Theorem opphllem4 25642

Description: Lemma for opphl 25646. (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 2622	. . . 4 ⊢ (pInvG‘𝐺) = (pInvG‘𝐺)
12		opphllem5.m	. . . 4 ⊢ (𝜑 → 𝑀 ∈ 𝑃)
13	1, 2, 3, 5, 11, 7, 12, 10, 9	mircl 25556	. . 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 25631	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐴𝑂𝐶 ↔ ((¬ 𝐴 ∈ 𝐷 ∧ ¬ 𝐶 ∈ 𝐷) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝐴𝐼𝐶))))
19	15, 18	mpbid 222	. . . . . . . . . 10 ⊢ (𝜑 → ((¬ 𝐴 ∈ 𝐷 ∧ ¬ 𝐶 ∈ 𝐷) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝐴𝐼𝐶)))
20	19	simpld 475	. . . . . . . . 9 ⊢ (𝜑 → (¬ 𝐴 ∈ 𝐷 ∧ ¬ 𝐶 ∈ 𝐷))
21	20	simpld 475	. . . . . . . 8 ⊢ (𝜑 → ¬ 𝐴 ∈ 𝐷)
22		opphllem5.r	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑅 ∈ 𝐷)
23	1, 5, 3, 7, 6, 22	tglnpt 25444	. . . . . . . . . . 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 25500	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑈 ≠ 𝑅)
27	26	necomd 2849	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑅 ≠ 𝑈)
28	1, 3, 24, 9, 16, 23, 7, 5, 25	hlln 25502	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑈 ∈ (𝐴𝐿𝑅))
29	1, 3, 24, 9, 16, 23, 7	ishlg 25497	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑈(𝐾‘𝑅)𝐴 ↔ (𝑈 ≠ 𝑅 ∧ 𝐴 ≠ 𝑅 ∧ (𝑈 ∈ (𝑅𝐼𝐴) ∨ 𝐴 ∈ (𝑅𝐼𝑈)))))
30	25, 29	mpbid 222	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑈 ≠ 𝑅 ∧ 𝐴 ≠ 𝑅 ∧ (𝑈 ∈ (𝑅𝐼𝐴) ∨ 𝐴 ∈ (𝑅𝐼𝑈))))
31	30	simp2d 1074	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐴 ≠ 𝑅)
32	1, 3, 5, 7, 23, 9, 16, 27, 28, 31	lnrot1 25518	. . . . . . . . . 10 ⊢ (𝜑 → 𝐴 ∈ (𝑅𝐿𝑈))
33	32	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑈 ∈ 𝐷) → 𝐴 ∈ (𝑅𝐿𝑈))
34	7	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑈 ∈ 𝐷) → 𝐺 ∈ TarskiG)
35	23	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑈 ∈ 𝐷) → 𝑅 ∈ 𝑃)
36	9	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑈 ∈ 𝐷) → 𝑈 ∈ 𝑃)
37	27	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑈 ∈ 𝐷) → 𝑅 ≠ 𝑈)
38	6	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑈 ∈ 𝐷) → 𝐷 ∈ ran 𝐿)
39	22	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑈 ∈ 𝐷) → 𝑅 ∈ 𝐷)
40		simpr 477	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑈 ∈ 𝐷) → 𝑈 ∈ 𝐷)
41	1, 3, 5, 34, 35, 36, 37, 37, 38, 39, 40	tglinethru 25531	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑈 ∈ 𝐷) → 𝐷 = (𝑅𝐿𝑈))
42	33, 41	eleqtrrd 2704	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑈 ∈ 𝐷) → 𝐴 ∈ 𝐷)
43	21, 42	mtand 691	. . . . . . 7 ⊢ (𝜑 → ¬ 𝑈 ∈ 𝐷)
44	7	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑁‘𝑈) ∈ 𝐷) → 𝐺 ∈ TarskiG)
45	12	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑁‘𝑈) ∈ 𝐷) → 𝑀 ∈ 𝑃)
46	9	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑁‘𝑈) ∈ 𝐷) → 𝑈 ∈ 𝑃)
47	1, 2, 3, 5, 11, 44, 45, 10, 46	mirmir 25557	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑁‘𝑈) ∈ 𝐷) → (𝑁‘(𝑁‘𝑈)) = 𝑈)
48	6	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑁‘𝑈) ∈ 𝐷) → 𝐷 ∈ ran 𝐿)
49	1, 5, 3, 7, 6, 14	tglnpt 25444	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑆 ∈ 𝑃)
50		opphllem3.t	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑅 ≠ 𝑆)
51	50	necomd 2849	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑆 ≠ 𝑅)
52	1, 2, 3, 5, 11, 7, 12, 10, 23	mirbtwn 25553	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑀 ∈ ((𝑁‘𝑅)𝐼𝑅))
53		opphllem3.v	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑁‘𝑅) = 𝑆)
