Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  cdlemg4b2 Structured version   Unicode version

Theorem cdlemg4b2 35283
Description: TODO: FIX COMMENT (Contributed by NM, 24-Apr-2013.)
Hypotheses
Ref Expression
cdlemg4.l  |-  .<_  =  ( le `  K )
cdlemg4.a  |-  A  =  ( Atoms `  K )
cdlemg4.h  |-  H  =  ( LHyp `  K
)
cdlemg4.t  |-  T  =  ( ( LTrn `  K
) `  W )
cdlemg4.r  |-  R  =  ( ( trL `  K
) `  W )
cdlemg4.j  |-  .\/  =  ( join `  K )
cdlemg4b.v  |-  V  =  ( R `  G
)
Assertion
Ref Expression
cdlemg4b2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  G  e.  T )  ->  (
( G `  P
)  .\/  V )  =  ( P  .\/  ( G `  P ) ) )

Proof of Theorem cdlemg4b2
StepHypRef Expression
1 cdlemg4b.v . . . 4  |-  V  =  ( R `  G
)
2 cdlemg4.l . . . . . 6  |-  .<_  =  ( le `  K )
3 cdlemg4.j . . . . . 6  |-  .\/  =  ( join `  K )
4 eqid 2462 . . . . . 6  |-  ( meet `  K )  =  (
meet `  K )
5 cdlemg4.a . . . . . 6  |-  A  =  ( Atoms `  K )
6 cdlemg4.h . . . . . 6  |-  H  =  ( LHyp `  K
)
7 cdlemg4.t . . . . . 6  |-  T  =  ( ( LTrn `  K
) `  W )
8 cdlemg4.r . . . . . 6  |-  R  =  ( ( trL `  K
) `  W )
92, 3, 4, 5, 6, 7, 8trlval2 34836 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  G  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( R `  G )  =  ( ( P  .\/  ( G `  P )
) ( meet `  K
) W ) )
1093com23 1197 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  G  e.  T )  ->  ( R `  G )  =  ( ( P 
.\/  ( G `  P ) ) (
meet `  K ) W ) )
111, 10syl5eq 2515 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  G  e.  T )  ->  V  =  ( ( P 
.\/  ( G `  P ) ) (
meet `  K ) W ) )
1211oveq2d 6293 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  G  e.  T )  ->  (
( G `  P
)  .\/  V )  =  ( ( G `
 P )  .\/  ( ( P  .\/  ( G `  P ) ) ( meet `  K
) W ) ) )
13 simp1 991 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  G  e.  T )  ->  ( K  e.  HL  /\  W  e.  H ) )
14 simp2l 1017 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  G  e.  T )  ->  P  e.  A )
152, 5, 6, 7ltrnel 34812 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  G  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( ( G `  P )  e.  A  /\  -.  ( G `  P )  .<_  W ) )
16153com23 1197 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  G  e.  T )  ->  (
( G `  P
)  e.  A  /\  -.  ( G `  P
)  .<_  W ) )
17 eqid 2462 . . . 4  |-  ( ( P  .\/  ( G `
 P ) ) ( meet `  K
) W )  =  ( ( P  .\/  ( G `  P ) ) ( meet `  K
) W )
182, 3, 4, 5, 6, 17cdleme0cq 34888 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  ( ( G `  P )  e.  A  /\  -.  ( G `  P ) 
.<_  W ) ) )  ->  ( ( G `
 P )  .\/  ( ( P  .\/  ( G `  P ) ) ( meet `  K
) W ) )  =  ( P  .\/  ( G `  P ) ) )
1913, 14, 16, 18syl12anc 1221 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  G  e.  T )  ->  (
( G `  P
)  .\/  ( ( P  .\/  ( G `  P ) ) (
meet `  K ) W ) )  =  ( P  .\/  ( G `  P )
) )
2012, 19eqtrd 2503 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  G  e.  T )  ->  (
( G `  P
)  .\/  V )  =  ( P  .\/  ( G `  P ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    /\ w3a 968    = wceq 1374    e. wcel 1762   class class class wbr 4442   ` cfv 5581  (class class class)co 6277   lecple 14553   joincjn 15422   meetcmee 15423   Atomscatm 33937   HLchlt 34024   LHypclh 34657   LTrncltrn 34774   trLctrl 34831
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-rep 4553  ax-sep 4563  ax-nul 4571  ax-pow 4620  ax-pr 4681  ax-un 6569
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-ral 2814  df-rex 2815  df-reu 2816  df-rab 2818  df-v 3110  df-sbc 3327  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3781  df-if 3935  df-pw 4007  df-sn 4023  df-pr 4025  df-op 4029  df-uni 4241  df-iun 4322  df-iin 4323  df-br 4443  df-opab 4501  df-mpt 4502  df-id 4790  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-rn 5005  df-res 5006  df-ima 5007  df-iota 5544  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-riota 6238  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-1st 6776  df-2nd 6777  df-map 7414  df-poset 15424  df-plt 15436  df-lub 15452  df-glb 15453  df-join 15454  df-meet 15455  df-p0 15517  df-p1 15518  df-lat 15524  df-clat 15586  df-oposet 33850  df-ol 33852  df-oml 33853  df-covers 33940  df-ats 33941  df-atl 33972  df-cvlat 33996  df-hlat 34025  df-psubsp 34176  df-pmap 34177  df-padd 34469  df-lhyp 34661  df-laut 34662  df-ldil 34777  df-ltrn 34778  df-trl 34832
This theorem is referenced by:  cdlemg4b12  35284  cdlemg4c  35285
  Copyright terms: Public domain W3C validator