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Theorem pmapglb2N 30630
Description: The projective map of the GLB of a set of lattice elements  S. Variant of Theorem 15.5.2 of [MaedaMaeda] p. 62. Allows  S  =  (/). (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
pmapglb2N  |-  ( ( K  e.  HL  /\  S  C_  B )  -> 
( M `  ( G `  S )
)  =  ( A  i^i  |^|_ x  e.  S  ( M `  x ) ) )
Distinct variable groups:    x, A    x, B    x, K    x, S
Allowed substitution hints:    G( x)    M( x)

Proof of Theorem pmapglb2N
StepHypRef Expression
1 hlop 30222 . . . . 5  |-  ( K  e.  HL  ->  K  e.  OP )
2 pmapglb2.g . . . . . . . 8  |-  G  =  ( glb `  K
)
3 eqid 2438 . . . . . . . 8  |-  ( 1.
`  K )  =  ( 1. `  K
)
42, 3glb0N 30053 . . . . . . 7  |-  ( K  e.  OP  ->  ( G `  (/) )  =  ( 1. `  K
) )
54fveq2d 5734 . . . . . 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 30626 . . . . . 6  |-  ( K  e.  OP  ->  ( M `  ( 1. `  K ) )  =  A )
95, 8eqtrd 2470 . . . . 5  |-  ( K  e.  OP  ->  ( M `  ( G `  (/) ) )  =  A )
101, 9syl 16 . . . 4  |-  ( K  e.  HL  ->  ( M `  ( G `  (/) ) )  =  A )
11 fveq2 5730 . . . . . 6  |-  ( S  =  (/)  ->  ( G `
 S )  =  ( G `  (/) ) )
1211fveq2d 5734 . . . . 5  |-  ( S  =  (/)  ->  ( M `
 ( G `  S ) )  =  ( M `  ( G `  (/) ) ) )
13 riin0 4166 . . . . 5  |-  ( S  =  (/)  ->  ( A  i^i  |^|_ x  e.  S  ( M `  x ) )  =  A )
1412, 13eqeq12d 2452 . . . 4  |-  ( S  =  (/)  ->  ( ( M `  ( G `
 S ) )  =  ( A  i^i  |^|_
x  e.  S  ( M `  x ) )  <->  ( M `  ( G `  (/) ) )  =  A ) )
1510, 14syl5ibrcom 215 . . 3  |-  ( K  e.  HL  ->  ( S  =  (/)  ->  ( M `  ( G `  S ) )  =  ( A  i^i  |^|_ x  e.  S  ( M `
 x ) ) ) )
1615adantr 453 . 2  |-  ( ( K  e.  HL  /\  S  C_  B )  -> 
( S  =  (/)  ->  ( M `  ( G `  S )
)  =  ( A  i^i  |^|_ x  e.  S  ( M `  x ) ) ) )
17 pmapglb2.b . . . . 5  |-  B  =  ( Base `  K
)
1817, 2, 7pmapglb 30629 . . . 4  |-  ( ( K  e.  HL  /\  S  C_  B  /\  S  =/=  (/) )  ->  ( M `  ( G `  S ) )  = 
|^|_ x  e.  S  ( M `  x ) )
19 simpr 449 . . . . . . . . . . 11  |-  ( ( ( K  e.  HL  /\  S  C_  B )  /\  x  e.  S
)  ->  x  e.  S )
20 simpll 732 . . . . . . . . . . . 12  |-  ( ( ( K  e.  HL  /\  S  C_  B )  /\  x  e.  S
)  ->  K  e.  HL )
21 ssel2 3345 . . . . . . . . . . . . 13  |-  ( ( S  C_  B  /\  x  e.  S )  ->  x  e.  B )
2221adantll 696 . . . . . . . . . . . 12  |-  ( ( ( K  e.  HL  /\  S  C_  B )  /\  x  e.  S
)  ->  x  e.  B )
2317, 6, 7pmapssat 30618 . . . . . . . . . . . 12  |-  ( ( K  e.  HL  /\  x  e.  B )  ->  ( M `  x
)  C_  A )
2420, 22, 23syl2anc 644 . . . . . . . . . . 11  |-  ( ( ( K  e.  HL  /\  S  C_  B )  /\  x  e.  S
