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Theorem pmapojoinN 35835
Description: For orthogonal elements, projective map of join equals projective sum. Compare pmapjoin 35719 where only one direction holds. (Contributed by NM, 11-Apr-2012.) (New usage is discouraged.)
Hypotheses
Ref Expression
pmapojoin.b  |-  B  =  ( Base `  K
)
pmapojoin.l  |-  .<_  =  ( le `  K )
pmapojoin.j  |-  .\/  =  ( join `  K )
pmapojoin.m  |-  M  =  ( pmap `  K
)
pmapojoin.o  |-  ._|_  =  ( oc `  K )
pmapojoin.p  |-  .+  =  ( +P `  K
)
Assertion
Ref Expression
pmapojoinN  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  X  .<_  (  ._|_  `  Y ) )  -> 
( M `  ( X  .\/  Y ) )  =  ( ( M `
 X )  .+  ( M `  Y ) ) )

Proof of Theorem pmapojoinN
StepHypRef Expression
1 pmapojoin.b . . . 4  |-  B  =  ( Base `  K
)
2 pmapojoin.j . . . 4  |-  .\/  =  ( join `  K )
3 pmapojoin.m . . . 4  |-  M  =  ( pmap `  K
)
4 pmapojoin.p . . . 4  |-  .+  =  ( +P `  K
)
5 eqid 2457 . . . 4  |-  ( _|_P `  K )  =  ( _|_P `  K )
61, 2, 3, 4, 5pmapj2N 35796 . . 3  |-  ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  ->  ( M `  ( X  .\/  Y ) )  =  ( ( _|_P `  K ) `
 ( ( _|_P `  K ) `
 ( ( M `
 X )  .+  ( M `  Y ) ) ) ) )
76adantr 465 . 2  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  X  .<_  (  ._|_  `  Y ) )  -> 
( M `  ( X  .\/  Y ) )  =  ( ( _|_P `  K ) `
 ( ( _|_P `  K ) `
 ( ( M `
 X )  .+  ( M `  Y ) ) ) ) )
8 simpl1 999 . . 3  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  X  .<_  (  ._|_  `  Y ) )  ->  K  e.  HL )
9 simpl2 1000 . . . . 5  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  X  .<_  (  ._|_  `  Y ) )  ->  X  e.  B )
10 eqid 2457 . . . . . 6  |-  ( PSubCl `  K )  =  (
PSubCl `  K )
111, 3, 10pmapsubclN 35813 . . . . 5  |-  ( ( K  e.  HL  /\  X  e.  B )  ->  ( M `  X
)  e.  ( PSubCl `  K ) )
128, 9, 11syl2anc 661 . . . 4  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  X  .<_  (  ._|_  `  Y ) )  -> 
( M `  X
)  e.  ( PSubCl `  K ) )
13 simpl3 1001 . . . . 5  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  X  .<_  (  ._|_  `  Y ) )  ->  Y  e.  B )
141, 3, 10pmapsubclN 35813 . . . . 5  |-  ( ( K  e.  HL  /\  Y  e.  B )  ->  ( M `  Y
)  e.  ( PSubCl `  K ) )
158, 13, 14syl2anc 661 . . . 4  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  X  .<_  (  ._|_  `  Y ) )  -> 
( M `  Y
)  e.  ( PSubCl `  K ) )
16 hlop 35230 . . . . . . . . 9  |-  ( K  e.  HL  ->  K  e.  OP )
17163ad2ant1 1017 . . . . . . . 8  |-  ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  ->  K  e.  OP )
18 simp3 998 . . . . . . . 8  |-  ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  ->  Y  e.  B )
19 pmapojoin.o . . . . . . . . 9  |-  ._|_  =  ( oc `  K )
201, 19opoccl 35062 . . . . . . . 8  |-  ( ( K  e.  OP  /\  Y  e.  B )  ->  (  ._|_  `  Y )  e.  B )
2117, 18, 20syl2anc 661 . . . . . . 7  |-  ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  ->  (  ._|_  `  Y )  e.  B )
22 pmapojoin.l . . . . . . . 8  |-  .<_  =  ( le `  K )
