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Theorem cdlemksv 29722
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 5377 . . . . . 6  |-  ( f  =  G  ->  ( R `  f )  =  ( R `  G ) )
21oveq2d 5726 . . . . 5  |-  ( f  =  G  ->  ( P  .\/  ( R `  f ) )  =  ( P  .\/  ( R `  G )
) )
3 coeq1 4748 . . . . . . 7  |-  ( f  =  G  ->  (
f  o.  `' F
)  =  ( G  o.  `' F ) )
43fveq2d 5381 . . . . . 6  |-  ( f  =  G  ->  ( R `  ( f  o.  `' F ) )  =  ( R `  ( G  o.  `' F
) ) )
54oveq2d 5726 . . . . 5  |-  ( f  =  G  ->  (
( N `  P
)  .\/  ( R `  ( f  o.  `' F ) ) )  =  ( ( N `
 P )  .\/  ( R `  ( G  o.  `' F ) ) ) )
62, 5oveq12d 5728 . . . 4  |-  ( f  =  G  ->  (
( P  .\/  ( R `  f )
)  ./\  ( ( N `  P )  .\/  ( R `  (
f  o.  `' F
) ) ) )  =  ( ( P 
.\/  ( R `  G ) )  ./\  ( ( N `  P )  .\/  ( R `  ( G  o.  `' F ) ) ) ) )
76eqeq2d 2264 . . 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 6192 . 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 6194 . 2  |-  ( iota_ i  e.  T ( i `
 P )  =  ( ( P  .\/  ( R `  G ) )  ./\  ( ( N `  P )  .\/  ( R `  ( G  o.  `' F
) ) ) ) )  e.  _V
118, 9, 10fvmpt 5454 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 1619    e. wcel 1621    e. cmpt 3974   `'ccnv 4579    o. ccom 4584   ` cfv 4592  (class class class)co 5710   iota_crio 6181   Basecbs 13022   lecple 13089   joincjn 13922   meetcmee 13923   Atomscatm 28142   LHypclh 28862   LTrncltrn 28979   trLctrl 29036
This theorem is referenced by:  cdlemksel  29723  cdlemksv2  29725  cdlemkuvN  29742  cdlemkuel  29743  cdlemkuv2  29745  cdlemkuv-2N  29761  cdlemkuu  29773
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234  ax-sep 4038  ax-nul 4046  ax-pr 4108  ax-un 4403
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2516  df-v 2729  df-sbc 2922  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-nul 3363  df-if 3471  df-sn 3550  df-pr 3551  df-op 3553  df-uni 3728  df-br 3921  df-opab 3975  df-mpt 3976  df-id 4202  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 4602  df-fv 4608  df-ov 5713  df-iota 6143  df-riota 6190
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