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Theorem paddasslem16 32852
Description: Lemma for paddass 32855. Use elpaddn0 32817 to eliminate  x and  r from paddasslem15 32851. (Contributed by NM, 11-Jan-2012.)
Hypotheses
Ref Expression
paddasslem.l  |-  .<_  =  ( le `  K )
paddasslem.j  |-  .\/  =  ( join `  K )
paddasslem.a  |-  A  =  ( Atoms `  K )
paddasslem.p  |-  .+  =  ( +P `  K
)
Assertion
Ref Expression
paddasslem16  |-  ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  C_  A )  /\  (
( X  =/=  (/)  /\  ( Y  .+  Z )  =/=  (/) )  /\  ( Y  =/=  (/)  /\  Z  =/=  (/) ) ) )  -> 
( X  .+  ( Y  .+  Z ) ) 
C_  ( ( X 
.+  Y )  .+  Z ) )

Proof of Theorem paddasslem16
Dummy variables  p  r  x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 hllat 32381 . . . . 5  |-  ( K  e.  HL  ->  K  e.  Lat )
213ad2ant1 1018 . . . 4  |-  ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  C_  A )  /\  (
( X  =/=  (/)  /\  ( Y  .+  Z )  =/=  (/) )  /\  ( Y  =/=  (/)  /\  Z  =/=  (/) ) ) )  ->  K  e.  Lat )
3 simp21 1030 . . . 4  |-  ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  C_  A )  /\  (
( X  =/=  (/)  /\  ( Y  .+  Z )  =/=  (/) )  /\  ( Y  =/=  (/)  /\  Z  =/=  (/) ) ) )  ->  X  C_  A )
4 simp1 997 . . . . 5  |-  ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  C_  A )  /\  (
( X  =/=  (/)  /\  ( Y  .+  Z )  =/=  (/) )  /\  ( Y  =/=  (/)  /\  Z  =/=  (/) ) ) )  ->  K  e.  HL )
5 simp22 1031 . . . . 5  |-  ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  C_  A )  /\  (
( X  =/=  (/)  /\  ( Y  .+  Z )  =/=  (/) )  /\  ( Y  =/=  (/)  /\  Z  =/=  (/) ) ) )  ->  Y  C_  A )
6 simp23 1032 . . . . 5  |-  ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  C_  A )  /\  (
( X  =/=  (/)  /\  ( Y  .+  Z )  =/=  (/) )  /\  ( Y  =/=  (/)  /\  Z  =/=  (/) ) ) )  ->  Z  C_  A )
7 paddasslem.a . . . . . 6  |-  A  =  ( Atoms `  K )
8 paddasslem.p . . . . . 6  |-  .+  =  ( +P `  K
)
97, 8paddssat 32831 . . . . 5  |-  ( ( K  e.  HL  /\  Y  C_  A  /\  Z  C_  A )  ->  ( Y  .+  Z )  C_  A )
104, 5, 6, 9syl3anc 1230 . . . 4  |-  ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  C_  A )  /\  (
( X  =/=  (/)  /\  ( Y  .+  Z )  =/=  (/) )  /\  ( Y  =/=  (/)  /\  Z  =/=  (/) ) ) )  -> 
( Y  .+  Z
)  C_  A )
11 simp3l 1025 . . . 4  |-  ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  C_  A )  /\  (
( X  =/=  (/)  /\  ( Y  .+  Z )  =/=  (/) )  /\  ( Y  =/=  (/)  /\  Z  =/=  (/) ) ) )  -> 
( X  =/=  (/)  /\  ( Y  .+  Z )  =/=  (/) ) )
12 paddasslem.l . . . . 5  |-  .<_  =  ( le `  K )
13 paddasslem.j . . . . 5  |-  .\/  =  ( join `  K )
1412, 13, 7, 8elpaddn0 32817 . . . 4  |-  ( ( ( K  e.  Lat  /\  X  C_  A  /\  ( Y  .+  Z ) 
C_  A )  /\  ( X  =/=  (/)  /\  ( Y  .+  Z )  =/=  (/) ) )  ->  (
p  e.  ( X 
.+  ( Y  .+  Z ) )  <->  ( p  e.  A  /\  E. x  e.  X  E. r  e.  ( Y  .+  Z
) p  .<_  ( x 
.\/  r ) ) ) )
152, 3, 10, 11, 14syl31anc 1233 . . 3  |-  ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  C_  A )  /\  (
( X  =/=  (/)  /\  ( Y  .+  Z )  =/=  (/) )  /\  ( Y  =/=  (/)  /\  Z  =/=  (/) ) ) )  -> 
( p  e.  ( X  .+  ( Y 
.+  Z ) )  <-> 
( p  e.  A  /\  E. x  e.  X  E. r  e.  ( Y  .+  Z ) p 
.<_  ( x  .\/  r
) ) ) )
16 simpr 459 . . . . . . . 8  |-  ( ( ( X  =/=  (/)  /\  ( Y  .+  Z )  =/=  (/) )  /\  ( Y  =/=  (/)  /\  Z  =/=  (/) ) )  ->  ( Y  =/=  (/)  /\  Z  =/=  (/) ) )
