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

Theorem mapdlsm 31927
Description: Subspace sum is preserved by the map defined by df-mapd 31888. 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 31855 . . . . . . . . . 10  |-  ( ph  ->  C  e.  LMod )
5 eqid 2285 . . . . . . . . . . 11  |-  ( LSubSp `  C )  =  (
LSubSp `  C )
65lsssssubg 15717 . . . . . . . . . 10  |-  ( C  e.  LMod  ->  ( LSubSp `  C )  C_  (SubGrp `  C ) )
74, 6syl 15 . . . . . . . . 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 31919 . . . . . . . . 9  |-  ( ph  ->  ( M `  X
)  e.  ( LSubSp `  C ) )
137, 12sseldd 3183 . . . . . . . 8  |-  ( ph  ->  ( M `  X
)  e.  (SubGrp `  C ) )
14 mapdlsm.y . . . . . . . . . 10  |-  ( ph  ->  Y  e.  S )
151, 8, 9, 10, 2, 5, 3, 14mapdcl2 31919 . . . . . . . . 9  |-  ( ph  ->  ( M `  Y
)  e.  ( LSubSp `  C ) )
167, 15sseldd 3183 . . . . . . . 8  |-  ( ph  ->  ( M `  Y
)  e.  (SubGrp `  C ) )
17 mapdlsm.q . . . . . . . . 9  |-  .+b  =  ( LSSum `  C )
1817lsmub1 14969 . . . . . . . 8  |-  ( ( ( M `  X
)  e.  (SubGrp `  C )  /\  ( M `  Y )  e.  (SubGrp `  C )
)  ->  ( M `  X )  C_  (
( M `  X
)  .+b  ( M `  Y ) ) )
1913, 16, 18syl2anc 642 . . . . . . 7  |-  ( ph  ->  ( M `  X
)  C_  ( ( M `  X )  .+b  ( M `  Y
) ) )
201, 8, 9, 10, 3, 11mapdcl 31916 . . . . . . . . 9  |-  ( ph  ->  ( M `  X
)  e.  ran  M
)
211, 8, 9, 10, 3, 14mapdcl 31916 . . . . . . . . 9  |-  ( ph  ->  ( M `  Y
)  e.  ran  M
)
221, 8, 9, 2, 17, 3, 20, 21mapdlsmcl 31926 . . . . . . . 8  |-  ( ph  ->  ( ( M `  X )  .+b  ( M `  Y )
)  e.  ran  M
)
231, 8, 3, 22mapdcnvid2 31920 . . . . . . 7  |-  ( ph  ->  ( M `  ( `' M `  ( ( M `  X ) 
.+b  ( M `  Y ) ) ) )  =  ( ( M `  X ) 
.+b  ( M `  Y ) ) )
2419, 23sseqtr4d 3217 . . . . . 6  |-  ( ph  ->  ( M `  X
)  C_  ( M `  ( `' M `  ( ( M `  X )  .+b  ( M `  Y )
) ) ) )
251, 8, 9, 10, 3, 22mapdcnvcl 31915 . . . . . . 7  |-  ( ph  ->  ( `' M `  ( ( M `  X )  .+b  ( M `  Y )
) )  e.  S
)
261, 9, 10, 8, 3, 11, 25mapdord 31901 . . . . . 6  |-  ( ph  ->  ( ( M `  X )  C_  ( M `  ( `' M `  ( ( M `  X )  .+b  ( M `  Y
) ) ) )  <-> 
X  C_  ( `' M `  ( ( M `  X )  .+b  ( M `  Y
) ) ) ) )
2724, 26mpbid 201 . . . . 5  |-  ( ph  ->  X  C_  ( `' M `  ( ( M `  X )  .+b  ( M `  Y
) ) ) )
2817lsmub2 14970 . . . . . . . 8  |-  ( ( ( M `  X
)  e.  (SubGrp `  C )  /\  ( M `  Y )  e.  (SubGrp `  C )
)  ->  ( M `  Y )  C_  (
( M `  X
)  .+b  ( M `  Y ) ) )
2913, 16, 28syl2anc 642 . . . . . . 7  |-  ( ph  ->  ( M `  Y
)  C_  ( ( M `  X )  .+b  ( M `  Y
) ) )
3029, 23sseqtr4d 3217 . . . . . 6  |-  ( ph  ->  ( M `  Y
)  C_  ( M `  ( `' M `  ( ( M `  X )  .+b  ( M `  Y )
) ) ) )
311, 9, 10, 8, 3, 14, 25mapdord 31901 . . . . . 6  |-  ( ph  ->  ( ( M `  Y )  C_  ( M `  ( `' M `  ( ( M `  X )  .+b  ( M `  Y
) ) ) )  <-> 
Y  C_  ( `' M `  ( ( M `  X )  .+b  ( M `  Y
) ) ) ) )
3230, 31mpbid 201 . . . . 5  |-  ( ph  ->  Y  C_  ( `' M `  ( ( M `  X )  .+b  ( M `  Y
) ) ) )
331, 9, 3dvhlmod 31373 . . . . . . . 8  |-  ( ph  ->  U  e.  LMod )
3410lsssssubg 15717 . . . . . . . 8  |-  ( U  e.  LMod  ->  S  C_  (SubGrp `  U ) )
3533, 34syl 15 . . . . . . 7  |-  ( ph  ->  S  C_  (SubGrp `  U
