Theorem opphllem4 28776

Description: Lemma for opphl 28780. (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 2740	. . . 4 ⊢ (pInvG‘𝐺) = (pInvG‘𝐺)
12		opphllem5.m	. . . 4 ⊢ (𝜑 → 𝑀 ∈ 𝑃)
13	1, 2, 3, 5, 11, 7, 12, 10, 9	mircl 28687	. . 3 ⊢ (𝜑 → (𝑁‘𝑈) ∈ 𝑃)
14		opphllem5.s	. . 3 ⊢ (𝜑 → 𝑆 ∈ 𝐷)
15	1, 5, 3, 7, 6, 14	tglnpt 28575	. . . . . 6 ⊢ (𝜑 → 𝑆 ∈ 𝑃)
16		opphllem5.r	. . . . . . 7 ⊢ (𝜑 → 𝑅 ∈ 𝐷)
17	1, 5, 3, 7, 6, 16	tglnpt 28575	. . . . . 6 ⊢ (𝜑 → 𝑅 ∈ 𝑃)
18		opphllem3.t	. . . . . . 7 ⊢ (𝜑 → 𝑅 ≠ 𝑆)
19	18	necomd 3002	. . . . . 6 ⊢ (𝜑 → 𝑆 ≠ 𝑅)
20	1, 2, 3, 5, 11, 7, 12, 10, 17	mirbtwn 28684	. . . . . . 7 ⊢ (𝜑 → 𝑀 ∈ ((𝑁‘𝑅)𝐼𝑅))
21		opphllem3.v	. . . . . . . 8 ⊢ (𝜑 → (𝑁‘𝑅) = 𝑆)
22	21	oveq1d 7463	. . . . . . 7 ⊢ (𝜑 → ((𝑁‘𝑅)𝐼𝑅) = (𝑆𝐼𝑅))
23	20, 22	eleqtrd 2846	. . . . . 6 ⊢ (𝜑 → 𝑀 ∈ (𝑆𝐼𝑅))
24	1, 3, 5, 7, 15, 17, 12, 19, 23	btwnlng1 28645	. . . . 5 ⊢ (𝜑 → 𝑀 ∈ (𝑆𝐿𝑅))
25	1, 3, 5, 7, 15, 17, 19, 19, 6, 14, 16	tglinethru 28662	. . . . 5 ⊢ (𝜑 → 𝐷 = (𝑆𝐿𝑅))
26	24, 25	eleqtrrd 2847	. . . 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 28767	. . . . . 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 28631	. . . . . . . . . 10 ⊢ (𝜑 → 𝑈 ≠ 𝑅)
34	33	necomd 3002	. . . . . . . . 9 ⊢ (𝜑 → 𝑅 ≠ 𝑈)
35	1, 3, 31, 9, 27, 17, 7, 5, 32	hlln 28633	. . . . . . . . 9 ⊢ (𝜑 → 𝑈 ∈ (𝐴𝐿𝑅))
36	1, 3, 31, 9, 27, 17, 7, 32	hlne2 28632	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 ≠ 𝑅)
37	1, 3, 5, 7, 17, 9, 27, 34, 35, 36	lnrot1 28649	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ (𝑅𝐿𝑈))
38	37	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑈 ∈ 𝐷) → 𝐴 ∈ (𝑅𝐿𝑈))
39	7	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑈 ∈ 𝐷) → 𝐺 ∈ TarskiG)
40	17	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑈 ∈ 𝐷) → 𝑅 ∈ 𝑃)
41	9	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑈 ∈ 𝐷) → 𝑈 ∈ 𝑃)
42	34	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑈 ∈ 𝐷) → 𝑅 ≠ 𝑈)
43	6	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑈 ∈ 𝐷) → 𝐷 ∈ ran 𝐿)
44	16	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑈 ∈ 𝐷) → 𝑅 ∈ 𝐷)
45		simpr 484	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑈 ∈ 𝐷) → 𝑈 ∈ 𝐷)
46	1, 3, 5, 39, 40, 41, 42, 42, 43, 44, 45	tglinethru 28662	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑈 ∈ 𝐷) → 𝐷 = (𝑅𝐿𝑈))
47	38, 46	eleqtrrd 2847	. . . . . 6 ⊢ ((𝜑 ∧ 𝑈 ∈ 𝐷) → 𝐴 ∈ 𝐷)
48	30, 47	mtand 815	. . . . 5 ⊢ (𝜑 → ¬ 𝑈 ∈ 𝐷)
49	7	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑁‘𝑈) ∈ 𝐷) → 𝐺 ∈ TarskiG)
50	12	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑁‘𝑈) ∈ 𝐷) → 𝑀 ∈ 𝑃)
51	9	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑁‘𝑈) ∈ 𝐷) → 𝑈 ∈ 𝑃)
52	1, 2, 3, 5, 11, 49, 50, 10, 51	mirmir 28688	. . . . . 6 ⊢ ((𝜑 ∧ (𝑁‘𝑈) ∈ 𝐷) → (𝑁‘(𝑁‘𝑈)) = 𝑈)
53	6	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑁‘𝑈) ∈ 𝐷) → 𝐷 ∈ ran 𝐿)
54	26	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑁‘𝑈) ∈ 𝐷) → 𝑀 ∈ 𝐷)
55		simpr 484	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑁‘𝑈) ∈ 𝐷) → (𝑁‘𝑈) ∈ 𝐷)
56	1, 2, 3, 5, 11, 49, 10, 53, 54, 55	mirln 28702	. . . . . 6 ⊢ ((𝜑 ∧ (𝑁‘𝑈) ∈ 𝐷) → (𝑁‘(𝑁‘𝑈)) ∈ 𝐷)
57	52, 56	eqeltrrd 2845	. . . . 5 ⊢ ((𝜑 ∧ (𝑁‘𝑈) ∈ 𝐷) → 𝑈 ∈ 𝐷)
58	48, 57	mtand 815	. . . 4 ⊢ (𝜑 → ¬ (𝑁‘𝑈) ∈ 𝐷)
59	1, 2, 3, 5, 11, 7, 12, 10, 9	mirbtwn 28684	. . . 4 ⊢ (𝜑 → 𝑀 ∈ ((𝑁‘𝑈)𝐼𝑈))
60	1, 2, 3, 4, 13, 9, 26, 58, 48, 59	islnoppd 28766	. . 3 ⊢ (𝜑 → (𝑁‘𝑈)𝑂𝑈)
61		eqidd 2741	. . 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 28775	. . . . . . 7 ⊢ (𝜑 → (𝑈(𝐾‘𝑅)𝐴 ↔ (𝑁‘𝑈)(𝐾‘𝑆)𝐶))
66	32, 65	mpbid 232	. . . . . 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 28630	. . . . . 6 ⊢ (𝜑 → 𝐶(𝐾‘𝑆)𝑉)
69	1, 3, 31, 13, 28, 8, 7, 15, 66, 68	hltr 28636	. . . . 5 ⊢ (𝜑 → (𝑁‘𝑈)(𝐾‘𝑆)𝑉)
70	1, 3, 31, 13, 8, 15, 7	ishlg 28628	. . . . 5 ⊢ (𝜑 → ((𝑁‘𝑈)(𝐾‘𝑆)𝑉 ↔ ((𝑁‘𝑈) ≠ 𝑆 ∧ 𝑉 ≠ 𝑆 ∧ ((𝑁‘𝑈) ∈ (𝑆𝐼𝑉) ∨ 𝑉 ∈ (𝑆𝐼(𝑁‘𝑈))))))
71	69, 70	mpbid 232	. . . 4 ⊢ (𝜑 → ((𝑁‘𝑈) ≠ 𝑆 ∧ 𝑉 ≠ 𝑆 ∧ ((𝑁‘𝑈) ∈ (𝑆𝐼𝑉) ∨ 𝑉 ∈ (𝑆𝐼(𝑁‘𝑈)))))
72	71	simp1d 1142	. . 3 ⊢ (𝜑 → (𝑁‘𝑈) ≠ 𝑆)
73	1, 3, 31, 28, 8, 15, 7, 68	hlne2 28632	. . 3 ⊢ (𝜑 → 𝑉 ≠ 𝑆)
74	71	simp3d 1144	. . 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 28774	. 2 ⊢ (𝜑 → 𝑉𝑂𝑈)
76	1, 2, 3, 4, 5, 6, 7, 8, 9, 75	oppcom 28770	1 ⊢ (𝜑 → 𝑈𝑂𝑉)

