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Theorem cdlemkuvN 29742
Description: Part of proof of Lemma K of [Crawley] p. 118. Value of the sigma1 (p) function  U. (Contributed by NM, 2-Jul-2013.) (New usage is discouraged.)
Hypotheses
Ref Expression
cdlemk1.b  |-  B  =  ( Base `  K
)
cdlemk1.l  |-  .<_  =  ( le `  K )
cdlemk1.j  |-  .\/  =  ( join `  K )
cdlemk1.m  |-  ./\  =  ( meet `  K )
cdlemk1.a  |-  A  =  ( Atoms `  K )
cdlemk1.h  |-  H  =  ( LHyp `  K
)
cdlemk1.t  |-  T  =  ( ( LTrn `  K
) `  W )
cdlemk1.r  |-  R  =  ( ( trL `  K
) `  W )
cdlemk1.s  |-  S  =  ( f  e.  T  |->  ( iota_ i  e.  T
( i `  P
)  =  ( ( P  .\/  ( R `
 f ) ) 
./\  ( ( N `
 P )  .\/  ( R `  ( f  o.  `' F ) ) ) ) ) )
cdlemk1.o  |-  O  =  ( S `  D
)
cdlemk1.u  |-  U  =  ( e  e.  T  |->  ( iota_ j  e.  T
( j `  P
)  =  ( ( P  .\/  ( R `
 e ) ) 
./\  ( ( O `
 P )  .\/  ( R `  ( e  o.  `' D ) ) ) ) ) )
Assertion
Ref Expression
cdlemkuvN  |-  ( G  e.  T  ->  ( U `  G )  =  ( iota_ j  e.  T ( j `  P )  =  ( ( P  .\/  ( R `  G )
)  ./\  ( ( O `  P )  .\/  ( R `  ( G  o.  `' D
) ) ) ) ) )
Distinct variable groups:    f, i,  ./\    .<_ , i    .\/ , f, i    A, i    D, f, i    f, F, i    i, H    i, K    f, N, i    P, f, i    R, f, i    T, f, i    f, W, i    ./\ , e    .\/ , e    D, e    e, j, G    e, O    P, e    R, e    T, e    e, W
Allowed substitution hints:    A( e, f, j)    B( e, f, i, j)    D( j)    P( j)    R( j)    S( e, f, i, j)    T( j)    U( e, f, i, j)    F( e, j)    G( f, i)    H( e, f, j)    .\/ ( j)    K( e, f, j)    .<_ ( e, f, j)    ./\ ( j)    N( e, j)    O( f, i, j)    W( j)

Proof of Theorem cdlemkuvN
StepHypRef Expression
1 cdlemk1.b . 2  |-  B  =  ( Base `  K
)
2 cdlemk1.l . 2  |-  .<_  =  ( le `  K )
3 cdlemk1.j . 2  |-  .\/  =  ( join `  K )
4 cdlemk1.a . 2  |-  A  =  ( Atoms `  K )
5 cdlemk1.h . 2  |-  H  =  ( LHyp `  K
)
6 cdlemk1.t . 2  |-  T  =  ( ( LTrn `  K
) `  W )
7 cdlemk1.r . 2  |-  R  =  ( ( trL `  K
) `  W )
8 cdlemk1.m . 2  |-  ./\  =  ( meet `  K )
9 cdlemk1.u . 2  |-  U  =  ( e  e.  T  |->  ( iota_ j  e.  T
( j `  P
)  =  ( ( P  .\/  ( R `
 e ) ) 
./\  ( ( O `
 P )  .\/  ( R `  ( e  o.  `' D ) ) ) ) ) )
101, 2, 3, 4, 5, 6, 7, 8, 9cdlemksv 29722 1  |-  ( G  e.  T  ->  ( U `  G )  =  ( iota_ j  e.  T ( j `  P )  =  ( ( P  .\/  ( R `  G )
)  ./\  ( ( O `  P )  .\/  ( R `  ( G  o.  `' D
) ) ) ) ) )
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 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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