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Theorem pmapj2N 30726
Description: The projective map of the join of two lattice elements. (Contributed by NM, 14-Mar-2012.) (New usage is discouraged.)
Hypotheses
Ref Expression
pmapj2.b  |-  B  =  ( Base `  K
)
pmapj2.j  |-  .\/  =  ( join `  K )
pmapj2.m  |-  M  =  ( pmap `  K
)
pmapj2.p  |-  .+  =  ( + P `  K
)
pmapj2.o  |-  ._|_  =  ( _|_ P `  K
)
Assertion
Ref Expression
pmapj2N  |-  ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  ->  ( M `  ( X  .\/  Y ) )  =  (  ._|_  `  (  ._|_  `  ( ( M `
 X )  .+  ( M `  Y ) ) ) ) )

Proof of Theorem pmapj2N
StepHypRef Expression
1 simp1 957 . . 3  |-  ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  ->  K  e.  HL )
2 hllat 30161 . . . . 5  |-  ( K  e.  HL  ->  K  e.  Lat )
323ad2ant1 978 . . . 4  |-  ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  ->  K  e.  Lat )
4 hlop 30160 . . . . . 6  |-  ( K  e.  HL  ->  K  e.  OP )
543ad2ant1 978 . . . . 5  |-  ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  ->  K  e.  OP )
6 simp2 958 . . . . 5  |-  ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  ->  X  e.  B )
7 pmapj2.b . . . . . 6  |-  B  =  ( Base `  K
)
8 eqid 2436 . . . . . 6  |-  ( oc
`  K )  =  ( oc `  K
)
97, 8opoccl 29992 . . . . 5  |-  ( ( K  e.  OP  /\  X  e.  B )  ->  ( ( oc `  K ) `  X
)  e.  B )
105, 6, 9syl2anc 643 . . . 4  |-  ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( oc `  K ) `  X
)  e.  B )
11 simp3 959 . . . . 5  |-  ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  ->  Y  e.  B )
127, 8opoccl 29992 . . . . 5  |-  ( ( K  e.  OP  /\  Y  e.  B )  ->  ( ( oc `  K ) `  Y
)  e.  B )
135, 11, 12syl2anc 643 . . . 4  |-  ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( oc `  K ) `  Y
)  e.  B )
14 eqid 2436 . . . . 5  |-  ( meet `  K )  =  (
meet `  K )
157, 14latmcl 14480 . . . 4  |-  ( ( K  e.  Lat  /\  ( ( oc `  K ) `  X
)  e.  B  /\  ( ( oc `  K ) `  Y
)  e.  B )  ->  ( ( ( oc `  K ) `
 X ) (
meet `  K )
( ( oc `  K ) `  Y
) )  e.  B
)
163, 10, 13, 15syl3anc 1184 . . 3  |-  ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( ( oc
`  K ) `  X ) ( meet `  K ) ( ( oc `  K ) `
 Y ) )  e.  B )
17 pmapj2.m . . . 4  |-  M  =  ( pmap `  K
)
18 pmapj2.o . . . 4  |-  ._|_  =  ( _|_ P `  K
)
197, 8, 17, 18polpmapN 30709 . . 3  |-  ( ( K  e.  HL  /\  ( ( ( oc
`  K ) `  X ) ( meet `  K ) ( ( oc `  K ) `
 Y ) )  e.  B )  -> 
(  ._|_  `  ( M `  ( ( ( oc
`  K ) `  X ) ( meet `  K ) ( ( oc `  K ) `
 Y ) ) ) )  =  ( M `  ( ( oc `  K ) `
 ( ( ( oc `  K ) `
 X ) (
meet `  K )
( ( oc `  K ) `  Y
) ) ) ) )
201, 16, 19syl2anc 643 . 2  |-  ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  ->  (  ._|_  `  ( M `
 ( ( ( oc `  K ) `
 X ) (
meet `  K )
( ( oc `  K ) `  Y
) ) ) )  =  ( M `  ( ( oc `  K ) `  (
( ( oc `  K ) `  X
) ( meet `  K
) ( ( oc
`  K ) `  Y ) ) ) ) )
217, 8, 17, 18polpmapN 30709 . . . . . 6  |-  ( ( K  e.  HL  /\  X  e.  B )  ->  (  ._|_  `  ( M `
 X ) )  =  ( M `  ( ( oc `  K ) `  X
) ) )
22213adant3 977 . . . . 5  |-  ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  ->  (  ._|_  `  ( M `
 X ) )  =  ( M `  ( ( oc `  K ) `  X
) ) )
237, 8, 17, 18polpmapN 30709 . . . . . 6  |-  ( ( K  e.  HL  /\  Y  e.  B )  ->  (  ._|_  `  ( M `
 Y ) )  =  ( M `  ( ( oc `  K ) `  Y
) ) )
24233adant2 976 . . . . 5  |-  ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  ->  (  ._|_  `  ( M `
 Y ) )  =  ( M `  ( ( oc `  K ) `  Y
) ) )
2522, 24ineq12d 3543 . . . 4  |-  ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  ->  ( (  ._|_  `  ( M `  X )
)  i^i  (  ._|_  `  ( M `  Y
) ) )  =  ( ( M `  ( ( oc `  K ) `  X
) )  i^i  ( M `  ( ( oc `  K ) `  Y ) ) ) )
26 eqid 2436 . . . . . . 7  |-  ( Atoms `  K )  =  (
Atoms `  K )
