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Theorem pmapglb2xN 30583
Description: The projective map of the GLB of a set of lattice elements. Index-set version of pmapglb2N 30582, 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 30174 . . . . 5  |-  ( K  e.  HL  ->  K  e.  OP )
2 pmapglb2.g . . . . . . . 8  |-  G  =  ( glb `  K
)
3 eqid 2296 . . . . . . . 8  |-  ( 1.
`  K )  =  ( 1. `  K
)
42, 3glb0N 30005 . . . . . . 7  |-  ( K  e.  OP  ->  ( G `  (/) )  =  ( 1. `  K
) )
54fveq2d 5545 . . . . . 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 30578 . . . . . 6  |-  ( K  e.  OP  ->  ( M `  ( 1. `  K ) )  =  A )
95, 8eqtrd 2328 . . . . 5  |-  ( K  e.  OP  ->  ( M `  ( G `  (/) ) )  =  A )
101, 9syl 15 . . . 4  |-  ( K  e.  HL  ->  ( M `  ( G `  (/) ) )  =  A )
11 rexeq 2750 . . . . . . . . 9  |-  ( I  =  (/)  ->  ( E. i  e.  I  y  =  S  <->  E. i  e.  (/)  y  =  S ) )
1211abbidv 2410 . . . . . . . 8  |-  ( I  =  (/)  ->  { y  |  E. i  e.  I  y  =  S }  =  { y  |  E. i  e.  (/)  y  =  S } )
13 rex0 3481 . . . . . . . . 9  |-  -.  E. i  e.  (/)  y  =  S
1413abf 3501 . . . . . . . 8  |-  { y  |  E. i  e.  (/)  y  =  S }  =  (/)
1512, 14syl6eq 2344 . . . . . . 7  |-  ( I  =  (/)  ->  { y  |  E. i  e.  I  y  =  S }  =  (/) )
1615fveq2d 5545 . . . . . 6  |-  ( I  =  (/)  ->  ( G `
 { y  |  E. i  e.  I 
y  =  S }
)  =  ( G `
 (/) ) )
1716fveq2d 5545 . . . . 5  |-  ( I  =  (/)  ->  ( M `
 ( G `  { y  |  E. i  e.  I  y  =  S } ) )  =  ( M `  ( G `  (/) ) ) )
18 riin0 3991 . . . . 5  |-  ( I  =  (/)  ->  ( A  i^i  |^|_ i  e.  I 
( M `  S
) )  =  A )
1917, 18eqeq12d 2310 . . . 4  |-  ( I  =  (/)  ->  ( ( M `  ( G `
 { y  |  E. i  e.  I 
y  =  S }
) )  =  ( A  i^i  |^|_ i  e.  I  ( M `  S ) )  <->  ( M `  ( G `  (/) ) )  =  A ) )
2010, 19syl5ibrcom 213 . . 3  |-  ( K  e.  HL  ->  (
I  =  (/)  ->  ( M `  ( G `  { y  |  E. i  e.  I  y  =  S } ) )  =  ( A  i^i  |^|_ i  e.  I  ( M `  S ) ) ) )
2120adantr 451 . 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 30580 . . . 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 1609 . . . . . . . . . 10  |-  F/ i  K  e.  HL
25 nfra1 2606 . . . . . . . . . 10  |-  F/ i A. i  e.  I  S  e.  B
2624, 25nfan 1783 . . . . . . . . 9  |-  F/ i ( K  e.  HL  /\ 
A. i  e.  I  S  e.  B )
27 simpr 447 . . . . . . . . . . 11  |-  ( ( ( K  e.  HL  /\ 
A. i  e.  I  S  e.  B )  /\  i  e.  I
)  ->  i  e.  I )
28 simpll 730 . . . . . . . . . . . 12  |-  ( ( ( K  e.  HL  /\ 
A. i  e.  I  S  e.  B )  /\  i  e.  I
)  ->  K  e.  HL )
29 rsp 2616 . . . . . . . . . . . . . 14  |-  ( A. i  e.  I  S  e.  B  ->  ( i  e.  I  ->  S  e.  B ) )
