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Theorem cdlemkfid2N 33955
Description: Lemma for cdlemkfid3N 33957. (Contributed by NM, 29-Jul-2013.) (New usage is discouraged.)
Hypotheses
Ref Expression
cdlemk5.b  |-  B  =  ( Base `  K
)
cdlemk5.l  |-  .<_  =  ( le `  K )
cdlemk5.j  |-  .\/  =  ( join `  K )
cdlemk5.m  |-  ./\  =  ( meet `  K )
cdlemk5.a  |-  A  =  ( Atoms `  K )
cdlemk5.h  |-  H  =  ( LHyp `  K
)
cdlemk5.t  |-  T  =  ( ( LTrn `  K
) `  W )
cdlemk5.r  |-  R  =  ( ( trL `  K
) `  W )
cdlemk5.z  |-  Z  =  ( ( P  .\/  ( R `  b ) )  ./\  ( ( N `  P )  .\/  ( R `  (
b  o.  `' F
) ) ) )
Assertion
Ref Expression
cdlemkfid2N  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  =  N )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B )  /\  b  e.  T
)  /\  ( ( R `  b )  =/=  ( R `  F
)  /\  ( P  e.  A  /\  -.  P  .<_  W ) ) )  ->  Z  =  ( b `  P ) )

Proof of Theorem cdlemkfid2N
StepHypRef Expression
1 cdlemk5.z . 2  |-  Z  =  ( ( P  .\/  ( R `  b ) )  ./\  ( ( N `  P )  .\/  ( R `  (
b  o.  `' F
) ) ) )
2 simp1r 1024 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  =  N )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B )  /\  b  e.  T
)  /\  ( ( R `  b )  =/=  ( R `  F
)  /\  ( P  e.  A  /\  -.  P  .<_  W ) ) )  ->  F  =  N )
32fveq1d 5853 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  =  N )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B )  /\  b  e.  T
)  /\  ( ( R `  b )  =/=  ( R `  F
)  /\  ( P  e.  A  /\  -.  P  .<_  W ) ) )  ->  ( F `  P )  =  ( N `  P ) )
43oveq1d 6295 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  =  N )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B )  /\  b  e.  T
)  /\  ( ( R `  b )  =/=  ( R `  F
)  /\  ( P  e.  A  /\  -.  P  .<_  W ) ) )  ->  ( ( F `
 P )  .\/  ( R `  ( b  o.  `' F ) ) )  =  ( ( N `  P
)  .\/  ( R `  ( b  o.  `' F ) ) ) )
54oveq2d 6296 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  =  N )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B )  /\  b  e.  T
)  /\  ( ( R `  b )  =/=  ( R `  F
)  /\  ( P  e.  A  /\  -.  P  .<_  W ) ) )  ->  ( ( P 
.\/  ( R `  b ) )  ./\  ( ( F `  P )  .\/  ( R `  ( b  o.  `' F ) ) ) )  =  ( ( P  .\/  ( R `
 b ) ) 
./\  ( ( N `
 P )  .\/  ( R `  ( b  o.  `' F ) ) ) ) )
6 cdlemk5.b . . . . 5  |-  B  =  ( Base `  K
)
7 cdlemk5.l . . . . 5  |-  .<_  =  ( le `  K )
8 cdlemk5.j . . . . 5  |-  .\/  =  ( join `  K )
9 cdlemk5.m . . . . 5  |-  ./\  =  ( meet `  K )
10 cdlemk5.a . . . . 5  |-  A  =  ( Atoms `  K )
11 cdlemk5.h . . . . 5  |-  H  =  ( LHyp `  K
)
12 cdlemk5.t . . . . 5  |-  T  =  ( ( LTrn `  K
) `  W )
13 cdlemk5.r . . . . 5  |-  R  =  ( ( trL `  K
) `  W )
146, 7, 8, 9, 10, 11, 12, 13cdlemkfid1N 33953 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B )  /\  b  e.  T
)  /\  ( ( R `  b )  =/=  ( R `  F
)  /\  ( P  e.  A  /\  -.  P  .<_  W ) ) )  ->  ( ( P 
.\/  ( R `  b ) )  ./\  ( ( F `  P )  .\/  ( R `  ( b  o.  `' F ) ) ) )  =  ( b `
 P ) )
