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Theorem sranlm 18195
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 18173 . . . . 5  |-  ( W  e. NrmRing  ->  W  e. NrmGrp )
21adantr 451 . . . 4  |-  ( ( W  e. NrmRing  /\  S  e.  (SubRing `  W )
)  ->  W  e. NrmGrp )
3 eqidd 2284 . . . . 5  |-  ( ( W  e. NrmRing  /\  S  e.  (SubRing `  W )
)  ->  ( Base `  W )  =  (
Base `  W )
)
4 sranlm.a . . . . . . 7  |-  A  =  ( ( subringAlg  `  W ) `
 S )
54a1i 10 . . . . . 6  |-  ( ( W  e. NrmRing  /\  S  e.  (SubRing `  W )
)  ->  A  =  ( ( subringAlg  `  W ) `
 S ) )
6 eqid 2283 . . . . . . . 8  |-  ( Base `  W )  =  (
Base `  W )
76subrgss 15546 . . . . . . 7  |-  ( S  e.  (SubRing `  W
)  ->  S  C_  ( Base `  W ) )
87adantl 452 . . . . . 6  |-  ( ( W  e. NrmRing  /\  S  e.  (SubRing `  W )
)  ->  S  C_  ( Base `  W ) )
95, 8srabase 15931 . . . . 5  |-  ( ( W  e. NrmRing  /\  S  e.  (SubRing `  W )
)  ->  ( Base `  W )  =  (
Base `  A )
)
105, 8sraaddg 15932 . . . . . 6  |-  ( ( W  e. NrmRing  /\  S  e.  (SubRing `  W )
)  ->  ( +g  `  W )  =  ( +g  `  A ) )
1110proplem3 13593 . . . . 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 15938 . . . . . 6  |-  ( ( W  e. NrmRing  /\  S  e.  (SubRing `  W )
)  ->  ( dist `  W )  =  (
dist `  A )
)
1312reseq1d 4954 . . . . 5  |-  ( ( W  e. NrmRing  /\  S  e.  (SubRing `  W )
)  ->  ( ( dist `  W )  |`  ( ( Base `  W
)  X.  ( Base `  W ) ) )  =  ( ( dist `  A )  |`  (
( Base `  W )  X.  ( Base `  W
) ) ) )
145, 8sratopn 15937 . . . . 5  |-  ( ( W  e. NrmRing  /\  S  e.  (SubRing `  W )
)  ->  ( TopOpen `  W )  =  (
TopOpen `  A ) )
153, 9, 11, 13, 14ngppropd 18153 . . . 4  |-  ( ( W  e. NrmRing  /\  S  e.  (SubRing `  W )
)  ->  ( W  e. NrmGrp  <-> 
A  e. NrmGrp ) )
162, 15mpbid 201 . . 3  |-  ( ( W  e. NrmRing  /\  S  e.  (SubRing `  W )
)  ->  A  e. NrmGrp )
174sralmod 15939 . . . 4  |-  ( S  e.  (SubRing `  W
)  ->  A  e.  LMod )
1817adantl 452 . . 3  |-  ( ( W  e. NrmRing  /\  S  e.  (SubRing `  W )
)  ->  A  e.  LMod )
195, 8srasca 15934 . . . 4  |-  ( ( W  e. NrmRing  /\  S  e.  (SubRing `  W )
)  ->  ( Ws  S
)  =  (Scalar `  A ) )
20 eqid 2283 . . . . 5  |-  ( Ws  S )  =  ( Ws  S )
2120subrgnrg 18184 . . . 4  |-  ( ( W  e. NrmRing  /\  S  e.  (SubRing `  W )
)  ->  ( Ws  S
)  e. NrmRing )
2219, 21eqeltrrd 2358 . . 3  |-  ( ( W  e. NrmRing  /\  S  e.  (SubRing `  W )
)  ->  (Scalar `  A
)  e. NrmRing )
2316, 18, 223jca 1132 . 2  |-  ( ( W  e. NrmRing  /\  S  e.  (SubRing `  W )
)  ->  ( A  e. NrmGrp  /\  A  e.  LMod  /\  (Scalar `  A )  e. NrmRing ) )
24 eqid 2283 . . . . . . . 8  |-  ( norm `  W )  =  (
norm `  W )
25 eqid 2283 . . . . . . . 8  |-  (AbsVal `  W )  =  (AbsVal `  W )
2624, 25nrgabv 18172 . . . . . . 7  |-  ( W  e. NrmRing  ->  ( norm `  W
)  e.  (AbsVal `  W ) )
2726ad2antrr 706 . . . . . 6  |-  ( ( ( W  e. NrmRing  /\  S  e.  (SubRing `  W )
)  /\  ( x  e.  ( Base `  (Scalar `  A ) )  /\  y  e.  ( Base `  A ) ) )  ->  ( norm `  W
)  e.  (AbsVal `  W ) )
288adantr 451 . . . . . . 7  |-  ( ( ( W  e. NrmRing  /\  S  e.  (SubRing `  W )
