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Theorem cdlemk30 31691
Description: Part of proof of Lemma K of [Crawley] p. 118. TODO: fix comment. Part of attempt to simplify hypotheses. (Contributed by NM, 17-Jul-2013.)
Hypotheses
Ref Expression
cdlemk3.b  |-  B  =  ( Base `  K
)
cdlemk3.l  |-  .<_  =  ( le `  K )
cdlemk3.j  |-  .\/  =  ( join `  K )
cdlemk3.m  |-  ./\  =  ( meet `  K )
cdlemk3.a  |-  A  =  ( Atoms `  K )
cdlemk3.h  |-  H  =  ( LHyp `  K
)
cdlemk3.t  |-  T  =  ( ( LTrn `  K
) `  W )
cdlemk3.r  |-  R  =  ( ( trL `  K
) `  W )
cdlemk3.s  |-  S  =  ( f  e.  T  |->  ( iota_ i  e.  T
( i `  P
)  =  ( ( P  .\/  ( R `
 f ) ) 
./\  ( ( N `
 P )  .\/  ( R `  ( f  o.  `' F ) ) ) ) ) )
Assertion
Ref Expression
cdlemk30  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( F  e.  T  /\  b  e.  T  /\  N  e.  T
)  /\  ( ( R `  b )  =/=  ( R `  F
)  /\  ( F  =/=  (  _I  |`  B )  /\  b  =/=  (  _I  |`  B ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) ) )  ->  ( ( S `
 b ) `  P )  =  ( ( P  .\/  ( R `  b )
)  ./\  ( ( N `  P )  .\/  ( R `  (
b  o.  `' F
) ) ) ) )
Distinct variable groups:    f, i,  ./\    .<_ , i    .\/ , f, i    A, i    f, F, i    i, H    i, K    f, N, i    P, f, i    R, f, i    T, f, i   
f, W, i    f,
b, i
Allowed substitution hints:    A( f, b)    B( f, i, b)    P( b)    R( b)    S( f, i, b)    T( b)    F( b)    H( f, b)    .\/ ( b)    K( f, b)    .<_ ( f, b)    ./\ ( b)    N( b)    W( b)

Proof of Theorem cdlemk30
StepHypRef Expression
1 simp1l 981 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( F  e.  T  /\  b  e.  T  /\  N  e.  T
)  /\  ( ( R `  b )  =/=  ( R `  F
)  /\  ( F  =/=  (  _I  |`  B )  /\  b  =/=  (  _I  |`  B ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) ) )  ->  ( K  e.  HL  /\  W  e.  H ) )
2 simp21 990 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( F  e.  T  /\  b  e.  T  /\  N  e.  T
)  /\  ( ( R `  b )  =/=  ( R `  F
)  /\  ( F  =/=  (  _I  |`  B )  /\  b  =/=  (  _I  |`  B ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) ) )  ->  F  e.  T
)
3 simp22 991 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( F  e.  T  /\  b  e.  T  /\  N  e.  T
)  /\  ( ( R `  b )  =/=  ( R `  F
)  /\  ( F  =/=  (  _I  |`  B )  /\  b  =/=  (  _I  |`  B ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) ) )  ->  b  e.  T
)
4 simp23 992 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( F  e.  T  /\  b  e.  T  /\  N  e.  T
)  /\  ( ( R `  b )  =/=  ( R `  F
)  /\  ( F  =/=  (  _I  |`  B )  /\  b  =/=  (  _I  |`  B ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) ) )  ->  N  e.  T
)
5 simp33 995 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( F  e.  T  /\  b  e.  T  /\  N  e.  T
)  /\  ( ( R `  b )  =/=  ( R `  F
)  /\  ( F  =/=  (  _I  |`  B )  /\  b  =/=  (  _I  |`  B ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) ) )  ->  ( P  e.  A  /\  -.  P  .<_  W ) )
6 simp1r 982 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( F  e.  T  /\  b  e.  T  /\  N  e.  T
)  /\  ( ( R `  b )  =/=  ( R `  F
)  /\  ( F  =/=  (  _I  |`  B )  /\  b  =/=  (  _I  |`  B ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) ) )  ->  ( R `  F )  =  ( R `  N ) )
7 simp32l 1082 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( F  e.  T  /\  b  e.  T  /\  N  e.  T
)  /\  ( ( R `  b )  =/=  ( R `  F
)  /\  ( F  =/=  (  _I  |`  B )  /\  b  =/=  (  _I  |`  B ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) ) )  ->  F  =/=  (  _I  |`  B ) )
8 simp32r 1083 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( F  e.  T  /\  b  e.  T  /\  N  e.  T
)  /\  ( ( R `  b )  =/=  ( R `  F
)  /\  ( F  =/=  (  _I  |`  B )  /\  b  =/=  (  _I  |`  B ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) ) )  ->  b  =/=  (  _I  |`  B ) )
9 simp31 993 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( F  e.  T  /\  b  e.  T  /\  N  e.  T
)  /\  ( ( R `  b )  =/=  ( R `  F
)  /\  ( F  =/=  (  _I  |`  B )  /\  b  =/=  (  _I  |`  B ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) ) )  ->  ( R `  b )  =/=  ( R `  F )
)
10 cdlemk3.b . . 3  |-  B  =  ( Base `  K
)
11 cdlemk3.l . . 3  |-  .<_  =  ( le `  K )
12 cdlemk3.j . . 3  |-  .\/  =  ( join `  K )
13 cdlemk3.a . . 3  |-  A  =  ( Atoms `  K )
14 cdlemk3.h . . 3  |-  H  =  ( LHyp `  K
)
15 cdlemk3.t . . 3  |-  T  =  ( ( LTrn `  K
) `  W )
16 cdlemk3.r . . 3  |-  R  =  ( ( trL `  K
) `  W )
17 cdlemk3.m . . 3  |-  ./\  =  ( meet `  K )
18 cdlemk3.s . . 3  |-  S  =  ( f  e.  T  |->  ( iota_ i  e.  T
( i `  P
)  =  ( ( P  .\/  ( R `
 f ) ) 
./\  ( ( N `
 P )  .\/  ( R `  ( f  o.  `' F ) ) ) ) ) )
1910, 11, 12, 13, 14, 15, 16, 17, 18cdlemksv2 31644 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  b  e.  T )  /\  ( N  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( F  =/=  (  _I  |`  B )  /\  b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F ) ) )  ->  ( ( S `
 b ) `  P )  =  ( ( P  .\/  ( R `  b )
)  ./\  ( ( N `  P )  .\/  ( R `  (
b  o.  `' F
) ) ) ) )
201, 2, 3, 4, 5, 6, 7, 8, 9, 19syl333anc 1216 1  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( F  e.  T  /\  b  e.  T  /\  N  e.  T
)  /\  ( ( R `  b )  =/=  ( R `  F
)  /\  ( F  =/=  (  _I  |`  B )  /\  b  =/=  (  _I  |`  B ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) ) )  ->  ( ( S `
 b ) `  P )  =  ( ( P  .\/  ( R `  b )
)  ./\  ( ( N `  P )  .\/  ( R `  (
b  o.  `' F
) ) ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725    =/= wne 2599   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:  cdlemk32  31694  cdlemky  31723  cdlemkyyN  31759
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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