) )
3635, 11sseldd 3183 . . . . . 6  |-  ( ph  ->  X  e.  (SubGrp `  U ) )
3735, 14sseldd 3183 . . . . . 6  |-  ( ph  ->  Y  e.  (SubGrp `  U ) )
3835, 25sseldd 3183 . . . . . 6  |-  ( ph  ->  ( `' M `  ( ( M `  X )  .+b  ( M `  Y )
) )  e.  (SubGrp `  U ) )
39 mapdlsm.p . . . . . . 7  |-  .(+)  =  (
LSSum `  U )
4039lsmlub 14976 . . . . . 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 1182 . . . . 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 887 . . . 4  |-  ( ph  ->  ( X  .(+)  Y ) 
C_  ( `' M `  ( ( M `  X )  .+b  ( M `  Y )
) ) )
4310, 39lsmcl 15838 . . . . . 6  |-  ( ( U  e.  LMod  /\  X  e.  S  /\  Y  e.  S )  ->  ( X  .(+)  Y )  e.  S )
4433, 11, 14, 43syl3anc 1182 . . . . 5  |-  ( ph  ->  ( X  .(+)  Y )  e.  S )
451, 9, 10, 8, 3, 44, 25mapdord 31901 . . . 4  |-  ( ph  ->  ( ( M `  ( X  .(+)  Y ) )  C_  ( M `  ( `' M `  ( ( M `  X )  .+b  ( M `  Y )
) ) )  <->  ( X  .(+) 
Y )  C_  ( `' M `  ( ( M `  X ) 
.+b  ( M `  Y ) ) ) ) )
4642, 45mpbird 223 . . 3  |-  ( ph  ->  ( M `  ( X  .(+)  Y ) ) 
C_  ( M `  ( `' M `  ( ( M `  X ) 
.+b  ( M `  Y ) ) ) ) )
4746, 23sseqtrd 3216 . 2  |-  ( ph  ->  ( M `  ( X  .(+)  Y ) ) 
C_  ( ( M `
 X )  .+b  ( M `  Y ) ) )
4839lsmub1 14969 . . . . 5  |-  ( ( X  e.  (SubGrp `  U )  /\  Y  e.  (SubGrp `  U )
)  ->  X  C_  ( X  .(+)  Y ) )
4936, 37, 48syl2anc 642 . . . 4  |-  ( ph  ->  X  C_  ( X  .(+) 
Y ) )
501, 9, 10, 8, 3, 11, 44mapdord 31901 . . . 4  |-  ( ph  ->  ( ( M `  X )  C_  ( M `  ( X  .(+) 
Y ) )  <->  X  C_  ( X  .(+)  Y ) ) )
5149, 50mpbird 223 . . 3  |-  ( ph  ->  ( M `  X
)  C_  ( M `  ( X  .(+)  Y ) ) )
5239lsmub2 14970 . . . . 5  |-  ( ( X  e.  (SubGrp `  U )  /\  Y  e.  (SubGrp `  U )
)  ->  Y  C_  ( X  .(+)  Y ) )
5336, 37, 52syl2anc 642 . . . 4  |-  ( ph  ->  Y  C_  ( X  .(+) 
Y ) )
541, 9, 10, 8, 3, 14, 44mapdord 31901 . . . 4  |-  ( ph  ->  ( ( M `  Y )  C_  ( M `  ( X  .(+) 
Y ) )  <->  Y  C_  ( X  .(+)  Y ) ) )
5553, 54mpbird 223 . . 3  |-  ( ph  ->  ( M `  Y
)  C_  ( M `  ( X  .(+)  Y ) ) )
561, 8, 9, 10, 2, 5, 3, 44mapdcl2 31919 . . . . 5  |-  ( ph  ->  ( M `  ( X  .(+)  Y ) )  e.  ( LSubSp `  C
) )
577, 56sseldd 3183 . . . 4  |-  ( ph  ->  ( M `  ( X  .(+)  Y ) )  e.  (SubGrp `  C
) )
5817lsmlub 14976 . . . 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 1182 . . 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 887 . 2  |-  ( ph  ->  ( ( M `  X )  .+b  ( M `  Y )
)  C_  ( M `  ( X  .(+)  Y ) ) )
6147, 60eqssd 3198 1  |-  ( ph  ->  ( M `  ( X  .(+)  Y ) )  =  ( ( M `
 X )  .+b  ( M `  Y ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1625    e. wcel 1686    C_ wss 3154   `'ccnv 4690   ` cfv 5257  (class class class)co 5860  SubGrpcsubg 14617   LSSumclsm 14947   LModclmod 15629   LSubSpclss 15691   HLchlt 29613   LHypclh 30246   DVecHcdvh 31341  LCDualclcd 31849  mapdcmpd 31887
This theorem is referenced by:  mapdindp  31934  mapdpglem1  31935  mapdheq4lem  31994  mapdh6lem1N  31996  mapdh6lem2N  31997  hdmap1l6lem1  32071  hdmap1l6lem2  32072  hdmaprnlem3eN  32124
