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Theorem dihffval 31725
Description: The isomorphism H for a lattice  K. Definition of isomorphism map in [Crawley] p. 122 line 3. (Contributed by NM, 28-Jan-2014.)
Hypotheses
Ref Expression
dihval.b  |-  B  =  ( Base `  K
)
dihval.l  |-  .<_  =  ( le `  K )
dihval.j  |-  .\/  =  ( join `  K )
dihval.m  |-  ./\  =  ( meet `  K )
dihval.a  |-  A  =  ( Atoms `  K )
dihval.h  |-  H  =  ( LHyp `  K
)
Assertion
Ref Expression
dihffval  |-  ( K  e.  V  ->  ( DIsoH `  K )  =  ( w  e.  H  |->  ( x  e.  B  |->  if ( x  .<_  w ,  ( ( (
DIsoB `  K ) `  w ) `  x
) ,  ( iota_ u  e.  ( LSubSp `  (
( DVecH `  K ) `  w ) ) A. q  e.  A  (
( -.  q  .<_  w  /\  ( q  .\/  ( x  ./\  w ) )  =  x )  ->  u  =  ( ( ( ( DIsoC `  K ) `  w
) `  q )
( LSSum `  ( ( DVecH `  K ) `  w ) ) ( ( ( DIsoB `  K
) `  w ) `  ( x  ./\  w
) ) ) ) ) ) ) ) )
Distinct variable groups:    A, q    w, H    u, q, w, x, K
Allowed substitution hints:    A( x, w, u)    B( x, w, u, q)    H( x, u, q)    .\/ ( x, w, u, q)    .<_ ( x, w, u, q)    ./\ (
x, w, u, q)    V( x, w, u, q)

Proof of Theorem dihffval
Dummy variable  k is distinct from all other variables.
StepHypRef Expression
1 elex 2932 . 2  |-  ( K  e.  V  ->  K  e.  _V )
2 fveq2 5695 . . . . 5  |-  ( k  =  K  ->  ( LHyp `  k )  =  ( LHyp `  K
) )
3 dihval.h . . . . 5  |-  H  =  ( LHyp `  K
)
42, 3syl6eqr 2462 . . . 4  |-  ( k  =  K  ->  ( LHyp `  k )  =  H )
5 fveq2 5695 . . . . . 6  |-  ( k  =  K  ->  ( Base `  k )  =  ( Base `  K
) )
6 dihval.b . . . . . 6  |-  B  =  ( Base `  K
)
75, 6syl6eqr 2462 . . . . 5  |-  ( k  =  K  ->  ( Base `  k )  =  B )
8 fveq2 5695 . . . . . . . 8  |-  ( k  =  K  ->  ( le `  k )  =  ( le `  K
) )
9 dihval.l . . . . . . . 8  |-  .<_  =  ( le `  K )
108, 9syl6eqr 2462 . . . . . . 7  |-  ( k  =  K  ->  ( le `  k )  = 
.<_  )
1110breqd 4191 . . . . . 6  |-  ( k  =  K  ->  (
x ( le `  k ) w  <->  x  .<_  w ) )
12 fveq2 5695 . . . . . . . 8  |-  ( k  =  K  ->  ( DIsoB `  k )  =  ( DIsoB `  K )
)
1312fveq1d 5697 . . . . . . 7  |-  ( k  =  K  ->  (
( DIsoB `  k ) `  w )  =  ( ( DIsoB `  K ) `  w ) )
1413fveq1d 5697 . . . . . 6  |-  ( k  =  K  ->  (
( ( DIsoB `  k
) `  w ) `  x )  =  ( ( ( DIsoB `  K
) `  w ) `  x ) )
15 fveq2 5695 . . . . . . . . 9  |-  ( k  =  K  ->  ( DVecH `  k )  =  ( DVecH `  K )
)
1615fveq1d 5697 . . . . . . . 8  |-  ( k  =  K  ->  (
( DVecH `  k ) `  w )  =  ( ( DVecH `  K ) `  w ) )
1716fveq2d 5699 . . . . . . 7  |-  ( k  =  K  ->  ( LSubSp `
