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Theorem lssnlm 20281
Description: A subspace of a normed module is a normed module. (Contributed by Mario Carneiro, 4-Oct-2015.)
Hypotheses
Ref Expression
lssnlm.x  |-  X  =  ( Ws  U )
lssnlm.s  |-  S  =  ( LSubSp `  W )
Assertion
Ref Expression
lssnlm  |-  ( ( W  e. NrmMod  /\  U  e.  S )  ->  X  e. NrmMod )

Proof of Theorem lssnlm
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nlmngp 20258 . . . . 5  |-  ( W  e. NrmMod  ->  W  e. NrmGrp )
21adantr 465 . . . 4  |-  ( ( W  e. NrmMod  /\  U  e.  S )  ->  W  e. NrmGrp )
3 nlmlmod 20259 . . . . 5  |-  ( W  e. NrmMod  ->  W  e.  LMod )
4 lssnlm.s . . . . . 6  |-  S  =  ( LSubSp `  W )
54lsssubg 17038 . . . . 5  |-  ( ( W  e.  LMod  /\  U  e.  S )  ->  U  e.  (SubGrp `  W )
)
63, 5sylan 471 . . . 4  |-  ( ( W  e. NrmMod  /\  U  e.  S )  ->  U  e.  (SubGrp `  W )
)
7 lssnlm.x . . . . 5  |-  X  =  ( Ws  U )
87subgngp 20221 . . . 4  |-  ( ( W  e. NrmGrp  /\  U  e.  (SubGrp `  W )
)  ->  X  e. NrmGrp )
92, 6, 8syl2anc 661 . . 3  |-  ( ( W  e. NrmMod  /\  U  e.  S )  ->  X  e. NrmGrp )
107, 4lsslmod 17041 . . . 4  |-  ( ( W  e.  LMod  /\  U  e.  S )  ->  X  e.  LMod )
113, 10sylan 471 . . 3  |-  ( ( W  e. NrmMod  /\  U  e.  S )  ->  X  e.  LMod )
12 eqid 2443 . . . . . 6  |-  (Scalar `  W )  =  (Scalar `  W )
137, 12resssca 14316 . . . . 5  |-  ( U  e.  S  ->  (Scalar `  W )  =  (Scalar `  X ) )
1413adantl 466 . . . 4  |-  ( ( W  e. NrmMod  /\  U  e.  S )  ->  (Scalar `  W )  =  (Scalar `  X ) )
1512nlmnrg 20260 . . . . 5  |-  ( W  e. NrmMod  ->  (Scalar `  W )  e. NrmRing )
1615adantr 465 . . . 4  |-  ( ( W  e. NrmMod  /\  U  e.  S )  ->  (Scalar `  W )  e. NrmRing )
1714, 16eqeltrrd 2518 . . 3  |-  ( ( W  e. NrmMod  /\  U  e.  S )  ->  (Scalar `  X )  e. NrmRing )
189, 11, 173jca 1168 . 2  |-  ( ( W  e. NrmMod  /\  U  e.  S )  ->  ( X  e. NrmGrp  /\  X  e. 
