Theorem opphllem4 25442

Description: Lemma for opphl 25446. (Contributed by Thierry Arnoux, 22-Feb-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))
opphllem3.t	⊢ (𝜑 → 𝑅 ≠ 𝑆)
opphllem3.l	⊢ (𝜑 → (𝑆 − 𝐶)(≤G‘𝐺)(𝑅 − 𝐴))
opphllem3.u	⊢ (𝜑 → 𝑈 ∈ 𝑃)
opphllem3.v	⊢ (𝜑 → (𝑁‘𝑅) = 𝑆)
opphllem4.u	⊢ (𝜑 → 𝑉 ∈ 𝑃)
opphllem4.1	⊢ (𝜑 → 𝑈(𝐾‘𝑅)𝐴)
opphllem4.2	⊢ (𝜑 → 𝑉(𝐾‘𝑆)𝐶)

Assertion

Ref	Expression
opphllem4	⊢ (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 opphllem4

Step	Hyp	Ref	Expression
1		hpg.p	. 2 ⊢ 𝑃 = (Base‘𝐺)
2		hpg.d	. 2 ⊢ − = (dist‘𝐺)
3		hpg.i	. 2 ⊢ 𝐼 = (Itv‘𝐺)
4		hpg.o	. 2 ⊢ 𝑂 = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝑃 ∖ 𝐷) ∧ 𝑏 ∈ (𝑃 ∖ 𝐷)) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝑎𝐼𝑏))}
5		opphl.l	. 2 ⊢ 𝐿 = (LineG‘𝐺)
6		opphl.d	. 2 ⊢ (𝜑 → 𝐷 ∈ ran 𝐿)
7		opphl.g	. 2 ⊢ (𝜑 → 𝐺 ∈ TarskiG)
8		opphllem4.u	. 2 ⊢ (𝜑 → 𝑉 ∈ 𝑃)
9		opphllem3.u	. 2 ⊢ (𝜑 → 𝑈 ∈ 𝑃)
10		opphllem5.n	. . 3 ⊢ 𝑁 = ((pInvG‘𝐺)‘𝑀)
11		eqid 2610	. . . 4 ⊢ (pInvG‘𝐺) = (pInvG‘𝐺)
12		opphllem5.m	. . . 4 ⊢ (𝜑 → 𝑀 ∈ 𝑃)
13	1, 2, 3, 5, 11, 7, 12, 10, 9	mircl 25356	. . 3 ⊢ (𝜑 → (𝑁‘𝑈) ∈ 𝑃)
14		opphllem5.s	. . 3 ⊢ (𝜑 → 𝑆 ∈ 𝐷)
15		opphllem5.o	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐴𝑂𝐶)
16		opphllem5.a	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐴 ∈ 𝑃)
17		opphllem5.c	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐶 ∈ 𝑃)
18	1, 2, 3, 4, 16, 17	islnopp 25431	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐴𝑂𝐶 ↔ ((¬ 𝐴 ∈ 𝐷 ∧ ¬ 𝐶 ∈ 𝐷) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝐴𝐼𝐶))))
19	15, 18	mpbid 221	. . . . . . . . . 10 ⊢ (𝜑 → ((¬ 𝐴 ∈ 𝐷 ∧ ¬ 𝐶 ∈ 𝐷) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝐴𝐼𝐶)))
20	19	simpld 474	. . . . . . . . 9 ⊢ (𝜑 → (¬ 𝐴 ∈ 𝐷 ∧ ¬ 𝐶 ∈ 𝐷))
21	20	simpld 474	. . . . . . . 8 ⊢ (𝜑 → ¬ 𝐴 ∈ 𝐷)
22		opphllem5.r	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑅 ∈ 𝐷)
23	1, 5, 3, 7, 6, 22	tglnpt 25244	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑅 ∈ 𝑃)
24		opphl.k	. . . . . . . . . . . . 13 ⊢ 𝐾 = (hlG‘𝐺)
25		opphllem4.1	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑈(𝐾‘𝑅)𝐴)
