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Theorem dihord2cN 31716
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 2413 . 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 29858 . . . . 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 30492 . . . . 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 14443 . . . 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 14469 . . . 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 31643 . . 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 1649    e. wcel 1721   <.cop 3785   class class class wbr 4180    e. cmpt 4234    _I cid 4461    |` cres 4847   ` cfv 5421  (class class class)co 6048   Basecbs 13432   lecple 13499   joincjn 14364   meetcmee 14365   Latclat 14437   LSSumclsm 15231   Atomscatm 29758   HLchlt 29845   LHypclh 30478   LTrncltrn 30595   trLctrl 30652   DVecHcdvh 31573   DIsoBcdib 31633   DIsoCcdic 31667
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-rep 4288  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-nel 2578  df-ral 2679  df-rex 2680  df-reu 2681  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-op 3791  df-uni 3984  df-iun 4063  df-br 4181  df-opab 4235  df-mpt 4236  df-id 4466  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-ov 6051  df-oprab 6052  df-mpt2 6053  df-1st 6316  df-2nd 6317  df-undef 6510  df-riota 6516  df-glb 14395  df-meet 14397  df-lat 14438  df-atl 29793  df-cvlat 29817  df-hlat 29846  df-lhyp 30482  df-disoa 31524  df-dib 31634
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