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Theorem cdlemk56w 31162
Description: Use a fixed element to eliminate  P in cdlemk56 31160. (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 955 . . 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 981 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  N  e.  T )  /\  ( R `  F )  =  ( R `  N ) )  ->  F  e.  T )
3 simp2r 982 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  N  e.  T )  /\  ( R `  F )  =  ( R `  N ) )  ->  N  e.  T )
4 simp3 957 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  N  e.  T )  /\  ( R `  F )  =  ( R `  N ) )  -> 
( R `  F
)  =  ( R `
 N ) )
5 eqid 2283 . . . . 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 5526 . . . . . 6  |-  (  ._|_  `  W )  =  ( ( oc `  K
) `  W )
118, 10eqtri 2303 . . . . 5  |-  P  =  ( ( oc `  K ) `  W
)
125, 6, 7, 11lhpocnel2 30208 . . . 4  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( P  e.  A  /\  -.  P ( le
`  K ) W ) )
13123ad2ant1 976 . . 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 31160 . . 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 1196 . 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 31161 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  N  e.  T )  /\  ( R `  F )  =  ( R `  N ) )  -> 
( U `  F
)  =  N )
2725, 26jca 518 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 358    /\ w3a 934    = wceq 1623    e. wcel 1684    =/= wne 2446   A.wral 2543   ifcif 3565   class class class wbr 4023    e. cmpt 4077    _I cid 4304   `'ccnv 4688    |` cres 4691    o. ccom 4693   ` cfv 5255  (class class class)co 5858   iota_crio 6297   Basecbs 13148   lecple 13215   occoc 13216   joincjn 14078   meetcmee 14079   Atomscatm 29453   HLchlt 29540   LHypclh 30173   LTrncltrn 30290   trLctrl 30347   TEndoctendo 30941
This theorem is referenced by:  cdlemk  31163  cdleml6  31170
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-iin 3908  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-undef 6298  df-riota 6304  df-map 6774  df-poset 14080  df-plt 14092  df-lub 14108  df-glb 14109  df-join 14110  df-meet 14111  df-p0 14145  df-p1 14146  df-lat 14152  df-clat 14214  df-oposet 29366  df-ol 29368  df-oml 29369  df-covers 29456  df-ats 29457  df-atl 29488  df-cvlat 29512  df-hlat 29541  df-llines 29687  df-lplanes 29688  df-lvols 29689  df-lines 29690  df-psubsp 29692  df-pmap 29693  df-padd 29985  df-lhyp 30177  df-laut 30178  df-ldil 30293  df-ltrn 30294  df-trl 30348  df-tendo 30944
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