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Theorem lssnlm 18689
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 18666 . . . . 5  |-  ( W  e. NrmMod  ->  W  e. NrmGrp )
21adantr 452 . . . 4  |-  ( ( W  e. NrmMod  /\  U  e.  S )  ->  W  e. NrmGrp )
3 nlmlmod 18667 . . . . 5  |-  ( W  e. NrmMod  ->  W  e.  LMod )
4 lssnlm.s . . . . . 6  |-  S  =  ( LSubSp `  W )
54lsssubg 15988 . . . . 5  |-  ( ( W  e.  LMod  /\  U  e.  S )  ->  U  e.  (SubGrp `  W )
)
63, 5sylan 458 . . . 4  |-  ( ( W  e. NrmMod  /\  U  e.  S )  ->  U  e.  (SubGrp `  W )
)
7 lssnlm.x . . . . 5  |-  X  =  ( Ws  U )
87subgngp 18629 . . . 4  |-  ( ( W  e. NrmGrp  /\  U  e.  (SubGrp `  W )
)  ->  X  e. NrmGrp )
92, 6, 8syl2anc 643 . . 3  |-  ( ( W  e. NrmMod  /\  U  e.  S )  ->  X  e. NrmGrp )
107, 4lsslmod 15991 . . . 4  |-  ( ( W  e.  LMod  /\  U  e.  S )  ->  X  e.  LMod )
113, 10sylan 458 . . 3  |-  ( ( W  e. NrmMod  /\  U  e.  S )  ->  X  e.  LMod )
12 eqid 2404 . . . . . 6  |-  (Scalar `  W )  =  (Scalar `  W )
137, 12resssca 13559 . . . . 5  |-  ( U  e.  S  ->  (Scalar `  W )  =  (Scalar `  X ) )
1413adantl 453 . . . 4  |-  ( ( W  e. NrmMod  /\  U  e.  S )  ->  (Scalar `  W )  =  (Scalar `  X ) )
1512nlmnrg 18668 . . . . 5  |-  ( W  e. NrmMod  ->  (Scalar `  W )  e. NrmRing )
1615adantr 452 . . . 4  |-  ( ( W  e. NrmMod  /\  U  e.  S )  ->  (Scalar `  W )  e. NrmRing )
1714, 16eqeltrrd 2479 . . 3  |-  ( ( W  e. NrmMod  /\  U  e.  S )  ->  (Scalar `  X )  e. NrmRing )
189, 11, 173jca 1134 . 2  |-  ( ( W  e. NrmMod  /\  U  e.  S )  ->  ( X  e. NrmGrp  /\  X  e. 
LMod  /\  (Scalar `  X
)  e. NrmRing ) )
19 simpll 731 . . . . 5  |-  ( ( ( W  e. NrmMod  /\  U  e.  S )  /\  (
x  e.  ( Base `  (Scalar `  X )
)  /\  y  e.  ( Base `  X )
) )  ->  W  e. NrmMod )
20 simprl 733 . . . . . 6  |-  ( ( ( W  e. NrmMod  /\  U  e.  S )  /\  (
x  e.  ( Base `  (Scalar `  X )
)  /\  y  e.  ( Base `  X )
) )  ->  x  e.  ( Base `  (Scalar `  X ) ) )
2114adantr 452 . . . . . . 7  |-  ( ( ( W  e. NrmMod  /\  U  e.  S )  /\  (
x  e.  ( Base `  (Scalar `  X )
)  /\  y  e.  ( Base `  X )
) )  ->  (Scalar `  W )  =  (Scalar `  X ) )
2221fveq2d 5691 . . . . . 6  |-  ( ( ( W  e. NrmMod  /\  U  e.  S )  /\  (
x  e.  ( Base `  (Scalar `  X )
)  /\  y  e.  ( Base `  X )
) )  ->  ( Base `  (Scalar `  W
) )  =  (
Base `  (Scalar `  X
) ) )
2320, 22eleqtrrd 2481 . . . . 5  |-  ( ( ( W  e. NrmMod  /\  U  e.  S )  /\  (
x  e.  ( Base `  (Scalar `  X )
)  /\  y  e.  ( Base `  X )
) )  ->  x  e.  ( Base `  (Scalar `  W ) ) )
246adantr 452 . . . . . . 7  |-  ( ( ( W  e. NrmMod  /\  U  e.  S )  /\  (
x  e.  ( Base `  (Scalar `  X )
)  /\  y  e.  ( Base `  X )
) )  ->  U  e.  (SubGrp `  W )
)
25 eqid 2404 . . . . . . . 8  |-  ( Base `  W )  =  (
Base `  W )
2625subgss 14900 . . . . . . 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 734 . . . . . . 7  |-  ( ( ( W  e. NrmMod  /\  U  e.  S )  /\  (
x  e.  ( Base `  (Scalar `  X )
)  /\  y  e.  ( Base `  X )
) )  ->  y  e.  ( Base `  X
) )
297subgbas 14903 . . . . . . . 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 2481 . . . . . 6  |-  ( ( ( W  e. NrmMod  /\  U  e.  S )  /\  (
x  e.  ( Base `  (Scalar `  X )
)  /\  y  e.  ( Base `  X )
) )  ->  y  e.  U )
3227, 31sseldd 3309 . . . . 5  |-  ( ( ( W  e. NrmMod  /\  U  e.  S )  /\  (
x  e.  ( Base `  (Scalar `  X )
)  /\  y  e.  ( Base `  X )
) )  ->  y  e.  ( Base `  W
) )
33 eqid 2404 . . . . . 6  |-  ( norm `  W )  =  (
norm `  W )
34 eqid 2404 . . . . . 6  |-  ( .s
`  W )  =  ( .s `  W
)
35 eqid 2404 . . . . . 6  |-  ( Base `  (Scalar `  W )
)  =  ( Base `  (Scalar `  W )
)
36 eqid 2404 . . . . . 6  |-  ( norm `  (Scalar `  W )
)  =  ( norm `  (Scalar `  W )
)
3725, 33, 34, 12, 35, 36nmvs 18665 . . . . 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 1184 . . . 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 732 . . . . . . . 8  |-  ( ( ( W  e. NrmMod  /\  U  e.  S )  /\  (
