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Theorem dihord2cN 30678
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 2285 . 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 28820 . . . . 5  |-  ( K  e.  HL  ->  K  e.  Lat )
64, 5syl 17 . . . 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 29454 . . . . 5  |-  ( W  e.  H  ->  W  e.  B )
128, 11syl 17 . . . 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 14151 . . . 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 14177 . . . 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 30605 . . 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 890 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 6    <-> wb 178    /\ wa 360    /\ w3a 936    = wceq 1624    e. wcel 1685   <.cop 3644   class class class wbr 4024    e. cmpt 4078    _I cid 4303    |` cres 4690   ` cfv 5221  (class class class)co 5819   Basecbs 13142   lecple 13209   joincjn 14072   meetcmee 14073   Latclat 14145   LSSumclsm 14939   Atomscatm 28720   HLchlt 28807   LHypclh 29440   LTrncltrn 29557   trLctrl 29614   DVecHcdvh 30535   DIsoBcdib 30595   DIsoCcdic 30629
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1867  ax-ext 2265  ax-rep 4132  ax-sep 4142  ax-nul 4150  ax-pow 4187  ax-pr 4213  ax-un 4511
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 938  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-nel 2450  df-ral 2549  df-rex 2550  df-reu 2551  df-rab 2553  df-v 2791  df-sbc 2993  df-csb 3083  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-op 3650  df-uni 3829  df-iun 3908  df-br 4025  df-opab 4079  df-mpt 4080  df-id 4308  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-fv 5229  df-ov 5822  df-oprab 5823  df-mpt2 5824  df-1st 6083  df-2nd 6084  df-iota 6252  df-undef 6291  df-riota 6299  df-glb 14103  df-meet 14105  df-lat 14146  df-atl 28755  df-cvlat 28779  df-hlat 28808  df-lhyp 29444  df-disoa 30486  df-dib 30596
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