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Theorem lssnlm 20181
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 20158 . . . . 5  |-  ( W  e. NrmMod  ->  W  e. NrmGrp )
21adantr 462 . . . 4  |-  ( ( W  e. NrmMod  /\  U  e.  S )  ->  W  e. NrmGrp )
3 nlmlmod 20159 . . . . 5  |-  ( W  e. NrmMod  ->  W  e.  LMod )
4 lssnlm.s . . . . . 6  |-  S  =  ( LSubSp `  W )
54lsssubg 17016 . . . . 5  |-  ( ( W  e.  LMod  /\  U  e.  S )  ->  U  e.  (SubGrp `  W )
)
63, 5sylan 468 . . . 4  |-  ( ( W  e. NrmMod  /\  U  e.  S )  ->  U  e.  (SubGrp `  W )
)
7 lssnlm.x . . . . 5  |-  X  =  ( Ws  U )
87subgngp 20121 . . . 4  |-  ( ( W  e. NrmGrp  /\  U  e.  (SubGrp `  W )
)  ->  X  e. NrmGrp )
92, 6, 8syl2anc 656 . . 3  |-  ( ( W  e. NrmMod  /\  U  e.  S )  ->  X  e. NrmGrp )
107, 4lsslmod 17019 . . . 4  |-  ( ( W  e.  LMod  /\  U  e.  S )  ->  X  e.  LMod )
113, 10sylan 468 . . 3  |-  ( ( W  e. NrmMod  /\  U  e.  S )  ->  X  e.  LMod )
12 eqid 2441 . . . . . 6  |-  (Scalar `  W )  =  (Scalar `  W )
137, 12resssca 14312 . . . . 5  |-  ( U  e.  S  ->  (Scalar `  W )  =  (Scalar `  X ) )
1413adantl 463 . . . 4  |-  ( ( W  e. NrmMod  /\  U  e.  S )  ->  (Scalar `  W )  =  (Scalar `  X ) )
1512nlmnrg 20160 . . . . 5  |-  ( W  e. NrmMod  ->  (Scalar `  W )  e. NrmRing )
1615adantr 462 . . . 4  |-  ( ( W  e. NrmMod  /\  U  e.  S )  ->  (Scalar `  W )  e. NrmRing )
1714, 16eqeltrrd 2516 . . 3  |-  ( ( W  e. NrmMod  /\  U  e.  S )  ->  (Scalar `  X )  e. NrmRing )
189, 11, 173jca 1163 . 2  |-  ( ( W  e. NrmMod  /\  U  e.  S )  ->  ( X  e. NrmGrp  /\  X  e. 
LMod  /\  (Scalar `  X
)  e. NrmRing ) )
19 simpll 748 . . . . 5  |-  ( ( ( W  e. NrmMod  /\  U  e.  S )  /\  (
x  e.  ( Base `  (Scalar `  X )
)  /\  y  e.  ( Base `  X )
) )  ->  W  e. NrmMod )
20 simprl 750 . . . . . 6  |-  ( ( ( W  e. NrmMod  /\  U  e.  S )  /\  (
x  e.  ( Base `  (Scalar `  X )
)  /\  y  e.  ( Base `  X )
) )  ->  x  e.  ( Base `  (Scalar `  X ) ) )
2114adantr 462 . . . . . . 7  |-  ( ( ( W  e. NrmMod  /\  U  e.  S )  /\  (
x  e.  ( Base `  (Scalar `  X )
)  /\  y  e.  ( Base `  X )
) )  ->  (Scalar `  W )  =  (Scalar `  X ) )
2221fveq2d 5692 . . . . . 6  |-  ( ( ( W  e. NrmMod  /\  U  e.  S )  /\  (
x  e.  ( Base `  (Scalar `  X )
)  /\  y  e.  ( Base `  X )
) )  ->  ( Base `  (Scalar `  W
) )  =  (
Base `  (Scalar `  X
) ) )
2320, 22eleqtrrd 2518 . . . . 5  |-  ( ( ( W  e. NrmMod  /\  U  e.  S )  /\  (
x  e.  ( Base `  (Scalar `  X )
)  /\  y  e.  ( Base `  X )
) )  ->  x  e.  ( Base `  (Scalar `  W ) ) )
246adantr 462 . . . . . . 7  |-  ( ( ( W  e. NrmMod  /\  U  e.  S )  /\  (
x  e.  ( Base `  (Scalar `  X )
)  /\  y  e.  ( Base `  X )
) )  ->  U  e.  (SubGrp `  W )
)
25 eqid 2441 . . . . . . . 8  |-  ( Base `  W )  =  (
Base `  W )
2625subgss 15675 . . . . . . 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 751 . . . . . . 7  |-  ( ( ( W  e. NrmMod  /\  U  e.  S )  /\  (
x  e.  ( Base `  (Scalar `  X )
)  /\  y  e.  ( Base `  X )
) )  ->  y  e.  ( Base `  X
) )
297subgbas 15678 . . . . . . . 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 2518 . . . . . 6  |-  ( ( ( W  e. NrmMod  /\  U  e.  S )  /\  (
x  e.  ( Base `  (Scalar `  X )
)  /\  y  e.  ( Base `  X )
) )  ->  y  e.  U )
3227, 31sseldd 3354 . . . . 5  |-  ( ( ( W  e. NrmMod  /\  U  e.  S )  /\  (
x  e.  ( Base `  (Scalar `  X )
)  /\  y  e.  ( Base `  X )
) )  ->  y  e.  ( Base `  W
) )
33 eqid 2441 . . . . . 6  |-  ( norm `  W )  =  (
norm `  W )
34 eqid 2441 . . . . . 6  |-  ( .s
`  W )  =  ( .s `  W
)
35 eqid 2441 . . . . . 6  |-  ( Base `  (Scalar `  W )
)  =  ( Base `  (Scalar `  W )
)
36 eqid 2441 . . . . . 6  |-  ( norm `  (Scalar `  W )
)  =  ( norm `  (Scalar `  W )
)
3725, 33, 34, 12, 35, 36nmvs 20157 . . . . 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 1213 . . . 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 749 . . . . . . . 8  |-  ( ( ( W  e. NrmMod  /\  U  e.  S )  /\  (
x  e.  ( Base `  (Scalar `  X )
)  /\  y  e.  ( Base `  X )
) )  ->  U  e.  S )
