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Theorem cdleme3c 30866
Description: Part of proof of Lemma E in [Crawley] p. 113. Lemma leading to cdleme3fa 30872 and cdleme3 30873. (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 731 . . . 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 30000 . . . . . 6  |-  ( K  e.  HL  ->  K  e.  Lat )
32ad2antrr 707 . . . . 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 1018 . . . . . 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 2435 . . . . . . 7  |-  ( Base `  K )  =  (
Base `  K )
6 cdleme1.a . . . . . . 7  |-  A  =  ( Atoms `  K )
75, 6atbase 29926 . . . . . 6  |-  ( R  e.  A  ->  R  e.  ( Base `  K
) )
84, 7syl 16 . . . . 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 29999 . . . . . . 7  |-  ( K  e.  HL  ->  K  e.  OP )
109ad2antrr 707 . . . . . 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 29821 . . . . . 6  |-  ( K  e.  OP  ->  .0.  e.  ( Base `  K
) )
1310, 12syl 16 . . . . 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 14467 . . . . 5  |-  ( ( K  e.  Lat  /\  R  e.  ( Base `  K )  /\  .0.  e.  ( Base `  K
) )  ->  ( R  .\/  .0.  )  e.  ( Base `  K
) )
163, 8, 13, 15syl3anc 1184 . . . 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 444 . . . . . 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 1014 . . . . . 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 1016 . . . . . 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 30862 . . . . . 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 1186 . . . . 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 14467 . . . . 5  |-  ( ( K  e.  Lat  /\  R  e.  ( Base `  K )  /\  F  e.  ( Base `  K
) )  ->  ( R  .\/  F )  e.  ( Base `  K
) )
283, 8, 26, 27syl3anc 1184 . . . 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 29926 . . . . . . . . . . . 12  |-  ( P  e.  A  ->  P  e.  ( Base `  K
) )
3018, 29syl 16 . . . . . . . . . . 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 29926 . . . . . . . . . . . 12  |-  ( Q  e.  A  ->  Q  e.  ( Base `  K
) )
3219, 31syl 16 . . . . . . . . . . 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 14467 . . . . . . . . . . 11  |-  ( ( K  e.  Lat  /\  P  e.  ( Base `  K )  /\  Q  e.  ( Base `  K
) )  ->  ( P  .\/  Q )  e.  ( Base `  K
) )
343, 30, 32, 33syl3anc 1184 . . . . . . . . . 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 30634 . . . . . . . . . . 11  |-  ( W  e.  H  ->  W  e.  ( Base `  K
) )
3635ad2antlr 708 . . . . . . . . . 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 14494 . . . . . . . . . 10  |-  ( ( K  e.  Lat  /\  ( P  .\/  Q )  e.  ( Base `  K
)  /\  W  e.  ( Base `  K )
)  ->  ( ( P  .\/  Q )  ./\  W )  .<_  W )
383, 34, 36, 37syl3anc 1184 . . . . . . . . 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 4237 . . . . . . . 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 1019 . . . . . . . 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 4222 . . . . . . . 8  |-  ( ( U  .<_  W  /\  -.  R  .<_  W )  ->  U  =/=  R
)
4239, 40, 41syl2anc 643 . . . . . . 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 2681 . . . . . 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 30681 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  P  =/=  Q ) )  ->  U  e.  A
)
45443adant3r3 1164 . . . . . . 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 2435 . . . . . . . 8  |-  (  <o  `  K )  =  ( 
<o  `  K )
4714, 46, 6atcvr1 30053 . . . . . . 7  |-  ( ( K  e.  HL  /\  R  e.  A  /\  U  e.  A )  ->  ( R  =/=  U  <->  R (  <o  `  K )
( R  .\/  U
) ) )
481, 4, 45, 47syl3anc 1184 . . . . . 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 202 . . . . 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 29998 . . . . . . 7  |-  ( K  e.  HL  ->  K  e.  OL )
5150ad2antrr 707 . . . . . 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 29862 . . . . . 6  |-  ( ( K  e.  OL  /\  R  e.  ( Base `  K ) )  -> 
( R  .\/  .0.  )  =  R )
5351, 8, 52syl2anc 643 . . . . 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 965 . . . . . 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 30863 . . . . . 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 1186 . . . . 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 4234 . . . 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 29918 . . . 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 1187 . . 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 6080 . . . 4  |-  (  .0.  =  F  ->  ( R  .\/  .0.  )  =  ( R  .\/  F
) )
6160necon3i 2637 . . 3  |-  ( ( R  .\/  .0.  )  =/=  ( R  .\/  F
)  ->  .0.  =/=  F )
6259, 61syl 16 . 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 2681 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 177    /\ 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   0.cp0 14454   Latclat 14462   OPcops 29809   OLcol 29811    <o ccvr 29899   Atomscatm 29900   HLchlt 29987   LHypclh 30620
This theorem is referenced by:  cdleme3h  30871
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-iin 4088  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-poset 14391  df-plt 14403  df-lub 14419  df-glb 14420  df-join 14421  df-meet 14422  df-p0 14456  df-p1 14457  df-lat 14463  df-clat 14525  df-oposet 29813  df-ol 29815  df-oml 29816  df-covers 29903  df-ats 29904  df-atl 29935  df-cvlat 29959  df-hlat 29988  df-psubsp 30139  df-pmap 30140  df-padd 30432  df-lhyp 30624
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