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Theorem pmapglb2N 28864
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 28456 . . . . 5  |-  ( K  e.  HL  ->  K  e.  OP )
2 pmapglb2.g . . . . . . . 8  |-  G  =  ( glb `  K
)
3 eqid 2253 . . . . . . . 8  |-  ( 1.
`  K )  =  ( 1. `  K
)
42, 3glb0N 28287 . . . . . . 7  |-  ( K  e.  OP  ->  ( G `  (/) )  =  ( 1. `  K
) )
54fveq2d 5381 . . . . . 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 28860 . . . . . 6  |-  ( K  e.  OP  ->  ( M `  ( 1. `  K ) )  =  A )
95, 8eqtrd 2285 . . . . 5  |-  ( K  e.  OP  ->  ( M `  ( G `  (/) ) )  =  A )
101, 9syl 17 . . . 4  |-  ( K  e.  HL  ->  ( M `  ( G `  (/) ) )  =  A )
11 fveq2 5377 . . . . . 6  |-  ( S  =  (/)  ->  ( G `
 S )  =  ( G `  (/) ) )
1211fveq2d 5381 . . . . 5  |-  ( S  =  (/)  ->  ( M `
 ( G `  S ) )  =  ( M `  ( G `  (/) ) ) )
13 riin0 3873 . . . . 5  |-  ( S  =  (/)  ->  ( A  i^i  |^|_ x  e.  S  ( M `  x ) )  =  A )
1412, 13eqeq12d 2267 . . . 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 28863 . . . 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 733 . . . . . . . . . . . 12  |-  ( ( ( K  e.  HL  /\  S  C_  B )  /\  x  e.  S
)  ->  K  e.  HL )
21 ssel2 3098 . . . . . . . . . . . . 13  |-  ( ( S  C_  B  /\  x  e.  S )  ->  x  e.  B )
2221adantll 697 . . . . . . . . . . . 12  |-  ( ( ( K  e.  HL  /\  S  C_  B )  /\  x  e.  S
)  ->  x  e.  B )
2317, 6, 7pmapssat 28852 . . . . . . . . . . . 12  |-  ( ( K  e.  HL  /\  x  e.  B )  ->  ( M `  x
)  C_  A )
2420, 22, 23syl2anc 645 . . . . . . . . . . 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 2018 . . . . . . . 8  |-  ( ( K  e.  HL  /\  S  C_  B )  -> 
( E. x  x  e.  S  ->  E. x
( x  e.  S  /\  ( M `  x
)  C_  A )
) )
28 n0 3371 . . . . . . . 8  |-  ( S  =/=  (/)  <->  E. x  x  e.  S )
29 df-rex 2514 . . . . . . . 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 1153 . . . . . 6  |-  ( ( K  e.  HL  /\  S  C_  B  /\  S  =/=  (/) )  ->  E. x  e.  S  ( M `  x )  C_  A
)
32 iinss 3851 . . . . . 6  |-  ( E. x  e.  S  ( M `  x ) 
C_  A  ->  |^|_ x  e.  S  ( M `  x )  C_  A
)
3331, 32syl 17 . . . . 5  |-  ( ( K  e.  HL  /\  S  C_  B  /\  S  =/=  (/) )  ->  |^|_ x  e.  S  ( M `  x )  C_  A
)
34 sseqin2 3295 . . . . 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 2288 . . 3  |-  ( ( K  e.  HL  /\  S  C_  B  /\  S  =/=  (/) )  ->  ( M `  ( G `  S ) )  =  ( A  i^i  |^|_ x  e.  S  ( M `
 x ) ) )
37363expia 1158 . 2  |-  ( ( K  e.  HL  /\  S  C_  B )  -> 
( S  =/=  (/)  ->  ( M `  ( G `  S ) )  =  ( A  i^i  |^|_ x  e.  S  ( M `
 x ) ) ) )
3816, 37pm2.61dne 2489 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 6    /\ wa 360    /\ w3a 939   E.wex 1537    = wceq 1619    e. wcel 1621    =/= wne 2412   E.wrex 2510    i^i cin 3077    C_ wss 3078   (/)c0 3362   |^|_ciin 3804   ` cfv 4592   Basecbs 13022   glbcglb 13921   1.cp1 13988   OPcops 28266   Atomscatm 28357   HLchlt 28444   pmapcpmap 28590
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 1926  ax-ext 2234  ax-rep 4028  ax-sep 4038  ax-nul 4046  ax-pow 4082  ax-pr 4108  ax-un 4403
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 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-nel 2415  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2516  df-v 2729  df-sbc 2922  df-csb 3010  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-nul 3363  df-if 3471  df-pw 3532  df-sn 3550  df-pr 3551  df-op 3553  df-uni 3728  df-iun 3805  df-iin 3806  df-br 3921  df-opab 3975  df-mpt 3976  df-id 4202  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 4602  df-fn 4603  df-f 4604  df-f1 4605  df-fo 4606  df-f1o 4607  df-fv 4608  df-ov 5713  df-oprab 5714  df-mpt2 5715  df-1st 5974  df-2nd 5975  df-iota 6143  df-undef 6182  df-riota 6190  df-poset 13924  df-lub 13952  df-glb 13953  df-join 13954  df-meet 13955  df-p1 13990  df-lat 13996  df-clat 14058  df-oposet 28270  df-ol 28272  df-oml 28273  df-ats 28361  df-hlat 28445  df-pmap 28597
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