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Theorem pmapglb2xN 29886
Description: The projective map of the GLB of a set of lattice elements. Index-set version of pmapglb2N 29885, where we read  S as  S ( i ). Extension of Theorem 15.5.2 of [MaedaMaeda] p. 62 that allows  I  =  (/). (Contributed by NM, 21-Jan-2012.) (New usage is discouraged.)
Hypotheses
Ref Expression
pmapglb2.b  |-  B  =  ( Base `  K
)
pmapglb2.g  |-  G  =  ( glb `  K
)
pmapglb2.a  |-  A  =  ( Atoms `  K )
pmapglb2.m  |-  M  =  ( pmap `  K
)
Assertion
Ref Expression
pmapglb2xN  |-  ( ( K  e.  HL  /\  A. i  e.  I  S  e.  B )  -> 
( M `  ( G `  { y  |  E. i  e.  I 
y  =  S }
) )  =  ( A  i^i  |^|_ i  e.  I  ( M `  S ) ) )
Distinct variable groups:    A, i    y, i, B    i, I,
y    i, K, y    y, S
Allowed substitution hints:    A( y)    S( i)    G( y, i)    M( y, i)

Proof of Theorem pmapglb2xN
StepHypRef Expression
1 hlop 29477 . . . . 5  |-  ( K  e.  HL  ->  K  e.  OP )
2 pmapglb2.g . . . . . . . 8  |-  G  =  ( glb `  K
)
3 eqid 2387 . . . . . . . 8  |-  ( 1.
`  K )  =  ( 1. `  K
)
42, 3glb0N 29308 . . . . . . 7  |-  ( K  e.  OP  ->  ( G `  (/) )  =  ( 1. `  K
) )
54fveq2d 5672 . . . . . 6  |-  ( K  e.  OP  ->  ( M `  ( G `  (/) ) )  =  ( M `  ( 1. `  K ) ) )
6 pmapglb2.a . . . . . . 7  |-  A  =  ( Atoms `  K )
7 pmapglb2.m . . . . . . 7  |-  M  =  ( pmap `  K
)
83, 6, 7pmap1N 29881 . . . . . 6  |-  ( K  e.  OP  ->  ( M `  ( 1. `  K ) )  =  A )
95, 8eqtrd 2419 . . . . 5  |-  ( K  e.  OP  ->  ( M `  ( G `  (/) ) )  =  A )
101, 9syl 16 . . . 4  |-  ( K  e.  HL  ->  ( M `  ( G `  (/) ) )  =  A )
11 rexeq 2848 . . . . . . . . 9  |-  ( I  =  (/)  ->  ( E. i  e.  I  y  =  S  <->  E. i  e.  (/)  y  =  S ) )
1211abbidv 2501 . . . . . . . 8  |-  ( I  =  (/)  ->  { y  |  E. i  e.  I  y  =  S }  =  { y  |  E. i  e.  (/)  y  =  S } )
13 rex0 3584 . . . . . . . . 9  |-  -.  E. i  e.  (/)  y  =  S
1413abf 3604 . . . . . . . 8  |-  { y  |  E. i  e.  (/)  y  =  S }  =  (/)
1512, 14syl6eq 2435 . . . . . . 7  |-  ( I  =  (/)  ->  { y  |  E. i  e.  I  y  =  S }  =  (/) )
1615fveq2d 5672 . . . . . 6  |-  ( I  =  (/)  ->  ( G `
 { y  |  E. i  e.  I 
y  =  S }
)  =  ( G `
 (/) ) )
1716fveq2d 5672 . . . . 5  |-  ( I  =  (/)  ->  ( M `
 ( G `  { y  |  E. i  e.  I  y  =  S } ) )  =  ( M `  ( G `  (/) ) ) )
18 riin0 4105 . . . . 5  |-  ( I  =  (/)  ->  ( A  i^i  |^|_ i  e.  I 
( M `  S
) )  =  A )
1917, 18eqeq12d 2401 . . . 4  |-  ( I  =  (/)  ->  ( ( M `  ( G `
 { y  |  E. i  e.  I 
