Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  cdlemb2 Unicode version

Theorem cdlemb2 30156
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 2388 . . . 4  |-  ( Base `  K )  =  (
Base `  K )
6 cdlemb2.h . . . 4  |-  H  =  ( LHyp `  K
)
75, 6lhpbase 30113 . . 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 2388 . . . 4  |-  ( 1.
`  K )  =  ( 1. `  K
)
11 eqid 2388 . . . 4  |-  (  <o  `  K )  =  ( 
<o  `  K )
1210, 11, 6lhp1cvr 30114 . . 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 29909 . 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 2551   E.wrex 2651   class class class wbr 4154   ` cfv 5395  (class class class)co 6021   Basecbs 13397   lecple 13464   joincjn 14329   1.cp1 14395    <o ccvr 29378   Atomscatm 29379   HLchlt 29466   LHypclh 30099
This theorem is referenced by:  cdlemd4  30316  cdlemd9  30321  cdleme25a  30468  cdleme25c  30470  cdleme25dN  30471  cdleme26ee  30475  cdlemefs32sn1aw  30529  cdleme43fsv1snlem  30535  cdleme41sn3a  30548  cdleme40m  30582  cdleme40n  30583  cdleme17d3  30611  cdlemeg46gfre  30647  cdleme50trn2  30666  cdlemb3  30721
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 2369  ax-rep 4262  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345  ax-un 4642
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 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-nel 2554  df-ral 2655  df-rex 2656  df-reu 2657  df-rab 2659  df-v 2902  df-sbc 3106  df-csb 3196  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-nul 3573  df-if 3684  df-pw 3745  df-sn 3764  df-pr 3765  df-op 3767  df-uni 3959  df-iun 4038  df-br 4155  df-opab 4209  df-mpt 4210  df-id 4440  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-f1 5400  df-fo 5401  df-f1o 5402  df-fv 5403  df-ov 6024  df-oprab 6025  df-mpt2 6026  df-1st 6289  df-2nd 6290  df-undef 6480  df-riota 6486  df-poset 14331  df-plt 14343  df-lub 14359  df-glb 14360  df-join 14361  df-meet 14362  df-p0 14396  df-p1 14397  df-lat 14403  df-clat 14465  df-oposet 29292  df-ol 29294  df-oml 29295  df-covers 29382  df-ats 29383  df-atl 29414  df-cvlat 29438  df-hlat 29467  df-lhyp 30103
  Copyright terms: Public domain W3C validator