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Theorem dihord2cN 35229
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 990 . 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 2455 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  B  /\  ( f  e.  T  /\  ( R `  f
)  .<_  ( X  ./\  W ) ) )  ->  O  =  O )
3 simp1 988 . . 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 1012 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  B  /\  ( f  e.  T  /\  ( R `  f
)  .<_  ( X  ./\  W ) ) )  ->  K  e.  HL )
5 hllat 33371 . . . . 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 989 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  B  /\  ( f  e.  T  /\  ( R `  f
)  .<_  ( X  ./\  W ) ) )  ->  X  e.  B )
8 simp1r 1013 . . . . 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 34005 . . . . 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 15345 . . . 4  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  W  e.  B )  ->  ( X  ./\  W
)  e.  B )
156, 7, 12, 14syl3anc 1219 . . 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 15370 . . . 4  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  W  e.  B )  ->  ( X  ./\  W
)  .<_  W )
186, 7, 12, 17syl3anc 1219 . . 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 35156 . . 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 1217 . 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 913 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 setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758   <.cop 3994   class class class wbr 4403    |-> cmpt 4461    _I cid 4742    |` cres 4953   ` cfv 5529  (class class class)co 6203   Basecbs 14296   lecple 14368   joincjn 15237   meetcmee 15238   Latclat 15338   LSSumclsm 16258   Atomscatm 33271   HLchlt 33358   LHypclh 33991   LTrncltrn 34108   trLctrl 34165   DVecHcdvh 35086   DIsoBcdib 35146   DIsoCcdic 35180
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4514  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-reu 2806  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-id 4747  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-riota 6164  df-ov 6206  df-oprab 6207  df-lub 15267  df-glb 15268  df-join 15269  df-meet 15270  df-lat 15339  df-atl 33306  df-cvlat 33330  df-hlat 33359  df-lhyp 33995  df-disoa 35037  df-dib 35147
This theorem is referenced by: (None)
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