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Theorem cdleme0b 33490
Description: Part of proof of Lemma E in [Crawley] p. 113. (Contributed by NM, 13-Jun-2012.)
Hypotheses
Ref Expression
cdleme0.l  |-  .<_  =  ( le `  K )
cdleme0.j  |-  .\/  =  ( join `  K )
cdleme0.m  |-  ./\  =  ( meet `  K )
cdleme0.a  |-  A  =  ( Atoms `  K )
cdleme0.h  |-  H  =  ( LHyp `  K
)
cdleme0.u  |-  U  =  ( ( P  .\/  Q )  ./\  W )
Assertion
Ref Expression
cdleme0b  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A )  ->  U  =/=  P )

Proof of Theorem cdleme0b
StepHypRef Expression
1 cdleme0.u . . 3  |-  U  =  ( ( P  .\/  Q )  ./\  W )
2 simp1l 1029 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A )  ->  K  e.  HL )
3 hllat 32641 . . . . 5  |-  ( K  e.  HL  ->  K  e.  Lat )
42, 3syl 17 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A )  ->  K  e.  Lat )
5 simp2l 1031 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A )  ->  P  e.  A )
6 eqid 2429 . . . . . . 7  |-  ( Base `  K )  =  (
Base `  K )
7 cdleme0.a . . . . . . 7  |-  A  =  ( Atoms `  K )
86, 7atbase 32567 . . . . . 6  |-  ( P  e.  A  ->  P  e.  ( Base `  K
) )
95, 8syl 17 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A )  ->  P  e.  ( Base `  K
) )
106, 7atbase 32567 . . . . . 6  |-  ( Q  e.  A  ->  Q  e.  ( Base `  K
) )
11103ad2ant3 1028 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A )  ->  Q  e.  ( Base `  K
) )
12 cdleme0.j . . . . . 6  |-  .\/  =  ( join `  K )
136, 12latjcl 16248 . . . . 5  |-  ( ( K  e.  Lat  /\  P  e.  ( Base `  K )  /\  Q  e.  ( Base `  K
) )  ->  ( P  .\/  Q )  e.  ( Base `  K
) )
144, 9, 11, 13syl3anc 1264 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A )  ->  ( P  .\/  Q )  e.  ( Base `  K
) )
15 simp1r 1030 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A )  ->  W  e.  H )
16 cdleme0.h . . . . . 6  |-  H  =  ( LHyp `  K
)
176, 16lhpbase 33275 . . . . 5  |-  ( W  e.  H  ->  W  e.  ( Base `  K
) )
1815, 17syl 17 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A )  ->  W  e.  ( Base `  K
) )
19 cdleme0.l . . . . 5  |-  .<_  =  ( le `  K )
20 cdleme0.m . . . . 5  |-  ./\  =  ( meet `  K )
216, 19, 20latmle2 16274 . . . 4  |-  ( ( K  e.  Lat  /\  ( P  .\/  Q )  e.  ( Base `  K
)  /\  W  e.  ( Base `  K )
)  ->  ( ( P  .\/  Q )  ./\  W )  .<_  W )
224, 14, 18, 21syl3anc 1264 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A )  ->  (
( P  .\/  Q
)  ./\  W )  .<_  W )
231, 22syl5eqbr 4459 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A )  ->  U  .<_  W )
24 simp2r 1032 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A )  ->  -.  P  .<_  W )
25 nbrne2 4444 . 2  |-  ( ( U  .<_  W  /\  -.  P  .<_  W )  ->  U  =/=  P
)
2623, 24, 25syl2anc 665 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A )  ->  U  =/=  P )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1870    =/= wne 2625   class class class wbr 4426   ` cfv 5601  (class class class)co 6305   Basecbs 15084   lecple 15159   joincjn 16140   meetcmee 16141   Latclat 16242   Atomscatm 32541   HLchlt 32628   LHypclh 33261
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-reu 2789  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-id 4769  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6267  df-ov 6308  df-oprab 6309  df-lub 16171  df-glb 16172  df-join 16173  df-meet 16174  df-lat 16243  df-ats 32545  df-atl 32576  df-cvlat 32600  df-hlat 32629  df-lhyp 33265
This theorem is referenced by:  cdleme11c  33539
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