277, 26, 17pmapssat 30556 . . . . . 6  |-  ( ( K  e.  HL  /\  X  e.  B )  ->  ( M `  X
)  C_  ( Atoms `  K ) )
28273adant3 977 . . . . 5  |-  ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  ->  ( M `  X
)  C_  ( Atoms `  K ) )
297, 26, 17pmapssat 30556 . . . . . 6  |-  ( ( K  e.  HL  /\  Y  e.  B )  ->  ( M `  Y
)  C_  ( Atoms `  K ) )
30293adant2 976 . . . . 5  |-  ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  ->  ( M `  Y
)  C_  ( Atoms `  K ) )
31 pmapj2.p . . . . . 6  |-  .+  =  ( + P `  K
)
3226, 31, 18poldmj1N 30725 . . . . 5  |-  ( ( K  e.  HL  /\  ( M `  X ) 
C_  ( Atoms `  K
)  /\  ( M `  Y )  C_  ( Atoms `  K ) )  ->  (  ._|_  `  (
( M `  X
)  .+  ( M `  Y ) ) )  =  ( (  ._|_  `  ( M `  X
) )  i^i  (  ._|_  `  ( M `  Y ) ) ) )
331, 28, 30, 32syl3anc 1184 . . . 4  |-  ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  ->  (  ._|_  `  ( ( M `  X ) 
.+  ( M `  Y ) ) )  =  ( (  ._|_  `  ( M `  X
) )  i^i  (  ._|_  `  ( M `  Y ) ) ) )
347, 14, 26, 17pmapmeet 30570 . . . . 5  |-  ( ( K  e.  HL  /\  ( ( oc `  K ) `  X
)  e.  B  /\  ( ( oc `  K ) `  Y
)  e.  B )  ->  ( M `  ( ( ( oc
`  K ) `  X ) ( meet `  K ) ( ( oc `  K ) `
 Y ) ) )  =  ( ( M `  ( ( oc `  K ) `
 X ) )  i^i  ( M `  ( ( oc `  K ) `  Y
) ) ) )
351, 10, 13, 34syl3anc 1184 . . . 4  |-  ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  ->  ( M `  (
( ( oc `  K ) `  X
) ( meet `  K
) ( ( oc
`  K ) `  Y ) ) )  =  ( ( M `
 ( ( oc
`  K ) `  X ) )  i^i  ( M `  (
( oc `  K
) `  Y )
) ) )
3625, 33, 353eqtr4rd 2479 . . 3  |-  ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  ->  ( M `  (
( ( oc `  K ) `  X
) ( meet `  K
) ( ( oc
`  K ) `  Y ) ) )  =  (  ._|_  `  (
( M `  X
)  .+  ( M `  Y ) ) ) )
3736fveq2d 5732 . 2  |-  ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  ->  (  ._|_  `  ( M `
 ( ( ( oc `  K ) `
 X ) (
meet `  K )
( ( oc `  K ) `  Y
) ) ) )  =  (  ._|_  `  (  ._|_  `  ( ( M `
 X )  .+  ( M `  Y ) ) ) ) )
38 hlol 30159 . . . 4  |-  ( K  e.  HL  ->  K  e.  OL )
39 pmapj2.j . . . . 5  |-  .\/  =  ( join `  K )
407, 39, 14, 8oldmm4 30018 . . . 4  |-  ( ( K  e.  OL  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( oc `  K ) `  (
( ( oc `  K ) `  X
) ( meet `  K
) ( ( oc
`  K ) `  Y ) ) )  =  ( X  .\/  Y ) )
4138, 40syl3an1 1217 . . 3  |-  ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( oc `  K ) `  (
( ( oc `  K ) `  X
) ( meet `  K
) ( ( oc
`  K ) `  Y ) ) )  =  ( X  .\/  Y ) )
4241fveq2d 5732 . 2  |-  ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  ->  ( M `  (
( oc `  K
) `  ( (
( oc `  K
) `  X )
( meet `  K )
( ( oc `  K ) `  Y
) ) ) )  =  ( M `  ( X  .\/  Y ) ) )
4320, 37, 423eqtr3rd 2477 1  |-  ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  ->  ( M `  ( X  .\/  Y ) )  =  (  ._|_  `  (  ._|_  `  ( ( M `
 X )  .+  ( M `  Y ) ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 936    = wceq 1652    e. wcel 1725    i^i cin 3319    C_ wss 3320   ` cfv 5454  (class class class)co 6081   Basecbs 13469   occoc 13537   joincjn 14401   meetcmee 14402   Latclat 14474   OPcops 29970   OLcol 29972   Atomscatm 30061   HLchlt 30148   pmapcpmap 30294   + Pcpadd 30592   _|_ PcpolN 30699
This theorem is referenced by:  pmapocjN  30727  pmapojoinN  30765
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-iun 4095  df-iin 4096  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-undef 6543  df-riota 6549  df-poset 14403  df-plt 14415  df-lub 14431  df-glb 14432  df-join 14433  df-meet 14434  df-p0 14468  df-p1 14469  df-lat 14475  df-clat 14537  df-oposet 29974  df-ol 29976  df-oml 29977  df-covers 30064  df-ats 30065  df-atl 30096  df-cvlat 30120  df-hlat 30149  df-psubsp 30300  df-pmap 30301  df-padd 30593  df-polarityN 30700
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