1712, 13, 7, 8paddasslem15 32851 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  C_  A )  /\  ( Y  =/=  (/)  /\  Z  =/=  (/) ) )  /\  ( p  e.  A  /\  ( x  e.  X  /\  r  e.  ( Y  .+  Z ) )  /\  p  .<_  ( x 
.\/  r ) ) )  ->  p  e.  ( ( X  .+  Y )  .+  Z
) )
1816, 17syl3anl3 1280 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  C_  A )  /\  ( ( X  =/=  (/)  /\  ( Y  .+  Z )  =/=  (/) )  /\  ( Y  =/=  (/)  /\  Z  =/=  (/) ) ) )  /\  ( p  e.  A  /\  ( x  e.  X  /\  r  e.  ( Y  .+  Z
) )  /\  p  .<_  ( x  .\/  r
) ) )  ->  p  e.  ( ( X  .+  Y )  .+  Z ) )
19183exp2 1215 . . . . . 6  |-  ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  C_  A )  /\  (
( X  =/=  (/)  /\  ( Y  .+  Z )  =/=  (/) )  /\  ( Y  =/=  (/)  /\  Z  =/=  (/) ) ) )  -> 
( p  e.  A  ->  ( ( x  e.  X  /\  r  e.  ( Y  .+  Z
) )  ->  (
p  .<_  ( x  .\/  r )  ->  p  e.  ( ( X  .+  Y )  .+  Z
) ) ) ) )
2019imp 427 . . . . 5  |-  ( ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  C_  A )  /\  ( ( X  =/=  (/)  /\  ( Y  .+  Z )  =/=  (/) )  /\  ( Y  =/=  (/)  /\  Z  =/=  (/) ) ) )  /\  p  e.  A
)  ->  ( (
x  e.  X  /\  r  e.  ( Y  .+  Z ) )  -> 
( p  .<_  ( x 
.\/  r )  ->  p  e.  ( ( X  .+  Y )  .+  Z ) ) ) )
2120rexlimdvv 2902 . . . 4  |-  ( ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  C_  A )  /\  ( ( X  =/=  (/)  /\  ( Y  .+  Z )  =/=  (/) )  /\  ( Y  =/=  (/)  /\  Z  =/=  (/) ) ) )  /\  p  e.  A
)  ->  ( E. x  e.  X  E. r  e.  ( Y  .+  Z ) p  .<_  ( x  .\/  r )  ->  p  e.  ( ( X  .+  Y
)  .+  Z )
) )
2221expimpd 601 . . 3  |-  ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  C_  A )  /\  (
( X  =/=  (/)  /\  ( Y  .+  Z )  =/=  (/) )  /\  ( Y  =/=  (/)  /\  Z  =/=  (/) ) ) )  -> 
( ( p  e.  A  /\  E. x  e.  X  E. r  e.  ( Y  .+  Z
) p  .<_  ( x 
.\/  r ) )  ->  p  e.  ( ( X  .+  Y
)  .+  Z )
) )
2315, 22sylbid 215 . 2  |-  ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  C_  A )  /\  (
( X  =/=  (/)  /\  ( Y  .+  Z )  =/=  (/) )  /\  ( Y  =/=  (/)  /\  Z  =/=  (/) ) ) )  -> 
( p  e.  ( X  .+  ( Y 
.+  Z ) )  ->  p  e.  ( ( X  .+  Y
)  .+  Z )
) )
2423ssrdv 3448 1  |-  ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  C_  A )  /\  (
( X  =/=  (/)  /\  ( Y  .+  Z )  =/=  (/) )  /\  ( Y  =/=  (/)  /\  Z  =/=  (/) ) ) )  -> 
( X  .+  ( Y  .+  Z ) ) 
C_  ( ( X 
.+  Y )  .+  Z ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    /\ w3a 974    = wceq 1405    e. wcel 1842    =/= wne 2598   E.wrex 2755    C_ wss 3414   (/)c0 3738   class class class wbr 4395   ` cfv 5569  (class class class)co 6278   lecple 14916   joincjn 15897   Latclat 15999   Atomscatm 32281   HLchlt 32368   +Pcpadd 32812
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4507  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2759  df-rex 2760  df-reu 2761  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4192  df-iun 4273  df-br 4396  df-opab 4454  df-mpt 4455  df-id 4738  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-riota 6240  df-ov 6281  df-oprab 6282  df-mpt2 6283  df-1st 6784  df-2nd 6785  df-preset 15881  df-poset 15899  df-plt 15912  df-lub 15928  df-glb 15929  df-join 15930  df-meet 15931  df-p0 15993  df-lat 16000  df-clat 16062  df-oposet 32194  df-ol 32196  df-oml 32197  df-covers 32284  df-ats 32285  df-atl 32316  df-cvlat 32340  df-hlat 32369  df-padd 32813
This theorem is referenced by:  paddasslem18  32854
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