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Theorem pmapjlln1 30666
Description: The projective map of the join of a lattice element and a lattice line (expressed as the join  Q  .\/  R of two atoms). (Contributed by NM, 16-Sep-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
pmapjlln1  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  Q  e.  A  /\  R  e.  A
) )  ->  ( M `  ( X  .\/  ( Q  .\/  R
) ) )  =  ( ( M `  X )  .+  ( M `  ( Q  .\/  R ) ) ) )

Proof of Theorem pmapjlln1
StepHypRef Expression
1 simpl 443 . . 3  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  Q  e.  A  /\  R  e.  A
) )  ->  K  e.  HL )
2 pmapjat.b . . . . 5  |-  B  =  ( Base `  K
)
3 pmapjat.a . . . . 5  |-  A  =  ( Atoms `  K )
4 pmapjat.m . . . . 5  |-  M  =  ( pmap `  K
)
52, 3, 4pmapssat 30570 . . . 4  |-  ( ( K  e.  HL  /\  X  e.  B )  ->  ( M `  X
)  C_  A )
653ad2antr1 1120 . . 3  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  Q  e.  A  /\  R  e.  A
) )  ->  ( M `  X )  C_  A )
7 simpr2 962 . . . . 5  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  Q  e.  A  /\  R  e.  A
) )  ->  Q  e.  A )
82, 3atbase 30101 . . . . 5  |-  ( Q  e.  A  ->  Q  e.  B )
97, 8syl 15 . . . 4  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  Q  e.  A  /\  R  e.  A
) )  ->  Q  e.  B )
102, 3, 4pmapssat 30570 . . . 4  |-  ( ( K  e.  HL  /\  Q  e.  B )  ->  ( M `  Q
)  C_  A )
119, 10syldan 456 . . 3  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  Q  e.  A  /\  R  e.  A
) )  ->  ( M `  Q )  C_  A )
12 simpr3 963 . . . . 5  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  Q  e.  A  /\  R  e.  A
) )  ->  R  e.  A )
132, 3atbase 30101 . . . . 5  |-  ( R  e.  A  ->  R  e.  B )
1412, 13syl 15 . . . 4  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  Q  e.  A  /\  R  e.  A
) )  ->  R  e.  B )
152, 3, 4pmapssat 30570 . . . 4  |-  ( ( K  e.  HL  /\  R  e.  B )  ->  ( M `  R
)  C_  A )
1614, 15syldan 456 . . 3  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  Q  e.  A  /\  R  e.  A
) )  ->  ( M `  R )  C_  A )
17 pmapjat.p . . . 4  |-  .+  =  ( + P `  K
)
183, 17paddass 30649 . . 3  |-  ( ( K  e.  HL  /\  ( ( M `  X )  C_  A  /\  ( M `  Q
)  C_  A  /\  ( M `  R ) 
C_  A ) )  ->  ( ( ( M `  X ) 
.+  ( M `  Q ) )  .+  ( M `  R ) )  =  ( ( M `  X ) 
.+  ( ( M `
 Q )  .+  ( M `  R ) ) ) )
191, 6, 11, 16, 18syl13anc 1184 . 2  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  Q  e.  A  /\  R  e.  A
) )  ->  (
( ( M `  X )  .+  ( M `  Q )
)  .+  ( M `  R ) )  =  ( ( M `  X )  .+  (
( M `  Q
)  .+  ( M `  R ) ) ) )
20 hllat 30175 . . . . . 6  |-  ( K  e.  HL  ->  K  e.  Lat )
2120adantr 451 . . . . 5  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  Q  e.  A  /\  R  e.  A
) )  ->  K  e.  Lat )
22 simpr1 961 . . . . 5  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  Q  e.  A  /\  R  e.  A
) )  ->  X  e.  B )
23 pmapjat.j . . . . . 6  |-  .\/  =  ( join `  K )
242, 23latjcl 14172 . . . . 5  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Q  e.  B )  ->  ( X  .\/  Q
)  e.  B )
2521, 22, 9, 24syl3anc 1182 . . . 4  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  Q  e.  A  /\  R  e.  A
