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Theorem cdlemkuu 31619
Description: Convert between function and operation forms of  Y. TODO: Use operation form everywhere. (Contributed by NM, 6-Jul-2013.)
Hypotheses
Ref Expression
cdlemk3.b  |-  B  =  ( Base `  K
)
cdlemk3.l  |-  .<_  =  ( le `  K )
cdlemk3.j  |-  .\/  =  ( join `  K )
cdlemk3.m  |-  ./\  =  ( meet `  K )
cdlemk3.a  |-  A  =  ( Atoms `  K )
cdlemk3.h  |-  H  =  ( LHyp `  K
)
cdlemk3.t  |-  T  =  ( ( LTrn `  K
) `  W )
cdlemk3.r  |-  R  =  ( ( trL `  K
) `  W )
cdlemk3.s  |-  S  =  ( f  e.  T  |->  ( iota_ i  e.  T
( i `  P
)  =  ( ( P  .\/  ( R `
 f ) ) 
./\  ( ( N `
 P )  .\/  ( R `  ( f  o.  `' F ) ) ) ) ) )
cdlemk3.u1  |-  Y  =  ( d  e.  T ,  e  e.  T  |->  ( iota_ j  e.  T
( j `  P
)  =  ( ( P  .\/  ( R `
 e ) ) 
./\  ( ( ( S `  d ) `
 P )  .\/  ( R `  ( e  o.  `' d ) ) ) ) ) )
cdlemk3.o2  |-  Q  =  ( S `  D
)
cdlemk3.u2  |-  Z  =  ( e  e.  T  |->  ( iota_ j  e.  T
( j `  P
)  =  ( ( P  .\/  ( R `
 e ) ) 
./\  ( ( Q `
 P )  .\/  ( R `  ( e  o.  `' D ) ) ) ) ) )
Assertion
Ref Expression
cdlemkuu  |-  ( ( D  e.  T  /\  G  e.  T )  ->  ( D Y G )  =  ( Z `
 G ) )
Distinct variable groups:    e, d,
f, i,  ./\    .<_ , i    .\/ , d, e, f, i    A, i    j, d, D, e, f, i    f, F, i    G, d, e, j   
i, H    i, K    f, N, i    P, d, e, f, i    Q, d, e    R, d, e, f, i    T, d, e, f, i    W, d, e, f, i
Allowed substitution hints:    A( e, f, j, d)    B( e, f, i, j, d)    P( j)    Q( f, i, j)    R( j)    S( e, f, i, j, d)    T( j)    F( e, j, d)    G( f, i)    H( e, f, j, d)    .\/ ( j)    K( e, f, j, d)    .<_ ( e, f, j, d)    ./\ ( j)    N( e, j, d)    W( j)    Y( e, f, i, j, d)    Z( e, f, i, j, d)

Proof of Theorem cdlemkuu
StepHypRef Expression
1 fveq2 5720 . . . . . . . . 9  |-  ( d  =  D  ->  ( S `  d )  =  ( S `  D ) )
2 cdlemk3.o2 . . . . . . . . 9  |-  Q  =  ( S `  D
)
31, 2syl6eqr 2485 . . . . . . . 8  |-  ( d  =  D  ->  ( S `  d )  =  Q )
43fveq1d 5722 . . . . . . 7  |-  ( d  =  D  ->  (
( S `  d
) `  P )  =  ( Q `  P ) )
5 cnveq 5038 . . . . . . . . 9  |-  ( d  =  D  ->  `' d  =  `' D
)
65coeq2d 5027 . . . . . . . 8  |-  ( d  =  D  ->  (
e  o.  `' d )  =  ( e  o.  `' D ) )
76fveq2d 5724 . . . . . . 7  |-  ( d  =  D  ->  ( R `  ( e  o.  `' d ) )  =  ( R `  ( e  o.  `' D ) ) )
84, 7oveq12d 6091 . . . . . 6  |-  ( d  =  D  ->  (
( ( S `  d ) `  P
)  .\/  ( R `  ( e  o.  `' d ) ) )  =  ( ( Q `
 P )  .\/  ( R `  ( e  o.  `' D ) ) ) )
98oveq2d 6089 . . . . 5  |-  ( d  =  D  ->  (
( P  .\/  ( R `  e )
)  ./\  ( (
( S `  d
) `  P )  .\/  ( R `  (
e  o.  `' d ) ) ) )  =  ( ( P 
.\/  ( R `  e ) )  ./\  ( ( Q `  P )  .\/  ( R `  ( e  o.  `' D ) ) ) ) )
109eqeq2d 2446 . . . 4  |-  ( d  =  D  ->  (
( j `  P
)  =  ( ( P  .\/  ( R `
 e ) ) 
./\  ( ( ( S `  d ) `
 P )  .\/  ( R `  ( e  o.  `' d ) ) ) )  <->  ( j `  P )  =  ( ( P  .\/  ( R `  e )
)  ./\  ( ( Q `  P )  .\/  ( R `  (
e  o.  `' D
) ) ) ) ) )
1110riotabidv 6543 . . 3  |-  ( d  =  D  ->  ( iota_ j  e.  T ( j `  P )  =  ( ( P 
.\/  ( R `  e ) )  ./\  ( ( ( S `
 d ) `  P )  .\/  ( R `  ( e  o.  `' d ) ) ) ) )  =  ( iota_ j  e.  T
( j `  P
)  =  ( ( P  .\/  ( R `
 e ) ) 
./\  ( ( Q `
 P )  .\/  ( R `  ( e  o.  `' D ) ) ) ) ) )
12 fveq2 5720 . . . . . . 7  |-  ( e  =  G  ->  ( R `  e )  =  ( R `  G ) )
1312oveq2d 6089 . . . . . 6  |-  ( e  =  G  ->  ( P  .\/  ( R `  e ) )  =  ( P  .\/  ( R `  G )
