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Theorem dihord2cN 32019
Description: Part of proof after Lemma N of [Crawley] p. 122. Reverse ordering property. TODO: needed? shorten other proof with it? (Contributed by NM, 3-Mar-2014.) (New usage is discouraged.)
Hypotheses
Ref Expression
dihjust.b  |-  B  =  ( Base `  K
)
dihjust.l  |-  .<_  =  ( le `  K )
dihjust.j  |-  .\/  =  ( join `  K )
dihjust.m  |-  ./\  =  ( meet `  K )
dihjust.a  |-  A  =  ( Atoms `  K )
dihjust.h  |-  H  =  ( LHyp `  K
)
dihjust.i  |-  I  =  ( ( DIsoB `  K
) `  W )
dihjust.J  |-  J  =  ( ( DIsoC `  K
) `  W )
dihjust.u  |-  U  =  ( ( DVecH `  K
) `  W )
dihjust.s  |-  .(+)  =  (
LSSum `  U )
dihord2c.t  |-  T  =  ( ( LTrn `  K
) `  W )
dihord2c.r  |-  R  =  ( ( trL `  K
) `  W )
dihord2c.o  |-  O  =  ( h  e.  T  |->  (  _I  |`  B ) )
Assertion
Ref Expression
dihord2cN  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  B  /\  ( f  e.  T  /\  ( R `  f
)  .<_  ( X  ./\  W ) ) )  ->  <. f ,  O >.  e.  ( I `  ( X  ./\  W ) ) )
Distinct variable groups:    .\/ , f    ./\ , f    .(+) ,
f    f, h, A    f, I    f, J    R, f    B, f, h    f, H, h    f, K, h    .<_ , f, h    T, f, h    f, W, h   
f, X
Allowed substitution hints:    .(+) ( h)    R( h)    U( f, h)    I( h)    J( h)    .\/ ( h)    ./\ ( h)    O( f, h)    X( h)

Proof of Theorem dihord2cN
StepHypRef Expression
1 simp3 959 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  B  /\  ( f  e.  T  /\  ( R `  f
)  .<_  ( X  ./\  W ) ) )  -> 
( f  e.  T  /\  ( R `  f
)  .<_  ( X  ./\  W ) ) )
2 eqidd 2437 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  B  /\  ( f  e.  T  /\  ( R `  f
)  .<_  ( X  ./\  W ) ) )  ->  O  =  O )
3 simp1 957 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  B  /\  ( f  e.  T  /\  ( R `  f
)  .<_  ( X  ./\  W ) ) )  -> 
( K  e.  HL  /\  W  e.  H ) )
4 simp1l 981 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  B  /\  ( f  e.  T  /\  ( R `  f
)  .<_  ( X  ./\  W ) ) )  ->  K  e.  HL )
5 hllat 30161 . . . . 5  |-  ( K  e.  HL  ->  K  e.  Lat )
64, 5syl 16 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  B  /\  ( f  e.  T  /\  ( R `  f
)  .<_  ( X  ./\  W ) ) )  ->  K  e.  Lat )
7 simp2 958 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  B  /\  ( f  e.  T  /\  ( R `  f
)  .<_  ( X  ./\  W ) ) )  ->  X  e.  B )
8 simp1r 982 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  B  /\  ( f  e.  T  /\  ( R `  f
)  .<_  ( X  ./\  W ) ) )  ->  W  e.  H )
9 dihjust.b . . . . . 6  |-  B  =  ( Base `  K
)
10 dihjust.h . . . . . 6  |-  H  =  ( LHyp `  K
)
119, 10lhpbase 30795 . . . . 5  |-  ( W  e.  H  ->  W  e.  B )
128, 11syl 16 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  B  /\  ( f  e.  T  /\  ( R `  f
)  .<_  ( X  ./\  W ) ) )  ->  W  e.  B )
13 dihjust.m . . . . 5  |-  ./\  =  ( meet `  K )
149, 13latmcl 14480 . . . 4  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  W  e.  B )  ->  ( X  ./\  W
)  e.  B )
156, 7, 12, 14syl3anc 1184 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  B  /\  ( f  e.  T  /\  ( R `  f
)  .<_  ( X  ./\  W ) ) )  -> 
( X  ./\  W
)  e.  B )
16 dihjust.l . . . . 5  |-  .<_  =  ( le `  K )
179, 16, 13latmle2 14506 . . . 4  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  W  e.  B )  ->  ( X  ./\  W
)  .<_  W )
186, 7, 12, 17syl3anc 1184 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  B  /\  ( f  e.  T  /\  ( R `  f
)  .<_  ( X  ./\  W ) ) )  -> 
( X  ./\  W
)  .<_  W )
19 dihord2c.t . . . 4  |-  T  =  ( ( LTrn `  K
) `  W )
20 dihord2c.r . . . 4  |-  R  =  ( ( trL `  K
) `  W )
21 dihord2c.o . . . 4  |-  O  =  ( h  e.  T  |->  (  _I  |`  B ) )
22 dihjust.i . . . 4  |-  I  =  ( ( DIsoB `  K
) `  W )
239, 16, 10, 19, 20, 21, 22dibopelval3 31946 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( X 
./\  W )  e.  B  /\  ( X 
./\  W )  .<_  W ) )  -> 
( <. f ,  O >.  e.  ( I `  ( X  ./\  W ) )  <->  ( ( f  e.  T  /\  ( R `  f )  .<_  ( X  ./\  W
) )  /\  O  =  O ) ) )
243, 15, 18, 23syl12anc 1182 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  B  /\  ( f  e.  T  /\  ( R `  f
)  .<_  ( X  ./\  W ) ) )  -> 
( <. f ,  O >.  e.  ( I `  ( X  ./\  W ) )  <->  ( ( f  e.  T  /\  ( R `  f )  .<_  ( X  ./\  W
) )  /\  O  =  O ) ) )
251, 2, 24mpbir2and 889 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  B  /\  ( f  e.  T  /\  ( R `  f
)  .<_  ( X  ./\  W ) ) )  ->  <. f ,  O >.  e.  ( I `  ( X  ./\  W ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   <.cop 3817   class class class wbr 4212    e. cmpt 4266    _I cid 4493    |` cres 4880   ` cfv 5454  (class class class)co 6081   Basecbs 13469   lecple 13536   joincjn 14401   meetcmee 14402   Latclat 14474   LSSumclsm 15268   Atomscatm 30061   HLchlt 30148   LHypclh 30781   LTrncltrn 30898   trLctrl 30955   DVecHcdvh 31876   DIsoBcdib 31936   DIsoCcdic 31970
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-undef 6543  df-riota 6549  df-glb 14432  df-meet 14434  df-lat 14475  df-atl 30096  df-cvlat 30120  df-hlat 30149  df-lhyp 30785  df-disoa 31827  df-dib 31937
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