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Theorem cdlemg4f 31109
Description: TODO: FIX COMMENT (Contributed by NM, 25-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
)
cdlemg4.m  |-  ./\  =  ( meet `  K )
Assertion
Ref Expression
cdlemg4f  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  F  e.  T
)  /\  ( G  e.  T  /\  -.  Q  .<_  ( P  .\/  V
)  /\  ( F `  ( G `  P
) )  =  P ) )  ->  ( F `  ( G `  Q ) )  =  ( ( Q  .\/  V )  ./\  ( P  .\/  ( ( P  .\/  Q )  ./\  W )
) ) )

Proof of Theorem cdlemg4f
StepHypRef Expression
1 cdlemg4.l . . 3  |-  .<_  =  ( le `  K )
2 cdlemg4.a . . 3  |-  A  =  ( Atoms `  K )
3 cdlemg4.h . . 3  |-  H  =  ( LHyp `  K
)
4 cdlemg4.t . . 3  |-  T  =  ( ( LTrn `  K
) `  W )
5 cdlemg4.r . . 3  |-  R  =  ( ( trL `  K
) `  W )
6 cdlemg4.j . . 3  |-  .\/  =  ( join `  K )
7 cdlemg4b.v . . 3  |-  V  =  ( R `  G
)
8 cdlemg4.m . . 3  |-  ./\  =  ( meet `  K )
91, 2, 3, 4, 5, 6, 7, 8cdlemg4e 31108 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  F  e.  T
)  /\  ( G  e.  T  /\  -.  Q  .<_  ( P  .\/  V
)  /\  ( F `  ( G `  P
) )  =  P ) )  ->  ( F `  ( G `  Q ) )  =  ( ( ( G `
 Q )  .\/  ( R `  F ) )  ./\  ( ( F `  ( G `  P ) )  .\/  ( ( ( G `
 P )  .\/  ( G `  Q ) )  ./\  W )
) ) )
10 simp1 957 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  F  e.  T
)  /\  ( G  e.  T  /\  -.  Q  .<_  ( P  .\/  V
)  /\  ( F `  ( G `  P
) )  =  P ) )  ->  ( K  e.  HL  /\  W  e.  H ) )
11 simp21 990 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  F  e.  T
)  /\  ( G  e.  T  /\  -.  Q  .<_  ( P  .\/  V
)  /\  ( F `  ( G `  P
) )  =  P ) )  ->  ( P  e.  A  /\  -.  P  .<_  W ) )
12 simp23 992 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  F  e.  T
)  /\  ( G  e.  T  /\  -.  Q  .<_  ( P  .\/  V
)  /\  ( F `  ( G `  P
) )  =  P ) )  ->  F  e.  T )
13 simp31 993 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  F  e.  T
)  /\  ( G  e.  T  /\  -.  Q  .<_  ( P  .\/  V
)  /\  ( F `  ( G `  P
) )  =  P ) )  ->  G  e.  T )
14 simp33 995 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  F  e.  T
)  /\  ( G  e.  T  /\  -.  Q  .<_  ( P  .\/  V
)  /\  ( F `  ( G `  P
) )  =  P ) )  ->  ( F `  ( G `  P ) )  =  P )
151, 2, 3, 4, 5cdlemg4a 31102 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  F  e.  T  /\  G  e.  T )  /\  ( F `  ( G `  P )
)  =  P )  ->  ( R `  F )  =  ( R `  G ) )
1610, 11, 12, 13, 14, 15syl131anc 1197 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  F  e.  T
)  /\  ( G  e.  T  /\  -.  Q  .<_  ( P  .\/  V
)  /\  ( F `  ( G `  P
) )  =  P ) )  ->  ( R `  F )  =  ( R `  G ) )
1716, 7syl6reqr 2463 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  F  e.  T
)  /\  ( G  e.  T  /\  -.  Q  .<_  ( P  .\/  V
)  /\  ( F `  ( G `  P
) )  =  P ) )  ->  V  =  ( R `  F ) )
1817oveq2d 6064 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  F  e.  T
)  /\  ( G  e.  T  /\  -.  Q  .<_  ( P  .\/  V
)  /\  ( F `  ( G `  P
) )  =  P ) )  ->  (
( G `  Q
)  .\/  V )  =  ( ( G `
 Q )  .\/  ( R `  F ) ) )
19 simp22 991 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  F  e.  T
)  /\  ( G  e.  T  /\  -.  Q  .<_  ( P  .\/  V
)  /\  ( F `  ( G `  P
) )  =  P ) )  ->  ( Q  e.  A  /\  -.  Q  .<_  W ) )
201, 2, 3, 4, 5, 6, 7cdlemg4b12 31105 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  G  e.  T )  ->  (
