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Theorem sranlm 20284
Description: The subring algebra over a normed ring is a normed left module. (Contributed by Mario Carneiro, 4-Oct-2015.)
Hypothesis
Ref Expression
sranlm.a  |-  A  =  ( (subringAlg  `  W ) `
 S )
Assertion
Ref Expression
sranlm  |-  ( ( W  e. NrmRing  /\  S  e.  (SubRing `  W )
)  ->  A  e. NrmMod )

Proof of Theorem sranlm
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nrgngp 20262 . . . . 5  |-  ( W  e. NrmRing  ->  W  e. NrmGrp )
21adantr 465 . . . 4  |-  ( ( W  e. NrmRing  /\  S  e.  (SubRing `  W )
)  ->  W  e. NrmGrp )
3 eqidd 2444 . . . . 5  |-  ( ( W  e. NrmRing  /\  S  e.  (SubRing `  W )
)  ->  ( Base `  W )  =  (
Base `  W )
)
4 sranlm.a . . . . . . 7  |-  A  =  ( (subringAlg  `  W ) `
 S )
54a1i 11 . . . . . 6  |-  ( ( W  e. NrmRing  /\  S  e.  (SubRing `  W )
)  ->  A  =  ( (subringAlg  `  W ) `  S ) )
6 eqid 2443 . . . . . . . 8  |-  ( Base `  W )  =  (
Base `  W )
76subrgss 16885 . . . . . . 7  |-  ( S  e.  (SubRing `  W
)  ->  S  C_  ( Base `  W ) )
87adantl 466 . . . . . 6  |-  ( ( W  e. NrmRing  /\  S  e.  (SubRing `  W )
)  ->  S  C_  ( Base `  W ) )
95, 8srabase 17278 . . . . 5  |-  ( ( W  e. NrmRing  /\  S  e.  (SubRing `  W )
)  ->  ( Base `  W )  =  (
Base `  A )
)
105, 8sraaddg 17279 . . . . . 6  |-  ( ( W  e. NrmRing  /\  S  e.  (SubRing `  W )
)  ->  ( +g  `  W )  =  ( +g  `  A ) )
1110proplem3 14648 . . . . 5  |-  ( ( ( W  e. NrmRing  /\  S  e.  (SubRing `  W )
)  /\  ( x  e.  ( Base `  W
)  /\  y  e.  ( Base `  W )
) )  ->  (
x ( +g  `  W
) y )  =  ( x ( +g  `  A ) y ) )
125, 8srads 17286 . . . . . 6  |-  ( ( W  e. NrmRing  /\  S  e.  (SubRing `  W )
)  ->  ( dist `  W )  =  (
dist `  A )
)
1312reseq1d 5128 . . . . 5  |-  ( ( W  e. NrmRing  /\  S  e.  (SubRing `  W )
)  ->  ( ( dist `  W )  |`  ( ( Base `  W
)  X.  ( Base `  W ) ) )  =  ( ( dist `  A )  |`  (
( Base `  W )  X.  ( Base `  W
) ) ) )
145, 8sratopn 17285 . . . . 5  |-  ( ( W  e. NrmRing  /\  S  e.  (SubRing `  W )
)  ->  ( TopOpen `  W )  =  (
TopOpen `  A ) )
153, 9, 11, 13, 14ngppropd 20242 . . . 4  |-  ( ( W  e. NrmRing  /\  S  e.  (SubRing `  W )
)  ->  ( W  e. NrmGrp  <-> 
A  e. NrmGrp ) )
162, 15mpbid 210 . . 3  |-  ( ( W  e. NrmRing  /\  S  e.  (SubRing `  W )
)  ->  A  e. NrmGrp )
174sralmod 17287 . . . 4  |-  ( S  e.  (SubRing `  W
)  ->  A  e.  LMod )
1817adantl 466 . . 3  |-  ( ( W  e. NrmRing  /\  S  e.  (SubRing `  W )
)  ->  A  e.  LMod )
195, 8srasca 17281 . . . 4  |-  ( ( W  e. NrmRing  /\  S  e.  (SubRing `  W )
)  ->  ( Ws  S
)  =  (Scalar `  A ) )
20 eqid 2443 . . . . 5  |-  ( Ws  S )  =  ( Ws  S )
2120subrgnrg 20273 . . . 4  |-  ( ( W  e. NrmRing  /\  S  e.  (SubRing `  W )
