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Theorem mapdlsm 31781
Description: Subspace sum is preserved by the map defined by df-mapd 31742. 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 31709 . . . . . . . . . 10  |-  ( ph  ->  C  e.  LMod )
5 eqid 2389 . . . . . . . . . . 11  |-  ( LSubSp `  C )  =  (
LSubSp `  C )
65lsssssubg 15963 . . . . . . . . . 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 31773 . . . . . . . . 9  |-  ( ph  ->  ( M `  X
)  e.  ( LSubSp `  C ) )
137, 12sseldd 3294 . . . . . . . 8  |-  ( ph  ->  ( M `  X
)  e.  (SubGrp `  C ) )
14 mapdlsm.y . . . . . . . . . 10  |-  ( ph  ->  Y  e.  S )
151, 8, 9, 10, 2, 5, 3, 14mapdcl2 31773 . . . . . . . . 9  |-  ( ph  ->  ( M `  Y
)  e.  ( LSubSp `  C ) )
167, 15sseldd 3294 . . . . . . . 8  |-  ( ph  ->  ( M `  Y
)  e.  (SubGrp `  C ) )
17 mapdlsm.q . . . . . . . . 9  |-  .+b  =  ( LSSum `  C )
1817lsmub1 15219 . . . . . . . 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 31770 . . . . . . . . 9  |-  ( ph  ->  ( M `  X
)  e.  ran  M
)
211, 8, 9, 10, 3, 14mapdcl 31770 . . . . . . . . 9  |-  ( ph  ->  ( M `  Y
)  e.  ran  M
)
221, 8, 9, 2, 17, 3, 20, 21mapdlsmcl 31780 . . . . . . . 8  |-  ( ph  ->  ( ( M `  X )  .+b  ( M `  Y )
)  e.  ran  M
)
231, 8, 3, 22mapdcnvid2 31774 . . . . . . 7  |-  ( ph  ->  ( M `  ( `' M `  ( ( M `  X ) 
.+b  ( M `  Y ) ) ) )  =  ( ( M `  X ) 
.+b  ( M `  Y ) ) )
2419, 23sseqtr4d 3330 . . . . . 6  |-  ( ph  ->  ( M `  X
)  C_  ( M `  ( `' M `  ( ( M `  X )  .+b  ( M `  Y )
) ) ) )
251, 8, 9, 10, 3, 22mapdcnvcl 31769 . . . . . . 7  |-  ( ph  ->  ( `' M `  ( ( M `  X )  .+b  ( M `  Y )
) )  e.  S
)
261, 9, 10, 8, 3, 11, 25mapdord 31755 . . . . . 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 15220 . . . . . . . 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 3330 . . . . . 6  |-  ( ph  ->  ( M `  Y
)  C_  ( M `  ( `' M `  ( ( M `  X )  .+b  ( M `  Y )
) ) ) )
311, 9, 10, 8, 3, 14, 25mapdord 31755 . . . . . 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 31227 . . . . . . . 8  |-  ( ph  ->  U  e.  LMod )
3410lsssssubg 15963 . . . . . . . 8  |-  ( U  e.  LMod  ->  S  C_  (SubGrp `  U ) )
3533, 34syl 16 . . . . . . 7  |-  ( ph  ->  S  C_  (SubGrp `  U
) )
3635, 11sseldd 3294 . . . . . 6  |-  ( ph  ->  X  e.  (SubGrp `  U ) )
3735, 14sseldd 3294 . . . . . 6  |-  ( ph  ->  Y  e.  (SubGrp `  U ) )
3835, 25sseldd 3294 . . . . . 6  |-  ( ph  ->  ( `' M `  ( ( M `  X )  .+b  ( M `  Y )
) )  e.  (SubGrp `  U ) )
39 mapdlsm.p . . . . . . 7  |-  .(+)  =  (
LSSum `  U )
4039lsmlub 15226 . . . . . 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 16084 . . . . . 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 31755 . . . 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 3329 . 2  |-  ( ph  ->  ( M `  ( X  .(+)  Y ) ) 
C_  ( ( M `
 X )  .+b  ( M `  Y ) ) )
4839lsmub1 15219 . . . . 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 31755 . . . 4  |-  ( ph  ->  ( ( M `  X )  C_  ( M `  ( X  .(+) 
Y ) )  <->  X  C_  ( X  .(+)  Y ) ) )
5149, 50mpbird 224 . . 3  |-  ( ph  ->  ( M `  X
)  C_  ( M `  ( X  .(+)  Y ) ) )
5239lsmub2 15220 . . . . 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 31755 . . . 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 31773 . . . . 5  |-  ( ph  ->  ( M `  ( X  .(+)  Y ) )  e.  ( LSubSp `  C
) )
577, 56sseldd 3294 . . . 4  |-  ( ph  ->  ( M `  ( X  .(+)  Y ) )  e.  (SubGrp `  C
) )
