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Theorem paddasslem6 30636
Description: Lemma for paddass 30649. (Contributed by NM, 8-Jan-2012.)
Hypotheses
Ref Expression
paddasslem.l  |-  .<_  =  ( le `  K )
paddasslem.j  |-  .\/  =  ( join `  K )
paddasslem.a  |-  A  =  ( Atoms `  K )
Assertion
Ref Expression
paddasslem6  |-  ( ( ( K  e.  HL  /\  ( p  e.  A  /\  s  e.  A
)  /\  z  e.  A )  /\  (
s  =/=  z  /\  s  .<_  ( p  .\/  z ) ) )  ->  p  .<_  ( s 
.\/  z ) )

Proof of Theorem paddasslem6
StepHypRef Expression
1 simpl1 958 . . 3  |-  ( ( ( K  e.  HL  /\  ( p  e.  A  /\  s  e.  A
)  /\  z  e.  A )  /\  (
s  =/=  z  /\  s  .<_  ( p  .\/  z ) ) )  ->  K  e.  HL )
2 simpl2r 1009 . . . 4  |-  ( ( ( K  e.  HL  /\  ( p  e.  A  /\  s  e.  A
)  /\  z  e.  A )  /\  (
s  =/=  z  /\  s  .<_  ( p  .\/  z ) ) )  ->  s  e.  A
)
3 simpl2l 1008 . . . 4  |-  ( ( ( K  e.  HL  /\  ( p  e.  A  /\  s  e.  A
)  /\  z  e.  A )  /\  (
s  =/=  z  /\  s  .<_  ( p  .\/  z ) ) )  ->  p  e.  A
)
4 simpl3 960 . . . 4  |-  ( ( ( K  e.  HL  /\  ( p  e.  A  /\  s  e.  A
)  /\  z  e.  A )  /\  (
s  =/=  z  /\  s  .<_  ( p  .\/  z ) ) )  ->  z  e.  A
)
52, 3, 43jca 1132 . . 3  |-  ( ( ( K  e.  HL  /\  ( p  e.  A  /\  s  e.  A
)  /\  z  e.  A )  /\  (
s  =/=  z  /\  s  .<_  ( p  .\/  z ) ) )  ->  ( s  e.  A  /\  p  e.  A  /\  z  e.  A ) )
6 simprl 732 . . 3  |-  ( ( ( K  e.  HL  /\  ( p  e.  A  /\  s  e.  A
)  /\  z  e.  A )  /\  (
s  =/=  z  /\  s  .<_  ( p  .\/  z ) ) )  ->  s  =/=  z
)
71, 5, 63jca 1132 . 2  |-  ( ( ( K  e.  HL  /\  ( p  e.  A  /\  s  e.  A
)  /\  z  e.  A )  /\  (
s  =/=  z  /\  s  .<_  ( p  .\/  z ) ) )  ->  ( K  e.  HL  /\  ( s  e.  A  /\  p  e.  A  /\  z  e.  A )  /\  s  =/=  z ) )
8 simprr 733 . 2  |-  ( ( ( K  e.  HL  /\  ( p  e.  A  /\  s  e.  A
)  /\  z  e.  A )  /\  (
s  =/=  z  /\  s  .<_  ( p  .\/  z ) ) )  ->  s  .<_  ( p 
.\/  z ) )
9 paddasslem.l . . 3  |-  .<_  =  ( le `  K )
10 paddasslem.j . . 3  |-  .\/  =  ( join `  K )
11 paddasslem.a . . 3  |-  A  =  ( Atoms `  K )
129, 10, 11hlatexch2 30207 . 2  |-  ( ( K  e.  HL  /\  ( s  e.  A  /\  p  e.  A  /\  z  e.  A
)  /\  s  =/=  z )  ->  (
s  .<_  ( p  .\/  z )  ->  p  .<_  ( s  .\/  z
) ) )
137, 8, 12sylc 56 1  |-  ( ( ( K  e.  HL  /\  ( p  e.  A  /\  s  e.  A
)  /\  z  e.  A )  /\  (
s  =/=  z  /\  s  .<_  ( p  .\/  z ) ) )  ->  p  .<_  ( s 
.\/  z ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696    =/= wne 2459   class class class wbr 4039   ` cfv 5271  (class class class)co 5874   lecple 13231   joincjn 14094   Atomscatm 30075   HLchlt 30162
This theorem is referenced by:  paddasslem7  30637
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-undef 6314  df-riota 6320  df-poset 14096  df-plt 14108  df-lub 14124  df-join 14126  df-lat 14168  df-covers 30078  df-ats 30079  df-atl 30110  df-cvlat 30134  df-hlat 30163
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