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Theorem sranlm 18722
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 18700 . . . . 5  |-  ( W  e. NrmRing  ->  W  e. NrmGrp )
21adantr 453 . . . 4  |-  ( ( W  e. NrmRing  /\  S  e.  (SubRing `  W )
)  ->  W  e. NrmGrp )
3 eqidd 2439 . . . . 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 2438 . . . . . . . 8  |-  ( Base `  W )  =  (
Base `  W )
76subrgss 15871 . . . . . . 7  |-  ( S  e.  (SubRing `  W
)  ->  S  C_  ( Base `  W ) )
87adantl 454 . . . . . 6  |-  ( ( W  e. NrmRing  /\  S  e.  (SubRing `  W )
)  ->  S  C_  ( Base `  W ) )
95, 8srabase 16252 . . . . 5  |-  ( ( W  e. NrmRing  /\  S  e.  (SubRing `  W )
)  ->  ( Base `  W )  =  (
Base `  A )
)
105, 8sraaddg 16253 . . . . . 6  |-  ( ( W  e. NrmRing  /\  S  e.  (SubRing `  W )
)  ->  ( +g  `  W )  =  ( +g  `  A ) )
1110proplem3 13918 . . . . 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 16259 . . . . . 6  |-  ( ( W  e. NrmRing  /\  S  e.  (SubRing `  W )
)  ->  ( dist `  W )  =  (
dist `  A )
)
1312reseq1d 5147 . . . . 5  |-  ( ( W  e. NrmRing  /\  S  e.  (SubRing `  W )
)  ->  ( ( dist `  W )  |`  ( ( Base `  W
)  X.  ( Base `  W ) ) )  =  ( ( dist `  A )  |`  (
( Base `  W )  X.  ( Base `  W
) ) ) )
145, 8sratopn 16258 . . . . 5  |-  ( ( W  e. NrmRing  /\  S  e.  (SubRing `  W )
)  ->  ( TopOpen `  W )  =  (
TopOpen `  A ) )
153, 9, 11, 13, 14ngppropd 18680 . . . 4  |-  ( ( W  e. NrmRing  /\  S  e.  (SubRing `  W )
)  ->  ( W  e. NrmGrp  <-> 
A  e. NrmGrp ) )
162, 15mpbid 203 . . 3  |-  ( ( W  e. NrmRing  /\  S  e.  (SubRing `  W )
)  ->  A  e. NrmGrp )
174sralmod 16260 . . . 4  |-  ( S  e.  (SubRing `  W
)  ->  A  e.  LMod )
1817adantl 454 . . 3  |-  ( ( W  e. NrmRing  /\  S  e.  (SubRing `  W )
)  ->  A  e.  LMod )
195, 8srasca 16255 . . . 4  |-  ( ( W  e. NrmRing  /\  S  e.  (SubRing `  W )
)  ->  ( Ws  S
)  =  (Scalar `  A ) )
20 eqid 2438 . . . . 5  |-  ( Ws  S )  =  ( Ws  S )
2120subrgnrg 18711 . . . 4  |-  ( ( W  e. NrmRing  /\  S  e.  (SubRing `  W )
)  ->  ( Ws  S
)  e. NrmRing )
2219, 21eqeltrrd 2513 . . 3  |-  ( ( W  e. NrmRing  /\  S  e.  (SubRing `  W )
)  ->  (Scalar `  A
)  e. NrmRing )
2316, 18, 223jca 1135 . 2  |-  ( ( W  e. NrmRing  /\  S  e.  (SubRing `  W )
)  ->  ( A  e. NrmGrp  /\  A  e.  LMod  /\  (Scalar `  A )  e. NrmRing ) )
24 eqid 2438 . . . . . . . 8  |-  ( norm `  W )  =  (
norm `  W )
25 eqid 2438 . . . . . . . 8  |-  (AbsVal `  W )  =  (AbsVal `  W )
2624, 25nrgabv 18699 . . . . . . 7  |-  ( W  e. NrmRing  ->  ( norm `  W
)  e.  (AbsVal `  W ) )
2726ad2antrr 708 . . . . . 6  |-  ( ( ( W  e. NrmRing  /\  S  e.  (SubRing `  W )
)  /\  ( x  e.  ( Base `  (Scalar `  A ) )  /\  y  e.  ( Base `  A ) ) )  ->  ( norm `  W
)  e.  (AbsVal `  W ) )
288adantr 453 . . . . . . 7  |-  ( ( ( W  e. NrmRing  /\  S  e.  (SubRing `  W )
)  /\  ( x  e.  ( Base `  (Scalar `  A ) )  /\  y  e.  ( Base `  A ) ) )  ->  S  C_  ( Base `  W ) )
29 simprl 734 . . . . . . . 8  |-  ( ( ( W  e. NrmRing  /\  S  e.  (SubRing `  W )
)  /\  ( x  e.  ( Base `  (Scalar `  A ) )  /\  y  e.  ( Base `  A ) ) )  ->  x  e.  (
Base `  (Scalar `  A
) ) )
3020subrgbas 15879 . . . . . . . . . . 11  |-  ( S  e.  (SubRing `  W
)  ->  S  =  ( Base `  ( Ws  S
) ) )
3130adantl 454 . . . . . . . . . 10  |-  ( ( W  e. NrmRing  /\  S  e.  (SubRing `  W )
