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

Theorem mapdlsm 32399
Description: Subspace sum is preserved by the map defined by df-mapd 32360. Part of property (e) in [Baer] p. 40. (Contributed by NM, 13-Mar-2015.)
Hypotheses
Ref Expression
mapdlsm.h  |-  H  =  ( LHyp `  K
)
mapdlsm.m  |-  M  =  ( (mapd `  K
) `  W )
mapdlsm.u  |-  U  =  ( ( DVecH `  K
) `  W )
mapdlsm.s  |-  S  =  ( LSubSp `  U )
mapdlsm.p  |-  .(+)  =  (
LSSum `  U )
mapdlsm.c  |-  C  =  ( (LCDual `  K
) `  W )
mapdlsm.q  |-  .+b  =  ( LSSum `  C )
mapdlsm.k  |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )
mapdlsm.x  |-  ( ph  ->  X  e.  S )
mapdlsm.y  |-  ( ph  ->  Y  e.  S )
Assertion
Ref Expression
mapdlsm  |-  ( ph  ->  ( M `  ( X  .(+)  Y ) )  =  ( ( M `
 X )  .+b  ( M `  Y ) ) )

Proof of Theorem mapdlsm
StepHypRef Expression
1 mapdlsm.h . . . . . . . . . . 11  |-  H  =  ( LHyp `  K
)
2 mapdlsm.c . . . . . . . . . . 11  |-  C  =  ( (LCDual `  K
) `  W )
3 mapdlsm.k . . . . . . . . . . 11  |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )
41, 2, 3lcdlmod 32327 . . . . . . . . . 10  |-  ( ph  ->  C  e.  LMod )
5 eqid 2435 . . . . . . . . . . 11  |-  ( LSubSp `  C )  =  (
LSubSp `  C )
65lsssssubg 16026 . . . . . . . . . 10  |-  ( C  e.  LMod  ->  ( LSubSp `  C )  C_  (SubGrp `  C ) )
74, 6syl 16 . . . . . . . . 9  |-  ( ph  ->  ( LSubSp `  C )  C_  (SubGrp `  C )
)
8 mapdlsm.m . . . . . . . . . 10  |-  M  =  ( (mapd `  K
) `  W )
9 mapdlsm.u . . . . . . . . . 10  |-  U  =  ( ( DVecH `  K
) `  W )
10 mapdlsm.s . . . . . . . . . 10  |-  S  =  ( LSubSp `  U )
11 mapdlsm.x . . . . . . . . . 10  |-  ( ph  ->  X  e.  S )
121, 8, 9, 10, 2, 5, 3, 11mapdcl2 32391 . . . . . . . . 9  |-  ( ph  ->  ( M `  X
)  e.  ( LSubSp `  C ) )
137, 12sseldd 3341 . . . . . . . 8  |-  ( ph  ->  ( M `  X
)  e.  (SubGrp `  C ) )
14 mapdlsm.y . . . . . . . . . 10  |-  ( ph  ->  Y  e.  S )
151, 8, 9, 10, 2, 5, 3, 14mapdcl2 32391 . . . . . . . . 9  |-  ( ph  ->  ( M `  Y
)  e.  ( LSubSp `  C ) )
167, 15sseldd 3341 . . . . . . . 8  |-  ( ph  ->  ( M `  Y
)  e.  (SubGrp `  C ) )
17 mapdlsm.q . . . . . . . . 9  |-  .+b  =  ( LSSum `  C )
1817lsmub1 15282 . . . . . . . 8  |-  ( ( ( M `  X
)  e.  (SubGrp `  C )  /\  ( M `  Y )  e.  (SubGrp `  C )
)  ->  ( M `  X )  C_  (
( M `  X
)  .+b  ( M `  Y ) ) )
