Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  dihord10 Structured version   Unicode version

Theorem dihord10 31958
Description: Part of proof after Lemma N of [Crawley] p. 122. Reverse ordering property. (Contributed by NM, 3-Mar-2014.)
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 ) )
dihord2.p  |-  P  =  ( ( oc `  K ) `  W
)
dihord2.e  |-  E  =  ( ( TEndo `  K
) `  W )
dihord2.d  |-  .+  =  ( +g  `  U )
dihord2.g  |-  G  =  ( iota_ h  e.  T
( h `  P
)  =  N )
Assertion
Ref Expression
dihord10  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( N  e.  A  /\  -.  N  .<_  W ) )  /\  ( f  e.  T  /\  ( R `  f
)  .<_  ( X  ./\  W ) )  /\  (
( s  e.  E  /\  g  e.  T
)  /\  ( R `  g )  .<_  ( Y 
./\  W )  /\  <.
f ,  O >.  =  ( <. ( s `  G ) ,  s
>.  .+  <. g ,  O >. ) ) )  -> 
( R `  f
)  .<_  ( Y  ./\  W ) )
Distinct variable groups:    f, g,
s,  .\/    ./\ , f, g, s    .(+) , f, g, s   
g, E, s    .+ , g,
s    f, h, A, g, s    f, I, g, s    f, J, g, s    g, G    g, O, s    P, h    Q, f, g, s    R, f, g, s    B, f, g, h, s    f, H, g, h, s    f, K, g, h, s    .<_ , f, g, h, s    f, N, g, h, s    T, f, g, h, s    f, W, g, h, s    f, X, g, s    f, Y, g, s
Allowed substitution hints:    P( f, g, s)    .+ ( f, h)    .(+) ( h)    Q( h)    R( h)    U( f,
g, h, s)    E( f, h)    G( f, h, s)    I( h)    J( h)    .\/ (
h)    ./\ ( h)    O( f, h)    X( h)    Y( h)

Proof of Theorem dihord10
StepHypRef Expression
1 simp11 987 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( N  e.  A  /\  -.  N  .<_  W ) )  /\  ( f  e.  T  /\  ( R `  f
)  .<_  ( X  ./\  W ) )  /\  (
( s  e.  E  /\  g  e.  T
)  /\  ( R `  g )  .<_  ( Y 
./\  W )  /\  <.
f ,  O >.  =  ( <. ( s `  G ) ,  s
>.  .+  <. g ,  O >. ) ) )  -> 
( K  e.  HL  /\  W  e.  H ) )
2 simp12 988 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( N  e.  A  /\  -.  N  .<_  W ) )  /\  ( f  e.  T  /\  ( R `  f
)  .<_  ( X  ./\  W ) )  /\  (
( s  e.  E  /\  g  e.  T
)  /\  ( R `  g )  .<_  ( Y 
./\  W )  /\  <.
f ,  O >.  =  ( <. ( s `  G ) ,  s
>.  .+  <. g ,  O >. ) ) )  -> 
( Q  e.  A  /\  -.  Q  .<_  W ) )
3 simp13 989 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( N  e.  A  /\  -.  N  .<_  W ) )  /\  ( f  e.  T  /\  ( R `  f
)  .<_  ( X  ./\  W ) )  /\  (
( s  e.  E  /\  g  e.  T
)  /\  ( R `  g )  .<_  ( Y 
./\  W )  /\  <.
f ,  O >.  =  ( <. ( s `  G ) ,  s
>.  .+  <. g ,  O >. ) ) )  -> 
( N  e.  A  /\  -.  N  .<_  W ) )
4 simp31l 1080 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( N  e.  A  /\  -.  N  .<_  W ) )  /\  ( f  e.  T  /\  ( R `  f
)  .<_  ( X  ./\  W ) )  /\  (
( s  e.  E  /\  g  e.  T
)  /\  ( R `  g )  .<_  ( Y 
./\  W )  /\  <.
f ,  O >.  =  ( <. ( s `  G ) ,  s
>.  .+  <. g ,  O >. ) ) )  -> 
s  e.  E )
5 simp31r 1081 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( N  e.  A  /\  -.  N  .<_  W ) )  /\  ( f  e.  T  /\  ( R `  f
)  .<_  ( X  ./\  W ) )  /\  (
( s  e.  E  /\  g  e.  T
)  /\  ( R `  g )  .<_  ( Y 
./\  W )  /\  <.
f ,  O >.  =  ( <. ( s `  G ) ,  s
>.  .+  <. g ,  O >. ) ) )  -> 
g  e.  T )
6 simp33 995 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( N  e.  A  /\  -.  N  .<_  W ) )  /\  ( f  e.  T  /\  ( R `  f
)  .<_  ( X  ./\  W ) )  /\  (
( s  e.  E  /\  g  e.  T
)  /\  ( R `  g )  .<_  ( Y 
./\  W )  /\  <.
