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Theorem pmapojoinN 30765
Description: For orthogonal elements, projective map of join equals projective sum. Compare pmapjoin 30649 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 2436 . . . 4  |-  ( _|_
P `  K )  =  ( _|_ P `  K )
61, 2, 3, 4, 5pmapj2N 30726 . . 3  |-  ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  ->  ( M `  ( X  .\/  Y ) )  =  ( ( _|_
P `  K ) `  ( ( _|_ P `  K ) `  (
( M `  X
)  .+  ( M `  Y ) ) ) ) )
76adantr 452 . 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 960 . . 3  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  X  .<_  (  ._|_  `  Y ) )  ->  K  e.  HL )
9 simpl2 961 . . . . 5  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  X  .<_  (  ._|_  `  Y ) )  ->  X  e.  B )
10 eqid 2436 . . . . . 6  |-  ( PSubCl `  K )  =  (
PSubCl `  K )
111, 3, 10pmapsubclN 30743 . . . . 5  |-  ( ( K  e.  HL  /\  X  e.  B )  ->  ( M `  X
)  e.  ( PSubCl `  K ) )
128, 9, 11syl2anc 643 . . . 4  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  X  .<_  (  ._|_  `  Y ) )  -> 
( M `  X
)  e.  ( PSubCl `  K ) )
13 simpl3 962 . . . . 5  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  X  .<_  (  ._|_  `  Y ) )  ->  Y  e.  B )
141, 3, 10pmapsubclN 30743 . . . . 5  |-  ( ( K  e.  HL  /\  Y  e.  B )  ->  ( M `  Y
)  e.  ( PSubCl `  K ) )
158, 13, 14syl2anc 643 . . . 4  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  X  .<_  (  ._|_  `  Y ) )  -> 
( M `  Y
)  e.  ( PSubCl `  K ) )
16 hlop 30160 . . . . . . . . 9  |-  ( K  e.  HL  ->  K  e.  OP )
17163ad2ant1 978 . . . . . . . 8  |-  ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  ->  K  e.  OP )
18 simp3 959 . . . . . . . 8  |-  ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  ->  Y  e.  B )
19 pmapojoin.o . . . . . . . . 9  |-  ._|_  =  ( oc `  K )
201, 19opoccl 29992 . . . . . . . 8  |-  ( ( K  e.  OP  /\  Y  e.  B )  ->  (  ._|_  `  Y )  e.  B )
2117, 18, 20syl2anc 643 . . . . . . 7  |-  ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  ->  (  ._|_  `  Y )  e.  B )
22 pmapojoin.l . . . . . . . 8  |-  .<_  =  ( le `  K )
231, 22, 3pmaple 30558 . . . . . . 7  |-  ( ( K  e.  HL  /\  X  e.  B  /\  (  ._|_  `  Y )  e.  B )  ->  ( X  .<_  (  ._|_  `  Y
)  <->  ( M `  X )  C_  ( M `  (  ._|_  `  Y ) ) ) )
2421, 23syld3an3 1229 . . . . . 6  |-  ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .<_  (  ._|_  `  Y )  <->  ( M `  X )  C_  ( M `  (  ._|_  `  Y ) ) ) )
2524biimpa 471 . . . . 5  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  X  .<_  (  ._|_  `  Y ) )  -> 
( M `  X
)  C_  ( M `  (  ._|_  `  Y
) ) )
261, 19, 3, 5polpmapN 30709 . . . . . 6  |-  ( ( K  e.  HL  /\  Y  e.  B )  ->  ( ( _|_ P `  K ) `  ( M `  Y )
)  =  ( M `
 (  ._|_  `  Y
) ) )
278, 13, 26syl2anc 643 . . . . 5  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  X  .<_  (  ._|_  `  Y ) )  -> 
( ( _|_ P `  K ) `  ( M `  Y )
)  =  ( M `
 (  ._|_  `  Y
) ) )
2825, 27sseqtr4d 3385 . . . 4  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  X  .<_  (  ._|_  `  Y ) )  -> 
( M `  X
)  C_  ( ( _|_ P `  K ) `
 ( M `  Y ) ) )
294, 5, 10osumclN 30764 . . . 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 1187 . . 3  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  X  .<_  (  ._|_  `  Y ) )  -> 
( ( M `  X )  .+  ( M `  Y )
)  e.  ( PSubCl `  K ) )
315, 10psubcli2N 30736 . . 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 643 . 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 2468 1  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  X  .<_  (  ._|_  `  Y ) )  -> 
( M `  ( X  .\/  Y ) )  =  ( ( M `
 X )  .+  ( M `  Y ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725    C_ wss 3320   class class class wbr 4212   ` cfv 5454  (class class class)co 6081   Basecbs 13469   lecple 13536   occoc 13537   joincjn 14401   OPcops 29970   HLchlt 30148   pmapcpmap 30294   + Pcpadd 30592   _|_ PcpolN 30699   PSubClcpscN 30731
This theorem is referenced by:  pl42lem1N  30776
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-pss 3336  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  df-psubclN 30732
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