1913, 16, 18syl2anc 643 . . . . . . 7  |-  ( ph  ->  ( M `  X
)  C_  ( ( M `  X )  .+b  ( M `  Y
) ) )
201, 8, 9, 10, 3, 11mapdcl 32388 . . . . . . . . 9  |-  ( ph  ->  ( M `  X
)  e.  ran  M
)
211, 8, 9, 10, 3, 14mapdcl 32388 . . . . . . . . 9  |-  ( ph  ->  ( M `  Y
)  e.  ran  M
)
221, 8, 9, 2, 17, 3, 20, 21mapdlsmcl 32398 . . . . . . . 8  |-  ( ph  ->  ( ( M `  X )  .+b  ( M `  Y )
)  e.  ran  M
)
231, 8, 3, 22mapdcnvid2 32392 . . . . . . 7  |-  ( ph  ->  ( M `  ( `' M `  ( ( M `  X ) 
.+b  ( M `  Y ) ) ) )  =  ( ( M `  X ) 
.+b  ( M `  Y ) ) )
2419, 23sseqtr4d 3377 . . . . . 6  |-  ( ph  ->  ( M `  X
)  C_  ( M `  ( `' M `  ( ( M `  X )  .+b  ( M `  Y )
) ) ) )
251, 8, 9, 10, 3, 22mapdcnvcl 32387 . . . . . . 7  |-  ( ph  ->  ( `' M `  ( ( M `  X )  .+b  ( M `  Y )
) )  e.  S
)
261, 9, 10, 8, 3, 11, 25mapdord 32373 . . . . . 6  |-  ( ph  ->  ( ( M `  X )  C_  ( M `  ( `' M `  ( ( M `  X )  .+b  ( M `  Y
) ) ) )  <-> 
X  C_  ( `' M `  ( ( M `  X )  .+b  ( M `  Y
) ) ) ) )
2724, 26mpbid 202 . . . . 5  |-  ( ph  ->  X  C_  ( `' M `  ( ( M `  X )  .+b  ( M `  Y
) ) ) )
2817lsmub2 15283 . . . . . . . 8  |-  ( ( ( M `  X
)  e.  (SubGrp `  C )  /\  ( M `  Y )  e.  (SubGrp `  C )
)  ->  ( M `  Y )  C_  (
( M `  X
)  .+b  ( M `  Y ) ) )
2913, 16, 28syl2anc 643 . . . . . . 7  |-  ( ph  ->  ( M `  Y
)  C_  ( ( M `  X )  .+b  ( M `  Y
) ) )
3029, 23sseqtr4d 3377 . . . . . 6  |-  ( ph  ->  ( M `  Y
)  C_  ( M `  ( `' M `  ( ( M `  X )  .+b  ( M `  Y )
) ) ) )
311, 9, 10, 8, 3, 14, 25mapdord 32373 . . . . . 6  |-  ( ph  ->  ( ( M `  Y )  C_  ( M `  ( `' M `  ( ( M `  X )  .+b  ( M `  Y
) ) ) )  <-> 
Y  C_  ( `' M `  ( ( M `  X )  .+b  ( M `  Y
) ) ) ) )
3230, 31mpbid 202 . . . . 5  |-  ( ph  ->  Y  C_  ( `' M `  ( ( M `  X )  .+b  ( M `  Y
) ) ) )
331, 9, 3dvhlmod 31845 . . . . . . . 8  |-  ( ph  ->  U  e.  LMod )
3410lsssssubg 16026 . . . . . . . 8  |-  ( U  e.  LMod  ->  S  C_  (SubGrp `  U ) )
3533, 34syl 16 . . . . . . 7  |-  ( ph  ->  S  C_  (SubGrp `  U
) )
3635, 11sseldd 3341 . . . . . 6  |-  ( ph  ->  X  e.  (SubGrp `  U ) )
3735, 14sseldd 3341 . . . . . 6  |-  ( ph  ->  Y  e.  (SubGrp `  U ) )
3835, 25sseldd 3341 . . . . . 6  |-  ( ph  ->  ( `' M `  ( ( M `  X )  .+b  ( M `  Y )
) )  e.  (SubGrp `  U ) )
39 mapdlsm.p . . . . . . 7  |-  .(+)  =  (
LSSum `  U )
