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Theorem cdleme0b 30474
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 979 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A )  ->  K  e.  HL )
3 hllat 29626 . . . . 5  |-  ( K  e.  HL  ->  K  e.  Lat )
42, 3syl 15 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A )  ->  K  e.  Lat )
5 simp2l 981 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A )  ->  P  e.  A )
6 eqid 2285 . . . . . . 7  |-  ( Base `  K )  =  (
Base `  K )
7 cdleme0.a . . . . . . 7  |-  A  =  ( Atoms `  K )
86, 7atbase 29552 . . . . . 6  |-  ( P  e.  A  ->  P  e.  ( Base `  K
) )
95, 8syl 15 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A )  ->  P  e.  ( Base `  K
) )
106, 7atbase 29552 . . . . . 6  |-  ( Q  e.  A  ->  Q  e.  ( Base `  K
) )
11103ad2ant3 978 . . . . 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 14158 . . . . 5  |-  ( ( K  e.  Lat  /\  P  e.  ( Base `  K )  /\  Q  e.  ( Base `  K
) )  ->  ( P  .\/  Q )  e.  ( Base `  K
) )
144, 9, 11, 13syl3anc 1182 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A )  ->  ( P  .\/  Q )  e.  ( Base `  K
) )
15 simp1r 980 . . . . 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 30260 . . . . 5  |-  ( W  e.  H  ->  W  e.  ( Base `  K
) )
1815, 17syl 15 . . . 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 14185 . . . 4  |-  ( ( K  e.  Lat  /\  ( P  .\/  Q )  e.  ( Base `  K
)  /\  W  e.  ( Base `  K )
)  ->  ( ( P  .\/  Q )  ./\  W )  .<_  W )
224, 14, 18, 21syl3anc 1182 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A )  ->  (
( P  .\/  Q
)  ./\  W )  .<_  W )
231, 22syl5eqbr 4058 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A )  ->  U  .<_  W )
24 simp2r 982 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A )  ->  -.  P  .<_  W )
25 nbrne2 4043 . 2  |-  ( ( U  .<_  W  /\  -.  P  .<_  W )  ->  U  =/=  P
)
2623, 24, 25syl2anc 642 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A )  ->  U  =/=  P )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1625    e. wcel 1686    =/= wne 2448   class class class wbr 4025   ` cfv 5257  (class class class)co 5860   Basecbs 13150   lecple 13217   joincjn 14080   meetcmee 14081   Latclat 14153   Atomscatm 29526   HLchlt 29613   LHypclh 30246
This theorem is referenced by:  cdleme11c  30523
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-13 1688  ax-14 1690  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266  ax-rep 4133  ax-sep 4143  ax-nul 4151  ax-pow 4190  ax-pr 4216  ax-un 4514
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1531  df-nf 1534  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-nel 2451  df-ral 2550  df-rex 2551  df-reu 2552  df-rab 2554  df-v 2792  df-sbc 2994  df-csb 3084  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-nul 3458  df-if 3568  df-pw 3629  df-sn 3648  df-pr 3649  df-op 3651  df-uni 3830  df-iun 3909  df-br 4026  df-opab 4080  df-mpt 4081  df-id 4311  df-xp 4697  df-rel 4698  df-cnv 4699  df-co 4700  df-dm 4701  df-rn 4702  df-res 4703  df-ima 4704  df-iota 5221  df-fun 5259  df-fn 5260  df-f 5261  df-f1 5262  df-fo 5263  df-f1o 5264  df-fv 5265  df-ov 5863  df-oprab 5864  df-mpt2 5865  df-1st 6124  df-2nd 6125  df-undef 6300  df-riota 6306  df-glb 14111  df-meet 14113  df-lat 14154  df-ats 29530  df-atl 29561  df-cvlat 29585  df-hlat 29614  df-lhyp 30250
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