26	1, 3, 24, 9, 16, 23, 7, 25	hlne1 25300	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑈 ≠ 𝑅)
27	26	necomd 2837	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑅 ≠ 𝑈)
28	1, 3, 24, 9, 16, 23, 7, 5, 25	hlln 25302	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑈 ∈ (𝐴𝐿𝑅))
29	1, 3, 24, 9, 16, 23, 7	ishlg 25297	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑈(𝐾‘𝑅)𝐴 ↔ (𝑈 ≠ 𝑅 ∧ 𝐴 ≠ 𝑅 ∧ (𝑈 ∈ (𝑅𝐼𝐴) ∨ 𝐴 ∈ (𝑅𝐼𝑈)))))
30	25, 29	mpbid 221	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑈 ≠ 𝑅 ∧ 𝐴 ≠ 𝑅 ∧ (𝑈 ∈ (𝑅𝐼𝐴) ∨ 𝐴 ∈ (𝑅𝐼𝑈))))
31	30	simp2d 1067	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐴 ≠ 𝑅)
32	1, 3, 5, 7, 23, 9, 16, 27, 28, 31	lnrot1 25318	. . . . . . . . . 10 ⊢ (𝜑 → 𝐴 ∈ (𝑅𝐿𝑈))
33	32	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑈 ∈ 𝐷) → 𝐴 ∈ (𝑅𝐿𝑈))
34	7	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑈 ∈ 𝐷) → 𝐺 ∈ TarskiG)
35	23	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑈 ∈ 𝐷) → 𝑅 ∈ 𝑃)
36	9	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑈 ∈ 𝐷) → 𝑈 ∈ 𝑃)
37	27	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑈 ∈ 𝐷) → 𝑅 ≠ 𝑈)
38	6	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑈 ∈ 𝐷) → 𝐷 ∈ ran 𝐿)
39	22	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑈 ∈ 𝐷) → 𝑅 ∈ 𝐷)
40		simpr 476	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑈 ∈ 𝐷) → 𝑈 ∈ 𝐷)
41	1, 3, 5, 34, 35, 36, 37, 37, 38, 39, 40	tglinethru 25331	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑈 ∈ 𝐷) → 𝐷 = (𝑅𝐿𝑈))
42	33, 41	eleqtrrd 2691	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑈 ∈ 𝐷) → 𝐴 ∈ 𝐷)
43	21, 42	mtand 689	. . . . . . 7 ⊢ (𝜑 → ¬ 𝑈 ∈ 𝐷)
44	7	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑁‘𝑈) ∈ 𝐷) → 𝐺 ∈ TarskiG)
45	12	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑁‘𝑈) ∈ 𝐷) → 𝑀 ∈ 𝑃)
46	9	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑁‘𝑈) ∈ 𝐷) → 𝑈 ∈ 𝑃)
47	1, 2, 3, 5, 11, 44, 45, 10, 46	mirmir 25357	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑁‘𝑈) ∈ 𝐷) → (𝑁‘(𝑁‘𝑈)) = 𝑈)
48	6	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑁‘𝑈) ∈ 𝐷) → 𝐷 ∈ ran 𝐿)
