MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  lssnlm Unicode version

Theorem lssnlm 18227
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 18204 . . . . 5  |-  ( W  e. NrmMod  ->  W  e. NrmGrp )
21adantr 451 . . . 4  |-  ( ( W  e. NrmMod  /\  U  e.  S )  ->  W  e. NrmGrp )
3 nlmlmod 18205 . . . . 5  |-  ( W  e. NrmMod  ->  W  e.  LMod )
4 lssnlm.s . . . . . 6  |-  S  =  ( LSubSp `  W )
54lsssubg 15730 . . . . 5  |-  ( ( W  e.  LMod  /\  U  e.  S )  ->  U  e.  (SubGrp `  W )
)
63, 5sylan 457 . . . 4  |-  ( ( W  e. NrmMod  /\  U  e.  S )  ->  U  e.  (SubGrp `  W )
)
7 lssnlm.x . . . . 5  |-  X  =  ( Ws  U )
87subgngp 18167 . . . 4  |-  ( ( W  e. NrmGrp  /\  U  e.  (SubGrp `  W )
)  ->  X  e. NrmGrp )
92, 6, 8syl2anc 642 . . 3  |-  ( ( W  e. NrmMod  /\  U  e.  S )  ->  X  e. NrmGrp )
107, 4lsslmod 15733 . . . 4  |-  ( ( W  e.  LMod  /\  U  e.  S )  ->  X  e.  LMod )
113, 10sylan 457 . . 3  |-  ( ( W  e. NrmMod  /\  U  e.  S )  ->  X  e.  LMod )
12 eqid 2296 . . . . . 6  |-  (Scalar `  W )  =  (Scalar `  W )
137, 12resssca 13299 . . . . 5  |-  ( U  e.  S  ->  (Scalar `  W )  =  (Scalar `  X ) )
1413adantl 452 . . . 4  |-  ( ( W  e. NrmMod  /\  U  e.  S )  ->  (Scalar `  W )  =  (Scalar `  X ) )
1512nlmnrg 18206 . . . . 5  |-  ( W  e. NrmMod  ->  (Scalar `  W )  e. NrmRing )
1615adantr 451 . . . 4  |-  ( ( W  e. NrmMod  /\  U  e.  S )  ->  (Scalar `  W )  e. NrmRing )
1714, 16eqeltrrd 2371 . . 3  |-  ( ( W  e. NrmMod  /\  U  e.  S )  ->  (Scalar `  X )  e. NrmRing )
189, 11, 173jca 1132 . 2  |-  ( ( W  e. NrmMod  /\  U  e.  S )  ->  ( X  e. NrmGrp  /\  X  e. 
LMod  /\  (Scalar `  X
)  e. NrmRing ) )
19 simpll 730 . . . . 5  |-  ( ( ( W  e. NrmMod  /\  U  e.  S )  /\  (
x  e.  ( Base `  (Scalar `  X )
)  /\  y  e.  ( Base `  X )
) )  ->  W  e. NrmMod )
20 simprl 732 . . . . . 6  |-  ( ( ( W  e. NrmMod  /\  U  e.  S )  /\  (
x  e.  ( Base `  (Scalar `  X )
)  /\  y  e.  ( Base `  X )
) )  ->  x  e.  ( Base `  (Scalar `  X ) ) )
2114adantr 451 . . . . . . 7  |-  ( ( ( W  e. NrmMod  /\  U  e.  S )  /\  (
x  e.  ( Base `  (Scalar `  X )
)  /\  y  e.  ( Base `  X )
) )  ->  (Scalar `  W )  =  (Scalar `  X ) )
2221fveq2d 5545 . . . . . 6  |-  ( ( ( W  e. NrmMod  /\  U  e.  S )  /\  (
x  e.  ( Base `  (Scalar `  X )
)  /\  y  e.  ( Base `  X )
) )  ->  ( Base `  (Scalar `  W
) )  =  (
Base `  (Scalar `  X
) ) )
2320, 22eleqtrrd 2373 . . . . 5  |-  ( ( ( W  e. NrmMod  /\  U  e.  S )  /\  (
x  e.  ( Base `  (Scalar `  X )
)  /\  y  e.  ( Base `  X )
) )  ->  x  e.  ( Base `  (Scalar `  W ) ) )
246adantr 451 . . . . . . 7  |-  ( ( ( W  e. NrmMod  /\  U  e.  S )  /\  (
x  e.  ( Base `  (Scalar `  X )
)  /\  y  e.  ( Base `  X )
) )  ->  U  e.  (SubGrp `  W )
)
25 eqid 2296 . . . . . . . 8  |-  ( Base `  W )  =  (