)  ->  S  =  ( Base `  ( Ws  S
) ) )
3219fveq2d 5734 . . . . . . . . . 10  |-  ( ( W  e. NrmRing  /\  S  e.  (SubRing `  W )
)  ->  ( Base `  ( Ws  S ) )  =  ( Base `  (Scalar `  A ) ) )
3331, 32eqtrd 2470 . . . . . . . . 9  |-  ( ( W  e. NrmRing  /\  S  e.  (SubRing `  W )
)  ->  S  =  ( Base `  (Scalar `  A
) ) )
3433adantr 453 . . . . . . . 8  |-  ( ( ( W  e. NrmRing  /\  S  e.  (SubRing `  W )
)  /\  ( x  e.  ( Base `  (Scalar `  A ) )  /\  y  e.  ( Base `  A ) ) )  ->  S  =  (
Base `  (Scalar `  A
) ) )
3529, 34eleqtrrd 2515 . . . . . . 7  |-  ( ( ( W  e. NrmRing  /\  S  e.  (SubRing `  W )
)  /\  ( x  e.  ( Base `  (Scalar `  A ) )  /\  y  e.  ( Base `  A ) ) )  ->  x  e.  S
)
3628, 35sseldd 3351 . . . . . 6  |-  ( ( ( W  e. NrmRing  /\  S  e.  (SubRing `  W )
)  /\  ( x  e.  ( Base `  (Scalar `  A ) )  /\  y  e.  ( Base `  A ) ) )  ->  x  e.  (
Base `  W )
)
37 simprr 735 . . . . . . 7  |-  ( ( ( W  e. NrmRing  /\  S  e.  (SubRing `  W )
)  /\  ( x  e.  ( Base `  (Scalar `  A ) )  /\  y  e.  ( Base `  A ) ) )  ->  y  e.  (
Base `  A )
)
389adantr 453 . . . . . . 7  |-  ( ( ( W  e. NrmRing  /\  S  e.  (SubRing `  W )
)  /\  ( x  e.  ( Base `  (Scalar `  A ) )  /\  y  e.  ( Base `  A ) ) )  ->  ( Base `  W
)  =  ( Base `  A ) )
3937, 38eleqtrrd 2515 . . . . . 6  |-  ( ( ( W  e. NrmRing  /\  S  e.  (SubRing `  W )
)  /\  ( x  e.  ( Base `  (Scalar `  A ) )  /\  y  e.  ( Base `  A ) ) )  ->  y  e.  (
Base `  W )
)
40 eqid 2438 . . . . . . 7  |-  ( .r
`  W )  =  ( .r `  W
)
4125, 6, 40abvmul 15919 . . . . . 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 1185 . . . . 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 18643 . . . . . . 7  |-  ( ( W  e. NrmRing  /\  S  e.  (SubRing `  W )
)  ->  ( norm `  W )  =  (
norm `  A )
)
4443adantr 453 . . . . . 6  |-  ( ( ( W  e. NrmRing  /\  S  e.  (SubRing `  W )
)  /\  ( x  e.  ( Base `  (Scalar `  A ) )  /\  y  e.  ( Base `  A ) ) )  ->  ( norm `  W
)  =  ( norm `  A ) )
455, 8sravsca 16256 . . . . . . 7  |-  ( ( W  e. NrmRing  /\  S  e.  (SubRing `  W )
)  ->  ( .r `  W )  =  ( .s `  A ) )
4645proplem3 13918 . . . . . 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 5736 . . . . 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 2472 . . . 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 15876 . . . . . . . 8  |-  ( S  e.  (SubRing `  W
)  ->  S  e.  (SubGrp `  W ) )
5049ad2antlr 709 . . . . . . 7  |-  ( ( ( W  e. NrmRing  /\  S  e.  (SubRing `  W )
)  /\  ( x  e.  ( Base `  (Scalar `  A ) )  /\  y  e.  ( Base `  A ) ) )  ->  S  e.  (SubGrp `  W ) )
51 eqid 2438 . . . . . . . 8  |-  ( norm `  ( Ws  S ) )  =  ( norm `  ( Ws  S ) )
5220, 24, 51subgnm2 18677 . . . . . . 7  |-  ( ( S  e.  (SubGrp `  W )  /\  x  e.  S )  ->  (
( norm `  ( Ws  S
) ) `  x
)  =  ( (
norm `  W ) `  x ) )
5350, 35, 52syl2anc 644 . . . . . 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 453 . . . . . . . 8  |-  ( ( ( W  e. NrmRing  /\  S  e.  (SubRing `  W )
)  /\  ( x  e.  ( Base `  (Scalar `  A ) )  /\  y  e.  ( Base `  A ) ) )  ->  ( Ws  S )  =  (Scalar `  A
) )
5554fveq2d 5734 . . . . . . 7  |-  ( ( ( W  e. NrmRing  /\  S  e.  (SubRing `  W )
)  /\  ( x  e.  ( Base `  (Scalar `  A ) )  /\  y  e.  ( Base `  A ) ) )  ->  ( norm `  ( Ws  S ) )  =  ( norm `  (Scalar `  A ) ) )