54	53	oveq1d 6665	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝑁‘𝑅)𝐼𝑅) = (𝑆𝐼𝑅))
55	52, 54	eleqtrd 2703	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑀 ∈ (𝑆𝐼𝑅))
56	1, 3, 5, 7, 49, 23, 12, 51, 55	btwnlng1 25514	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑀 ∈ (𝑆𝐿𝑅))
57	1, 3, 5, 7, 49, 23, 51, 51, 6, 14, 22	tglinethru 25531	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐷 = (𝑆𝐿𝑅))
58	56, 57	eleqtrrd 2704	. . . . . . . . . 10 ⊢ (𝜑 → 𝑀 ∈ 𝐷)
59	58	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑁‘𝑈) ∈ 𝐷) → 𝑀 ∈ 𝐷)
60		simpr 477	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑁‘𝑈) ∈ 𝐷) → (𝑁‘𝑈) ∈ 𝐷)
61	1, 2, 3, 5, 11, 44, 10, 48, 59, 60	mirln 25571	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑁‘𝑈) ∈ 𝐷) → (𝑁‘(𝑁‘𝑈)) ∈ 𝐷)
62	47, 61	eqeltrrd 2702	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑁‘𝑈) ∈ 𝐷) → 𝑈 ∈ 𝐷)
63	43, 62	mtand 691	. . . . . 6 ⊢ (𝜑 → ¬ (𝑁‘𝑈) ∈ 𝐷)
64	63, 43	jca 554	. . . . 5 ⊢ (𝜑 → (¬ (𝑁‘𝑈) ∈ 𝐷 ∧ ¬ 𝑈 ∈ 𝐷))
65	1, 2, 3, 5, 11, 7, 12, 10, 9	mirbtwn 25553	. . . . . 6 ⊢ (𝜑 → 𝑀 ∈ ((𝑁‘𝑈)𝐼𝑈))
66		eleq1 2689	. . . . . . 7 ⊢ (𝑡 = 𝑀 → (𝑡 ∈ ((𝑁‘𝑈)𝐼𝑈) ↔ 𝑀 ∈ ((𝑁‘𝑈)𝐼𝑈)))
67	66	rspcev 3309	. . . . . 6 ⊢ ((𝑀 ∈ 𝐷 ∧ 𝑀 ∈ ((𝑁‘𝑈)𝐼𝑈)) → ∃𝑡 ∈ 𝐷 𝑡 ∈ ((𝑁‘𝑈)𝐼𝑈))
68	58, 65, 67	syl2anc 693	. . . . 5 ⊢ (𝜑 → ∃𝑡 ∈ 𝐷 𝑡 ∈ ((𝑁‘𝑈)𝐼𝑈))
69	64, 68	jca 554	. . . 4 ⊢ (𝜑 → ((¬ (𝑁‘𝑈) ∈ 𝐷 ∧ ¬ 𝑈 ∈ 𝐷) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ ((𝑁‘𝑈)𝐼𝑈)))
70	1, 2, 3, 4, 13, 9	islnopp 25631	. . . 4 ⊢ (𝜑 → ((𝑁‘𝑈)𝑂𝑈 ↔ ((¬ (𝑁‘𝑈) ∈ 𝐷 ∧ ¬ 𝑈 ∈ 𝐷) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ ((𝑁‘𝑈)𝐼𝑈))))
71	69, 70	mpbird 247	. . 3 ⊢ (𝜑 → (𝑁‘𝑈)𝑂𝑈)
72		eqidd 2623	. . 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 25641	. . . . . . 7 ⊢ (𝜑 → (𝑈(𝐾‘𝑅)𝐴 ↔ (𝑁‘𝑈)(𝐾‘𝑆)𝐶))
77	25, 76	mpbid 222	. . . . . 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 25499	. . . . . 6 ⊢ (𝜑 → 𝐶(𝐾‘𝑆)𝑉)
80	1, 3, 24, 13, 17, 8, 7, 49, 77, 79	hltr 25505	. . . . 5 ⊢ (𝜑 → (𝑁‘𝑈)(𝐾‘𝑆)𝑉)
81	1, 3, 24, 13, 8, 49, 7	ishlg 25497	. . . . 5 ⊢ (𝜑 → ((𝑁‘𝑈)(𝐾‘𝑆)𝑉 ↔ ((𝑁‘𝑈) ≠ 𝑆 ∧ 𝑉 ≠ 𝑆 ∧ ((𝑁‘𝑈) ∈ (𝑆𝐼𝑉) ∨ 𝑉 ∈ (𝑆𝐼(𝑁‘𝑈))))))
82	80, 81	mpbid 222	. . . 4 ⊢ (𝜑 → ((𝑁‘𝑈) ≠ 𝑆 ∧ 𝑉 ≠ 𝑆 ∧ ((𝑁‘𝑈) ∈ (𝑆𝐼𝑉) ∨ 𝑉 ∈ (𝑆𝐼(𝑁‘𝑈)))))
83	82	simp1d 1073	. . 3 ⊢ (𝜑 → (𝑁‘𝑈) ≠ 𝑆)
84	82	simp2d 1074	. . 3 ⊢ (𝜑 → 𝑉 ≠ 𝑆)
85	82	simp3d 1075	. . 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 25640	. 2 ⊢ (𝜑 → 𝑉𝑂𝑈)
87	1, 2, 3, 4, 5, 6, 7, 8, 9, 86	oppcom 25636	1 ⊢ (𝜑 → 𝑈𝑂𝑉)

Syntax hints:  ¬ wn 3   → wi 4   ∨ wo 383   ∧ wa 384   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990   ≠ wne 2794  ∃wrex 2913   ∖ cdif 3571   class class class wbr 4653  {copab 4712  ran crn 5115  ‘cfv 5888  (class class class)co 6650  Basecbs 15857  distcds 15950  TarskiGcstrkg 25329  Itvcitv 25335  LineGclng 25336  ≤Gcleg 25477  hlGchlg 25495  pInvGcmir 25547  ⟂Gcperpg 25590

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-map 7859  df-pm 7860  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-card 8765  df-cda 8990  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-3 11080  df-n0 11293  df-xnn0 11364  df-z 11378  df-uz 11688  df-fz 12327  df-fzo 12466  df-hash 13118  df-word 13299  df-concat 13301  df-s1 13302  df-s2 13593  df-s3 13594  df-trkgc 25347  df-trkgb 25348  df-trkgcb 25349  df-trkg 25352  df-cgrg 25406  df-leg 25478  df-hlg 25496  df-mir 25548  df-rag 25589  df-perpg 25591

This theorem is referenced by:  opphllem5  25643