( G `  Q
)  .\/  V )  =  ( Q  .\/  V ) )
2110, 19, 13, 20syl3anc 1184 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  F  e.  T
)  /\  ( G  e.  T  /\  -.  Q  .<_  ( P  .\/  V
)  /\  ( F `  ( G `  P
) )  =  P ) )  ->  (
( G `  Q
)  .\/  V )  =  ( Q  .\/  V ) )
2218, 21eqtr3d 2446 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  F  e.  T
)  /\  ( G  e.  T  /\  -.  Q  .<_  ( P  .\/  V
)  /\  ( F `  ( G `  P
) )  =  P ) )  ->  (
( G `  Q
)  .\/  ( R `  F ) )  =  ( Q  .\/  V
) )
23 eqid 2412 . . . . . 6  |-  ( ( P  .\/  Q ) 
./\  W )  =  ( ( P  .\/  Q )  ./\  W )
243, 4, 1, 6, 2, 8, 23cdlemg2m 31098 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  G  e.  T )  ->  (
( ( G `  P )  .\/  ( G `  Q )
)  ./\  W )  =  ( ( P 
.\/  Q )  ./\  W ) )
2510, 11, 19, 13, 24syl121anc 1189 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  F  e.  T
)  /\  ( G  e.  T  /\  -.  Q  .<_  ( P  .\/  V
)  /\  ( F `  ( G `  P
) )  =  P ) )  ->  (
( ( G `  P )  .\/  ( G `  Q )
)  ./\  W )  =  ( ( P 
.\/  Q )  ./\  W ) )
2614, 25oveq12d 6066 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  F  e.  T
)  /\  ( G  e.  T  /\  -.  Q  .<_  ( P  .\/  V
)  /\  ( F `  ( G `  P
) )  =  P ) )  ->  (
( F `  ( G `  P )
)  .\/  ( (
( G `  P
)  .\/  ( G `  Q ) )  ./\  W ) )  =  ( P  .\/  ( ( P  .\/  Q ) 
./\  W ) ) )
2722, 26oveq12d 6066 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  F  e.  T
)  /\  ( G  e.  T  /\  -.  Q  .<_  ( P  .\/  V
)  /\  ( F `  ( G `  P
) )  =  P ) )  ->  (
( ( G `  Q )  .\/  ( R `  F )
)  ./\  ( ( F `  ( G `  P ) )  .\/  ( ( ( G `
 P )  .\/  ( G `  Q ) )  ./\  W )
) )  =  ( ( Q  .\/  V
)  ./\  ( P  .\/  ( ( P  .\/  Q )  ./\  W )
) ) )
289, 27eqtrd 2444 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  F  e.  T
)  /\  ( G  e.  T  /\  -.  Q  .<_  ( P  .\/  V
)  /\  ( F `  ( G `  P
) )  =  P ) )  ->  ( F `  ( G `  Q ) )  =  ( ( Q  .\/  V )  ./\  ( P  .\/  ( ( P  .\/  Q )  ./\  W )
) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721   class class class wbr 4180   ` cfv 5421  (class class class)co 6048   lecple 13499   joincjn 14364   meetcmee 14365   Atomscatm 29758   HLchlt 29845   LHypclh 30478   LTrncltrn 30595   trLctrl 30652
This theorem is referenced by:  cdlemg4g  31110
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-rep 4288  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-nel 2578  df-ral 2679  df-rex 2680  df-reu 2681  df-rmo 2682  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-op 3791  df-uni 3984  df-iun 4063  df-iin 4064  df-br 4181  df-opab 4235  df-mpt 4236  df-id 4466  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-ov 6051  df-oprab 6052  df-mpt2 6053  df-1st 6316  df-2nd 6317  df-undef 6510  df-riota 6516  df-map 6987  df-poset 14366  df-plt 14378  df-lub 14394  df-glb 14395  df-join 14396  df-meet 14397  df-p0 14431  df-p1 14432  df-lat 14438  df-clat 14500  df-oposet 29671  df-ol 29673  df-oml 29674  df-covers 29761  df-ats 29762  df-atl 29793  df-cvlat 29817  df-hlat 29846  df-llines 29992  df-lplanes 29993  df-lvols 29994  df-lines 29995  df-psubsp 29997  df-pmap 29998  df-padd 30290  df-lhyp 30482  df-laut 30483  df-ldil 30598  df-ltrn 30599  df-trl 30653
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