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Theorem pmapojoinN 34764
Description: For orthogonal elements, projective map of join equals projective sum. Compare pmapjoin 34648 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 2467 . . . 4  |-  ( _|_P `  K )  =  ( _|_P `  K )
61, 2, 3, 4, 5pmapj2N 34725 . . 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 2467 . . . . . 6  |-  ( PSubCl `  K )  =  (
PSubCl `  K )
111, 3, 10pmapsubclN 34742 . . . . 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 34742 . . . . 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 34159 . . . . . . . . 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 33991 . . . . . . . 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 34557 . . . . . . 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 34708 . . . . . 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 3541 . . . 4  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  X  .<_  (  ._|_  `  Y ) )  -> 
( M `  X
)  C_  ( ( _|_P `  K ) `
 ( M `  Y ) ) )
294, 5, 10osumclN 34763 . . . 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 34735 . . 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 2508 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 1379    e. wcel 1767    C_ wss 3476   class class class wbr 4447   ` cfv 5586  (class class class)co 6282   Basecbs 14483   lecple 14555   occoc 14556   joincjn 15424   OPcops 33969   HLchlt 34147   pmapcpmap 34293   +Pcpadd 34591   _|_PcpolN 34698   PSubClcpscN 34730
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-riotaBAD 33756
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-iin 4328  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-1st 6781  df-2nd 6782  df-undef 6999  df-poset 15426  df-plt 15438  df-lub 15454  df-glb 15455  df-join 15456  df-meet 15457  df-p0 15519  df-p1 15520  df-lat 15526  df-clat 15588  df-oposet 33973  df-ol 33975  df-oml 33976  df-covers 34063  df-ats 34064  df-atl 34095  df-cvlat 34119  df-hlat 34148  df-psubsp 34299  df-pmap 34300  df-padd 34592  df-polarityN 34699  df-psubclN 34731
This theorem is referenced by:  pl42lem1N  34775
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