Theorem opphllem4 29137

Description: Lemma for opphl 29141. (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 2760	. . . 4 ⊢ (pInvG‘𝐺) = (pInvG‘𝐺)
12		opphllem5.m	. . . 4 ⊢ (𝜑 → 𝑀 ∈ 𝑃)
13	1, 2, 3, 5, 11, 7, 12, 10, 9	mircl 29044	. . 3 ⊢ (𝜑 → (𝑁‘𝑈) ∈ 𝑃)
14		opphllem5.s	. . 3 ⊢ (𝜑 → 𝑆 ∈ 𝐷)
15	1, 5, 3, 7, 6, 14	tglnpt 28923	. . . . . 6 ⊢ (𝜑 → 𝑆 ∈ 𝑃)
16		opphllem5.r	. . . . . . 7 ⊢ (𝜑 → 𝑅 ∈ 𝐷)
17	1, 5, 3, 7, 6, 16	tglnpt 28923	. . . . . 6 ⊢ (𝜑 → 𝑅 ∈ 𝑃)
18		opphllem3.t	. . . . . . 7 ⊢ (𝜑 → 𝑅 ≠ 𝑆)
19	18	necomd 3010	. . . . . 6 ⊢ (𝜑 → 𝑆 ≠ 𝑅)
20	1, 2, 3, 5, 11, 7, 12, 10, 17	mirbtwn 29041	. . . . . . 7 ⊢ (𝜑 → 𝑀 ∈ ((𝑁‘𝑅)𝐼𝑅))
21		opphllem3.v	. . . . . . . 8 ⊢ (𝜑 → (𝑁‘𝑅) = 𝑆)
22	21	oveq1d 7431	. . . . . . 7 ⊢ (𝜑 → ((𝑁‘𝑅)𝐼𝑅) = (𝑆𝐼𝑅))
23	20, 22	eleqtrd 2862	. . . . . 6 ⊢ (𝜑 → 𝑀 ∈ (𝑆𝐼𝑅))
24	1, 3, 5, 7, 15, 17, 12, 19, 23	btwnlng1 28998	. . . . 5 ⊢ (𝜑 → 𝑀 ∈ (𝑆𝐿𝑅))
25	1, 3, 5, 7, 15, 17, 19, 19, 6, 14, 16	tglinethru 29015	. . . . 5 ⊢ (𝜑 → 𝐷 = (𝑆𝐿𝑅))
26	24, 25	eleqtrrd 2863	. . . 4 ⊢ (𝜑 → 𝑀 ∈ 𝐷)
27		opphllem5.a	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ 𝑃)
28		opphllem5.c	. . . . . . 7 ⊢ (𝜑 → 𝐶 ∈ 𝑃)
29		opphllem5.o	. . . . . . 7 ⊢ (𝜑 → 𝐴𝑂𝐶)
30	1, 2, 3, 4, 5, 6, 7, 27, 28, 29	oppne1 29128	. . . . . 6 ⊢ (𝜑 → ¬ 𝐴 ∈ 𝐷)
31		opphl.k	. . . . . . . . . . 11 ⊢ 𝐾 = (hlG‘𝐺)
32		opphllem4.1	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑈(𝐾‘𝑅)𝐴)
33	1, 3, 31, 9, 27, 17, 7, 32	hlne1 28982	. . . . . . . . . 10 ⊢ (𝜑 → 𝑈 ≠ 𝑅)
34	33	necomd 3010	. . . . . . . . 9 ⊢ (𝜑 → 𝑅 ≠ 𝑈)
35	1, 3, 31, 9, 27, 17, 7, 5, 32	hlln 28984	. . . . . . . . 9 ⊢ (𝜑 → 𝑈 ∈ (𝐴𝐿𝑅))
36	1, 3, 31, 9, 27, 17, 7, 32	hlne2 28983	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 ≠ 𝑅)
37	1, 3, 5, 7, 17, 9, 27, 34, 35, 36	lnrot1 29002	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ (𝑅𝐿𝑈))
38	37	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑈 ∈ 𝐷) → 𝐴 ∈ (𝑅𝐿𝑈))
39	7	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑈 ∈ 𝐷) → 𝐺 ∈ TarskiG)
40	17	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑈 ∈ 𝐷) → 𝑅 ∈ 𝑃)
41	9	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑈 ∈ 𝐷) → 𝑈 ∈ 𝑃)
42	34	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑈 ∈ 𝐷) → 𝑅 ≠ 𝑈)
43	6	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑈 ∈ 𝐷) → 𝐷 ∈ ran 𝐿)
