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Theorem cdlemkuu 31084
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 5525 . . . . . . . . 9  |-  ( d  =  D  ->  ( S `  d )  =  ( S `  D ) )
2 cdlemk3.o2 . . . . . . . . 9  |-  Q  =  ( S `  D
)
31, 2syl6eqr 2333 . . . . . . . 8  |-  ( d  =  D  ->  ( S `  d )  =  Q )
43fveq1d 5527 . . . . . . 7  |-  ( d  =  D  ->  (
( S `  d
) `  P )  =  ( Q `  P ) )
5 cnveq 4855 . . . . . . . . 9  |-  ( d  =  D  ->  `' d  =  `' D
)
65coeq2d 4846 . . . . . . . 8  |-  ( d  =  D  ->  (
e  o.  `' d )  =  ( e  o.  `' D ) )
76fveq2d 5529 . . . . . . 7  |-  ( d  =  D  ->  ( R `  ( e  o.  `' d ) )  =  ( R `  ( e  o.  `' D ) ) )
84, 7oveq12d 5876 . . . . . 6  |-  ( d  =  D  ->  (
( ( S `  d ) `  P
)  .\/  ( R `  ( e  o.  `' d ) ) )  =  ( ( Q `
 P )  .\/  ( R `  ( e  o.  `' D ) ) ) )
98oveq2d 5874 . . . . 5  |-  ( d  =  D  ->  (
( P  .\/  ( R `  e )
)  ./\  ( (
( S `  d
) `  P )  .\/  ( R `  (
e  o.  `' d ) ) ) )  =  ( ( P 
.\/  ( R `  e ) )  ./\  ( ( Q `  P )  .\/  ( R `  ( e  o.  `' D ) ) ) ) )
109eqeq2d 2294 . . . 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 6306 . . 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 5525 . . . . . . 7  |-  ( e  =  G  ->  ( R `  e )  =  ( R `  G ) )
1312oveq2d 5874 . . . . . 6  |-  ( e  =  G  ->  ( P  .\/  ( R `  e ) )  =  ( P  .\/  ( R `  G )
) )
14 coeq1 4841 . . . . . . . 8  |-  ( e  =  G  ->  (
e  o.  `' D
)  =  ( G  o.  `' D ) )
1514fveq2d 5529 . . . . . . 7  |-  ( e  =  G  ->  ( R `  ( e  o.  `' D ) )  =  ( R `  ( G  o.  `' D
) ) )
1615oveq2d 5874 . . . . . 6  |-  ( e  =  G  ->  (
( Q `  P
)  .\/  ( R `  ( e  o.  `' D ) ) )  =  ( ( Q `
 P )  .\/  ( R `  ( G  o.  `' D ) ) ) )
1713, 16oveq12d 5876 . . . . 5  |-  ( e  =  G  ->  (
( P  .\/  ( R `  e )
)  ./\  ( ( Q `  P )  .\/  ( R `  (
e  o.  `' D
) ) ) )  =  ( ( P 
.\/  ( R `  G ) )  ./\  ( ( Q `  P )  .\/  ( R `  ( G  o.  `' D ) ) ) ) )
1817eqeq2d 2294 . . . 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 6306 . . 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 6308 . . 3  |-  ( iota_ j  e.  T ( j `
 P )  =  ( ( P  .\/  ( R `  G ) )  ./\  ( ( Q `  P )  .\/  ( R `  ( G  o.  `' D
) ) ) ) )  e.  _V
2211, 19, 20, 21ovmpt2 5983 . 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 31033 . . 3  |-  ( G  e.  T  ->  ( Z `  G )  =  ( iota_ j  e.  T ( j `  P )  =  ( ( P  .\/  ( R `  G )
)  ./\  ( ( Q `  P )  .\/  ( R `  ( G  o.  `' D
) ) ) ) ) )
3332adantl 452 . 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 2318 1  |-  ( ( D  e.  T  /\  G  e.  T )  ->  ( D Y G )  =  ( Z `
 G ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684    e. cmpt 4077   `'ccnv 4688    o. ccom 4693   ` cfv 5255  (class class class)co 5858    e. cmpt2 5860   iota_crio 6297   Basecbs 13148   lecple 13215   joincjn 14078   meetcmee 14079   Atomscatm 29453   LHypclh 30173   LTrncltrn 30290   trLctrl 30347
This theorem is referenced by:  cdlemk31  31085  cdlemkuel-3  31087  cdlemkuv2-3N  31088  cdlemk18-3N  31089  cdlemk22-3  31090  cdlemkyu  31116
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-iota 5219  df-fun 5257  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-riota 6304
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