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Theorem dihvalcqat 36392
Description: Value of isomorphism H for a lattice  K at an atom not under  W. (Contributed by NM, 27-Mar-2014.)
Hypotheses
Ref Expression
dihvalcqat.l  |-  .<_  =  ( le `  K )
dihvalcqat.a  |-  A  =  ( Atoms `  K )
dihvalcqat.h  |-  H  =  ( LHyp `  K
)
dihvalcqat.j  |-  J  =  ( ( DIsoC `  K
) `  W )
dihvalcqat.i  |-  I  =  ( ( DIsoH `  K
) `  W )
Assertion
Ref Expression
dihvalcqat  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  -> 
( I `  Q
)  =  ( J `
 Q ) )

Proof of Theorem dihvalcqat
StepHypRef Expression
1 simpl 457 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  -> 
( K  e.  HL  /\  W  e.  H ) )
2 eqid 2467 . . . . 5  |-  ( Base `  K )  =  (
Base `  K )
3 dihvalcqat.a . . . . 5  |-  A  =  ( Atoms `  K )
42, 3atbase 34442 . . . 4  |-  ( Q  e.  A  ->  Q  e.  ( Base `  K
) )
54ad2antrl 727 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  ->  Q  e.  ( Base `  K ) )
6 simprr 756 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  ->  -.  Q  .<_  W )
7 simpr 461 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  -> 
( Q  e.  A  /\  -.  Q  .<_  W ) )
8 dihvalcqat.l . . . . . 6  |-  .<_  =  ( le `  K )
9 eqid 2467 . . . . . 6  |-  ( meet `  K )  =  (
meet `  K )
10 eqid 2467 . . . . . 6  |-  ( 0.
`  K )  =  ( 0. `  K
)
11 dihvalcqat.h . . . . . 6  |-  H  =  ( LHyp `  K
)
128, 9, 10, 3, 11lhpmat 35182 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  -> 
( Q ( meet `  K ) W )  =  ( 0. `  K ) )
1312oveq2d 6311 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  -> 
( Q ( join `  K ) ( Q ( meet `  K
) W ) )  =  ( Q (
join `  K )
( 0. `  K
) ) )
14 hlol 34514 . . . . . 6  |-  ( K  e.  HL  ->  K  e.  OL )
1514ad2antrr 725 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  ->  K  e.  OL )
16 eqid 2467 . . . . . 6  |-  ( join `  K )  =  (
join `  K )
172, 16, 10olj01 34378 . . . . 5  |-  ( ( K  e.  OL  /\  Q  e.  ( Base `  K ) )  -> 
( Q ( join `  K ) ( 0.
`  K ) )  =  Q )
1815, 5, 17syl2anc 661 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  -> 
( Q ( join `  K ) ( 0.
`  K ) )  =  Q )
1913, 18eqtrd 2508 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  -> 
( Q ( join `  K ) ( Q ( meet `  K
) W ) )  =  Q )
20 dihvalcqat.i . . . 4  |-  I  =  ( ( DIsoH `  K
) `  W )
21 eqid 2467 . . . 4  |-  ( (
DIsoB `  K ) `  W )  =  ( ( DIsoB `  K ) `  W )
22 dihvalcqat.j . . . 4  |-  J  =  ( ( DIsoC `  K
) `  W )
23 eqid 2467 . . . 4  |-  ( (
DVecH `  K ) `  W )  =  ( ( DVecH `  K ) `  W )
24 eqid 2467 . . . 4  |-  ( LSSum `  ( ( DVecH `  K
) `  W )
)  =  ( LSSum `  ( ( DVecH `  K
) `  W )
)
252, 8, 16, 9, 3, 11, 20, 21, 22, 23, 24dihvalcq 36389 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  ( Base `  K
)  /\  -.  Q  .<_  W )  /\  (
( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( Q (
join `  K )
( Q ( meet `  K ) W ) )  =  Q ) )  ->  ( I `  Q )  =  ( ( J `  Q
) ( LSSum `  (
( DVecH `  K ) `  W ) ) ( ( ( DIsoB `  K
) `  W ) `  ( Q ( meet `  K ) W ) ) ) )
261, 5, 6, 7, 19, 25syl122anc 1237 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  -> 
( I `  Q
)  =  ( ( J `  Q ) ( LSSum `  ( ( DVecH `  K ) `  W ) ) ( ( ( DIsoB `  K
) `  W ) `  ( Q ( meet `  K ) W ) ) ) )
2712fveq2d 5876 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  -> 
( ( ( DIsoB `  K ) `  W
) `  ( Q
( meet `  K ) W ) )  =  ( ( ( DIsoB `  K ) `  W
) `  ( 0. `  K ) ) )
28 eqid 2467 . . . . . . 7  |-  ( 0g
`  ( ( DVecH `  K ) `  W
) )  =  ( 0g `  ( (
DVecH `  K ) `  W ) )
2910, 11, 21, 23, 28dib0 36317 . . . . . 6  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( ( ( DIsoB `  K ) `  W
) `  ( 0. `  K ) )  =  { ( 0g `  ( ( DVecH `  K
) `  W )
) } )
3029adantr 465 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  -> 
( ( ( DIsoB `  K ) `  W
) `  ( 0. `  K ) )  =  { ( 0g `  ( ( DVecH `  K
) `  W )
) } )
3127, 30eqtrd 2508 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  -> 
( ( ( DIsoB `  K ) `  W
) `  ( Q
