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Theorem dihord2b 35228
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 )
Assertion
Ref Expression
dihord2b  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( X  e.  B  /\  Y  e.  B
)  /\  ( ( J `  Q )  .(+)  ( I `  ( X  ./\  W ) ) )  C_  ( ( J `  R )  .(+)  ( I `  ( Y  ./\  W ) ) ) )  ->  (
I `  ( X  ./\ 
W ) )  C_  ( ( J `  R )  .(+)  ( I `
 ( Y  ./\  W ) ) ) )

Proof of Theorem dihord2b
StepHypRef Expression
1 dihjust.h . . . . . 6  |-  H  =  ( LHyp `  K
)
2 dihjust.u . . . . . 6  |-  U  =  ( ( DVecH `  K
) `  W )
3 simp11 1018 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( X  e.  B  /\  Y  e.  B
)  /\  ( ( J `  Q )  .(+)  ( I `  ( X  ./\  W ) ) )  C_  ( ( J `  R )  .(+)  ( I `  ( Y  ./\  W ) ) ) )  ->  ( K  e.  HL  /\  W  e.  H ) )
41, 2, 3dvhlmod 35118 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( X  e.  B  /\  Y  e.  B
)  /\  ( ( J `  Q )  .(+)  ( I `  ( X  ./\  W ) ) )  C_  ( ( J `  R )  .(+)  ( I `  ( Y  ./\  W ) ) ) )  ->  U  e.  LMod )
5 eqid 2454 . . . . . 6  |-  ( LSubSp `  U )  =  (
LSubSp `  U )
65lsssssubg 17172 . . . . 5  |-  ( U  e.  LMod  ->  ( LSubSp `  U )  C_  (SubGrp `  U ) )
74, 6syl 16 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( X  e.  B  /\  Y  e.  B
)  /\  ( ( J `  Q )  .(+)  ( I `  ( X  ./\  W ) ) )  C_  ( ( J `  R )  .(+)  ( I `  ( Y  ./\  W ) ) ) )  ->  ( LSubSp `
 U )  C_  (SubGrp `  U ) )
8 simp12 1019 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( X  e.  B  /\  Y  e.  B
)  /\  ( ( J `  Q )  .(+)  ( I `  ( X  ./\  W ) ) )  C_  ( ( J `  R )  .(+)  ( I `  ( Y  ./\  W ) ) ) )  ->  ( Q  e.  A  /\  -.  Q  .<_  W ) )
9 dihjust.l . . . . . 6  |-  .<_  =  ( le `  K )
10 dihjust.a . . . . . 6  |-  A  =  ( Atoms `  K )
11 dihjust.J . . . . . 6  |-  J  =  ( ( DIsoC `  K
) `  W )
129, 10, 1, 2, 11, 5diclss 35201 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  -> 
( J `  Q
)  e.  ( LSubSp `  U ) )
133, 8, 12syl2anc 661 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( X  e.  B  /\  Y  e.  B
)  /\  ( ( J `  Q )  .(+)  ( I `  ( X  ./\  W ) ) )  C_  ( ( J `  R )  .(+)  ( I `  ( Y  ./\  W ) ) ) )  ->  ( J `  Q )  e.  ( LSubSp `  U )
)
147, 13sseldd 3468 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( X  e.  B  /\  Y  e.  B
)  /\  ( ( J `  Q )  .(+)  ( I `  ( X  ./\  W ) ) )  C_  ( ( J `  R )  .(+)  ( I `  ( Y  ./\  W ) ) ) )  ->  ( J `  Q )  e.  (SubGrp `  U )
)
15 simp11l 1099 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( X  e.  B  /\  Y  e.  B
)  /\  ( ( J `  Q )  .(+)  ( I `  ( X  ./\  W ) ) )  C_  ( ( J `  R )  .(+)  ( I `  ( Y  ./\  W ) ) ) )  ->  K  e.  HL )
16 hllat 33371 . . . . . . 7  |-  ( K  e.  HL  ->  K  e.  Lat )
1715, 16syl 16 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( X  e.  B  /\  Y  e.  B
)  /\  ( ( J `  Q )  .(+)  ( I `  ( X  ./\  W ) ) )  C_  ( ( J `  R )  .(+)  ( I `  ( Y  ./\  W ) ) ) )  ->  K  e.  Lat )
18 simp2l 1014 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( X  e.  B  /\  Y  e.  B
