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Theorem paddasslem16 30024
Description: Lemma for paddass 30027. Use elpaddn0 29989 to eliminate  x and  r from paddasslem15 30023. (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 29553 . . . . 5  |-  ( K  e.  HL  ->  K  e.  Lat )
213ad2ant1 976 . . . 4  |-  ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  C_  A )  /\  (
( X  =/=  (/)  /\  ( Y  .+  Z )  =/=  (/) )  /\  ( Y  =/=  (/)  /\  Z  =/=  (/) ) ) )  ->  K  e.  Lat )
3 simp21 988 . . . 4  |-  ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  C_  A )  /\  (
( X  =/=  (/)  /\  ( Y  .+  Z )  =/=  (/) )  /\  ( Y  =/=  (/)  /\  Z  =/=  (/) ) ) )  ->  X  C_  A )
4 simp1 955 . . . . 5  |-  ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  C_  A )  /\  (
( X  =/=  (/)  /\  ( Y  .+  Z )  =/=  (/) )  /\  ( Y  =/=  (/)  /\  Z  =/=  (/) ) ) )  ->  K  e.  HL )
5 simp22 989 . . . . 5  |-  ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  C_  A )  /\  (
( X  =/=  (/)  /\  ( Y  .+  Z )  =/=  (/) )  /\  ( Y  =/=  (/)  /\  Z  =/=  (/) ) ) )  ->  Y  C_  A )
6 simp23 990 . . . . 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 30003 . . . . 5  |-  ( ( K  e.  HL  /\  Y  C_  A  /\  Z  C_  A )  ->  ( Y  .+  Z )  C_  A )
104, 5, 6, 9syl3anc 1182 . . . 4  |-  ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  C_  A )  /\  (
( X  =/=  (/)  /\  ( Y  .+  Z )  =/=  (/) )  /\  ( Y  =/=  (/)  /\  Z  =/=  (/) ) ) )  -> 
( Y  .+  Z
)  C_  A )
11 simp3l 983 . . . 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 29989 . . . 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 1185 . . 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 447 . . . . . . . 8  |-  ( ( ( X  =/=  (/)  /\  ( Y  .+  Z )  =/=  (/) )  /\  ( Y  =/=  (/)  /\  Z  =/=  (/) ) )  ->  ( Y  =/=  (/)  /\  Z  =/=  (/) ) )
1712, 13, 7, 8paddasslem15 30023 . . . . . . . 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 1232 . . . . . . 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 1169 . . . . . 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 418 . . . . 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 2673 . . . 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 586 . . 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 206 . 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 3185 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 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684    =/= wne 2446   E.wrex 2544    C_ wss 3152   (/)c0 3455   class class class wbr 4023   ` cfv 5255  (class class class)co 5858   lecple 13215   joincjn 14078   Latclat 14151   Atomscatm 29453   HLchlt 29540   + Pcpadd 29984
This theorem is referenced by:  paddasslem18  30026
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-padd 29985
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