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Theorem cdleme3c 31041
Description: Part of proof of Lemma E in [Crawley] p. 113. Lemma leading to cdleme3fa 31047 and cdleme3 31048. (Contributed by NM, 6-Jun-2012.)
Hypotheses
Ref Expression
cdleme1.l  |-  .<_  =  ( le `  K )
cdleme1.j  |-  .\/  =  ( join `  K )
cdleme1.m  |-  ./\  =  ( meet `  K )
cdleme1.a  |-  A  =  ( Atoms `  K )
cdleme1.h  |-  H  =  ( LHyp `  K
)
cdleme1.u  |-  U  =  ( ( P  .\/  Q )  ./\  W )
cdleme1.f  |-  F  =  ( ( R  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  R )  ./\  W )
) )
cdleme3c.z  |-  .0.  =  ( 0. `  K )
Assertion
Ref Expression
cdleme3c  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  P  =/=  Q
)  /\  ( R  e.  A  /\  -.  R  .<_  W ) ) )  ->  F  =/=  .0.  )

Proof of Theorem cdleme3c
StepHypRef Expression
1 simpll 730 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  P  =/=  Q
)  /\  ( R  e.  A  /\  -.  R  .<_  W ) ) )  ->  K  e.  HL )
2 hllat 30175 . . . . . 6  |-  ( K  e.  HL  ->  K  e.  Lat )
32ad2antrr 706 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  P  =/=  Q
)  /\  ( R  e.  A  /\  -.  R  .<_  W ) ) )  ->  K  e.  Lat )
4 simpr3l 1016 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  P  =/=  Q
)  /\  ( R  e.  A  /\  -.  R  .<_  W ) ) )  ->  R  e.  A
)
5 eqid 2296 . . . . . . 7  |-  ( Base `  K )  =  (
Base `  K )
6 cdleme1.a . . . . . . 7  |-  A  =  ( Atoms `  K )
75, 6atbase 30101 . . . . . 6  |-  ( R  e.  A  ->  R  e.  ( Base `  K
) )
84, 7syl 15 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  P  =/=  Q
)  /\  ( R  e.  A  /\  -.  R  .<_  W ) ) )  ->  R  e.  (
Base `  K )
)
9 hlop 30174 . . . . . . 7  |-  ( K  e.  HL  ->  K  e.  OP )
109ad2antrr 706 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  P  =/=  Q
)  /\  ( R  e.  A  /\  -.  R  .<_  W ) ) )  ->  K  e.  OP )
11 cdleme3c.z . . . . . . 7  |-  .0.  =  ( 0. `  K )
125, 11op0cl 29996 . . . . . 6  |-  ( K  e.  OP  ->  .0.  e.  ( Base `  K
) )
1310, 12syl 15 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  P  =/=  Q
)  /\  ( R  e.  A  /\  -.  R  .<_  W ) ) )  ->  .0.  e.  ( Base `  K ) )
14 cdleme1.j . . . . . 6  |-  .\/  =  ( join `  K )
155, 14latjcl 14172 . . . . 5  |-  ( ( K  e.  Lat  /\  R  e.  ( Base `  K )  /\  .0.  e.  ( Base `  K
) )  ->  ( R  .\/  .0.  )  e.  ( Base `  K
) )
163, 8, 13, 15syl3anc 1182 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  P  =/=  Q
)  /\  ( R  e.  A  /\  -.  R  .<_  W ) ) )  ->  ( R  .\/  .0.  )  e.  ( Base `  K ) )
17 simpl 443 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  P  =/=  Q
)  /\  ( R  e.  A  /\  -.  R  .<_  W ) ) )  ->  ( K  e.  HL  /\  W  e.  H ) )
18 simpr1l 1012 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  P  =/=  Q
)  /\  ( R  e.  A  /\  -.  R  .<_  W ) ) )  ->  P  e.  A
)
19 simpr2l 1014 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  P  =/=  Q
)  /\  ( R  e.  A  /\  -.  R  .<_  W ) ) )  ->  Q  e.  A
)
20 cdleme1.l . . . . . . 7  |-  .<_  =  ( le `  K )
21 cdleme1.m . . . . . . 7  |-  ./\  =  ( meet `  K )
22 cdleme1.h . . . . . . 7  |-  H  =  ( LHyp `  K
)
23 cdleme1.u . . . . . . 7  |-  U  =  ( ( P  .\/  Q )  ./\  W )
24 cdleme1.f . . . . . . 7  |-  F  =  ( ( R  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  R )  ./\  W )
) )