)  /\  ( ( J `  Q )  .(+)  ( I `  ( X  ./\  W ) ) )  C_  ( ( J `  R )  .(+)  ( I `  ( Y  ./\  W ) ) ) )  ->  X  e.  B )
19 simp11r 1100 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( X  e.  B  /\  Y  e.  B
)  /\  ( ( J `  Q )  .(+)  ( I `  ( X  ./\  W ) ) )  C_  ( ( J `  R )  .(+)  ( I `  ( Y  ./\  W ) ) ) )  ->  W  e.  H )
20 dihjust.b . . . . . . . 8  |-  B  =  ( Base `  K
)
2120, 1lhpbase 34005 . . . . . . 7  |-  ( W  e.  H  ->  W  e.  B )
2219, 21syl 16 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( X  e.  B  /\  Y  e.  B
)  /\  ( ( J `  Q )  .(+)  ( I `  ( X  ./\  W ) ) )  C_  ( ( J `  R )  .(+)  ( I `  ( Y  ./\  W ) ) ) )  ->  W  e.  B )
23 dihjust.m . . . . . . 7  |-  ./\  =  ( meet `  K )
2420, 23latmcl 15345 . . . . . 6  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  W  e.  B )  ->  ( X  ./\  W
)  e.  B )
2517, 18, 22, 24syl3anc 1219 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( X  e.  B  /\  Y  e.  B
)  /\  ( ( J `  Q )  .(+)  ( I `  ( X  ./\  W ) ) )  C_  ( ( J `  R )  .(+)  ( I `  ( Y  ./\  W ) ) ) )  ->  ( X  ./\  W )  e.  B )
2620, 9, 23latmle2 15370 . . . . . 6  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  W  e.  B )  ->  ( X  ./\  W
)  .<_  W )
2717, 18, 22, 26syl3anc 1219 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( X  e.  B  /\  Y  e.  B
)  /\  ( ( J `  Q )  .(+)  ( I `  ( X  ./\  W ) ) )  C_  ( ( J `  R )  .(+)  ( I `  ( Y  ./\  W ) ) ) )  ->  ( X  ./\  W )  .<_  W )
28 dihjust.i . . . . . 6  |-  I  =  ( ( DIsoB `  K
) `  W )
2920, 9, 1, 2, 28, 5diblss 35178 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( X 
./\  W )  e.  B  /\  ( X 
./\  W )  .<_  W ) )  -> 
( I `  ( X  ./\  W ) )  e.  ( LSubSp `  U
) )
303, 25, 27, 29syl12anc 1217 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( X  e.  B  /\  Y  e.  B
)  /\  ( ( J `  Q )  .(+)  ( I `  ( X  ./\  W ) ) )  C_  ( ( J `  R )  .(+)  ( I `  ( Y  ./\  W ) ) ) )  ->  (
I `  ( X  ./\ 
W ) )  e.  ( LSubSp `  U )
)
317, 30sseldd 3468 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( X  e.  B  /\  Y  e.  B
)  /\  ( ( J `  Q )  .(+)  ( I `  ( X  ./\  W ) ) )  C_  ( ( J `  R )  .(+)  ( I `  ( Y  ./\  W ) ) ) )  ->  (
I `  ( X  ./\ 
W ) )  e.  (SubGrp `  U )
)
32 dihjust.s . . . 4  |-  .(+)  =  (
LSSum `  U )
3332lsmub2 16281 . . 3  |-  ( ( ( J `  Q
)  e.  (SubGrp `  U )  /\  (
I `  ( X  ./\ 
W ) )  e.  (SubGrp `  U )
)  ->  ( I `  ( X  ./\  W
) )  C_  (
( J `  Q
)  .(+)  ( I `  ( X  ./\  W ) ) ) )
3414, 31, 33syl2anc 661 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( X  e.  B  /\  Y  e.  B
)  /\  ( ( J `  Q )  .(+)  ( I `  ( X  ./\  W ) ) )  C_  ( ( J `  R )  .(+)  ( I `  ( Y  ./\  W ) ) ) )  ->  (
I `  ( X  ./\ 
W ) )  C_  ( ( J `  Q )  .(+)  ( I `
 ( X  ./\  W ) ) ) )
35 simp3 990 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( X  e.  B  /\  Y  e.  B
)  /\  ( ( J `  Q )  .(+)  ( I `  ( X  ./\  W ) ) )  C_  ( ( J `  R )  .(+)  ( I `  ( Y  ./\  W ) ) ) )  ->  (
( J `  Q
)  .(+)  ( I `  ( X  ./\  W ) ) )  C_  (
( J `  R
)  .(+)  ( I `  ( Y  ./\  W ) ) ) )
3634, 35sstrd 3477 1  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( X  e.  B  /\  Y  e.  B
