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Theorem lssnlm 21082
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 21059 . . . . 5  |-  ( W  e. NrmMod  ->  W  e. NrmGrp )
21adantr 465 . . . 4  |-  ( ( W  e. NrmMod  /\  U  e.  S )  ->  W  e. NrmGrp )
3 nlmlmod 21060 . . . . 5  |-  ( W  e. NrmMod  ->  W  e.  LMod )
4 lssnlm.s . . . . . 6  |-  S  =  ( LSubSp `  W )
54lsssubg 17477 . . . . 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 21022 . . . 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 17480 . . . 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 14652 . . . . 5  |-  ( U  e.  S  ->  (Scalar `  W )  =  (Scalar `  X ) )
1413adantl 466 . . . 4  |-  ( ( W  e. NrmMod  /\  U  e.  S )  ->  (Scalar `  W )  =  (Scalar `  X ) )
1512nlmnrg 21061 . . . . 5  |-  ( W  e. NrmMod  ->  (Scalar `  W )  e. NrmRing )
1615adantr 465 . . . 4  |-  ( ( W  e. NrmMod  /\  U  e.  S )  ->  (Scalar `  W )  e. NrmRing )
1714, 16eqeltrrd 2532 . . 3  |-  ( ( W  e. NrmMod  /\  U  e.  S )  ->  (Scalar `  X )  e. NrmRing )
189, 11, 173jca 1177 . 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 756 . . . . . 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 5860 . . . . . 6  |-  ( ( ( W  e. NrmMod  /\  U  e.  S )  /\  (
x  e.  ( Base `  (Scalar `  X )
)  /\  y  e.  ( Base `  X )
) )  ->  ( Base `  (Scalar `  W
) )  =  (
Base `  (Scalar `  X
) ) )
2320, 22eleqtrrd 2534 . . . . 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 16076 . . . . . . 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 757 . . . . . . 7  |-  ( ( ( W  e. NrmMod  /\  U  e.  S )  /\  (
x  e.  ( Base `  (Scalar `  X )
)  /\  y  e.  ( Base `  X )
) )  ->  y  e.  ( Base `  X
) )
297subgbas 16079 . . . . . . . 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 2534 . . . . . 6  |-  ( ( ( W  e. NrmMod  /\  U  e.  S )  /\  (
x  e.  ( Base `  (Scalar `  X )
)  /\  y  e.  ( Base `  X )
) )  ->  y  e.  U )
3227, 31sseldd 3490 . . . . 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 21058 . . . . 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 1229 . . . 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 755 . . . . . . . 8  |-  ( ( ( W  e. NrmMod  /\  U  e.  S )  /\  (
x  e.  ( Base `  (Scalar `  X )
)  /\  y  e.  ( Base `  X )
) )  ->  U  e.  S )
407, 34ressvsca 14653 . . . . . . . 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 6298 . . . . . 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 5860 . . . . 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 17475 . . . . . . 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 1230 . . . . . 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 21021 . . . . . 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 2486 . . . 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 2451 . . . . . . 7  |-  ( ( ( W  e. NrmMod  /\  U  e.  S )  /\  (
x  e.  ( Base `  (Scalar `  X )
)  /\  y  e.  ( Base `  X )
) )  ->  (Scalar `  X )  =  (Scalar `  W ) )
5251fveq2d 5860 . . . . . 6  |-  ( ( ( W  e. NrmMod  /\  U  e.  S )  /\  (
x  e.  ( Base `  (Scalar `  X )
)  /\  y  e.  ( Base `  X )
) )  ->  ( norm `  (Scalar `  X
) )  =  (
norm `  (Scalar `  W
) ) )
5352fveq1d 5858 . . . . 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 21021 . . . . . 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 6299 . . . 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 2494 . . 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 2864 . 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 21057 . 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 974    = wceq 1383    e. wcel 1804   A.wral 2793    C_ wss 3461   ` cfv 5578  (class class class)co 6281    x. cmul 9500   Basecbs 14509   ↾s cress 14510  Scalarcsca 14577   .scvsca 14578  SubGrpcsubg 16069   LModclmod 17386   LSubSpclss 17452   normcnm 20970  NrmGrpcngp 20971  NrmRingcnrg 20973  NrmModcnlm 20974
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577  ax-cnex 9551  ax-resscn 9552  ax-1cn 9553  ax-icn 9554  ax-addcl 9555  ax-addrcl 9556  ax-mulcl 9557  ax-mulrcl 9558  ax-mulcom 9559  ax-addass 9560  ax-mulass 9561  ax-distr 9562  ax-i2m1 9563  ax-1ne0 9564  ax-1rid 9565  ax-rnegex 9566  ax-rrecex 9567  ax-cnre 9568  ax-pre-lttri 9569  ax-pre-lttrn 9570  ax-pre-ltadd 9571  ax-pre-mulgt0 9572  ax-pre-sup 9573
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-nel 2641  df-ral 2798  df-rex 2799  df-reu 2800  df-rmo 2801  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-uni 4235  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-tr 4531  df-eprel 4781  df-id 4785  df-po 4790  df-so 4791  df-fr 4828  df-we 4830  df-ord 4871  df-on 4872  df-lim 4873  df-suc 4874  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-riota 6242  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-om 6686  df-1st 6785  df-2nd 6786  df-recs 7044  df-rdg 7078  df-er 7313  df-map 7424  df-en 7519  df-dom 7520  df-sdom 7521  df-sup 7903  df-pnf 9633  df-mnf 9634  df-xr 9635  df-ltxr 9636  df-le 9637  df-sub 9812  df-neg 9813  df-div 10213  df-nn 10543  df-2 10600  df-3 10601  df-4 10602  df-5 10603  df-6 10604  df-7 10605  df-8 10606  df-9 10607  df-10 10608  df-n0 10802  df-z 10871  df-dec 10985  df-uz 11091  df-q 11192  df-rp 11230  df-xneg 11327  df-xadd 11328  df-xmul 11329  df-ndx 14512  df-slot 14513  df-base 14514  df-sets 14515  df-ress 14516  df-plusg 14587  df-sca 14590  df-vsca 14591  df-tset 14593  df-ds 14596  df-rest 14697  df-topn 14698  df-0g 14716  df-topgen 14718  df-mgm 15746  df-sgrp 15785  df-mnd 15795  df-grp 15931  df-minusg 15932  df-sbg 15933  df-subg 16072  df-mgp 17016  df-ur 17028  df-ring 17074  df-lmod 17388  df-lss 17453  df-psmet 18285  df-xmet 18286  df-met 18287  df-bl 18288  df-mopn 18289  df-top 19272  df-bases 19274  df-topon 19275  df-topsp 19276  df-xms 20696  df-ms 20697  df-nm 20976  df-ngp 20977  df-nlm 20980
This theorem is referenced by:  lssnvc  21083
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