opphllem4 - Metamath Proof Explorer

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Theorem opphllem4 26098

Description: Lemma for opphl 26102. (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 2777	. . . 4 ⊢ (pInvG‘𝐺) = (pInvG‘𝐺)
12		opphllem5.m	. . . 4 ⊢ (𝜑 → 𝑀 ∈ 𝑃)
13	1, 2, 3, 5, 11, 7, 12, 10, 9	mircl 26012	. . 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 26087	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐴𝑂𝐶 ↔ ((¬ 𝐴 ∈ 𝐷 ∧ ¬ 𝐶 ∈ 𝐷) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝐴𝐼𝐶))))
19	15, 18	mpbid 224	. . . . . . . . . 10 ⊢ (𝜑 → ((¬ 𝐴 ∈ 𝐷 ∧ ¬ 𝐶 ∈ 𝐷) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝐴𝐼𝐶)))
20	19	simpld 490	. . . . . . . . 9 ⊢ (𝜑 → (¬ 𝐴 ∈ 𝐷 ∧ ¬ 𝐶 ∈ 𝐷))
21	20	simpld 490	. . . . . . . 8 ⊢ (𝜑 → ¬ 𝐴 ∈ 𝐷)
22		opphllem5.r	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑅 ∈ 𝐷)
23	1, 5, 3, 7, 6, 22	tglnpt 25900	. . . . . . . . . . 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 25956	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑈 ≠ 𝑅)
27	26	necomd 3023	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑅 ≠ 𝑈)
28	1, 3, 24, 9, 16, 23, 7, 5, 25	hlln 25958	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑈 ∈ (𝐴𝐿𝑅))
29	1, 3, 24, 9, 16, 23, 7	ishlg 25953	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑈(𝐾‘𝑅)𝐴 ↔ (𝑈 ≠ 𝑅 ∧ 𝐴 ≠ 𝑅 ∧ (𝑈 ∈ (𝑅𝐼𝐴) ∨ 𝐴 ∈ (𝑅𝐼𝑈)))))
30	25, 29	mpbid 224	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑈 ≠ 𝑅 ∧ 𝐴 ≠ 𝑅 ∧ (𝑈 ∈ (𝑅𝐼𝐴) ∨ 𝐴 ∈ (𝑅𝐼𝑈))))
31	30	simp2d 1134	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐴 ≠ 𝑅)
32	1, 3, 5, 7, 23, 9, 16, 27, 28, 31	lnrot1 25974	. . . . . . . . . 10 ⊢ (𝜑 → 𝐴 ∈ (𝑅𝐿𝑈))
33	32	adantr 474	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑈 ∈ 𝐷) → 𝐴 ∈ (𝑅𝐿𝑈))
34	7	adantr 474	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑈 ∈ 𝐷) → 𝐺 ∈ TarskiG)
35	23	adantr 474	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑈 ∈ 𝐷) → 𝑅 ∈ 𝑃)
36	9	adantr 474	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑈 ∈ 𝐷) → 𝑈 ∈ 𝑃)
37	27	adantr 474	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑈 ∈ 𝐷) → 𝑅 ≠ 𝑈)
38	6	adantr 474	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑈 ∈ 𝐷) → 𝐷 ∈ ran 𝐿)
39	22	adantr 474	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑈 ∈ 𝐷) → 𝑅 ∈ 𝐷)
40		simpr 479	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑈 ∈ 𝐷) → 𝑈 ∈ 𝐷)
41	1, 3, 5, 34, 35, 36, 37, 37, 38, 39, 40	tglinethru 25987	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑈 ∈ 𝐷) → 𝐷 = (𝑅𝐿𝑈))
42	33, 41	eleqtrrd 2861	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑈 ∈ 𝐷) → 𝐴 ∈ 𝐷)
43	21, 42	mtand 806	. . . . . . 7 ⊢ (𝜑 → ¬ 𝑈 ∈ 𝐷)
44	7	adantr 474	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑁‘𝑈) ∈ 𝐷) → 𝐺 ∈ TarskiG)
45	12	adantr 474	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑁‘𝑈) ∈ 𝐷) → 𝑀 ∈ 𝑃)
46	9	adantr 474	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑁‘𝑈) ∈ 𝐷) → 𝑈 ∈ 𝑃)
47	1, 2, 3, 5, 11, 44, 45, 10, 46	mirmir 26013	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑁‘𝑈) ∈ 𝐷) → (𝑁‘(𝑁‘𝑈)) = 𝑈)
48	6	adantr 474	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑁‘𝑈) ∈ 𝐷) → 𝐷 ∈ ran 𝐿)