5817lsmlub 15226 . . . 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 3310 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 1649    e. wcel 1717    C_ wss 3265   `'ccnv 4819   ` cfv 5396  (class class class)co 6022  SubGrpcsubg 14867   LSSumclsm 15197   LModclmod 15879   LSubSpclss 15937   HLchlt 29467   LHypclh 30100   DVecHcdvh 31195  LCDualclcd 31703  mapdcmpd 31741
This theorem is referenced by:  mapdindp  31788  mapdpglem1  31789  mapdheq4lem  31848  mapdh6lem1N  31850  mapdh6lem2N  31851  hdmap1l6lem1  31925  hdmap1l6lem2  31926  hdmaprnlem3eN  31978
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-rep 4263  ax-sep 4273  ax-nul 4281  ax-pow 4320  ax-pr 4346  ax-un 4643  ax-cnex 8981  ax-resscn 8982  ax-1cn 8983  ax-icn 8984  ax-addcl 8985  ax-addrcl 8986  ax-mulcl 8987  ax-mulrcl 8988  ax-mulcom 8989  ax-addass 8990  ax-mulass 8991  ax-distr 8992  ax-i2m1 8993  ax-1ne0 8994  ax-1rid 8995  ax-rnegex 8996  ax-rrecex 8997  ax-cnre 8998  ax-pre-lttri 8999  ax-pre-lttrn 9000  ax-pre-ltadd 9001  ax-pre-mulgt0 9002
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-fal 1326  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-nel 2555  df-ral 2656  df-rex 2657  df-reu 2658  df-rmo 2659  df-rab 2660  df-v 2903  df-sbc 3107  df-csb 3197  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-pss 3281  df-nul 3574  df-if 3685  df-pw 3746  df-sn 3765  df-pr 3766  df-tp 3767  df-op 3768  df-uni 3960  df-int 3995  df-iun 4039  df-iin 4040  df-br 4156  df-opab 4210  df-mpt 4211  df-tr 4246  df-eprel 4437  df-id 4441  df-po 4446  df-so 4447  df-fr 4484  df-we 4486  df-ord 4527  df-on 4528  df-lim 4529  df-suc 4530  df-om 4788  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-res 4832  df-ima 4833  df-iota 5360  df-fun 5398  df-fn 5399  df-f 5400  df-f1 5401  df-fo 5402  df-f1o 5403  df-fv 5404  df-ov 6025  df-oprab 6026  df-mpt2 6027  df-of 6246  df-1st 6290  df-2nd 6291  df-tpos 6417  df-undef 6481  df-riota 6487  df-recs 6571  df-rdg 6606  df-1o 6662  df-oadd 6666  df-er 6843  df-map 6958  df-en 7048  df-dom 7049  df-sdom 7050  df-fin 7051  df-pnf 9057  df-mnf 9058  df-xr 9059  df-ltxr 9060  df-le 9061  df-sub 9227  df-neg 9228  df-nn 9935  df-2 9992  df-3 9993  df-4 9994  df-5 9995  df-6 9996  df-n0 10156  df-z 10217  df-uz 10423  df-fz 10978  df-struct 13400  df-ndx 13401  df-slot 13402  df-base 13403  df-sets 13404  df-ress 13405  df-plusg 13471  df-mulr 13472  df-sca 13474  df-vsca 13475  df-0g 13656  df-mre 13740  df-mrc 13741  df-acs 13743  df-poset 14332  df-plt 14344  df-lub 14360  df-glb 14361  df-join 14362  df-meet 14363  df-p0 14397  df-p1 14398  df-lat 14404  df-clat 14466  df-mnd 14619  df-submnd 14668  df-grp 14741  df-minusg 14742  df-sbg 14743  df-subg 14870  df-cntz 15045  df-oppg 15071  df-lsm 15199  df-cmn 15343  df-abl 15344  df-mgp 15578  df-rng 15592  df-ur 15594  df-oppr 15657  df-dvdsr 15675  df-unit 15676  df-invr 15706  df-dvr 15717  df-drng 15766  df-lmod 15881  df-lss 15938  df-lsp 15977  df-lvec 16104  df-lsatoms 29093  df-lshyp 29094  df-lcv 29136  df-lfl 29175  df-lkr 29203  df-ldual 29241  df-oposet 29293  df-ol 29295  df-oml 29296  df-covers 29383  df-ats 29384  df-atl 29415  df-cvlat 29439  df-hlat 29468  df-llines 29614  df-lplanes 29615  df-lvols 29616  df-lines 29617  df-psubsp 29619  df-pmap 29620  df-padd 29912  df-lhyp 30104  df-laut 30105  df-ldil 30220  df-ltrn 30221  df-trl 30275  df-tgrp 30859  df-tendo 30871  df-edring 30873  df-dveca 31119  df-disoa 31146  df-dvech 31196  df-dib 31256  df-dic 31290  df-dih 31346  df-doch 31465  df-djh 31512  df-lcdual 31704  df-mapd 31742
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