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Theorem cdleme3c 29108
Description: Part of proof of Lemma E in [Crawley] p. 113. Lemma leading to cdleme3fa 29114 and cdleme3 29115. (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 733 . . . 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 28242 . . . . . 6  |-  ( K  e.  HL  ->  K  e.  Lat )
32ad2antrr 709 . . . . 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 1021 . . . . . 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 2253 . . . . . . 7  |-  ( Base `  K )  =  (
Base `  K )
6 cdleme1.a . . . . . . 7  |-  A  =  ( Atoms `  K )
75, 6atbase 28168 . . . . . 6  |-  ( R  e.  A  ->  R  e.  ( Base `  K
) )
84, 7syl 17 . . . . 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 28241 . . . . . . 7  |-  ( K  e.  HL  ->  K  e.  OP )
109ad2antrr 709 . . . . . 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 28063 . . . . . 6  |-  ( K  e.  OP  ->  .0.  e.  ( Base `  K
) )
1310, 12syl 17 . . . . 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 14000 . . . . 5  |-  ( ( K  e.  Lat  /\  R  e.  ( Base `  K )  /\  .0.  e.  ( Base `  K
) )  ->  ( R  .\/  .0.  )  e.  ( Base `  K
) )
163, 8, 13, 15syl3anc 1187 . . . 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 445 . . . . . 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 1017 . . . . . 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 1019 . . . . . 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 29104 . . . . . 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 1189 . . . . 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 14000 . . . . 5  |-  ( ( K  e.  Lat  /\  R  e.  ( Base `  K )  /\  F  e.  ( Base `  K
) )  ->  ( R  .\/  F )  e.  ( Base `  K
) )
283, 8, 26, 27syl3anc 1187 . . . 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 28168 . . . . . . . . . . . 12  |-  ( P  e.  A  ->  P  e.  ( Base `  K
) )
3018, 29syl 17 . . . . . . . . . . 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 28168 . . . . . . . . . . . 12  |-  ( Q  e.  A  ->  Q  e.  ( Base `  K
) )
3219, 31syl 17 . . . . . . . . . . 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 14000 . . . . . . . . . . 11  |-  ( ( K  e.  Lat  /\  P  e.  ( Base `  K )  /\  Q  e.  ( Base `  K
) )  ->  ( P  .\/  Q )  e.  ( Base `  K
) )
343, 30, 32, 33syl3anc 1187 . . . . . . . . . 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 28876 . . . . . . . . . . 11  |-  ( W  e.  H  ->  W  e.  ( Base `  K
) )
3635ad2antlr 710 . . . . . . . . . 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 14027 . . . . . . . . . 10  |-  ( ( K  e.  Lat  /\  ( P  .\/  Q )  e.  ( Base `  K
)  /\  W  e.  ( Base `  K )
)  ->  ( ( P  .\/  Q )  ./\  W )  .<_  W )
383, 34, 36, 37syl3anc 1187 . . . . . . . . 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 3953 . . . . . . . 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 1022 . . . . . . . 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 3938 . . . . . . . 8  |-  ( ( U  .<_  W  /\  -.  R  .<_  W )  ->  U  =/=  R
)
4239, 40, 41syl2anc 645 . . . . . . 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 2495 . . . . . 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 28923 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  P  =/=  Q ) )  ->  U  e.  A
)
45443adant3r3 1167 . . . . . . 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 2253 . . . . . . . 8  |-  (  <o  `  K )  =  ( 
<o  `  K )
4714, 46, 6atcvr1 28295 . . . . . . 7  |-  ( ( K  e.  HL  /\  R  e.  A  /\  U  e.  A )  ->  ( R  =/=  U  <->  R (  <o  `  K )
( R  .\/  U
) ) )
481, 4, 45, 47syl3anc 1187 . . . . . 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 203 . . . . 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 28240 . . . . . . 7  |-  ( K  e.  HL  ->  K  e.  OL )
5150ad2antrr 709 . . . . . 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 28104 . . . . . 6  |-  ( ( K  e.  OL  /\  R  e.  ( Base `  K ) )  -> 
( R  .\/  .0.  )  =  R )
5351, 8, 52syl2anc 645 . . . . 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 968 . . . . . 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 29105 . . . . . 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 1189 . . . . 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 3950 . . . 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 28160 . . . 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 1190 . . 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 5718 . . . 4  |-  (  .0.  =  F  ->  ( R  .\/  .0.  )  =  ( R  .\/  F
) )
6160necon3i 2451 . . 3  |-  ( ( R  .\/  .0.  )  =/=  ( R  .\/  F
)  ->  .0.  =/=  F )
6259, 61syl 17 . 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 2495 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 5    -> wi 6    <-> wb 178    /\ wa 360    /\ w3a 939    = wceq 1619    e. wcel 1621    =/= wne 2412   class class class wbr 3920   ` cfv 4592  (class class class)co 5710   Basecbs 13022   lecple 13089   joincjn 13922   meetcmee 13923   0.cp0 13987   Latclat 13995   OPcops 28051   OLcol 28053    <o ccvr 28141   Atomscatm 28142   HLchlt 28229   LHypclh 28862
This theorem is referenced by:  cdleme3h  29113
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-iin 3806  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-plt 13936  df-lub 13952  df-glb 13953  df-join 13954  df-meet 13955  df-p0 13989  df-p1 13990  df-lat 13996  df-clat 14058  df-oposet 28055  df-ol 28057  df-oml 28058  df-covers 28145  df-ats 28146  df-atl 28177  df-cvlat 28201  df-hlat 28230  df-psubsp 28381  df-pmap 28382  df-padd 28674  df-lhyp 28866
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