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Theorem pmapjlln1 34526
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 457 . . 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 34430 . . . 4  |-  ( ( K  e.  HL  /\  X  e.  B )  ->  ( M `  X
)  C_  A )
653ad2antr1 1156 . . 3  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  Q  e.  A  /\  R  e.  A
) )  ->  ( M `  X )  C_  A )
7 simpr2 998 . . . . 5  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  Q  e.  A  /\  R  e.  A
) )  ->  Q  e.  A )
82, 3atbase 33961 . . . . 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 34430 . . . 4  |-  ( ( K  e.  HL  /\  Q  e.  B )  ->  ( M `  Q
)  C_  A )
119, 10syldan 470 . . 3  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  Q  e.  A  /\  R  e.  A
) )  ->  ( M `  Q )  C_  A )
12 simpr3 999 . . . . 5  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  Q  e.  A  /\  R  e.  A
) )  ->  R  e.  A )
132, 3atbase 33961 . . . . 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 34430 . . . 4  |-  ( ( K  e.  HL  /\  R  e.  B )  ->  ( M `  R
)  C_  A )
1614, 15syldan 470 . . 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 34509 . . 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 1225 . 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 34035 . . . . . 6  |-  ( K  e.  HL  ->  K  e.  Lat )
2120adantr 465 . . . . 5  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  Q  e.  A  /\  R  e.  A
) )  ->  K  e.  Lat )
22 simpr1 997 . . . . 5  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  Q  e.  A  /\  R  e.  A
) )  ->  X  e.  B )
23 pmapjat.j . . . . . 6  |-  .\/  =  ( join `  K )
242, 23latjcl 15527 . . . . 5  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Q  e.  B )  ->  ( X  .\/  Q
)  e.  B )
2521, 22, 9, 24syl3anc 1223 . . . 4  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  Q  e.  A  /\  R  e.  A
) )  ->  ( X  .\/  Q )  e.  B )
262, 23, 3, 4, 17pmapjat1 34524 . . . 4  |-  ( ( K  e.  HL  /\  ( X  .\/  Q )  e.  B  /\  R  e.  A )  ->  ( M `  ( ( X  .\/  Q )  .\/  R ) )  =  ( ( M `  ( X  .\/  Q ) ) 
.+  ( M `  R ) ) )
271, 25, 12, 26syl3anc 1223 . . 3  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  Q  e.  A  /\  R  e.  A
) )  ->  ( M `  ( ( X  .\/  Q )  .\/  R ) )  =  ( ( M `  ( X  .\/  Q ) ) 
.+  ( M `  R ) ) )
282, 23latjass 15571 . . . . 5  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Q  e.  B  /\  R  e.  B
) )  ->  (
( X  .\/  Q
)  .\/  R )  =  ( X  .\/  ( Q  .\/  R ) ) )
2921, 22, 9, 14, 28syl13anc 1225 . . . 4  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  Q  e.  A  /\  R  e.  A
) )  ->  (
( X  .\/  Q
)  .\/  R )  =  ( X  .\/  ( Q  .\/  R ) ) )
3029fveq2d 5861 . . 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 34524 . . . . 5  |-  ( ( K  e.  HL  /\  X  e.  B  /\  Q  e.  A )  ->  ( M `  ( X  .\/  Q ) )  =  ( ( M `
 X )  .+  ( M `  Q ) ) )
32313adant3r3 1202 . . . 4  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  Q  e.  A  /\  R  e.  A
) )  ->  ( M `  ( X  .\/  Q ) )  =  ( ( M `  X )  .+  ( M `  Q )
) )
3332oveq1d 6290 . . 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 2509 . 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 34524 . . . 4  |-  ( ( K  e.  HL  /\  Q  e.  B  /\  R  e.  A )  ->  ( M `  ( Q  .\/  R ) )  =  ( ( M `
 Q )  .+  ( M `  R ) ) )
361, 9, 12, 35syl3anc 1223 . . 3  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  Q  e.  A  /\  R  e.  A
) )  ->  ( M `  ( Q  .\/  R ) )  =  ( ( M `  Q )  .+  ( M `  R )
) )
3736oveq2d 6291 . 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 2511 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 setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 968    = wceq 1374    e. wcel 1762    C_ wss 3469   ` cfv 5579  (class class class)co 6275   Basecbs 14479   joincjn 15420   Latclat 15521   Atomscatm 33935   HLchlt 34022   pmapcpmap 34168   +Pcpadd 34466
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-id 4788  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-riota 6236  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-1st 6774  df-2nd 6775  df-poset 15422  df-plt 15434  df-lub 15450  df-glb 15451  df-join 15452  df-meet 15453  df-p0 15515  df-lat 15522  df-clat 15584  df-oposet 33848  df-ol 33850  df-oml 33851  df-covers 33938  df-ats 33939  df-atl 33970  df-cvlat 33994  df-hlat 34023  df-pmap 34175  df-padd 34467
This theorem is referenced by:  llnmod1i2  34531
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