49	1, 5, 3, 7, 6, 14	tglnpt 25900	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑆 ∈ 𝑃)
50		opphllem3.t	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑅 ≠ 𝑆)
51	50	necomd 3023	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑆 ≠ 𝑅)
52	1, 2, 3, 5, 11, 7, 12, 10, 23	mirbtwn 26009	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑀 ∈ ((𝑁‘𝑅)𝐼𝑅))
53		opphllem3.v	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑁‘𝑅) = 𝑆)
54	53	oveq1d 6937	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝑁‘𝑅)𝐼𝑅) = (𝑆𝐼𝑅))
55	52, 54	eleqtrd 2860	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑀 ∈ (𝑆𝐼𝑅))
56	1, 3, 5, 7, 49, 23, 12, 51, 55	btwnlng1 25970	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑀 ∈ (𝑆𝐿𝑅))
57	1, 3, 5, 7, 49, 23, 51, 51, 6, 14, 22	tglinethru 25987	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐷 = (𝑆𝐿𝑅))
58	56, 57	eleqtrrd 2861	. . . . . . . . . 10 ⊢ (𝜑 → 𝑀 ∈ 𝐷)
59	58	adantr 474	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑁‘𝑈) ∈ 𝐷) → 𝑀 ∈ 𝐷)
60		simpr 479	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑁‘𝑈) ∈ 𝐷) → (𝑁‘𝑈) ∈ 𝐷)
61	1, 2, 3, 5, 11, 44, 10, 48, 59, 60	mirln 26027	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑁‘𝑈) ∈ 𝐷) → (𝑁‘(𝑁‘𝑈)) ∈ 𝐷)
62	47, 61	eqeltrrd 2859	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑁‘𝑈) ∈ 𝐷) → 𝑈 ∈ 𝐷)
63	43, 62	mtand 806	. . . . . 6 ⊢ (𝜑 → ¬ (𝑁‘𝑈) ∈ 𝐷)
64	63, 43	jca 507	. . . . 5 ⊢ (𝜑 → (¬ (𝑁‘𝑈) ∈ 𝐷 ∧ ¬ 𝑈 ∈ 𝐷))
65	1, 2, 3, 5, 11, 7, 12, 10, 9	mirbtwn 26009	. . . . . 6 ⊢ (𝜑 → 𝑀 ∈ ((𝑁‘𝑈)𝐼𝑈))
66		eleq1 2846	. . . . . . 7 ⊢ (𝑡 = 𝑀 → (𝑡 ∈ ((𝑁‘𝑈)𝐼𝑈) ↔ 𝑀 ∈ ((𝑁‘𝑈)𝐼𝑈)))
67	66	rspcev 3510	. . . . . 6 ⊢ ((𝑀 ∈ 𝐷 ∧ 𝑀 ∈ ((𝑁‘𝑈)𝐼𝑈)) → ∃𝑡 ∈ 𝐷 𝑡 ∈ ((𝑁‘𝑈)𝐼𝑈))
68	58, 65, 67	syl2anc 579	. . . . 5 ⊢ (𝜑 → ∃𝑡 ∈ 𝐷 𝑡 ∈ ((𝑁‘𝑈)𝐼𝑈))
69	64, 68	jca 507	. . . 4 ⊢ (𝜑 → ((¬ (𝑁‘𝑈) ∈ 𝐷 ∧ ¬ 𝑈 ∈ 𝐷) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ ((𝑁‘𝑈)𝐼𝑈)))
70	1, 2, 3, 4, 13, 9	islnopp 26087	. . . 4 ⊢ (𝜑 → ((𝑁‘𝑈)𝑂𝑈 ↔ ((¬ (𝑁‘𝑈) ∈ 𝐷 ∧ ¬ 𝑈 ∈ 𝐷) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ ((𝑁‘𝑈)𝐼𝑈))))
71	69, 70	mpbird 249	. . 3 ⊢ (𝜑 → (𝑁‘𝑈)𝑂𝑈)
72		eqidd 2778	. . 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 26097	. . . . . . 7 ⊢ (𝜑 → (𝑈(𝐾‘𝑅)𝐴 ↔ (𝑁‘𝑈)(𝐾‘𝑆)𝐶))
77	25, 76	mpbid 224	. . . . . 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 25955	. . . . . 6 ⊢ (𝜑 → 𝐶(𝐾‘𝑆)𝑉)
80	1, 3, 24, 13, 17, 8, 7, 49, 77, 79	hltr 25961	. . . . 5 ⊢ (𝜑 → (𝑁‘𝑈)(𝐾‘𝑆)𝑉)
81	1, 3, 24, 13, 8, 49, 7	ishlg 25953	. . . . 5 ⊢ (𝜑 → ((𝑁‘𝑈)(𝐾‘𝑆)𝑉 ↔ ((𝑁‘𝑈) ≠ 𝑆 ∧ 𝑉 ≠ 𝑆 ∧ ((𝑁‘𝑈) ∈ (𝑆𝐼𝑉) ∨ 𝑉 ∈ (𝑆𝐼(𝑁‘𝑈))))))
82	80, 81	mpbid 224	. . . 4 ⊢ (𝜑 → ((𝑁‘𝑈) ≠ 𝑆 ∧ 𝑉 ≠ 𝑆 ∧ ((𝑁‘𝑈) ∈ (𝑆𝐼𝑉) ∨ 𝑉 ∈ (𝑆𝐼(𝑁‘𝑈)))))
83	82	simp1d 1133	. . 3 ⊢ (𝜑 → (𝑁‘𝑈) ≠ 𝑆)
84	82	simp2d 1134	. . 3 ⊢ (𝜑 → 𝑉 ≠ 𝑆)
85	82	simp3d 1135	. . 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 26096	. 2 ⊢ (𝜑 → 𝑉𝑂𝑈)
87	1, 2, 3, 4, 5, 6, 7, 8, 9, 86	oppcom 26092	1 ⊢ (𝜑 → 𝑈𝑂𝑉)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 386 ∨ wo 836 ∧ w3a 1071 = wceq 1601 ∈ wcel 2106 ≠ wne 2968 ∃wrex 3090 ∖ cdif 3788 class class class wbr 4886 {copab 4948 ran crn 5356 ‘cfv 6135 (class class class)co 6922 Basecbs 16255 distcds 16347 TarskiGcstrkg 25781 Itvcitv 25787 LineGclng 25788 ≤Gcleg 25933 hlGchlg 25951 pInvGcmir 26003 ⟂Gcperpg 26046

