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Theorem cdlemkfid2N 35594
Description: Lemma for cdlemkfid3N 35596. (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 1016 . . . . . 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 5859 . . . . 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 6290 . . . 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 6291 . . 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 35592 . . . 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 1216 . . 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 2503 . 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 2513 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 968    = wceq 1374    e. wcel 1762    =/= wne 2655   class class class wbr 4440    _I cid 4783   `'ccnv 4991    |` cres 4994    o. ccom 4996   ` cfv 5579  (class class class)co 6275   Basecbs 14479   lecple 14551   joincjn 15420   meetcmee 15421   Atomscatm 33935   HLchlt 34022   LHypclh 34655   LTrncltrn 34772   trLctrl 34829
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 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567  ax-riotaBAD 33631
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-nel 2658  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-iun 4320  df-iin 4321  df-br 4441  df-opab 4499  df-mpt 4500  df-id 4788  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-riota 6236  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-1st 6774  df-2nd 6775  df-undef 6992  df-map 7412  df-poset 15422  df-plt 15434  df-lub 15450  df-glb 15451  df-join 15452  df-meet 15453  df-p0 15515  df-p1 15516  df-lat 15522  df-clat 15584  df-oposet 33848  df-ol 33850  df-oml 33851  df-covers 33938  df-ats 33939  df-atl 33970  df-cvlat 33994  df-hlat 34023  df-llines 34169  df-lplanes 34170  df-lvols 34171  df-lines 34172  df-psubsp 34174  df-pmap 34175  df-padd 34467  df-lhyp 34659  df-laut 34660  df-ldil 34775  df-ltrn 34776  df-trl 34830
This theorem is referenced by:  cdlemkfid3N  35596
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