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Theorem pmapjat2 35679
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 35678 . 2  |-  ( ( K  e.  HL  /\  X  e.  B  /\  Q  e.  A )  ->  ( M `  ( X  .\/  Q ) )  =  ( ( M `
 X )  .+  ( M `  Q ) ) )
7 hllat 35189 . . . . 5  |-  ( K  e.  HL  ->  K  e.  Lat )
873ad2ant1 1017 . . . 4  |-  ( ( K  e.  HL  /\  X  e.  B  /\  Q  e.  A )  ->  K  e.  Lat )
91, 3atbase 35115 . . . . 5  |-  ( Q  e.  A  ->  Q  e.  B )
1093ad2ant3 1019 . . . 4  |-  ( ( K  e.  HL  /\  X  e.  B  /\  Q  e.  A )  ->  Q  e.  B )
11 simp2 997 . . . 4  |-  ( ( K  e.  HL  /\  X  e.  B  /\  Q  e.  A )  ->  X  e.  B )
121, 2latjcom 15815 . . . 4  |-  ( ( K  e.  Lat  /\  Q  e.  B  /\  X  e.  B )  ->  ( Q  .\/  X
)  =  ( X 
.\/  Q ) )
138, 10, 11, 12syl3anc 1228 . . 3  |-  ( ( K  e.  HL  /\  X  e.  B  /\  Q  e.  A )  ->  ( Q  .\/  X
)  =  ( X 
.\/  Q ) )
1413fveq2d 5876 . 2  |-  ( ( K  e.  HL  /\  X  e.  B  /\  Q  e.  A )  ->  ( M `  ( Q  .\/  X ) )  =  ( M `  ( X  .\/  Q ) ) )
15 simp1 996 . . . 4  |-  ( ( K  e.  HL  /\  X  e.  B  /\  Q  e.  A )  ->  K  e.  HL )
161, 3, 4pmapssat 35584 . . . 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 35584 . . . 4  |-  ( ( K  e.  HL  /\  X  e.  B )  ->  ( M `  X
)  C_  A )
19183adant3 1016 . . 3  |-  ( ( K  e.  HL  /\  X  e.  B  /\  Q  e.  A )  ->  ( M `  X
)  C_  A )
203, 5paddcom 35638 . . 3  |-  ( ( K  e.  Lat  /\  ( M `  Q ) 
C_  A  /\  ( M `  X )  C_  A )  ->  (
( M `  Q
)  .+  ( M `  X ) )  =  ( ( M `  X )  .+  ( M `  Q )
) )
218, 17, 19, 20syl3anc 1228 . 2  |-  ( ( K  e.  HL  /\  X  e.  B  /\  Q  e.  A )  ->  ( ( M `  Q )  .+  ( M `  X )
)  =  ( ( M `  X ) 
.+  ( M `  Q ) ) )
226, 14, 213eqtr4d 2508 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 973    = wceq 1395    e. wcel 1819    C_ wss 3471   ` cfv 5594  (class class class)co 6296   Basecbs 14643   joincjn 15699   Latclat 15801   Atomscatm 35089   HLchlt 35176   pmapcpmap 35322   +Pcpadd 35620
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-1st 6799  df-2nd 6800  df-preset 15683  df-poset 15701  df-plt 15714  df-lub 15730  df-glb 15731  df-join 15732  df-meet 15733  df-p0 15795  df-lat 15802  df-clat 15864  df-oposet 35002  df-ol 35004  df-oml 35005  df-covers 35092  df-ats 35093  df-atl 35124  df-cvlat 35148  df-hlat 35177  df-pmap 35329  df-padd 35621
This theorem is referenced by:  atmod1i1  35682
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