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Theorem cdlemkfid2N 31794
Description: Lemma for cdlemkfid3N 31796. (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 983 . . . . . 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 5733 . . . . 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 6099 . . . 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 6100 . . 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 31792 . . . 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 1178 . . 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 2472 . 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 2482 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 set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726    =/= wne 2601   class class class wbr 4215    _I cid 4496   `'ccnv 4880    |` cres 4883    o. ccom 4885   ` cfv 5457  (class class class)co 6084   Basecbs 13474   lecple 13541   joincjn 14406   meetcmee 14407   Atomscatm 30135   HLchlt 30222   LHypclh 30855   LTrncltrn 30972   trLctrl 31029
This theorem is referenced by:  cdlemkfid3N  31796
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-iin 4098  df-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-1st 6352  df-2nd 6353  df-undef 6546  df-riota 6552  df-map 7023  df-poset 14408  df-plt 14420  df-lub 14436  df-glb 14437  df-join 14438  df-meet 14439  df-p0 14473  df-p1 14474  df-lat 14480  df-clat 14542  df-oposet 30048  df-ol 30050  df-oml 30051  df-covers 30138  df-ats 30139  df-atl 30170  df-cvlat 30194  df-hlat 30223  df-llines 30369  df-lplanes 30370  df-lvols 30371  df-lines 30372  df-psubsp 30374  df-pmap 30375  df-padd 30667  df-lhyp 30859  df-laut 30860  df-ldil 30975  df-ltrn 30976  df-trl 31030
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