝑡) → 𝑅 ≠ 𝑡)
60	18	ad4antr 732	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) ∧ 𝑅 ≠ 𝑡) → 𝐷 ∈ ran 𝐿)
61	19	ad4antr 732	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) ∧ 𝑅 ≠ 𝑡) → 𝑅 ∈ 𝐷)
62		simpllr 775	. . . . . . . . . . . 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 28563	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) ∧ 𝑅 ≠ 𝑡) → 𝐷 = (𝑅𝐿𝑡))
64	58, 63	breqtrd 5133	. . . . . . . . . 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 28653	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) ∧ 𝑅 ≠ 𝑡) → ⟨“𝐶𝑅𝑡”⟩ ∈ (∟G‘𝐺))
66	1, 3, 4, 6, 12, 20, 23	tglinerflx2 28561	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑅 ∈ (𝐴𝐿𝑅))
67	66	ad4antr 732	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) ∧ 𝑅 ≠ 𝑡) → 𝑅 ∈ (𝐴𝐿𝑅))
68	1, 2, 3, 4, 6, 18, 22, 21	perpcom 28640	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐴𝐿𝑅)(⟂G‘𝐺)𝐷)
69	68	ad4antr 732	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) ∧ 𝑅 ≠ 𝑡) → (𝐴𝐿𝑅)(⟂G‘𝐺)𝐷)
70	69, 63	breqtrd 5133	. . . . . . . . . 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 28653	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) ∧ 𝑅 ≠ 𝑡) → ⟨“𝐴𝑅𝑡”⟩ ∈ (∟G‘𝐺))
72		simplr 768	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) ∧ 𝑅 ≠ 𝑡) → 𝑡 ∈ (𝐴𝐼𝐶))
73	1, 2, 3, 42, 50, 49, 43, 72	tgbtwncom 28415	. . . . . . . . 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 28630	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) ∧ 𝑅 ≠ 𝑡) → 𝑅 = 𝑡)
75	41, 74	pm2.61dane 3012	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) → 𝑅 = 𝑡)
76		simpr 484	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) → 𝑡 ∈ (𝐴𝐼𝐶))
77	75, 76	eqeltrd 2828	. . . . . 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 28666	. . . . . . . . 9 ⊢ (𝜑 → (𝐴𝑂𝐶 ↔ ((¬ 𝐴 ∈ 𝐷 ∧ ¬ 𝐶 ∈ 𝐷) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝐴𝐼𝐶))))
81	78, 80	mpbid 232	. . . . . . . 8 ⊢ (𝜑 → ((¬ 𝐴 ∈ 𝐷 ∧ ¬ 𝐶 ∈ 𝐷) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝐴𝐼𝐶)))
82	81	simprd 495	. . . . . . 7 ⊢ (𝜑 → ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝐴𝐼𝐶))
83	82	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑅 = 𝑆) → ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝐴𝐼𝐶))
84	77, 83	r19.29a 3141	. . . . 5 ⊢ ((𝜑 ∧ 𝑅 = 𝑆) → 𝑅 ∈ (𝐴𝐼𝐶))
85	31, 84	eqeltrd 2828	. . . 4 ⊢ ((𝜑 ∧ 𝑅 = 𝑆) → 𝑀 ∈ (𝐴𝐼𝐶))
86	1, 2, 3, 4, 5, 7, 8, 9, 11, 13, 15, 17, 32, 40, 85	mirbtwnhl 28607	. . 3 ⊢ ((𝜑 ∧ 𝑅 = 𝑆) → (𝑈(𝐾‘𝑀)𝐴 ↔ (𝑁‘𝑈)(𝐾‘𝑀)𝐶))
87	31	fveq2d 6862	. . . 4 ⊢ ((𝜑 ∧ 𝑅 = 𝑆) → (𝐾‘𝑀) = (𝐾‘𝑅))
88	87	breqd 5118	. . 3 ⊢ ((𝜑 ∧ 𝑅 = 𝑆) → (𝑈(𝐾‘𝑀)𝐴 ↔ 𝑈(𝐾‘𝑅)𝐴))
89	39	fveq2d 6862	. . . 4 ⊢ ((𝜑 ∧ 𝑅 = 𝑆) → (𝐾‘𝑀) = (𝐾‘𝑆))