49	1, 5, 3, 7, 6, 14	tglnpt 25244	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑆 ∈ 𝑃)
50		opphllem3.t	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑅 ≠ 𝑆)
51	50	necomd 2837	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑆 ≠ 𝑅)
52	1, 2, 3, 5, 11, 7, 12, 10, 23	mirbtwn 25353	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑀 ∈ ((𝑁‘𝑅)𝐼𝑅))
53		opphllem3.v	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑁‘𝑅) = 𝑆)
54	53	oveq1d 6564	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝑁‘𝑅)𝐼𝑅) = (𝑆𝐼𝑅))
55	52, 54	eleqtrd 2690	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑀 ∈ (𝑆𝐼𝑅))
56	1, 3, 5, 7, 49, 23, 12, 51, 55	btwnlng1 25314	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑀 ∈ (𝑆𝐿𝑅))
57	1, 3, 5, 7, 49, 23, 51, 51, 6, 14, 22	tglinethru 25331	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐷 = (𝑆𝐿𝑅))
58	56, 57	eleqtrrd 2691	. . . . . . . . . 10 ⊢ (𝜑 → 𝑀 ∈ 𝐷)
59	58	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑁‘𝑈) ∈ 𝐷) → 𝑀 ∈ 𝐷)
60		simpr 476	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑁‘𝑈) ∈ 𝐷) → (𝑁‘𝑈) ∈ 𝐷)
61	1, 2, 3, 5, 11, 44, 10, 48, 59, 60	mirln 25371	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑁‘𝑈) ∈ 𝐷) → (𝑁‘(𝑁‘𝑈)) ∈ 𝐷)
62	47, 61	eqeltrrd 2689	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑁‘𝑈) ∈ 𝐷) → 𝑈 ∈ 𝐷)
63	43, 62	mtand 689	. . . . . 6 ⊢ (𝜑 → ¬ (𝑁‘𝑈) ∈ 𝐷)
64	63, 43	jca 553	. . . . 5 ⊢ (𝜑 → (¬ (𝑁‘𝑈) ∈ 𝐷 ∧ ¬ 𝑈 ∈ 𝐷))
65	1, 2, 3, 5, 11, 7, 12, 10, 9	mirbtwn 25353	. . . . . 6 ⊢ (𝜑 → 𝑀 ∈ ((𝑁‘𝑈)𝐼𝑈))
66		eleq1 2676	. . . . . . 7 ⊢ (𝑡 = 𝑀 → (𝑡 ∈ ((𝑁‘𝑈)𝐼𝑈) ↔ 𝑀 ∈ ((𝑁‘𝑈)𝐼𝑈)))
67	66	rspcev 3282	. . . . . 6 ⊢ ((𝑀 ∈ 𝐷 ∧ 𝑀 ∈ ((𝑁‘𝑈)𝐼𝑈)) → ∃𝑡 ∈ 𝐷 𝑡 ∈ ((𝑁‘𝑈)𝐼𝑈))
68	58, 65, 67	syl2anc 691	. . . . 5 ⊢ (𝜑 → ∃𝑡 ∈ 𝐷 𝑡 ∈ ((𝑁‘𝑈)𝐼𝑈))
69	64, 68	jca 553	. . . 4 ⊢ (𝜑 → ((¬ (𝑁‘𝑈) ∈ 𝐷 ∧ ¬ 𝑈 ∈ 𝐷) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ ((𝑁‘𝑈)𝐼𝑈)))
70	1, 2, 3, 4, 13, 9	islnopp 25431	. . . 4 ⊢ (𝜑 → ((𝑁‘𝑈)𝑂𝑈 ↔ ((¬ (𝑁‘𝑈) ∈ 𝐷 ∧ ¬ 𝑈 ∈ 𝐷) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ ((𝑁‘𝑈)𝐼𝑈))))
71	69, 70	mpbird 246	. . 3 ⊢ (𝜑 → (𝑁‘𝑈)𝑂𝑈)
