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Theorem dihord11b 30563
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
dihord11b  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( N  e.  A  /\  -.  N  .<_  W ) )  /\  ( X  e.  B  /\  Y  e.  B
)  /\  ( ( J `  Q )  .(+)  ( I `  ( X  ./\  W ) ) )  C_  ( ( J `  N )  .(+)  ( I `  ( Y  ./\  W ) ) ) )  /\  (
f  e.  T  /\  ( R `  f ) 
.<_  ( X  ./\  W
) ) )  ->  <. f ,  O >.  e.  ( ( J `  N )  .(+)  ( I `
 ( Y  ./\  W ) ) ) )
Distinct variable groups:    .\/ , f    ./\ , f    .(+) ,
f    f, h, A    f, I    f, J    P, h    Q, f    R, f    B, f, h    f, H, h   
f, K, h    .<_ , f, h    f, N, h    T, f, h    f, W, h    f, X    f, Y
Allowed substitution hints:    P( f)    .+ ( f, h)   
.(+) ( h)    Q( h)    R( h)    U( f, h)    E( f, h)    G( f, h)    I( h)    J( h)    .\/ ( h)    ./\ (
h)    O( f, h)    X( h)    Y( h)

Proof of Theorem dihord11b
StepHypRef Expression
1 dihjust.b . . . 4  |-  B  =  ( Base `  K
)
2 dihjust.l . . . 4  |-  .<_  =  ( le `  K )
3 dihjust.j . . . 4  |-  .\/  =  ( join `  K )
4 dihjust.m . . . 4  |-  ./\  =  ( meet `  K )
5 dihjust.a . . . 4  |-  A  =  ( Atoms `  K )
6 dihjust.h . . . 4  |-  H  =  ( LHyp `  K
)
7 dihjust.i . . . 4  |-  I  =  ( ( DIsoB `  K
) `  W )
8 dihjust.J . . . 4  |-  J  =  ( ( DIsoC `  K
) `  W )
9 dihjust.u . . . 4  |-  U  =  ( ( DVecH `  K
) `  W )
10 dihjust.s . . . 4  |-  .(+)  =  (
LSSum `  U )
111, 2, 3, 4, 5, 6, 7, 8, 9, 10dihord2b 30561 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( N  e.  A  /\  -.  N  .<_  W ) )  /\  ( X  e.  B  /\  Y  e.  B
)  /\  ( ( J `  Q )  .(+)  ( I `  ( X  ./\  W ) ) )  C_  ( ( J `  N )  .(+)  ( I `  ( Y  ./\  W ) ) ) )  ->  (
I `  ( X  ./\ 
W ) )  C_  ( ( J `  N )  .(+)  ( I `
 ( Y  ./\  W ) ) ) )
1211adantr 453 . 2  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( N  e.  A  /\  -.  N  .<_  W ) )  /\  ( X  e.  B  /\  Y  e.  B
)  /\  ( ( J `  Q )  .(+)  ( I `  ( X  ./\  W ) ) )  C_  ( ( J `  N )  .(+)  ( I `  ( Y  ./\  W ) ) ) )  /\  (
f  e.  T  /\  ( R `  f ) 
.<_  ( X  ./\  W
) ) )  -> 
( I `  ( X  ./\  W ) ) 
C_  ( ( J `
 N )  .(+)  ( I `  ( Y 
./\  W ) ) ) )
13 simpr 449 . . 3  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( N  e.  A  /\  -.  N  .<_  W ) )  /\  ( X  e.  B  /\  Y  e.  B
)  /\  ( ( J `  Q )  .(+)  ( I `  ( X  ./\  W ) ) )  C_  ( ( J `  N )  .(+)  ( I `  ( Y  ./\  W ) ) ) )  /\  (
f  e.  T  /\  ( R `  f ) 
.<_  ( X  ./\  W
) ) )  -> 
( f  e.  T  /\  ( R `  f
)  .<_  ( X  ./\  W ) ) )
14 eqidd 2257 . . 3  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( N  e.  A  /\  -.  N  .<_  W ) )  /\  ( X  e.  B  /\  Y  e.  B
)  /\  ( ( J `  Q )  .(+)  ( I `  ( X  ./\  W ) ) )  C_  ( ( J `  N )  .(+)  ( I `  ( Y  ./\  W ) ) ) )  /\  (
f  e.  T  /\  ( R `  f ) 
.<_  ( X  ./\  W
