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Theorem cdlemg41 34197
Description: Convert cdlemg40 34196 to function composition. TODO: Fix comment. (Contributed by NM, 31-May-2013.)
Hypotheses
Ref Expression
cdlemg35.l  |-  .<_  =  ( le `  K )
cdlemg35.j  |-  .\/  =  ( join `  K )
cdlemg35.m  |-  ./\  =  ( meet `  K )
cdlemg35.a  |-  A  =  ( Atoms `  K )
cdlemg35.h  |-  H  =  ( LHyp `  K
)
cdlemg35.t  |-  T  =  ( ( LTrn `  K
) `  W )
Assertion
Ref Expression
cdlemg41  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( F  e.  T  /\  G  e.  T ) )  -> 
( ( P  .\/  ( ( F  o.  G ) `  P
) )  ./\  W
)  =  ( ( Q  .\/  ( ( F  o.  G ) `
 Q ) ) 
./\  W ) )

Proof of Theorem cdlemg41
StepHypRef Expression
1 cdlemg35.l . . 3  |-  .<_  =  ( le `  K )
2 cdlemg35.j . . 3  |-  .\/  =  ( join `  K )
3 cdlemg35.m . . 3  |-  ./\  =  ( meet `  K )
4 cdlemg35.a . . 3  |-  A  =  ( Atoms `  K )
5 cdlemg35.h . . 3  |-  H  =  ( LHyp `  K
)
6 cdlemg35.t . . 3  |-  T  =  ( ( LTrn `  K
) `  W )
71, 2, 3, 4, 5, 6cdlemg40 34196 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( F  e.  T  /\  G  e.  T ) )  -> 
( ( P  .\/  ( F `  ( G `
 P ) ) )  ./\  W )  =  ( ( Q 
.\/  ( F `  ( G `  Q ) ) )  ./\  W
) )
8 simp1 1005 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( F  e.  T  /\  G  e.  T ) )  -> 
( K  e.  HL  /\  W  e.  H ) )
9 simp3 1007 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( F  e.  T  /\  G  e.  T ) )  -> 
( F  e.  T  /\  G  e.  T
) )
10 simp2ll 1072 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( F  e.  T  /\  G  e.  T ) )  ->  P  e.  A )
111, 4, 5, 6ltrncoval 33622 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  P  e.  A )  ->  (
( F  o.  G
) `  P )  =  ( F `  ( G `  P ) ) )
128, 9, 10, 11syl3anc 1264 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( F  e.  T  /\  G  e.  T ) )  -> 
( ( F  o.  G ) `  P
)  =  ( F `
 ( G `  P ) ) )
1312oveq2d 6265 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( F  e.  T  /\  G  e.  T ) )  -> 
( P  .\/  (
( F  o.  G
) `  P )
)  =  ( P 
.\/  ( F `  ( G `  P ) ) ) )
1413oveq1d 6264 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( F  e.  T  /\  G  e.  T ) )  -> 
( ( P  .\/  ( ( F  o.  G ) `  P
) )  ./\  W
)  =  ( ( P  .\/  ( F `
 ( G `  P ) ) ) 
./\  W ) )
15 simp2rl 1074 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( F  e.  T  /\  G  e.  T ) )  ->  Q  e.  A )
161, 4, 5, 6ltrncoval 33622 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  Q  e.  A )  ->  (
( F  o.  G
) `  Q )  =  ( F `  ( G `  Q ) ) )
178, 9, 15, 16syl3anc 1264 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( F  e.  T  /\  G  e.  T ) )  -> 
( ( F  o.  G ) `  Q
)  =  ( F `
 ( G `  Q ) ) )
1817oveq2d 6265 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( F  e.  T  /\  G  e.  T ) )  -> 
( Q  .\/  (
( F  o.  G
) `  Q )
)  =  ( Q 
.\/  ( F `  ( G `  Q ) ) ) )
1918oveq1d 6264 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( F  e.  T  /\  G  e.  T ) )  -> 
( ( Q  .\/  ( ( F  o.  G ) `  Q
) )  ./\  W
)  =  ( ( Q  .\/  ( F `
 ( G `  Q ) ) ) 
./\  W ) )
207, 14, 193eqtr4d 2472 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( F  e.  T  /\  G  e.  T ) )  -> 
( ( P  .\/  ( ( F  o.  G ) `  P
) )  ./\  W
)  =  ( ( Q  .\/  ( ( F  o.  G ) `
 Q ) ) 
./\  W ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1872   class class class wbr 4366    o. ccom 4800   ` cfv 5544  (class class class)co 6249   lecple 15140   joincjn 16132   meetcmee 16133   Atomscatm 32741   HLchlt 32828   LHypclh 33461   LTrncltrn 33578
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2063  ax-ext 2408  ax-rep 4479  ax-sep 4489  ax-nul 4498  ax-pow 4545  ax-pr 4603  ax-un 6541  ax-riotaBAD 32437
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2280  df-mo 2281  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2558  df-ne 2601  df-nel 2602  df-ral 2719  df-rex 2720  df-reu 2721  df-rmo 2722  df-rab 2723  df-v 3024  df-sbc 3243  df-csb 3339  df-dif 3382  df-un 3384  df-in 3386  df-ss 3393  df-nul 3705  df-if 3855  df-pw 3926  df-sn 3942  df-pr 3944  df-op 3948  df-uni 4163  df-iun 4244  df-iin 4245  df-br 4367  df-opab 4426  df-mpt 4427  df-id 4711  df-xp 4802  df-rel 4803  df-cnv 4804  df-co 4805  df-dm 4806  df-rn 4807  df-res 4808  df-ima 4809  df-iota 5508  df-fun 5546  df-fn 5547  df-f 5548  df-f1 5549  df-fo 5550  df-f1o 5551  df-fv 5552  df-riota 6211  df-ov 6252  df-oprab 6253  df-mpt2 6254  df-1st 6751  df-2nd 6752  df-undef 6975  df-map 7429  df-preset 16116  df-poset 16134  df-plt 16147  df-lub 16163  df-glb 16164  df-join 16165  df-meet 16166  df-p0 16228  df-p1 16229  df-lat 16235  df-clat 16297  df-oposet 32654  df-ol 32656  df-oml 32657  df-covers 32744  df-ats 32745  df-atl 32776  df-cvlat 32800  df-hlat 32829  df-llines 32975  df-lplanes 32976  df-lvols 32977  df-lines 32978  df-psubsp 32980  df-pmap 32981  df-padd 33273  df-lhyp 33465  df-laut 33466  df-ldil 33581  df-ltrn 33582  df-trl 33637
This theorem is referenced by:  ltrnco  34198
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