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Theorem pmapojoinN 30157
Description: For orthogonal elements, projective map of join equals projective sum. Compare pmapjoin 30041 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 2283 . . . 4  |-  ( _|_
P `  K )  =  ( _|_ P `  K )
61, 2, 3, 4, 5pmapj2N 30118 . . 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 2283 . . . . . 6  |-  ( PSubCl `  K )  =  (
PSubCl `  K )
111, 3, 10pmapsubclN 30135 . . . . 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 30135 . . . . 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 29552 . . . . . . . . 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 29384 . . . . . . . 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 29950 . . . . . . 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 30101 . . . . . 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 3215 . . . 4  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  X  .<_  (  ._|_  `  Y ) )  -> 
( M `  X
)  C_  ( ( _|_ P `  K ) `
 ( M `  Y ) ) )
294, 5, 10osumclN 30156 . . . 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 30128 . . 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 2315 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 1623    e. wcel 1684    C_ wss 3152   class class class wbr 4023   ` cfv 5255  (class class class)co 5858   Basecbs 13148   lecple 13215   occoc 13216   joincjn 14078   OPcops 29362   HLchlt 29540   pmapcpmap 29686   + Pcpadd 29984   _|_ PcpolN 30091   PSubClcpscN 30123
This theorem is referenced by:  pl42lem1N  30168
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-iin 3908  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-undef 6298  df-riota 6304  df-poset 14080  df-plt 14092  df-lub 14108  df-glb 14109  df-join 14110  df-meet 14111  df-p0 14145  df-p1 14146  df-lat 14152  df-clat 14214  df-oposet 29366  df-ol 29368  df-oml 29369  df-covers 29456  df-ats 29457  df-atl 29488  df-cvlat 29512  df-hlat 29541  df-psubsp 29692  df-pmap 29693  df-padd 29985  df-polarityN 30092  df-psubclN 30124
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