 ( ( DVecH `  k ) `  w
) )  =  (
LSubSp `  ( ( DVecH `  K ) `  w
) ) )
18 fveq2 5695 . . . . . . . . 9  |-  ( k  =  K  ->  ( Atoms `  k )  =  ( Atoms `  K )
)
19 dihval.a . . . . . . . . 9  |-  A  =  ( Atoms `  K )
2018, 19syl6eqr 2462 . . . . . . . 8  |-  ( k  =  K  ->  ( Atoms `  k )  =  A )
2110breqd 4191 . . . . . . . . . . 11  |-  ( k  =  K  ->  (
q ( le `  k ) w  <->  q  .<_  w ) )
2221notbid 286 . . . . . . . . . 10  |-  ( k  =  K  ->  ( -.  q ( le `  k ) w  <->  -.  q  .<_  w ) )
23 fveq2 5695 . . . . . . . . . . . . 13  |-  ( k  =  K  ->  ( join `  k )  =  ( join `  K
) )
24 dihval.j . . . . . . . . . . . . 13  |-  .\/  =  ( join `  K )
2523, 24syl6eqr 2462 . . . . . . . . . . . 12  |-  ( k  =  K  ->  ( join `  k )  = 
.\/  )
26 eqidd 2413 . . . . . . . . . . . 12  |-  ( k  =  K  ->  q  =  q )
27 fveq2 5695 . . . . . . . . . . . . . 14  |-  ( k  =  K  ->  ( meet `  k )  =  ( meet `  K
) )
28 dihval.m . . . . . . . . . . . . . 14  |-  ./\  =  ( meet `  K )
2927, 28syl6eqr 2462 . . . . . . . . . . . . 13  |-  ( k  =  K  ->  ( meet `  k )  = 
./\  )
3029oveqd 6065 . . . . . . . . . . . 12  |-  ( k  =  K  ->  (
x ( meet `  k
) w )  =  ( x  ./\  w
) )
3125, 26, 30oveq123d 6069 . . . . . . . . . . 11  |-  ( k  =  K  ->  (
q ( join `  k
) ( x (
meet `  k )
w ) )  =  ( q  .\/  (
x  ./\  w )
) )
3231eqeq1d 2420 . . . . . . . . . 10  |-  ( k  =  K  ->  (
( q ( join `  k ) ( x ( meet `  k
) w ) )  =  x  <->  ( q  .\/  ( x  ./\  w
) )  =  x ) )
3322, 32anbi12d 692 . . . . . . . . 9  |-  ( k  =  K  ->  (
( -.  q ( le `  k ) w  /\  ( q ( join `  k
) ( x (
meet `  k )
w ) )  =  x )  <->  ( -.  q  .<_  w  /\  (
q  .\/  ( x  ./\  w ) )  =  x ) ) )
3416fveq2d 5699 . . . . . . . . . . 11  |-  ( k  =  K  ->  ( LSSum `  ( ( DVecH `  k ) `  w
) )  =  (
LSSum `  ( ( DVecH `  K ) `  w
) ) )
35 fveq2 5695 . . . . . . . . . . . . 13  |-  ( k  =  K  ->  ( DIsoC `  k )  =  ( DIsoC `  K )
)
3635fveq1d 5697 . . . . . . . . . . . 12  |-  ( k  =  K  ->  (
( DIsoC `  k ) `  w )  =  ( ( DIsoC `  K ) `  w ) )
3736fveq1d 5697 . . . . . . . . . . 11  |-  ( k  =  K  ->  (
( ( DIsoC `  k
) `  w ) `  q )  =  ( ( ( DIsoC `  K
) `  w ) `  q ) )
3813, 30fveq12d 5701 . . . . . . . . . . 11  |-  ( k  =  K  ->  (
( ( DIsoB `  k
) `  w ) `  ( x ( meet `  k ) w ) )  =  ( ( ( DIsoB `  K ) `  w ) `  (
x  ./\  w )
) )
3934, 37, 38oveq123d 6069 . . . . . . . . . 10  |-  ( k  =  K  ->  (
( ( ( DIsoC `  k ) `  w
) `  q )
( LSSum `  ( ( DVecH `  k ) `  w ) ) ( ( ( DIsoB `  k