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2054 ax-8 2108 ax-9 2115 ax-10 2134 ax-11 2149 ax-12 2162 ax-13 2333 ax-ext 2753 ax-rep 5006 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 ax-cnex 10328 ax-resscn 10329 ax-1cn 10330 ax-icn 10331 ax-addcl 10332 ax-addrcl 10333 ax-mulcl 10334 ax-mulrcl 10335 ax-mulcom 10336 ax-addass 10337 ax-mulass 10338 ax-distr 10339 ax-i2m1 10340 ax-1ne0 10341 ax-1rid 10342 ax-rnegex 10343 ax-rrecex 10344 ax-cnre 10345 ax-pre-lttri 10346 ax-pre-lttrn 10347 ax-pre-ltadd 10348 ax-pre-mulgt0 10349

This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2550 df-eu 2586 df-clab 2763 df-cleq 2769 df-clel 2773 df-nfc 2920 df-ne 2969 df-nel 3075 df-ral 3094 df-rex 3095 df-reu 3096 df-rmo 3097 df-rab 3098 df-v 3399 df-sbc 3652 df-csb 3751 df-dif 3794 df-un 3796 df-in 3798 df-ss 3805 df-pss 3807 df-nul 4141 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-tp 4402 df-op 4404 df-uni 4672 df-int 4711 df-iun 4755 df-br 4887 df-opab 4949 df-mpt 4966 df-tr 4988 df-id 5261 df-eprel 5266 df-po 5274 df-so 5275 df-fr 5314 df-we 5316 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-pred 5933 df-ord 5979 df-on 5980 df-lim 5981 df-suc 5982 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-riota 6883 df-ov 6925 df-oprab 6926 df-mpt2 6927 df-om 7344 df-1st 7445 df-2nd 7446 df-wrecs 7689 df-recs 7751 df-rdg 7789 df-1o 7843 df-oadd 7847 df-er 8026 df-map 8142 df-pm 8143 df-en 8242 df-dom 8243 df-sdom 8244 df-fin 8245 df-card 9098 df-cda 9325 df-pnf 10413 df-mnf 10414 df-xr 10415 df-ltxr 10416 df-le 10417 df-sub 10608 df-neg 10609 df-nn 11375 df-2 11438 df-3 11439 df-n0 11643 df-xnn0 11715 df-z 11729 df-uz 11993 df-fz 12644 df-fzo 12785 df-hash 13436 df-word 13600 df-concat 13661 df-s1 13686 df-s2 13999 df-s3 14000 df-trkgc 25799 df-trkgb 25800 df-trkgcb 25801 df-trkg 25804 df-cgrg 25862 df-leg 25934 df-hlg 25952 df-mir 26004 df-rag 26045 df-perpg 26047

This theorem is referenced by: opphllem5 26099

W3C validator