y  =  S }
) )  =  ( A  i^i  |^|_ i  e.  I  ( M `  S ) )  <->  ( M `  ( G `  (/) ) )  =  A ) )
2010, 19syl5ibrcom 214 . . 3  |-  ( K  e.  HL  ->  (
I  =  (/)  ->  ( M `  ( G `  { y  |  E. i  e.  I  y  =  S } ) )  =  ( A  i^i  |^|_ i  e.  I  ( M `  S ) ) ) )
2120adantr 452 . 2  |-  ( ( K  e.  HL  /\  A. i  e.  I  S  e.  B )  -> 
( I  =  (/)  ->  ( M `  ( G `  { y  |  E. i  e.  I 
y  =  S }
) )  =  ( A  i^i  |^|_ i  e.  I  ( M `  S ) ) ) )
22 pmapglb2.b . . . . 5  |-  B  =  ( Base `  K
)
2322, 2, 7pmapglbx 29883 . . . 4  |-  ( ( K  e.  HL  /\  A. i  e.  I  S  e.  B  /\  I  =/=  (/) )  ->  ( M `  ( G `  { y  |  E. i  e.  I  y  =  S } ) )  =  |^|_ i  e.  I 
( M `  S
) )
24 nfv 1626 . . . . . . . . . 10  |-  F/ i  K  e.  HL
25 nfra1 2699 . . . . . . . . . 10  |-  F/ i A. i  e.  I  S  e.  B
2624, 25nfan 1836 . . . . . . . . 9  |-  F/ i ( K  e.  HL  /\ 
A. i  e.  I  S  e.  B )
27 simpr 448 . . . . . . . . . . 11  |-  ( ( ( K  e.  HL  /\ 
A. i  e.  I  S  e.  B )  /\  i  e.  I
)  ->  i  e.  I )
28 simpll 731 . . . . . . . . . . . 12  |-  ( ( ( K  e.  HL  /\ 
A. i  e.  I  S  e.  B )  /\  i  e.  I
)  ->  K  e.  HL )
29 rsp 2709 . . . . . . . . . . . . . 14  |-  ( A. i  e.  I  S  e.  B  ->  ( i  e.  I  ->  S  e.  B ) )
3029imp 419 . . . . . . . . . . . . 13  |-  ( ( A. i  e.  I  S  e.  B  /\  i  e.  I )  ->  S  e.  B )
3130adantll 695 . . . . . . . . . . . 12  |-  ( ( ( K  e.  HL  /\ 
A. i  e.  I  S  e.  B )  /\  i  e.  I
)  ->  S  e.  B )
3222, 6, 7pmapssat 29873 . . . . . . . . . . . 12  |-  ( ( K  e.  HL  /\  S  e.  B )  ->  ( M `  S
)  C_  A )
3328, 31, 32syl2anc 643 . . . . . . . . . . 11  |-  ( ( ( K  e.  HL  /\ 
A. i  e.  I  S  e.  B )  /\  i  e.  I
)  ->  ( M `  S )  C_  A
)
3427, 33jca 519 . . . . . . . . . 10  |-  ( ( ( K  e.  HL  /\ 
A. i  e.  I  S  e.  B )  /\  i  e.  I
)  ->  ( i  e.  I  /\  ( M `  S )  C_  A ) )
3534ex 424 . . . . . . . . 9  |-  ( ( K  e.  HL  /\  A. i  e.  I  S  e.  B )  -> 
( i  e.  I  ->  ( i  e.  I  /\  ( M `  S
)  C_  A )
) )
3626, 35eximd 1778 . . . . . . . 8  |-  ( ( K  e.  HL  /\  A. i  e.  I  S  e.  B )  -> 
( E. i  i  e.  I  ->  E. i
( i  e.  I  /\  ( M `  S
)  C_  A )
) )
37 n0 3580 . . . . . . . 8  |-  ( I  =/=  (/)  <->  E. i  i  e.  I )
38 df-rex 2655 . . . . . . . 8  |-  ( E. i  e.  I  ( M `  S ) 
C_  A  <->  E. i
( i  e.  I  /\  ( M `  S
)  C_  A )
)
3936, 37, 383imtr4g 262 . . . . . . 7  |-  ( ( K  e.  HL  /\  A. i  e.  I  S  e.  B )  -> 
( I  =/=  (/)  ->  E. i  e.  I  ( M `  S )  C_  A
) )