72		eqidd 2611	. . 3 ⊢ (𝜑 → (𝑁‘𝑈) = (𝑁‘𝑈))
73		opphllem5.p	. . . . . . . 8 ⊢ (𝜑 → 𝐷(⟂G‘𝐺)(𝐴𝐿𝑅))
74		opphllem5.q	. . . . . . . 8 ⊢ (𝜑 → 𝐷(⟂G‘𝐺)(𝐶𝐿𝑆))
75		opphllem3.l	. . . . . . . 8 ⊢ (𝜑 → (𝑆 − 𝐶)(≤G‘𝐺)(𝑅 − 𝐴))
76	1, 2, 3, 4, 5, 6, 7, 24, 10, 16, 17, 22, 14, 12, 15, 73, 74, 50, 75, 9, 53	opphllem3 25441	. . . . . . 7 ⊢ (𝜑 → (𝑈(𝐾‘𝑅)𝐴 ↔ (𝑁‘𝑈)(𝐾‘𝑆)𝐶))
77	25, 76	mpbid 221	. . . . . 6 ⊢ (𝜑 → (𝑁‘𝑈)(𝐾‘𝑆)𝐶)
78		opphllem4.2	. . . . . . 7 ⊢ (𝜑 → 𝑉(𝐾‘𝑆)𝐶)
79	1, 3, 24, 8, 17, 49, 7, 78	hlcomd 25299	. . . . . 6 ⊢ (𝜑 → 𝐶(𝐾‘𝑆)𝑉)
80	1, 3, 24, 13, 17, 8, 7, 49, 77, 79	hltr 25305	. . . . 5 ⊢ (𝜑 → (𝑁‘𝑈)(𝐾‘𝑆)𝑉)
81	1, 3, 24, 13, 8, 49, 7	ishlg 25297	. . . . 5 ⊢ (𝜑 → ((𝑁‘𝑈)(𝐾‘𝑆)𝑉 ↔ ((𝑁‘𝑈) ≠ 𝑆 ∧ 𝑉 ≠ 𝑆 ∧ ((𝑁‘𝑈) ∈ (𝑆𝐼𝑉) ∨ 𝑉 ∈ (𝑆𝐼(𝑁‘𝑈))))))
82	80, 81	mpbid 221	. . . 4 ⊢ (𝜑 → ((𝑁‘𝑈) ≠ 𝑆 ∧ 𝑉 ≠ 𝑆 ∧ ((𝑁‘𝑈) ∈ (𝑆𝐼𝑉) ∨ 𝑉 ∈ (𝑆𝐼(𝑁‘𝑈)))))
83	82	simp1d 1066	. . 3 ⊢ (𝜑 → (𝑁‘𝑈) ≠ 𝑆)
84	82	simp2d 1067	. . 3 ⊢ (𝜑 → 𝑉 ≠ 𝑆)
85	82	simp3d 1068	. . 3 ⊢ (𝜑 → ((𝑁‘𝑈) ∈ (𝑆𝐼𝑉) ∨ 𝑉 ∈ (𝑆𝐼(𝑁‘𝑈))))
86	1, 2, 3, 4, 5, 6, 7, 10, 13, 8, 9, 14, 71, 58, 72, 83, 84, 85	opphllem2 25440	. 2 ⊢ (𝜑 → 𝑉𝑂𝑈)
87	1, 2, 3, 4, 5, 6, 7, 8, 9, 86	oppcom 25436	1 ⊢ (𝜑 → 𝑈𝑂𝑉)

Syntax hints:  ¬ wn 3   → wi 4   ∨ wo 382   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∃wrex 2897   ∖ cdif 3537   class class class wbr 4583  {copab 4642  ran crn 5039  ‘cfv 5804  (class class class)co 6549  Basecbs 15695  distcds 15777  TarskiGcstrkg 25129  Itvcitv 25135  LineGclng 25136  ≤Gcleg 25277  hlGchlg 25295  pInvGcmir 25347  ⟂Gcperpg 25390

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-map 7746  df-pm 7747  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-card 8648  df-cda 8873  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-3 10957  df-n0 11170  df-xnn0 11241  df-z 11255  df-uz 11564  df-fz 12198  df-fzo 12335  df-hash 12980  df-word 13154  df-concat 13156  df-s1 13157  df-s2 13444  df-s3 13445  df-trkgc 25147  df-trkgb 25148  df-trkgcb 25149  df-trkg 25152  df-cgrg 25206  df-leg 25278  df-hlg 25296  df-mir 25348  df-rag 25389  df-perpg 25391

This theorem is referenced by:  opphllem5  25443