( meet `  K ) W ) )  =  { ( 0g `  ( ( DVecH `  K
) `  W )
) } )
3231oveq2d 6311 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  -> 
( ( J `  Q ) ( LSSum `  ( ( DVecH `  K
) `  W )
) ( ( (
DIsoB `  K ) `  W ) `  ( Q ( meet `  K
) W ) ) )  =  ( ( J `  Q ) ( LSSum `  ( ( DVecH `  K ) `  W ) ) { ( 0g `  (
( DVecH `  K ) `  W ) ) } ) )
3311, 23, 1dvhlmod 36263 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  -> 
( ( DVecH `  K
) `  W )  e.  LMod )
34 eqid 2467 . . . . . 6  |-  ( LSubSp `  ( ( DVecH `  K
) `  W )
)  =  ( LSubSp `  ( ( DVecH `  K
) `  W )
)
358, 3, 11, 23, 22, 34diclss 36346 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  -> 
( J `  Q
)  e.  ( LSubSp `  ( ( DVecH `  K
) `  W )
) )
3634lsssubg 17474 . . . . 5  |-  ( ( ( ( DVecH `  K
) `  W )  e.  LMod  /\  ( J `  Q )  e.  (
LSubSp `  ( ( DVecH `  K ) `  W
) ) )  -> 
( J `  Q
)  e.  (SubGrp `  ( ( DVecH `  K
) `  W )
) )
3733, 35, 36syl2anc 661 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  -> 
( J `  Q
)  e.  (SubGrp `  ( ( DVecH `  K
) `  W )
) )
3828, 24lsm01 16562 . . . 4  |-  ( ( J `  Q )  e.  (SubGrp `  (
( DVecH `  K ) `  W ) )  -> 
( ( J `  Q ) ( LSSum `  ( ( DVecH `  K
) `  W )
) { ( 0g
`  ( ( DVecH `  K ) `  W
) ) } )  =  ( J `  Q ) )
3937, 38syl 16 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  -> 
( ( J `  Q ) ( LSSum `  ( ( DVecH `  K
) `  W )
) { ( 0g
`  ( ( DVecH `  K ) `  W
) ) } )  =  ( J `  Q ) )
4032, 39eqtrd 2508 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  -> 
( ( J `  Q ) ( LSSum `  ( ( DVecH `  K
) `  W )
) ( ( (
DIsoB `  K ) `  W ) `  ( Q ( meet `  K
) W ) ) )  =  ( J `
 Q ) )
4126, 40eqtrd 2508 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  -> 
( I `  Q
)  =  ( J `
 Q ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767   {csn 4033   class class class wbr 4453   ` cfv 5594  (class class class)co 6295   Basecbs 14507   lecple 14579   0gc0g 14712   joincjn 15448   meetcmee 15449   0.cp0 15541  SubGrpcsubg 16067   LSSumclsm 16527   LModclmod 17383   LSubSpclss 17449   OLcol 34327   Atomscatm 34416   HLchlt 34503   LHypclh 35136   DVecHcdvh 36231   DIsoBcdib 36291   DIsoCcdic 36325   DIsoHcdih 36381
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581  ax-riotaBAD 34112
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-int 4289  df-iun 4333  df-iin 4334  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6696  df-1st 6795  df-2nd 6796  df-tpos 6967  df-undef 7014  df-recs 7054  df-rdg 7088  df-1o 7142  df-oadd 7146  df-er 7323  df-map 7434  df-en 7529  df-dom 7530  df-sdom 7531  df-fin 7532  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-nn 10549  df-2 10606  df-3 10607  df-4 10608  df-5 10609  df-6 10610  df-n0 10808  df-z 10877  df-uz 11095  df-fz 11685  df-struct 14509  df-ndx 14510  df-slot 14511  df-base 14512  df-sets 14513  df-ress 14514  df-plusg 14585  df-mulr 14586  df-sca 14588  df-vsca 14589  df-0g 14714  df-poset 15450  df-plt 15462  df-lub 15478  df-glb 15479  df-join 15480  df-meet 15481  df-p0 15543  df-p1 15544  df-lat 15550  df-clat 15612  df-mgm 15746  df-sgrp 15785  df-mnd 15795  df-submnd 15840  df-grp 15929  df-minusg 15930  df-sbg 15931  df-subg 16070  df-cntz 16227  df-lsm 16529  df-cmn 16673  df-abl 16674  df-mgp 17014  df-ur 17026  df-ring 17072  df-oppr 17144  df-dvdsr 17162  df-unit 17163  df-invr 17193  df-dvr 17204  df-drng 17269  df-lmod 17385  df-lss 17450  df-lsp 17489  df-lvec 17620  df-oposet 34329  df-ol 34331  df-oml 34332  df-covers 34419  df-ats 34420  df-atl 34451  df-cvlat 34475  df-hlat 34504  df-llines 34650  df-lplanes 34651  df-lvols 34652  df-lines 34653  df-psubsp 34655  df-pmap 34656  df-padd 34948  df-lhyp 35140  df-laut 35141  df-ldil 35256  df-ltrn 35257  df-trl 35311  df-tendo 35907  df-edring 35909  df-disoa 36182  df-dvech 36232  df-dib 36292  df-dic 36326  df-dih 36382
This theorem is referenced by:  dih1dimc  36395  dihopelvalcqat  36399  dihvalcq2  36400  dih1dimatlem  36482  dihpN  36489
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