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Theorem 4atexlempsb 30696
Description: Lemma for 4atexlem7 30711. (Contributed by NM, 23-Nov-2012.)
Hypotheses
Ref Expression
4thatlem.ph  |-  ( ph  <->  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( S  e.  A  /\  ( R  e.  A  /\  -.  R  .<_  W  /\  ( P  .\/  R )  =  ( Q  .\/  R ) )  /\  ( T  e.  A  /\  ( U  .\/  T )  =  ( V  .\/  T ) ) )  /\  ( P  =/=  Q  /\  -.  S  .<_  ( P 
.\/  Q ) ) ) )
4thatlempqb.j  |-  .\/  =  ( join `  K )
4thatlempqb.a  |-  A  =  ( Atoms `  K )
Assertion
Ref Expression
4atexlempsb  |-  ( ph  ->  ( P  .\/  S
)  e.  ( Base `  K ) )

Proof of Theorem 4atexlempsb
StepHypRef Expression
1 4thatlem.ph . . 3  |-  ( ph  <->  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( S  e.  A  /\  ( R  e.  A  /\  -.  R  .<_  W  /\  ( P  .\/  R )  =  ( Q  .\/  R ) )  /\  ( T  e.  A  /\  ( U  .\/  T )  =  ( V  .\/  T ) ) )  /\  ( P  =/=  Q  /\  -.  S  .<_  ( P 
.\/  Q ) ) ) )
214atexlemk 30683 . 2  |-  ( ph  ->  K  e.  HL )
314atexlemp 30686 . 2  |-  ( ph  ->  P  e.  A )
414atexlems 30688 . 2  |-  ( ph  ->  S  e.  A )
5 eqid 2435 . . 3  |-  ( Base `  K )  =  (
Base `  K )
6 4thatlempqb.j . . 3  |-  .\/  =  ( join `  K )
7 4thatlempqb.a . . 3  |-  A  =  ( Atoms `  K )
85, 6, 7hlatjcl 30003 . 2  |-  ( ( K  e.  HL  /\  P  e.  A  /\  S  e.  A )  ->  ( P  .\/  S
)  e.  ( Base `  K ) )
92, 3, 4, 8syl3anc 1184 1  |-  ( ph  ->  ( P  .\/  S
)  e.  ( Base `  K ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725    =/= wne 2598   class class class wbr 4204   ` cfv 5445  (class class class)co 6072   Basecbs 13457   joincjn 14389   Atomscatm 29900   HLchlt 29987
This theorem is referenced by:  4atexlemunv  30702  4atexlemtlw  30703  4atexlemc  30705  4atexlemnclw  30706  4atexlemex2  30707  4atexlemcnd  30708
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-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395
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-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4875  df-rel 4876  df-cnv 4877  df-co 4878  df-dm 4879  df-iota 5409  df-fun 5447  df-fv 5453  df-ov 6075  df-lat 14463  df-ats 29904  df-atl 29935  df-cvlat 29959  df-hlat 29988
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