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Theorem cdlemb2 30155
Description: Given two atoms not under the fiducial (reference) co-atom  W, there is a third. Lemma B in [Crawley] p. 112. (Contributed by NM, 30-May-2012.)
Hypotheses
Ref Expression
cdlemb2.l  |-  .<_  =  ( le `  K )
cdlemb2.j  |-  .\/  =  ( join `  K )
cdlemb2.a  |-  A  =  ( Atoms `  K )
cdlemb2.h  |-  H  =  ( LHyp `  K
)
Assertion
Ref Expression
cdlemb2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  P  =/= 
Q )  ->  E. r  e.  A  ( -.  r  .<_  W  /\  -.  r  .<_  ( P  .\/  Q ) ) )
Distinct variable groups:    A, r    .\/ , r    K, r    .<_ , r    P, r    Q, r    W, r
Allowed substitution hint:    H( r)

Proof of Theorem cdlemb2
StepHypRef Expression
1 simp1l 981 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  P  =/= 
Q )  ->  K  e.  HL )
2 simp2ll 1024 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  P  =/= 
Q )  ->  P  e.  A )
3 simp2rl 1026 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  P  =/= 
Q )  ->  Q  e.  A )
4 simp1r 982 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  P  =/= 
Q )  ->  W  e.  H )
5 eqid 2387 . . . 4  |-  ( Base `  K )  =  (
Base `  K )
6 cdlemb2.h . . . 4  |-  H  =  ( LHyp `  K
)
75, 6lhpbase 30112 . . 3  |-  ( W  e.  H  ->  W  e.  ( Base `  K
) )
84, 7syl 16 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  P  =/= 
Q )  ->  W  e.  ( Base `  K
) )
9 simp3 959 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  P  =/= 
Q )  ->  P  =/=  Q )
10 eqid 2387 . . . 4  |-  ( 1.
`  K )  =  ( 1. `  K
)
11 eqid 2387 . . . 4  |-  (  <o  `  K )  =  ( 
<o  `  K )
1210, 11, 6lhp1cvr 30113 . . 3  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  W (  <o  `  K
) ( 1. `  K ) )
13123ad2ant1 978 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  P  =/= 
Q )  ->  W
(  <o  `  K )
( 1. `  K
) )
14 simp2lr 1025 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  P  =/= 
Q )  ->  -.  P  .<_  W )
15 simp2rr 1027 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  P  =/= 
Q )  ->  -.  Q  .<_  W )
16 cdlemb2.l . . 3  |-  .<_  =  ( le `  K )
17 cdlemb2.j . . 3  |-  .\/  =  ( join `  K )
18 cdlemb2.a . . 3  |-  A  =  ( Atoms `  K )
195, 16, 17, 10, 11, 18cdlemb 29908 . 2  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( W  e.  (
Base `  K )  /\  P  =/=  Q
)  /\  ( W
(  <o  `  K )
( 1. `  K
)  /\  -.  P  .<_  W  /\  -.  Q  .<_  W ) )  ->  E. r  e.  A  ( -.  r  .<_  W  /\  -.  r  .<_  ( P  .\/  Q ) ) )
201, 2, 3, 8, 9, 13, 14, 15, 19syl323anc 1214 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  P  =/= 
Q )  ->  E. r  e.  A  ( -.  r  .<_  W  /\  -.  r  .<_  ( P  .\/  Q ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717    =/= wne 2550   E.wrex 2650   class class class wbr 4153   ` cfv 5394  (class class class)co 6020   Basecbs 13396   lecple 13463   joincjn 14328   1.cp1 14394    <o ccvr 29377   Atomscatm 29378   HLchlt 29465   LHypclh 30098
This theorem is referenced by:  cdlemd4  30315  cdlemd9  30320  cdleme25a  30467  cdleme25c  30469  cdleme25dN  30470  cdleme26ee  30474  cdlemefs32sn1aw  30528  cdleme43fsv1snlem  30534  cdleme41sn3a  30547  cdleme40m  30581  cdleme40n  30582  cdleme17d3  30610  cdlemeg46gfre  30646  cdleme50trn2  30665  cdlemb3  30720
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 2368  ax-rep 4261  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641
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 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-nel 2553  df-ral 2654  df-rex 2655  df-reu 2656  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-op 3766  df-uni 3958  df-iun 4037  df-br 4154  df-opab 4208  df-mpt 4209  df-id 4439  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-ov 6023  df-oprab 6024  df-mpt2 6025  df-1st 6288  df-2nd 6289  df-undef 6479  df-riota 6485  df-poset 14330  df-plt 14342  df-lub 14358  df-glb 14359  df-join 14360  df-meet 14361  df-p0 14395  df-p1 14396  df-lat 14402  df-clat 14464  df-oposet 29291  df-ol 29293  df-oml 29294  df-covers 29381  df-ats 29382  df-atl 29413  df-cvlat 29437  df-hlat 29466  df-lhyp 30102
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