40393impia 1150 . . . . . 6  |-  ( ( K  e.  HL  /\  A. i  e.  I  S  e.  B  /\  I  =/=  (/) )  ->  E. i  e.  I  ( M `  S )  C_  A
)
41 iinss 4083 . . . . . 6  |-  ( E. i  e.  I  ( M `  S ) 
C_  A  ->  |^|_ i  e.  I  ( M `  S )  C_  A
)
4240, 41syl 16 . . . . 5  |-  ( ( K  e.  HL  /\  A. i  e.  I  S  e.  B  /\  I  =/=  (/) )  ->  |^|_ i  e.  I  ( M `  S )  C_  A
)
43 sseqin2 3503 . . . . 5  |-  ( |^|_ i  e.  I  ( M `  S )  C_  A  <->  ( A  i^i  |^|_ i  e.  I  ( M `  S ) )  =  |^|_ i  e.  I  ( M `  S ) )
4442, 43sylib 189 . . . 4  |-  ( ( K  e.  HL  /\  A. i  e.  I  S  e.  B  /\  I  =/=  (/) )  ->  ( A  i^i  |^|_ i  e.  I 
( M `  S
) )  =  |^|_ i  e.  I  ( M `  S )
)
4523, 44eqtr4d 2422 . . 3  |-  ( ( K  e.  HL  /\  A. i  e.  I  S  e.  B  /\  I  =/=  (/) )  ->  ( M `  ( G `  { y  |  E. i  e.  I  y  =  S } ) )  =  ( A  i^i  |^|_ i  e.  I  ( M `  S ) ) )
46453expia 1155 . 2  |-  ( ( K  e.  HL  /\  A. i  e.  I  S  e.  B )  -> 
( I  =/=  (/)  ->  ( M `  ( G `  { y  |  E. i  e.  I  y  =  S } ) )  =  ( A  i^i  |^|_ i  e.  I  ( M `  S ) ) ) )
4721, 46pm2.61dne 2627 1  |-  ( ( K  e.  HL  /\  A. i  e.  I  S  e.  B )  -> 
( M `  ( G `  { y  |  E. i  e.  I 
y  =  S }
) )  =  ( A  i^i  |^|_ i  e.  I  ( M `  S ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936   E.wex 1547    = wceq 1649    e. wcel 1717   {cab 2373    =/= wne 2550   A.wral 2649   E.wrex 2650    i^i cin 3262    C_ wss 3263   (/)c0 3571   |^|_ciin 4036   ` cfv 5394   Basecbs 13396   glbcglb 14327   1.cp1 14394   OPcops 29287   Atomscatm 29378   HLchlt 29465   pmapcpmap 29611
This theorem is referenced by:  polval2N  30020
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 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-rep 4261  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641
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 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-nel 2553  df-ral 2654  df-rex 2655  df-reu 2656  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-op 3766  df-uni 3958  df-iun 4037  df-iin 4038  df-br 4154  df-opab 4208  df-mpt 4209  df-id 4439  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-ov 6023  df-oprab 6024  df-mpt2 6025  df-1st 6288  df-2nd 6289  df-undef 6479  df-riota 6485  df-poset 14330  df-lub 14358  df-glb 14359  df-join 14360  df-meet 14361  df-p1 14396  df-lat 14402  df-clat 14464  df-oposet 29291  df-ol 29293  df-oml 29294  df-ats 29382  df-hlat 29466  df-pmap 29618
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