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Theorem cdlemksv 31102
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 5608 . . . . . 6  |-  ( f  =  G  ->  ( R `  f )  =  ( R `  G ) )
21oveq2d 5961 . . . . 5  |-  ( f  =  G  ->  ( P  .\/  ( R `  f ) )  =  ( P  .\/  ( R `  G )
) )
3 coeq1 4923 . . . . . . 7  |-  ( f  =  G  ->  (
f  o.  `' F
)  =  ( G  o.  `' F ) )
43fveq2d 5612 . . . . . 6  |-  ( f  =  G  ->  ( R `  ( f  o.  `' F ) )  =  ( R `  ( G  o.  `' F
) ) )
54oveq2d 5961 . . . . 5  |-  ( f  =  G  ->  (
( N `  P
)  .\/  ( R `  ( f  o.  `' F ) ) )  =  ( ( N `
 P )  .\/  ( R `  ( G  o.  `' F ) ) ) )
62, 5oveq12d 5963 . . . 4  |-  ( f  =  G  ->  (
( P  .\/  ( R `  f )
)  ./\  ( ( N `  P )  .\/  ( R `  (
f  o.  `' F
) ) ) )  =  ( ( P 
.\/  ( R `  G ) )  ./\  ( ( N `  P )  .\/  ( R `  ( G  o.  `' F ) ) ) ) )
76eqeq2d 2369 . . 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 6393 . 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 6395 . 2  |-  ( iota_ i  e.  T ( i `
 P )  =  ( ( P  .\/  ( R `  G ) )  ./\  ( ( N `  P )  .\/  ( R `  ( G  o.  `' F
) ) ) ) )  e.  _V
118, 9, 10fvmpt 5685 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 1642    e. wcel 1710    e. cmpt 4158   `'ccnv 4770    o. ccom 4775   ` cfv 5337  (class class class)co 5945   iota_crio 6384   Basecbs 13245   lecple 13312   joincjn 14177   meetcmee 14178   Atomscatm 29522   LHypclh 30242   LTrncltrn 30359   trLctrl 30416
This theorem is referenced by:  cdlemksel  31103  cdlemksv2  31105  cdlemkuvN  31122  cdlemkuel  31123  cdlemkuv2  31125  cdlemkuv-2N  31141  cdlemkuu  31153
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4222  ax-nul 4230  ax-pr 4295
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-reu 2626  df-rab 2628  df-v 2866  df-sbc 3068  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3909  df-br 4105  df-opab 4159  df-mpt 4160  df-id 4391  df-xp 4777  df-rel 4778  df-cnv 4779  df-co 4780  df-dm 4781  df-iota 5301  df-fun 5339  df-fv 5345  df-ov 5948  df-riota 6391
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