Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  pmapjat2 Unicode version

Theorem pmapjat2 30043
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 30042 . 2  |-  ( ( K  e.  HL  /\  X  e.  B  /\  Q  e.  A )  ->  ( M `  ( X  .\/  Q ) )  =  ( ( M `
 X )  .+  ( M `  Q ) ) )
7 hllat 29553 . . . . 5  |-  ( K  e.  HL  ->  K  e.  Lat )
873ad2ant1 976 . . . 4  |-  ( ( K  e.  HL  /\  X  e.  B  /\  Q  e.  A )  ->  K  e.  Lat )
91, 3atbase 29479 . . . . 5  |-  ( Q  e.  A  ->  Q  e.  B )
1093ad2ant3 978 . . . 4  |-  ( ( K  e.  HL  /\  X  e.  B  /\  Q  e.  A )  ->  Q  e.  B )
11 simp2 956 . . . 4  |-  ( ( K  e.  HL  /\  X  e.  B  /\  Q  e.  A )  ->  X  e.  B )
121, 2latjcom 14165 . . . 4  |-  ( ( K  e.  Lat  /\  Q  e.  B  /\  X  e.  B )  ->  ( Q  .\/  X
)  =  ( X 
.\/  Q ) )
138, 10, 11, 12syl3anc 1182 . . 3  |-  ( ( K  e.  HL  /\  X  e.  B  /\  Q  e.  A )  ->  ( Q  .\/  X
)  =  ( X 
.\/  Q ) )
1413fveq2d 5529 . 2  |-  ( ( K  e.  HL  /\  X  e.  B  /\  Q  e.  A )  ->  ( M `  ( Q  .\/  X ) )  =  ( M `  ( X  .\/  Q ) ) )
15 simp1 955 . . . 4  |-  ( ( K  e.  HL  /\  X  e.  B  /\  Q  e.  A )  ->  K  e.  HL )
161, 3, 4pmapssat 29948 . . . 4  |-  ( ( K  e.  HL  /\  Q  e.  B )  ->  ( M `  Q
)  C_  A )
1715, 10, 16syl2anc 642 . . 3  |-  ( ( K  e.  HL  /\  X  e.  B  /\  Q  e.  A )  ->  ( M `  Q
)  C_  A )
181, 3, 4pmapssat 29948 . . . 4  |-  ( ( K  e.  HL  /\  X  e.  B )  ->  ( M `  X
)  C_  A )
19183adant3 975 . . 3  |-  ( ( K  e.  HL  /\  X  e.  B  /\  Q  e.  A )  ->  ( M `  X
)  C_  A )
203, 5paddcom 30002 . . 3  |-  ( ( K  e.  Lat  /\  ( M `  Q ) 
C_  A  /\  ( M `  X )  C_  A )  ->  (
( M `  Q
)  .+  ( M `  X ) )  =  ( ( M `  X )  .+  ( M `  Q )
) )
218, 17, 19, 20syl3anc 1182 . 2  |-  ( ( K  e.  HL  /\  X  e.  B  /\  Q  e.  A )  ->  ( ( M `  Q )  .+  ( M `  X )
)  =  ( ( M `  X ) 
.+  ( M `  Q ) ) )
226, 14, 213eqtr4d 2325 1  |-  ( ( K  e.  HL  /\  X  e.  B  /\  Q  e.  A )  ->  ( M `  ( Q  .\/  X ) )  =  ( ( M `
 Q )  .+  ( M `  X ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 934    = wceq 1623    e. wcel 1684    C_ wss 3152   ` cfv 5255  (class class class)co 5858   Basecbs 13148   joincjn 14078   Latclat 14151   Atomscatm 29453   HLchlt 29540   pmapcpmap 29686   + Pcpadd 29984
This theorem is referenced by:  atmod1i1  30046
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-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  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-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-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-pmap 29693  df-padd 29985
  Copyright terms: Public domain W3C validator