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

Theorem pmapglb2xN 34969
Description: The projective map of the GLB of a set of lattice elements. Index-set version of pmapglb2N 34968, 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 34560 . . . . 5  |-  ( K  e.  HL  ->  K  e.  OP )
2 pmapglb2.g . . . . . . . 8  |-  G  =  ( glb `  K
)
3 eqid 2467 . . . . . . . 8  |-  ( 1.
`  K )  =  ( 1. `  K
)
42, 3glb0N 34391 . . . . . . 7  |-  ( K  e.  OP  ->  ( G `  (/) )  =  ( 1. `  K
) )
54fveq2d 5876 . . . . . 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 34964 . . . . . 6  |-  ( K  e.  OP  ->  ( M `  ( 1. `  K ) )  =  A )
95, 8eqtrd 2508 . . . . 5  |-  ( K  e.  OP  ->  ( M `  ( G `  (/) ) )  =  A )
101, 9syl 16 . . . 4  |-  ( K  e.  HL  ->  ( M `  ( G `  (/) ) )  =  A )
11 rexeq 3064 . . . . . . . . 9  |-  ( I  =  (/)  ->  ( E. i  e.  I  y  =  S  <->  E. i  e.  (/)  y  =  S ) )
1211abbidv 2603 . . . . . . . 8  |-  ( I  =  (/)  ->  { y  |  E. i  e.  I  y  =  S }  =  { y  |  E. i  e.  (/)  y  =  S } )
13 rex0 3804 . . . . . . . . 9  |-  -.  E. i  e.  (/)  y  =  S
1413abf 3824 . . . . . . . 8  |-  { y  |  E. i  e.  (/)  y  =  S }  =  (/)
1512, 14syl6eq 2524 . . . . . . 7  |-  ( I  =  (/)  ->  { y  |  E. i  e.  I  y  =  S }  =  (/) )
1615fveq2d 5876 . . . . . 6  |-  ( I  =  (/)  ->  ( G `
 { y  |  E. i  e.  I 
y  =  S }
)  =  ( G `
 (/) ) )
1716fveq2d 5876 . . . . 5  |-  ( I  =  (/)  ->  ( M `
 ( G `  { y  |  E. i  e.  I  y  =  S } ) )  =  ( M `  ( G `  (/) ) ) )
18 riin0 4405 . . . . 5  |-  ( I  =  (/)  ->  ( A  i^i  |^|_ i  e.  I 
( M `  S
) )  =  A )
1917, 18eqeq12d 2489 . . . 4  |-  ( I  =  (/)  ->  ( ( M `  ( G `
 { y  |  E. i  e.  I 
y  =  S }
) )  =  ( A  i^i  |^|_ i  e.  I  ( M `  S ) )  <->  ( M `  ( G `  (/) ) )  =  A ) )
2010, 19syl5ibrcom 222 . . 3  |-  ( K  e.  HL  ->  (
I  =  (/)  ->  ( M `  ( G `  { y  |  E. i  e.  I  y  =  S } ) )  =  ( A  i^i  |^|_ i  e.  I  ( M `  S ) ) ) )
2120adantr 465 . 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 34966 . . . 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 1683 . . . . . . . . . 10  |-  F/ i  K  e.  HL
25 nfra1 2848 . . . . . . . . . 10  |-  F/ i A. i  e.  I  S  e.  B
2624, 25nfan 1875 . . . . . . . . 9  |-  F/ i ( K  e.  HL  /\ 
A. i  e.  I  S  e.  B )
27 simpr 461 . . . . . . . . . . 11  |-  ( ( ( K  e.  HL  /\ 
A. i  e.  I  S  e.  B )  /\  i  e.  I
)  ->  i  e.  I )
28 simpll 753 . . . . . . . . . . . 12  |-  ( ( ( K  e.  HL  /\ 
A. i  e.  I  S  e.  B )  /\  i  e.  I
)  ->  K  e.  HL )
29 rsp 2833 . . . . . . . . . . . . . 14  |-  ( A. i  e.  I  S  e.  B  ->  ( i  e.  I  ->  S  e.  B ) )
3029imp 429 . . . . . . . . . . . . 13  |-  ( ( A. i  e.  I  S  e.  B  /\  i  e.  I )  ->  S  e.  B )
3130adantll 713 . . . . . . . . . . . 12  |-  ( ( ( K  e.  HL  /\ 
A. i  e.  I  S  e.  B )  /\  i  e.  I
)  ->  S  e.  B )
3222, 6, 7pmapssat 34956 . . . . . . . . . . . 12  |-  ( ( K  e.  HL  /\  S  e.  B )  ->  ( M `  S
)  C_  A )
3328, 31, 32syl2anc 661 . . . . . . . . . . 11  |-  ( ( ( K  e.  HL  /\ 
A. i  e.  I  S  e.  B )  /\  i  e.  I
)  ->  ( M `  S )  C_  A
)
3427, 33jca 532 . . . . . . . . . 10  |-  ( ( ( K  e.  HL  /\ 
A. i  e.  I  S  e.  B )  /\  i  e.  I
)  ->  ( i  e.  I  /\  ( M `  S )  C_  A ) )
3534ex 434 . . . . . . . . 9  |-  ( ( K  e.  HL  /\  A. i  e.  I  S  e.  B )  -> 
( i  e.  I  ->  ( i  e.  I  /\  ( M `  S
)  C_  A )
) )
3626, 35eximd 1830 . . . . . . . 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 3799 . . . . . . . 8  |-  ( I  =/=  (/)  <->  E. i  i  e.  I )
38 df-rex 2823 . . . . . . . 8  |-  ( E. i  e.  I  ( M `  S ) 
C_  A  <->  E. i
( i  e.  I  /\  ( M `  S
)  C_  A )
)
3936, 37, 383imtr4g 270 . . . . . . 7  |-  ( ( K  e.  HL  /\  A. i  e.  I  S  e.  B )  -> 
( I  =/=  (/)  ->  E. i  e.  I  ( M `  S )  C_  A
) )
40393impia 1193 . . . . . 6  |-  ( ( K  e.  HL  /\  A. i  e.  I  S  e.  B  /\  I  =/=  (/) )  ->  E. i  e.  I  ( M `  S )  C_  A
)
41 iinss 4382 . . . . . 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 3722 . . . . 5  |-  ( |^|_ i  e.  I  ( M `  S )  C_  A  <->  ( A  i^i  |^|_ i  e.  I  ( M `  S ) )  =  |^|_ i  e.  I  ( M `  S ) )
4442, 43sylib 196 . . . 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 2511 . . 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 1198 . 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 2784 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 setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 973    = wceq 1379   E.wex 1596    e. wcel 1767   {cab 2452    =/= wne 2662   A.wral 2817   E.wrex 2818    i^i cin 3480    C_ wss 3481   (/)c0 3790   |^|_ciin 4332   ` cfv 5594   Basecbs 14507   glbcglb 15447   1.cp1 15542   OPcops 34370   Atomscatm 34461   HLchlt 34548   pmapcpmap 34694
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-reu 2824  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-iun 4333  df-iin 4334  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6256  df-ov 6298  df-oprab 6299  df-poset 15450  df-lub 15478  df-glb 15479  df-join 15480  df-meet 15481  df-p1 15544  df-lat 15550  df-clat 15612  df-oposet 34374  df-ol 34376  df-oml 34377  df-ats 34465  df-hlat 34549  df-pmap 34701
This theorem is referenced by:  polval2N  35103
  Copyright terms: Public domain W3C validator