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Theorem cdlemb3 29484
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 28883? Then replace cdlemb2 28919 with it. This is a more general version of cdlemb2 28919 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 963 . . . 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 964 . . . 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 29483 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  E. r  e.  A  ( P  =/=  r  /\  -.  r  .<_  W ) )
81, 2, 7syl2anc 645 . . 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 439 . . . . . 6  |-  ( ( P  =/=  r  /\  -.  r  .<_  W )  <-> 
( -.  r  .<_  W  /\  P  =/=  r
) )
10 eqcom 2255 . . . . . . . . 9  |-  ( P  =  r  <->  r  =  P )
11 simp2 961 . . . . . . . . . . . . 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 5726 . . . . . . . . . . . 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 1071 . . . . . . . . . . . . 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 1073 . . . . . . . . . . . . 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 28247 . . . . . . . . . . . . 13  |-  ( ( K  e.  HL  /\  P  e.  A )  ->  ( P  .\/  P
)  =  P )
1613, 14, 15syl2anc 645 . . . . . . . . . . . 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 2287 . . . . . . . . . . 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 3932 . . . . . . . . . 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 28239 . . . . . . . . . . . 12  |-  ( K  e.  HL  ->  K  e.  AtLat )
2013, 19syl 17 . . . . . . . . . . 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 962 . . . . . . . . . . 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 28190 . . . . . . . . . . 11  |-  ( ( K  e.  AtLat  /\  r  e.  A  /\  P  e.  A )  ->  (
r  .<_  P  <->  r  =  P ) )
2320, 21, 14, 22syl3anc 1187 . . . . . . . . . 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 247 . . . . . . . . 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 250 . . . . . . . 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 2445 . . . . . . 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 687 . . . . . 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 250 . . . . 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 1156 . . . 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 2524 . . 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 203 . 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 963 . . 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 964 . . 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 965 . . 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 449 . . 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 28919 . . 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 1192 . 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 2490 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 5    -> wi 6    <-> wb 178    /\ wa 360    /\ w3a 939    = wceq 1619    e. wcel 1621    =/= wne 2412   E.wrex 2510   class class class wbr 3920   ` cfv 4592  (class class class)co 5710   lecple 13089   joincjn 13922   Atomscatm 28142   AtLatcal 28143   HLchlt 28229   LHypclh 28862
This theorem is referenced by:  cdlemg6e  29500
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234  ax-rep 4028  ax-sep 4038  ax-nul 4046  ax-pow 4082  ax-pr 4108  ax-un 4403
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-nel 2415  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2516  df-v 2729  df-sbc 2922  df-csb 3010  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-nul 3363  df-if 3471  df-pw 3532  df-sn 3550  df-pr 3551  df-op 3553  df-uni 3728  df-iun 3805  df-br 3921  df-opab 3975  df-mpt 3976  df-id 4202  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 4602  df-fn 4603  df-f 4604  df-f1 4605  df-fo 4606  df-f1o 4607  df-fv 4608  df-ov 5713  df-oprab 5714  df-mpt2 5715  df-1st 5974  df-2nd 5975  df-iota 6143  df-undef 6182  df-riota 6190  df-poset 13924  df-plt 13936  df-lub 13952  df-glb 13953  df-join 13954  df-meet 13955  df-p0 13989  df-p1 13990  df-lat 13996  df-clat 14058  df-oposet 28055  df-ol 28057  df-oml 28058  df-covers 28145  df-ats 28146  df-atl 28177  df-cvlat 28201  df-hlat 28230  df-lhyp 28866
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