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Theorem cdlemc2 29511
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 984 . . . . 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 1031 . . . . 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 1033 . . . . 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 28695 . . . . 5  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  Q  .<_  ( P  .\/  Q ) )
81, 2, 3, 7syl3anc 1187 . . . 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 960 . . . . 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 2256 . . . . . . 7  |-  ( Base `  K )  =  (
Base `  K )
1110, 6atbase 28609 . . . . . 6  |-  ( Q  e.  A  ->  Q  e.  ( Base `  K
) )
123, 11syl 17 . . . . 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 988 . . . . 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 29510 . . . . 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 1187 . . . 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 3989 . . 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 961 . . . 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 28683 . . . . . 6  |-  ( K  e.  HL  ->  K  e.  Lat )
211, 20syl 17 . . . . 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 28609 . . . . . 6  |-  ( P  e.  A  ->  P  e.  ( Base `  K
) )
232, 22syl 17 . . . . 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 14083 . . . . . . 7  |-  ( ( K  e.  Lat  /\  P  e.  ( Base `  K )  /\  Q  e.  ( Base `  K
) )  ->  ( P  .\/  Q )  e.  ( Base `  K
) )
2521, 23, 12, 24syl3anc 1187 . . . . . 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 985 . . . . . . 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 29317 . . . . . . 7  |-  ( W  e.  H  ->  W  e.  ( Base `  K
) )
2826, 27syl 17 . . . . . 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 14084 . . . . . 6  |-  ( ( K  e.  Lat  /\  ( P  .\/  Q )  e.  ( Base `  K
)  /\  W  e.  ( Base `  K )
)  ->  ( ( P  .\/  Q )  ./\  W )  e.  ( Base `  K ) )
3021, 25, 28, 29syl3anc 1187 . . . . 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 14083 . . . . 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 1187 . . . 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 29448 . . . 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 1191 . . 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 203 . 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 29451 . . . 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 1191 . . 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 14110 . . . . . 6  |-  ( ( K  e.  Lat  /\  ( P  .\/  Q )  e.  ( Base `  K
)  /\  W  e.  ( Base `  K )
)  ->  ( ( P  .\/  Q )  ./\  W )  .<_  W )
4021, 25, 28, 39syl3anc 1187 . . . . 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 29453 . . . . 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 1191 . . . 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 5773 . . 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 2288 . 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 3987 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 5    -> wi 6    <-> wb 178    /\ wa 360    /\ w3a 939    = wceq 1619    e. wcel 1621   class class class wbr 3963   ` cfv 4638  (class class class)co 5757   Basecbs 13075   lecple 13142   joincjn 14005   meetcmee 14006   Latclat 14078   Atomscatm 28583   HLchlt 28670   LHypclh 29303   LTrncltrn 29420
This theorem is referenced by:  cdlemc5  29514
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-map 6707  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  df-laut 29308  df-ldil 29423  df-ltrn 29424
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