x  e.  ( Base `  (Scalar `  X )
)  /\  y  e.  ( Base `  X )
) )  ->  U  e.  S )
407, 34ressvsca 13560 . . . . . . . 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 6057 . . . . . 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 5691 . . . . 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 707 . . . . . . 7  |-  ( ( ( W  e. NrmMod  /\  U  e.  S )  /\  (
x  e.  ( Base `  (Scalar `  X )
)  /\  y  e.  ( Base `  X )
) )  ->  W  e.  LMod )
4512, 34, 35, 4lssvscl 15986 . . . . . . 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 1185 . . . . . 6  |-  ( ( ( W  e. NrmMod  /\  U  e.  S )  /\  (
x  e.  ( Base `  (Scalar `  X )
)  /\  y  e.  ( Base `  X )
) )  ->  (
x ( .s `  W ) y )  e.  U )
47 eqid 2404 . . . . . . 7  |-  ( norm `  X )  =  (
norm `  X )
487, 33, 47subgnm2 18628 . . . . . 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 643 . . . . 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 2438 . . . 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 2409 . . . . . . 7  |-  ( ( ( W  e. NrmMod  /\  U  e.  S )  /\  (
x  e.  ( Base `  (Scalar `  X )
)  /\  y  e.  ( Base `  X )
) )  ->  (Scalar `  X )  =  (Scalar `  W ) )
5251fveq2d 5691 . . . . . 6  |-  ( ( ( W  e. NrmMod  /\  U  e.  S )  /\  (
x  e.  ( Base `  (Scalar `  X )
)  /\  y  e.  ( Base `  X )
) )  ->  ( norm `  (Scalar `  X
) )  =  (
norm `  (Scalar `  W
) ) )
5352fveq1d 5689 . . . . 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 18628 . . . . . 6  |-  ( ( U  e.  (SubGrp `  W )  /\  y  e.  U )  ->  (
( norm `  X ) `  y )  =  ( ( norm `  W
) `  y )
)
5524, 31, 54syl2anc 643 . . . . 5  |-  ( ( ( W  e. NrmMod  /\  U  e.  S )  /\  (
x  e.  ( Base `  (Scalar `  X )
)  /\  y  e.  ( Base `  X )
) )  ->  (
( norm `  X ) `  y )  =  ( ( norm `  W
) `  y )
)
5653, 55oveq12d 6058 . . . 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 2446 . . 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 2758 . 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 2404 . . 3  |-  ( Base `  X )  =  (
Base `  X )
60 eqid 2404 . . 3  |-  ( .s
`  X )  =  ( .s `  X
)
61 eqid 2404 . . 3  |-  (Scalar `  X )  =  (Scalar `  X )
62 eqid 2404 . . 3  |-  ( Base `  (Scalar `  X )
)  =  ( Base `  (Scalar `  X )
)
63 eqid 2404 . . 3  |-  ( norm `  (Scalar `  X )
)  =  ( norm `  (Scalar `  X )
)
6459, 47, 60, 61, 62, 63isnlm 18664 . 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 646 1  |-  ( ( W  e. NrmMod  /\  U  e.  S )  ->  X  e. NrmMod )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721   A.wral 2666    C_ wss 3280   ` cfv 5413  (class class class)co 6040    x. cmul 8951   Basecbs 13424   ↾s cress 13425  Scalarcsca 13487   .scvsca 13488  SubGrpcsubg 14893   LModclmod 15905   LSubSpclss 15963   normcnm 18577  NrmGrpcngp 18578  NrmRingcnrg 18580  NrmModcnlm 18581
This theorem is referenced by:  lssnvc  18690
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023  ax-pre-sup 9024
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-er 6864  df-map 6979  df-en 7069  df-dom 7070  df-sdom 7071  df-sup 7404  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-nn 9957  df-2 10014  df-3 10015  df-4 10016  df-5 10017  df-6 10018  df-7 10019  df-8 10020  df-9 10021  df-10 10022  df-n0 10178  df-z 10239  df-dec 10339  df-uz 10445  df-q 10531  df-rp 10569  df-xneg 10666  df-xadd 10667  df-xmul 10668  df-ndx 13427  df-slot 13428  df-base 13429  df-sets 13430  df-ress 13431  df-plusg 13497  df-sca 13500  df-vsca 13501  df-tset 13503  df-ds 13506  df-rest 13605  df-topn 13606  df-topgen 13622  df-0g 13682  df-mnd 14645  df-grp 14767  df-minusg 14768  df-sbg 14769  df-subg 14896  df-mgp 15604  df-rng 15618  df-ur 15620  df-lmod 15907  df-lss 15964  df-psmet 16649  df-xmet 16650  df-met 16651  df-bl 16652  df-mopn 16653  df-top 16918  df-bases 16920  df-topon 16921  df-topsp 16922  df-xms 18303  df-ms 18304  df-nm 18583  df-ngp 18584  df-nlm 18587
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