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Theorem cdlemb3 36729
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 36126? Then replace cdlemb2 36162 with it. This is a more general version of cdlemb2 36162 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 997 . . . 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 998 . . . 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 36728 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  E. r  e.  A  ( P  =/=  r  /\  -.  r  .<_  W ) )
81, 2, 7syl2anc 659 . . 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 448 . . . . . 6  |-  ( ( P  =/=  r  /\  -.  r  .<_  W )  <-> 
( -.  r  .<_  W  /\  P  =/=  r
) )
10 eqcom 2463 . . . . . . . . 9  |-  ( P  =  r  <->  r  =  P )
11 simp2 995 . . . . . . . . . . . . 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 6286 . . . . . . . . . . . 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 1105 . . . . . . . . . . . . 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 1107 . . . . . . . . . . . . 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 35490 . . . . . . . . . . . . 13  |-  ( ( K  e.  HL  /\  P  e.  A )  ->  ( P  .\/  P
)  =  P )
1613, 14, 15syl2anc 659 . . . . . . . . . . . 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 2497 . . . . . . . . . . 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 4451 . . . . . . . . . 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 35482 . . . . . . . . . . . 12  |-  ( K  e.  HL  ->  K  e.  AtLat )
2013, 19syl 16 . . . . . . . . . . 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 996 . . . . . . . . . . 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 35433 . . . . . . . . . . 11  |-  ( ( K  e.  AtLat  /\  r  e.  A  /\  P  e.  A )  ->  (
r  .<_  P  <->  r  =  P ) )
2320, 21, 14, 22syl3anc 1226 . . . . . . . . . 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 254 . . . . . . . . 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 257 . . . . . . . 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 2700 . . . . . . 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 701 . . . . . 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 257 . . . . 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 1194 . . . 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 2962 . . 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 210 . 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 997 . . 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 998 . . 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 999 . . 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 459 . . 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 36162 . . 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 1231 . 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 2772 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 setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 367    /\ w3a 971    = wceq 1398    e. wcel 1823    =/= wne 2649   E.wrex 2805   class class class wbr 4439   ` cfv 5570  (class class class)co 6270   lecple 14791   joincjn 15772   Atomscatm 35385   AtLatcal 35386   HLchlt 35472   LHypclh 36105
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-preset 15756  df-poset 15774  df-plt 15787  df-lub 15803  df-glb 15804  df-join 15805  df-meet 15806  df-p0 15868  df-p1 15869  df-lat 15875  df-clat 15937  df-oposet 35298  df-ol 35300  df-oml 35301  df-covers 35388  df-ats 35389  df-atl 35420  df-cvlat 35444  df-hlat 35473  df-lhyp 36109
This theorem is referenced by:  cdlemg6e  36745
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