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Theorem pmapjlln1 33807
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 33711 . . . 4  |-  ( ( K  e.  HL  /\  X  e.  B )  ->  ( M `  X
)  C_  A )
653ad2antr1 1153 . . 3  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  Q  e.  A  /\  R  e.  A
) )  ->  ( M `  X )  C_  A )
7 simpr2 995 . . . . 5  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  Q  e.  A  /\  R  e.  A
) )  ->  Q  e.  A )
82, 3atbase 33242 . . . . 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 33711 . . . 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 996 . . . . 5  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  Q  e.  A  /\  R  e.  A
) )  ->  R  e.  A )
132, 3atbase 33242 . . . . 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 33711 . . . 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 33790 . . 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 1221 . 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 33316 . . . . . 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 994 . . . . 5  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  Q  e.  A  /\  R  e.  A
) )  ->  X  e.  B )
23 pmapjat.j . . . . . 6  |-  .\/  =  ( join `  K )
242, 23latjcl 15325 . . . . 5  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Q  e.  B )  ->  ( X  .\/  Q
)  e.  B )
2521, 22, 9, 24syl3anc 1219 . . . 4  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  Q  e.  A  /\  R  e.  A
) )  ->  ( X  .\/  Q )  e.  B )
262, 23, 3, 4, 17pmapjat1 33805 . . . 4  |-  ( ( K  e.  HL  /\  ( X  .\/  Q )  e.  B  /\  R  e.  A )  ->  ( M `  ( ( X  .\/  Q )  .\/  R ) )  =  ( ( M `  ( X  .\/  Q ) ) 
.+  ( M `  R ) ) )
271, 25, 12, 26syl3anc 1219 . . 3  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  Q  e.  A  /\  R  e.  A
) )  ->  ( M `  ( ( X  .\/  Q )  .\/  R ) )  =  ( ( M `  ( X  .\/  Q ) ) 
.+  ( M `  R ) ) )
282, 23latjass 15369 . . . . 5  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Q  e.  B  /\  R  e.  B
) )  ->  (
( X  .\/  Q
)  .\/  R )  =  ( X  .\/  ( Q  .\/  R ) ) )
2921, 22, 9, 14, 28syl13anc 1221 . . . 4  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  Q  e.  A  /\  R  e.  A
) )  ->  (
( X  .\/  Q
)  .\/  R )  =  ( X  .\/  ( Q  .\/  R ) ) )
3029fveq2d 5795 . . 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 33805 . . . . 5  |-  ( ( K  e.  HL  /\  X  e.  B  /\  Q  e.  A )  ->  ( M `  ( X  .\/  Q ) )  =  ( ( M `
 X )  .+  ( M `  Q ) ) )
32313adant3r3 1199 . . . 4  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  Q  e.  A  /\  R  e.  A
) )  ->  ( M `  ( X  .\/  Q ) )  =  ( ( M `  X )  .+  ( M `  Q )
) )
3332oveq1d 6207 . . 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 2500 . 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 33805 . . . 4  |-  ( ( K  e.  HL  /\  Q  e.  B  /\  R  e.  A )  ->  ( M `  ( Q  .\/  R ) )  =  ( ( M `
 Q )  .+  ( M `  R ) ) )
361, 9, 12, 35syl3anc 1219 . . 3  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  Q  e.  A  /\  R  e.  A
) )  ->  ( M `  ( Q  .\/  R ) )  =  ( ( M `  Q )  .+  ( M `  R )
) )
3736oveq2d 6208 . 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 2502 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 965    = wceq 1370    e. wcel 1758    C_ wss 3428   ` cfv 5518  (class class class)co 6192   Basecbs 14278   joincjn 15218   Latclat 15319   Atomscatm 33216   HLchlt 33303   pmapcpmap 33449   +Pcpadd 33747
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4503  ax-sep 4513  ax-nul 4521  ax-pow 4570  ax-pr 4631  ax-un 6474
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-reu 2802  df-rab 2804  df-v 3072  df-sbc 3287  df-csb 3389  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-nul 3738  df-if 3892  df-pw 3962  df-sn 3978  df-pr 3980  df-op 3984  df-uni 4192  df-iun 4273  df-br 4393  df-opab 4451  df-mpt 4452  df-id 4736  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5481  df-fun 5520  df-fn 5521  df-f 5522  df-f1 5523  df-fo 5524  df-f1o 5525  df-fv 5526  df-riota 6153  df-ov 6195  df-oprab 6196  df-mpt2 6197  df-1st 6679  df-2nd 6680  df-poset 15220  df-plt 15232  df-lub 15248  df-glb 15249  df-join 15250  df-meet 15251  df-p0 15313  df-lat 15320  df-clat 15382  df-oposet 33129  df-ol 33131  df-oml 33132  df-covers 33219  df-ats 33220  df-atl 33251  df-cvlat 33275  df-hlat 33304  df-pmap 33456  df-padd 33748
This theorem is referenced by:  llnmod1i2  33812
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