) ) )  ->  O  =  O )
15 simpl11 1035 . . . 4  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( N  e.  A  /\  -.  N  .<_  W ) )  /\  ( X  e.  B  /\  Y  e.  B
)  /\  ( ( J `  Q )  .(+)  ( I `  ( X  ./\  W ) ) )  C_  ( ( J `  N )  .(+)  ( I `  ( Y  ./\  W ) ) ) )  /\  (
f  e.  T  /\  ( R `  f ) 
.<_  ( X  ./\  W
) ) )  -> 
( K  e.  HL  /\  W  e.  H ) )
16 simp11l 1071 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( N  e.  A  /\  -.  N  .<_  W ) )  /\  ( X  e.  B  /\  Y  e.  B
)  /\  ( ( J `  Q )  .(+)  ( I `  ( X  ./\  W ) ) )  C_  ( ( J `  N )  .(+)  ( I `  ( Y  ./\  W ) ) ) )  ->  K  e.  HL )
1716adantr 453 . . . . . 6  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( N  e.  A  /\  -.  N  .<_  W ) )  /\  ( X  e.  B  /\  Y  e.  B
)  /\  ( ( J `  Q )  .(+)  ( I `  ( X  ./\  W ) ) )  C_  ( ( J `  N )  .(+)  ( I `  ( Y  ./\  W ) ) ) )  /\  (
f  e.  T  /\  ( R `  f ) 
.<_  ( X  ./\  W
) ) )  ->  K  e.  HL )
18 hllat 28704 . . . . . 6  |-  ( K  e.  HL  ->  K  e.  Lat )
1917, 18syl 17 . . . . 5  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( N  e.  A  /\  -.  N  .<_  W ) )  /\  ( X  e.  B  /\  Y  e.  B
)  /\  ( ( J `  Q )  .(+)  ( I `  ( X  ./\  W ) ) )  C_  ( ( J `  N )  .(+)  ( I `  ( Y  ./\  W ) ) ) )  /\  (
f  e.  T  /\  ( R `  f ) 
.<_  ( X  ./\  W
) ) )  ->  K  e.  Lat )
20 simpl2l 1013 . . . . 5  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( N  e.  A  /\  -.  N  .<_  W ) )  /\  ( X  e.  B  /\  Y  e.  B
)  /\  ( ( J `  Q )  .(+)  ( I `  ( X  ./\  W ) ) )  C_  ( ( J `  N )  .(+)  ( I `  ( Y  ./\  W ) ) ) )  /\  (
f  e.  T  /\  ( R `  f ) 
.<_  ( X  ./\  W
) ) )  ->  X  e.  B )
21 simp11r 1072 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( N  e.  A  /\  -.  N  .<_  W ) )  /\  ( X  e.  B  /\  Y  e.  B
)  /\  ( ( J `  Q )  .(+)  ( I `  ( X  ./\  W ) ) )  C_  ( ( J `  N )  .(+)  ( I `  ( Y  ./\  W ) ) ) )  ->  W  e.  H )
2221adantr 453 . . . . . 6  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( N  e.  A  /\  -.  N  .<_  W ) )  /\  ( X  e.  B  /\  Y  e.  B
)  /\  ( ( J `  Q )  .(+)  ( I `  ( X  ./\  W ) ) )  C_  ( ( J `  N )  .(+)  ( I `  ( Y  ./\  W ) ) ) )  /\  (
f  e.  T  /\  ( R `  f ) 
.<_  ( X  ./\  W
) ) )  ->  W  e.  H )
231, 6lhpbase 29338 . . . . . 6  |-  ( W  e.  H  ->  W  e.  B )
2422, 23syl 17 . . . . 5  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( N  e.  A  /\  -.  N  .<_  W ) )  /\  ( X  e.  B  /\  Y  e.  B
)  /\  ( ( J `  Q )  .(+)  ( I `  ( X  ./\  W ) ) )  C_  ( ( J `  N )  .(+)  ( I `  ( Y  ./\  W ) ) ) )  /\  (
f  e.  T  /\  ( R `  f ) 
.<_  ( X  ./\  W
) ) )  ->  W  e.  B )
251, 4latmcl 14105 . . . . 5  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  W  e.  B )  ->  ( X  ./\  W
)  e.  B )
2619, 20, 24, 25syl3anc 1187 . . . 4  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( N  e.  A  /\  -.  N  .<_  W ) )  /\  ( X  e.  B  /\  Y  e.  B
