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Theorem 4atexlempsb 30320
Description: Lemma for 4atexlem7 30335. (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 30307 . 2  |-  ( ph  ->  K  e.  HL )
314atexlemp 30310 . 2  |-  ( ph  ->  P  e.  A )
414atexlems 30312 . 2  |-  ( ph  ->  S  e.  A )
5 eqid 2366 . . 3  |-  ( Base `  K )  =  (
Base `  K )
6 4thatlempqb.j . . 3  |-  .\/  =  ( join `  K )
7 4thatlempqb.a . . 3  |-  A  =  ( Atoms `  K )
85, 6, 7hlatjcl 29627 . 2  |-  ( ( K  e.  HL  /\  P  e.  A  /\  S  e.  A )  ->  ( P  .\/  S
)  e.  ( Base `  K ) )
92, 3, 4, 8syl3anc 1183 1  |-  ( ph  ->  ( P  .\/  S
)  e.  ( Base `  K ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 935    = wceq 1647    e. wcel 1715    =/= wne 2529   class class class wbr 4125   ` cfv 5358  (class class class)co 5981   Basecbs 13356   joincjn 14288   Atomscatm 29524   HLchlt 29611
This theorem is referenced by:  4atexlemunv  30326  4atexlemtlw  30327  4atexlemc  30329  4atexlemnclw  30330  4atexlemex2  30331  4atexlemcnd  30332
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-13 1717  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-sep 4243  ax-nul 4251  ax-pow 4290  ax-pr 4316
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-ral 2633  df-rex 2634  df-rab 2637  df-v 2875  df-sbc 3078  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-nul 3544  df-if 3655  df-sn 3735  df-pr 3736  df-op 3738  df-uni 3930  df-br 4126  df-opab 4180  df-mpt 4181  df-id 4412  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-iota 5322  df-fun 5360  df-fv 5366  df-ov 5984  df-lat 14362  df-ats 29528  df-atl 29559  df-cvlat 29583  df-hlat 29612
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