) `  w ) `  ( x ( meet `  k ) w ) ) )  =  ( ( ( ( DIsoC `  K ) `  w
) `  q )
( LSSum `  ( ( DVecH `  K ) `  w ) ) ( ( ( DIsoB `  K
) `  w ) `  ( x  ./\  w
) ) ) )
4039eqeq2d 2423 . . . . . . . . 9  |-  ( k  =  K  ->  (
u  =  ( ( ( ( DIsoC `  k
) `  w ) `  q ) ( LSSum `  ( ( DVecH `  k
) `  w )
) ( ( (
DIsoB `  k ) `  w ) `  (
x ( meet `  k
) w ) ) )  <->  u  =  (
( ( ( DIsoC `  K ) `  w
) `  q )
( LSSum `  ( ( DVecH `  K ) `  w ) ) ( ( ( DIsoB `  K
) `  w ) `  ( x  ./\  w
) ) ) ) )
4133, 40imbi12d 312 . . . . . . . 8  |-  ( k  =  K  ->  (
( ( -.  q
( le `  k
) w  /\  (
q ( join `  k
) ( x (
meet `  k )
w ) )  =  x )  ->  u  =  ( ( ( ( DIsoC `  k ) `  w ) `  q
) ( LSSum `  (
( DVecH `  k ) `  w ) ) ( ( ( DIsoB `  k
) `  w ) `  ( x ( meet `  k ) w ) ) ) )  <->  ( ( -.  q  .<_  w  /\  ( q  .\/  (
x  ./\  w )
)  =  x )  ->  u  =  ( ( ( ( DIsoC `  K ) `  w
) `  q )
( LSSum `  ( ( DVecH `  K ) `  w ) ) ( ( ( DIsoB `  K
) `  w ) `  ( x  ./\  w
) ) ) ) ) )
4220, 41raleqbidv 2884 . . . . . . 7  |-  ( k  =  K  ->  ( A. q  e.  ( Atoms `  k ) ( ( -.  q ( le `  k ) w  /\  ( q ( join `  k
) ( x (
meet `  k )
w ) )  =  x )  ->  u  =  ( ( ( ( DIsoC `  k ) `  w ) `  q
) ( LSSum `  (
( DVecH `  k ) `  w ) ) ( ( ( DIsoB `  k
) `  w ) `  ( x ( meet `  k ) w ) ) ) )  <->  A. q  e.  A  ( ( -.  q  .<_  w  /\  ( q  .\/  (
x  ./\  w )
)  =  x )  ->  u  =  ( ( ( ( DIsoC `  K ) `  w
) `  q )
( LSSum `  ( ( DVecH `  K ) `  w ) ) ( ( ( DIsoB `  K
) `  w ) `  ( x  ./\  w
) ) ) ) ) )
4317, 42riotaeqbidv 6519 . . . . . 6  |-  ( k  =  K  ->  ( iota_ u  e.  ( LSubSp `  ( ( DVecH `  k
) `  w )
) A. q  e.  ( Atoms `  k )
( ( -.  q
( le `  k
) w  /\  (
q ( join `  k
) ( x (
meet `  k )
w ) )  =  x )  ->  u  =  ( ( ( ( DIsoC `  k ) `  w ) `  q
) ( LSSum `  (
( DVecH `  k ) `  w ) ) ( ( ( DIsoB `  k
) `  w ) `  ( x ( meet `  k ) w ) ) ) ) )  =  ( iota_ u  e.  ( LSubSp `  ( ( DVecH `  K ) `  w ) ) A. q  e.  A  (
( -.  q  .<_  w  /\  ( q  .\/  ( x  ./\  w ) )  =  x )  ->  u  =  ( ( ( ( DIsoC `  K ) `  w
) `  q )
( LSSum `  ( ( DVecH `  K ) `  w ) ) ( ( ( DIsoB `  K
) `  w ) `  ( x  ./\  w
) ) ) ) ) )
4411, 14, 43ifbieq12d 3729 . . . . 5  |-  ( k  =  K  ->  if ( x ( le
`  k ) w ,  ( ( (
DIsoB `  k ) `  w ) `  x
) ,  ( iota_ u  e.  ( LSubSp `  (
( DVecH `  k ) `  w ) ) A. q  e.  ( Atoms `  k ) ( ( -.  q ( le
`  k ) w  /\  ( q (
join `  k )
( x ( meet `  k ) w ) )  =  x )  ->  u  =  ( ( ( ( DIsoC `  k ) `  w
) `  q )
( LSSum `  ( ( DVecH `  k ) `  w ) ) ( ( ( DIsoB `  k
) `  w ) `  ( x ( meet `  k ) w ) ) ) ) ) )  =  if ( x  .<_  w , 