LMod  /\  (Scalar `  X
)  e. NrmRing ) )
19 simpll 753 . . . . 5  |-  ( ( ( W  e. NrmMod  /\  U  e.  S )  /\  (
x  e.  ( Base `  (Scalar `  X )
)  /\  y  e.  ( Base `  X )
) )  ->  W  e. NrmMod )
20 simprl 755 . . . . . 6  |-  ( ( ( W  e. NrmMod  /\  U  e.  S )  /\  (
x  e.  ( Base `  (Scalar `  X )
)  /\  y  e.  ( Base `  X )
) )  ->  x  e.  ( Base `  (Scalar `  X ) ) )
2114adantr 465 . . . . . . 7  |-  ( ( ( W  e. NrmMod  /\  U  e.  S )  /\  (
x  e.  ( Base `  (Scalar `  X )
)  /\  y  e.  ( Base `  X )
) )  ->  (Scalar `  W )  =  (Scalar `  X ) )
2221fveq2d 5695 . . . . . 6  |-  ( ( ( W  e. NrmMod  /\  U  e.  S )  /\  (
x  e.  ( Base `  (Scalar `  X )
)  /\  y  e.  ( Base `  X )
) )  ->  ( Base `  (Scalar `  W
) )  =  (
Base `  (Scalar `  X
) ) )
2320, 22eleqtrrd 2520 . . . . 5  |-  ( ( ( W  e. NrmMod  /\  U  e.  S )  /\  (
x  e.  ( Base `  (Scalar `  X )
)  /\  y  e.  ( Base `  X )
) )  ->  x  e.  ( Base `  (Scalar `  W ) ) )
246adantr 465 . . . . . . 7  |-  ( ( ( W  e. NrmMod  /\  U  e.  S )  /\  (
x  e.  ( Base `  (Scalar `  X )
)  /\  y  e.  ( Base `  X )
) )  ->  U  e.  (SubGrp `  W )
)
25 eqid 2443 . . . . . . . 8  |-  ( Base `  W )  =  (
Base `  W )
2625subgss 15682 . . . . . . 7  |-  ( U  e.  (SubGrp `  W
)  ->  U  C_  ( Base `  W ) )
2724, 26syl 16 . . . . . 6  |-  ( ( ( W  e. NrmMod  /\  U  e.  S )  /\  (
x  e.  ( Base `  (Scalar `  X )
)  /\  y  e.  ( Base `  X )
) )  ->  U  C_  ( Base `  W
) )
28 simprr 756 . . . . . . 7  |-  ( ( ( W  e. NrmMod  /\  U  e.  S )  /\  (
x  e.  ( Base `  (Scalar `  X )
)  /\  y  e.  ( Base `  X )
) )  ->  y  e.  ( Base `  X
) )
297subgbas 15685 . . . . . . . 8  |-  ( U  e.  (SubGrp `  W
)  ->  U  =  ( Base `  X )
)
3024, 29syl 16 . . . . . . 7  |-  ( ( ( W  e. NrmMod  /\  U  e.  S )  /\  (
x  e.  ( Base `  (Scalar `  X )
)  /\  y  e.  ( Base `  X )
) )  ->  U  =  ( Base `  X
) )
3128, 30eleqtrrd 2520 . . . . . 6  |-  ( ( ( W  e. NrmMod  /\  U  e.  S )  /\  (
x  e.  ( Base `  (Scalar `  X )
)  /\  y  e.  ( Base `  X )
) )  ->  y  e.  U )
3227, 31sseldd 3357 . . . . 5  |-  ( ( ( W  e. NrmMod  /\  U  e.  S )  /\  (
x  e.  ( Base `  (Scalar `  X )
)  /\  y  e.  ( Base `  X )
) )  ->  y  e.  ( Base `  W
) )
33 eqid 2443 . . . . . 6  |-  ( norm `  W )  =  (
norm `  W )
34 eqid 2443 . . . . . 6  |-  ( .s
`  W )  =  ( .s `  W
)
35 eqid 2443 . . . . . 6  |-  ( Base `  (Scalar `  W )
)  =  ( Base `  (Scalar `  W )
)
36 eqid 2443 . . . . . 6  |-  ( norm `  (Scalar `  W )
)  =  ( norm `  (Scalar `  W )
)