)  /\  ( ( J `  Q )  .(+)  ( I `  ( X  ./\  W ) ) )  C_  ( ( J `  R )  .(+)  ( I `  ( Y  ./\  W ) ) ) )  ->  (
I `  ( X  ./\ 
W ) )  C_  ( ( J `  R )  .(+)  ( I `
 ( Y  ./\  W ) ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758    C_ wss 3439   class class class wbr 4403   ` cfv 5529  (class class class)co 6203   Basecbs 14296   lecple 14368   joincjn 15237   meetcmee 15238   Latclat 15338  SubGrpcsubg 15798   LSSumclsm 16258   LModclmod 17081   LSubSpclss 17146   Atomscatm 33271   HLchlt 33358   LHypclh 33991   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  ax-cnex 9453  ax-resscn 9454  ax-1cn 9455  ax-icn 9456  ax-addcl 9457  ax-addrcl 9458  ax-mulcl 9459  ax-mulrcl 9460  ax-mulcom 9461  ax-addass 9462  ax-mulass 9463  ax-distr 9464  ax-i2m1 9465  ax-1ne0 9466  ax-1rid 9467  ax-rnegex 9468  ax-rrecex 9469  ax-cnre 9470  ax-pre-lttri 9471  ax-pre-lttrn 9472  ax-pre-ltadd 9473  ax-pre-mulgt0 9474  ax-riotaBAD 32967
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-fal 1376  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-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rmo 2807  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-pss 3455  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-tp 3993  df-op 3995  df-uni 4203  df-int 4240  df-iun 4284  df-iin 4285  df-br 4404  df-opab 4462  df-mpt 4463  df-tr 4497  df-eprel 4743  df-id 4747  df-po 4752  df-so 4753  df-fr 4790  df-we 4792  df-ord 4833  df-on 4834  df-lim 4835  df-suc 4836  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-mpt2 6208  df-om 6590  df-1st 6690  df-2nd 6691  df-tpos 6858  df-undef 6905  df-recs 6945  df-rdg 6979  df-1o 7033  df-oadd 7037  df-er 7214  df-map 7329  df-en 7424  df-dom 7425  df-sdom 7426  df-fin 7427  df-pnf 9535  df-mnf 9536  df-xr 9537  df-ltxr 9538  df-le 9539  df-sub 9712  df-neg 9713  df-nn 10438  df-2 10495  df-3 10496  df-4 10497  df-5 10498  df-6 10499  df-n0 10695  df-z 10762  df-uz 10977  df-fz 11559  df-struct 14298  df-ndx 14299  df-slot 14300  df-base 14301  df-sets 14302  df-ress 14303  df-plusg 14374  df-mulr 14375  df-sca 14377  df-vsca 14378  df-0g 14503  df-poset 15239  df-plt 15251  df-lub 15267  df-glb 15268  df-join 15269  df-meet 15270  df-p0 15332  df-p1 15333  df-lat 15339  df-clat 15401  df-mnd 15538  df-submnd 15588  df-grp 15668  df-minusg 15669  df-sbg 15670  df-subg 15801  df-lsm 16260  df-mgp 16724  df-ur 16736  df-rng 16780  df-oppr 16848  df-dvdsr 16866  df-unit 16867  df-invr 16897  df-dvr 16908  df-drng 16967  df-lmod 17083  df-lss 17147  df-lvec 17317  df-oposet 33184  df-ol 33186  df-oml 33187  df-covers 33274  df-ats 33275  df-atl 33306  df-cvlat 33330  df-hlat 33359  df-llines 33505  df-lplanes 33506  df-lvols 33507  df-lines 33508  df-psubsp 33510  df-pmap 33511  df-padd 33803  df-lhyp 33995  df-laut 33996  df-ldil 34111  df-ltrn 34112  df-trl 34166  df-tendo 34762  df-edring 34764  df-disoa 35037  df-dvech 35087  df-dib 35147  df-dic 35181
This theorem is referenced by:  dihord11b  35230
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