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-13 1688  ax-14 1690  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266  ax-rep 4133  ax-sep 4143  ax-nul 4151  ax-pow 4190  ax-pr 4216  ax-un 4514  ax-cnex 8795  ax-resscn 8796  ax-1cn 8797  ax-icn 8798  ax-addcl 8799  ax-addrcl 8800  ax-mulcl 8801  ax-mulrcl 8802  ax-mulcom 8803  ax-addass 8804  ax-mulass 8805  ax-distr 8806  ax-i2m1 8807  ax-1ne0 8808  ax-1rid 8809  ax-rnegex 8810  ax-rrecex 8811  ax-cnre 8812  ax-pre-lttri 8813  ax-pre-lttrn 8814  ax-pre-ltadd 8815  ax-pre-mulgt0 8816
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-fal 1311  df-ex 1531  df-nf 1534  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-nel 2451  df-ral 2550  df-rex 2551  df-reu 2552  df-rmo 2553  df-rab 2554  df-v 2792  df-sbc 2994  df-csb 3084  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-pss 3170  df-nul 3458  df-if 3568  df-pw 3629  df-sn 3648  df-pr 3649  df-tp 3650  df-op 3651  df-uni 3830  df-int 3865  df-iun 3909  df-iin 3910  df-br 4026  df-opab 4080  df-mpt 4081  df-tr 4116  df-eprel 4307  df-id 4311  df-po 4316  df-so 4317  df-fr 4354  df-we 4356  df-ord 4397  df-on 4398  df-lim 4399  df-suc 4400  df-om 4659  df-xp 4697  df-rel 4698  df-cnv 4699  df-co 4700  df-dm 4701  df-rn 4702  df-res 4703  df-ima 4704  df-iota 5221  df-fun 5259  df-fn 5260  df-f 5261  df-f1 5262  df-fo 5263  df-f1o 5264  df-fv 5265  df-ov 5863  df-oprab 5864  df-mpt2 5865  df-of 6080  df-1st 6124  df-2nd 6125  df-tpos 6236  df-undef 6300  df-riota 6306  df-recs 6390  df-rdg 6425  df-1o 6481  df-oadd 6485  df-er 6662  df-map 6776  df-en 6866  df-dom 6867  df-sdom 6868  df-fin 6869  df-pnf 8871  df-mnf 8872  df-xr 8873  df-ltxr 8874  df-le 8875  df-sub 9041  df-neg 9042  df-nn 9749  df-2 9806  df-3 9807  df-4 9808  df-5 9809  df-6 9810  df-n0 9968  df-z 10027  df-uz 10233  df-fz 10785  df-struct 13152  df-ndx 13153  df-slot 13154  df-base 13155  df-sets 13156  df-ress 13157  df-plusg 13223  df-mulr 13224  df-sca 13226  df-vsca 13227  df-0g 13406  df-mre 13490  df-mrc 13491  df-acs 13493  df-poset 14082  df-plt 14094  df-lub 14110  df-glb 14111  df-join 14112  df-meet 14113  df-p0 14147  df-p1 14148  df-lat 14154  df-clat 14216  df-mnd 14369  df-submnd 14418  df-grp 14491  df-minusg 14492  df-sbg 14493  df-subg 14620  df-cntz 14795  df-oppg 14821  df-lsm 14949  df-cmn 15093  df-abl 15094  df-mgp 15328  df-rng 15342  df-ur 15344  df-oppr 15407  df-dvdsr 15425  df-unit 15426  df-invr 15456  df-dvr 15467  df-drng 15516  df-lmod 15631  df-lss 15692  df-lsp 15731  df-lvec 15858  df-lsatoms 29239  df-lshyp 29240  df-lcv 29282  df-lfl 29321  df-lkr 29349  df-ldual 29387  df-oposet 29439  df-ol 29441  df-oml 29442  df-covers 29529  df-ats 29530  df-atl 29561  df-cvlat 29585  df-hlat 29614  df-llines 29760  df-lplanes 29761  df-lvols 29762  df-lines 29763  df-psubsp 29765  df-pmap 29766  df-padd 30058  df-lhyp 30250  df-laut 30251  df-ldil 30366  df-ltrn 30367  df-trl 30421  df-tgrp 31005  df-tendo 31017  df-edring 31019  df-dveca 31265  df-disoa 31292  df-dvech 31342  df-dib 31402  df-dic 31436  df-dih 31492  df-doch 31611  df-djh 31658  df-lcdual 31850  df-mapd 31888
  Copyright terms: Public domain W3C validator