)  ->  ( Ws  S
)  e. NrmRing )
2219, 21eqeltrrd 2518 . . 3  |-  ( ( W  e. NrmRing  /\  S  e.  (SubRing `  W )
)  ->  (Scalar `  A
)  e. NrmRing )
2316, 18, 223jca 1168 . 2  |-  ( ( W  e. NrmRing  /\  S  e.  (SubRing `  W )
)  ->  ( A  e. NrmGrp  /\  A  e.  LMod  /\  (Scalar `  A )  e. NrmRing ) )
24 eqid 2443 . . . . . . . 8  |-  ( norm `  W )  =  (
norm `  W )
25 eqid 2443 . . . . . . . 8  |-  (AbsVal `  W )  =  (AbsVal `  W )
2624, 25nrgabv 20261 . . . . . . 7  |-  ( W  e. NrmRing  ->  ( norm `  W
)  e.  (AbsVal `  W ) )
2726ad2antrr 725 . . . . . 6  |-  ( ( ( W  e. NrmRing  /\  S  e.  (SubRing `  W )
)  /\  ( x  e.  ( Base `  (Scalar `  A ) )  /\  y  e.  ( Base `  A ) ) )  ->  ( norm `  W
)  e.  (AbsVal `  W ) )
288adantr 465 . . . . . . 7  |-  ( ( ( W  e. NrmRing  /\  S  e.  (SubRing `  W )
)  /\  ( x  e.  ( Base `  (Scalar `  A ) )  /\  y  e.  ( Base `  A ) ) )  ->  S  C_  ( Base `  W ) )
29 simprl 755 . . . . . . . 8  |-  ( ( ( W  e. NrmRing  /\  S  e.  (SubRing `  W )
)  /\  ( x  e.  ( Base `  (Scalar `  A ) )  /\  y  e.  ( Base `  A ) ) )  ->  x  e.  (
Base `  (Scalar `  A
) ) )
3020subrgbas 16893 . . . . . . . . . . 11  |-  ( S  e.  (SubRing `  W
)  ->  S  =  ( Base `  ( Ws  S
) ) )
3130adantl 466 . . . . . . . . . 10  |-  ( ( W  e. NrmRing  /\  S  e.  (SubRing `  W )
)  ->  S  =  ( Base `  ( Ws  S
) ) )
3219fveq2d 5714 . . . . . . . . . 10  |-  ( ( W  e. NrmRing  /\  S  e.  (SubRing `  W )
)  ->  ( Base `  ( Ws  S ) )  =  ( Base `  (Scalar `  A ) ) )
3331, 32eqtrd 2475 . . . . . . . . 9  |-  ( ( W  e. NrmRing  /\  S  e.  (SubRing `  W )
)  ->  S  =  ( Base `  (Scalar `  A
) ) )
3433adantr 465 . . . . . . . 8  |-  ( ( ( W  e. NrmRing  /\  S  e.  (SubRing `  W )
)  /\  ( x  e.  ( Base `  (Scalar `  A ) )  /\  y  e.  ( Base `  A ) ) )  ->  S  =  (
Base `  (Scalar `  A
) ) )
3529, 34eleqtrrd 2520 . . . . . . 7  |-  ( ( ( W  e. NrmRing  /\  S  e.  (SubRing `  W )
)  /\  ( x  e.  ( Base `  (Scalar `  A ) )  /\  y  e.  ( Base `  A ) ) )  ->  x  e.  S
)
3628, 35sseldd 3376 . . . . . 6  |-  ( ( ( W  e. NrmRing  /\  S  e.  (SubRing `  W )
)  /\  ( x  e.  ( Base `  (Scalar `  A ) )  /\  y  e.  ( Base `  A ) ) )  ->  x  e.  (
Base `  W )
)
37 simprr 756 . . . . . . 7  |-  ( ( ( W  e. NrmRing  /\  S  e.  (SubRing `  W )
)  /\  ( x  e.  ( Base `  (Scalar `  A ) )  /\  y  e.  ( Base `  A ) ) )  ->  y  e.  (
Base `  A )
)
389adantr 465 . . . . . . 7  |-  ( ( ( W  e. NrmRing  /\  S  e.  (SubRing `  W )
)  /\  ( x  e.  ( Base `  (Scalar `  A ) )  /\  y  e.  ( Base `  A ) ) )  ->  ( Base `  W
)  =  ( Base `  A ) )
3937, 38eleqtrrd 2520 . . . . . 6  |-  ( ( ( W  e. NrmRing  /\  S  e.  (SubRing `  W )
)  /\  ( x  e.  ( Base `  (Scalar `  A ) )  /\  y  e.  ( Base `  A ) ) )  ->  y  e.  (