90	89	breqd 5118	. . 3 ⊢ ((𝜑 ∧ 𝑅 = 𝑆) → ((𝑁‘𝑈)(𝐾‘𝑀)𝐶 ↔ (𝑁‘𝑈)(𝐾‘𝑆)𝐶))
91	86, 88, 90	3bitr3d 309	. 2 ⊢ ((𝜑 ∧ 𝑅 = 𝑆) → (𝑈(𝐾‘𝑅)𝐴 ↔ (𝑁‘𝑈)(𝐾‘𝑆)𝐶))
92	18	ad2antrr 726	. . . 4 ⊢ (((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ (𝑆 − 𝐶)(≤G‘𝐺)(𝑅 − 𝐴)) → 𝐷 ∈ ran 𝐿)
93	6	ad2antrr 726	. . . 4 ⊢ (((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ (𝑆 − 𝐶)(≤G‘𝐺)(𝑅 − 𝐴)) → 𝐺 ∈ TarskiG)
94	12	ad2antrr 726	. . . 4 ⊢ (((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ (𝑆 − 𝐶)(≤G‘𝐺)(𝑅 − 𝐴)) → 𝐴 ∈ 𝑃)
95	14	ad2antrr 726	. . . 4 ⊢ (((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ (𝑆 − 𝐶)(≤G‘𝐺)(𝑅 − 𝐴)) → 𝐶 ∈ 𝑃)
96	19	ad2antrr 726	. . . 4 ⊢ (((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ (𝑆 − 𝐶)(≤G‘𝐺)(𝑅 − 𝐴)) → 𝑅 ∈ 𝐷)
97	33	ad2antrr 726	. . . 4 ⊢ (((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ (𝑆 − 𝐶)(≤G‘𝐺)(𝑅 − 𝐴)) → 𝑆 ∈ 𝐷)
98	10	ad2antrr 726	. . . 4 ⊢ (((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ (𝑆 − 𝐶)(≤G‘𝐺)(𝑅 − 𝐴)) → 𝑀 ∈ 𝑃)
99	78	ad2antrr 726	. . . 4 ⊢ (((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ (𝑆 − 𝐶)(≤G‘𝐺)(𝑅 − 𝐴)) → 𝐴𝑂𝐶)
100	21	ad2antrr 726	. . . 4 ⊢ (((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ (𝑆 − 𝐶)(≤G‘𝐺)(𝑅 − 𝐴)) → 𝐷(⟂G‘𝐺)(𝐴𝐿𝑅))
101	35	ad2antrr 726	. . . 4 ⊢ (((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ (𝑆 − 𝐶)(≤G‘𝐺)(𝑅 − 𝐴)) → 𝐷(⟂G‘𝐺)(𝐶𝐿𝑆))
102		simplr 768	. . . 4 ⊢ (((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ (𝑆 − 𝐶)(≤G‘𝐺)(𝑅 − 𝐴)) → 𝑅 ≠ 𝑆)
103		simpr 484	. . . 4 ⊢ (((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ (𝑆 − 𝐶)(≤G‘𝐺)(𝑅 − 𝐴)) → (𝑆 − 𝐶)(≤G‘𝐺)(𝑅 − 𝐴))
104	16	ad2antrr 726	. . . 4 ⊢ (((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ (𝑆 − 𝐶)(≤G‘𝐺)(𝑅 − 𝐴)) → 𝑈 ∈ 𝑃)
105	25	ad2antrr 726	. . . 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 28676	. . 3 ⊢ (((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ (𝑆 − 𝐶)(≤G‘𝐺)(𝑅 − 𝐴)) → (𝑈(𝐾‘𝑅)𝐴 ↔ (𝑁‘𝑈)(𝐾‘𝑆)𝐶))
107	18	ad2antrr 726	. . . . 5 ⊢ (((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ (𝑅 − 𝐴)(≤G‘𝐺)(𝑆 − 𝐶)) → 𝐷 ∈ ran 𝐿)
108	6	ad2antrr 726	. . . . 5 ⊢ (((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ (𝑅 − 𝐴)(≤G‘𝐺)(𝑆 − 𝐶)) → 𝐺 ∈ TarskiG)
109	14	ad2antrr 726	. . . . 5 ⊢ (((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ (𝑅 − 𝐴)(≤G‘𝐺)(𝑆 − 𝐶)) → 𝐶 ∈ 𝑃)
110	12	ad2antrr 726	. . . . 5 ⊢ (((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ (𝑅 − 𝐴)(≤G‘𝐺)(𝑆 − 𝐶)) → 𝐴 ∈ 𝑃)
111	33	ad2antrr 726	. . . . 5 ⊢ (((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ (𝑅 − 𝐴)(≤G‘𝐺)(𝑆 − 𝐶)) → 𝑆 ∈ 𝐷)
112	19	ad2antrr 726	. . . . 5 ⊢ (((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ (𝑅 − 𝐴)(≤G‘𝐺)(𝑆 − 𝐶)) → 𝑅 ∈ 𝐷)