) )
14 coeq1 5022 . . . . . . . 8  |-  ( e  =  G  ->  (
e  o.  `' D
)  =  ( G  o.  `' D ) )
1514fveq2d 5724 . . . . . . 7  |-  ( e  =  G  ->  ( R `  ( e  o.  `' D ) )  =  ( R `  ( G  o.  `' D
) ) )
1615oveq2d 6089 . . . . . 6  |-  ( e  =  G  ->  (
( Q `  P
)  .\/  ( R `  ( e  o.  `' D ) ) )  =  ( ( Q `
 P )  .\/  ( R `  ( G  o.  `' D ) ) ) )
1713, 16oveq12d 6091 . . . . 5  |-  ( e  =  G  ->  (
( P  .\/  ( R `  e )
)  ./\  ( ( Q `  P )  .\/  ( R `  (
e  o.  `' D
) ) ) )  =  ( ( P 
.\/  ( R `  G ) )  ./\  ( ( Q `  P )  .\/  ( R `  ( G  o.  `' D ) ) ) ) )
1817eqeq2d 2446 . . . 4  |-  ( e  =  G  ->  (
( j `  P
)  =  ( ( P  .\/  ( R `
 e ) ) 
./\  ( ( Q `
 P )  .\/  ( R `  ( e  o.  `' D ) ) ) )  <->  ( j `  P )  =  ( ( P  .\/  ( R `  G )
)  ./\  ( ( Q `  P )  .\/  ( R `  ( G  o.  `' D
) ) ) ) ) )
1918riotabidv 6543 . . 3  |-  ( e  =  G  ->  ( iota_ j  e.  T ( j `  P )  =  ( ( P 
.\/  ( R `  e ) )  ./\  ( ( Q `  P )  .\/  ( R `  ( e  o.  `' D ) ) ) ) )  =  (
iota_ j  e.  T
( j `  P
)  =  ( ( P  .\/  ( R `
 G ) ) 
./\  ( ( Q `
 P )  .\/  ( R `  ( G  o.  `' D ) ) ) ) ) )
20 cdlemk3.u1 . . 3  |-  Y  =  ( d  e.  T ,  e  e.  T  |->  ( iota_ j  e.  T
( j `  P
)  =  ( ( P  .\/  ( R `
 e ) ) 
./\  ( ( ( S `  d ) `
 P )  .\/  ( R `  ( e  o.  `' d ) ) ) ) ) )
21 riotaex 6545 . . 3  |-  ( iota_ j  e.  T ( j `
 P )  =  ( ( P  .\/  ( R `  G ) )  ./\  ( ( Q `  P )  .\/  ( R `  ( G  o.  `' D
) ) ) ) )  e.  _V
2211, 19, 20, 21ovmpt2 6201 . 2  |-  ( ( D  e.  T  /\  G  e.  T )  ->  ( D Y G )  =  ( iota_ j  e.  T ( j `
 P )  =  ( ( P  .\/  ( R `  G ) )  ./\  ( ( Q `  P )  .\/  ( R `  ( G  o.  `' D
) ) ) ) ) )
23 cdlemk3.b . . . 4  |-  B  =  ( Base `  K
)
24 cdlemk3.l . . . 4  |-  .<_  =  ( le `  K )
25 cdlemk3.j . . . 4  |-  .\/  =  ( join `  K )
26 cdlemk3.a . . . 4  |-  A  =  ( Atoms `  K )
27 cdlemk3.h . . . 4  |-  H  =  ( LHyp `  K
)
28 cdlemk3.t . . . 4  |-  T  =  ( ( LTrn `  K
) `  W )
29 cdlemk3.r . . . 4  |-  R  =  ( ( trL `  K
) `  W )
30 cdlemk3.m . . . 4  |-  ./\  =  ( meet `  K )
31 cdlemk3.u2 . . . 4  |-  Z  =  ( e  e.  T  |->  ( iota_ j  e.  T
( j `  P
)  =  ( ( P  .\/  ( R `
 e ) ) 
./\  ( ( Q `
 P )  .\/  ( R `  ( e  o.  `' D ) ) ) ) ) )
3223, 24, 25, 26, 27, 28, 29, 30, 31cdlemksv 31568 . . 3  |-  ( G  e.  T  ->  ( Z `  G )  =  ( iota_ j  e.  T ( j `  P )  =  ( ( P  .\/  ( R `  G )
)  ./\  ( ( Q `  P )  .\/  ( R `  ( G  o.  `' D
) ) ) ) ) )
3332adantl 453 . 2  |-  ( ( D  e.  T  /\  G  e.  T )  ->  ( Z `  G
)  =  ( iota_ j  e.  T ( j `
 P )  =  ( ( P  .\/  ( R `  G ) )  ./\  ( ( Q `  P )  .\/  ( R `  ( G  o.  `' D
) ) ) ) ) )
3422, 33eqtr4d 2470 1  |-  ( ( D  e.  T  /\  G  e.  T )  ->  ( D Y G )  =  ( Z `
 G ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725    e. cmpt 4258   `'ccnv 4869    o. ccom 4874   ` cfv 5446  (class class class)co 6073    e. cmpt2 6075   iota_crio 6534   Basecbs 13461   lecple 13528   joincjn 14393   meetcmee 14394   Atomscatm 29988   LHypclh 30708   LTrncltrn 30825   trLctrl 30882
This theorem is referenced by:  cdlemk31  31620  cdlemkuel-3  31622  cdlemkuv2-3N  31623  cdlemk18-3N  31624  cdlemk22-3  31625  cdlemkyu  31651
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 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-iota 5410  df-fun 5448  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-riota 6541
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