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Theorem sranlm 18211
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 18189 . . . . 5  |-  ( W  e. NrmRing  ->  W  e. NrmGrp )
21adantr 451 . . . 4  |-  ( ( W  e. NrmRing  /\  S  e.  (SubRing `  W )
)  ->  W  e. NrmGrp )
3 eqidd 2297 . . . . 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 2296 . . . . . . . 8  |-  ( Base `  W )  =  (
Base `  W )
76subrgss 15562 . . . . . . 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 15947 . . . . 5  |-  ( ( W  e. NrmRing  /\  S  e.  (SubRing `  W )
)  ->  ( Base `  W )  =  (
Base `  A )
)
105, 8sraaddg 15948 . . . . . 6  |-  ( ( W  e. NrmRing  /\  S  e.  (SubRing `  W )
)  ->  ( +g  `  W )  =  ( +g  `  A ) )
1110proplem3 13609 . . . . 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 15954 . . . . . 6  |-  ( ( W  e. NrmRing  /\  S  e.  (SubRing `  W )
)  ->  ( dist `  W )  =  (
dist `  A )
)
1312reseq1d 4970 . . . . 5  |-  ( ( W  e. NrmRing  /\  S  e.  (SubRing `  W )
)  ->  ( ( dist `  W )  |`  ( ( Base `  W
)  X.  ( Base `  W ) ) )  =  ( ( dist `  A )  |`  (
( Base `  W )  X.  ( Base `  W
) ) ) )
145, 8sratopn 15953 . . . . 5  |-  ( ( W  e. NrmRing  /\  S  e.  (SubRing `  W )
)  ->  ( TopOpen `  W )  =  (
TopOpen `  A ) )
153, 9, 11, 13, 14ngppropd 18169 . . . 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 15955 . . . 4  |-  ( S  e.  (SubRing `  W
)  ->  A  e.  LMod )
1817adantl 452 . . 3  |-  ( ( W  e. NrmRing  /\  S  e.  (SubRing `  W )
)  ->  A  e.  LMod )
195, 8srasca 15950 . . . 4  |-  ( ( W  e. NrmRing  /\  S  e.  (SubRing `  W )
)  ->  ( Ws  S
)  =  (Scalar `  A ) )
20 eqid 2296 . . . . 5  |-  ( Ws  S )  =  ( Ws  S )
2120subrgnrg 18200 . . . 4  |-  ( ( W  e. NrmRing  /\  S  e.  (SubRing `  W )
)  ->  ( Ws  S
)  e. NrmRing )
2219, 21eqeltrrd 2371 . . 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 2296 . . . . . . . 8  |-  ( norm `  W )  =  (
norm `  W )
25 eqid 2296 . . . . . . . 8  |-  (AbsVal `  W )  =  (AbsVal `  W )
2624, 25nrgabv 18188 . . . . . . 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 15570 . . . . . . . . . . 11  |-  ( S  e.  (SubRing `  W
)  ->  S  =  ( Base `  ( Ws  S
) ) )
3130adantl 452 . . . . . . . . . 10  |-  ( ( W  e. NrmRing  /\  S  e.  (SubRing `  W )
)  ->  S  =  ( Base `  ( Ws  S
) ) )
3219fveq2d 5545 . . . . . . . . . 10  |-  ( ( W  e. NrmRing  /\  S  e.  (SubRing `  W )
)  ->  ( Base `  ( Ws  S ) )  =  ( Base `  (Scalar `  A ) ) )
3331, 32eqtrd 2328 . . . . . . . . 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 2373 . . . . . . 7  |-  ( ( ( W  e. NrmRing  /\  S  e.  (SubRing `  W )
)  /\  ( x  e.  ( Base `  (Scalar `  A ) )  /\  y  e.  ( Base `  A ) ) )  ->  x  e.  S
)
3628, 35sseldd 3194 . . . . . 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 2373 . . . . . 6  |-  ( ( ( W  e. NrmRing  /\  S  e.  (SubRing `  W )
)  /\  ( x  e.  ( Base `  (Scalar `  A ) )  /\  y  e.  ( Base `  A ) ) )  ->  y  e.  (
Base `  W )
)
40 eqid 2296 . . . . . . 7  |-  ( .r
`  W )  =  ( .r `  W
)
4125, 6, 40abvmul 15610 . . . . . 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 18132 . . . . . . 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 15951 . . . . . . 7  |-  ( ( W  e. NrmRing  /\  S  e.  (SubRing `  W )
