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Theorem cdlemkuu 34551
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 5703 . . . . . . . . 9  |-  ( d  =  D  ->  ( S `  d )  =  ( S `  D ) )
2 cdlemk3.o2 . . . . . . . . 9  |-  Q  =  ( S `  D
)
31, 2syl6eqr 2493 . . . . . . . 8  |-  ( d  =  D  ->  ( S `  d )  =  Q )
43fveq1d 5705 . . . . . . 7  |-  ( d  =  D  ->  (
( S `  d
) `  P )  =  ( Q `  P ) )
5 cnveq 5025 . . . . . . . . 9  |-  ( d  =  D  ->  `' d  =  `' D
)
65coeq2d 5014 . . . . . . . 8  |-  ( d  =  D  ->  (
e  o.  `' d )  =  ( e  o.  `' D ) )
76fveq2d 5707 . . . . . . 7  |-  ( d  =  D  ->  ( R `  ( e  o.  `' d ) )  =  ( R `  ( e  o.  `' D ) ) )
84, 7oveq12d 6121 . . . . . 6  |-  ( d  =  D  ->  (
( ( S `  d ) `  P
)  .\/  ( R `  ( e  o.  `' d ) ) )  =  ( ( Q `
 P )  .\/  ( R `  ( e  o.  `' D ) ) ) )
98oveq2d 6119 . . . . 5  |-  ( d  =  D  ->  (
( P  .\/  ( R `  e )
)  ./\  ( (
( S `  d
) `  P )  .\/  ( R `  (
e  o.  `' d ) ) ) )  =  ( ( P 
.\/  ( R `  e ) )  ./\  ( ( Q `  P )  .\/  ( R `  ( e  o.  `' D ) ) ) ) )
109eqeq2d 2454 . . . 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 6066 . . 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 5703 . . . . . . 7  |-  ( e  =  G  ->  ( R `  e )  =  ( R `  G ) )
1312oveq2d 6119 . . . . . 6  |-  ( e  =  G  ->  ( P  .\/  ( R `  e ) )  =  ( P  .\/  ( R `  G )
) )
14 coeq1 5009 . . . . . . . 8  |-  ( e  =  G  ->  (
e  o.  `' D
)  =  ( G  o.  `' D ) )
1514fveq2d 5707 . . . . . . 7  |-  ( e  =  G  ->  ( R `  ( e  o.  `' D ) )  =  ( R `  ( G  o.  `' D
) ) )
1615oveq2d 6119 . . . . . 6  |-  ( e  =  G  ->  (
( Q `  P
)  .\/  ( R `  ( e  o.  `' D ) ) )  =  ( ( Q `
 P )  .\/  ( R `  ( G  o.  `' D ) ) ) )
1713, 16oveq12d 6121 . . . . 5  |-  ( e  =  G  ->  (
( P  .\/  ( R `  e )
)  ./\  ( ( Q `  P )  .\/  ( R `  (
e  o.  `' D
) ) ) )  =  ( ( P 
.\/  ( R `  G ) )  ./\  ( ( Q `  P )  .\/  ( R `  ( G  o.  `' D ) ) ) ) )
1817eqeq2d 2454 . . . 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 6066 . . 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 6068 . . 3  |-  ( iota_ j  e.  T  ( j `
 P )  =  ( ( P  .\/  ( R `  G ) )  ./\  ( ( Q `  P )  .\/  ( R `  ( G  o.  `' D
) ) ) ) )  e.  _V
2211, 19, 20, 21ovmpt2 6238 . 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 34500 . . 3  |-  ( G  e.  T  ->  ( Z `  G )  =  ( iota_ j  e.  T  ( j `  P )  =  ( ( P  .\/  ( R `  G )
)  ./\  ( ( Q `  P )  .\/  ( R `  ( G  o.  `' D
) ) ) ) ) )
3332adantl 466 . 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 2478 1  |-  ( ( D  e.  T  /\  G  e.  T )  ->  ( D Y G )  =  ( Z `
 G ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756    e. cmpt 4362   `'ccnv 4851    o. ccom 4856   ` cfv 5430   iota_crio 6063  (class class class)co 6103    e. cmpt2 6105   Basecbs 14186   lecple 14257   joincjn 15126   meetcmee 15127   Atomscatm 32920   LHypclh 33640   LTrncltrn 33757   trLctrl 33814
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4425  ax-nul 4433  ax-pr 4543
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2620  df-ral 2732  df-rex 2733  df-rab 2736  df-v 2986  df-sbc 3199  df-dif 3343  df-un 3345  df-in 3347  df-ss 3354  df-nul 3650  df-if 3804  df-sn 3890  df-pr 3892  df-op 3896  df-uni 4104  df-br 4305  df-opab 4363  df-mpt 4364  df-id 4648  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-iota 5393  df-fun 5432  df-fv 5438  df-riota 6064  df-ov 6106  df-oprab 6107  df-mpt2 6108
This theorem is referenced by:  cdlemk31  34552  cdlemkuel-3  34554  cdlemkuv2-3N  34555  cdlemk18-3N  34556  cdlemk22-3  34557  cdlemkyu  34583
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