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Theorem cdlemk19u1 31766
Description: cdlemk19 31666 with simpler hypotheses. TODO: Clean all this up. (Contributed by NM, 31-Jul-2013.)
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
) ) ) )
cdlemk5.y  |-  Y  =  ( ( P  .\/  ( R `  g ) )  ./\  ( Z  .\/  ( R `  (
g  o.  `' b ) ) ) )
cdlemk5.x  |-  X  =  ( iota_ z  e.  T A. b  e.  T  ( ( b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F )  /\  ( R `  b
)  =/=  ( R `
 g ) )  ->  ( z `  P )  =  Y ) )
cdlemk5.u  |-  U  =  ( g  e.  T  |->  if ( F  =  N ,  g ,  X ) )
Assertion
Ref Expression
cdlemk19u1  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( F  e.  T  /\  F  =/=  N  /\  N  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  -> 
( ( U `  F ) `  P
)  =  ( N `
 P ) )
Distinct variable groups:    ./\ , g    .\/ , g    B, g    P, g    R, g    T, g    g, Z    g, b, z,  ./\    .<_ , b   
z, g,  .<_    .\/ , b,
z    A, b, g, z    B, b, z    F, b, g, z    H, b, g, z    K, b, g, z    N, b, g, z    P, b, z    R, b, z    T, b, z    W, b, g, z    z, Y
Allowed substitution hints:    U( z, g, b)    X( z, g, b)    Y( g, b)    Z( z, b)

Proof of Theorem cdlemk19u1
StepHypRef Expression
1 simp22 991 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( F  e.  T  /\  F  =/=  N  /\  N  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  F  =/=  N )
2 simp21 990 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( F  e.  T  /\  F  =/=  N  /\  N  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  F  e.  T )
3 cdlemk5.x . . . . 5  |-  X  =  ( iota_ z  e.  T A. b  e.  T  ( ( b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F )  /\  ( R `  b
)  =/=  ( R `
 g ) )  ->  ( z `  P )  =  Y ) )
4 cdlemk5.u . . . . 5  |-  U  =  ( g  e.  T  |->  if ( F  =  N ,  g ,  X ) )
53, 4cdlemk40f 31716 . . . 4  |-  ( ( F  =/=  N  /\  F  e.  T )  ->  ( U `  F
)  =  [_ F  /  g ]_ X
)
61, 2, 5syl2anc 643 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( F  e.  T  /\  F  =/=  N  /\  N  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  -> 
( U `  F
)  =  [_ F  /  g ]_ X
)
76fveq1d 5730 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( F  e.  T  /\  F  =/=  N  /\  N  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  -> 
( ( U `  F ) `  P
)  =  ( [_ F  /  g ]_ X `  P ) )
8 simp1l 981 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( F  e.  T  /\  F  =/=  N  /\  N  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  -> 
( K  e.  HL  /\  W  e.  H ) )
9 simp23 992 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( F  e.  T  /\  F  =/=  N  /\  N  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  N  e.  T )
10 simp1r 982 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( F  e.  T  /\  F  =/=  N  /\  N  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  -> 
( R `  F
)  =  ( R `
 N ) )
11 cdlemk5.b . . . . . 6  |-  B  =  ( Base `  K
)
12 cdlemk5.h . . . . . 6  |-  H  =  ( LHyp `  K
)
13 cdlemk5.t . . . . . 6  |-  T  =  ( ( LTrn `  K
) `  W )
14 cdlemk5.r . . . . . 6  |-  R  =  ( ( trL `  K
) `  W )
1511, 12, 13, 14trlnid 30976 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  N  e.  T )  /\  ( F  =/=  N  /\  ( R `  F )  =  ( R `  N ) ) )  ->  F  =/=  (  _I  |`  B ) )
168, 2, 9, 1, 10, 15syl122anc 1193 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( F  e.  T  /\  F  =/=  N  /\  N  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  F  =/=  (  _I  |`  B ) )
172, 16, 93jca 1134 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( F  e.  T  /\  F  =/=  N  /\  N  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  -> 
( F  e.  T  /\  F  =/=  (  _I  |`  B )  /\  N  e.  T )
)
18 cdlemk5.l . . . 4  |-  .<_  =  ( le `  K )
19 cdlemk5.j . . . 4  |-  .\/  =  ( join `  K )
20 cdlemk5.m . . . 4  |-  ./\  =  ( meet `  K )
21 cdlemk5.a . . . 4  |-  A  =  ( Atoms `  K )
22 cdlemk5.z . . . 4  |-  Z  =  ( ( P  .\/  ( R `  b ) )  ./\  ( ( N `  P )  .\/  ( R `  (
b  o.  `' F
) ) ) )
23 cdlemk5.y . . . 4  |-  Y  =  ( ( P  .\/  ( R `  g ) )  ./\  ( Z  .\/  ( R `  (
g  o.  `' b ) ) ) )
2411, 18, 19, 20, 21, 12, 13, 14, 22, 23, 3cdlemk19x 31740 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B )  /\  N  e.  T )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( [_ F  /  g ]_ X `  P )  =  ( N `  P ) )
2517, 24syld3an2 1231 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( F  e.  T  /\  F  =/=  N  /\  N  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  -> 
( [_ F  /  g ]_ X `  P )  =  ( N `  P ) )
267, 25eqtrd 2468 1  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( F  e.  T  /\  F  =/=  N  /\  N  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  -> 
( ( U `  F ) `  P
)  =  ( N `
 P ) )
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   [_csb 3251   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   joincjn 14401   meetcmee 14402   Atomscatm 30061   HLchlt 30148   LHypclh 30781   LTrncltrn 30898   trLctrl 30955
This theorem is referenced by:  cdlemk19u  31767
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
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