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Theorem paddasslem16 34631
Description: Lemma for paddass 34634. Use elpaddn0 34596 to eliminate  x and  r from paddasslem15 34630. (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 34160 . . . . 5  |-  ( K  e.  HL  ->  K  e.  Lat )
213ad2ant1 1017 . . . 4  |-  ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  C_  A )  /\  (
( X  =/=  (/)  /\  ( Y  .+  Z )  =/=  (/) )  /\  ( Y  =/=  (/)  /\  Z  =/=  (/) ) ) )  ->  K  e.  Lat )
3 simp21 1029 . . . 4  |-  ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  C_  A )  /\  (
( X  =/=  (/)  /\  ( Y  .+  Z )  =/=  (/) )  /\  ( Y  =/=  (/)  /\  Z  =/=  (/) ) ) )  ->  X  C_  A )
4 simp1 996 . . . . 5  |-  ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  C_  A )  /\  (
( X  =/=  (/)  /\  ( Y  .+  Z )  =/=  (/) )  /\  ( Y  =/=  (/)  /\  Z  =/=  (/) ) ) )  ->  K  e.  HL )
5 simp22 1030 . . . . 5  |-  ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  C_  A )  /\  (
( X  =/=  (/)  /\  ( Y  .+  Z )  =/=  (/) )  /\  ( Y  =/=  (/)  /\  Z  =/=  (/) ) ) )  ->  Y  C_  A )
6 simp23 1031 . . . . 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 34610 . . . . 5  |-  ( ( K  e.  HL  /\  Y  C_  A  /\  Z  C_  A )  ->  ( Y  .+  Z )  C_  A )
104, 5, 6, 9syl3anc 1228 . . . 4  |-  ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  C_  A )  /\  (
( X  =/=  (/)  /\  ( Y  .+  Z )  =/=  (/) )  /\  ( Y  =/=  (/)  /\  Z  =/=  (/) ) ) )  -> 
( Y  .+  Z
)  C_  A )
11 simp3l 1024 . . . 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 34596 . . . 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 1231 . . 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 34630 . . . . . . . 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 1278 . . . . . . 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 1214 . . . . . 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 2961 . . . 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 3510 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 973    = wceq 1379    e. wcel 1767    =/= wne 2662   E.wrex 2815    C_ wss 3476   (/)c0 3785   class class class wbr 4447   ` cfv 5586  (class class class)co 6282   lecple 14558   joincjn 15427   Latclat 15528   Atomscatm 34060   HLchlt 34147   +Pcpadd 34591
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-1st 6781  df-2nd 6782  df-poset 15429  df-plt 15441  df-lub 15457  df-glb 15458  df-join 15459  df-meet 15460  df-p0 15522  df-lat 15529  df-clat 15591  df-oposet 33973  df-ol 33975  df-oml 33976  df-covers 34063  df-ats 34064  df-atl 34095  df-cvlat 34119  df-hlat 34148  df-padd 34592
This theorem is referenced by:  paddasslem18  34633
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