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Theorem cdlemk56w 31770
Description: Use a fixed element to eliminate  P in cdlemk56 31768. (Contributed by NM, 1-Aug-2013.)
Hypotheses
Ref Expression
cdlemk6.b  |-  B  =  ( Base `  K
)
cdlemk6.j  |-  .\/  =  ( join `  K )
cdlemk6.m  |-  ./\  =  ( meet `  K )
cdlemk6.o  |-  ._|_  =  ( oc `  K )
cdlemk6.a  |-  A  =  ( Atoms `  K )
cdlemk6.h  |-  H  =  ( LHyp `  K
)
cdlemk6.t  |-  T  =  ( ( LTrn `  K
) `  W )
cdlemk6.r  |-  R  =  ( ( trL `  K
) `  W )
cdlemk6.p  |-  P  =  (  ._|_  `  W )
cdlemk6.z  |-  Z  =  ( ( P  .\/  ( R `  b ) )  ./\  ( ( N `  P )  .\/  ( R `  (
b  o.  `' F
) ) ) )
cdlemk6.y  |-  Y  =  ( ( P  .\/  ( R `  g ) )  ./\  ( Z  .\/  ( R `  (
g  o.  `' b ) ) ) )
cdlemk6.x  |-  X  =  ( iota_ z  e.  T A. b  e.  T  ( ( b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F )  /\  ( R `  b
)  =/=  ( R `
 g ) )  ->  ( z `  P )  =  Y ) )
cdlemk6.u  |-  U  =  ( g  e.  T  |->  if ( F  =  N ,  g ,  X ) )
cdlemk6.e  |-  E  =  ( ( TEndo `  K
) `  W )
Assertion
Ref Expression
cdlemk56w  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  N  e.  T )  /\  ( R `  F )  =  ( R `  N ) )  -> 
( U  e.  E  /\  ( U `  F
)  =  N ) )
Distinct variable groups:    g, b,
z,  ./\    .\/ , b, g, z    A, b, g, z    B, b, g, z    F, b, g, z    H, b, g, z    K, b, g, z    N, b, g, z    P, b, g, z    R, b, g, z    T, b, g, z    W, b, g, z    z, Y   
g, Z
Allowed substitution hints:    U( z, g, b)    E( z, g, b)    ._|_ ( z, g, b)    X( z, g, b)    Y( g, b)    Z( z, b)

Proof of Theorem cdlemk56w
StepHypRef Expression
1 simp1 957 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  N  e.  T )  /\  ( R `  F )  =  ( R `  N ) )  -> 
( K  e.  HL  /\  W  e.  H ) )
2 simp2l 983 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  N  e.  T )  /\  ( R `  F )  =  ( R `  N ) )  ->  F  e.  T )
3 simp2r 984 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  N  e.  T )  /\  ( R `  F )  =  ( R `  N ) )  ->  N  e.  T )
4 simp3 959 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  N  e.  T )  /\  ( R `  F )  =  ( R `  N ) )  -> 
( R `  F
)  =  ( R `
 N ) )
5 eqid 2436 . . . . 5  |-  ( le
`  K )  =  ( le `  K
)
6 cdlemk6.a . . . . 5  |-  A  =  ( Atoms `  K )
7 cdlemk6.h . . . . 5  |-  H  =  ( LHyp `  K
)
8 cdlemk6.p . . . . . 6  |-  P  =  (  ._|_  `  W )
9 cdlemk6.o . . . . . . 7  |-  ._|_  =  ( oc `  K )
109fveq1i 5729 . . . . . 6  |-  (  ._|_  `  W )  =  ( ( oc `  K
) `  W )
118, 10eqtri 2456 . . . . 5  |-  P  =  ( ( oc `  K ) `  W
)
125, 6, 7, 11lhpocnel2 30816 . . . 4  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( P  e.  A  /\  -.  P ( le
`  K ) W ) )
13123ad2ant1 978 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  N  e.  T )  /\  ( R `  F )  =  ( R `  N ) )  -> 
( P  e.  A  /\  -.  P ( le
`  K ) W ) )
14 cdlemk6.b . . . 4  |-  B  =  ( Base `  K
)
15 cdlemk6.j . . . 4  |-  .\/  =  ( join `  K )
16 cdlemk6.m . . . 4  |-  ./\  =  ( meet `  K )
17 cdlemk6.t . . . 4  |-  T  =  ( ( LTrn `  K
) `  W )
18 cdlemk6.r . . . 4  |-  R  =  ( ( trL `  K
) `  W )
19 cdlemk6.z . . . 4  |-  Z  =  ( ( P  .\/  ( R `  b ) )  ./\  ( ( N `  P )  .\/  ( R `  (
b  o.  `' F
) ) ) )
20 cdlemk6.y . . . 4  |-  Y  =  ( ( P  .\/  ( R `  g ) )  ./\  ( Z  .\/  ( R `  (
g  o.  `' b ) ) ) )
21 cdlemk6.x . . . 4  |-  X  =  ( iota_ z  e.  T A. b  e.  T  ( ( b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F )  /\  ( R `  b
)  =/=  ( R `
 g ) )  ->  ( z `  P )  =  Y ) )
22 cdlemk6.u . . . 4  |-  U  =  ( g  e.  T  |->  if ( F  =  N ,  g ,  X ) )
23 cdlemk6.e . . . 4  |-  E  =  ( ( TEndo `  K
) `  W )
2414, 5, 15, 16, 6, 7, 17, 18, 19, 20, 21, 22, 23cdlemk56 31768 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  N  e.  T )  /\  ( R `  F )  =  ( R `  N )  /\  ( P  e.  A  /\  -.  P ( le `  K ) W ) )  ->  U  e.  E )
251, 2, 3, 4, 13, 24syl311anc 1198 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  N  e.  T )  /\  ( R `  F )  =  ( R `  N ) )  ->  U  e.  E )
2614, 15, 16, 9, 6, 7, 17, 18, 8, 19, 20, 21, 22cdlemk19w 31769 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  N  e.  T )  /\  ( R `  F )  =  ( R `  N ) )  -> 
( U `  F
)  =  N )
2725, 26jca 519 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  N  e.  T )  /\  ( R `  F )  =  ( R `  N ) )  -> 
( U  e.  E  /\  ( U `  F
)  =  N ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725    =/= wne 2599   A.wral 2705   ifcif 3739   class class class wbr 4212    e. cmpt 4266    _I cid 4493   `'ccnv 4877    |` cres 4880    o. ccom 4882   ` cfv 5454  (class class class)co 6081   iota_crio 6542   Basecbs 13469   lecple 13536   occoc 13537   joincjn 14401   meetcmee 14402   Atomscatm 30061   HLchlt 30148   LHypclh 30781   LTrncltrn 30898   trLctrl 30955   TEndoctendo 31549
This theorem is referenced by:  cdlemk  31771  cdleml6  31778
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-iun 4095  df-iin 4096  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-undef 6543  df-riota 6549  df-map 7020  df-poset 14403  df-plt 14415  df-lub 14431  df-glb 14432  df-join 14433  df-meet 14434  df-p0 14468  df-p1 14469  df-lat 14475  df-clat 14537  df-oposet 29974  df-ol 29976  df-oml 29977  df-covers 30064  df-ats 30065  df-atl 30096  df-cvlat 30120  df-hlat 30149  df-llines 30295  df-lplanes 30296  df-lvols 30297  df-lines 30298  df-psubsp 30300  df-pmap 30301  df-padd 30593  df-lhyp 30785  df-laut 30786  df-ldil 30901  df-ltrn 30902  df-trl 30956  df-tendo 31552
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