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Theorem pmapojoinN 30779
Description: For orthogonal elements, projective map of join equals projective sum. Compare pmapjoin 30663 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 2296 . . . 4  |-  ( _|_
P `  K )  =  ( _|_ P `  K )
61, 2, 3, 4, 5pmapj2N 30740 . . 3  |-  ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  ->  ( M `  ( X  .\/  Y ) )  =  ( ( _|_
P `  K ) `  ( ( _|_ P `  K ) `  (
( M `  X
)  .+  ( M `  Y ) ) ) ) )
76adantr 451 . 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 958 . . 3  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  X  .<_  (  ._|_  `  Y ) )  ->  K  e.  HL )
9 simpl2 959 . . . . 5  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  X  .<_  (  ._|_  `  Y ) )  ->  X  e.  B )
10 eqid 2296 . . . . . 6  |-  ( PSubCl `  K )  =  (
PSubCl `  K )
111, 3, 10pmapsubclN 30757 . . . . 5  |-  ( ( K  e.  HL  /\  X  e.  B )  ->  ( M `  X
)  e.  ( PSubCl `  K ) )
128, 9, 11syl2anc 642 . . . 4  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  X  .<_  (  ._|_  `  Y ) )  -> 
( M `  X
)  e.  ( PSubCl `  K ) )
13 simpl3 960 . . . . 5  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  X  .<_  (  ._|_  `  Y ) )  ->  Y  e.  B )
141, 3, 10pmapsubclN 30757 . . . . 5  |-  ( ( K  e.  HL  /\  Y  e.  B )  ->  ( M `  Y
)  e.  ( PSubCl `  K ) )
158, 13, 14syl2anc 642 . . . 4  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  X  .<_  (  ._|_  `  Y ) )  -> 
( M `  Y
)  e.  ( PSubCl `  K ) )
16 hlop 30174 . . . . . . . . 9  |-  ( K  e.  HL  ->  K  e.  OP )
17163ad2ant1 976 . . . . . . . 8  |-  ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  ->  K  e.  OP )
18 simp3 957 . . . . . . . 8  |-  ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  ->  Y  e.  B )
19 pmapojoin.o . . . . . . . . 9  |-  ._|_  =  ( oc `  K )
201, 19opoccl 30006 . . . . . . . 8  |-  ( ( K  e.  OP  /\  Y  e.  B )  ->  (  ._|_  `  Y )  e.  B )
2117, 18, 20syl2anc 642 . . . . . . 7  |-  ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  ->  (  ._|_  `  Y )  e.  B )
22 pmapojoin.l . . . . . . . 8  |-  .<_  =  ( le `  K )
231, 22, 3pmaple 30572 . . . . . . 7  |-  ( ( K  e.  HL  /\  X  e.  B  /\  (  ._|_  `  Y )  e.  B )  ->  ( X  .<_  (  ._|_  `  Y
)  <->  ( M `  X )  C_  ( M `  (  ._|_  `  Y ) ) ) )
2421, 23syld3an3 1227 . . . . . 6  |-  ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .<_  (  ._|_  `  Y )  <->  ( M `  X )  C_  ( M `  (  ._|_  `  Y ) ) ) )
2524biimpa 470 . . . . 5  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  X  .<_  (  ._|_  `  Y ) )  -> 
( M `  X
)  C_  ( M `  (  ._|_  `  Y
) ) )
261, 19, 3, 5polpmapN 30723 . . . . . 6  |-  ( ( K  e.  HL  /\  Y  e.  B )  ->  ( ( _|_ P `  K ) `  ( M `  Y )
)  =  ( M `
 (  ._|_  `  Y
) ) )
278, 13, 26syl2anc 642 . . . . 5  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  X  .<_  (  ._|_  `  Y ) )  -> 
( ( _|_ P `  K ) `  ( M `  Y )
)  =  ( M `
 (  ._|_  `  Y
) ) )
2825, 27sseqtr4d 3228 . . . 4  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  X  .<_  (  ._|_  `  Y ) )  -> 
( M `  X
)  C_  ( ( _|_ P `  K ) `
 ( M `  Y ) ) )
294, 5, 10osumclN 30778 . . . 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 1185 . . 3  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  X  .<_  (  ._|_  `  Y ) )  -> 
( ( M `  X )  .+  ( M `  Y )
)  e.  ( PSubCl `  K ) )
315, 10psubcli2N 30750 . . 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 642 . 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 2328 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 176    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696    C_ wss 3165   class class class wbr 4039   ` cfv 5271  (class class class)co 5874   Basecbs 13164   lecple 13231   occoc 13232   joincjn 14094   OPcops 29984   HLchlt 30162   pmapcpmap 30308   + Pcpadd 30606   _|_ PcpolN 30713   PSubClcpscN 30745
This theorem is referenced by:  pl42lem1N  30790
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-iin 3924  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-undef 6314  df-riota 6320  df-poset 14096  df-plt 14108  df-lub 14124  df-glb 14125  df-join 14126  df-meet 14127  df-p0 14161  df-p1 14162  df-lat 14168  df-clat 14230  df-oposet 29988  df-ol 29990  df-oml 29991  df-covers 30078  df-ats 30079  df-atl 30110  df-cvlat 30134  df-hlat 30163  df-psubsp 30314  df-pmap 30315  df-padd 30607  df-polarityN 30714  df-psubclN 30746
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