)  ->  ( .r `  W )  =  ( .s `  A ) )
4645proplem3 13609 . . . . . 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 5547 . . . . 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 2330 . . . 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 15567 . . . . . . . 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 2296 . . . . . . . 8  |-  ( norm `  ( Ws  S ) )  =  ( norm `  ( Ws  S ) )
5220, 24, 51subgnm2 18166 . . . . . . 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 5545 . . . . . . 7  |-  ( ( ( W  e. NrmRing  /\  S  e.  (SubRing `  W )
)  /\  ( x  e.  ( Base `  (Scalar `  A ) )  /\  y  e.  ( Base `  A ) ) )  ->  ( norm `  ( Ws  S ) )  =  ( norm `  (Scalar `  A ) ) )
5655fveq1d 5543 . . . . . 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 2330 . . . . 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 5543 . . . . 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 5892 . . . 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 2330 . . 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 2648 . 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 2296 . . 3  |-  ( Base `  A )  =  (
Base `  A )
63 eqid 2296 . . 3  |-  ( norm `  A )  =  (
norm `  A )
64 eqid 2296 . . 3  |-  ( .s
`  A )  =  ( .s `  A
)
65 eqid 2296 . . 3  |-  (Scalar `  A )  =  (Scalar `  A )
66 eqid 2296 . . 3  |-  ( Base `  (Scalar `  A )
)  =  ( Base `  (Scalar `  A )
)
67 eqid 2296 . . 3  |-  ( norm `  (Scalar `  A )
)  =  ( norm `  (Scalar `  A )
)
6862, 63, 64, 65, 66, 67isnlm 18202 . 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 1632    e. wcel 1696   A.wral 2556    C_ wss 3165    X. cxp 4703   ` cfv 5271  (class class class)co 5874    x. cmul 8758   Basecbs 13164   ↾s cress 13165   +g cplusg 13224   .rcmulr 13225  Scalarcsca 13227   .scvsca 13228   distcds 13233  SubGrpcsubg 14631  SubRingcsubrg 15557  AbsValcabv 15597   LModclmod 15643   subringAlg csra 15937   normcnm 18115  NrmGrpcngp 18116  NrmRingcnrg 18118  NrmModcnlm 18119
This theorem is referenced by:  rlmnlm  18215  srabn  18793
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-er 6676  df-map 6790  df-en 6880  df-dom 6881  df-sdom 6882  df-sup 7210  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-3 9821  df-4 9822  df-5 9823  df-6 9824  df-7 9825  df-8 9826  df-9 9827  df-10 9828  df-n0 9982  df-z 10041  df-dec 10141  df-uz 10247  df-q 10333  df-rp 10371  df-xneg 10468  df-xadd 10469  df-xmul 10470  df-ico 10678  df-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-ress 13171  df-plusg 13237  df-mulr 13238  df-sca 13240  df-vsca 13241  df-tset 13243  df-ds 13246  df-rest 13343  df-topn 13344  df-topgen 13360  df-0g 13420  df-mnd 14383  df-grp 14505  df-minusg 14506  df-sbg 14507  df-subg 14634  df-mgp 15342  df-rng 15356  df-ur 15358  df-subrg 15559  df-abv 15598  df-lmod 15645  df-sra 15941  df-xmet 16389  df-met 16390  df-bl 16391  df-mopn 16392  df-top 16652  df-bases 16654  df-topon 16655  df-topsp 16656  df-xms 17901  df-ms 17902  df-nm 18121  df-ngp 18122  df-nrg 18124  df-nlm 18125
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