Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  paddasslem11 Unicode version

Theorem paddasslem11 30019
Description: Lemma for paddass 30027. The case when  p  =  z. (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
paddasslem11  |-  ( ( ( ( K  e.  HL  /\  p  =  z )  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  C_  A ) )  /\  z  e.  Z )  ->  p  e.  ( ( X  .+  Y ) 
.+  Z ) )

Proof of Theorem paddasslem11
StepHypRef Expression
1 simplll 734 . . 3  |-  ( ( ( ( K  e.  HL  /\  p  =  z )  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  C_  A ) )  /\  z  e.  Z )  ->  K  e.  HL )
2 simplr3 999 . . 3  |-  ( ( ( ( K  e.  HL  /\  p  =  z )  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  C_  A ) )  /\  z  e.  Z )  ->  Z  C_  A )
3 simplr1 997 . . . 4  |-  ( ( ( ( K  e.  HL  /\  p  =  z )  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  C_  A ) )  /\  z  e.  Z )  ->  X  C_  A )
4 simplr2 998 . . . 4  |-  ( ( ( ( K  e.  HL  /\  p  =  z )  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  C_  A ) )  /\  z  e.  Z )  ->  Y  C_  A )
5 paddasslem.a . . . . 5  |-  A  =  ( Atoms `  K )
6 paddasslem.p . . . . 5  |-  .+  =  ( + P `  K
)
75, 6paddssat 30003 . . . 4  |-  ( ( K  e.  HL  /\  X  C_  A  /\  Y  C_  A )  ->  ( X  .+  Y )  C_  A )
81, 3, 4, 7syl3anc 1182 . . 3  |-  ( ( ( ( K  e.  HL  /\  p  =  z )  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  C_  A ) )  /\  z  e.  Z )  ->  ( X  .+  Y
)  C_  A )
95, 6sspadd2 30005 . . 3  |-  ( ( K  e.  HL  /\  Z  C_  A  /\  ( X  .+  Y )  C_  A )  ->  Z  C_  ( ( X  .+  Y )  .+  Z
) )
101, 2, 8, 9syl3anc 1182 . 2  |-  ( ( ( ( K  e.  HL  /\  p  =  z )  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  C_  A ) )  /\  z  e.  Z )  ->  Z  C_  ( ( X  .+  Y )  .+  Z ) )
11 simpllr 735 . . 3  |-  ( ( ( ( K  e.  HL  /\  p  =  z )  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  C_  A ) )  /\  z  e.  Z )  ->  p  =  z )
12 simpr 447 . . 3  |-  ( ( ( ( K  e.  HL  /\  p  =  z )  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  C_  A ) )  /\  z  e.  Z )  ->  z  e.  Z )
1311, 12eqeltrd 2357 . 2  |-  ( ( ( ( K  e.  HL  /\  p  =  z )  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  C_  A ) )  /\  z  e.  Z )  ->  p  e.  Z )
1410, 13sseldd 3181 1  |-  ( ( ( ( K  e.  HL  /\  p  =  z )  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  C_  A ) )  /\  z  e.  Z )  ->  p  e.  ( ( X  .+  Y ) 
.+  Z ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684    C_ wss 3152   ` cfv 5255  (class class class)co 5858   lecple 13215   joincjn 14078   Atomscatm 29453   HLchlt 29540   + Pcpadd 29984
This theorem is referenced by:  paddasslem14  30022
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-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-padd 29985
  Copyright terms: Public domain W3C validator