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Theorem pmapglb2xN 29091
Description: The projective map of the GLB of a set of lattice elements. Index-set version of pmapglb2N 29090, 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 28682 . . . . 5  |-  ( K  e.  HL  ->  K  e.  OP )
2 pmapglb2.g . . . . . . . 8  |-  G  =  ( glb `  K
)
3 eqid 2256 . . . . . . . 8  |-  ( 1.
`  K )  =  ( 1. `  K
)
42, 3glb0N 28513 . . . . . . 7  |-  ( K  e.  OP  ->  ( G `  (/) )  =  ( 1. `  K
) )
54fveq2d 5427 . . . . . 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 29086 . . . . . 6  |-  ( K  e.  OP  ->  ( M `  ( 1. `  K ) )  =  A )
95, 8eqtrd 2288 . . . . 5  |-  ( K  e.  OP  ->  ( M `  ( G `  (/) ) )  =  A )
101, 9syl 17 . . . 4  |-  ( K  e.  HL  ->  ( M `  ( G `  (/) ) )  =  A )
11 rexeq 2699 . . . . . . . . 9  |-  ( I  =  (/)  ->  ( E. i  e.  I  y  =  S  <->  E. i  e.  (/)  y  =  S ) )
1211abbidv 2370 . . . . . . . 8  |-  ( I  =  (/)  ->  { y  |  E. i  e.  I  y  =  S }  =  { y  |  E. i  e.  (/)  y  =  S } )
13 rex0 3410 . . . . . . . . 9  |-  -.  E. i  e.  (/)  y  =  S
1413abf 3430 . . . . . . . 8  |-  { y  |  E. i  e.  (/)  y  =  S }  =  (/)
1512, 14syl6eq 2304 . . . . . . 7  |-  ( I  =  (/)  ->  { y  |  E. i  e.  I  y  =  S }  =  (/) )
1615fveq2d 5427 . . . . . 6  |-  ( I  =  (/)  ->  ( G `
 { y  |  E. i  e.  I 
y  =  S }
)  =  ( G `
 (/) ) )
1716fveq2d 5427 . . . . 5  |-  ( I  =  (/)  ->  ( M `
 ( G `  { y  |  E. i  e.  I  y  =  S } ) )  =  ( M `  ( G `  (/) ) ) )
18 riin0 3916 . . . . 5  |-  ( I  =  (/)  ->  ( A  i^i  |^|_ i  e.  I 
( M `  S
) )  =  A )
1917, 18eqeq12d 2270 . . . 4  |-  ( I  =  (/)  ->  ( ( M `  ( G `
 { y  |  E. i  e.  I 
y  =  S }
) )  =  ( A  i^i  |^|_ i  e.  I  ( M `  S ) )  <->  ( M `  ( G `  (/) ) )  =  A ) )
2010, 19syl5ibrcom 215 . . 3  |-  ( K  e.  HL  ->  (
I  =  (/)  ->  ( M `  ( G `  { y  |  E. i  e.  I  y  =  S } ) )  =  ( A  i^i  |^|_ i  e.  I  ( M `  S ) ) ) )
2120adantr 453 . 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 29088 . . . 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 1629 . . . . . . . . . 10  |-  F/ i  K  e.  HL
25 nfra1 2564 . . . . . . . . . 10  |-  F/ i A. i  e.  I  S  e.  B
2624, 25nfan 1737 . . . . . . . . 9  |-  F/ i ( K  e.  HL  /\ 
A. i  e.  I  S  e.  B )
27 simpr 449 . . . . . . . . . . 11  |-  ( ( ( K  e.  HL  /\ 
A. i  e.  I  S  e.  B )  /\  i  e.  I
)  ->  i  e.  I )
28 simpll 733 . . . . . . . . . . . 12  |-  ( ( ( K  e.  HL  /\ 
A. i  e.  I  S  e.  B )  /\  i  e.  I
)  ->  K  e.  HL )
29 ra4 2574 . . . . . . . . . . . . . 14  |-  ( A. i  e.  I  S  e.  B  ->  ( i  e.  I  ->  S  e.  B ) )
3029imp 420 . . . . . . . . . . . . 13  |-  ( ( A. i  e.  I  S  e.  B  /\  i  e.  I )  ->  S  e.  B )
3130adantll 697 . . . . . . . . . . . 12  |-  ( ( ( K  e.  HL  /\ 
A. i  e.  I  S  e.  B )  /\  i  e.  I
)  ->  S  e.  B )
3222, 6, 7pmapssat 29078 . . . . . . . . . . . 12  |-  ( ( K  e.  HL  /\  S  e.  B )  ->  ( M `  S
)  C_  A )
3328, 31, 32syl2anc 645 . . . . . . . . . . 11  |-  ( ( ( K  e.  HL  /\ 