Base `  W )
)
40 eqid 2443 . . . . . . 7  |-  ( .r
`  W )  =  ( .r `  W
)
4125, 6, 40abvmul 16933 . . . . . 6  |-  ( ( ( norm `  W
)  e.  (AbsVal `  W )  /\  x  e.  ( Base `  W
)  /\  y  e.  ( Base `  W )
)  ->  ( ( norm `  W ) `  ( x ( .r
`  W ) y ) )  =  ( ( ( norm `  W
) `  x )  x.  ( ( norm `  W
) `  y )
) )
4227, 36, 39, 41syl3anc 1218 . . . . 5  |-  ( ( ( W  e. NrmRing  /\  S  e.  (SubRing `  W )
)  /\  ( x  e.  ( Base `  (Scalar `  A ) )  /\  y  e.  ( Base `  A ) ) )  ->  ( ( norm `  W ) `  (
x ( .r `  W ) y ) )  =  ( ( ( norm `  W
) `  x )  x.  ( ( norm `  W
) `  y )
) )
439, 10, 12nmpropd 20205 . . . . . . 7  |-  ( ( W  e. NrmRing  /\  S  e.  (SubRing `  W )
)  ->  ( norm `  W )  =  (
norm `  A )
)
4443adantr 465 . . . . . 6  |-  ( ( ( W  e. NrmRing  /\  S  e.  (SubRing `  W )
)  /\  ( x  e.  ( Base `  (Scalar `  A ) )  /\  y  e.  ( Base `  A ) ) )  ->  ( norm `  W
)  =  ( norm `  A ) )
455, 8sravsca 17282 . . . . . . 7  |-  ( ( W  e. NrmRing  /\  S  e.  (SubRing `  W )
)  ->  ( .r `  W )  =  ( .s `  A ) )
4645proplem3 14648 . . . . . 6  |-  ( ( ( W  e. NrmRing  /\  S  e.  (SubRing `  W )
)  /\  ( x  e.  ( Base `  (Scalar `  A ) )  /\  y  e.  ( Base `  A ) ) )  ->  ( x ( .r `  W ) y )  =  ( x ( .s `  A ) y ) )
4744, 46fveq12d 5716 . . . . 5  |-  ( ( ( W  e. NrmRing  /\  S  e.  (SubRing `  W )
)  /\  ( x  e.  ( Base `  (Scalar `  A ) )  /\  y  e.  ( Base `  A ) ) )  ->  ( ( norm `  W ) `  (
x ( .r `  W ) y ) )  =  ( (
norm `  A ) `  ( x ( .s
`  A ) y ) ) )
4842, 47eqtr3d 2477 . . . 4  |-  ( ( ( W  e. NrmRing  /\  S  e.  (SubRing `  W )
)  /\  ( x  e.  ( Base `  (Scalar `  A ) )  /\  y  e.  ( Base `  A ) ) )  ->  ( ( (
norm `  W ) `  x )  x.  (
( norm `  W ) `  y ) )  =  ( ( norm `  A
) `  ( x
( .s `  A
) y ) ) )
49 subrgsubg 16890 . . . . . . . 8  |-  ( S  e.  (SubRing `  W
)  ->  S  e.  (SubGrp `  W ) )
5049ad2antlr 726 . . . . . . 7  |-  ( ( ( W  e. NrmRing  /\  S  e.  (SubRing `  W )
)  /\  ( x  e.  ( Base `  (Scalar `  A ) )  /\  y  e.  ( Base `  A ) ) )  ->  S  e.  (SubGrp `  W ) )
51 eqid 2443 . . . . . . . 8  |-  ( norm `  ( Ws  S ) )  =  ( norm `  ( Ws  S ) )
5220, 24, 51subgnm2 20239 . . . . . . 7  |-  ( ( S  e.  (SubGrp `  W )  /\  x  e.  S )  ->  (
( norm `  ( Ws  S
) ) `  x
)  =  ( (
norm `  W ) `  x ) )
5350, 35, 52syl2anc 661 . . . . . 6  |-  ( ( ( W  e. NrmRing  /\  S  e.  (SubRing `  W )
)  /\  ( x  e.  ( Base `  (Scalar `  A ) )  /\  y  e.  ( Base `  A ) ) )  ->  ( ( norm `  ( Ws  S ) ) `  x )  =  ( ( norm `  W
) `  x )
)
5419adantr 465 . . . . . . . 8  |-  ( ( ( W  e. NrmRing  /\  S  e.  (SubRing `  W )
)  /\  ( x  e.  ( Base `  (Scalar `  A ) )  /\  y  e.  ( Base `  A ) ) )  ->  ( Ws  S )  =  (Scalar `  A
) )
