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Theorem cdleme11dN 36384
Description: Part of proof of Lemma E in [Crawley] p. 113. Lemma leading to cdleme11 36392. (Contributed by NM, 13-Jun-2012.) (New usage is discouraged.)
Hypotheses
Ref Expression
cdleme11.l  |-  .<_  =  ( le `  K )
cdleme11.j  |-  .\/  =  ( join `  K )
cdleme11.m  |-  ./\  =  ( meet `  K )
cdleme11.a  |-  A  =  ( Atoms `  K )
cdleme11.h  |-  H  =  ( LHyp `  K
)
cdleme11.u  |-  U  =  ( ( P  .\/  Q )  ./\  W )
Assertion
Ref Expression
cdleme11dN  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A
)  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  T  e.  A  /\  P  =/=  Q
)  /\  ( S  =/=  T  /\  -.  S  .<_  ( P  .\/  Q
)  /\  U  .<_  ( S  .\/  T ) ) )  ->  ( P  .\/  S )  =/=  ( P  .\/  T
) )

Proof of Theorem cdleme11dN
StepHypRef Expression
1 simp1 994 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A
)  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  T  e.  A  /\  P  =/=  Q
)  /\  ( S  =/=  T  /\  -.  S  .<_  ( P  .\/  Q
)  /\  U  .<_  ( S  .\/  T ) ) )  ->  (
( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A ) )
2 simp2 995 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A
)  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  T  e.  A  /\  P  =/=  Q
)  /\  ( S  =/=  T  /\  -.  S  .<_  ( P  .\/  Q
)  /\  U  .<_  ( S  .\/  T ) ) )  ->  (
( S  e.  A  /\  -.  S  .<_  W )  /\  T  e.  A  /\  P  =/=  Q
) )
3 simp32 1031 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A
)  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  T  e.  A  /\  P  =/=  Q
)  /\  ( S  =/=  T  /\  -.  S  .<_  ( P  .\/  Q
)  /\  U  .<_  ( S  .\/  T ) ) )  ->  -.  S  .<_  ( P  .\/  Q ) )
4 simp33 1032 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A
)  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  T  e.  A  /\  P  =/=  Q
)  /\  ( S  =/=  T  /\  -.  S  .<_  ( P  .\/  Q
)  /\  U  .<_  ( S  .\/  T ) ) )  ->  U  .<_  ( S  .\/  T
) )
5 cdleme11.l . . . 4  |-  .<_  =  ( le `  K )
6 cdleme11.j . . . 4  |-  .\/  =  ( join `  K )
7 cdleme11.m . . . 4  |-  ./\  =  ( meet `  K )
8 cdleme11.a . . . 4  |-  A  =  ( Atoms `  K )
9 cdleme11.h . . . 4  |-  H  =  ( LHyp `  K
)
10 cdleme11.u . . . 4  |-  U  =  ( ( P  .\/  Q )  ./\  W )
115, 6, 7, 8, 9, 10cdleme11c 36383 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A
)  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  T  e.  A  /\  P  =/=  Q
)  /\  ( -.  S  .<_  ( P  .\/  Q )  /\  U  .<_  ( S  .\/  T ) ) )  ->  -.  P  .<_  ( S  .\/  T ) )
121, 2, 3, 4, 11syl112anc 1230 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A
)  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  T  e.  A  /\  P  =/=  Q
)  /\  ( S  =/=  T  /\  -.  S  .<_  ( P  .\/  Q
)  /\  U  .<_  ( S  .\/  T ) ) )  ->  -.  P  .<_  ( S  .\/  T ) )
13 simp11l 1105 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A
)  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  T  e.  A  /\  P  =/=  Q
)  /\  ( S  =/=  T  /\  -.  S  .<_  ( P  .\/  Q
)  /\  U  .<_  ( S  .\/  T ) ) )  ->  K  e.  HL )
14 simp12l 1107 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A
)  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  T  e.  A  /\  P  =/=  Q
)  /\  ( S  =/=  T  /\  -.  S  .<_  ( P  .\/  Q
)  /\  U  .<_  ( S  .\/  T ) ) )  ->  P  e.  A )
15 simp21l 1111 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A
)  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  T  e.  A  /\  P  =/=  Q