4039lsmlub 15289 . . . . . 6  |-  ( ( X  e.  (SubGrp `  U )  /\  Y  e.  (SubGrp `  U )  /\  ( `' M `  ( ( M `  X )  .+b  ( M `  Y )
) )  e.  (SubGrp `  U ) )  -> 
( ( X  C_  ( `' M `  ( ( M `  X ) 
.+b  ( M `  Y ) ) )  /\  Y  C_  ( `' M `  ( ( M `  X ) 
.+b  ( M `  Y ) ) ) )  <->  ( X  .(+)  Y )  C_  ( `' M `  ( ( M `  X )  .+b  ( M `  Y
) ) ) ) )
4136, 37, 38, 40syl3anc 1184 . . . . 5  |-  ( ph  ->  ( ( X  C_  ( `' M `  ( ( M `  X ) 
.+b  ( M `  Y ) ) )  /\  Y  C_  ( `' M `  ( ( M `  X ) 
.+b  ( M `  Y ) ) ) )  <->  ( X  .(+)  Y )  C_  ( `' M `  ( ( M `  X )  .+b  ( M `  Y
) ) ) ) )
4227, 32, 41mpbi2and 888 . . . 4  |-  ( ph  ->  ( X  .(+)  Y ) 
C_  ( `' M `  ( ( M `  X )  .+b  ( M `  Y )
) ) )
4310, 39lsmcl 16147 . . . . . 6  |-  ( ( U  e.  LMod  /\  X  e.  S  /\  Y  e.  S )  ->  ( X  .(+)  Y )  e.  S )
4433, 11, 14, 43syl3anc 1184 . . . . 5  |-  ( ph  ->  ( X  .(+)  Y )  e.  S )
451, 9, 10, 8, 3, 44, 25mapdord 32373 . . . 4  |-  ( ph  ->  ( ( M `  ( X  .(+)  Y ) )  C_  ( M `  ( `' M `  ( ( M `  X )  .+b  ( M `  Y )
) ) )  <->  ( X  .(+) 
Y )  C_  ( `' M `  ( ( M `  X ) 
.+b  ( M `  Y ) ) ) ) )
4642, 45mpbird 224 . . 3  |-  ( ph  ->  ( M `  ( X  .(+)  Y ) ) 
C_  ( M `  ( `' M `  ( ( M `  X ) 
.+b  ( M `  Y ) ) ) ) )
4746, 23sseqtrd 3376 . 2  |-  ( ph  ->  ( M `  ( X  .(+)  Y ) ) 
C_  ( ( M `
 X )  .+b  ( M `  Y ) ) )
4839lsmub1 15282 . . . . 5  |-  ( ( X  e.  (SubGrp `  U )  /\  Y  e.  (SubGrp `  U )
)  ->  X  C_  ( X  .(+)  Y ) )
4936, 37, 48syl2anc 643 . . . 4  |-  ( ph  ->  X  C_  ( X  .(+) 
Y ) )
501, 9, 10, 8, 3, 11, 44mapdord 32373 . . . 4  |-  ( ph  ->  ( ( M `  X )  C_  ( M `  ( X  .(+) 
Y ) )  <->  X  C_  ( X  .(+)  Y ) ) )
5149, 50mpbird 224 . . 3  |-  ( ph  ->  ( M `  X
)  C_  ( M `  ( X  .(+)  Y ) ) )
5239lsmub2 15283 . . . . 5  |-  ( ( X  e.  (SubGrp `  U )  /\  Y  e.  (SubGrp `  U )
)  ->  Y  C_  ( X  .(+)  Y ) )
5336, 37, 52syl2anc 643 . . . 4  |-  ( ph  ->  Y  C_  ( X  .(+) 
Y ) )
541, 9, 10, 8, 3, 14, 44mapdord 32373 . . . 4  |-  ( ph  ->  ( ( M `  Y )  C_  ( M `  ( X  .(+) 
Y ) )  <->  Y  C_  ( X  .(+)  Y ) ) )
5553, 54mpbird 224 . . 3  |-  ( ph  ->  ( M `  Y
)  C_  ( M `  ( X  .(+)  Y ) ) )
561, 8, 9, 10, 2, 5, 3, 44mapdcl2 32391 . . . . 5  |-  ( ph  ->  ( M `  ( X  .(+)  Y ) )  e.  ( LSubSp `  C
) )
577, 56sseldd 3341 . . . 4  |-  ( ph  ->  ( M `  ( X  .(+)  Y ) )  e.  (SubGrp `  C
) )
5817lsmlub 15289 . . . 4  |-  ( ( ( M `  X
