Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  cdleme50trn3 Structured version   Visualization version   GIF version

Theorem cdleme50trn3 36362
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 1227 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃 = 𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊))) → (𝐾 ∈ HL ∧ 𝑊𝐻))
2 simprr 756 . . . . 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 2771 . . . . . 6 (0.‘𝐾) = (0.‘𝐾)
6 cdlemef50.a . . . . . 6 𝐴 = (Atoms‘𝐾)
7 cdlemef50.h . . . . . 6 𝐻 = (LHyp‘𝐾)
83, 4, 5, 6, 7lhpmat 35838 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) → (𝑅 𝑊) = (0.‘𝐾))
91, 2, 8syl2anc 573 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃 = 𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊))) → (𝑅 𝑊) = (0.‘𝐾))
10 simprrl 766 . . . . . . . . 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 35098 . . . . . . . . 9 (𝑅𝐴𝑅𝐵)
1310, 12syl 17 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃 = 𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊))) → 𝑅𝐵)
14 simprl 754 . . . . . . . 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 36203 . . . . . . . 8 ((𝑅𝐵𝑃 = 𝑄) → (𝐹𝑅) = 𝑅)
1713, 14, 16syl2anc 573 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃 = 𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊))) → (𝐹𝑅) = 𝑅)
1817oveq2d 6809 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃 = 𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊))) → (𝑅 (𝐹𝑅)) = (𝑅 𝑅))
19 simpl1l 1278 . . . . . . 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 35177 . . . . . . 7 ((𝐾 ∈ HL ∧ 𝑅𝐴) → (𝑅 𝑅) = 𝑅)
2219, 10, 21syl2anc 573 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃 = 𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊))) → (𝑅 𝑅) = 𝑅)
2318, 22eqtrd 2805 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃 = 𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊))) → (𝑅 (𝐹𝑅)) = 𝑅)
2423oveq1d 6808 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃 = 𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊))) → ((𝑅 (𝐹𝑅)) 𝑊) = (𝑅 𝑊))
25 simpl2 1229 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃 = 𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊))) → (𝑃𝐴 ∧ ¬ 𝑃 𝑊))
263, 4, 5, 6, 7lhpmat 35838 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → (𝑃 𝑊) = (0.‘𝐾))
271, 25, 26syl2anc 573 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃 = 𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊))) → (𝑃 𝑊) = (0.‘𝐾))
289, 24, 273eqtr4d 2815 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃 = 𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊))) → ((𝑅 (𝐹𝑅)) 𝑊) = (𝑃 𝑊))
29 simpl2l 1282 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃 = 𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊))) → 𝑃𝐴)
3020, 6hlatjidm 35177 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑃𝐴) → (𝑃 𝑃) = 𝑃)
3119, 29, 30syl2anc 573 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃 = 𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊))) → (𝑃 𝑃) = 𝑃)
3214oveq2d 6809 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃 = 𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊))) → (𝑃 𝑃) = (𝑃 𝑄))
3331, 32eqtr3d 2807 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃 = 𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊))) → 𝑃 = (𝑃 𝑄))
3433oveq1d 6808 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃 = 𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊))) → (𝑃 𝑊) = ((𝑃 𝑄) 𝑊))
3528, 34eqtrd 2805 . 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 2823 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 382  w3a 1071   = wceq 1631  wcel 2145  wne 2943  wral 3061  csb 3682  ifcif 4225   class class class wbr 4786  cmpt 4863  cfv 6031  crio 6753  (class class class)co 6793  Basecbs 16064  lecple 16156  joincjn 17152  meetcmee 17153  0.cp0 17245  Atomscatm 35072  HLchlt 35159  LHypclh 35792
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-riota 6754  df-ov 6796  df-oprab 6797  df-preset 17136  df-poset 17154  df-plt 17166  df-lub 17182  df-glb 17183  df-join 17184  df-meet 17185  df-p0 17247  df-lat 17254  df-covers 35075  df-ats 35076  df-atl 35107  df-cvlat 35131  df-hlat 35160  df-lhyp 35796
This theorem is referenced by:  cdleme50trn123  36363
  Copyright terms: Public domain W3C validator