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Theorem pmapjlln1 30044
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 29948 . . . 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 29479 . . . . 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 29948 . . . 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 29479 . . . . 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 29948 . . . 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 30027 . . 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 29553 . . . . . 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 14156 . . . . 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 30042 . . . 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 14201 . . . . 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 5529 . . 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 30042 . . . . 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 5873 . . 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 2323 . 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 30042 . . . 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 5874 . 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 2325 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 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:  llnmod1i2  30049
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
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