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Theorem cdlemkuv-2N 30340
Description: Part of proof of Lemma K of [Crawley] p. 118. Value of the sigma2 (p) function, given  V. (Contributed by NM, 2-Jul-2013.) (New usage is discouraged.)
Hypotheses
Ref Expression
cdlemk2.b  |-  B  =  ( Base `  K
)
cdlemk2.l  |-  .<_  =  ( le `  K )
cdlemk2.j  |-  .\/  =  ( join `  K )
cdlemk2.m  |-  ./\  =  ( meet `  K )
cdlemk2.a  |-  A  =  ( Atoms `  K )
cdlemk2.h  |-  H  =  ( LHyp `  K
)
cdlemk2.t  |-  T  =  ( ( LTrn `  K
) `  W )
cdlemk2.r  |-  R  =  ( ( trL `  K
) `  W )
cdlemk2.s  |-  S  =  ( f  e.  T  |->  ( iota_ i  e.  T
( i `  P
)  =  ( ( P  .\/  ( R `
 f ) ) 
./\  ( ( N `
 P )  .\/  ( R `  ( f  o.  `' F ) ) ) ) ) )
cdlemk2.q  |-  Q  =  ( S `  C
)
cdlemk2.v  |-  V  =  ( d  e.  T  |->  ( iota_ k  e.  T
( k `  P
)  =  ( ( P  .\/  ( R `
 d ) ) 
./\  ( ( Q `
 P )  .\/  ( R `  ( d  o.  `' C ) ) ) ) ) )
Assertion
Ref Expression
cdlemkuv-2N  |-  ( G  e.  T  ->  ( V `  G )  =  ( iota_ k  e.  T ( k `  P )  =  ( ( P  .\/  ( R `  G )
)  ./\  ( ( Q `  P )  .\/  ( R `  ( G  o.  `' C
) ) ) ) ) )
Distinct variable groups:    f, i,  ./\    .<_ , i    .\/ , f, i    A, i    C, f, i    f, F, i    i, H    i, K    f, N, i    P, f, i    R, f, i    T, f, i    f, W, i    ./\ , d    .\/ , d    C, d    k, d, G    Q, d    P, d    R, d    T, d    W, d
Allowed substitution hints:    A( f, k, d)    B( f, i, k, d)    C( k)    P( k)    Q( f, i, k)    R( k)    S( f, i, k, d)    T( k)    F( k, d)    G( f, i)    H( f, k, d)    .\/ ( k)    K( f, k, d)    .<_ ( f, k, d)    ./\ ( k)    N( k,
d)    V( f, i, k, d)    W( k)

Proof of Theorem cdlemkuv-2N
StepHypRef Expression
1 cdlemk2.b . 2  |-  B  =  ( Base `  K
)
2 cdlemk2.l . 2  |-  .<_  =  ( le `  K )
3 cdlemk2.j . 2  |-  .\/  =  ( join `  K )
4 cdlemk2.a . 2  |-  A  =  ( Atoms `  K )
5 cdlemk2.h . 2  |-  H  =  ( LHyp `  K
)
6 cdlemk2.t . 2  |-  T  =  ( ( LTrn `  K
) `  W )
7 cdlemk2.r . 2  |-  R  =  ( ( trL `  K
) `  W )
8 cdlemk2.m . 2  |-  ./\  =  ( meet `  K )
9 cdlemk2.v . 2  |-  V  =  ( d  e.  T  |->  ( iota_ k  e.  T
( k `  P
)  =  ( ( P  .\/  ( R `
 d ) ) 
./\  ( ( Q `
 P )  .\/  ( R `  ( d  o.  `' C ) ) ) ) ) )
101, 2, 3, 4, 5, 6, 7, 8, 9cdlemksv 30301 1  |-  ( G  e.  T  ->  ( V `  G )  =  ( iota_ k  e.  T ( k `  P )  =  ( ( P  .\/  ( R `  G )
)  ./\  ( ( Q `  P )  .\/  ( R `  ( G  o.  `' C
) ) ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    = wceq 1624    e. wcel 1685    e. cmpt 4079   `'ccnv 4688    o. ccom 4693   ` cfv 5222  (class class class)co 5820   iota_crio 6291   Basecbs 13143   lecple 13210   joincjn 14073   meetcmee 14074   Atomscatm 28721   LHypclh 29441   LTrncltrn 29558   trLctrl 29615
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 1868  ax-ext 2266  ax-sep 4143  ax-nul 4151  ax-pr 4214  ax-un 4512
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 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-ral 2550  df-rex 2551  df-reu 2552  df-rab 2554  df-v 2792  df-sbc 2994  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-nul 3458  df-if 3568  df-sn 3648  df-pr 3649  df-op 3651  df-uni 3830  df-br 4026  df-opab 4080  df-mpt 4081  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-fun 5224  df-fv 5230  df-ov 5823  df-iota 6253  df-riota 6300
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