44	16	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑈 ∈ 𝐷) → 𝑅 ∈ 𝐷)
45		simpr 490	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑈 ∈ 𝐷) → 𝑈 ∈ 𝐷)
46	1, 3, 5, 39, 40, 41, 42, 42, 43, 44, 45	tglinethru 29015	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑈 ∈ 𝐷) → 𝐷 = (𝑅𝐿𝑈))
47	38, 46	eleqtrrd 2863	. . . . . 6 ⊢ ((𝜑 ∧ 𝑈 ∈ 𝐷) → 𝐴 ∈ 𝐷)
48	30, 47	mtand 828	. . . . 5 ⊢ (𝜑 → ¬ 𝑈 ∈ 𝐷)
49	7	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑁‘𝑈) ∈ 𝐷) → 𝐺 ∈ TarskiG)
50	12	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑁‘𝑈) ∈ 𝐷) → 𝑀 ∈ 𝑃)
51	9	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑁‘𝑈) ∈ 𝐷) → 𝑈 ∈ 𝑃)
52	1, 2, 3, 5, 11, 49, 50, 10, 51	mirmir 29045	. . . . . 6 ⊢ ((𝜑 ∧ (𝑁‘𝑈) ∈ 𝐷) → (𝑁‘(𝑁‘𝑈)) = 𝑈)
53	6	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑁‘𝑈) ∈ 𝐷) → 𝐷 ∈ ran 𝐿)
54	26	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑁‘𝑈) ∈ 𝐷) → 𝑀 ∈ 𝐷)
55		simpr 490	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑁‘𝑈) ∈ 𝐷) → (𝑁‘𝑈) ∈ 𝐷)
56	1, 2, 3, 5, 11, 49, 10, 53, 54, 55	mirln 29059	. . . . . 6 ⊢ ((𝜑 ∧ (𝑁‘𝑈) ∈ 𝐷) → (𝑁‘(𝑁‘𝑈)) ∈ 𝐷)
57	52, 56	eqeltrrd 2861	. . . . 5 ⊢ ((𝜑 ∧ (𝑁‘𝑈) ∈ 𝐷) → 𝑈 ∈ 𝐷)
58	48, 57	mtand 828	. . . 4 ⊢ (𝜑 → ¬ (𝑁‘𝑈) ∈ 𝐷)
59	1, 2, 3, 5, 11, 7, 12, 10, 9	mirbtwn 29041	. . . 4 ⊢ (𝜑 → 𝑀 ∈ ((𝑁‘𝑈)𝐼𝑈))
60	1, 2, 3, 4, 13, 9, 26, 58, 48, 59	islnoppd 29127	. . 3 ⊢ (𝜑 → (𝑁‘𝑈)𝑂𝑈)
61		eqidd 2761	. . 3 ⊢ (𝜑 → (𝑁‘𝑈) = (𝑁‘𝑈))
62		opphllem5.p	. . . . . . . 8 ⊢ (𝜑 → 𝐷(⟂G‘𝐺)(𝐴𝐿𝑅))
63		opphllem5.q	. . . . . . . 8 ⊢ (𝜑 → 𝐷(⟂G‘𝐺)(𝐶𝐿𝑆))
64		opphllem3.l	. . . . . . . 8 ⊢ (𝜑 → (𝑆 − 𝐶)(≤G‘𝐺)(𝑅 − 𝐴))
65	1, 2, 3, 4, 5, 6, 7, 31, 10, 27, 28, 16, 14, 12, 29, 62, 63, 18, 64, 9, 21	opphllem3 29136	. . . . . . 7 ⊢ (𝜑 → (𝑈(𝐾‘𝑅)𝐴 ↔ (𝑁‘𝑈)(𝐾‘𝑆)𝐶))
66	32, 65	mpbid 235	. . . . . 6 ⊢ (𝜑 → (𝑁‘𝑈)(𝐾‘𝑆)𝐶)
67		opphllem4.2	. . . . . . 7 ⊢ (𝜑 → 𝑉(𝐾‘𝑆)𝐶)
68	1, 3, 31, 8, 28, 15, 7, 67	hlcomd 28981	. . . . . 6 ⊢ (𝜑 → 𝐶(𝐾‘𝑆)𝑉)
69	1, 3, 31, 13, 28, 8, 7, 15, 66, 68	hltr 28987	. . . . 5 ⊢ (𝜑 → (𝑁‘𝑈)(𝐾‘𝑆)𝑉)
70	1, 3, 31, 13, 8, 15, 7	ishlg 28979	. . . . 5 ⊢ (𝜑 → ((𝑁‘𝑈)(𝐾‘𝑆)𝑉 ↔ ((𝑁‘𝑈) ≠ 𝑆 ∧ 𝑉 ≠ 𝑆 ∧ ((𝑁‘𝑈) ∈ (𝑆𝐼𝑉) ∨ 𝑉 ∈ (𝑆𝐼(𝑁‘𝑈))))))
71	69, 70	mpbid 235	. . . 4 ⊢ (𝜑 → ((𝑁‘𝑈) ≠ 𝑆 ∧ 𝑉 ≠ 𝑆 ∧ ((𝑁‘𝑈) ∈ (𝑆𝐼𝑉) ∨ 𝑉 ∈ (𝑆𝐼(𝑁‘𝑈)))))
72	71	simp1d 1160	. . 3 ⊢ (𝜑 → (𝑁‘𝑈) ≠ 𝑆)
73	1, 3, 31, 28, 8, 15, 7, 68	hlne2 28983	. . 3 ⊢ (𝜑 → 𝑉 ≠ 𝑆)