)  /\  ( x  e.  ( Base `  (Scalar `  A ) )  /\  y  e.  ( Base `  A ) ) )  ->  S  C_  ( Base `  W ) )
29 simprl 732 . . . . . . . 8  |-  ( ( ( W  e. NrmRing  /\  S  e.  (SubRing `  W )
)  /\  ( x  e.  ( Base `  (Scalar `  A ) )  /\  y  e.  ( Base `  A ) ) )  ->  x  e.  (
Base `  (Scalar `  A
) ) )
3020subrgbas 15554 . . . . . . . . . . 11  |-  ( S  e.  (SubRing `  W
)  ->  S  =  ( Base `  ( Ws  S
) ) )
3130adantl 452 . . . . . . . . . 10  |-  ( ( W  e. NrmRing  /\  S  e.  (SubRing `  W )
)  ->  S  =  ( Base `  ( Ws  S
) ) )
3219fveq2d 5529 . . . . . . . . . 10  |-  ( ( W  e. NrmRing  /\  S  e.  (SubRing `  W )
)  ->  ( Base `  ( Ws  S ) )  =  ( Base `  (Scalar `  A ) ) )
3331, 32eqtrd 2315 . . . . . . . . 9  |-  ( ( W  e. NrmRing  /\  S  e.  (SubRing `  W )
)  ->  S  =  ( Base `  (Scalar `  A
) ) )
3433adantr 451 . . . . . . . 8  |-  ( ( ( W  e. NrmRing  /\  S  e.  (SubRing `  W )
)  /\  ( x  e.  ( Base `  (Scalar `  A ) )  /\  y  e.  ( Base `  A ) ) )  ->  S  =  (
Base `  (Scalar `  A
) ) )
3529, 34eleqtrrd 2360 . . . . . . 7  |-  ( ( ( W  e. NrmRing  /\  S  e.  (SubRing `  W )
)  /\  ( x  e.  ( Base `  (Scalar `  A ) )  /\  y  e.  ( Base `  A ) ) )  ->  x  e.  S
)
3628, 35sseldd 3181 . . . . . 6  |-  ( ( ( W  e. NrmRing  /\  S  e.  (SubRing `  W )
)  /\  ( x  e.  ( Base `  (Scalar `  A ) )  /\  y  e.  ( Base `  A ) ) )  ->  x  e.  (
Base `  W )
)
37 simprr 733 . . . . . . 7  |-  ( ( ( W  e. NrmRing  /\  S  e.  (SubRing `  W )
)  /\  ( x  e.  ( Base `  (Scalar `  A ) )  /\  y  e.  ( Base `  A ) ) )  ->  y  e.  (
Base `  A )
)
389adantr 451 . . . . . . 7  |-  ( ( ( W  e. NrmRing  /\  S  e.  (SubRing `  W )
)  /\  ( x  e.  ( Base `  (Scalar `  A ) )  /\  y  e.  ( Base `  A ) ) )  ->  ( Base `  W
)  =  ( Base `  A ) )
3937, 38eleqtrrd 2360 . . . . . 6  |-  ( ( ( W  e. NrmRing  /\  S  e.  (SubRing `  W )
)  /\  ( x  e.  ( Base `  (Scalar `  A ) )  /\  y  e.  ( Base `  A ) ) )  ->  y  e.  (
Base `  W )
)
40 eqid 2283 . . . . . . 7  |-  ( .r
`  W )  =  ( .r `  W
)
4125, 6, 40abvmul 15594 . . . . . 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 1182 . . . . 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 18116 . . . . . . 7  |-  ( ( W  e. NrmRing  /\  S  e.  (SubRing `  W )
)  ->  ( norm `  W )  =  (
norm `  A )
)
4443adantr 451 . . . . . 6  |-  ( ( ( W  e. NrmRing  /\  S  e.  (SubRing `  W )
)  /\  ( x  e.  ( Base `  (Scalar `  A ) )  /\  y  e.  ( Base `  A ) ) )  ->  ( norm `  W
)  =  ( norm `  A ) )
455, 8sravsca 15935 . . . . . . 7  |-  ( ( W  e. NrmRing  /\  S  e.  (SubRing `  W )
)  ->  ( .r `  W )  =  ( .s `  A ) )
4645proplem3 13593 . . . . . 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 5531 . . . . 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 2317 . . . 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 15551 . . . . . . . 8  |-  ( S  e.  (SubRing `  W
)  ->  S  e.  (SubGrp `  W ) )
5049ad2antlr 707 . . . . . . 7  |-  ( ( ( W  e. NrmRing  /\  S  e.  (SubRing `  W )
)  /\  ( x  e.  ( Base `  (Scalar `  A ) )  /\  y  e.  ( Base `  A ) ) )  ->  S  e.  (SubGrp `  W ) )
51 eqid 2283 . . . . . . . 8  |-  ( norm `  ( Ws  S ) )  =  ( norm `  ( Ws  S ) )