3725, 33, 34, 12, 35, 36nmvs 20257 . . . . 5  |-  ( ( W  e. NrmMod  /\  x  e.  ( Base `  (Scalar `  W ) )  /\  y  e.  ( Base `  W ) )  -> 
( ( norm `  W
) `  ( x
( .s `  W
) y ) )  =  ( ( (
norm `  (Scalar `  W
) ) `  x
)  x.  ( (
norm `  W ) `  y ) ) )
3819, 23, 32, 37syl3anc 1218 . . . 4  |-  ( ( ( W  e. NrmMod  /\  U  e.  S )  /\  (
x  e.  ( Base `  (Scalar `  X )
)  /\  y  e.  ( Base `  X )
) )  ->  (
( norm `  W ) `  ( x ( .s
`  W ) y ) )  =  ( ( ( norm `  (Scalar `  W ) ) `  x )  x.  (
( norm `  W ) `  y ) ) )
39 simplr 754 . . . . . . . 8  |-  ( ( ( W  e. NrmMod  /\  U  e.  S )  /\  (
x  e.  ( Base `  (Scalar `  X )
)  /\  y  e.  ( Base `  X )
) )  ->  U  e.  S )
407, 34ressvsca 14317 . . . . . . . 8  |-  ( U  e.  S  ->  ( .s `  W )  =  ( .s `  X
) )
4139, 40syl 16 . . . . . . 7  |-  ( ( ( W  e. NrmMod  /\  U  e.  S )  /\  (
x  e.  ( Base `  (Scalar `  X )
)  /\  y  e.  ( Base `  X )
) )  ->  ( .s `  W )  =  ( .s `  X
) )
4241oveqd 6108 . . . . . 6  |-  ( ( ( W  e. NrmMod  /\  U  e.  S )  /\  (
x  e.  ( Base `  (Scalar `  X )
)  /\  y  e.  ( Base `  X )
) )  ->  (
x ( .s `  W ) y )  =  ( x ( .s `  X ) y ) )
4342fveq2d 5695 . . . . 5  |-  ( ( ( W  e. NrmMod  /\  U  e.  S )  /\  (
x  e.  ( Base `  (Scalar `  X )
)  /\  y  e.  ( Base `  X )
) )  ->  (
( norm `  X ) `  ( x ( .s
`  W ) y ) )  =  ( ( norm `  X
) `  ( x
( .s `  X
) y ) ) )
443ad2antrr 725 . . . . . . 7  |-  ( ( ( W  e. NrmMod  /\  U  e.  S )  /\  (
x  e.  ( Base `  (Scalar `  X )
)  /\  y  e.  ( Base `  X )
) )  ->  W  e.  LMod )
4512, 34, 35, 4lssvscl 17036 . . . . . . 7  |-  ( ( ( W  e.  LMod  /\  U  e.  S )  /\  ( x  e.  ( Base `  (Scalar `  W ) )  /\  y  e.  U )
)  ->  ( x
( .s `  W
) y )  e.  U )
4644, 39, 23, 31, 45syl22anc 1219 . . . . . 6  |-  ( ( ( W  e. NrmMod  /\  U  e.  S )  /\  (
x  e.  ( Base `  (Scalar `  X )
)  /\  y  e.  ( Base `  X )
) )  ->  (
x ( .s `  W ) y )  e.  U )
47 eqid 2443 . . . . . . 7  |-  ( norm `  X )  =  (
norm `  X )
487, 33, 47subgnm2 20220 . . . . . 6  |-  ( ( U  e.  (SubGrp `  W )  /\  (
x ( .s `  W ) y )  e.  U )  -> 
( ( norm `  X
) `  ( x
( .s `  W
) y ) )  =  ( ( norm `  W ) `  (
x ( .s `  W ) y ) ) )
4924, 46, 48syl2anc 661 . . . . 5  |-  ( ( ( W  e. NrmMod  /\  U  e.  S )  /\  (
x  e.  ( Base `  (Scalar `  X )
)  /\  y  e.  ( Base `  X )
) )  ->  (
( norm `  X ) `  ( x ( .s
`  W ) y ) )  =  ( ( norm `  W
) `  ( x
( .s `  W
) y ) ) )
5043, 49eqtr3d 2477 . . . 4  |-  ( ( ( W  e. NrmMod  /\  U  e.  S )  /\  (
x  e.  ( Base `  (Scalar `  X )
)  /\  y  e.  ( Base `  X )
) )  ->  (
( norm `  X ) `  ( x ( .s