A. i  e.  I  S  e.  B )  /\  i  e.  I
)  ->  ( M `  S )  C_  A
)
3427, 33jca 520 . . . . . . . . . 10  |-  ( ( ( K  e.  HL  /\ 
A. i  e.  I  S  e.  B )  /\  i  e.  I
)  ->  ( i  e.  I  /\  ( M `  S )  C_  A ) )
3534ex 425 . . . . . . . . 9  |-  ( ( K  e.  HL  /\  A. i  e.  I  S  e.  B )  -> 
( i  e.  I  ->  ( i  e.  I  /\  ( M `  S
)  C_  A )
) )
3626, 35eximd 1711 . . . . . . . 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 3406 . . . . . . . 8  |-  ( I  =/=  (/)  <->  E. i  i  e.  I )
38 df-rex 2521 . . . . . . . 8  |-  ( E. i  e.  I  ( M `  S ) 
C_  A  <->  E. i
( i  e.  I  /\  ( M `  S
)  C_  A )
)
3936, 37, 383imtr4g 263 . . . . . . 7  |-  ( ( K  e.  HL  /\  A. i  e.  I  S  e.  B )  -> 
( I  =/=  (/)  ->  E. i  e.  I  ( M `  S )  C_  A
) )
40393impia 1153 . . . . . 6  |-  ( ( K  e.  HL  /\  A. i  e.  I  S  e.  B  /\  I  =/=  (/) )  ->  E. i  e.  I  ( M `  S )  C_  A
)
41 iinss 3894 . . . . . 6  |-  ( E. i  e.  I  ( M `  S ) 
C_  A  ->  |^|_ i  e.  I  ( M `  S )  C_  A
)
4240, 41syl 17 . . . . 5  |-  ( ( K  e.  HL  /\  A. i  e.  I  S  e.  B  /\  I  =/=  (/) )  ->  |^|_ i  e.  I  ( M `  S )  C_  A
)
43 sseqin2 3330 . . . . 5  |-  ( |^|_ i  e.  I  ( M `  S )  C_  A  <->  ( A  i^i  |^|_ i  e.  I  ( M `  S ) )  =  |^|_ i  e.  I  ( M `  S ) )
4442, 43sylib 190 . . . 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 2291 . . 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 1158 . 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 2496 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 6    /\ wa 360    /\ w3a 939   E.wex 1537    = wceq 1619    e. wcel 1621   {cab 2242    =/= wne 2419   A.wral 2516   E.wrex 2517    i^i cin 3093    C_ wss 3094   (/)c0 3397   |^|_ciin 3847   ` cfv 4638   Basecbs 13075   glbcglb 14004   1.cp1 14071   OPcops 28492   Atomscatm 28583   HLchlt 28670   pmapcpmap 28816
This theorem is referenced by:  polval2N  29225
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237  ax-rep 4071  ax-sep 4081  ax-nul 4089  ax-pow 4126  ax-pr 4152  ax-un 4449
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-nel 2422  df-ral 2520  df-rex 2521  df-reu 2522  df-rab 2523  df-v 2742  df-sbc 2936  df-csb 3024  df-dif 3097  df-un 3099  df-in 3101  df-ss 3108  df-nul 3398  df-if 3507  df-pw 3568  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3769  df-iun 3848  df-iin 3849  df-br 3964  df-opab 4018  df-mpt 4019  df-id 4246  df-xp 4640  df-rel 4641  df-cnv 4642  df-co 4643  df-dm 4644  df-rn 4645  df-res 4646  df-ima 4647  df-fun 4648  df-fn 4649  df-f 4650  df-f1 4651  df-fo 4652  df-f1o 4653  df-fv 4654  df-ov 5760  df-oprab 5761  df-mpt2 5762  df-1st 6021  df-2nd 6022  df-iota 6190  df-undef 6229  df-riota 6237  df-poset 14007  df-lub 14035  df-glb 14036  df-join 14037  df-meet 14038  df-p1 14073  df-lat 14079  df-clat 14141  df-oposet 28496  df-ol 28498  df-oml 28499  df-ats 28587  df-hlat 28671  df-pmap 28823
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