5655fveq1d 5732 . . . . . 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 2472 . . . . 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 5732 . . . . 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 6101 . . . 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 2472 . . 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 2800 . 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 2438 . . 3  |-  ( Base `  A )  =  (
Base `  A )
63 eqid 2438 . . 3  |-  ( norm `  A )  =  (
norm `  A )
64 eqid 2438 . . 3  |-  ( .s
`  A )  =  ( .s `  A
)
65 eqid 2438 . . 3  |-  (Scalar `  A )  =  (Scalar `  A )
66 eqid 2438 . . 3  |-  ( Base `  (Scalar `  A )
)  =  ( Base `  (Scalar `  A )
)
67 eqid 2438 . . 3  |-  ( norm `  (Scalar `  A )
)  =  ( norm `  (Scalar `  A )
)
6862, 63, 64, 65, 66, 67isnlm 18713 . 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 647 1  |-  ( ( W  e. NrmRing  /\  S  e.  (SubRing `  W )
)  ->  A  e. NrmMod )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726   A.wral 2707    C_ wss 3322    X. cxp 4878   ` cfv 5456  (class class class)co 6083    x. cmul 8997   Basecbs 13471   ↾s cress 13472   +g cplusg 13531   .rcmulr 13532  Scalarcsca 13534   .scvsca 13535   distcds 13540  SubGrpcsubg 14940  SubRingcsubrg 15866  AbsValcabv 15906   LModclmod 15952   subringAlg csra 16242   normcnm 18626  NrmGrpcngp 18627  NrmRingcnrg 18629  NrmModcnlm 18630
This theorem is referenced by:  rlmnlm  18726  srabn  19316
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703  ax-cnex 9048  ax-resscn 9049  ax-1cn 9050  ax-icn 9051  ax-addcl 9052  ax-addrcl 9053  ax-mulcl 9054  ax-mulrcl 9055  ax-mulcom 9056  ax-addass 9057  ax-mulass 9058  ax-distr 9059  ax-i2m1 9060  ax-1ne0 9061  ax-1rid 9062  ax-rnegex 9063  ax-rrecex 9064  ax-cnre 9065  ax-pre-lttri 9066  ax-pre-lttrn 9067  ax-pre-ltadd 9068  ax-pre-mulgt0 9069  ax-pre-sup 9070
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-tr 4305  df-eprel 4496  df-id 4500  df-po 4505  df-so 4506  df-fr 4543  df-we 4545  df-ord 4586  df-on 4587  df-lim 4588  df-suc 4589  df-om 4848  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-1st 6351  df-2nd 6352  df-riota 6551  df-recs 6635  df-rdg 6670  df-er 6907  df-map 7022  df-en 7112  df-dom 7113  df-sdom 7114  df-sup 7448  df-pnf 9124  df-mnf 9125  df-xr 9126  df-ltxr 9127  df-le 9128  df-sub 9295  df-neg 9296  df-div 9680  df-nn 10003  df-2 10060  df-3 10061  df-4 10062  df-5 10063  df-6 10064  df-7 10065  df-8 10066  df-9 10067  df-10 10068  df-n0 10224  df-z 10285  df-dec 10385  df-uz 10491  df-q 10577  df-rp 10615  df-xneg 10712  df-xadd 10713  df-xmul 10714  df-ico 10924  df-ndx 13474  df-slot 13475  df-base 13476  df-sets 13477  df-ress 13478  df-plusg 13544  df-mulr 13545  df-sca 13547  df-vsca 13548  df-tset 13550  df-ds 13553  df-rest 13652  df-topn 13653  df-topgen 13669  df-0g 13729  df-mnd 14692  df-grp 14814  df-minusg 14815  df-sbg 14816  df-subg 14943  df-mgp 15651  df-rng 15665  df-ur 15667  df-subrg 15868  df-abv 15907  df-lmod 15954  df-sra 16246  df-psmet 16696  df-xmet 16697  df-met 16698  df-bl 16699  df-mopn 16700  df-top 16965  df-bases 16967  df-topon 16968  df-topsp 16969  df-xms 18352  df-ms 18353  df-nm 18632  df-ngp 18633  df-nrg 18635  df-nlm 18636
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