Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  cdlemkuv-2N Unicode version

Theorem cdlemkuv-2N 31519
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 31480 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 4    = wceq 1652    e. wcel 1725    e. cmpt 4258   `'ccnv 4868    o. ccom 4873   ` cfv 5445  (class class class)co 6072   iota_crio 6533   Basecbs 13457   lecple 13524   joincjn 14389   meetcmee 14390   Atomscatm 29900   LHypclh 30620   LTrncltrn 30737   trLctrl 30794
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4875  df-rel 4876  df-cnv 4877  df-co 4878  df-dm 4879  df-iota 5409  df-fun 5447  df-fv 5453  df-ov 6075  df-riota 6540
  Copyright terms: Public domain W3C validator