Syntax hints:   → wi 4   ∧ wa 395   ∨ wo 846   ∧ w3a 1087   = wceq 1537   ∈ wcel 2108   ≠ wne 2946  ∃wrex 3076   ∖ cdif 3973   class class class wbr 5166  {copab 5228  ran crn 5701  ‘cfv 6573  (class class class)co 7448  Basecbs 17258  distcds 17320  TarskiGcstrkg 28453  Itvcitv 28459  LineGclng 28460  ≤Gcleg 28608  hlGchlg 28626  pInvGcmir 28678  ⟂Gcperpg 28721

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770  ax-cnex 11240  ax-resscn 11241  ax-1cn 11242  ax-icn 11243  ax-addcl 11244  ax-addrcl 11245  ax-mulcl 11246  ax-mulrcl 11247  ax-mulcom 11248  ax-addass 11249  ax-mulass 11250  ax-distr 11251  ax-i2m1 11252  ax-1ne0 11253  ax-1rid 11254  ax-rnegex 11255  ax-rrecex 11256  ax-cnre 11257  ax-pre-lttri 11258  ax-pre-lttrn 11259  ax-pre-ltadd 11260  ax-pre-mulgt0 11261

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-nel 3053  df-ral 3068  df-rex 3077  df-rmo 3388  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-tp 4653  df-op 4655  df-uni 4932  df-int 4971  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-riota 7404  df-ov 7451  df-oprab 7452  df-mpo 7453  df-om 7904  df-1st 8030  df-2nd 8031  df-frecs 8322  df-wrecs 8353  df-recs 8427  df-rdg 8466  df-1o 8522  df-oadd 8526  df-er 8763  df-map 8886  df-pm 8887  df-en 9004  df-dom 9005  df-sdom 9006  df-fin 9007  df-dju 9970  df-card 10008  df-pnf 11326  df-mnf 11327  df-xr 11328  df-ltxr 11329  df-le 11330  df-sub 11522  df-neg 11523  df-nn 12294  df-2 12356  df-3 12357  df-n0 12554  df-xnn0 12626  df-z 12640  df-uz 12904  df-fz 13568  df-fzo 13712  df-hash 14380  df-word 14563  df-concat 14619  df-s1 14644  df-s2 14897  df-s3 14898  df-trkgc 28474  df-trkgb 28475  df-trkgcb 28476  df-trkg 28479  df-cgrg 28537  df-leg 28609  df-hlg 28627  df-mir 28679  df-rag 28720  df-perpg 28722

This theorem is referenced by:  opphllem5  28777