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Theorem mapdlsm 31104
Description: Subspace sum is preserved by the map defined by df-mapd 31065. 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 31032 . . . . . . . . . 10  |-  ( ph  ->  C  e.  LMod )
5 eqid 2258 . . . . . . . . . . 11  |-  ( LSubSp `  C )  =  (
LSubSp `  C )
65lsssssubg 15678 . . . . . . . . . 10  |-  ( C  e.  LMod  ->  ( LSubSp `  C )  C_  (SubGrp `  C ) )
74, 6syl 17 . . . . . . . . 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 31096 . . . . . . . . 9  |-  ( ph  ->  ( M `  X
)  e.  ( LSubSp `  C ) )
137, 12sseldd 3156 . . . . . . . 8  |-  ( ph  ->  ( M `  X
)  e.  (SubGrp `  C ) )
14 mapdlsm.y . . . . . . . . . 10  |-  ( ph  ->  Y  e.  S )
151, 8, 9, 10, 2, 5, 3, 14mapdcl2 31096 . . . . . . . . 9  |-  ( ph  ->  ( M `  Y
)  e.  ( LSubSp `  C ) )
167, 15sseldd 3156 . . . . . . . 8  |-  ( ph  ->  ( M `  Y
)  e.  (SubGrp `  C ) )
17 mapdlsm.q . . . . . . . . 9  |-  .+b  =  ( LSSum `  C )
1817lsmub1 14930 . . . . . . . 8  |-  ( ( ( M `  X
)  e.  (SubGrp `  C )  /\  ( M `  Y )  e.  (SubGrp `  C )
)  ->  ( M `  X )  C_  (
( M `  X
)  .+b  ( M `  Y ) ) )
1913, 16, 18syl2anc 645 . . . . . . 7  |-  ( ph  ->  ( M `  X
)  C_  ( ( M `  X )  .+b  ( M `  Y
) ) )
201, 8, 9, 10, 3, 11mapdcl 31093 . . . . . . . . 9  |-  ( ph  ->  ( M `  X
)  e.  ran  M
)
211, 8, 9, 10, 3, 14mapdcl 31093 . . . . . . . . 9  |-  ( ph  ->  ( M `  Y
)  e.  ran  M
)
221, 8, 9, 2, 17, 3, 20, 21mapdlsmcl 31103 . . . . . . . 8  |-  ( ph  ->  ( ( M `  X )  .+b  ( M `  Y )
)  e.  ran  M
)
231, 8, 3, 22mapdcnvid2 31097 . . . . . . 7  |-  ( ph  ->  ( M `  ( `' M `  ( ( M `  X ) 
.+b  ( M `  Y ) ) ) )  =  ( ( M `  X ) 
.+b  ( M `  Y ) ) )
2419, 23sseqtr4d 3190 . . . . . 6  |-  ( ph  ->  ( M `  X
)  C_  ( M `  ( `' M `  ( ( M `  X )  .+b  ( M `  Y )
) ) ) )
251, 8, 9, 10, 3, 22mapdcnvcl 31092 . . . . . . 7  |-  ( ph  ->  ( `' M `  ( ( M `  X )  .+b  ( M `  Y )
) )  e.  S
)
261, 9, 10, 8, 3, 11, 25mapdord 31078 . . . . . 6  |-  ( ph  ->  ( ( M `  X )  C_  ( M `  ( `' M `  ( ( M `  X )  .+b  ( M `  Y
) ) ) )  <-> 
X  C_  ( `' M `  ( ( M `  X )  .+b  ( M `  Y
) ) ) ) )
2724, 26mpbid 203 . . . . 5  |-  ( ph  ->  X  C_  ( `' M `  ( ( M `  X )  .+b  ( M `  Y
) ) ) )
2817lsmub2 14931 . . . . . . . 8  |-  ( ( ( M `  X
)  e.  (SubGrp `  C )  /\  ( M `  Y )  e.  (SubGrp `  C )
)  ->  ( M `  Y )  C_  (
( M `  X
)  .+b  ( M `  Y ) ) )
2913, 16, 28syl2anc 645 . . . . . . 7  |-  ( ph  ->  ( M `  Y
)  C_  ( ( M `  X )  .+b  ( M `  Y
) ) )
3029, 23sseqtr4d 3190 . . . . . 6  |-  ( ph  ->  ( M `  Y
)  C_  ( M `  ( `' M `  ( ( M `  X )  .+b  ( M `  Y )
) ) ) )
311, 9, 10, 8, 3, 14, 25mapdord 31078 . . . . . 6  |-  ( ph  ->  ( ( M `  Y )  C_  ( M `  ( `' M `  ( ( M `  X )  .+b  ( M `  Y
) ) ) )  <-> 
Y  C_  ( `' M `  ( ( M `  X )  .+b  ( M `  Y
) ) ) ) )
3230, 31mpbid 203 . . . . 5  |-  ( ph  ->  Y  C_  ( `' M `  ( ( M `  X )  .+b  ( M `  Y
) ) ) )
331, 9, 3dvhlmod 30550 . . . . . . . 8  |-  ( ph  ->  U  e.  LMod )
3410lsssssubg 15678 . . . . . . . 8  |-  ( U  e.  LMod  ->  S  C_  (SubGrp `  U ) )
3533, 34syl 17 . . . . . . 7  |-  ( ph  ->  S  C_  (SubGrp `  U
