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Theorem cdlemksv 31033
Description: Part of proof of Lemma K of [Crawley] p. 118. Value of the sigma(p) function. (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
cdlemksv  |-  ( G  e.  T  ->  ( S `  G )  =  ( iota_ i  e.  T ( i `  P )  =  ( ( P  .\/  ( R `  G )
)  ./\  ( ( N `  P )  .\/  ( R `  ( G  o.  `' F
) ) ) ) ) )
Distinct variable groups:    ./\ , f    .\/ , f    f, F    f, i, G    f, N    P, f    R, f    T, f    f, W
Allowed substitution hints:    A( f, i)    B( f, i)    P( i)    R( i)    S( f, i)    T( i)    F( i)    H( f, i)    .\/ ( i)    K( f, i)    .<_ ( f, i)    ./\ ( i)    N( i)    W( i)

Proof of Theorem cdlemksv
StepHypRef Expression
1 fveq2 5525 . . . . . 6  |-  ( f  =  G  ->  ( R `  f )  =  ( R `  G ) )
21oveq2d 5874 . . . . 5  |-  ( f  =  G  ->  ( P  .\/  ( R `  f ) )  =  ( P  .\/  ( R `  G )
) )
3 coeq1 4841 . . . . . . 7  |-  ( f  =  G  ->  (
f  o.  `' F
)  =  ( G  o.  `' F ) )
43fveq2d 5529 . . . . . 6  |-  ( f  =  G  ->  ( R `  ( f  o.  `' F ) )  =  ( R `  ( G  o.  `' F
) ) )
54oveq2d 5874 . . . . 5  |-  ( f  =  G  ->  (
( N `  P
)  .\/  ( R `  ( f  o.  `' F ) ) )  =  ( ( N `
 P )  .\/  ( R `  ( G  o.  `' F ) ) ) )
62, 5oveq12d 5876 . . . 4  |-  ( f  =  G  ->  (
( P  .\/  ( R `  f )
)  ./\  ( ( N `  P )  .\/  ( R `  (
f  o.  `' F
) ) ) )  =  ( ( P 
.\/  ( R `  G ) )  ./\  ( ( N `  P )  .\/  ( R `  ( G  o.  `' F ) ) ) ) )
76eqeq2d 2294 . . 3  |-  ( f  =  G  ->  (
( i `  P
)  =  ( ( P  .\/  ( R `
 f ) ) 
./\  ( ( N `
 P )  .\/  ( R `  ( f  o.  `' F ) ) ) )  <->  ( i `  P )  =  ( ( P  .\/  ( R `  G )
)  ./\  ( ( N `  P )  .\/  ( R `  ( G  o.  `' F
) ) ) ) ) )
87riotabidv 6306 . 2  |-  ( f  =  G  ->  ( iota_ i  e.  T ( i `  P )  =  ( ( P 
.\/  ( R `  f ) )  ./\  ( ( N `  P )  .\/  ( R `  ( f  o.  `' F ) ) ) ) )  =  (
iota_ i  e.  T
( i `  P
)  =  ( ( P  .\/  ( R `
 G ) ) 
./\  ( ( N `
 P )  .\/  ( R `  ( G  o.  `' F ) ) ) ) ) )
9 cdlemk.s . 2  |-  S  =  ( f  e.  T  |->  ( iota_ i  e.  T
( i `  P
)  =  ( ( P  .\/  ( R `
 f ) ) 
./\  ( ( N `
 P )  .\/  ( R `  ( f  o.  `' F ) ) ) ) ) )
10 riotaex 6308 . 2  |-  ( iota_ i  e.  T ( i `
 P )  =  ( ( P  .\/  ( R `  G ) )  ./\  ( ( N `  P )  .\/  ( R `  ( G  o.  `' F
) ) ) ) )  e.  _V
118, 9, 10fvmpt 5602 1  |-  ( G  e.  T  ->  ( S `  G )  =  ( iota_ i  e.  T ( i `  P )  =  ( ( P  .\/  ( R `  G )
)  ./\  ( ( N `  P )  .\/  ( R `  ( G  o.  `' F
) ) ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623    e. wcel 1684    e. cmpt 4077   `'ccnv 4688    o. ccom 4693   ` cfv 5255  (class class class)co 5858   iota_crio 6297   Basecbs 13148   lecple 13215   joincjn 14078   meetcmee 14079   Atomscatm 29453   LHypclh 30173   LTrncltrn 30290   trLctrl 30347
This theorem is referenced by:  cdlemksel  31034  cdlemksv2  31036  cdlemkuvN  31053  cdlemkuel  31054  cdlemkuv2  31056  cdlemkuv-2N  31072  cdlemkuu  31084
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-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  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-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  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-iota 5219  df-fun 5257  df-fv 5263  df-ov 5861  df-riota 6304
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