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Theorem pmapjat2 33498
Description: The projective map of the join of an atom with a lattice element. (Contributed by NM, 12-May-2012.)
Hypotheses
Ref Expression
pmapjat.b  |-  B  =  ( Base `  K
)
pmapjat.j  |-  .\/  =  ( join `  K )
pmapjat.a  |-  A  =  ( Atoms `  K )
pmapjat.m  |-  M  =  ( pmap `  K
)
pmapjat.p  |-  .+  =  ( +P `  K
)
Assertion
Ref Expression
pmapjat2  |-  ( ( K  e.  HL  /\  X  e.  B  /\  Q  e.  A )  ->  ( M `  ( Q  .\/  X ) )  =  ( ( M `
 Q )  .+  ( M `  X ) ) )

Proof of Theorem pmapjat2
StepHypRef Expression
1 pmapjat.b . . 3  |-  B  =  ( Base `  K
)
2 pmapjat.j . . 3  |-  .\/  =  ( join `  K )
3 pmapjat.a . . 3  |-  A  =  ( Atoms `  K )
4 pmapjat.m . . 3  |-  M  =  ( pmap `  K
)
5 pmapjat.p . . 3  |-  .+  =  ( +P `  K
)
61, 2, 3, 4, 5pmapjat1 33497 . 2  |-  ( ( K  e.  HL  /\  X  e.  B  /\  Q  e.  A )  ->  ( M `  ( X  .\/  Q ) )  =  ( ( M `
 X )  .+  ( M `  Q ) ) )
7 hllat 33008 . . . . 5  |-  ( K  e.  HL  ->  K  e.  Lat )
873ad2ant1 1009 . . . 4  |-  ( ( K  e.  HL  /\  X  e.  B  /\  Q  e.  A )  ->  K  e.  Lat )
91, 3atbase 32934 . . . . 5  |-  ( Q  e.  A  ->  Q  e.  B )
1093ad2ant3 1011 . . . 4  |-  ( ( K  e.  HL  /\  X  e.  B  /\  Q  e.  A )  ->  Q  e.  B )
11 simp2 989 . . . 4  |-  ( ( K  e.  HL  /\  X  e.  B  /\  Q  e.  A )  ->  X  e.  B )
121, 2latjcom 15229 . . . 4  |-  ( ( K  e.  Lat  /\  Q  e.  B  /\  X  e.  B )  ->  ( Q  .\/  X
)  =  ( X 
.\/  Q ) )
138, 10, 11, 12syl3anc 1218 . . 3  |-  ( ( K  e.  HL  /\  X  e.  B  /\  Q  e.  A )  ->  ( Q  .\/  X
)  =  ( X 
.\/  Q ) )
1413fveq2d 5695 . 2  |-  ( ( K  e.  HL  /\  X  e.  B  /\  Q  e.  A )  ->  ( M `  ( Q  .\/  X ) )  =  ( M `  ( X  .\/  Q ) ) )
15 simp1 988 . . . 4  |-  ( ( K  e.  HL  /\  X  e.  B  /\  Q  e.  A )  ->  K  e.  HL )
161, 3, 4pmapssat 33403 . . . 4  |-  ( ( K  e.  HL  /\  Q  e.  B )  ->  ( M `  Q
)  C_  A )
1715, 10, 16syl2anc 661 . . 3  |-  ( ( K  e.  HL  /\  X  e.  B  /\  Q  e.  A )  ->  ( M `  Q
)  C_  A )
181, 3, 4pmapssat 33403 . . . 4  |-  ( ( K  e.  HL  /\  X  e.  B )  ->  ( M `  X
)  C_  A )
19183adant3 1008 . . 3  |-  ( ( K  e.  HL  /\  X  e.  B  /\  Q  e.  A )  ->  ( M `  X
)  C_  A )
203, 5paddcom 33457 . . 3  |-  ( ( K  e.  Lat  /\  ( M `  Q ) 
C_  A  /\  ( M `  X )  C_  A )  ->  (
( M `  Q
)  .+  ( M `  X ) )  =  ( ( M `  X )  .+  ( M `  Q )
) )
218, 17, 19, 20syl3anc 1218 . 2  |-  ( ( K  e.  HL  /\  X  e.  B  /\  Q  e.  A )  ->  ( ( M `  Q )  .+  ( M `  X )
)  =  ( ( M `  X ) 
.+  ( M `  Q ) ) )
226, 14, 213eqtr4d 2485 1  |-  ( ( K  e.  HL  /\  X  e.  B  /\  Q  e.  A )  ->  ( M `  ( Q  .\/  X ) )  =  ( ( M `
 Q )  .+  ( M `  X ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ w3a 965    = wceq 1369    e. wcel 1756    C_ wss 3328   ` cfv 5418  (class class class)co 6091   Basecbs 14174   joincjn 15114   Latclat 15215   Atomscatm 32908   HLchlt 32995   pmapcpmap 33141   +Pcpadd 33439
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4403  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2720  df-rex 2721  df-reu 2722  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-op 3884  df-uni 4092  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-id 4636  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-riota 6052  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-1st 6577  df-2nd 6578  df-poset 15116  df-plt 15128  df-lub 15144  df-glb 15145  df-join 15146  df-meet 15147  df-p0 15209  df-lat 15216  df-clat 15278  df-oposet 32821  df-ol 32823  df-oml 32824  df-covers 32911  df-ats 32912  df-atl 32943  df-cvlat 32967  df-hlat 32996  df-pmap 33148  df-padd 33440
This theorem is referenced by:  atmod1i1  33501
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