) )
3635, 11sseldd 3156 . . . . . 6  |-  ( ph  ->  X  e.  (SubGrp `  U ) )
3735, 14sseldd 3156 . . . . . 6  |-  ( ph  ->  Y  e.  (SubGrp `  U ) )
3835, 25sseldd 3156 . . . . . 6  |-  ( ph  ->  ( `' M `  ( ( M `  X )  .+b  ( M `  Y )
) )  e.  (SubGrp `  U ) )
39 mapdlsm.p . . . . . . 7  |-  .(+)  =  (
LSSum `  U )
4039lsmlub 14937 . . . . . 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 1187 . . . . 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 892 . . . 4  |-  ( ph  ->  ( X  .(+)  Y ) 
C_  ( `' M `  ( ( M `  X )  .+b  ( M `  Y )
) ) )
4310, 39lsmcl 15799 . . . . . 6  |-  ( ( U  e.  LMod  /\  X  e.  S  /\  Y  e.  S )  ->  ( X  .(+)  Y )  e.  S )
4433, 11, 14, 43syl3anc 1187 . . . . 5  |-  ( ph  ->  ( X  .(+)  Y )  e.  S )
451, 9, 10, 8, 3, 44, 25mapdord 31078 . . . 4  |-  ( ph  ->  ( ( M `  ( X  .(+)  Y ) )  C_  ( M `  ( `' M `  ( ( M `  X )  .+b  ( M `  Y )
) ) )  <->  ( X  .(+) 
Y )  C_  ( `' M `  ( ( M `  X ) 
.+b  ( M `  Y ) ) ) ) )
4642, 45mpbird 225 . . 3  |-  ( ph  ->  ( M `  ( X  .(+)  Y ) ) 
C_  ( M `  ( `' M `  ( ( M `  X ) 
.+b  ( M `  Y ) ) ) ) )
4746, 23sseqtrd 3189 . 2  |-  ( ph  ->  ( M `  ( X  .(+)  Y ) ) 
C_  ( ( M `
 X )  .+b  ( M `  Y ) ) )
4839lsmub1 14930 . . . . 5  |-  ( ( X  e.  (SubGrp `  U )  /\  Y  e.  (SubGrp `  U )
)  ->  X  C_  ( X  .(+)  Y ) )
4936, 37, 48syl2anc 645 . . . 4  |-  ( ph  ->  X  C_  ( X  .(+) 
Y ) )
501, 9, 10, 8, 3, 11, 44mapdord 31078 . . . 4  |-  ( ph  ->  ( ( M `  X )  C_  ( M `  ( X  .(+) 
Y ) )  <->  X  C_  ( X  .(+)  Y ) ) )
5149, 50mpbird 225 . . 3  |-  ( ph  ->  ( M `  X
)  C_  ( M `  ( X  .(+)  Y ) ) )
5239lsmub2 14931 . . . . 5  |-  ( ( X  e.  (SubGrp `  U )  /\  Y  e.  (SubGrp `  U )
)  ->  Y  C_  ( X  .(+)  Y ) )
5336, 37, 52syl2anc 645 . . . 4  |-  ( ph  ->  Y  C_  ( X  .(+) 
Y ) )
541, 9, 10, 8, 3, 14, 44mapdord 31078 . . . 4  |-  ( ph  ->  ( ( M `  Y )  C_  ( M `  ( X  .(+) 
Y ) )  <->  Y  C_  ( X  .(+)  Y ) ) )
5553, 54mpbird 225 . . 3  |-  ( ph  ->  ( M `  Y
)  C_  ( M `  ( X  .(+)  Y ) ) )
561, 8, 9, 10, 2, 5, 3, 44mapdcl2 31096 . . . . 5  |-  ( ph  ->  ( M `  ( X  .(+)  Y ) )  e.  ( LSubSp `  C
) )
577, 56sseldd 3156 . . . 4  |-  ( ph  ->  ( M `  ( X  .(+)  Y ) )  e.  (SubGrp `  C
) )
5817lsmlub 14937 . . . 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 1187 . . 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 892 . 2  |-  ( ph  ->  ( ( M `  X )  .+b  ( M `  Y )
)  C_  ( M `  ( X  .(+)  Y ) ) )
6147, 60eqssd 3171 1  |-  ( ph  ->  ( M `  ( X  .(+)  Y ) )  =  ( ( M `
 X )  .+b  ( M `  Y ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    <-> wb 178    /\ wa 360    = wceq 1619    e. wcel 1621    C_ wss 3127   `'ccnv 4660   ` cfv 4673  (class class class)co 5792  SubGrpcsubg 14578   LSSumclsm 14908   LModclmod 15590   LSubSpclss 15652   HLchlt 28790   LHypclh 29423   DVecHcdvh 30518  LCDualclcd 31026  mapdcmpd 31064
This theorem is referenced by:  mapdindp  31111  mapdpglem1  31112  mapdheq4lem  31171  mapdh6lem1N  31173  mapdh6lem2N  31174  hdmap1l6lem1  31248  hdmap1l6lem2  31249  hdmaprnlem3eN  31301
