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Theorem cdleme42c 30661
Description: Part of proof of Lemma E in [Crawley] p. 113. Match  -.  x  .<_  W. (Contributed by NM, 6-Mar-2013.)
Hypotheses
Ref Expression
cdleme42.b  |-  B  =  ( Base `  K
)
cdleme42.l  |-  .<_  =  ( le `  K )
cdleme42.j  |-  .\/  =  ( join `  K )
cdleme42.m  |-  ./\  =  ( meet `  K )
cdleme42.a  |-  A  =  ( Atoms `  K )
cdleme42.h  |-  H  =  ( LHyp `  K
)
cdleme42.v  |-  V  =  ( ( R  .\/  S )  ./\  W )
Assertion
Ref Expression
cdleme42c  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R  e.  A  /\  -.  R  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  ->  -.  ( R  .\/  V )  .<_  W )

Proof of Theorem cdleme42c
StepHypRef Expression
1 simp2r 982 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R  e.  A  /\  -.  R  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  ->  -.  R  .<_  W )
2 simp1l 979 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R  e.  A  /\  -.  R  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  ->  K  e.  HL )
3 hllat 29553 . . . . 5  |-  ( K  e.  HL  ->  K  e.  Lat )
42, 3syl 15 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R  e.  A  /\  -.  R  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  ->  K  e.  Lat )
5 simp2l 981 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R  e.  A  /\  -.  R  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  ->  R  e.  A )
6 cdleme42.b . . . . . 6  |-  B  =  ( Base `  K
)
7 cdleme42.a . . . . . 6  |-  A  =  ( Atoms `  K )
86, 7atbase 29479 . . . . 5  |-  ( R  e.  A  ->  R  e.  B )
95, 8syl 15 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R  e.  A  /\  -.  R  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  ->  R  e.  B )
10 cdleme42.v . . . . 5  |-  V  =  ( ( R  .\/  S )  ./\  W )
11 simp3l 983 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R  e.  A  /\  -.  R  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  ->  S  e.  A )
12 cdleme42.j . . . . . . . 8  |-  .\/  =  ( join `  K )
136, 12, 7hlatjcl 29556 . . . . . . 7  |-  ( ( K  e.  HL  /\  R  e.  A  /\  S  e.  A )  ->  ( R  .\/  S
)  e.  B )
142, 5, 11, 13syl3anc 1182 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R  e.  A  /\  -.  R  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  ->  ( R  .\/  S )  e.  B
)
15 simp1r 980 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R  e.  A  /\  -.  R  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  ->  W  e.  H )
16 cdleme42.h . . . . . . . 8  |-  H  =  ( LHyp `  K
)
176, 16lhpbase 30187 . . . . . . 7  |-  ( W  e.  H  ->  W  e.  B )
1815, 17syl 15 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R  e.  A  /\  -.  R  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  ->  W  e.  B )
19 cdleme42.m . . . . . . 7  |-  ./\  =  ( meet `  K )
206, 19latmcl 14157 . . . . . 6  |-  ( ( K  e.  Lat  /\  ( R  .\/  S )  e.  B  /\  W  e.  B )  ->  (
( R  .\/  S
)  ./\  W )  e.  B )
214, 14, 18, 20syl3anc 1182 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R  e.  A  /\  -.  R  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  ->  ( ( R  .\/  S )  ./\  W )  e.  B )
2210, 21syl5eqel 2367 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R  e.  A  /\  -.  R  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  ->  V  e.  B )
23 cdleme42.l . . . . 5  |-  .<_  =  ( le `  K )
246, 23, 12latjle12 14168 . . . 4  |-  ( ( K  e.  Lat  /\  ( R  e.  B  /\  V  e.  B  /\  W  e.  B
) )  ->  (
( R  .<_  W  /\  V  .<_  W )  <->  ( R  .\/  V )  .<_  W ) )
254, 9, 22, 18, 24syl13anc 1184 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R  e.  A  /\  -.  R  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  ->  ( ( R  .<_  W  /\  V  .<_  W )  <->  ( R  .\/  V )  .<_  W ) )
26 simpl 443 . . 3  |-  ( ( R  .<_  W  /\  V  .<_  W )  ->  R  .<_  W )
2725, 26syl6bir 220 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R  e.  A  /\  -.  R  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  ->  ( ( R  .\/  V )  .<_  W  ->  R  .<_  W ) )
281, 27mtod 168 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R  e.  A  /\  -.  R  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  ->  -.  ( R  .\/  V )  .<_  W )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   class class class wbr 4023   ` cfv 5255  (class class class)co 5858   Basecbs 13148   lecple 13215   joincjn 14078   meetcmee 14079   Latclat 14151   Atomscatm 29453   HLchlt 29540   LHypclh 30173
This theorem is referenced by:  cdleme42e  30668
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-undef 6298  df-riota 6304  df-poset 14080  df-lub 14108  df-join 14110  df-lat 14152  df-ats 29457  df-atl 29488  df-cvlat 29512  df-hlat 29541  df-lhyp 30177
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