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Theorem paddasslem16 30321
Description: Lemma for paddass 30324. Use elpaddn0 30286 to eliminate  x and  r from paddasslem15 30320. (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 29850 . . . . 5  |-  ( K  e.  HL  ->  K  e.  Lat )
213ad2ant1 978 . . . 4  |-  ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  C_  A )  /\  (
( X  =/=  (/)  /\  ( Y  .+  Z )  =/=  (/) )  /\  ( Y  =/=  (/)  /\  Z  =/=  (/) ) ) )  ->  K  e.  Lat )
3 simp21 990 . . . 4  |-  ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  C_  A )  /\  (
( X  =/=  (/)  /\  ( Y  .+  Z )  =/=  (/) )  /\  ( Y  =/=  (/)  /\  Z  =/=  (/) ) ) )  ->  X  C_  A )
4 simp1 957 . . . . 5  |-  ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  C_  A )  /\  (
( X  =/=  (/)  /\  ( Y  .+  Z )  =/=  (/) )  /\  ( Y  =/=  (/)  /\  Z  =/=  (/) ) ) )  ->  K  e.  HL )
5 simp22 991 . . . . 5  |-  ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  C_  A )  /\  (
( X  =/=  (/)  /\  ( Y  .+  Z )  =/=  (/) )  /\  ( Y  =/=  (/)  /\  Z  =/=  (/) ) ) )  ->  Y  C_  A )
6 simp23 992 . . . . 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 30300 . . . . 5  |-  ( ( K  e.  HL  /\  Y  C_  A  /\  Z  C_  A )  ->  ( Y  .+  Z )  C_  A )
104, 5, 6, 9syl3anc 1184 . . . 4  |-  ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  C_  A )  /\  (
( X  =/=  (/)  /\  ( Y  .+  Z )  =/=  (/) )  /\  ( Y  =/=  (/)  /\  Z  =/=  (/) ) ) )  -> 
( Y  .+  Z
)  C_  A )
11 simp3l 985 . . . 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 30286 . . . 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 1187 . . 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 448 . . . . . . . 8  |-  ( ( ( X  =/=  (/)  /\  ( Y  .+  Z )  =/=  (/) )  /\  ( Y  =/=  (/)  /\  Z  =/=  (/) ) )  ->  ( Y  =/=  (/)  /\  Z  =/=  (/) ) )
1712, 13, 7, 8paddasslem15 30320 . . . . . . . 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 1234 . . . . . . 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 1171 . . . . . 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 419 . . . . 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 2800 . . . 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 587 . . 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 207 . 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 3318 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 set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721    =/= wne 2571   E.wrex 2671    C_ wss 3284   (/)c0 3592   class class class wbr 4176   ` cfv 5417  (class class class)co 6044   lecple 13495   joincjn 14360   Latclat 14433   Atomscatm 29750   HLchlt 29837   + Pcpadd 30281
This theorem is referenced by:  paddasslem18  30323
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389  ax-rep 4284  ax-sep 4294  ax-nul 4302  ax-pow 4341  ax-pr 4367  ax-un 4664
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-nel 2574  df-ral 2675  df-rex 2676  df-reu 2677  df-rab 2679  df-v 2922  df-sbc 3126  df-csb 3216  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-nul 3593  df-if 3704  df-pw 3765  df-sn 3784  df-pr 3785  df-op 3787  df-uni 3980  df-iun 4059  df-br 4177  df-opab 4231  df-mpt 4232  df-id 4462  df-xp 4847  df-rel 4848  df-cnv 4849  df-co 4850  df-dm 4851  df-rn 4852  df-res 4853  df-ima 4854  df-iota 5381  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-ov 6047  df-oprab 6048  df-mpt2 6049  df-1st 6312  df-2nd 6313  df-undef 6506  df-riota 6512  df-poset 14362  df-plt 14374  df-lub 14390  df-glb 14391  df-join 14392  df-meet 14393  df-p0 14427  df-lat 14434  df-clat 14496  df-oposet 29663  df-ol 29665  df-oml 29666  df-covers 29753  df-ats 29754  df-atl 29785  df-cvlat 29809  df-hlat 29838  df-padd 30282
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