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2239  ax-rep 4105  ax-sep 4115  ax-nul 4123  ax-pow 4160  ax-pr 4186  ax-un 4484  ax-cnex 8761  ax-resscn 8762  ax-1cn 8763  ax-icn 8764  ax-addcl 8765  ax-addrcl 8766  ax-mulcl 8767  ax-mulrcl 8768  ax-mulcom 8769  ax-addass 8770  ax-mulass 8771  ax-distr 8772  ax-i2m1 8773  ax-1ne0 8774  ax-1rid 8775  ax-rnegex 8776  ax-rrecex 8777  ax-cnre 8778  ax-pre-lttri 8779  ax-pre-lttrn 8780  ax-pre-ltadd 8781  ax-pre-mulgt0 8782
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-fal 1316  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2122  df-mo 2123  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ne 2423  df-nel 2424  df-ral 2523  df-rex 2524  df-reu 2525  df-rmo 2526  df-rab 2527  df-v 2765  df-sbc 2967  df-csb 3057  df-dif 3130  df-un 3132  df-in 3134  df-ss 3141  df-pss 3143  df-nul 3431  df-if 3540  df-pw 3601  df-sn 3620  df-pr 3621  df-tp 3622  df-op 3623  df-uni 3802  df-int 3837  df-iun 3881  df-iin 3882  df-br 3998  df-opab 4052  df-mpt 4053  df-tr 4088  df-eprel 4277  df-id 4281  df-po 4286  df-so 4287  df-fr 4324  df-we 4326  df-ord 4367  df-on 4368  df-lim 4369  df-suc 4370  df-om 4629  df-xp 4675  df-rel 4676  df-cnv 4677  df-co 4678  df-dm 4679  df-rn 4680  df-res 4681  df-ima 4682  df-fun 4683  df-fn 4684  df-f 4685  df-f1 4686  df-fo 4687  df-f1o 4688  df-fv 4689  df-ov 5795  df-oprab 5796  df-mpt2 5797  df-of 6012  df-1st 6056  df-2nd 6057  df-tpos 6168  df-iota 6225  df-undef 6264  df-riota 6272  df-recs 6356  df-rdg 6391  df-1o 6447  df-oadd 6451  df-er 6628  df-map 6742  df-en 6832  df-dom 6833  df-sdom 6834  df-fin 6835  df-pnf 8837  df-mnf 8838  df-xr 8839  df-ltxr 8840  df-le 8841  df-sub 9007  df-neg 9008  df-n 9715  df-2 9772  df-3 9773  df-4 9774  df-5 9775  df-6 9776  df-n0 9934  df-z 9993  df-uz 10199  df-fz 10750  df-struct 13113  df-ndx 13114  df-slot 13115  df-base 13116  df-sets 13117  df-ress 13118  df-plusg 13184  df-mulr 13185  df-sca 13187  df-vsca 13188  df-0g 13367  df-mre 13451  df-mrc 13452  df-acs 13454  df-poset 14043  df-plt 14055  df-lub 14071  df-glb 14072  df-join 14073  df-meet 14074  df-p0 14108  df-p1 14109  df-lat 14115  df-clat 14177  df-mnd 14330  df-submnd 14379  df-grp 14452  df-minusg 14453  df-sbg 14454  df-subg 14581  df-cntz 14756  df-oppg 14782  df-lsm 14910  df-cmn 15054  df-abl 15055  df-mgp 15289  df-ring 15303  df-ur 15305  df-oppr 15368  df-dvdsr 15386  df-unit 15387  df-invr 15417  df-dvr 15428  df-drng 15477  df-lmod 15592  df-lss 15653  df-lsp 15692  df-lvec 15819  df-lsatoms 28416  df-lshyp 28417  df-lcv 28459  df-lfl 28498  df-lkr 28526  df-ldual 28564  df-oposet 28616  df-ol 28618  df-oml 28619  df-covers 28706  df-ats 28707  df-atl 28738  df-cvlat 28762  df-hlat 28791  df-llines 28937  df-lplanes 28938  df-lvols 28939  df-lines 28940  df-psubsp 28942  df-pmap 28943  df-padd 29235  df-lhyp 29427  df-laut 29428  df-ldil 29543  df-ltrn 29544  df-trl 29598  df-tgrp 30182  df-tendo 30194  df-edring 30196  df-dveca 30442  df-disoa 30469  df-dvech 30519  df-dib 30579  df-dic 30613  df-dih 30669  df-doch 30788  df-djh 30835  df-lcdual 31027  df-mapd 31065
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