Base `  W )
2625subgss 14638 . . . . . . 7  |-  ( U  e.  (SubGrp `  W
)  ->  U  C_  ( Base `  W ) )
2724, 26syl 15 . . . . . 6  |-  ( ( ( W  e. NrmMod  /\  U  e.  S )  /\  (
x  e.  ( Base `  (Scalar `  X )
)  /\  y  e.  ( Base `  X )
) )  ->  U  C_  ( Base `  W
) )
28 simprr 733 . . . . . . 7  |-  ( ( ( W  e. NrmMod  /\  U  e.  S )  /\  (
x  e.  ( Base `  (Scalar `  X )
)  /\  y  e.  ( Base `  X )
) )  ->  y  e.  ( Base `  X
) )
297subgbas 14641 . . . . . . . 8  |-  ( U  e.  (SubGrp `  W
)  ->  U  =  ( Base `  X )
)
3024, 29syl 15 . . . . . . 7  |-  ( ( ( W  e. NrmMod  /\  U  e.  S )  /\  (
x  e.  ( Base `  (Scalar `  X )
)  /\  y  e.  ( Base `  X )
) )  ->  U  =  ( Base `  X
) )
3128, 30eleqtrrd 2373 . . . . . 6  |-  ( ( ( W  e. NrmMod  /\  U  e.  S )  /\  (
x  e.  ( Base `  (Scalar `  X )
)  /\  y  e.  ( Base `  X )
) )  ->  y  e.  U )
3227, 31sseldd 3194 . . . . 5  |-  ( ( ( W  e. NrmMod  /\  U  e.  S )  /\  (
x  e.  ( Base `  (Scalar `  X )
)  /\  y  e.  ( Base `  X )
) )  ->  y  e.  ( Base `  W
) )
33 eqid 2296 . . . . . 6  |-  ( norm `  W )  =  (
norm `  W )
34 eqid 2296 . . . . . 6  |-  ( .s
`  W )  =  ( .s `  W
)
35 eqid 2296 . . . . . 6  |-  ( Base `  (Scalar `  W )
)  =  ( Base `  (Scalar `  W )
)
36 eqid 2296 . . . . . 6  |-  ( norm `  (Scalar `  W )
)  =  ( norm `  (Scalar `  W )
)
3725, 33, 34, 12, 35, 36nmvs 18203 . . . . 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 1182 . . . 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 731 . . . . . . . 8  |-  ( ( ( W  e. NrmMod  /\  U  e.  S )  /\  (
x  e.  ( Base `  (Scalar `  X )
)  /\  y  e.  ( Base `  X )
) )  ->  U  e.  S )
407, 34ressvsca 13300 . . . . . . . 8  |-  ( U  e.  S  ->  ( .s `  W )  =  ( .s `  X
) )
4139, 40syl 15 . . . . . . 7  |-  ( ( ( W  e. NrmMod  /\  U  e.  S )  /\  (
x  e.  ( Base `  (Scalar `  X )
)  /\  y  e.  ( Base `  X )
) )  ->  ( .s `  W )  =  ( .s `  X
) )
4241oveqd 5891 . . . . . 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 5545 . . . . 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 706 . . . . . . 7  |-  ( ( ( W  e. NrmMod  /\  U  e.  S )  /\  (
x  e.  ( Base `  (Scalar `  X )
)  /\  y  e.  ( Base `  X )
) )  ->  W  e.  LMod )
4512, 34, 35, 4lssvscl 15728 . . . . . . 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 1183 . . . . . 6  |-  ( ( ( W  e. NrmMod  /\  U  e.  S )  /\  (
x  e.  ( Base `  (Scalar `  X )
)  /\  y  e.  ( Base `  X )
) )  ->  (
x ( .s `  W ) y )  e.  U )
47 eqid 2296 . . . . . . 7  |-  ( norm `  X )  =  (
norm `  X )
487, 33, 47subgnm2 18166 . . . . . 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 642 . . . . 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 2330 . . . 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 2301 . . . . . . 7  |-  ( ( ( W  e. NrmMod  /\  U  e.  S )  /\  (
