Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  pmapglb2xN Unicode version

Theorem pmapglb2xN 29961
Description: The projective map of the GLB of a set of lattice elements. Index-set version of pmapglb2N 29960, 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 29552 . . . . 5  |-  ( K  e.  HL  ->  K  e.  OP )
2 pmapglb2.g . . . . . . . 8  |-  G  =  ( glb `  K
)
3 eqid 2283 . . . . . . . 8  |-  ( 1.
`  K )  =  ( 1. `  K
)
42, 3glb0N 29383 . . . . . . 7  |-  ( K  e.  OP  ->  ( G `  (/) )  =  ( 1. `  K
) )
54fveq2d 5529 . . . . . 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 29956 . . . . . 6  |-  ( K  e.  OP  ->  ( M `  ( 1. `  K ) )  =  A )
95, 8eqtrd 2315 . . . . 5  |-  ( K  e.  OP  ->  ( M `  ( G `  (/) ) )  =  A )
101, 9syl 15 . . . 4  |-  ( K  e.  HL  ->  ( M `  ( G `  (/) ) )  =  A )
11 rexeq 2737 . . . . . . . . 9  |-  ( I  =  (/)  ->  ( E. i  e.  I  y  =  S  <->  E. i  e.  (/)  y  =  S ) )
1211abbidv 2397 . . . . . . . 8  |-  ( I  =  (/)  ->  { y  |  E. i  e.  I  y  =  S }  =  { y  |  E. i  e.  (/)  y  =  S } )
13 rex0 3468 . . . . . . . . 9  |-  -.  E. i  e.  (/)  y  =  S
1413abf 3488 . . . . . . . 8  |-  { y  |  E. i  e.  (/)  y  =  S }  =  (/)
1512, 14syl6eq 2331 . . . . . . 7  |-  ( I  =  (/)  ->  { y  |  E. i  e.  I  y  =  S }  =  (/) )
1615fveq2d 5529 . . . . . 6  |-  ( I  =  (/)  ->  ( G `
 { y  |  E. i  e.  I 
y  =  S }
)  =  ( G `
 (/) ) )
1716fveq2d 5529 . . . . 5  |-  ( I  =  (/)  ->  ( M `
 ( G `  { y  |  E. i  e.  I  y  =  S } ) )  =  ( M `  ( G `  (/) ) ) )
18 riin0 3975 . . . . 5  |-  ( I  =  (/)  ->  ( A  i^i  |^|_ i  e.  I 
( M `  S
) )  =  A )
1917, 18eqeq12d 2297 . . . 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 29958 . . . 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 1605 . . . . . . . . . 10  |-  F/ i  K  e.  HL
25 nfra1 2593 . . . . . . . . . 10  |-  F/ i A. i  e.  I  S  e.  B
2624, 25nfan 1771 . . . . . . . . 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 2603 . . . . . . . . . . . . . 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 29948 . . . . . . . . . . . 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 1750 . . . . . . . 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 3464 . . . . . . . 8  |-  ( I  =/=  (/)  <->  E. i  i  e.  I )
38 df-rex 2549 . . . . . . . 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 3953 . . . . . 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 3388 . . . . 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 2318 . . 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 2523 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 1528    = wceq 1623    e. wcel 1684   {cab 2269    =/= wne 2446   A.wral 2543   E.wrex 2544    i^i cin 3151    C_ wss 3152   (/)c0 3455   |^|_ciin 3906   ` cfv 5255   Basecbs 13148   glbcglb 14077   1.cp1 14144   OPcops 29362   Atomscatm 29453   HLchlt 29540   pmapcpmap 29686
This theorem is referenced by:  polval2N  30095
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-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
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-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-iin 3908  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-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-undef 6298  df-riota 6304  df-poset 14080  df-lub 14108  df-glb 14109  df-join 14110  df-meet 14111  df-p1 14146  df-lat 14152  df-clat 14214  df-oposet 29366  df-ol 29368  df-oml 29369  df-ats 29457  df-hlat 29541  df-pmap 29693
  Copyright terms: Public domain W3C validator