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Theorem cdleme50trn3 36702
Description: Part of proof that 𝐹 is a translation. 𝑃 = 𝑄 case. TODO: fix comment. (Contributed by NM, 10-Apr-2013.)
Hypotheses
Ref Expression
cdlemef50.b 𝐵 = (Base‘𝐾)
cdlemef50.l = (le‘𝐾)
cdlemef50.j = (join‘𝐾)
cdlemef50.m = (meet‘𝐾)
cdlemef50.a 𝐴 = (Atoms‘𝐾)
cdlemef50.h 𝐻 = (LHyp‘𝐾)
cdlemef50.u 𝑈 = ((𝑃 𝑄) 𝑊)
cdlemef50.d 𝐷 = ((𝑡 𝑈) (𝑄 ((𝑃 𝑡) 𝑊)))
cdlemefs50.e 𝐸 = ((𝑃 𝑄) (𝐷 ((𝑠 𝑡) 𝑊)))
cdlemef50.f 𝐹 = (𝑥𝐵 ↦ if((𝑃𝑄 ∧ ¬ 𝑥 𝑊), (𝑧𝐵𝑠𝐴 ((¬ 𝑠 𝑊 ∧ (𝑠 (𝑥 𝑊)) = 𝑥) → 𝑧 = (if(𝑠 (𝑃 𝑄), (𝑦𝐵𝑡𝐴 ((¬ 𝑡 𝑊 ∧ ¬ 𝑡 (𝑃 𝑄)) → 𝑦 = 𝐸)), 𝑠 / 𝑡𝐷) (𝑥 𝑊)))), 𝑥))
Assertion
Ref Expression
cdleme50trn3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃 = 𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊))) → ((𝑅 (𝐹𝑅)) 𝑊) = 𝑈)
Distinct variable groups:   𝑡,𝑠,𝑥,𝑦,𝑧,   ,𝑠,𝑡,𝑥,𝑦,𝑧   ,𝑠,𝑡,𝑥,𝑦,𝑧   𝐴,𝑠,𝑡,𝑥,𝑦,𝑧   𝐵,𝑠,𝑡,𝑥,𝑦,𝑧   𝐷,𝑠,𝑥,𝑦,𝑧   𝑥,𝐸,𝑦,𝑧   𝐻,𝑠,𝑡,𝑥,𝑦,𝑧   𝐾,𝑠,𝑡,𝑥,𝑦,𝑧   𝑃,𝑠,𝑡,𝑥,𝑦,𝑧   𝑄,𝑠,𝑡,𝑥,𝑦,𝑧   𝑅,𝑠,𝑡,𝑥,𝑦,𝑧   𝑈,𝑠,𝑡,𝑥,𝑦,𝑧   𝑊,𝑠,𝑡,𝑥,𝑦,𝑧
Allowed substitution hints:   𝐷(𝑡)   𝐸(𝑡,𝑠)   𝐹(𝑥,𝑦,𝑧,𝑡,𝑠)

Proof of Theorem cdleme50trn3
StepHypRef Expression
1 simpl1 1199 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃 = 𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊))) → (𝐾 ∈ HL ∧ 𝑊𝐻))
2 simprr 763 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃 = 𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊))) → (𝑅𝐴 ∧ ¬ 𝑅 𝑊))
3 cdlemef50.l . . . . . 6 = (le‘𝐾)
4 cdlemef50.m . . . . . 6 = (meet‘𝐾)
5 eqid 2777 . . . . . 6 (0.‘𝐾) = (0.‘𝐾)
6 cdlemef50.a . . . . . 6 𝐴 = (Atoms‘𝐾)
7 cdlemef50.h . . . . . 6 𝐻 = (LHyp‘𝐾)
83, 4, 5, 6, 7lhpmat 36179 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) → (𝑅 𝑊) = (0.‘𝐾))
91, 2, 8syl2anc 579 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃 = 𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊))) → (𝑅 𝑊) = (0.‘𝐾))
10 simprrl 771 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃 = 𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊))) → 𝑅𝐴)
11 cdlemef50.b . . . . . . . . . 10 𝐵 = (Base‘𝐾)
1211, 6atbase 35438 . . . . . . . . 9 (𝑅𝐴𝑅𝐵)
1310, 12syl 17 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃 = 𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊))) → 𝑅𝐵)
14 simprl 761 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃 = 𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊))) → 𝑃 = 𝑄)
15 cdlemef50.f . . . . . . . . 9 𝐹 = (𝑥𝐵 ↦ if((𝑃𝑄 ∧ ¬ 𝑥 𝑊), (𝑧𝐵𝑠𝐴 ((¬ 𝑠 𝑊 ∧ (𝑠 (𝑥 𝑊)) = 𝑥) → 𝑧 = (if(𝑠 (𝑃 𝑄), (𝑦𝐵𝑡𝐴 ((¬ 𝑡 𝑊 ∧ ¬ 𝑡 (𝑃 𝑄)) → 𝑦 = 𝐸)), 𝑠 / 𝑡𝐷) (𝑥 𝑊)))), 𝑥))
1615cdleme31id 36543 . . . . . . . 8 ((𝑅𝐵𝑃 = 𝑄) → (𝐹𝑅) = 𝑅)
1713, 14, 16syl2anc 579 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃 = 𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊))) → (𝐹𝑅) = 𝑅)
1817oveq2d 6938 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃 = 𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊))) → (𝑅 (𝐹𝑅)) = (𝑅 𝑅))
19 simpl1l 1250 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃 = 𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊))) → 𝐾 ∈ HL)
20 cdlemef50.j . . . . . . . 8 = (join‘𝐾)
2120, 6hlatjidm 35518 . . . . . . 7 ((𝐾 ∈ HL ∧ 𝑅𝐴) → (𝑅 𝑅) = 𝑅)
2219, 10, 21syl2anc 579 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃 = 𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊))) → (𝑅 𝑅) = 𝑅)
