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Theorem pmapjlln1 30652
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 444 . . 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 30556 . . . 4  |-  ( ( K  e.  HL  /\  X  e.  B )  ->  ( M `  X
)  C_  A )
653ad2antr1 1122 . . 3  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  Q  e.  A  /\  R  e.  A
) )  ->  ( M `  X )  C_  A )
7 simpr2 964 . . . . 5  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  Q  e.  A  /\  R  e.  A
) )  ->  Q  e.  A )
82, 3atbase 30087 . . . . 5  |-  ( Q  e.  A  ->  Q  e.  B )
97, 8syl 16 . . . 4  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  Q  e.  A  /\  R  e.  A
) )  ->  Q  e.  B )
102, 3, 4pmapssat 30556 . . . 4  |-  ( ( K  e.  HL  /\  Q  e.  B )  ->  ( M `  Q
)  C_  A )
119, 10syldan 457 . . 3  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  Q  e.  A  /\  R  e.  A
) )  ->  ( M `  Q )  C_  A )
12 simpr3 965 . . . . 5  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  Q  e.  A  /\  R  e.  A
) )  ->  R  e.  A )
132, 3atbase 30087 . . . . 5  |-  ( R  e.  A  ->  R  e.  B )
1412, 13syl 16 . . . 4  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  Q  e.  A  /\  R  e.  A
) )  ->  R  e.  B )
152, 3, 4pmapssat 30556 . . . 4  |-  ( ( K  e.  HL  /\  R  e.  B )  ->  ( M `  R
)  C_  A )
1614, 15syldan 457 . . 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 30635 . . 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 1186 . 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 30161 . . . . . 6  |-  ( K  e.  HL  ->  K  e.  Lat )
2120adantr 452 . . . . 5  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  Q  e.  A  /\  R  e.  A
) )  ->  K  e.  Lat )
22 simpr1 963 . . . . 5  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  Q  e.  A  /\  R  e.  A
) )  ->  X  e.  B )
23 pmapjat.j . . . . . 6  |-  .\/  =  ( join `  K )
242, 23latjcl 14479 . . . . 5  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Q  e.  B )  ->  ( X  .\/  Q
)  e.  B )
2521, 22, 9, 24syl3anc 1184 . . . 4  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  Q  e.  A  /\  R  e.  A
) )  ->  ( X  .\/  Q )  e.  B )
262, 23, 3, 4, 17pmapjat1 30650 . . . 4  |-  ( ( K  e.  HL  /\  ( X  .\/  Q )  e.  B  /\  R  e.  A )  ->  ( M `  ( ( X  .\/  Q )  .\/  R ) )  =  ( ( M `  ( X  .\/  Q ) ) 
.+  ( M `  R ) ) )
271, 25, 12, 26syl3anc 1184 . . 3  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  Q  e.  A  /\  R  e.  A
) )  ->  ( M `  ( ( X  .\/  Q )  .\/  R ) )  =  ( ( M `  ( X  .\/  Q ) ) 
.+  ( M `  R ) ) )
282, 23latjass 14524 . . . . 5  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Q  e.  B  /\  R  e.  B
) )  ->  (
( X  .\/  Q
)  .\/  R )  =  ( X  .\/  ( Q  .\/  R ) ) )
2921, 22, 9, 14, 28syl13anc 1186 . . . 4  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  Q  e.  A  /\  R  e.  A
) )  ->  (
( X  .\/  Q
)  .\/  R )  =  ( X  .\/  ( Q  .\/  R ) ) )
3029fveq2d 5732 . . 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 30650 . . . . 5  |-  ( ( K  e.  HL  /\  X  e.  B  /\  Q  e.  A )  ->  ( M `  ( X  .\/  Q ) )  =  ( ( M `
 X )  .+  ( M `  Q ) ) )
32313adant3r3 1164 . . . 4  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  Q  e.  A  /\  R  e.  A
) )  ->  ( M `  ( X  .\/  Q ) )  =  ( ( M `  X )  .+  ( M `  Q )
) )
3332oveq1d 6096 . . 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 2476 . 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 30650 . . . 4  |-  ( ( K  e.  HL  /\  Q  e.  B  /\  R  e.  A )  ->  ( M `  ( Q  .\/  R ) )  =  ( ( M `
 Q )  .+  ( M `  R ) ) )
361, 9, 12, 35syl3anc 1184 . . 3  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  Q  e.  A  /\  R  e.  A
) )  ->  ( M `  ( Q  .\/  R ) )  =  ( ( M `  Q )  .+  ( M `  R )
) )
3736oveq2d 6097 . 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 2478 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 359    /\ w3a 936    = wceq 1652    e. wcel 1725    C_ wss 3320   ` cfv 5454  (class class class)co 6081   Basecbs 13469   joincjn 14401   Latclat 14474   Atomscatm 30061   HLchlt 30148   pmapcpmap 30294   + Pcpadd 30592
This theorem is referenced by:  llnmod1i2  30657
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-undef 6543  df-riota 6549  df-poset 14403  df-plt 14415  df-lub 14431  df-glb 14432  df-join 14433  df-meet 14434  df-p0 14468  df-lat 14475  df-clat 14537  df-oposet 29974  df-ol 29976  df-oml 29977  df-covers 30064  df-ats 30065  df-atl 30096  df-cvlat 30120  df-hlat 30149  df-pmap 30301  df-padd 30593
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