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Theorem cdlemksel 34797
Description: Part of proof of Lemma K of [Crawley] p. 118. Conditions for the sigma(p) function to be a translation. TODO: combine cdlemki 34793? (Contributed by NM, 26-Jun-2013.)
Hypotheses
Ref Expression
cdlemk.b  |-  B  =  ( Base `  K
)
cdlemk.l  |-  .<_  =  ( le `  K )
cdlemk.j  |-  .\/  =  ( join `  K )
cdlemk.a  |-  A  =  ( Atoms `  K )
cdlemk.h  |-  H  =  ( LHyp `  K
)
cdlemk.t  |-  T  =  ( ( LTrn `  K
) `  W )
cdlemk.r  |-  R  =  ( ( trL `  K
) `  W )
cdlemk.m  |-  ./\  =  ( meet `  K )
cdlemk.s  |-  S  =  ( f  e.  T  |->  ( iota_ i  e.  T  ( i `  P
)  =  ( ( P  .\/  ( R `
 f ) ) 
./\  ( ( N `
 P )  .\/  ( R `  ( f  o.  `' F ) ) ) ) ) )
Assertion
Ref Expression
cdlemksel  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  G  e.  T )  /\  ( N  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( F  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B )  /\  ( R `  G )  =/=  ( R `  F ) ) )  ->  ( S `  G )  e.  T
)
Distinct variable groups:    ./\ , f    .\/ , f    f, F, i    f, G, i    f, N    P, f    R, f    T, f   
f, W    ./\ , i    .<_ , i    .\/ , i    A, i    i, F   
i, H    i, K    i, N    P, i    R, i    T, i    i, W
Allowed substitution hints:    A( f)    B( f, i)    S( f, i)    H( f)    K( f)    .<_ ( f)

Proof of Theorem cdlemksel
StepHypRef Expression
1 simp13 1020 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  G  e.  T )  /\  ( N  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( F  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B )  /\  ( R `  G )  =/=  ( R `  F ) ) )  ->  G  e.  T
)
2 cdlemk.b . . . 4  |-  B  =  ( Base `  K
)
3 cdlemk.l . . . 4  |-  .<_  =  ( le `  K )
4 cdlemk.j . . . 4  |-  .\/  =  ( join `  K )
5 cdlemk.a . . . 4  |-  A  =  ( Atoms `  K )
6 cdlemk.h . . . 4  |-  H  =  ( LHyp `  K
)
7 cdlemk.t . . . 4  |-  T  =  ( ( LTrn `  K
) `  W )
8 cdlemk.r . . . 4  |-  R  =  ( ( trL `  K
) `  W )
9 cdlemk.m . . . 4  |-  ./\  =  ( meet `  K )
10 cdlemk.s . . . 4  |-  S  =  ( f  e.  T  |->  ( iota_ i  e.  T  ( i `  P
)  =  ( ( P  .\/  ( R `
 f ) ) 
./\  ( ( N `
 P )  .\/  ( R `  ( f  o.  `' F ) ) ) ) ) )
112, 3, 4, 5, 6, 7, 8, 9, 10cdlemksv 34796 . . 3  |-  ( G  e.  T  ->  ( S `  G )  =  ( iota_ i  e.  T  ( i `  P )  =  ( ( P  .\/  ( R `  G )
)  ./\  ( ( N `  P )  .\/  ( R `  ( G  o.  `' F
) ) ) ) ) )
121, 11syl 16 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  G  e.  T )  /\  ( N  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( F  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B )  /\  ( R `  G )  =/=  ( R `  F ) ) )  ->  ( S `  G )  =  (
iota_ i  e.  T  ( i `  P
)  =  ( ( P  .\/  ( R `
 G ) ) 
./\  ( ( N `
 P )  .\/  ( R `  ( G  o.  `' F ) ) ) ) ) )
13 eqid 2451 . . 3  |-  ( iota_ i  e.  T  ( i `
 P )  =  ( ( P  .\/  ( R `  G ) )  ./\  ( ( N `  P )  .\/  ( R `  ( G  o.  `' F
) ) ) ) )  =  ( iota_ i  e.  T  ( i `
 P )  =  ( ( P  .\/  ( R `  G ) )  ./\  ( ( N `  P )  .\/  ( R `  ( G  o.  `' F
) ) ) ) )
142, 3, 4, 5, 6, 7, 8, 9, 13cdlemki 34793 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  G  e.  T )  /\  ( N  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( F  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B )  /\  ( R `  G )  =/=  ( R `  F ) ) )  ->  ( iota_ i  e.  T  ( i `  P )  =  ( ( P  .\/  ( R `  G )
)  ./\  ( ( N `  P )  .\/  ( R `  ( G  o.  `' F
) ) ) ) )  e.  T )
1512, 14eqeltrd 2539 1  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  G  e.  T )  /\  ( N  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( F  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B )  /\  ( R `  G )  =/=  ( R `  F ) ) )  ->  ( S `  G )  e.  T
)
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758    =/= wne 2644   class class class wbr 4392    |-> cmpt 4450    _I cid 4731   `'ccnv 4939    |` cres 4942    o. ccom 4944   ` cfv 5518   iota_crio 6152  (class class class)co 6192   Basecbs 14278   lecple 14349   joincjn 15218   meetcmee 15219   Atomscatm 33216   HLchlt 33303   LHypclh 33936   LTrncltrn 34053   trLctrl 34110
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4503  ax-sep 4513  ax-nul 4521  ax-pow 4570  ax-pr 4631  ax-un 6474  ax-riotaBAD 32912
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rmo 2803  df-rab 2804  df-v 3072  df-sbc 3287  df-csb 3389  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-nul 3738  df-if 3892  df-pw 3962  df-sn 3978  df-pr 3980  df-op 3984  df-uni 4192  df-iun 4273  df-iin 4274  df-br 4393  df-opab 4451  df-mpt 4452  df-id 4736  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5481  df-fun 5520  df-fn 5521  df-f 5522  df-f1 5523  df-fo 5524  df-f1o 5525  df-fv 5526  df-riota 6153  df-ov 6195  df-oprab 6196  df-mpt2 6197  df-1st 6679  df-2nd 6680  df-undef 6894  df-map 7318  df-poset 15220  df-plt 15232  df-lub 15248  df-glb 15249  df-join 15250  df-meet 15251  df-p0 15313  df-p1 15314  df-lat 15320  df-clat 15382  df-oposet 33129  df-ol 33131  df-oml 33132  df-covers 33219  df-ats 33220  df-atl 33251  df-cvlat 33275  df-hlat 33304  df-llines 33450  df-lplanes 33451  df-lvols 33452  df-lines 33453  df-psubsp 33455  df-pmap 33456  df-padd 33748  df-lhyp 33940  df-laut 33941  df-ldil 34056  df-ltrn 34057  df-trl 34111
This theorem is referenced by:  cdlemksat  34798  cdlemksv2  34799  cdlemk12  34802  cdlemkoatnle  34803
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