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Theorem paddasslem16 33787
Description: Lemma for paddass 33790. Use elpaddn0 33752 to eliminate  x and  r from paddasslem15 33786. (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 33316 . . . . 5  |-  ( K  e.  HL  ->  K  e.  Lat )
213ad2ant1 1009 . . . 4  |-  ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  C_  A )  /\  (
( X  =/=  (/)  /\  ( Y  .+  Z )  =/=  (/) )  /\  ( Y  =/=  (/)  /\  Z  =/=  (/) ) ) )  ->  K  e.  Lat )
3 simp21 1021 . . . 4  |-  ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  C_  A )  /\  (
( X  =/=  (/)  /\  ( Y  .+  Z )  =/=  (/) )  /\  ( Y  =/=  (/)  /\  Z  =/=  (/) ) ) )  ->  X  C_  A )
4 simp1 988 . . . . 5  |-  ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  C_  A )  /\  (
( X  =/=  (/)  /\  ( Y  .+  Z )  =/=  (/) )  /\  ( Y  =/=  (/)  /\  Z  =/=  (/) ) ) )  ->  K  e.  HL )
5 simp22 1022 . . . . 5  |-  ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  C_  A )  /\  (
( X  =/=  (/)  /\  ( Y  .+  Z )  =/=  (/) )  /\  ( Y  =/=  (/)  /\  Z  =/=  (/) ) ) )  ->  Y  C_  A )
6 simp23 1023 . . . . 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 33766 . . . . 5  |-  ( ( K  e.  HL  /\  Y  C_  A  /\  Z  C_  A )  ->  ( Y  .+  Z )  C_  A )
104, 5, 6, 9syl3anc 1219 . . . 4  |-  ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  C_  A )  /\  (
( X  =/=  (/)  /\  ( Y  .+  Z )  =/=  (/) )  /\  ( Y  =/=  (/)  /\  Z  =/=  (/) ) ) )  -> 
( Y  .+  Z
)  C_  A )
11 simp3l 1016 . . . 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 33752 . . . 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 1222 . . 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 461 . . . . . . . 8  |-  ( ( ( X  =/=  (/)  /\  ( Y  .+  Z )  =/=  (/) )  /\  ( Y  =/=  (/)  /\  Z  =/=  (/) ) )  ->  ( Y  =/=  (/)  /\  Z  =/=  (/) ) )
1712, 13, 7, 8paddasslem15 33786 . . . . . . . 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 1269 . . . . . . 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 1206 . . . . . 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 429 . . . . 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 2945 . . . 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 603 . . 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 3462 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 369    /\ w3a 965    = wceq 1370    e. wcel 1758    =/= wne 2644   E.wrex 2796    C_ wss 3428   (/)c0 3737   class class class wbr 4392   ` cfv 5518  (class class class)co 6192   lecple 14349   joincjn 15218   Latclat 15319   Atomscatm 33216   HLchlt 33303   +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-padd 33748
This theorem is referenced by:  paddasslem18  33789
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