15143adant1r 1225 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  =  N )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B )  /\  b  e.  T
)  /\  ( ( R `  b )  =/=  ( R `  F
)  /\  ( P  e.  A  /\  -.  P  .<_  W ) ) )  ->  ( ( P 
.\/  ( R `  b ) )  ./\  ( ( F `  P )  .\/  ( R `  ( b  o.  `' F ) ) ) )  =  ( b `
 P ) )
165, 15eqtr3d 2447 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  =  N )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B )  /\  b  e.  T
)  /\  ( ( R `  b )  =/=  ( R `  F
)  /\  ( P  e.  A  /\  -.  P  .<_  W ) ) )  ->  ( ( P 
.\/  ( R `  b ) )  ./\  ( ( N `  P )  .\/  ( R `  ( b  o.  `' F ) ) ) )  =  ( b `
 P ) )
171, 16syl5eq 2457 1  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  =  N )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B )  /\  b  e.  T
)  /\  ( ( R `  b )  =/=  ( R `  F
)  /\  ( P  e.  A  /\  -.  P  .<_  W ) ) )  ->  Z  =  ( b `  P ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    /\ w3a 976    = wceq 1407    e. wcel 1844    =/= wne 2600   class class class wbr 4397    _I cid 4735   `'ccnv 4824    |` cres 4827    o. ccom 4829   ` cfv 5571  (class class class)co 6280   Basecbs 14843   lecple 14918   joincjn 15899   meetcmee 15900   Atomscatm 32294   HLchlt 32381   LHypclh 33014   LTrncltrn 33131   trLctrl 33189
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1641  ax-4 1654  ax-5 1727  ax-6 1773  ax-7 1816  ax-8 1846  ax-9 1848  ax-10 1863  ax-11 1868  ax-12 1880  ax-13 2028  ax-ext 2382  ax-rep 4509  ax-sep 4519  ax-nul 4527  ax-pow 4574  ax-pr 4632  ax-un 6576  ax-riotaBAD 31990
This theorem depends on definitions:  df-bi 187  df-or 370  df-an 371  df-3or 977  df-3an 978  df-tru 1410  df-ex 1636  df-nf 1640  df-sb 1766  df-eu 2244  df-mo 2245  df-clab 2390  df-cleq 2396  df-clel 2399  df-nfc 2554  df-ne 2602  df-nel 2603  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3063  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3741  df-if 3888  df-pw 3959  df-sn 3975  df-pr 3977  df-op 3981  df-uni 4194  df-iun 4275  df-iin 4276  df-br 4398  df-opab 4456  df-mpt 4457  df-id 4740  df-xp 4831  df-rel 4832  df-cnv 4833  df-co 4834  df-dm 4835  df-rn 4836  df-res 4837  df-ima 4838  df-iota 5535  df-fun 5573  df-fn 5574  df-f 5575  df-f1 5576  df-fo 5577  df-f1o 5578  df-fv 5579  df-riota 6242  df-ov 6283  df-oprab 6284  df-mpt2 6285  df-1st 6786  df-2nd 6787  df-undef 7007  df-map 7461  df-preset 15883  df-poset 15901  df-plt 15914  df-lub 15930  df-glb 15931  df-join 15932  df-meet 15933  df-p0 15995  df-p1 15996  df-lat 16002  df-clat 16064  df-oposet 32207  df-ol 32209  df-oml 32210  df-covers 32297  df-ats 32298  df-atl 32329  df-cvlat 32353  df-hlat 32382  df-llines 32528  df-lplanes 32529  df-lvols 32530  df-lines 32531  df-psubsp 32533  df-pmap 32534  df-padd 32826  df-lhyp 33018  df-laut 33019  df-ldil 33134  df-ltrn 33135  df-trl 33190
This theorem is referenced by:  cdlemkfid3N  33957
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