2520, 14, 21, 6, 22, 23, 24, 5cdleme1b 31037 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A ) )  ->  F  e.  ( Base `  K ) )
2617, 18, 19, 4, 25syl13anc 1184 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  P  =/=  Q
)  /\  ( R  e.  A  /\  -.  R  .<_  W ) ) )  ->  F  e.  (
Base `  K )
)
275, 14latjcl 14172 . . . . 5  |-  ( ( K  e.  Lat  /\  R  e.  ( Base `  K )  /\  F  e.  ( Base `  K
) )  ->  ( R  .\/  F )  e.  ( Base `  K
) )
283, 8, 26, 27syl3anc 1182 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  P  =/=  Q
)  /\  ( R  e.  A  /\  -.  R  .<_  W ) ) )  ->  ( R  .\/  F )  e.  ( Base `  K ) )
295, 6atbase 30101 . . . . . . . . . . . 12  |-  ( P  e.  A  ->  P  e.  ( Base `  K
) )
3018, 29syl 15 . . . . . . . . . . 11  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  P  =/=  Q
)  /\  ( R  e.  A  /\  -.  R  .<_  W ) ) )  ->  P  e.  (
Base `  K )
)
315, 6atbase 30101 . . . . . . . . . . . 12  |-  ( Q  e.  A  ->  Q  e.  ( Base `  K
) )
3219, 31syl 15 . . . . . . . . . . 11  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  P  =/=  Q
)  /\  ( R  e.  A  /\  -.  R  .<_  W ) ) )  ->  Q  e.  (
Base `  K )
)
335, 14latjcl 14172 . . . . . . . . . . 11  |-  ( ( K  e.  Lat  /\  P  e.  ( Base `  K )  /\  Q  e.  ( Base `  K
) )  ->  ( P  .\/  Q )  e.  ( Base `  K
) )
343, 30, 32, 33syl3anc 1182 . . . . . . . . . 10  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  P  =/=  Q
)  /\  ( R  e.  A  /\  -.  R  .<_  W ) ) )  ->  ( P  .\/  Q )  e.  ( Base `  K ) )
355, 22lhpbase 30809 . . . . . . . . . . 11  |-  ( W  e.  H  ->  W  e.  ( Base `  K
) )
3635ad2antlr 707 . . . . . . . . . 10  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  P  =/=  Q
)  /\  ( R  e.  A  /\  -.  R  .<_  W ) ) )  ->  W  e.  (
Base `  K )
)
375, 20, 21latmle2 14199 . . . . . . . . . 10  |-  ( ( K  e.  Lat  /\  ( P  .\/  Q )  e.  ( Base `  K
)  /\  W  e.  ( Base `  K )
)  ->  ( ( P  .\/  Q )  ./\  W )  .<_  W )
383, 34, 36, 37syl3anc 1182 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  P  =/=  Q
)  /\  ( R  e.  A  /\  -.  R  .<_  W ) ) )  ->  ( ( P 
.\/  Q )  ./\  W )  .<_  W )
3923, 38syl5eqbr 4072 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  P  =/=  Q
)  /\  ( R  e.  A  /\  -.  R  .<_  W ) ) )  ->  U  .<_  W )
40 simpr3r 1017 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  P  =/=  Q
)  /\  ( R  e.  A  /\  -.  R  .<_  W ) ) )  ->  -.  R  .<_  W )
41 nbrne2 4057 . . . . . . . 8  |-  ( ( U  .<_  W  /\  -.  R  .<_  W )  ->  U  =/=  R
)
4239, 40, 41syl2anc 642 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  P  =/=  Q
)  /\  ( R  e.  A  /\  -.  R  .<_  W ) ) )  ->  U  =/=  R
)
4342necomd 2542 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  P  =/=  Q
)  /\  ( R  e.  A  /\  -.  R  .<_  W ) ) )  ->  R  =/=  U
)
4420, 14, 21, 6, 22, 23lhpat2 30856 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  P  =/=  Q ) )  ->  U  e.  A
)
45443adant3r3 1162 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  P  =/=  Q
)  /\  ( R  e.  A  /\  -.  R  .<_  W ) ) )  ->  U  e.  A
)
46 eqid 2296 . . . . . . . 8  |-  (  <o  `  K )  =  ( 
<o  `  K )
4714, 46, 6atcvr1 30228 . . . . . . 7  |-  ( ( K  e.  HL  /\  R  e.  A  /\  U  e.  A )  ->  ( R  =/=  U  <->  R (  <o  `  K )
( R  .\/  U
) ) )
