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Theorem cdleme3c 29549
Description: Part of proof of Lemma E in [Crawley] p. 113. Lemma leading to cdleme3fa 29555 and cdleme3 29556. (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 28683 . . . . . 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 2256 . . . . . . 7  |-  ( Base `  K )  =  (
Base `  K )
6 cdleme1.a . . . . . . 7  |-  A  =  ( Atoms `  K )
75, 6atbase 28609 . . . . . 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 28682 . . . . . . 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 28504 . . . . . 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 14083 . . . . 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 29545 . . . . . 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 14083 . . . . 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 28609 . . . . . . . . . . . 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 28609 . . . . . . . . . . . 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 14083 . . . . . . . . . . 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 29317 . . . . . . . . . . 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 14110 . . . . . . . . . 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 3996 . . . . . . . 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 3981 . . . . . . . 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 2502 . . . . . 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 29364 . . . . . . . 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 2256 . . . . . . . 8  |-  (  <o  `  K )  =  ( 
<o  `  K )
4714, 46, 6atcvr1 28736 . . . . . . 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 28681 . . . . . . 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 28545 . . . . . 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 29546 . . . . . 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 3993 . . . 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 28601 . . . 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 5765 . . . 4  |-  (  .0.  =  F  ->  ( R  .\/  .0.  )  =  ( R  .\/  F
) )
6160necon3i 2458 . . 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 2502 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 2419   class class class wbr 3963   ` cfv 4638  (class class class)co 5757   Basecbs 13075   lecple 13142   joincjn 14005   meetcmee 14006   0.cp0 14070   Latclat 14078   OPcops 28492   OLcol 28494    <o ccvr 28582   Atomscatm 28583   HLchlt 28670   LHypclh 29303
This theorem is referenced by:  cdleme3h  29554
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 1927  ax-ext 2237  ax-rep 4071  ax-sep 4081  ax-nul 4089  ax-pow 4126  ax-pr 4152  ax-un 4449
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 1884  df-eu 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-nel 2422  df-ral 2520  df-rex 2521  df-reu 2522  df-rab 2523  df-v 2742  df-sbc 2936  df-csb 3024  df-dif 3097  df-un 3099  df-in 3101  df-ss 3108  df-nul 3398  df-if 3507  df-pw 3568  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3769  df-iun 3848  df-iin 3849  df-br 3964  df-opab 4018  df-mpt 4019  df-id 4246  df-xp 4640  df-rel 4641  df-cnv 4642  df-co 4643  df-dm 4644  df-rn 4645  df-res 4646  df-ima 4647  df-fun 4648  df-fn 4649  df-f 4650  df-f1 4651  df-fo 4652  df-f1o 4653  df-fv 4654  df-ov 5760  df-oprab 5761  df-mpt2 5762  df-1st 6021  df-2nd 6022  df-iota 6190  df-undef 6229  df-riota 6237  df-poset 14007  df-plt 14019  df-lub 14035  df-glb 14036  df-join 14037  df-meet 14038  df-p0 14072  df-p1 14073  df-lat 14079  df-clat 14141  df-oposet 28496  df-ol 28498  df-oml 28499  df-covers 28586  df-ats 28587  df-atl 28618  df-cvlat 28642  df-hlat 28671  df-psubsp 28822  df-pmap 28823  df-padd 29115  df-lhyp 29307
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