113	10	ad2antrr 726	. . . . 5 ⊢ (((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ (𝑅 − 𝐴)(≤G‘𝐺)(𝑆 − 𝐶)) → 𝑀 ∈ 𝑃)
114	78	ad2antrr 726	. . . . . 6 ⊢ (((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ (𝑅 − 𝐴)(≤G‘𝐺)(𝑆 − 𝐶)) → 𝐴𝑂𝐶)
115	1, 2, 3, 79, 4, 107, 108, 110, 109, 114	oppcom 28671	. . . . 5 ⊢ (((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ (𝑅 − 𝐴)(≤G‘𝐺)(𝑆 − 𝐶)) → 𝐶𝑂𝐴)
116	35	ad2antrr 726	. . . . 5 ⊢ (((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ (𝑅 − 𝐴)(≤G‘𝐺)(𝑆 − 𝐶)) → 𝐷(⟂G‘𝐺)(𝐶𝐿𝑆))
117	21	ad2antrr 726	. . . . 5 ⊢ (((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ (𝑅 − 𝐴)(≤G‘𝐺)(𝑆 − 𝐶)) → 𝐷(⟂G‘𝐺)(𝐴𝐿𝑅))
118		simpr 484	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑅 ≠ 𝑆) → 𝑅 ≠ 𝑆)
119	118	necomd 2980	. . . . . 6 ⊢ ((𝜑 ∧ 𝑅 ≠ 𝑆) → 𝑆 ≠ 𝑅)
120	119	adantr 480	. . . . 5 ⊢ (((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ (𝑅 − 𝐴)(≤G‘𝐺)(𝑆 − 𝐶)) → 𝑆 ≠ 𝑅)
121		simpr 484	. . . . 5 ⊢ (((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ (𝑅 − 𝐴)(≤G‘𝐺)(𝑆 − 𝐶)) → (𝑅 − 𝐴)(≤G‘𝐺)(𝑆 − 𝐶))
122	16	ad2antrr 726	. . . . . 6 ⊢ (((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ (𝑅 − 𝐴)(≤G‘𝐺)(𝑆 − 𝐶)) → 𝑈 ∈ 𝑃)
123	1, 2, 3, 4, 5, 108, 113, 8, 122	mircl 28588	. . . . 5 ⊢ (((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ (𝑅 − 𝐴)(≤G‘𝐺)(𝑆 − 𝐶)) → (𝑁‘𝑈) ∈ 𝑃)
124	20	ad2antrr 726	. . . . . 6 ⊢ (((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ (𝑅 − 𝐴)(≤G‘𝐺)(𝑆 − 𝐶)) → 𝑅 ∈ 𝑃)
125	25	ad2antrr 726	. . . . . 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 28590	. . . . 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 28676	. . . 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 28589	. . . . 5 ⊢ (((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ (𝑅 − 𝐴)(≤G‘𝐺)(𝑆 − 𝐶)) → (𝑁‘(𝑁‘𝑈)) = 𝑈)
129	128	breq1d 5117	. . . 4 ⊢ (((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ (𝑅 − 𝐴)(≤G‘𝐺)(𝑆 − 𝐶)) → ((𝑁‘(𝑁‘𝑈))(𝐾‘𝑅)𝐴 ↔ 𝑈(𝐾‘𝑅)𝐴))
130	127, 129	bitr2d 280	. . 3 ⊢ (((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ (𝑅 − 𝐴)(≤G‘𝐺)(𝑆 − 𝐶)) → (𝑈(𝐾‘𝑅)𝐴 ↔ (𝑁‘𝑈)(𝐾‘𝑆)𝐶))
131		eqid 2729	. . . . 5 ⊢ (≤G‘𝐺) = (≤G‘𝐺)
132	1, 2, 3, 131, 6, 34, 14, 20, 12	legtrid 28518	. . . 4 ⊢ (𝜑 → ((𝑆 − 𝐶)(≤G‘𝐺)(𝑅 − 𝐴) ∨ (𝑅 − 𝐴)(≤G‘𝐺)(𝑆 − 𝐶)))
133	132	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝑅 ≠ 𝑆) → ((𝑆 − 𝐶)(≤G‘𝐺)(𝑅 − 𝐴) ∨ (𝑅 − 𝐴)(≤G‘𝐺)(𝑆 − 𝐶)))
134	106, 130, 133	mpjaodan 960	. 2 ⊢ ((𝜑 ∧ 𝑅 ≠ 𝑆) → (𝑈(𝐾‘𝑅)𝐴 ↔ (𝑁‘𝑈)(𝐾‘𝑆)𝐶))
135	91, 134	pm2.61dane 3012	1 ⊢ (𝜑 → (𝑈(𝐾‘𝑅)𝐴 ↔ (𝑁‘𝑈)(𝐾‘𝑆)𝐶))

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 Basecbs 17179 distcds 17229 TarskiGcstrkg 28354 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-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