( ( ( DIsoB `  K ) `  w
) `  x ) ,  ( iota_ u  e.  ( LSubSp `  ( ( DVecH `  K ) `  w ) ) A. q  e.  A  (
( -.  q  .<_  w  /\  ( q  .\/  ( x  ./\  w ) )  =  x )  ->  u  =  ( ( ( ( DIsoC `  K ) `  w
) `  q )
( LSSum `  ( ( DVecH `  K ) `  w ) ) ( ( ( DIsoB `  K
) `  w ) `  ( x  ./\  w
) ) ) ) ) ) )
457, 44mpteq12dv 4255 . . . 4  |-  ( k  =  K  ->  (
x  e.  ( Base `  k )  |->  if ( x ( le `  k ) w ,  ( ( ( DIsoB `  k ) `  w
) `  x ) ,  ( iota_ u  e.  ( LSubSp `  ( ( DVecH `  k ) `  w ) ) A. q  e.  ( Atoms `  k ) ( ( -.  q ( le
`  k ) w  /\  ( q (
join `  k )
( x ( meet `  k ) w ) )  =  x )  ->  u  =  ( ( ( ( DIsoC `  k ) `  w
) `  q )
( LSSum `  ( ( DVecH `  k ) `  w ) ) ( ( ( DIsoB `  k
) `  w ) `  ( x ( meet `  k ) w ) ) ) ) ) ) )  =  ( x  e.  B  |->  if ( x  .<_  w ,  ( ( ( DIsoB `  K ) `  w
) `  x ) ,  ( iota_ u  e.  ( LSubSp `  ( ( DVecH `  K ) `  w ) ) A. q  e.  A  (
( -.  q  .<_  w  /\  ( q  .\/  ( x  ./\  w ) )  =  x )  ->  u  =  ( ( ( ( DIsoC `  K ) `  w
) `  q )
( LSSum `  ( ( DVecH `  K ) `  w ) ) ( ( ( DIsoB `  K
) `  w ) `  ( x  ./\  w
) ) ) ) ) ) ) )
464, 45mpteq12dv 4255 . . 3  |-  ( k  =  K  ->  (
w  e.  ( LHyp `  k )  |->  ( x  e.  ( Base `  k
)  |->  if ( x ( le `  k
) w ,  ( ( ( DIsoB `  k
) `  w ) `  x ) ,  (
iota_ u  e.  ( LSubSp `
 ( ( DVecH `  k ) `  w
) ) A. q  e.  ( Atoms `  k )
( ( -.  q
( le `  k
) w  /\  (
q ( join `  k
) ( x (
meet `  k )
w ) )  =  x )  ->  u  =  ( ( ( ( DIsoC `  k ) `  w ) `  q
) ( LSSum `  (
( DVecH `  k ) `  w ) ) ( ( ( DIsoB `  k
) `  w ) `  ( x ( meet `  k ) w ) ) ) ) ) ) ) )  =  ( w  e.  H  |->  ( x  e.  B  |->  if ( x  .<_  w ,  ( ( (
DIsoB `  K ) `  w ) `  x
) ,  ( iota_ u  e.  ( LSubSp `  (
( DVecH `  K ) `  w ) ) A. q  e.  A  (
( -.  q  .<_  w  /\  ( q  .\/  ( x  ./\  w ) )  =  x )  ->  u  =  ( ( ( ( DIsoC `  K ) `  w
) `  q )
( LSSum `  ( ( DVecH `  K ) `  w ) ) ( ( ( DIsoB `  K
) `  w ) `  ( x  ./\  w
) ) ) ) ) ) ) ) )
47 df-dih 31724 . . 3  |-  DIsoH  =  ( k  e.  _V  |->  ( w  e.  ( LHyp `  k )  |->  ( x  e.  ( Base `  k
)  |->  if ( x ( le `  k
) w ,  ( ( ( DIsoB `  k
) `  w ) `  x ) ,  (
iota_ u  e.  ( LSubSp `
 ( ( DVecH `  k ) `  w
) ) A. q  e.  ( Atoms `  k )
( ( -.  q
( le `  k
) w  /\  (
q ( join `  k
) ( x (
meet `  k )
w ) )  =  x )  ->  u  =  ( ( ( ( DIsoC `  k ) `  w ) `  q
) ( LSSum `  (
( DVecH `  k ) `  w ) ) ( ( ( DIsoB `  k
) `  w ) `  ( x ( meet `  k ) w ) ) ) ) ) ) ) ) )
48 fvex 5709 . . . . 5  |-  ( LHyp `  K )  e.  _V
493, 48eqeltri 2482 . . . 4  |-  H  e. 