)  e.  (SubGrp `  C )  /\  ( M `  Y )  e.  (SubGrp `  C )  /\  ( M `  ( X  .(+)  Y ) )  e.  (SubGrp `  C
) )  ->  (
( ( M `  X )  C_  ( M `  ( X  .(+) 
Y ) )  /\  ( M `  Y ) 
C_  ( M `  ( X  .(+)  Y ) ) )  <->  ( ( M `  X )  .+b  ( M `  Y
) )  C_  ( M `  ( X  .(+) 
Y ) ) ) )
5913, 16, 57, 58syl3anc 1184 . . 3  |-  ( ph  ->  ( ( ( M `
 X )  C_  ( M `  ( X 
.(+)  Y ) )  /\  ( M `  Y ) 
C_  ( M `  ( X  .(+)  Y ) ) )  <->  ( ( M `  X )  .+b  ( M `  Y
) )  C_  ( M `  ( X  .(+) 
Y ) ) ) )
6051, 55, 59mpbi2and 888 . 2  |-  ( ph  ->  ( ( M `  X )  .+b  ( M `  Y )
)  C_  ( M `  ( X  .(+)  Y ) ) )
6147, 60eqssd 3357 1  |-  ( ph  ->  ( M `  ( X  .(+)  Y ) )  =  ( ( M `
 X )  .+b  ( M `  Y ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725    C_ wss 3312   `'ccnv 4869   ` cfv 5446  (class class class)co 6073  SubGrpcsubg 14930   LSSumclsm 15260   LModclmod 15942   LSubSpclss 16000   HLchlt 30085   LHypclh 30718   DVecHcdvh 31813  LCDualclcd 32321  mapdcmpd 32359
This theorem is referenced by:  mapdindp  32406  mapdpglem1  32407  mapdheq4lem  32466  mapdh6lem1N  32468  mapdh6lem2N  32469  hdmap1l6lem1  32543  hdmap1l6lem2  32544  hdmaprnlem3eN  32596
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-fal 1329  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-of 6297  df-1st 6341  df-2nd 6342  df-tpos 6471  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-sets 13467  df-ress 13468  df-plusg 13534  df-mulr 13535  df-sca 13537  df-vsca 13538  df-0g 13719  df-mre 13803  df-mrc 13804  df-acs 13806  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-mnd 14682  df-submnd 14731  df-grp 14804  df-minusg 14805  df-sbg 14806  df-subg 14933  df-cntz 15108  df-oppg 15134  df-lsm 15262  df-cmn 15406  df-abl 15407  df-mgp 15641  df-rng 15655  df-ur 15657  df-oppr 15720  df-dvdsr 15738  df-unit 15739  df-invr 15769  df-dvr 15780  df-drng 15829  df-lmod 15944  df-lss 16001  df-lsp 16040  df-lvec 16167  df-lsatoms 29711  df-lshyp 29712  df-lcv 29754  df-lfl 29793  df-lkr 29821  df-ldual 29859  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-tgrp 31477  df-tendo 31489  df-edring 31491  df-dveca 31737  df-disoa 31764  df-dvech 31814  df-dib 31874  df-dic 31908  df-dih 31964  df-doch 32083  df-djh 32130  df-lcdual 32322  df-mapd 32360
  Copyright terms: Public domain W3C validator