f ,  O >.  =  ( <. ( s `  G ) ,  s
>.  .+  <. g ,  O >. ) ) )  ->  <. f ,  O >.  =  ( <. ( s `  G ) ,  s
>.  .+  <. g ,  O >. ) )
7 dihjust.b . . . . . 6  |-  B  =  ( Base `  K
)
8 dihjust.l . . . . . 6  |-  .<_  =  ( le `  K )
9 dihjust.a . . . . . 6  |-  A  =  ( Atoms `  K )
10 dihjust.h . . . . . 6  |-  H  =  ( LHyp `  K
)
11 dihord2.p . . . . . 6  |-  P  =  ( ( oc `  K ) `  W
)
12 dihord2c.o . . . . . 6  |-  O  =  ( h  e.  T  |->  (  _I  |`  B ) )
13 dihord2c.t . . . . . 6  |-  T  =  ( ( LTrn `  K
) `  W )
14 dihord2.e . . . . . 6  |-  E  =  ( ( TEndo `  K
) `  W )
15 dihjust.u . . . . . 6  |-  U  =  ( ( DVecH `  K
) `  W )
16 dihord2.d . . . . . 6  |-  .+  =  ( +g  `  U )
17 dihord2.g . . . . . 6  |-  G  =  ( iota_ h  e.  T
( h `  P
)  =  N )
187, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17dihordlem7b 31950 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( N  e.  A  /\  -.  N  .<_  W ) )  /\  ( s  e.  E  /\  g  e.  T  /\  <. f ,  O >.  =  ( <. ( s `  G
) ,  s >.  .+  <. g ,  O >. ) ) )  -> 
( f  =  g  /\  O  =  s ) )
1918simpld 446 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( N  e.  A  /\  -.  N  .<_  W ) )  /\  ( s  e.  E  /\  g  e.  T  /\  <. f ,  O >.  =  ( <. ( s `  G
) ,  s >.  .+  <. g ,  O >. ) ) )  -> 
f  =  g )
201, 2, 3, 4, 5, 6, 19syl123anc 1201 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( N  e.  A  /\  -.  N  .<_  W ) )  /\  ( f  e.  T  /\  ( R `  f
)  .<_  ( X  ./\  W ) )  /\  (
( s  e.  E  /\  g  e.  T
)  /\  ( R `  g )  .<_  ( Y 
./\  W )  /\  <.
f ,  O >.  =  ( <. ( s `  G ) ,  s
>.  .+  <. g ,  O >. ) ) )  -> 
f  =  g )
2120fveq2d 5724 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( N  e.  A  /\  -.  N  .<_  W ) )  /\  ( f  e.  T  /\  ( R `  f
)  .<_  ( X  ./\  W ) )  /\  (
( s  e.  E  /\  g  e.  T
)  /\  ( R `  g )  .<_  ( Y 
./\  W )  /\  <.
f ,  O >.  =  ( <. ( s `  G ) ,  s
>.  .+  <. g ,  O >. ) ) )  -> 
( R `  f
)  =  ( R `
 g ) )
22 simp32 994 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( N  e.  A  /\  -.  N  .<_  W ) )  /\  ( f  e.  T  /\  ( R `  f
)  .<_  ( X  ./\  W ) )  /\  (
( s  e.  E  /\  g  e.  T
)  /\  ( R `  g )  .<_  ( Y 
./\  W )  /\  <.
f ,  O >.  =  ( <. ( s `  G ) ,  s
>.  .+  <. g ,  O >. ) ) )  -> 
( R `  g
)  .<_  ( Y  ./\  W ) )
2321, 22eqbrtrd 4224 1  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( N  e.  A  /\  -.  N  .<_  W ) )  /\  ( f  e.  T  /\  ( R `  f
)  .<_  ( X  ./\  W ) )  /\  (
( s  e.  E  /\  g  e.  T
)  /\  ( R `  g )  .<_  ( Y 
./\  W )  /\  <.
f ,  O >.  =  ( <. ( s `  G ) ,  s
>.  .+  <. g ,  O >. ) ) )  -> 
( R `  f
)  .<_  ( Y  ./\  W ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   <.cop 3809   class class class wbr 4204    e. cmpt 4258    _I cid 4485    |` cres 4872   ` cfv 5446  (class class class)co 6073   iota_crio 6534   Basecbs 13461   +g cplusg 13521   lecple 13528   occoc 13529   joincjn 14393   meetcmee 14394   LSSumclsm 15260   Atomscatm 29998   HLchlt 30085   LHypclh 30718   LTrncltrn 30835   trLctrl 30892   TEndoctendo 31486   DVecHcdvh 31813   DIsoBcdib 31873   DIsoCcdic 31907
This theorem is referenced by:  dihord2pre  31960
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 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-iin 4088  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-undef 6535  df-riota 6541  df-recs 6625  df-rdg 6660  df-1o 6716  df-oadd 6720  df-er 6897  df-map 7012  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-nn 9993  df-2 10050  df-3 10051  df-4 10052  df-5 10053  df-6 10054  df-n0 10214  df-z 10275  df-uz 10481  df-fz 11036  df-struct 13463  df-ndx 13464  df-slot 13465  df-base 13466  df-plusg 13534  df-mulr 13535  df-sca 13537  df-vsca 13538  df-poset 14395  df-plt 14407  df-lub 14423  df-glb 14424  df-join 14425  df-meet 14426  df-p0 14460  df-p1 14461  df-lat 14467  df-clat 14529  df-oposet 29911  df-ol 29913  df-oml 29914  df-covers 30001  df-ats 30002  df-atl 30033  df-cvlat 30057  df-hlat 30086  df-llines 30232  df-lplanes 30233  df-lvols 30234  df-lines 30235  df-psubsp 30237  df-pmap 30238  df-padd 30530  df-lhyp 30722  df-laut 30723  df-ldil 30838  df-ltrn 30839  df-trl 30893  df-tendo 31489  df-edring 31491  df-dvech 31814
  Copyright terms: Public domain W3C validator