5554fveq2d 5714 . . . . . . 7  |-  ( ( ( W  e. NrmRing  /\  S  e.  (SubRing `  W )
)  /\  ( x  e.  ( Base `  (Scalar `  A ) )  /\  y  e.  ( Base `  A ) ) )  ->  ( norm `  ( Ws  S ) )  =  ( norm `  (Scalar `  A ) ) )
5655fveq1d 5712 . . . . . 6  |-  ( ( ( W  e. NrmRing  /\  S  e.  (SubRing `  W )
)  /\  ( x  e.  ( Base `  (Scalar `  A ) )  /\  y  e.  ( Base `  A ) ) )  ->  ( ( norm `  ( Ws  S ) ) `  x )  =  ( ( norm `  (Scalar `  A ) ) `  x ) )
5753, 56eqtr3d 2477 . . . . 5  |-  ( ( ( W  e. NrmRing  /\  S  e.  (SubRing `  W )
)  /\  ( x  e.  ( Base `  (Scalar `  A ) )  /\  y  e.  ( Base `  A ) ) )  ->  ( ( norm `  W ) `  x
)  =  ( (
norm `  (Scalar `  A
) ) `  x
) )
5844fveq1d 5712 . . . . 5  |-  ( ( ( W  e. NrmRing  /\  S  e.  (SubRing `  W )
)  /\  ( x  e.  ( Base `  (Scalar `  A ) )  /\  y  e.  ( Base `  A ) ) )  ->  ( ( norm `  W ) `  y
)  =  ( (
norm `  A ) `  y ) )
5957, 58oveq12d 6128 . . . 4  |-  ( ( ( W  e. NrmRing  /\  S  e.  (SubRing `  W )
)  /\  ( x  e.  ( Base `  (Scalar `  A ) )  /\  y  e.  ( Base `  A ) ) )  ->  ( ( (
norm `  W ) `  x )  x.  (
( norm `  W ) `  y ) )  =  ( ( ( norm `  (Scalar `  A )
) `  x )  x.  ( ( norm `  A
) `  y )
) )
6048, 59eqtr3d 2477 . . 3  |-  ( ( ( W  e. NrmRing  /\  S  e.  (SubRing `  W )
)  /\  ( x  e.  ( Base `  (Scalar `  A ) )  /\  y  e.  ( Base `  A ) ) )  ->  ( ( norm `  A ) `  (
x ( .s `  A ) y ) )  =  ( ( ( norm `  (Scalar `  A ) ) `  x )  x.  (
( norm `  A ) `  y ) ) )
6160ralrimivva 2827 . 2  |-  ( ( W  e. NrmRing  /\  S  e.  (SubRing `  W )
)  ->  A. x  e.  ( Base `  (Scalar `  A ) ) A. y  e.  ( Base `  A ) ( (
norm `  A ) `  ( x ( .s
`  A ) y ) )  =  ( ( ( norm `  (Scalar `  A ) ) `  x )  x.  (
( norm `  A ) `  y ) ) )
62 eqid 2443 . . 3  |-  ( Base `  A )  =  (
Base `  A )
63 eqid 2443 . . 3  |-  ( norm `  A )  =  (
norm `  A )
64 eqid 2443 . . 3  |-  ( .s
`  A )  =  ( .s `  A
)
65 eqid 2443 . . 3  |-  (Scalar `  A )  =  (Scalar `  A )
66 eqid 2443 . . 3  |-  ( Base `  (Scalar `  A )
)  =  ( Base `  (Scalar `  A )
)
67 eqid 2443 . . 3  |-  ( norm `  (Scalar `  A )
)  =  ( norm `  (Scalar `  A )
)
6862, 63, 64, 65, 66, 67isnlm 20275 . 2  |-  ( A  e. NrmMod 
<->  ( ( A  e. NrmGrp  /\  A  e.  LMod  /\  (Scalar `  A )  e. NrmRing )  /\  A. x  e.  ( Base `  (Scalar `  A ) ) A. y  e.  ( Base `  A ) ( (
norm `  A ) `  ( x ( .s
`  A ) y ) )  =  ( ( ( norm `  (Scalar `  A ) ) `  x )  x.  (
( norm `  A ) `  y ) ) ) )
6923, 61, 68sylanbrc 664 1  |-  ( ( W  e. NrmRing  /\  S  e.  (SubRing `  W )
)  ->  A  e. NrmMod )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756   A.wral 2734    C_ wss 3347    X. cxp 4857   ` cfv 5437  (class class class)co 6110    x. cmul 9306   Basecbs 14193   ↾s cress 14194   +g cplusg 14257   .rcmulr 14258  Scalarcsca 14260   .scvsca 14261   distcds 14266  SubGrpcsubg 15694  SubRingcsubrg 16880  AbsValcabv 16920   LModclmod 16967  subringAlg csra 17268   normcnm 20188  NrmGrpcngp 20189  NrmRingcnrg 20191  NrmModcnlm 20192