)  /\  ( S  =/=  T  /\  -.  S  .<_  ( P  .\/  Q
)  /\  U  .<_  ( S  .\/  T ) ) )  ->  S  e.  A )
165, 6, 8hlatlej2 35497 . . . . . 6  |-  ( ( K  e.  HL  /\  P  e.  A  /\  S  e.  A )  ->  S  .<_  ( P  .\/  S ) )
1713, 14, 15, 16syl3anc 1226 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A
)  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  T  e.  A  /\  P  =/=  Q
)  /\  ( S  =/=  T  /\  -.  S  .<_  ( P  .\/  Q
)  /\  U  .<_  ( S  .\/  T ) ) )  ->  S  .<_  ( P  .\/  S
) )
18 breq2 4443 . . . . 5  |-  ( ( P  .\/  S )  =  ( P  .\/  T )  ->  ( S  .<_  ( P  .\/  S
)  <->  S  .<_  ( P 
.\/  T ) ) )
1917, 18syl5ibcom 220 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A
)  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  T  e.  A  /\  P  =/=  Q
)  /\  ( S  =/=  T  /\  -.  S  .<_  ( P  .\/  Q
)  /\  U  .<_  ( S  .\/  T ) ) )  ->  (
( P  .\/  S
)  =  ( P 
.\/  T )  ->  S  .<_  ( P  .\/  T ) ) )
20 simp22 1028 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A
)  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  T  e.  A  /\  P  =/=  Q
)  /\  ( S  =/=  T  /\  -.  S  .<_  ( P  .\/  Q
)  /\  U  .<_  ( S  .\/  T ) ) )  ->  T  e.  A )
21 simp31 1030 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A
)  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  T  e.  A  /\  P  =/=  Q
)  /\  ( S  =/=  T  /\  -.  S  .<_  ( P  .\/  Q
)  /\  U  .<_  ( S  .\/  T ) ) )  ->  S  =/=  T )
225, 6, 8hlatexch2 35517 . . . . 5  |-  ( ( K  e.  HL  /\  ( S  e.  A  /\  P  e.  A  /\  T  e.  A
)  /\  S  =/=  T )  ->  ( S  .<_  ( P  .\/  T
)  ->  P  .<_  ( S  .\/  T ) ) )
2313, 15, 14, 20, 21, 22syl131anc 1239 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A
)  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  T  e.  A  /\  P  =/=  Q
)  /\  ( S  =/=  T  /\  -.  S  .<_  ( P  .\/  Q
)  /\  U  .<_  ( S  .\/  T ) ) )  ->  ( S  .<_  ( P  .\/  T )  ->  P  .<_  ( S  .\/  T ) ) )
2419, 23syld 44 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A
)  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  T  e.  A  /\  P  =/=  Q
)  /\  ( S  =/=  T  /\  -.  S  .<_  ( P  .\/  Q
)  /\  U  .<_  ( S  .\/  T ) ) )  ->  (
( P  .\/  S
)  =  ( P 
.\/  T )  ->  P  .<_  ( S  .\/  T ) ) )
2524necon3bd 2666 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A
)  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  T  e.  A  /\  P  =/=  Q
)  /\  ( S  =/=  T  /\  -.  S  .<_  ( P  .\/  Q
)  /\  U  .<_  ( S  .\/  T ) ) )  ->  ( -.  P  .<_  ( S 
.\/  T )  -> 
( P  .\/  S
)  =/=  ( P 
.\/  T ) ) )
2612, 25mpd 15 1  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A
)  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  T  e.  A  /\  P  =/=  Q
)  /\  ( S  =/=  T  /\  -.  S  .<_  ( P  .\/  Q
)  /\  U  .<_  ( S  .\/  T ) ) )  ->  ( P  .\/  S )  =/=  ( P  .\/  T
) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 367    /\ w3a 971    = wceq 1398    e. wcel 1823    =/= wne 2649   class class class wbr 4439   ` cfv 5570  (class class class)co 6270   lecple 14791   joincjn 15772   meetcmee 15773   Atomscatm 35385   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-iin 4318  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-mpt2 6275  df-1st 6773  df-2nd 6774  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-psubsp 35624  df-pmap 35625  df-padd 35917  df-lhyp 36109
This theorem is referenced by: (None)
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