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Theorem mapdlsm 30613
Description: Subspace sum is preserved by the map defined by df-mapd 30574. 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 30541 . . . . . . . . . 10  |-  ( ph  ->  C  e.  LMod )
5 eqid 2253 . . . . . . . . . . 11  |-  ( LSubSp `  C )  =  (
LSubSp `  C )
65lsssssubg 15550 . . . . . . . . . 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 30605 . . . . . . . . 9  |-  ( ph  ->  ( M `  X
)  e.  ( LSubSp `  C ) )
137, 12sseldd 3104 . . . . . . . 8  |-  ( ph  ->  ( M `  X
)  e.  (SubGrp `  C ) )
14 mapdlsm.y . . . . . . . . . 10  |-  ( ph  ->  Y  e.  S )
151, 8, 9, 10, 2, 5, 3, 14mapdcl2 30605 . . . . . . . . 9  |-  ( ph  ->  ( M `  Y
)  e.  ( LSubSp `  C ) )
167, 15sseldd 3104 . . . . . . . 8  |-  ( ph  ->  ( M `  Y
)  e.  (SubGrp `  C ) )
17 mapdlsm.q . . . . . . . . 9  |-  .+b  =  ( LSSum `  C )
1817lsmub1 14802 . . . . . . . 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 30602 . . . . . . . . 9  |-  ( ph  ->  ( M `  X
)  e.  ran  M
)
211, 8, 9, 10, 3, 14mapdcl 30602 . . . . . . . . 9  |-  ( ph  ->  ( M `  Y
)  e.  ran  M
)
221, 8, 9, 2, 17, 3, 20, 21mapdlsmcl 30612 . . . . . . . 8  |-  ( ph  ->  ( ( M `  X )  .+b  ( M `  Y )
)  e.  ran  M
)
231, 8, 3, 22mapdcnvid2 30606 . . . . . . 7  |-  ( ph  ->  ( M `  ( `' M `  ( ( M `  X ) 
.+b  ( M `  Y ) ) ) )  =  ( ( M `  X ) 
.+b  ( M `  Y ) ) )
2419, 23sseqtr4d 3136 . . . . . 6  |-  ( ph  ->  ( M `  X
)  C_  ( M `  ( `' M `  ( ( M `  X )  .+b  ( M `  Y )
) ) ) )
251, 8, 9, 10, 3, 22mapdcnvcl 30601 . . . . . . 7  |-  ( ph  ->  ( `' M `  ( ( M `  X )  .+b  ( M `  Y )
) )  e.  S
)
261, 9, 10, 8, 3, 11, 25mapdord 30587 . . . . . 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 14803 . . . . . . . 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 3136 . . . . . 6  |-  ( ph  ->  ( M `  Y
)  C_  ( M `  ( `' M `  ( ( M `  X )  .+b  ( M `  Y )
) ) ) )
311, 9, 10, 8, 3, 14, 25mapdord 30587 . . . . . 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 30059 . . . . . . . 8  |-  ( ph  ->  U  e.  LMod )
3410lsssssubg 15550 . . . . . . . 8  |-  ( U  e.  LMod  ->  S  C_  (SubGrp `  U ) )
3533, 34syl 17 . . . . . . 7  |-  ( ph  ->  S  C_  (SubGrp `  U
) )
3635, 11sseldd 3104 . . . . . 6  |-  ( ph  ->  X  e.  (SubGrp `  U ) )
3735, 14sseldd 3104 . . . . . 6  |-  ( ph  ->  Y  e.  (SubGrp `  U ) )
3835, 25sseldd 3104 . . . . . 6  |-  ( ph  ->  ( `' M `  ( ( M `  X )  .+b  ( M `  Y )
) )  e.  (SubGrp `  U ) )
39 mapdlsm.p . . . . . . 7  |-  .(+)  =  (
LSSum `  U )
4039lsmlub 14809 . . . . . 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 15671 . . . . . 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 30587 . . . 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 3135 . 2  |-  ( ph  ->  ( M `  ( X  .(+)  Y ) ) 
C_  ( ( M `
 X )  .+b  ( M `  Y ) ) )
4839lsmub1 14802 . . . . 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 30587 . . . 4  |-  ( ph  ->  ( ( M `  X )  C_  ( M `  ( X  .(+) 
Y ) )  <->  X  C_  ( X  .(+)  Y ) ) )
5149, 50mpbird 225 . . 3  |-  ( ph  ->  ( M `  X
)  C_  ( M `  ( X  .(+)  Y ) ) )
5239lsmub2 14803 . . . . 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 30587 . . . 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 30605 . . . . 5  |-  ( ph  ->  ( M `  ( X  .(+)  Y ) )  e.  ( LSubSp `  C
) )
577, 56sseldd 3104 . . . 4  |-  ( ph  ->  ( M `  ( X  .(+)  Y ) )  e.  (SubGrp `  C
) )
