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Theorem cdlemksv 30300
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 5485 . . . . . 6  |-  ( f  =  G  ->  ( R `  f )  =  ( R `  G ) )
21oveq2d 5835 . . . . 5  |-  ( f  =  G  ->  ( P  .\/  ( R `  f ) )  =  ( P  .\/  ( R `  G )
) )
3 coeq1 4840 . . . . . . 7  |-  ( f  =  G  ->  (
f  o.  `' F
)  =  ( G  o.  `' F ) )
43fveq2d 5489 . . . . . 6  |-  ( f  =  G  ->  ( R `  ( f  o.  `' F ) )  =  ( R `  ( G  o.  `' F
) ) )
54oveq2d 5835 . . . . 5  |-  ( f  =  G  ->  (
( N `  P
)  .\/  ( R `  ( f  o.  `' F ) ) )  =  ( ( N `
 P )  .\/  ( R `  ( G  o.  `' F ) ) ) )
62, 5oveq12d 5837 . . . 4  |-  ( f  =  G  ->  (
( P  .\/  ( R `  f )
)  ./\  ( ( N `  P )  .\/  ( R `  (
f  o.  `' F
) ) ) )  =  ( ( P 
.\/  ( R `  G ) )  ./\  ( ( N `  P )  .\/  ( R `  ( G  o.  `' F ) ) ) ) )
76eqeq2d 2295 . . 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 6301 . 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 6303 . 2  |-  ( iota_ i  e.  T ( i `
 P )  =  ( ( P  .\/  ( R `  G ) )  ./\  ( ( N `  P )  .\/  ( R `  ( G  o.  `' F
) ) ) ) )  e.  _V
118, 9, 10fvmpt 5563 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 6    = wceq 1624    e. wcel 1685    e. cmpt 4078   `'ccnv 4687    o. ccom 4692   ` cfv 5221  (class class class)co 5819   iota_crio 6290   Basecbs 13142   lecple 13209   joincjn 14072   meetcmee 14073   Atomscatm 28720   LHypclh 29440   LTrncltrn 29557   trLctrl 29614
This theorem is referenced by:  cdlemksel  30301  cdlemksv2  30303  cdlemkuvN  30320  cdlemkuel  30321  cdlemkuv2  30323  cdlemkuv-2N  30339  cdlemkuu  30351
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1867  ax-ext 2265  ax-sep 4142  ax-nul 4150  ax-pr 4213  ax-un 4511
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 938  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-ral 2549  df-rex 2550  df-reu 2551  df-rab 2553  df-v 2791  df-sbc 2993  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3457  df-if 3567  df-sn 3647  df-pr 3648  df-op 3650  df-uni 3829  df-br 4025  df-opab 4079  df-mpt 4080  df-id 4308  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fv 5229  df-ov 5822  df-iota 6252  df-riota 6299
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