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Theorem cdleme42c 29350
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 987 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R  e.  A  /\  -.  R  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  ->  -.  R  .<_  W )
2 simp1l 984 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R  e.  A  /\  -.  R  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  ->  K  e.  HL )
3 hllat 28242 . . . . 5  |-  ( K  e.  HL  ->  K  e.  Lat )
42, 3syl 17 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R  e.  A  /\  -.  R  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  ->  K  e.  Lat )
5 simp2l 986 . . . . 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 28168 . . . . 5  |-  ( R  e.  A  ->  R  e.  B )
95, 8syl 17 . . . 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 988 . . . . . . 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 28245 . . . . . . 7  |-  ( ( K  e.  HL  /\  R  e.  A  /\  S  e.  A )  ->  ( R  .\/  S
)  e.  B )
142, 5, 11, 13syl3anc 1187 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R  e.  A  /\  -.  R  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  ->  ( R  .\/  S )  e.  B
)
15 simp1r 985 . . . . . . 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 28876 . . . . . . 7  |-  ( W  e.  H  ->  W  e.  B )
1815, 17syl 17 . . . . . 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 14001 . . . . . 6  |-  ( ( K  e.  Lat  /\  ( R  .\/  S )  e.  B  /\  W  e.  B )  ->  (
( R  .\/  S
)  ./\  W )  e.  B )
214, 14, 18, 20syl3anc 1187 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R  e.  A  /\  -.  R  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  ->  ( ( R  .\/  S )  ./\  W )  e.  B )
2210, 21syl5eqel 2337 . . . 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 14012 . . . 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 1189 . . 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 445 . . 3  |-  ( ( R  .<_  W  /\  V  .<_  W )  ->  R  .<_  W )
2725, 26syl6bir 222 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R  e.  A  /\  -.  R  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  ->  ( ( R  .\/  V )  .<_  W  ->  R  .<_  W ) )
281, 27mtod 170 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 5    -> wi 6    <-> wb 178    /\ wa 360    /\ w3a 939    = wceq 1619    e. wcel 1621   class class class wbr 3920   ` cfv 4592  (class class class)co 5710   Basecbs 13022   lecple 13089   joincjn 13922   meetcmee 13923   Latclat 13995   Atomscatm 28142   HLchlt 28229   LHypclh 28862
This theorem is referenced by:  cdleme42e  29357
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234  ax-rep 4028  ax-sep 4038  ax-nul 4046  ax-pow 4082  ax-pr 4108  ax-un 4403
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-nel 2415  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2516  df-v 2729  df-sbc 2922  df-csb 3010  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-nul 3363  df-if 3471  df-pw 3532  df-sn 3550  df-pr 3551  df-op 3553  df-uni 3728  df-iun 3805  df-br 3921  df-opab 3975  df-mpt 3976  df-id 4202  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 4602  df-fn 4603  df-f 4604  df-f1 4605  df-fo 4606  df-f1o 4607  df-fv 4608  df-ov 5713  df-oprab 5714  df-mpt2 5715  df-1st 5974  df-2nd 5975  df-iota 6143  df-undef 6182  df-riota 6190  df-poset 13924  df-lub 13952  df-join 13954  df-lat 13996  df-ats 28146  df-atl 28177  df-cvlat 28201  df-hlat 28230  df-lhyp 28866
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