)  /\  ( ( J `  Q )  .(+)  ( I `  ( X  ./\  W ) ) )  C_  ( ( J `  N )  .(+)  ( I `  ( Y  ./\  W ) ) ) )  /\  (
f  e.  T  /\  ( R `  f ) 
.<_  ( X  ./\  W
) ) )  -> 
( X  ./\  W
)  e.  B )
271, 2, 4latmle2 14131 . . . . 5  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  W  e.  B )  ->  ( X  ./\  W
)  .<_  W )
2819, 20, 24, 27syl3anc 1187 . . . 4  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( N  e.  A  /\  -.  N  .<_  W ) )  /\  ( X  e.  B  /\  Y  e.  B
)  /\  ( ( J `  Q )  .(+)  ( I `  ( X  ./\  W ) ) )  C_  ( ( J `  N )  .(+)  ( I `  ( Y  ./\  W ) ) ) )  /\  (
f  e.  T  /\  ( R `  f ) 
.<_  ( X  ./\  W
) ) )  -> 
( X  ./\  W
)  .<_  W )
29 dihord2c.t . . . . 5  |-  T  =  ( ( LTrn `  K
) `  W )
30 dihord2c.r . . . . 5  |-  R  =  ( ( trL `  K
) `  W )
31 dihord2c.o . . . . 5  |-  O  =  ( h  e.  T  |->  (  _I  |`  B ) )
321, 2, 6, 29, 30, 31, 7dibopelval3 30489 . . . 4  |-  ( ( ( 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 ) ) )
3315, 26, 28, 32syl12anc 1185 . . 3  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( N  e.  A  /\  -.  N  .<_  W ) )  /\  ( X  e.  B  /\  Y  e.  B
)  /\  ( ( J `  Q )  .(+)  ( I `  ( X  ./\  W ) ) )  C_  ( ( J `  N )  .(+)  ( I `  ( Y  ./\  W ) ) ) )  /\  (
f  e.  T  /\  ( R `  f ) 
.<_  ( X  ./\  W
) ) )  -> 
( <. f ,  O >.  e.  ( I `  ( X  ./\  W ) )  <->  ( ( f  e.  T  /\  ( R `  f )  .<_  ( X  ./\  W
) )  /\  O  =  O ) ) )
3413, 14, 33mpbir2and 893 . 2  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( N  e.  A  /\  -.  N  .<_  W ) )  /\  ( X  e.  B  /\  Y  e.  B
)  /\  ( ( J `  Q )  .(+)  ( I `  ( X  ./\  W ) ) )  C_  ( ( J `  N )  .(+)  ( I `  ( Y  ./\  W ) ) ) )  /\  (
f  e.  T  /\  ( R `  f ) 
.<_  ( X  ./\  W
) ) )  ->  <. f ,  O >.  e.  ( I `  ( X  ./\  W ) ) )
3512, 34sseldd 3142 1  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( N  e.  A  /\  -.  N  .<_  W ) )  /\  ( X  e.  B  /\  Y  e.  B
)  /\  ( ( J `  Q )  .(+)  ( I `  ( X  ./\  W ) ) )  C_  ( ( J `  N )  .(+)  ( I `  ( Y  ./\  W ) ) ) )  /\  (
f  e.  T  /\  ( R `  f ) 
.<_  ( X  ./\  W
) ) )  ->  <. f ,  O >.  e.  ( ( J `  N )  .(+)  ( I `
 ( Y  ./\  W ) ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 5    -> wi 6    <-> wb 178    /\ wa 360    /\ w3a 939    = wceq 1619    e. wcel 1621    C_ wss 3113   <.cop 3603   class class class wbr 3983    e. cmpt 4037    _I cid 4262    |` cres 4649   ` cfv 4659  (class class class)co 5778   iota_crio 6249   Basecbs 13096   +g cplusg 13156   lecple 13163   occoc 13164   joincjn 14026   meetcmee 14027   Latclat 14099   LSSumclsm 14893   Atomscatm 28604   HLchlt 28691   LHypclh 29324   LTrncltrn 29441   trLctrl 29498   TEndoctendo 30092   DVecHcdvh 30419   DIsoBcdib 30479   DIsoCcdic 30513
This theorem is referenced by:  dihord11c  30565