`  X ) y ) )  =  ( ( norm `  W
) `  ( x
( .s `  W
) y ) ) )
5121eqcomd 2448 . . . . . . 7  |-  ( ( ( W  e. NrmMod  /\  U  e.  S )  /\  (
x  e.  ( Base `  (Scalar `  X )
)  /\  y  e.  ( Base `  X )
) )  ->  (Scalar `  X )  =  (Scalar `  W ) )
5251fveq2d 5695 . . . . . 6  |-  ( ( ( W  e. NrmMod  /\  U  e.  S )  /\  (
x  e.  ( Base `  (Scalar `  X )
)  /\  y  e.  ( Base `  X )
) )  ->  ( norm `  (Scalar `  X
) )  =  (
norm `  (Scalar `  W
) ) )
5352fveq1d 5693 . . . . 5  |-  ( ( ( W  e. NrmMod  /\  U  e.  S )  /\  (
x  e.  ( Base `  (Scalar `  X )
)  /\  y  e.  ( Base `  X )
) )  ->  (
( norm `  (Scalar `  X
) ) `  x
)  =  ( (
norm `  (Scalar `  W
) ) `  x
) )
547, 33, 47subgnm2 20220 . . . . . 6  |-  ( ( U  e.  (SubGrp `  W )  /\  y  e.  U )  ->  (
( norm `  X ) `  y )  =  ( ( norm `  W
) `  y )
)
5524, 31, 54syl2anc 661 . . . . 5  |-  ( ( ( W  e. NrmMod  /\  U  e.  S )  /\  (
x  e.  ( Base `  (Scalar `  X )
)  /\  y  e.  ( Base `  X )
) )  ->  (
( norm `  X ) `  y )  =  ( ( norm `  W
) `  y )
)
5653, 55oveq12d 6109 . . . 4  |-  ( ( ( W  e. NrmMod  /\  U  e.  S )  /\  (
x  e.  ( Base `  (Scalar `  X )
)  /\  y  e.  ( Base `  X )
) )  ->  (
( ( norm `  (Scalar `  X ) ) `  x )  x.  (
( norm `  X ) `  y ) )  =  ( ( ( norm `  (Scalar `  W )
) `  x )  x.  ( ( norm `  W
) `  y )
) )
5738, 50, 563eqtr4d 2485 . . 3  |-  ( ( ( W  e. NrmMod  /\  U  e.  S )  /\  (
x  e.  ( Base `  (Scalar `  X )
)  /\  y  e.  ( Base `  X )
) )  ->  (
( norm `  X ) `  ( x ( .s
`  X ) y ) )  =  ( ( ( norm `  (Scalar `  X ) ) `  x )  x.  (
( norm `  X ) `  y ) ) )
5857ralrimivva 2808 . 2  |-  ( ( W  e. NrmMod  /\  U  e.  S )  ->  A. x  e.  ( Base `  (Scalar `  X ) ) A. y  e.  ( Base `  X ) ( (
norm `  X ) `  ( x ( .s
`  X ) y ) )  =  ( ( ( norm `  (Scalar `  X ) ) `  x )  x.  (
( norm `  X ) `  y ) ) )
59 eqid 2443 . . 3  |-  ( Base `  X )  =  (
Base `  X )
60 eqid 2443 . . 3  |-  ( .s
`  X )  =  ( .s `  X
)
61 eqid 2443 . . 3  |-  (Scalar `  X )  =  (Scalar `  X )
62 eqid 2443 . . 3  |-  ( Base `  (Scalar `  X )
)  =  ( Base `  (Scalar `  X )
)
63 eqid 2443 . . 3  |-  ( norm `  (Scalar `  X )
)  =  ( norm `  (Scalar `  X )
)
6459, 47, 60, 61, 62, 63isnlm 20256 . 2  |-  ( X  e. NrmMod 
<->  ( ( X  e. NrmGrp  /\  X  e.  LMod  /\  (Scalar `  X )  e. NrmRing )  /\  A. x  e.  ( Base `  (Scalar `  X ) ) A. y  e.  ( Base `  X ) ( (
norm `  X ) `  ( x ( .s
`  X ) y ) )  =  ( ( ( norm `  (Scalar `  X ) ) `  x )  x.  (
( norm `  X ) `  y ) ) ) )