_V
5049mptex 5933 . . 3  |-  ( w  e.  H  |->  ( x  e.  B  |->  if ( x  .<_  w , 
( ( ( DIsoB `  K ) `  w
) `  x ) ,  ( iota_ u  e.  ( LSubSp `  ( ( DVecH `  K ) `  w ) ) A. q  e.  A  (
( -.  q  .<_  w  /\  ( q  .\/  ( x  ./\  w ) )  =  x )  ->  u  =  ( ( ( ( DIsoC `  K ) `  w
) `  q )
( LSSum `  ( ( DVecH `  K ) `  w ) ) ( ( ( DIsoB `  K
) `  w ) `  ( x  ./\  w
) ) ) ) ) ) ) )  e.  _V
5146, 47, 50fvmpt 5773 . 2  |-  ( K  e.  _V  ->  ( DIsoH `  K )  =  ( w  e.  H  |->  ( x  e.  B  |->  if ( x  .<_  w ,  ( ( (
DIsoB `  K ) `  w ) `  x
) ,  ( iota_ u  e.  ( LSubSp `  (
( DVecH `  K ) `  w ) ) A. q  e.  A  (
( -.  q  .<_  w  /\  ( q  .\/  ( x  ./\  w ) )  =  x )  ->  u  =  ( ( ( ( DIsoC `  K ) `  w
) `  q )
( LSSum `  ( ( DVecH `  K ) `  w ) ) ( ( ( DIsoB `  K
) `  w ) `  ( x  ./\  w
) ) ) ) ) ) ) ) )
521, 51syl 16 1  |-  ( K  e.  V  ->  ( DIsoH `  K )  =  ( w  e.  H  |->  ( x  e.  B  |->  if ( x  .<_  w ,  ( ( (
DIsoB `  K ) `  w ) `  x
) ,  ( iota_ u  e.  ( LSubSp `  (
( DVecH `  K ) `  w ) ) A. q  e.  A  (
( -.  q  .<_  w  /\  ( q  .\/  ( x  ./\  w ) )  =  x )  ->  u  =  ( ( ( ( DIsoC `  K ) `  w
) `  q )
( LSSum `  ( ( DVecH `  K ) `  w ) ) ( ( ( DIsoB `  K
) `  w ) `  ( x  ./\  w
) ) ) ) ) ) ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1721   A.wral 2674   _Vcvv 2924   ifcif 3707   class class class wbr 4180    e. cmpt 4234   ` cfv 5421  (class class class)co 6048   iota_crio 6509   Basecbs 13432   lecple 13499   joincjn 14364   meetcmee 14365   LSSumclsm 15231   LSubSpclss 15971   Atomscatm 29758   LHypclh 30478   DVecHcdvh 31573   DIsoBcdib 31633   DIsoCcdic 31667   DIsoHcdih 31723
This theorem is referenced by:  dihfval  31726
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-rep 4288  ax-sep 4298  ax-nul 4306  ax-pr 4371
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-ral 2679  df-rex 2680  df-reu 2681  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-sn 3788  df-pr 3789  df-op 3791  df-uni 3984  df-iun 4063  df-br 4181  df-opab 4235  df-mpt 4236  df-id 4466  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-ov 6051  df-riota 6516  df-dih 31724
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