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237  ax-rep 4091  ax-sep 4101  ax-nul 4109  ax-pow 4146  ax-pr 4172  ax-un 4470  ax-cnex 8747  ax-resscn 8748  ax-1cn 8749  ax-icn 8750  ax-addcl 8751  ax-addrcl 8752  ax-mulcl 8753  ax-mulrcl 8754  ax-mulcom 8755  ax-addass 8756  ax-mulass 8757  ax-distr 8758  ax-i2m1 8759  ax-1ne0 8760  ax-1rid 8761  ax-rnegex 8762  ax-rrecex 8763  ax-cnre 8764  ax-pre-lttri 8765  ax-pre-lttrn 8766  ax-pre-ltadd 8767  ax-pre-mulgt0 8768
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-fal 1316  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-nel 2422  df-ral 2521  df-rex 2522  df-reu 2523  df-rmo 2524  df-rab 2525  df-v 2759  df-sbc 2953  df-csb 3043  df-dif 3116  df-un 3118  df-in 3120  df-ss 3127  df-pss 3129  df-nul 3417  df-if 3526  df-pw 3587  df-sn 3606  df-pr 3607  df-tp 3608  df-op 3609  df-uni 3788  df-int 3823  df-iun 3867  df-iin 3868  df-br 3984  df-opab 4038  df-mpt 4039  df-tr 4074  df-eprel 4263  df-id 4267  df-po 4272  df-so 4273  df-fr 4310  df-we 4312  df-ord 4353  df-on 4354  df-lim 4355  df-suc 4356  df-om 4615  df-xp 4661  df-rel 4662  df-cnv 4663  df-co 4664  df-dm 4665  df-rn 4666  df-res 4667  df-ima 4668  df-fun 4669  df-fn 4670  df-f 4671  df-f1 4672  df-fo 4673  df-f1o 4674  df-fv 4675  df-ov 5781  df-oprab 5782  df-mpt2 5783  df-1st 6042  df-2nd 6043  df-tpos 6154  df-iota 6211  df-undef 6250  df-riota 6258  df-recs 6342  df-rdg 6377  df-1o 6433  df-oadd 6437  df-er 6614  df-map 6728  df-en 6818  df-dom 6819  df-sdom 6820  df-fin 6821  df-pnf 8823  df-mnf 8824  df-xr 8825  df-ltxr 8826  df-le 8827  df-sub 8993  df-neg 8994  df-n 9701  df-2 9758  df-3 9759  df-4 9760  df-5 9761  df-6 9762  df-n0 9919  df-z 9978  df-uz 10184  df-fz 10735  df-struct 13098  df-ndx 13099  df-slot 13100  df-base 13101  df-sets 13102  df-ress 13103  df-plusg 13169  df-mulr 13170  df-sca 13172  df-vsca 13173  df-0g 13352  df-poset 14028  df-plt 14040  df-lub 14056  df-glb 14057  df-join 14058  df-meet 14059  df-p0 14093  df-p1 14094  df-lat 14100  df-clat 14162  df-mnd 14315  df-submnd 14364  df-grp 14437  df-minusg 14438  df-sbg 14439  df-subg 14566  df-lsm 14895  df-mgp 15274  df-ring 15288  df-ur 15290  df-oppr 15353  df-dvdsr 15371  df-unit 15372  df-invr 15402  df-dvr 15413  df-drng 15462  df-lmod 15577  df-lss 15638  df-lvec 15804  df-oposet 28517  df-ol 28519  df-oml 28520  df-covers 28607  df-ats 28608  df-atl 28639  df-cvlat 28663  df-hlat 28692  df-llines 28838  df-lplanes 28839  df-lvols 28840  df-lines 28841  df-psubsp 28843  df-pmap 28844  df-padd 29136  df-lhyp 29328  df-laut 29329  df-ldil 29444  df-ltrn 29445  df-trl 29499  df-tendo 30095  df-edring 30097  df-disoa 30370  df-dvech 30420  df-dib 30480  df-dic 30514
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