3029imp 418 . . . . . . . . . . . . 13  |-  ( ( A. i  e.  I  S  e.  B  /\  i  e.  I )  ->  S  e.  B )
3130adantll 694 . . . . . . . . . . . 12  |-  ( ( ( K  e.  HL  /\ 
A. i  e.  I  S  e.  B )  /\  i  e.  I
)  ->  S  e.  B )
3222, 6, 7pmapssat 30570 . . . . . . . . . . . 12  |-  ( ( K  e.  HL  /\  S  e.  B )  ->  ( M `  S
)  C_  A )
3328, 31, 32syl2anc 642 . . . . . . . . . . 11  |-  ( ( ( K  e.  HL  /\ 
A. i  e.  I  S  e.  B )  /\  i  e.  I
)  ->  ( M `  S )  C_  A
)
3427, 33jca 518 . . . . . . . . . 10  |-  ( ( ( K  e.  HL  /\ 
A. i  e.  I  S  e.  B )  /\  i  e.  I
)  ->  ( i  e.  I  /\  ( M `  S )  C_  A ) )
3534ex 423 . . . . . . . . 9  |-  ( ( K  e.  HL  /\  A. i  e.  I  S  e.  B )  -> 
( i  e.  I  ->  ( i  e.  I  /\  ( M `  S
)  C_  A )
) )
3626, 35eximd 1762 . . . . . . . 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 3477 . . . . . . . 8  |-  ( I  =/=  (/)  <->  E. i  i  e.  I )
38 df-rex 2562 . . . . . . . 8  |-  ( E. i  e.  I  ( M `  S ) 
C_  A  <->  E. i
( i  e.  I  /\  ( M `  S
)  C_  A )
)
3936, 37, 383imtr4g 261 . . . . . . 7  |-  ( ( K  e.  HL  /\  A. i  e.  I  S  e.  B )  -> 
( I  =/=  (/)  ->  E. i  e.  I  ( M `  S )  C_  A
) )
40393impia 1148 . . . . . 6  |-  ( ( K  e.  HL  /\  A. i  e.  I  S  e.  B  /\  I  =/=  (/) )  ->  E. i  e.  I  ( M `  S )  C_  A
)
41 iinss 3969 . . . . . 6  |-  ( E. i  e.  I  ( M `  S ) 
C_  A  ->  |^|_ i  e.  I  ( M `  S )  C_  A
)
4240, 41syl 15 . . . . 5  |-  ( ( K  e.  HL  /\  A. i  e.  I  S  e.  B  /\  I  =/=  (/) )  ->  |^|_ i  e.  I  ( M `  S )  C_  A
)
43 sseqin2 3401 . . . . 5  |-  ( |^|_ i  e.  I  ( M `  S )  C_  A  <->  ( A  i^i  |^|_ i  e.  I  ( M `  S ) )  =  |^|_ i  e.  I  ( M `  S ) )
4442, 43sylib 188 . . . 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 2331 . . 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 1153 . 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 2536 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 358    /\ w3a 934   E.wex 1531    = wceq 1632    e. wcel 1696   {cab 2282    =/= wne 2459   A.wral 2556   E.wrex 2557    i^i cin 3164    C_ wss 3165   (/)c0 3468   |^|_ciin 3922   ` cfv 5271   Basecbs 13164   glbcglb 14093   1.cp1 14160   OPcops 29984   Atomscatm 30075   HLchlt 30162   pmapcpmap 30308
This theorem is referenced by:  polval2N  30717
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-iin 3924  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-undef 6314  df-riota 6320  df-poset 14096  df-lub 14124  df-glb 14125  df-join 14126  df-meet 14127  df-p1 14162  df-lat 14168  df-clat 14230  df-oposet 29988  df-ol 29990  df-oml 29991  df-ats 30079  df-hlat 30163  df-pmap 30315
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