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Theorem pmapglb2N 33249
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 32841 . . . . 5  |-  ( K  e.  HL  ->  K  e.  OP )
2 pmapglb2.g . . . . . . . 8  |-  G  =  ( glb `  K
)
3 eqid 2420 . . . . . . . 8  |-  ( 1.
`  K )  =  ( 1. `  K
)
42, 3glb0N 32672 . . . . . . 7  |-  ( K  e.  OP  ->  ( G `  (/) )  =  ( 1. `  K
) )
54fveq2d 5877 . . . . . 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 33245 . . . . . 6  |-  ( K  e.  OP  ->  ( M `  ( 1. `  K ) )  =  A )
95, 8eqtrd 2461 . . . . 5  |-  ( K  e.  OP  ->  ( M `  ( G `  (/) ) )  =  A )
101, 9syl 17 . . . 4  |-  ( K  e.  HL  ->  ( M `  ( G `  (/) ) )  =  A )
11 fveq2 5873 . . . . . 6  |-  ( S  =  (/)  ->  ( G `
 S )  =  ( G `  (/) ) )
1211fveq2d 5877 . . . . 5  |-  ( S  =  (/)  ->  ( M `
 ( G `  S ) )  =  ( M `  ( G `  (/) ) ) )
13 riin0 4367 . . . . 5  |-  ( S  =  (/)  ->  ( A  i^i  |^|_ x  e.  S  ( M `  x ) )  =  A )
1412, 13eqeq12d 2442 . . . 4  |-  ( S  =  (/)  ->  ( ( M `  ( G `
 S ) )  =  ( A  i^i  |^|_
x  e.  S  ( M `  x ) )  <->  ( M `  ( G `  (/) ) )  =  A ) )
1510, 14syl5ibrcom 225 . . 3  |-  ( K  e.  HL  ->  ( S  =  (/)  ->  ( M `  ( G `  S ) )  =  ( A  i^i  |^|_ x  e.  S  ( M `
 x ) ) ) )
1615adantr 466 . 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 33248 . . . 4  |-  ( ( K  e.  HL  /\  S  C_  B  /\  S  =/=  (/) )  ->  ( M `  ( G `  S ) )  = 
|^|_ x  e.  S  ( M `  x ) )
19 simpr 462 . . . . . . . . . . 11  |-  ( ( ( K  e.  HL  /\  S  C_  B )  /\  x  e.  S
)  ->  x  e.  S )
20 simpll 758 . . . . . . . . . . . 12  |-  ( ( ( K  e.  HL  /\  S  C_  B )  /\  x  e.  S
)  ->  K  e.  HL )
21 ssel2 3456 . . . . . . . . . . . . 13  |-  ( ( S  C_  B  /\  x  e.  S )  ->  x  e.  B )
2221adantll 718 . . . . . . . . . . . 12  |-  ( ( ( K  e.  HL  /\  S  C_  B )  /\  x  e.  S
)  ->  x  e.  B )
2317, 6, 7pmapssat 33237 . . . . . . . . . . . 12  |-  ( ( K  e.  HL  /\  x  e.  B )  ->  ( M `  x
)  C_  A )
2420, 22, 23syl2anc 665 . . . . . . . . . . 11  |-  ( ( ( K  e.  HL  /\  S  C_  B )  /\  x  e.  S
)  ->  ( M `  x )  C_  A
)
2519, 24jca 534 . . . . . . . . . 10  |-  ( ( ( K  e.  HL  /\  S  C_  B )  /\  x  e.  S
)  ->  ( x  e.  S  /\  ( M `  x )  C_  A ) )
2625ex 435 . . . . . . . . 9  |-  ( ( K  e.  HL  /\  S  C_  B )  -> 
( x  e.  S  ->  ( x  e.  S  /\  ( M `  x
)  C_  A )
) )
2726eximdv 1754 . . . . . . . 8  |-  ( ( K  e.  HL  /\  S  C_  B )  -> 
( E. x  x  e.  S  ->  E. x
( x  e.  S  /\  ( M `  x
)  C_  A )
) )
28 n0 3768 . . . . . . . 8  |-  ( S  =/=  (/)  <->  E. x  x  e.  S )
29 df-rex 2779 . . . . . . . 8  |-  ( E. x  e.  S  ( M `  x ) 
C_  A  <->  E. x
( x  e.  S  /\  ( M `  x
)  C_  A )
)
3027, 28, 293imtr4g 273 . . . . . . 7  |-  ( ( K  e.  HL  /\  S  C_  B )  -> 
( S  =/=  (/)  ->  E. x  e.  S  ( M `  x )  C_  A
) )
31303impia 1202 . . . . . 6  |-  ( ( K  e.  HL  /\  S  C_  B  /\  S  =/=  (/) )  ->  E. x  e.  S  ( M `  x )  C_  A
)
32 iinss 4344 . . . . . 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 3678 . . . . 5  |-  ( |^|_ x  e.  S  ( M `
 x )  C_  A 
<->  ( A  i^i  |^|_ x  e.  S  ( M `
 x ) )  =  |^|_ x  e.  S  ( M `  x ) )
3533, 34sylib 199 . . . 4  |-  ( ( K  e.  HL  /\  S  C_  B  /\  S  =/=  (/) )  ->  ( A  i^i  |^|_ x  e.  S  ( M `  x ) )  =  |^|_ x  e.  S  ( M `  x ) )
3618, 35eqtr4d 2464 . . 3  |-  ( ( K  e.  HL  /\  S  C_  B  /\  S  =/=  (/) )  ->  ( M `  ( G `  S ) )  =  ( A  i^i  |^|_ x  e.  S  ( M `
 x ) ) )
37363expia 1207 . 2  |-  ( ( K  e.  HL  /\  S  C_  B )  -> 
( S  =/=  (/)  ->  ( M `  ( G `  S ) )  =  ( A  i^i  |^|_ x  e.  S  ( M `
 x ) ) ) )
3816, 37pm2.61dne 2739 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 370    /\ w3a 982    = wceq 1437   E.wex 1659    e. wcel 1867    =/= wne 2616   E.wrex 2774    i^i cin 3432    C_ wss 3433   (/)c0 3758   |^|_ciin 4294   ` cfv 5593   Basecbs 15099   glbcglb 16166   1.cp1 16262   OPcops 32651   Atomscatm 32742   HLchlt 32829   pmapcpmap 32975
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-8 1869  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-rep 4530  ax-sep 4540  ax-nul 4548  ax-pow 4595  ax-pr 4653  ax-un 6589
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-ral 2778  df-rex 2779  df-reu 2780  df-rab 2782  df-v 3080  df-sbc 3297  df-csb 3393  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-nul 3759  df-if 3907  df-pw 3978  df-sn 3994  df-pr 3996  df-op 4000  df-uni 4214  df-iun 4295  df-iin 4296  df-br 4418  df-opab 4477  df-mpt 4478  df-id 4761  df-xp 4852  df-rel 4853  df-cnv 4854  df-co 4855  df-dm 4856  df-rn 4857  df-res 4858  df-ima 4859  df-iota 5557  df-fun 5595  df-fn 5596  df-f 5597  df-f1 5598  df-fo 5599  df-f1o 5600  df-fv 5601  df-riota 6259  df-ov 6300  df-oprab 6301  df-preset 16151  df-poset 16169  df-lub 16198  df-glb 16199  df-join 16200  df-meet 16201  df-p1 16264  df-lat 16270  df-clat 16332  df-oposet 32655  df-ol 32657  df-oml 32658  df-ats 32746  df-hlat 32830  df-pmap 32982
This theorem is referenced by: (None)
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