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Theorem pmapglb2N 33255
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 32847 . . . . 5  |-  ( K  e.  HL  ->  K  e.  OP )
2 pmapglb2.g . . . . . . . 8  |-  G  =  ( glb `  K
)
3 eqid 2438 . . . . . . . 8  |-  ( 1.
`  K )  =  ( 1. `  K
)
42, 3glb0N 32678 . . . . . . 7  |-  ( K  e.  OP  ->  ( G `  (/) )  =  ( 1. `  K
) )
54fveq2d 5690 . . . . . 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 33251 . . . . . 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 5686 . . . . . 6  |-  ( S  =  (/)  ->  ( G `
 S )  =  ( G `  (/) ) )
1211fveq2d 5690 . . . . 5  |-  ( S  =  (/)  ->  ( M `
 ( G `  S ) )  =  ( M `  ( G `  (/) ) ) )
13 riin0 4239 . . . . 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 222 . . 3  |-  ( K  e.  HL  ->  ( S  =  (/)  ->  ( M `  ( G `  S ) )  =  ( A  i^i  |^|_ x  e.  S  ( M `
 x ) ) ) )
1615adantr 465 . 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 33254 . . . 4  |-  ( ( K  e.  HL  /\  S  C_  B  /\  S  =/=  (/) )  ->  ( M `  ( G `  S ) )  = 
|^|_ x  e.  S  ( M `  x ) )
19 simpr 461 . . . . . . . . . . 11  |-  ( ( ( K  e.  HL  /\  S  C_  B )  /\  x  e.  S
)  ->  x  e.  S )
20 simpll 753 . . . . . . . . . . . 12  |-  ( ( ( K  e.  HL  /\  S  C_  B )  /\  x  e.  S
)  ->  K  e.  HL )
21 ssel2 3346 . . . . . . . . . . . . 13  |-  ( ( S  C_  B  /\  x  e.  S )  ->  x  e.  B )
2221adantll 713 . . . . . . . . . . . 12  |-  ( ( ( K  e.  HL  /\  S  C_  B )  /\  x  e.  S
)  ->  x  e.  B )
2317, 6, 7pmapssat 33243 . . . . . . . . . . . 12  |-  ( ( K  e.  HL  /\  x  e.  B )  ->  ( M `  x
)  C_  A )
2420, 22, 23syl2anc 661 . . . . . . . . . . 11  |-  ( ( ( K  e.  HL  /\  S  C_  B )  /\  x  e.  S
)  ->  ( M `  x )  C_  A
)
2519, 24jca 532 . . . . . . . . . 10  |-  ( ( ( K  e.  HL  /\  S  C_  B )  /\  x  e.  S
)  ->  ( x  e.  S  /\  ( M `  x )  C_  A ) )
2625ex 434 . . . . . . . . 9  |-  ( ( K  e.  HL  /\  S  C_  B )  -> 
( x  e.  S  ->  ( x  e.  S  /\  ( M `  x
)  C_  A )
) )
2726eximdv 1676 . . . . . . . 8  |-  ( ( K  e.  HL  /\  S  C_  B )  -> 
( E. x  x  e.  S  ->  E. x
( x  e.  S  /\  ( M `  x
)  C_  A )
) )
28 n0 3641 . . . . . . . 8  |-  ( S  =/=  (/)  <->  E. x  x  e.  S )
29 df-rex 2716 . . . . . . . 8  |-  ( E. x  e.  S  ( M `  x ) 
C_  A  <->  E. x
( x  e.  S  /\  ( M `  x
)  C_  A )
)
3027, 28, 293imtr4g 270 . . . . . . 7  |-  ( ( K  e.  HL  /\  S  C_  B )  -> 
( S  =/=  (/)  ->  E. x  e.  S  ( M `  x )  C_  A
) )
31303impia 1184 . . . . . 6  |-  ( ( K  e.  HL  /\  S  C_  B  /\  S  =/=  (/) )  ->  E. x  e.  S  ( M `  x )  C_  A
)
32 iinss 4216 . . . . . 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 3564 . . . . 5  |-  ( |^|_ x  e.  S  ( M `
 x )  C_  A 
<->  ( A  i^i  |^|_ x  e.  S  ( M `
 x ) )  =  |^|_ x  e.  S  ( M `  x ) )
3533, 34sylib 196 . . . 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 1189 . 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 setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1369   E.wex 1586    e. wcel 1756    =/= wne 2601   E.wrex 2711    i^i cin 3322    C_ wss 3323   (/)c0 3632   |^|_ciin 4167   ` cfv 5413   Basecbs 14166   glbcglb 15105   1.cp1 15200   OPcops 32657   Atomscatm 32748   HLchlt 32835   pmapcpmap 32981
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-rep 4398  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2715  df-rex 2716  df-reu 2717  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-op 3879  df-uni 4087  df-iun 4168  df-iin 4169  df-br 4288  df-opab 4346  df-mpt 4347  df-id 4631  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-riota 6047  df-ov 6089  df-oprab 6090  df-poset 15108  df-lub 15136  df-glb 15137  df-join 15138  df-meet 15139  df-p1 15202  df-lat 15208  df-clat 15270  df-oposet 32661  df-ol 32663  df-oml 32664  df-ats 32752  df-hlat 32836  df-pmap 32988
This theorem is referenced by: (None)
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