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Theorem pmapglb2N 33773
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 33365 . . . . 5  |-  ( K  e.  HL  ->  K  e.  OP )
2 pmapglb2.g . . . . . . . 8  |-  G  =  ( glb `  K
)
3 eqid 2454 . . . . . . . 8  |-  ( 1.
`  K )  =  ( 1. `  K
)
42, 3glb0N 33196 . . . . . . 7  |-  ( K  e.  OP  ->  ( G `  (/) )  =  ( 1. `  K
) )
54fveq2d 5806 . . . . . 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 33769 . . . . . 6  |-  ( K  e.  OP  ->  ( M `  ( 1. `  K ) )  =  A )
95, 8eqtrd 2495 . . . . 5  |-  ( K  e.  OP  ->  ( M `  ( G `  (/) ) )  =  A )
101, 9syl 16 . . . 4  |-  ( K  e.  HL  ->  ( M `  ( G `  (/) ) )  =  A )
11 fveq2 5802 . . . . . 6  |-  ( S  =  (/)  ->  ( G `
 S )  =  ( G `  (/) ) )
1211fveq2d 5806 . . . . 5  |-  ( S  =  (/)  ->  ( M `
 ( G `  S ) )  =  ( M `  ( G `  (/) ) ) )
13 riin0 4355 . . . . 5  |-  ( S  =  (/)  ->  ( A  i^i  |^|_ x  e.  S  ( M `  x ) )  =  A )
1412, 13eqeq12d 2476 . . . 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 33772 . . . 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 3462 . . . . . . . . . . . . 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 33761 . . . . . . . . . . . 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 1677 . . . . . . . 8  |-  ( ( K  e.  HL  /\  S  C_  B )  -> 
( E. x  x  e.  S  ->  E. x
( x  e.  S  /\  ( M `  x
)  C_  A )
) )
28 n0 3757 . . . . . . . 8  |-  ( S  =/=  (/)  <->  E. x  x  e.  S )
29 df-rex 2805 . . . . . . . 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 1185 . . . . . 6  |-  ( ( K  e.  HL  /\  S  C_  B  /\  S  =/=  (/) )  ->  E. x  e.  S  ( M `  x )  C_  A
)
32 iinss 4332 . . . . . 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 3680 . . . . 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 2498 . . 3  |-  ( ( K  e.  HL  /\  S  C_  B  /\  S  =/=  (/) )  ->  ( M `  ( G `  S ) )  =  ( A  i^i  |^|_ x  e.  S  ( M `
 x ) ) )
37363expia 1190 . 2  |-  ( ( K  e.  HL  /\  S  C_  B )  -> 
( S  =/=  (/)  ->  ( M `  ( G `  S ) )  =  ( A  i^i  |^|_ x  e.  S  ( M `
 x ) ) ) )
3816, 37pm2.61dne 2769 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 1370   E.wex 1587    e. wcel 1758    =/= wne 2648   E.wrex 2800    i^i cin 3438    C_ wss 3439   (/)c0 3748   |^|_ciin 4283   ` cfv 5529   Basecbs 14295   glbcglb 15235   1.cp1 15330   OPcops 33175   Atomscatm 33266   HLchlt 33353   pmapcpmap 33499
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4514  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-reu 2806  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-iun 4284  df-iin 4285  df-br 4404  df-opab 4462  df-mpt 4463  df-id 4747  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-riota 6164  df-ov 6206  df-oprab 6207  df-poset 15238  df-lub 15266  df-glb 15267  df-join 15268  df-meet 15269  df-p1 15332  df-lat 15338  df-clat 15400  df-oposet 33179  df-ol 33181  df-oml 33182  df-ats 33270  df-hlat 33354  df-pmap 33506
This theorem is referenced by: (None)
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