231, 22, 3pmaple 35628 . . . . . . 7  |-  ( ( K  e.  HL  /\  X  e.  B  /\  (  ._|_  `  Y )  e.  B )  ->  ( X  .<_  (  ._|_  `  Y
)  <->  ( M `  X )  C_  ( M `  (  ._|_  `  Y ) ) ) )
2421, 23syld3an3 1273 . . . . . 6  |-  ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .<_  (  ._|_  `  Y )  <->  ( M `  X )  C_  ( M `  (  ._|_  `  Y ) ) ) )
2524biimpa 484 . . . . 5  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  X  .<_  (  ._|_  `  Y ) )  -> 
( M `  X
)  C_  ( M `  (  ._|_  `  Y
) ) )
261, 19, 3, 5polpmapN 35779 . . . . . 6  |-  ( ( K  e.  HL  /\  Y  e.  B )  ->  ( ( _|_P `  K ) `  ( M `  Y )
)  =  ( M `
 (  ._|_  `  Y
) ) )
278, 13, 26syl2anc 661 . . . . 5  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  X  .<_  (  ._|_  `  Y ) )  -> 
( ( _|_P `  K ) `  ( M `  Y )
)  =  ( M `
 (  ._|_  `  Y
) ) )
2825, 27sseqtr4d 3536 . . . 4  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  X  .<_  (  ._|_  `  Y ) )  -> 
( M `  X
)  C_  ( ( _|_P `  K ) `
 ( M `  Y ) ) )
294, 5, 10osumclN 35834 . . . 4  |-  ( ( ( K  e.  HL  /\  ( M `  X
)  e.  ( PSubCl `  K )  /\  ( M `  Y )  e.  ( PSubCl `  K )
)  /\  ( M `  X )  C_  (
( _|_P `  K ) `  ( M `  Y )
) )  ->  (
( M `  X
)  .+  ( M `  Y ) )  e.  ( PSubCl `  K )
)
308, 12, 15, 28, 29syl31anc 1231 . . 3  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  X  .<_  (  ._|_  `  Y ) )  -> 
( ( M `  X )  .+  ( M `  Y )
)  e.  ( PSubCl `  K ) )
315, 10psubcli2N 35806 . . 3  |-  ( ( K  e.  HL  /\  ( ( M `  X )  .+  ( M `  Y )
)  e.  ( PSubCl `  K ) )  -> 
( ( _|_P `  K ) `  (
( _|_P `  K ) `  (
( M `  X
)  .+  ( M `  Y ) ) ) )  =  ( ( M `  X ) 
.+  ( M `  Y ) ) )
328, 30, 31syl2anc 661 . 2  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  X  .<_  (  ._|_  `  Y ) )  -> 
( ( _|_P `  K ) `  (
( _|_P `  K ) `  (
( M `  X
)  .+  ( M `  Y ) ) ) )  =  ( ( M `  X ) 
.+  ( M `  Y ) ) )
337, 32eqtrd 2498 1  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  X  .<_  (  ._|_  `  Y ) )  -> 
( M `  ( X  .\/  Y ) )  =  ( ( M `
 X )  .+  ( M `  Y ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1395    e. wcel 1819    C_ wss 3471   class class class wbr 4456   ` cfv 5594  (class class class)co 6296   Basecbs 14644   lecple 14719   occoc 14720   joincjn 15700   OPcops 35040   HLchlt 35218   pmapcpmap 35364   +Pcpadd 35662   _|_PcpolN 35769   PSubClcpscN 35801
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-riotaBAD 34827
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-fal 1401  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-iun 4334  df-iin 4335  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-1st 6799  df-2nd 6800  df-undef 7020  df-preset 15684  df-poset 15702  df-plt 15715  df-lub 15731  df-glb 15732  df-join 15733  df-meet 15734  df-p0 15796  df-p1 15797  df-lat 15803  df-clat 15865  df-oposet 35044  df-ol 35046  df-oml 35047  df-covers 35134  df-ats 35135  df-atl 35166  df-cvlat 35190  df-hlat 35219  df-psubsp 35370  df-pmap 35371  df-padd 35663  df-polarityN 35770  df-psubclN 35802
This theorem is referenced by:  pl42lem1N  35846
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