2318, 22eqtrd 2813 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃 = 𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊))) → (𝑅 (𝐹𝑅)) = 𝑅)
2423oveq1d 6937 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃 = 𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊))) → ((𝑅 (𝐹𝑅)) 𝑊) = (𝑅 𝑊))
25 simpl2 1201 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃 = 𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊))) → (𝑃𝐴 ∧ ¬ 𝑃 𝑊))
263, 4, 5, 6, 7lhpmat 36179 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → (𝑃 𝑊) = (0.‘𝐾))
271, 25, 26syl2anc 579 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃 = 𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊))) → (𝑃 𝑊) = (0.‘𝐾))
289, 24, 273eqtr4d 2823 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃 = 𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊))) → ((𝑅 (𝐹𝑅)) 𝑊) = (𝑃 𝑊))
29 simpl2l 1254 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃 = 𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊))) → 𝑃𝐴)
3020, 6hlatjidm 35518 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑃𝐴) → (𝑃 𝑃) = 𝑃)
3119, 29, 30syl2anc 579 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃 = 𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊))) → (𝑃 𝑃) = 𝑃)
3214oveq2d 6938 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃 = 𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊))) → (𝑃 𝑃) = (𝑃 𝑄))
3331, 32eqtr3d 2815 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃 = 𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊))) → 𝑃 = (𝑃 𝑄))
3433oveq1d 6937 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃 = 𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊))) → (𝑃 𝑊) = ((𝑃 𝑄) 𝑊))
3528, 34eqtrd 2813 . 2 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃 = 𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊))) → ((𝑅 (𝐹𝑅)) 𝑊) = ((𝑃 𝑄) 𝑊))
36 cdlemef50.u . 2 𝑈 = ((𝑃 𝑄) 𝑊)
3735, 36syl6eqr 2831 1 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃 = 𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊))) → ((𝑅 (𝐹𝑅)) 𝑊) = 𝑈)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 386  w3a 1071   = wceq 1601  wcel 2106  wne 2968  wral 3089  csb 3750  ifcif 4306   class class class wbr 4886  cmpt 4965  cfv 6135  crio 6882  (class class class)co 6922  Basecbs 16255  lecple 16345  joincjn 17330  meetcmee 17331  0.cp0 17423  Atomscatm 35412  HLchlt 35499  LHypclh 36133
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
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  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-ral 3094  df-rex 3095  df-reu 3096  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-nul 4141  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4672  df-iun 4755  df-br 4887  df-opab 4949  df-mpt 4966  df-id 5261  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-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-proset 17314  df-poset 17332  df-plt 17344  df-lub 17360  df-glb 17361  df-join 17362  df-meet 17363  df-p0 17425  df-lat 17432  df-covers 35415  df-ats 35416  df-atl 35447  df-cvlat 35471  df-hlat 35500  df-lhyp 36137
This theorem is referenced by:  cdleme50trn123  36703
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