Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  cdleme1b Unicode version

Theorem cdleme1b 29545
Description: Part of proof of Lemma E in [Crawley] p. 113. Utility lemma showing  F is a lattice element.  F represents their f(r). (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 )
) )
cdleme1.b  |-  B  =  ( Base `  K
)
Assertion
Ref Expression
cdleme1b  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A ) )  ->  F  e.  B )

Proof of Theorem cdleme1b
StepHypRef Expression
1 cdleme1.f . 2  |-  F  =  ( ( R  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  R )  ./\  W )
) )
2 hllat 28683 . . . 4  |-  ( K  e.  HL  ->  K  e.  Lat )
32ad2antrr 709 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A ) )  ->  K  e.  Lat )
4 simpr3 968 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A ) )  ->  R  e.  A )
5 cdleme1.b . . . . . 6  |-  B  =  ( Base `  K
)
6 cdleme1.a . . . . . 6  |-  A  =  ( Atoms `  K )
75, 6atbase 28609 . . . . 5  |-  ( R  e.  A  ->  R  e.  B )
84, 7syl 17 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A ) )  ->  R  e.  B )
9 cdleme1.l . . . . . 6  |-  .<_  =  ( le `  K )
10 cdleme1.j . . . . . 6  |-  .\/  =  ( join `  K )
11 cdleme1.m . . . . . 6  |-  ./\  =  ( meet `  K )
12 cdleme1.h . . . . . 6  |-  H  =  ( LHyp `  K
)
13 cdleme1.u . . . . . 6  |-  U  =  ( ( P  .\/  Q )  ./\  W )
149, 10, 11, 6, 12, 13, 5cdleme0aa 29529 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  P  e.  A  /\  Q  e.  A
)  ->  U  e.  B )
15143adant3r3 1167 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A ) )  ->  U  e.  B )
165, 10latjcl 14083 . . . 4  |-  ( ( K  e.  Lat  /\  R  e.  B  /\  U  e.  B )  ->  ( R  .\/  U
)  e.  B )
173, 8, 15, 16syl3anc 1187 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A ) )  -> 
( R  .\/  U
)  e.  B )
18 simpr2 967 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A ) )  ->  Q  e.  A )
195, 6atbase 28609 . . . . 5  |-  ( Q  e.  A  ->  Q  e.  B )
2018, 19syl 17 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A ) )  ->  Q  e.  B )
21 simpr1 966 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A ) )  ->  P  e.  A )
225, 6atbase 28609 . . . . . . 7  |-  ( P  e.  A  ->  P  e.  B )
2321, 22syl 17 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A ) )  ->  P  e.  B )
245, 10latjcl 14083 . . . . . 6  |-  ( ( K  e.  Lat  /\  P  e.  B  /\  R  e.  B )  ->  ( P  .\/  R
)  e.  B )
253, 23, 8, 24syl3anc 1187 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A ) )  -> 
( P  .\/  R
)  e.  B )
265, 12lhpbase 29317 . . . . . 6  |-  ( W  e.  H  ->  W  e.  B )
2726ad2antlr 710 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A ) )  ->  W  e.  B )
285, 11latmcl 14084 . . . . 5  |-  ( ( K  e.  Lat  /\  ( P  .\/  R )  e.  B  /\  W  e.  B )  ->  (
( P  .\/  R
)  ./\  W )  e.  B )
293, 25, 27, 28syl3anc 1187 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A ) )  -> 
( ( P  .\/  R )  ./\  W )  e.  B )
305, 10latjcl 14083 . . . 4  |-  ( ( K  e.  Lat  /\  Q  e.  B  /\  ( ( P  .\/  R )  ./\  W )  e.  B )  ->  ( Q  .\/  ( ( P 
.\/  R )  ./\  W ) )  e.  B
)
313, 20, 29, 30syl3anc 1187 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A ) )  -> 
( Q  .\/  (
( P  .\/  R
)  ./\  W )
)  e.  B )
325, 11latmcl 14084 . . 3  |-  ( ( K  e.  Lat  /\  ( R  .\/  U )  e.  B  /\  ( Q  .\/  ( ( P 
.\/  R )  ./\  W ) )  e.  B
)  ->  ( ( R  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  R ) 
./\  W ) ) )  e.  B )
333, 17, 31, 32syl3anc 1187 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A ) )  -> 
( ( R  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  R )  ./\  W )
) )  e.  B
)
341, 33syl5eqel 2340 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A ) )  ->  F  e.  B )
Colors of variables: wff set class
Syntax hints:    -> wi 6    /\ wa 360    /\ w3a 939    = wceq 1619    e. wcel 1621   ` cfv 4638  (class class class)co 5757   Basecbs 13075   lecple 13142   joincjn 14005   meetcmee 14006   Latclat 14078   Atomscatm 28583   HLchlt 28670   LHypclh 29303
This theorem is referenced by:  cdleme3c  29549  cdleme4a  29558  cdleme5  29559  cdleme7e  29566  cdleme11  29589  cdleme15  29597  cdleme22gb  29613  cdleme19b  29623  cdleme19e  29626  cdleme20d  29631  cdleme20j  29637  cdleme20k  29638  cdleme20l2  29640  cdleme20l  29641  cdleme20m  29642  cdleme22e  29663  cdleme22eALTN  29664  cdleme22f  29665  cdleme27cl  29685  cdlemefr27cl  29722  cdleme35fnpq  29768
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-sep 4081  ax-nul 4089  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-ral 2520  df-rex 2521  df-rab 2523  df-v 2742  df-sbc 2936  df-dif 3097  df-un 3099  df-in 3101  df-ss 3108  df-nul 3398  df-if 3507  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3769  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-fv 4654  df-ov 5760  df-lat 14079  df-ats 28587  df-atl 28618  df-cvlat 28642  df-hlat 28671  df-lhyp 29307
  Copyright terms: Public domain W3C validator