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Theorem pmapj2N 30118
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 955 . . 3  |-  ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  ->  K  e.  HL )
2 hllat 29553 . . . . 5  |-  ( K  e.  HL  ->  K  e.  Lat )
323ad2ant1 976 . . . 4  |-  ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  ->  K  e.  Lat )
4 hlop 29552 . . . . . 6  |-  ( K  e.  HL  ->  K  e.  OP )
543ad2ant1 976 . . . . 5  |-  ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  ->  K  e.  OP )
6 simp2 956 . . . . 5  |-  ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  ->  X  e.  B )
7 pmapj2.b . . . . . 6  |-  B  =  ( Base `  K
)
8 eqid 2283 . . . . . 6  |-  ( oc
`  K )  =  ( oc `  K
)
97, 8opoccl 29384 . . . . 5  |-  ( ( K  e.  OP  /\  X  e.  B )  ->  ( ( oc `  K ) `  X
)  e.  B )
105, 6, 9syl2anc 642 . . . 4  |-  ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( oc `  K ) `  X
)  e.  B )
11 simp3 957 . . . . 5  |-  ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  ->  Y  e.  B )
127, 8opoccl 29384 . . . . 5  |-  ( ( K  e.  OP  /\  Y  e.  B )  ->  ( ( oc `  K ) `  Y
)  e.  B )
135, 11, 12syl2anc 642 . . . 4  |-  ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( oc `  K ) `  Y
)  e.  B )
14 eqid 2283 . . . . 5  |-  ( meet `  K )  =  (
meet `  K )
157, 14latmcl 14157 . . . 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 1182 . . 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 30101 . . 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 642 . 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 30101 . . . . . 6  |-  ( ( K  e.  HL  /\  X  e.  B )  ->  (  ._|_  `  ( M `
 X ) )  =  ( M `  ( ( oc `  K ) `  X
) ) )
22213adant3 975 . . . . 5  |-  ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  ->  (  ._|_  `  ( M `
 X ) )  =  ( M `  ( ( oc `  K ) `  X
) ) )
237, 8, 17, 18polpmapN 30101 . . . . . 6  |-  ( ( K  e.  HL  /\  Y  e.  B )  ->  (  ._|_  `  ( M `
 Y ) )  =  ( M `  ( ( oc `  K ) `  Y
) ) )
24233adant2 974 . . . . 5  |-  ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  ->  (  ._|_  `  ( M `
 Y ) )  =  ( M `  ( ( oc `  K ) `  Y
) ) )
2522, 24ineq12d 3371 . . . 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 2283 . . . . . . 7  |-  ( Atoms `  K )  =  (
Atoms `  K )
277, 26, 17pmapssat 29948 . . . . . 6  |-  ( ( K  e.  HL  /\  X  e.  B )  ->  ( M `  X
)  C_  ( Atoms `  K ) )
28273adant3 975 . . . . 5  |-  ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  ->  ( M `  X
)  C_  ( Atoms `  K ) )
297, 26, 17pmapssat 29948 . . . . . 6  |-  ( ( K  e.  HL  /\  Y  e.  B )  ->  ( M `  Y
)  C_  ( Atoms `  K ) )
30293adant2 974 . . . . 5  |-  ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  ->  ( M `  Y
)  C_  ( Atoms `  K ) )
31 pmapj2.p . . . . . 6  |-  .+  =  ( + P `  K
)
3226, 31, 18poldmj1N 30117 . . . . 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 1182 . . . 4  |-  ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  ->  (  ._|_  `  ( ( M `  X ) 
.+  ( M `  Y ) ) )  =  ( (  ._|_  `  ( M `  X
) )  i^i  (  ._|_  `  ( M `  Y ) ) ) )
347, 14, 26, 17pmapmeet 29962 . . . . 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 1182 . . . 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 2326 . . 3  |-  ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  ->  ( M `  (
( ( oc `  K ) `  X
) ( meet `  K
) ( ( oc
`  K ) `  Y ) ) )  =  (  ._|_  `  (
( M `  X
)  .+  ( M `  Y ) ) ) )
3736fveq2d 5529 . 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 29551 . . . 4  |-  ( K  e.  HL  ->  K  e.  OL )
39 pmapj2.j . . . . 5  |-  .\/  =  ( join `  K )
407, 39, 14, 8oldmm4 29410 . . . 4  |-  ( ( K  e.  OL  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( oc `  K ) `  (
( ( oc `  K ) `  X
) ( meet `  K
) ( ( oc
`  K ) `  Y ) ) )  =  ( X  .\/  Y ) )
4138, 40syl3an1 1215 . . 3  |-  ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( oc `  K ) `  (
( ( oc `  K ) `  X
) ( meet `  K
) ( ( oc
`  K ) `  Y ) ) )  =  ( X  .\/  Y ) )
4241fveq2d 5529 . 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 2324 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 934    = wceq 1623    e. wcel 1684    i^i cin 3151    C_ wss 3152   ` cfv 5255  (class class class)co 5858   Basecbs 13148   occoc 13216   joincjn 14078   meetcmee 14079   Latclat 14151   OPcops 29362   OLcol 29364   Atomscatm 29453   HLchlt 29540   pmapcpmap 29686   + Pcpadd 29984   _|_ PcpolN 30091
This theorem is referenced by:  pmapocjN  30119  pmapojoinN  30157
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-iin 3908  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-undef 6298  df-riota 6304  df-poset 14080  df-plt 14092  df-lub 14108  df-glb 14109  df-join 14110  df-meet 14111  df-p0 14145  df-p1 14146  df-lat 14152  df-clat 14214  df-oposet 29366  df-ol 29368  df-oml 29369  df-covers 29456  df-ats 29457  df-atl 29488  df-cvlat 29512  df-hlat 29541  df-psubsp 29692  df-pmap 29693  df-padd 29985  df-polarityN 30092
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