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Theorem paddasslem11 30554
Description: Lemma for paddass 30562. 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 735 . . 3  |-  ( ( ( ( K  e.  HL  /\  p  =  z )  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  C_  A ) )  /\  z  e.  Z )  ->  K  e.  HL )
2 simplr3 1001 . . 3  |-  ( ( ( ( K  e.  HL  /\  p  =  z )  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  C_  A ) )  /\  z  e.  Z )  ->  Z  C_  A )
3 simplr1 999 . . . 4  |-  ( ( ( ( K  e.  HL  /\  p  =  z )  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  C_  A ) )  /\  z  e.  Z )  ->  X  C_  A )
4 simplr2 1000 . . . 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 30538 . . . 4  |-  ( ( K  e.  HL  /\  X  C_  A  /\  Y  C_  A )  ->  ( X  .+  Y )  C_  A )
81, 3, 4, 7syl3anc 1184 . . 3  |-  ( ( ( ( K  e.  HL  /\  p  =  z )  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  C_  A ) )  /\  z  e.  Z )  ->  ( X  .+  Y
)  C_  A )
95, 6sspadd2 30540 . . 3  |-  ( ( K  e.  HL  /\  Z  C_  A  /\  ( X  .+  Y )  C_  A )  ->  Z  C_  ( ( X  .+  Y )  .+  Z
) )
101, 2, 8, 9syl3anc 1184 . 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 736 . . 3  |-  ( ( ( ( K  e.  HL  /\  p  =  z )  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  C_  A ) )  /\  z  e.  Z )  ->  p  =  z )
12 simpr 448 . . 3  |-  ( ( ( ( K  e.  HL  /\  p  =  z )  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  C_  A ) )  /\  z  e.  Z )  ->  z  e.  Z )
1311, 12eqeltrd 2509 . 2  |-  ( ( ( ( K  e.  HL  /\  p  =  z )  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  C_  A ) )  /\  z  e.  Z )  ->  p  e.  Z )
1410, 13sseldd 3341 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 359    /\ w3a 936    = wceq 1652    e. wcel 1725    C_ wss 3312   ` cfv 5446  (class class class)co 6073   lecple 13528   joincjn 14393   Atomscatm 29988   HLchlt 30075   + Pcpadd 30519
This theorem is referenced by:  paddasslem14  30557
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-padd 30520
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