407, 34ressvsca 14313 . . . . . . . 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 6107 . . . . . 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 5692 . . . . 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 720 . . . . . . 7  |-  ( ( ( W  e. NrmMod  /\  U  e.  S )  /\  (
x  e.  ( Base `  (Scalar `  X )
)  /\  y  e.  ( Base `  X )
) )  ->  W  e.  LMod )
4512, 34, 35, 4lssvscl 17014 . . . . . . 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 1214 . . . . . 6  |-  ( ( ( W  e. NrmMod  /\  U  e.  S )  /\  (
x  e.  ( Base `  (Scalar `  X )
)  /\  y  e.  ( Base `  X )
) )  ->  (
x ( .s `  W ) y )  e.  U )
47 eqid 2441 . . . . . . 7  |-  ( norm `  X )  =  (
norm `  X )
487, 33, 47subgnm2 20120 . . . . . 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 656 . . . . 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 2475 . . . 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 2446 . . . . . . 7  |-  ( ( ( W  e. NrmMod  /\  U  e.  S )  /\  (
x  e.  ( Base `  (Scalar `  X )
)  /\  y  e.  ( Base `  X )
) )  ->  (Scalar `  X )  =  (Scalar `  W ) )
5251fveq2d 5692 . . . . . 6  |-  ( ( ( W  e. NrmMod  /\  U  e.  S )  /\  (
x  e.  ( Base `  (Scalar `  X )
)  /\  y  e.  ( Base `  X )
) )  ->  ( norm `  (Scalar `  X
) )  =  (
norm `  (Scalar `  W
) ) )
5352fveq1d 5690 . . . . 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 20120 . . . . . 6  |-  ( ( U  e.  (SubGrp `  W )  /\  y  e.  U )  ->  (
( norm `  X ) `  y )  =  ( ( norm `  W
) `  y )
)
5524, 31, 54syl2anc 656 . . . . 5  |-  ( ( ( W  e. NrmMod  /\  U  e.  S )  /\  (
x  e.  ( Base `  (Scalar `  X )
)  /\  y  e.  ( Base `  X )
) )  ->  (
( norm `  X ) `  y )  =  ( ( norm `  W
) `  y )
)
5653, 55oveq12d 6108 . . . 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 2483 . . 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 2806 . 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 2441 . . 3  |-  ( Base `  X )  =  (
Base `  X )
60 eqid 2441 . . 3  |-  ( .s
`  X )  =  ( .s `  X
)
61 eqid 2441 . . 3  |-  (Scalar `  X )  =  (Scalar `  X )
62 eqid 2441 . . 3  |-  ( Base `  (Scalar `  X )
)  =  ( Base `  (Scalar `  X )
)
63 eqid 2441 . . 3  |-  ( norm `  (Scalar `  X )
)  =  ( norm `  (Scalar `  X )
)
6459, 47, 60, 61, 62, 63isnlm 20156 . 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 659 1  |-  ( ( W  e. NrmMod  /\  U  e.  S )  ->  X  e. NrmMod )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 960    = wceq 1364    e. wcel 1761   A.wral 2713    C_ wss 3325   ` cfv 5415  (class class class)co 6090    x. cmul 9283   Basecbs 14170   ↾s cress 14171  Scalarcsca 14237   .scvsca 14238  SubGrpcsubg 15668   LModclmod 16928   LSubSpclss 16991   normcnm 20069  NrmGrpcngp 20070  NrmRingcnrg 20072  NrmModcnlm 20073
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355  ax-pre-sup 9356
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2263  df-mo 2264  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-1st 6576  df-2nd 6577  df-recs 6828  df-rdg 6862  df-er 7097  df-map 7212  df-en 7307  df-dom 7308  df-sdom 7309  df-sup 7687  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-div 9990  df-nn 10319  df-2 10376  df-3 10377  df-4 10378  df-5 10379  df-6 10380  df-7 10381  df-8 10382  df-9 10383  df-10 10384  df-n0 10576  df-z 10643  df-dec 10752  df-uz 10858  df-q 10950  df-rp 10988  df-xneg 11085  df-xadd 11086  df-xmul 11087  df-ndx 14173  df-slot 14174  df-base 14175  df-sets 14176  df-ress 14177  df-plusg 14247  df-sca 14250  df-vsca 14251  df-tset 14253  df-ds 14256  df-rest 14357  df-topn 14358  df-0g 14376  df-topgen 14378  df-mnd 15411  df-grp 15538  df-minusg 15539  df-sbg 15540  df-subg 15671  df-mgp 16582  df-ur 16594  df-rng 16637  df-lmod 16930  df-lss 16992  df-psmet 17709  df-xmet 17710  df-met 17711  df-bl 17712  df-mopn 17713  df-top 18403  df-bases 18405  df-topon 18406  df-topsp 18407  df-xms 19795  df-ms 19796  df-nm 20075  df-ngp 20076  df-nlm 20079
This theorem is referenced by:  lssnvc  20182
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