) )  ->  ( X  .\/  Q )  e.  B )
262, 23, 3, 4, 17pmapjat1 30664 . . . 4  |-  ( ( K  e.  HL  /\  ( X  .\/  Q )  e.  B  /\  R  e.  A )  ->  ( M `  ( ( X  .\/  Q )  .\/  R ) )  =  ( ( M `  ( X  .\/  Q ) ) 
.+  ( M `  R ) ) )
271, 25, 12, 26syl3anc 1182 . . 3  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  Q  e.  A  /\  R  e.  A
) )  ->  ( M `  ( ( X  .\/  Q )  .\/  R ) )  =  ( ( M `  ( X  .\/  Q ) ) 
.+  ( M `  R ) ) )
282, 23latjass 14217 . . . . 5  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Q  e.  B  /\  R  e.  B
) )  ->  (
( X  .\/  Q
)  .\/  R )  =  ( X  .\/  ( Q  .\/  R ) ) )
2921, 22, 9, 14, 28syl13anc 1184 . . . 4  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  Q  e.  A  /\  R  e.  A
) )  ->  (
( X  .\/  Q
)  .\/  R )  =  ( X  .\/  ( Q  .\/  R ) ) )
3029fveq2d 5545 . . 3  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  Q  e.  A  /\  R  e.  A
) )  ->  ( M `  ( ( X  .\/  Q )  .\/  R ) )  =  ( M `  ( X 
.\/  ( Q  .\/  R ) ) ) )
312, 23, 3, 4, 17pmapjat1 30664 . . . . 5  |-  ( ( K  e.  HL  /\  X  e.  B  /\  Q  e.  A )  ->  ( M `  ( X  .\/  Q ) )  =  ( ( M `
 X )  .+  ( M `  Q ) ) )
32313adant3r3 1162 . . . 4  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  Q  e.  A  /\  R  e.  A
) )  ->  ( M `  ( X  .\/  Q ) )  =  ( ( M `  X )  .+  ( M `  Q )
) )
3332oveq1d 5889 . . 3  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  Q  e.  A  /\  R  e.  A
) )  ->  (
( M `  ( X  .\/  Q ) ) 
.+  ( M `  R ) )  =  ( ( ( M `
 X )  .+  ( M `  Q ) )  .+  ( M `
 R ) ) )
3427, 30, 333eqtr3d 2336 . 2  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  Q  e.  A  /\  R  e.  A
) )  ->  ( M `  ( X  .\/  ( Q  .\/  R
) ) )  =  ( ( ( M `
 X )  .+  ( M `  Q ) )  .+  ( M `
 R ) ) )
352, 23, 3, 4, 17pmapjat1 30664 . . . 4  |-  ( ( K  e.  HL  /\  Q  e.  B  /\  R  e.  A )  ->  ( M `  ( Q  .\/  R ) )  =  ( ( M `
 Q )  .+  ( M `  R ) ) )
361, 9, 12, 35syl3anc 1182 . . 3  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  Q  e.  A  /\  R  e.  A
) )  ->  ( M `  ( Q  .\/  R ) )  =  ( ( M `  Q )  .+  ( M `  R )
) )
3736oveq2d 5890 . 2  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  Q  e.  A  /\  R  e.  A
) )  ->  (
( M `  X
)  .+  ( M `  ( Q  .\/  R
) ) )  =  ( ( M `  X )  .+  (
( M `  Q
)  .+  ( M `  R ) ) ) )
3819, 34, 373eqtr4d 2338 1  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  Q  e.  A  /\  R  e.  A
) )  ->  ( M `  ( X  .\/  ( Q  .\/  R
) ) )  =  ( ( M `  X )  .+  ( M `  ( Q  .\/  R ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696    C_ wss 3165   ` cfv 5271  (class class class)co 5874   Basecbs 13164   joincjn 14094   Latclat 14167   Atomscatm 30075   HLchlt 30162   pmapcpmap 30308   + Pcpadd 30606
This theorem is referenced by:  llnmod1i2  30671
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-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  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-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-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-pmap 30315  df-padd 30607
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