5220, 24, 51subgnm2 18150 . . . . . . 7  |-  ( ( S  e.  (SubGrp `  W )  /\  x  e.  S )  ->  (
( norm `  ( Ws  S
) ) `  x
)  =  ( (
norm `  W ) `  x ) )
5350, 35, 52syl2anc 642 . . . . . 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 451 . . . . . . . 8  |-  ( ( ( W  e. NrmRing  /\  S  e.  (SubRing `  W )
)  /\  ( x  e.  ( Base `  (Scalar `  A ) )  /\  y  e.  ( Base `  A ) ) )  ->  ( Ws  S )  =  (Scalar `  A
) )
5554fveq2d 5529 . . . . . . 7  |-  ( ( ( W  e. NrmRing  /\  S  e.  (SubRing `  W )
)  /\  ( x  e.  ( Base `  (Scalar `  A ) )  /\  y  e.  ( Base `  A ) ) )  ->  ( norm `  ( Ws  S ) )  =  ( norm `  (Scalar `  A ) ) )
5655fveq1d 5527 . . . . . 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 2317 . . . . 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 5527 . . . . 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 5876 . . . 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 2317 . . 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 2635 . 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 2283 . . 3  |-  ( Base `  A )  =  (
Base `  A )
63 eqid 2283 . . 3  |-  ( norm `  A )  =  (
norm `  A )
64 eqid 2283 . . 3  |-  ( .s
`  A )  =  ( .s `  A
)
65 eqid 2283 . . 3  |-  (Scalar `  A )  =  (Scalar `  A )
66 eqid 2283 . . 3  |-  ( Base `  (Scalar `  A )
)  =  ( Base `  (Scalar `  A )
)
67 eqid 2283 . . 3  |-  ( norm `  (Scalar `  A )
)  =  ( norm `  (Scalar `  A )
)
6862, 63, 64, 65, 66, 67isnlm 18186 . 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 645 1  |-  ( ( W  e. NrmRing  /\  S  e.  (SubRing `  W )
)  ->  A  e. NrmMod )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   A.wral 2543    C_ wss 3152    X. cxp 4687   ` cfv 5255  (class class class)co 5858    x. cmul 8742   Basecbs 13148   ↾s cress 13149   +g cplusg 13208   .rcmulr 13209  Scalarcsca 13211   .scvsca 13212   distcds 13217  SubGrpcsubg 14615  SubRingcsubrg 15541  AbsValcabv 15581   LModclmod 15627   subringAlg csra 15921   normcnm 18099  NrmGrpcngp 18100  NrmRingcnrg 18102  NrmModcnlm 18103
This theorem is referenced by:  rlmnlm  18199  srabn  18777
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-map 6774  df-en 6864  df-dom 6865  df-sdom 6866  df-sup 7194  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-7 9809  df-8 9810  df-9 9811  df-10 9812  df-n0 9966  df-z 10025  df-dec 10125  df-uz 10231  df-q 10317  df-rp 10355  df-xneg 10452  df-xadd 10453  df-xmul 10454  df-ico 10662  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-mulr 13222  df-sca 13224  df-vsca 13225  df-tset 13227  df-ds 13230  df-rest 13327  df-topn 13328  df-topgen 13344  df-0g 13404  df-mnd 14367  df-grp 14489  df-minusg 14490  df-sbg 14491  df-subg 14618  df-mgp 15326  df-rng 15340  df-ur 15342  df-subrg 15543  df-abv 15582  df-lmod 15629  df-sra 15925  df-xmet 16373  df-met 16374  df-bl 16375  df-mopn 16376  df-top 16636  df-bases 16638  df-topon 16639  df-topsp 16640  df-xms 17885  df-ms 17886  df-nm 18105  df-ngp 18106  df-nrg 18108  df-nlm 18109
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