481, 4, 45, 47syl3anc 1182 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  P  =/=  Q
)  /\  ( R  e.  A  /\  -.  R  .<_  W ) ) )  ->  ( R  =/= 
U  <->  R (  <o  `  K
) ( R  .\/  U ) ) )
4943, 48mpbid 201 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  P  =/=  Q
)  /\  ( R  e.  A  /\  -.  R  .<_  W ) ) )  ->  R (  <o  `  K ) ( R 
.\/  U ) )
50 hlol 30173 . . . . . . 7  |-  ( K  e.  HL  ->  K  e.  OL )
5150ad2antrr 706 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  P  =/=  Q
)  /\  ( R  e.  A  /\  -.  R  .<_  W ) ) )  ->  K  e.  OL )
525, 14, 11olj01 30037 . . . . . 6  |-  ( ( K  e.  OL  /\  R  e.  ( Base `  K ) )  -> 
( R  .\/  .0.  )  =  R )
5351, 8, 52syl2anc 642 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  P  =/=  Q
)  /\  ( R  e.  A  /\  -.  R  .<_  W ) ) )  ->  ( R  .\/  .0.  )  =  R
)
54 simpr3 963 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  P  =/=  Q
)  /\  ( R  e.  A  /\  -.  R  .<_  W ) ) )  ->  ( R  e.  A  /\  -.  R  .<_  W ) )
5520, 14, 21, 6, 22, 23, 24cdleme1 31038 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A  /\  ( R  e.  A  /\  -.  R  .<_  W ) ) )  ->  ( R  .\/  F )  =  ( R  .\/  U ) )
5617, 18, 19, 54, 55syl13anc 1184 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  P  =/=  Q
)  /\  ( R  e.  A  /\  -.  R  .<_  W ) ) )  ->  ( R  .\/  F )  =  ( R 
.\/  U ) )
5749, 53, 563brtr4d 4069 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  P  =/=  Q
)  /\  ( R  e.  A  /\  -.  R  .<_  W ) ) )  ->  ( R  .\/  .0.  ) (  <o  `  K
) ( R  .\/  F ) )
585, 46cvrne 30093 . . . 4  |-  ( ( ( K  e.  HL  /\  ( R  .\/  .0.  )  e.  ( Base `  K )  /\  ( R  .\/  F )  e.  ( Base `  K
) )  /\  ( R  .\/  .0.  ) ( 
<o  `  K ) ( R  .\/  F ) )  ->  ( R  .\/  .0.  )  =/=  ( R  .\/  F ) )
591, 16, 28, 57, 58syl31anc 1185 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  P  =/=  Q
)  /\  ( R  e.  A  /\  -.  R  .<_  W ) ) )  ->  ( R  .\/  .0.  )  =/=  ( R  .\/  F ) )
60 oveq2 5882 . . . 4  |-  (  .0.  =  F  ->  ( R  .\/  .0.  )  =  ( R  .\/  F
) )
6160necon3i 2498 . . 3  |-  ( ( R  .\/  .0.  )  =/=  ( R  .\/  F
)  ->  .0.  =/=  F )
6259, 61syl 15 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  P  =/=  Q
)  /\  ( R  e.  A  /\  -.  R  .<_  W ) ) )  ->  .0.  =/=  F
)
6362necomd 2542 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  P  =/=  Q
)  /\  ( R  e.  A  /\  -.  R  .<_  W ) ) )  ->  F  =/=  .0.  )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696    =/= wne 2459   class class class wbr 4039   ` cfv 5271  (class class class)co 5874   Basecbs 13164   lecple 13231   joincjn 14094   meetcmee 14095   0.cp0 14159   Latclat 14167   OPcops 29984   OLcol 29986    <o ccvr 30074   Atomscatm 30075   HLchlt 30162   LHypclh 30795
This theorem is referenced by:  cdleme3h  31046
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-iin 3924  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-undef 6314  df-riota 6320  df-poset 14096  df-plt 14108  df-lub 14124  df-glb 14125  df-join 14126  df-meet 14127  df-p0 14161  df-p1 14162  df-lat 14168  df-clat 14230  df-oposet 29988  df-ol 29990  df-oml 29991  df-covers 30078  df-ats 30079  df-atl 30110  df-cvlat 30134  df-hlat 30163  df-psubsp 30314  df-pmap 30315  df-padd 30607  df-lhyp 30799
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