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 4422  ax-sep 4432  ax-nul 4440  ax-pow 4489  ax-pr 4550  ax-un 6391  ax-cnex 9357  ax-resscn 9358  ax-1cn 9359  ax-icn 9360  ax-addcl 9361  ax-addrcl 9362  ax-mulcl 9363  ax-mulrcl 9364  ax-mulcom 9365  ax-addass 9366  ax-mulass 9367  ax-distr 9368  ax-i2m1 9369  ax-1ne0 9370  ax-1rid 9371  ax-rnegex 9372  ax-rrecex 9373  ax-cnre 9374  ax-pre-lttri 9375  ax-pre-lttrn 9376  ax-pre-ltadd 9377  ax-pre-mulgt0 9378  ax-pre-sup 9379
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 2577  df-ne 2622  df-nel 2623  df-ral 2739  df-rex 2740  df-reu 2741  df-rmo 2742  df-rab 2743  df-v 2993  df-sbc 3206  df-csb 3308  df-dif 3350  df-un 3352  df-in 3354  df-ss 3361  df-pss 3363  df-nul 3657  df-if 3811  df-pw 3881  df-sn 3897  df-pr 3899  df-tp 3901  df-op 3903  df-uni 4111  df-iun 4192  df-br 4312  df-opab 4370  df-mpt 4371  df-tr 4405  df-eprel 4651  df-id 4655  df-po 4660  df-so 4661  df-fr 4698  df-we 4700  df-ord 4741  df-on 4742  df-lim 4743  df-suc 4744  df-xp 4865  df-rel 4866  df-cnv 4867  df-co 4868  df-dm 4869  df-rn 4870  df-res 4871  df-ima 4872  df-iota 5400  df-fun 5439  df-fn 5440  df-f 5441  df-f1 5442  df-fo 5443  df-f1o 5444  df-fv 5445  df-riota 6071  df-ov 6113  df-oprab 6114  df-mpt2 6115  df-om 6496  df-1st 6596  df-2nd 6597  df-recs 6851  df-rdg 6885  df-er 7120  df-map 7235  df-en 7330  df-dom 7331  df-sdom 7332  df-sup 7710  df-pnf 9439  df-mnf 9440  df-xr 9441  df-ltxr 9442  df-le 9443  df-sub 9616  df-neg 9617  df-div 10013  df-nn 10342  df-2 10399  df-3 10400  df-4 10401  df-5 10402  df-6 10403  df-7 10404  df-8 10405  df-9 10406  df-10 10407  df-n0 10599  df-z 10666  df-dec 10775  df-uz 10881  df-q 10973  df-rp 11011  df-xneg 11108  df-xadd 11109  df-xmul 11110  df-ico 11325  df-ndx 14196  df-slot 14197  df-base 14198  df-sets 14199  df-ress 14200  df-plusg 14270  df-mulr 14271  df-sca 14273  df-vsca 14274  df-ip 14275  df-tset 14276  df-ds 14279  df-rest 14380  df-topn 14381  df-0g 14399  df-topgen 14401  df-mnd 15434  df-grp 15564  df-minusg 15565  df-sbg 15566  df-subg 15697  df-mgp 16611  df-ur 16623  df-rng 16666  df-subrg 16882  df-abv 16921  df-lmod 16969  df-sra 17272  df-psmet 17828  df-xmet 17829  df-met 17830  df-bl 17831  df-mopn 17832  df-top 18522  df-bases 18524  df-topon 18525  df-topsp 18526  df-xms 19914  df-ms 19915  df-nm 20194  df-ngp 20195  df-nrg 20197  df-nlm 20198
This theorem is referenced by:  rlmnlm  20288  srabn  20891
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