)  ->  ( M `  x )  C_  A
)
2519, 24jca 520 . . . . . . . . . 10  |-  ( ( ( K  e.  HL  /\  S  C_  B )  /\  x  e.  S
)  ->  ( x  e.  S  /\  ( M `  x )  C_  A ) )
2625ex 425 . . . . . . . . 9  |-  ( ( K  e.  HL  /\  S  C_  B )  -> 
( x  e.  S  ->  ( x  e.  S  /\  ( M `  x
)  C_  A )
) )
2726eximdv 1633 . . . . . . . 8  |-  ( ( K  e.  HL  /\  S  C_  B )  -> 
( E. x  x  e.  S  ->  E. x
( x  e.  S  /\  ( M `  x
)  C_  A )
) )
28 n0 3639 . . . . . . . 8  |-  ( S  =/=  (/)  <->  E. x  x  e.  S )
29 df-rex 2713 . . . . . . . 8  |-  ( E. x  e.  S  ( M `  x ) 
C_  A  <->  E. x
( x  e.  S  /\  ( M `  x
)  C_  A )
)
3027, 28, 293imtr4g 263 . . . . . . 7  |-  ( ( K  e.  HL  /\  S  C_  B )  -> 
( S  =/=  (/)  ->  E. x  e.  S  ( M `  x )  C_  A
) )
31303impia 1151 . . . . . 6  |-  ( ( K  e.  HL  /\  S  C_  B  /\  S  =/=  (/) )  ->  E. x  e.  S  ( M `  x )  C_  A
)
32 iinss 4144 . . . . . 6  |-  ( E. x  e.  S  ( M `  x ) 
C_  A  ->  |^|_ x  e.  S  ( M `  x )  C_  A
)
3331, 32syl 16 . . . . 5  |-  ( ( K  e.  HL  /\  S  C_  B  /\  S  =/=  (/) )  ->  |^|_ x  e.  S  ( M `  x )  C_  A
)
34 sseqin2 3562 . . . . 5  |-  ( |^|_ x  e.  S  ( M `
 x )  C_  A 
<->  ( A  i^i  |^|_ x  e.  S  ( M `
 x ) )  =  |^|_ x  e.  S  ( M `  x ) )
3533, 34sylib 190 . . . 4  |-  ( ( K  e.  HL  /\  S  C_  B  /\  S  =/=  (/) )  ->  ( A  i^i  |^|_ x  e.  S  ( M `  x ) )  =  |^|_ x  e.  S  ( M `  x ) )
3618, 35eqtr4d 2473 . . 3  |-  ( ( K  e.  HL  /\  S  C_  B  /\  S  =/=  (/) )  ->  ( M `  ( G `  S ) )  =  ( A  i^i  |^|_ x  e.  S  ( M `
 x ) ) )
37363expia 1156 . 2  |-  ( ( K  e.  HL  /\  S  C_  B )  -> 
( S  =/=  (/)  ->  ( M `  ( G `  S ) )  =  ( A  i^i  |^|_ x  e.  S  ( M `
 x ) ) ) )
3816, 37pm2.61dne 2683 1  |-  ( ( K  e.  HL  /\  S  C_  B )  -> 
( M `  ( G `  S )
)  =  ( A  i^i  |^|_ x  e.  S  ( M `  x ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    /\ w3a 937   E.wex 1551    = wceq 1653    e. wcel 1726    =/= wne 2601   E.wrex 2708    i^i cin 3321    C_ wss 3322   (/)c0 3630   |^|_ciin 4096   ` cfv 5456   Basecbs 13471   glbcglb 14402   1.cp1 14469   OPcops 30032   Atomscatm 30123   HLchlt 30210   pmapcpmap 30356
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-iin 4098  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-1st 6351  df-2nd 6352  df-undef 6545  df-riota 6551  df-poset 14405  df-lub 14433  df-glb 14434  df-join 14435  df-meet 14436  df-p1 14471  df-lat 14477  df-clat 14539  df-oposet 30036  df-ol 30038  df-oml 30039  df-ats 30127  df-hlat 30211  df-pmap 30363
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