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Theorem cdlemc2 29754
Description: Part of proof of Lemma C in [Crawley] p. 112. (Contributed by NM, 25-May-2012.)
Hypotheses
Ref Expression
cdlemc2.l  |-  .<_  =  ( le `  K )
cdlemc2.j  |-  .\/  =  ( join `  K )
cdlemc2.m  |-  ./\  =  ( meet `  K )
cdlemc2.a  |-  A  =  ( Atoms `  K )
cdlemc2.h  |-  H  =  ( LHyp `  K
)
cdlemc2.t  |-  T  =  ( ( LTrn `  K
) `  W )
Assertion
Ref Expression
cdlemc2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) ) )  ->  ( F `  Q )  .<_  ( ( F `  P )  .\/  (
( P  .\/  Q
)  ./\  W )
) )

Proof of Theorem cdlemc2
StepHypRef Expression
1 simp1l 979 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) ) )  ->  K  e.  HL )
2 simp3ll 1026 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) ) )  ->  P  e.  A )
3 simp3rl 1028 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) ) )  ->  Q  e.  A )
4 cdlemc2.l . . . . . 6  |-  .<_  =  ( le `  K )
5 cdlemc2.j . . . . . 6  |-  .\/  =  ( join `  K )
6 cdlemc2.a . . . . . 6  |-  A  =  ( Atoms `  K )
74, 5, 6hlatlej2 28938 . . . . 5  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  Q  .<_  ( P  .\/  Q ) )
81, 2, 3, 7syl3anc 1182 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) ) )  ->  Q  .<_  ( P  .\/  Q
) )
9 simp1 955 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) ) )  ->  ( K  e.  HL  /\  W  e.  H ) )
10 eqid 2283 . . . . . . 7  |-  ( Base `  K )  =  (
Base `  K )
1110, 6atbase 28852 . . . . . 6  |-  ( Q  e.  A  ->  Q  e.  ( Base `  K
) )
123, 11syl 15 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) ) )  ->  Q  e.  ( Base `  K
) )
13 simp3l 983 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) ) )  ->  ( P  e.  A  /\  -.  P  .<_  W ) )
14 cdlemc2.m . . . . . 6  |-  ./\  =  ( meet `  K )
15 cdlemc2.h . . . . . 6  |-  H  =  ( LHyp `  K
)
1610, 4, 5, 14, 6, 15cdlemc1 29753 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  Q  e.  (
Base `  K )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( P  .\/  ( ( P  .\/  Q )  ./\  W )
)  =  ( P 
.\/  Q ) )
179, 12, 13, 16syl3anc 1182 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) ) )  ->  ( P  .\/  ( ( P 
.\/  Q )  ./\  W ) )  =  ( P  .\/  Q ) )
188, 17breqtrrd 4049 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) ) )  ->  Q  .<_  ( P  .\/  (
( P  .\/  Q
)  ./\  W )
) )
19 simp2 956 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) ) )  ->  F  e.  T )
20 hllat 28926 . . . . . 6  |-  ( K  e.  HL  ->  K  e.  Lat )
211, 20syl 15 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) ) )  ->  K  e.  Lat )
2210, 6atbase 28852 . . . . . 6  |-  ( P  e.  A  ->  P  e.  ( Base `  K
) )
232, 22syl 15 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) ) )  ->  P  e.  ( Base `  K
) )
2410, 5latjcl 14156 . . . . . . 7  |-  ( ( K  e.  Lat  /\  P  e.  ( Base `  K )  /\  Q  e.  ( Base `  K
) )  ->  ( P  .\/  Q )  e.  ( Base `  K
) )
2521, 23, 12, 24syl3anc 1182 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) ) )  ->  ( P  .\/  Q )  e.  ( Base `  K
) )
26 simp1r 980 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) ) )  ->  W  e.  H )
2710, 15lhpbase 29560 . . . . . . 7  |-  ( W  e.  H  ->  W  e.  ( Base `  K
) )
2826, 27syl 15 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) ) )  ->  W  e.  ( Base `  K
) )
2910, 14latmcl 14157 . . . . . 6  |-  ( ( K  e.  Lat  /\  ( P  .\/  Q )  e.  ( Base `  K
)  /\  W  e.  ( Base `  K )
)  ->  ( ( P  .\/  Q )  ./\  W )  e.  ( Base `  K ) )
3021, 25, 28, 29syl3anc 1182 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) ) )  ->  (
( P  .\/  Q
)  ./\  W )  e.  ( Base `  K
) )
