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Theorem cdlemb3 30866
Description: Given two atoms not under the fiducial co-atom  W, there is a third. Lemma B in [Crawley] p. 112. TODO: Is there a simpler more direct proof, that could be placed earlier e.g. near lhpexle 30265? Then replace cdlemb2 30301 with it. This is a more general version of cdlemb2 30301 without  P  =/=  Q condition. (Contributed by NM, 27-Apr-2013.)
Hypotheses
Ref Expression
cdlemg5.l  |-  .<_  =  ( le `  K )
cdlemg5.j  |-  .\/  =  ( join `  K )
cdlemg5.a  |-  A  =  ( Atoms `  K )
cdlemg5.h  |-  H  =  ( LHyp `  K
)
Assertion
Ref Expression
cdlemb3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  ->  E. r  e.  A  ( -.  r  .<_  W  /\  -.  r  .<_  ( P  .\/  Q ) ) )
Distinct variable groups:    A, r    H, r    K, r    .<_ , r    P, r    W, r    .\/ , r    Q, r

Proof of Theorem cdlemb3
StepHypRef Expression
1 simpl1 959 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  P  =  Q )  ->  ( K  e.  HL  /\  W  e.  H ) )
2 simpl2 960 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  P  =  Q )  ->  ( P  e.  A  /\  -.  P  .<_  W ) )
3 cdlemg5.l . . . . 5  |-  .<_  =  ( le `  K )
4 cdlemg5.j . . . . 5  |-  .\/  =  ( join `  K )
5 cdlemg5.a . . . . 5  |-  A  =  ( Atoms `  K )
6 cdlemg5.h . . . . 5  |-  H  =  ( LHyp `  K
)
73, 4, 5, 6cdlemg5 30865 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  E. r  e.  A  ( P  =/=  r  /\  -.  r  .<_  W ) )
81, 2, 7syl2anc 642 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  P  =  Q )  ->  E. r  e.  A  ( P  =/=  r  /\  -.  r  .<_  W ) )
9 ancom 437 . . . . . 6  |-  ( ( P  =/=  r  /\  -.  r  .<_  W )  <-> 
( -.  r  .<_  W  /\  P  =/=  r
) )
10 eqcom 2368 . . . . . . . . 9  |-  ( P  =  r  <->  r  =  P )
11 simp2 957 . . . . . . . . . . . . 13  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  P  =  Q  /\  r  e.  A )  ->  P  =  Q )
1211oveq2d 5997 . . . . . . . . . . . 12  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  P  =  Q  /\  r  e.  A )  ->  ( P  .\/  P
)  =  ( P 
.\/  Q ) )
13 simp11l 1067 . . . . . . . . . . . . 13  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  P  =  Q  /\  r  e.  A )  ->  K  e.  HL )
14 simp12l 1069 . . . . . . . . . . . . 13  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  P  =  Q  /\  r  e.  A )  ->  P  e.  A )
154, 5hlatjidm 29629 . . . . . . . . . . . . 13  |-  ( ( K  e.  HL  /\  P  e.  A )  ->  ( P  .\/  P
)  =  P )
1613, 14, 15syl2anc 642 . . . . . . . . . . . 12  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  P  =  Q  /\  r  e.  A )  ->  ( P  .\/  P
)  =  P )
1712, 16eqtr3d 2400 . . . . . . . . . . 11  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  P  =  Q  /\  r  e.  A )  ->  ( P  .\/  Q
)  =  P )
1817breq2d 4137 . . . . . . . . . 10  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  P  =  Q  /\  r  e.  A )  ->  ( r  .<_  ( P 
.\/  Q )  <->  r  .<_  P ) )
19 hlatl 29621 . . . . . . . . . . . 12  |-  ( K  e.  HL  ->  K  e.  AtLat )
2013, 19syl 15 . . . . . . . . . . 11  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  P  =  Q  /\  r  e.  A )  ->  K  e.  AtLat )
21 simp3 958 . . . . . . . . . . 11  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  P  =  Q  /\  r  e.  A )  ->  r  e.  A )
223, 5atcmp 29572 . . . . . . . . . . 11  |-  ( ( K  e.  AtLat  /\  r  e.  A  /\  P  e.  A )  ->  (
r  .<_  P  <->  r  =  P ) )
2320, 21, 14, 22syl3anc 1183 . . . . . . . . . 10  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  P  =  Q  /\  r  e.  A )  ->  ( r  .<_  P  <->  r  =  P ) )
