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Theorem cdleme0fN 30854
Description: Part of proof of Lemma E in [Crawley] p. 113. (Contributed by NM, 14-Jun-2012.) (New usage is discouraged.)
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 )
cdleme0c.3  |-  V  =  ( ( P  .\/  R )  ./\  W )
Assertion
Ref Expression
cdleme0fN  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  R  e.  A )
)  ->  V  =/=  P )

Proof of Theorem cdleme0fN
StepHypRef Expression
1 cdleme0c.3 . . 3  |-  V  =  ( ( P  .\/  R )  ./\  W )
2 simp1l 981 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  R  e.  A )
)  ->  K  e.  HL )
3 hllat 30000 . . . . 5  |-  ( K  e.  HL  ->  K  e.  Lat )
42, 3syl 16 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  R  e.  A )
)  ->  K  e.  Lat )
5 simp2l 983 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  R  e.  A )
)  ->  P  e.  A )
6 eqid 2435 . . . . . . 7  |-  ( Base `  K )  =  (
Base `  K )
7 cdleme0.a . . . . . . 7  |-  A  =  ( Atoms `  K )
86, 7atbase 29926 . . . . . 6  |-  ( P  e.  A  ->  P  e.  ( Base `  K
) )
95, 8syl 16 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  R  e.  A )
)  ->  P  e.  ( Base `  K )
)
10 simp3r 986 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  R  e.  A )
)  ->  R  e.  A )
116, 7atbase 29926 . . . . . 6  |-  ( R  e.  A  ->  R  e.  ( Base `  K
) )
1210, 11syl 16 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  R  e.  A )
)  ->  R  e.  ( Base `  K )
)
13 cdleme0.j . . . . . 6  |-  .\/  =  ( join `  K )
146, 13latjcl 14467 . . . . 5  |-  ( ( K  e.  Lat  /\  P  e.  ( Base `  K )  /\  R  e.  ( Base `  K
) )  ->  ( P  .\/  R )  e.  ( Base `  K
) )
154, 9, 12, 14syl3anc 1184 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  R  e.  A )
)  ->  ( P  .\/  R )  e.  (
Base `  K )
)
16 simp1r 982 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  R  e.  A )
)  ->  W  e.  H )
17 cdleme0.h . . . . . 6  |-  H  =  ( LHyp `  K
)
186, 17lhpbase 30634 . . . . 5  |-  ( W  e.  H  ->  W  e.  ( Base `  K
) )
1916, 18syl 16 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  R  e.  A )
)  ->  W  e.  ( Base `  K )
)
20 cdleme0.l . . . . 5  |-  .<_  =  ( le `  K )
21 cdleme0.m . . . . 5  |-  ./\  =  ( meet `  K )
226, 20, 21latmle2 14494 . . . 4  |-  ( ( K  e.  Lat  /\  ( P  .\/  R )  e.  ( Base `  K
)  /\  W  e.  ( Base `  K )
)  ->  ( ( P  .\/  R )  ./\  W )  .<_  W )
234, 15, 19, 22syl3anc 1184 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  R  e.  A )
)  ->  ( ( P  .\/  R )  ./\  W )  .<_  W )
241, 23syl5eqbr 4237 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  R  e.  A )
)  ->  V  .<_  W )
25 simp2r 984 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  R  e.  A )
)  ->  -.  P  .<_  W )
26 nbrne2 4222 . 2  |-  ( ( V  .<_  W  /\  -.  P  .<_  W )  ->  V  =/=  P
)
2724, 25, 26syl2anc 643 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  R  e.  A )
)  ->  V  =/=  P )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725    =/= wne 2598   class class class wbr 4204   ` cfv 5445  (class class class)co 6072   Basecbs 13457   lecple 13524   joincjn 14389   meetcmee 14390   Latclat 14462   Atomscatm 29900   HLchlt 29987   LHypclh 30620
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4692
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4875  df-rel 4876  df-cnv 4877  df-co 4878  df-dm 4879  df-rn 4880  df-res 4881  df-ima 4882  df-iota 5409  df-fun 5447  df-fn 5448  df-f 5449  df-f1 5450  df-fo 5451  df-f1o 5452  df-fv 5453  df-ov 6075  df-oprab 6076  df-mpt2 6077  df-1st 6340  df-2nd 6341  df-undef 6534  df-riota 6540  df-glb 14420  df-meet 14422  df-lat 14463  df-ats 29904  df-atl 29935  df-cvlat 29959  df-hlat 29988  df-lhyp 30624
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