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Theorem 4atexlempsb 30225
Description: Lemma for 4atexlem7 30240. (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 30212 . 2  |-  ( ph  ->  K  e.  HL )
314atexlemp 30215 . 2  |-  ( ph  ->  P  e.  A )
414atexlems 30217 . 2  |-  ( ph  ->  S  e.  A )
5 eqid 2380 . . 3  |-  ( Base `  K )  =  (
Base `  K )
6 4thatlempqb.j . . 3  |-  .\/  =  ( join `  K )
7 4thatlempqb.a . . 3  |-  A  =  ( Atoms `  K )
85, 6, 7hlatjcl 29532 . 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 1649    e. wcel 1717    =/= wne 2543   class class class wbr 4146   ` cfv 5387  (class class class)co 6013   Basecbs 13389   joincjn 14321   Atomscatm 29429   HLchlt 29516
This theorem is referenced by:  4atexlemunv  30231  4atexlemtlw  30232  4atexlemc  30234  4atexlemnclw  30235  4atexlemex2  30236  4atexlemcnd  30237
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-ral 2647  df-rex 2648  df-rab 2651  df-v 2894  df-sbc 3098  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-nul 3565  df-if 3676  df-sn 3756  df-pr 3757  df-op 3759  df-uni 3951  df-br 4147  df-opab 4201  df-mpt 4202  df-id 4432  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-iota 5351  df-fun 5389  df-fv 5395  df-ov 6016  df-lat 14395  df-ats 29433  df-atl 29464  df-cvlat 29488  df-hlat 29517
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