6518, 58, 64sylanbrc 664 1  |-  ( ( W  e. NrmMod  /\  U  e.  S )  ->  X  e. NrmMod )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756   A.wral 2715    C_ wss 3328   ` cfv 5418  (class class class)co 6091    x. cmul 9287   Basecbs 14174   ↾s cress 14175  Scalarcsca 14241   .scvsca 14242  SubGrpcsubg 15675   LModclmod 16948   LSubSpclss 17013   normcnm 20169  NrmGrpcngp 20170  NrmRingcnrg 20172  NrmModcnlm 20173
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4403  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372  ax-cnex 9338  ax-resscn 9339  ax-1cn 9340  ax-icn 9341  ax-addcl 9342  ax-addrcl 9343  ax-mulcl 9344  ax-mulrcl 9345  ax-mulcom 9346  ax-addass 9347  ax-mulass 9348  ax-distr 9349  ax-i2m1 9350  ax-1ne0 9351  ax-1rid 9352  ax-rnegex 9353  ax-rrecex 9354  ax-cnre 9355  ax-pre-lttri 9356  ax-pre-lttrn 9357  ax-pre-ltadd 9358  ax-pre-mulgt0 9359  ax-pre-sup 9360
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-nel 2609  df-ral 2720  df-rex 2721  df-reu 2722  df-rmo 2723  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-pss 3344  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-tp 3882  df-op 3884  df-uni 4092  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-tr 4386  df-eprel 4632  df-id 4636  df-po 4641  df-so 4642  df-fr 4679  df-we 4681  df-ord 4722  df-on 4723  df-lim 4724  df-suc 4725  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-riota 6052  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-om 6477  df-1st 6577  df-2nd 6578  df-recs 6832  df-rdg 6866  df-er 7101  df-map 7216  df-en 7311  df-dom 7312  df-sdom 7313  df-sup 7691  df-pnf 9420  df-mnf 9421  df-xr 9422  df-ltxr 9423  df-le 9424  df-sub 9597  df-neg 9598  df-div 9994  df-nn 10323  df-2 10380  df-3 10381  df-4 10382  df-5 10383  df-6 10384  df-7 10385  df-8 10386  df-9 10387  df-10 10388  df-n0 10580  df-z 10647  df-dec 10756  df-uz 10862  df-q 10954  df-rp 10992  df-xneg 11089  df-xadd 11090  df-xmul 11091  df-ndx 14177  df-slot 14178  df-base 14179  df-sets 14180  df-ress 14181  df-plusg 14251  df-sca 14254  df-vsca 14255  df-tset 14257  df-ds 14260  df-rest 14361  df-topn 14362  df-0g 14380  df-topgen 14382  df-mnd 15415  df-grp 15545  df-minusg 15546  df-sbg 15547  df-subg 15678  df-mgp 16592  df-ur 16604  df-rng 16647  df-lmod 16950  df-lss 17014  df-psmet 17809  df-xmet 17810  df-met 17811  df-bl 17812  df-mopn 17813  df-top 18503  df-bases 18505  df-topon 18506  df-topsp 18507  df-xms 19895  df-ms 19896  df-nm 20175  df-ngp 20176  df-nlm 20179
This theorem is referenced by:  lssnvc  20282
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