x  e.  ( Base `  (Scalar `  X )
)  /\  y  e.  ( Base `  X )
) )  ->  (Scalar `  X )  =  (Scalar `  W ) )
5251fveq2d 5545 . . . . . 6  |-  ( ( ( W  e. NrmMod  /\  U  e.  S )  /\  (
x  e.  ( Base `  (Scalar `  X )
)  /\  y  e.  ( Base `  X )
) )  ->  ( norm `  (Scalar `  X
) )  =  (
norm `  (Scalar `  W
) ) )
5352fveq1d 5543 . . . . 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 18166 . . . . . 6  |-  ( ( U  e.  (SubGrp `  W )  /\  y  e.  U )  ->  (
( norm `  X ) `  y )  =  ( ( norm `  W
) `  y )
)
5524, 31, 54syl2anc 642 . . . . 5  |-  ( ( ( W  e. NrmMod  /\  U  e.  S )  /\  (
x  e.  ( Base `  (Scalar `  X )
)  /\  y  e.  ( Base `  X )
) )  ->  (
( norm `  X ) `  y )  =  ( ( norm `  W
) `  y )
)
5653, 55oveq12d 5892 . . . 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 2338 . . 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 2648 . 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 2296 . . 3  |-  ( Base `  X )  =  (
Base `  X )
60 eqid 2296 . . 3  |-  ( .s
`  X )  =  ( .s `  X
)
61 eqid 2296 . . 3  |-  (Scalar `  X )  =  (Scalar `  X )
62 eqid 2296 . . 3  |-  ( Base `  (Scalar `  X )
)  =  ( Base `  (Scalar `  X )
)
63 eqid 2296 . . 3  |-  ( norm `  (Scalar `  X )
)  =  ( norm `  (Scalar `  X )
)
6459, 47, 60, 61, 62, 63isnlm 18202 . 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 645 1  |-  ( ( W  e. NrmMod  /\  U  e.  S )  ->  X  e. NrmMod )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696   A.wral 2556    C_ wss 3165   ` cfv 5271  (class class class)co 5874    x. cmul 8758   Basecbs 13164   ↾s cress 13165  Scalarcsca 13227   .scvsca 13228  SubGrpcsubg 14631   LModclmod 15643   LSubSpclss 15705   normcnm 18115  NrmGrpcngp 18116  NrmRingcnrg 18118  NrmModcnlm 18119
This theorem is referenced by:  lssnvc  18228
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-er 6676  df-map 6790  df-en 6880  df-dom 6881  df-sdom 6882  df-sup 7210  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-3 9821  df-4 9822  df-5 9823  df-6 9824  df-7 9825  df-8 9826  df-9 9827  df-10 9828  df-n0 9982  df-z 10041  df-dec 10141  df-uz 10247  df-q 10333  df-rp 10371  df-xneg 10468  df-xadd 10469  df-xmul 10470  df-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-ress 13171  df-plusg 13237  df-sca 13240  df-vsca 13241  df-tset 13243  df-ds 13246  df-rest 13343  df-topn 13344  df-topgen 13360  df-0g 13420  df-mnd 14383  df-grp 14505  df-minusg 14506  df-sbg 14507  df-subg 14634  df-mgp 15342  df-rng 15356  df-ur 15358  df-lmod 15645  df-lss 15706  df-xmet 16389  df-met 16390  df-bl 16391  df-mopn 16392  df-top 16652  df-bases 16654  df-topon 16655  df-topsp 16656  df-xms 17901  df-ms 17902  df-nm 18121  df-ngp 18122  df-nlm 18125
  Copyright terms: Public domain W3C validator