74	71	simp3d 1162	. . 3 ⊢ (𝜑 → ((𝑁‘𝑈) ∈ (𝑆𝐼𝑉) ∨ 𝑉 ∈ (𝑆𝐼(𝑁‘𝑈))))
75	1, 2, 3, 4, 5, 6, 7, 10, 13, 8, 9, 14, 60, 26, 61, 72, 73, 74	opphllem2 29135	. 2 ⊢ (𝜑 → 𝑉𝑂𝑈)
76	1, 2, 3, 4, 5, 6, 7, 8, 9, 75	oppcom 29131	1 ⊢ (𝜑 → 𝑈𝑂𝑉)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∨ wo 861   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145   ≠ wne 2955  ∃wrex 3086   ∖ cdif 3896   class class class wbr 5103  {copab 5167  ran crn 5656  ‘cfv 6535  (class class class)co 7416  Basecbs 17326  distcds 17376  TarskiGcstrkg 28800  Itvcitv 28806  LineGclng 28807  ≤Gcleg 28956  hlGchlg 28974  pInvGcmir 29035  ⟂Gcperpg 29081

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-rep 5232  ax-sep 5251  ax-nul 5263  ax-pow 5330  ax-pr 5398  ax-un 7742  ax-cnex 11205  ax-resscn 11206  ax-1cn 11207  ax-icn 11208  ax-addcl 11209  ax-addrcl 11210  ax-mulcl 11211  ax-mulrcl 11212  ax-mulcom 11213  ax-addass 11214  ax-mulass 11215  ax-distr 11216  ax-i2m1 11217  ax-1ne0 11218  ax-1rid 11219  ax-rnegex 11220  ax-rrecex 11221  ax-cnre 11222  ax-pre-lttri 11223  ax-pre-lttrn 11224  ax-pre-ltadd 11225  ax-pre-mulgt0 11226

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-nel 3062  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-tp 4589  df-op 4591  df-uni 4868  df-int 4908  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6301  df-ord 6362  df-on 6363  df-lim 6364  df-suc 6365  df-iota 6491  df-fun 6537  df-fn 6538  df-f 6539  df-f1 6540  df-fo 6541  df-f1o 6542  df-fv 6543  df-riota 7373  df-ov 7419  df-oprab 7420  df-mpo 7421  df-om 7869  df-1st 7992  df-2nd 7993  df-frecs 8285  df-wrecs 8316  df-recs 8365  df-rdg 8404  df-1o 8462  df-oadd 8466  df-er 8703  df-map 8835  df-pm 8836  df-en 8960  df-dom 8961  df-sdom 8962  df-fin 8963  df-dju 9931  df-card 9969  df-pnf 11294  df-mnf 11295  df-xr 11296  df-ltxr 11297  df-le 11298  df-sub 11492  df-neg 11493  df-nn 12283  df-2 12352  df-3 12353  df-n0 12554  df-xnn0 12627  df-z 12641  df-uz 12913  df-fz 13587  df-fzo 13735  df-hash 14420  df-word 14604  df-concat 14661  df-s1 14688  df-s2 14944  df-s3 14945  df-trkgc 28821  df-trkgb 28822  df-trkgcb 28823  df-trkg 28826  df-cgrg 28885  df-leg 28957  df-hlg 28975  df-mir 29036  df-rag 29080  df-perpg 29082

This theorem is used by:  opphllem5  29138