5817lsmlub 14809 . . . 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 3117 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 3078   `'ccnv 4579   ` cfv 4592  (class class class)co 5710  SubGrpcsubg 14450   LSSumclsm 14780   LModclmod 15462   LSubSpclss 15524   HLchlt 28299   LHypclh 28932   DVecHcdvh 30027  LCDualclcd 30535  mapdcmpd 30573
This theorem is referenced by:  mapdindp  30620  mapdpglem1  30621  mapdheq4lem  30680  mapdh6lem1N  30682  mapdh6lem2N  30683  hdmap1l6lem1  30757  hdmap1l6lem2  30758  hdmaprnlem3eN  30810
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 1926  ax-ext 2234  ax-rep 4028  ax-sep 4038  ax-nul 4046  ax-pow 4082  ax-pr 4108  ax-un 4403  ax-cnex 8673  ax-resscn 8674  ax-1cn 8675  ax-icn 8676  ax-addcl 8677  ax-addrcl 8678  ax-mulcl 8679  ax-mulrcl 8680  ax-mulcom 8681  ax-addass 8682  ax-mulass 8683  ax-distr 8684  ax-i2m1 8685  ax-1ne0 8686  ax-1rid 8687  ax-rnegex 8688  ax-rrecex 8689  ax-cnre 8690  ax-pre-lttri 8691  ax-pre-lttrn 8692  ax-pre-ltadd 8693  ax-pre-mulgt0 8694
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 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-nel 2415  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2516  df-v 2729  df-sbc 2922  df-csb 3010  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-pss 3091  df-nul 3363  df-if 3471  df-pw 3532  df-sn 3550  df-pr 3551  df-tp 3552  df-op 3553  df-uni 3728  df-int 3761  df-iun 3805  df-iin 3806  df-br 3921  df-opab 3975  df-mpt 3976  df-tr 4011  df-eprel 4198  df-id 4202  df-po 4207  df-so 4208  df-fr 4245  df-we 4247  df-ord 4288  df-on 4289  df-lim 4290  df-suc 4291  df-om 4548  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 4602  df-fn 4603  df-f 4604  df-f1 4605  df-fo 4606  df-f1o 4607  df-fv 4608  df-ov 5713  df-oprab 5714  df-mpt2 5715  df-of 5930  df-1st 5974  df-2nd 5975  df-tpos 6086  df-iota 6143  df-undef 6182  df-riota 6190  df-recs 6274  df-rdg 6309  df-1o 6365  df-oadd 6369  df-er 6546  df-map 6660  df-en 6750  df-dom 6751  df-sdom 6752  df-fin 6753  df-pnf 8749  df-mnf 8750  df-xr 8751  df-ltxr 8752  df-le 8753  df-sub 8919  df-neg 8920  df-n 9627  df-2 9684  df-3 9685  df-4 9686  df-5 9687  df-6 9688  df-n0 9845  df-z 9904  df-uz 10110  df-fz 10661  df-struct 13024  df-ndx 13025  df-slot 13026  df-base 13027  df-sets 13028  df-ress 13029  df-plusg 13095  df-mulr 13096  df-sca 13098  df-vsca 13099  df-0g 13278  df-mre 13361  df-mrc 13362  df-acs 13363  df-poset 13924  df-plt 13936  df-lub 13952  df-glb 13953  df-join 13954  df-meet 13955  df-p0 13989  df-p1 13990  df-lat 13996  df-clat 14058  df-mnd 14202  df-submnd 14251  df-grp 14324  df-minusg 14325  df-sbg 14326  df-subg 14453  df-cntz 14628  df-oppg 14654  df-lsm 14782  df-cmn 14926  df-abl 14927  df-mgp 15161  df-ring 15175  df-ur 15177  df-oppr 15240  df-dvdsr 15258  df-unit 15259  df-invr 15289  df-dvr 15300  df-drng 15349  df-lmod 15464  df-lss 15525  df-lsp 15564  df-lvec 15691  df-lsatoms 27925  df-lshyp 27926  df-lcv 27968  df-lfl 28007  df-lkr 28035  df-ldual 28073  df-oposet 28125  df-ol 28127  df-oml 28128  df-covers 28215  df-ats 28216  df-atl 28247  df-cvlat 28271  df-hlat 28300  df-llines 28446  df-lplanes 28447  df-lvols 28448  df-lines 28449  df-psubsp 28451  df-pmap 28452  df-padd 28744  df-lhyp 28936  df-laut 28937  df-ldil 29052  df-ltrn 29053  df-trl 29107  df-tgrp 29691  df-tendo 29703  df-edring 29705  df-dveca 29951  df-disoa 29978  df-dvech 30028  df-dib 30088  df-dic 30122  df-dih 30178  df-doch 30297  df-djh 30344  df-lcdual 30536  df-mapd 30574
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