Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  cdleme9tN Unicode version

Theorem cdleme9tN 30785
Description: Part of proof of Lemma E in [Crawley] p. 113, 2nd paragraph on p. 114.  X and  F represent t1 and f(t) respectively. In their notation, we prove f(t)  \/ t1 = q  \/ t1. (Contributed by NM, 8-Oct-2012.) (New usage is discouraged.)
Hypotheses
Ref Expression
cdleme9t.l  |-  .<_  =  ( le `  K )
cdleme9t.j  |-  .\/  =  ( join `  K )
cdleme9t.m  |-  ./\  =  ( meet `  K )
cdleme9t.a  |-  A  =  ( Atoms `  K )
cdleme9t.h  |-  H  =  ( LHyp `  K
)
cdleme9t.u  |-  U  =  ( ( P  .\/  Q )  ./\  W )
cdleme9t.g  |-  F  =  ( ( T  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  T )  ./\  W )
) )
cdleme9t.x  |-  X  =  ( ( P  .\/  T )  ./\  W )
Assertion
Ref Expression
cdleme9tN  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A  /\  ( T  e.  A  /\  -.  T  .<_  W ) )  /\  -.  T  .<_  ( P  .\/  Q
) )  ->  ( F  .\/  X )  =  ( Q  .\/  X
) )

Proof of Theorem cdleme9tN
StepHypRef Expression
1 cdleme9t.l . 2  |-  .<_  =  ( le `  K )
2 cdleme9t.j . 2  |-  .\/  =  ( join `  K )
3 cdleme9t.m . 2  |-  ./\  =  ( meet `  K )
4 cdleme9t.a . 2  |-  A  =  ( Atoms `  K )
5 cdleme9t.h . 2  |-  H  =  ( LHyp `  K
)
6 cdleme9t.u . 2  |-  U  =  ( ( P  .\/  Q )  ./\  W )
7 cdleme9t.g . 2  |-  F  =  ( ( T  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  T )  ./\  W )
) )
8 cdleme9t.x . 2  |-  X  =  ( ( P  .\/  T )  ./\  W )
91, 2, 3, 4, 5, 6, 7, 8cdleme9 30781 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A  /\  ( T  e.  A  /\  -.  T  .<_  W ) )  /\  -.  T  .<_  ( P  .\/  Q
) )  ->  ( F  .\/  X )  =  ( Q  .\/  X
) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   class class class wbr 4199   ` cfv 5440  (class class class)co 6067   lecple 13519   joincjn 14384   meetcmee 14385   Atomscatm 29792   HLchlt 29879   LHypclh 30512
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2411  ax-rep 4307  ax-sep 4317  ax-nul 4325  ax-pow 4364  ax-pr 4390  ax-un 4687
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2417  df-cleq 2423  df-clel 2426  df-nfc 2555  df-ne 2595  df-nel 2596  df-ral 2697  df-rex 2698  df-reu 2699  df-rab 2701  df-v 2945  df-sbc 3149  df-csb 3239  df-dif 3310  df-un 3312  df-in 3314  df-ss 3321  df-nul 3616  df-if 3727  df-pw 3788  df-sn 3807  df-pr 3808  df-op 3810  df-uni 4003  df-iun 4082  df-iin 4083  df-br 4200  df-opab 4254  df-mpt 4255  df-id 4485  df-xp 4870  df-rel 4871  df-cnv 4872  df-co 4873  df-dm 4874  df-rn 4875  df-res 4876  df-ima 4877  df-iota 5404  df-fun 5442  df-fn 5443  df-f 5444  df-f1 5445  df-fo 5446  df-f1o 5447  df-fv 5448  df-ov 6070  df-oprab 6071  df-mpt2 6072  df-1st 6335  df-2nd 6336  df-undef 6529  df-riota 6535  df-poset 14386  df-plt 14398  df-lub 14414  df-glb 14415  df-join 14416  df-meet 14417  df-p0 14451  df-p1 14452  df-lat 14458  df-clat 14520  df-oposet 29705  df-ol 29707  df-oml 29708  df-covers 29795  df-ats 29796  df-atl 29827  df-cvlat 29851  df-hlat 29880  df-psubsp 30031  df-pmap 30032  df-padd 30324  df-lhyp 30516
  Copyright terms: Public domain W3C validator