2418, 23bitr2d 245 . . . . . . . . 9  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  P  =  Q  /\  r  e.  A )  ->  ( r  =  P  <-> 
r  .<_  ( P  .\/  Q ) ) )
2510, 24syl5bb 248 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  P  =  Q  /\  r  e.  A )  ->  ( P  =  r  <-> 
r  .<_  ( P  .\/  Q ) ) )
2625necon3abid 2562 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  P  =  Q  /\  r  e.  A )  ->  ( P  =/=  r  <->  -.  r  .<_  ( P  .\/  Q ) ) )
2726anbi2d 684 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  P  =  Q  /\  r  e.  A )  ->  ( ( -.  r  .<_  W  /\  P  =/=  r )  <->  ( -.  r  .<_  W  /\  -.  r  .<_  ( P  .\/  Q ) ) ) )
289, 27syl5bb 248 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  P  =  Q  /\  r  e.  A )  ->  ( ( P  =/=  r  /\  -.  r  .<_  W )  <->  ( -.  r  .<_  W  /\  -.  r  .<_  ( P  .\/  Q ) ) ) )
29283expa 1152 . . . 4  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  P  =  Q )  /\  r  e.  A
)  ->  ( ( P  =/=  r  /\  -.  r  .<_  W )  <->  ( -.  r  .<_  W  /\  -.  r  .<_  ( P  .\/  Q ) ) ) )
3029rexbidva 2645 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  P  =  Q )  ->  ( E. r  e.  A  ( P  =/=  r  /\  -.  r  .<_  W )  <->  E. r  e.  A  ( -.  r  .<_  W  /\  -.  r  .<_  ( P  .\/  Q ) ) ) )
318, 30mpbid 201 . 2  |-  ( ( ( ( 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 ) ) )
32 simpl1 959 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  P  =/=  Q )  -> 
( K  e.  HL  /\  W  e.  H ) )
33 simpl2 960 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  P  =/=  Q )  -> 
( P  e.  A  /\  -.  P  .<_  W ) )
34 simpl3 961 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  P  =/=  Q )  -> 
( Q  e.  A  /\  -.  Q  .<_  W ) )
35 simpr 447 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  P  =/=  Q )  ->  P  =/=  Q )
363, 4, 5, 6cdlemb2 30301 . . 3  |-  ( ( ( 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 ) ) )
3732, 33, 34, 35, 36syl121anc 1188 . 2  |-  ( ( ( ( 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 ) ) )
3831, 37pm2.61dane 2607 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  ->  E. r  e.  A  ( -.  r  .<_  W  /\  -.  r  .<_  ( P  .\/  Q ) ) )
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   E.wrex 2629   class class class wbr 4125   ` cfv 5358  (class class class)co 5981   lecple 13423   joincjn 14288   Atomscatm 29524   AtLatcal 29525   HLchlt 29611   LHypclh 30244
This theorem is referenced by:  cdlemg6e  30882
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-rep 4233  ax-sep 4243  ax-nul 4251  ax-pow 4290  ax-pr 4316  ax-un 4615
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-nel 2532  df-ral 2633  df-rex 2634  df-reu 2635  df-rab 2637  df-v 2875  df-sbc 3078  df-csb 3168  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-nul 3544  df-if 3655  df-pw 3716  df-sn 3735  df-pr 3736  df-op 3738  df-uni 3930  df-iun 4009  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-rn 4803  df-res 4804  df-ima 4805  df-iota 5322  df-fun 5360  df-fn 5361  df-f 5362  df-f1 5363  df-fo 5364  df-f1o 5365  df-fv 5366  df-ov 5984  df-oprab 5985  df-mpt2 5986  df-1st 6249  df-2nd 6250  df-undef 6440  df-riota 6446  df-poset 14290  df-plt 14302  df-lub 14318  df-glb 14319  df-join 14320  df-meet 14321  df-p0 14355  df-p1 14356  df-lat 14362  df-clat 14424  df-oposet 29437  df-ol 29439  df-oml 29440  df-covers 29527  df-ats 29528  df-atl 29559  df-cvlat 29583  df-hlat 29612  df-lhyp 30248
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