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Theorem pmapj2N 29936
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 29371 . . . . 5  |-  ( K  e.  HL  ->  K  e.  Lat )
323ad2ant1 976 . . . 4  |-  ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  ->  K  e.  Lat )
4 hlop 29370 . . . . . 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 2316 . . . . . 6  |-  ( oc
`  K )  =  ( oc `  K
)
97, 8opoccl 29202 . . . . 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 29202 . . . . 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 2316 . . . . 5  |-  ( meet `  K )  =  (
meet `  K )
157, 14latmcl 14206 . . . 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 29919 . . 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 29919 . . . . . 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 29919 . . . . . 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 3405 . . . 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 2316 . . . . . . 7  |-  ( Atoms `  K )  =  (
Atoms `  K )
277, 26, 17pmapssat 29766 . . . . . 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 29766 . . . . . 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 29935 . . . . 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 29780 . . . . 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 2359 . . 3  |-  ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  ->  ( M `  (
( ( oc `  K ) `  X
) ( meet `  K
) ( ( oc
`  K ) `  Y ) ) )  =  (  ._|_  `  (
( M `  X
)  .+  ( M `  Y ) ) ) )
3736fveq2d 5567 . 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 29369 . . . 4  |-  ( K  e.  HL  ->  K  e.  OL )
39 pmapj2.j . . . . 5  |-  .\/  =  ( join `  K )
407, 39, 14, 8oldmm4 29228 . . . 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 5567 . 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 2357 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 1633    e. wcel 1701    i^i cin 3185    C_ wss 3186   ` cfv 5292  (class class class)co 5900   Basecbs 13195   occoc 13263   joincjn 14127   meetcmee 14128   Latclat 14200   OPcops 29180   OLcol 29182   Atomscatm 29271   HLchlt 29358   pmapcpmap 29504   + Pcpadd 29802   _|_ PcpolN 29909
This theorem is referenced by:  pmapocjN  29937  pmapojoinN  29975
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-rep 4168  ax-sep 4178  ax-nul 4186  ax-pow 4225  ax-pr 4251  ax-un 4549
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-nel 2482  df-ral 2582  df-rex 2583  df-reu 2584  df-rmo 2585  df-rab 2586  df-v 2824  df-sbc 3026  df-csb 3116  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-nul 3490  df-if 3600  df-pw 3661  df-sn 3680  df-pr 3681  df-op 3683  df-uni 3865  df-iun 3944  df-iin 3945  df-br 4061  df-opab 4115  df-mpt 4116  df-id 4346  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-res 4738  df-ima 4739  df-iota 5256  df-fun 5294  df-fn 5295  df-f 5296  df-f1 5297  df-fo 5298  df-f1o 5299  df-fv 5300  df-ov 5903  df-oprab 5904  df-mpt2 5905  df-1st 6164  df-2nd 6165  df-undef 6340  df-riota 6346  df-poset 14129  df-plt 14141  df-lub 14157  df-glb 14158  df-join 14159  df-meet 14160  df-p0 14194  df-p1 14195  df-lat 14201  df-clat 14263  df-oposet 29184  df-ol 29186  df-oml 29187  df-covers 29274  df-ats 29275  df-atl 29306  df-cvlat 29330  df-hlat 29359  df-psubsp 29510  df-pmap 29511  df-padd 29803  df-polarityN 29910
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