3110, 5latjcl 14156 . . . . 5  |-  ( ( K  e.  Lat  /\  P  e.  ( Base `  K )  /\  (
( P  .\/  Q
)  ./\  W )  e.  ( Base `  K
) )  ->  ( P  .\/  ( ( P 
.\/  Q )  ./\  W ) )  e.  (
Base `  K )
)
3221, 23, 30, 31syl3anc 1182 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) ) )  ->  ( P  .\/  ( ( P 
.\/  Q )  ./\  W ) )  e.  (
Base `  K )
)
33 cdlemc2.t . . . . 5  |-  T  =  ( ( LTrn `  K
) `  W )
3410, 4, 15, 33ltrnle 29691 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( Q  e.  (
Base `  K )  /\  ( P  .\/  (
( P  .\/  Q
)  ./\  W )
)  e.  ( Base `  K ) ) )  ->  ( Q  .<_  ( P  .\/  ( ( P  .\/  Q ) 
./\  W ) )  <-> 
( F `  Q
)  .<_  ( F `  ( P  .\/  ( ( P  .\/  Q ) 
./\  W ) ) ) ) )
359, 19, 12, 32, 34syl112anc 1186 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) ) )  ->  ( Q  .<_  ( P  .\/  ( ( P  .\/  Q )  ./\  W )
)  <->  ( F `  Q )  .<_  ( F `
 ( P  .\/  ( ( P  .\/  Q )  ./\  W )
) ) ) )
3618, 35mpbid 201 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) ) )  ->  ( F `  Q )  .<_  ( F `  ( P  .\/  ( ( P 
.\/  Q )  ./\  W ) ) ) )
3710, 5, 15, 33ltrnj 29694 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  (
Base `  K )  /\  ( ( P  .\/  Q )  ./\  W )  e.  ( Base `  K
) ) )  -> 
( F `  ( P  .\/  ( ( P 
.\/  Q )  ./\  W ) ) )  =  ( ( F `  P )  .\/  ( F `  ( ( P  .\/  Q )  ./\  W ) ) ) )
389, 19, 23, 30, 37syl112anc 1186 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) ) )  ->  ( F `  ( P  .\/  ( ( P  .\/  Q )  ./\  W )
) )  =  ( ( F `  P
)  .\/  ( F `  ( ( P  .\/  Q )  ./\  W )
) ) )
3910, 4, 14latmle2 14183 . . . . . 6  |-  ( ( K  e.  Lat  /\  ( P  .\/  Q )  e.  ( Base `  K
)  /\  W  e.  ( Base `  K )
)  ->  ( ( P  .\/  Q )  ./\  W )  .<_  W )
4021, 25, 28, 39syl3anc 1182 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) ) )  ->  (
( P  .\/  Q
)  ./\  W )  .<_  W )
4110, 4, 15, 33ltrnval1 29696 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( ( ( P 
.\/  Q )  ./\  W )  e.  ( Base `  K )  /\  (
( P  .\/  Q
)  ./\  W )  .<_  W ) )  -> 
( F `  (
( P  .\/  Q
)  ./\  W )
)  =  ( ( P  .\/  Q ) 
./\  W ) )
429, 19, 30, 40, 41syl112anc 1186 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) ) )  ->  ( F `  ( ( P  .\/  Q )  ./\  W ) )  =  ( ( P  .\/  Q
)  ./\  W )
)
4342oveq2d 5874 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) ) )  ->  (
( F `  P
)  .\/  ( F `  ( ( P  .\/  Q )  ./\  W )
) )  =  ( ( F `  P
)  .\/  ( ( P  .\/  Q )  ./\  W ) ) )
4438, 43eqtrd 2315 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) ) )  ->  ( F `  ( P  .\/  ( ( P  .\/  Q )  ./\  W )
) )  =  ( ( F `  P
)  .\/  ( ( P  .\/  Q )  ./\  W ) ) )
4536, 44breqtrd 4047 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) ) )  ->  ( F `  Q )  .<_  ( ( F `  P )  .\/  (
( P  .\/  Q
)  ./\  W )
) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   class class class wbr 4023   ` cfv 5255  (class class class)co 5858   Basecbs 13148   lecple 13215   joincjn 14078   meetcmee 14079   Latclat 14151   Atomscatm 28826   HLchlt 28913   LHypclh 29546   LTrncltrn 29663
This theorem is referenced by:  cdlemc5  29757
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-iin 3908  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-undef 6298  df-riota 6304  df-map 6774  df-poset 14080  df-plt 14092  df-lub 14108  df-glb 14109  df-join 14110  df-meet 14111  df-p0 14145  df-p1 14146  df-lat 14152  df-clat 14214  df-oposet 28739  df-ol 28741  df-oml 28742  df-covers 28829  df-ats 28830  df-atl 28861  df-cvlat 28885  df-hlat 28914  df-psubsp 29065  df-pmap 29066  df-padd 29358  df-lhyp 29550  df-laut 29551  df-ldil 29666  df-ltrn 29667
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