MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  sranlm Structured version   Unicode version

Theorem sranlm 20921
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 20899 . . . . 5  |-  ( W  e. NrmRing  ->  W  e. NrmGrp )
21adantr 465 . . . 4  |-  ( ( W  e. NrmRing  /\  S  e.  (SubRing `  W )
)  ->  W  e. NrmGrp )
3 eqidd 2461 . . . . 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 2460 . . . . . . . 8  |-  ( Base `  W )  =  (
Base `  W )
76subrgss 17206 . . . . . . 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 17600 . . . . 5  |-  ( ( W  e. NrmRing  /\  S  e.  (SubRing `  W )
)  ->  ( Base `  W )  =  (
Base `  A )
)
105, 8sraaddg 17601 . . . . . 6  |-  ( ( W  e. NrmRing  /\  S  e.  (SubRing `  W )
)  ->  ( +g  `  W )  =  ( +g  `  A ) )
1110proplem3 14935 . . . . 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 17608 . . . . . 6  |-  ( ( W  e. NrmRing  /\  S  e.  (SubRing `  W )
)  ->  ( dist `  W )  =  (
dist `  A )
)
1312reseq1d 5263 . . . . 5  |-  ( ( W  e. NrmRing  /\  S  e.  (SubRing `  W )
)  ->  ( ( dist `  W )  |`  ( ( Base `  W
)  X.  ( Base `  W ) ) )  =  ( ( dist `  A )  |`  (
( Base `  W )  X.  ( Base `  W
) ) ) )
145, 8sratopn 17607 . . . . 5  |-  ( ( W  e. NrmRing  /\  S  e.  (SubRing `  W )
)  ->  ( TopOpen `  W )  =  (
TopOpen `  A ) )
153, 9, 11, 13, 14ngppropd 20879 . . . 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 17609 . . . 4  |-  ( S  e.  (SubRing `  W
)  ->  A  e.  LMod )
1817adantl 466 . . 3  |-  ( ( W  e. NrmRing  /\  S  e.  (SubRing `  W )
)  ->  A  e.  LMod )
195, 8srasca 17603 . . . 4  |-  ( ( W  e. NrmRing  /\  S  e.  (SubRing `  W )
)  ->  ( Ws  S
)  =  (Scalar `  A ) )
20 eqid 2460 . . . . 5  |-  ( Ws  S )  =  ( Ws  S )
2120subrgnrg 20910 . . . 4  |-  ( ( W  e. NrmRing  /\  S  e.  (SubRing `  W )
)  ->  ( Ws  S
)  e. NrmRing )
2219, 21eqeltrrd 2549 . . 3  |-  ( ( W  e. NrmRing  /\  S  e.  (SubRing `  W )
)  ->  (Scalar `  A
)  e. NrmRing )
2316, 18, 223jca 1171 . 2  |-  ( ( W  e. NrmRing  /\  S  e.  (SubRing `  W )
)  ->  ( A  e. NrmGrp  /\  A  e.  LMod  /\  (Scalar `  A )  e. NrmRing ) )
24 eqid 2460 . . . . . . . 8  |-  ( norm `  W )  =  (
norm `  W )
25 eqid 2460 . . . . . . . 8  |-  (AbsVal `  W )  =  (AbsVal `  W )
2624, 25nrgabv 20898 . . . . . . 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 17214 . . . . . . . . . . 11  |-  ( S  e.  (SubRing `  W
)  ->  S  =  ( Base `  ( Ws  S
) ) )
3130adantl 466 . . . . . . . . . 10  |-  ( ( W  e. NrmRing  /\  S  e.  (SubRing `  W )
)  ->  S  =  ( Base `  ( Ws  S
) ) )
3219fveq2d 5861 . . . . . . . . . 10  |-  ( ( W  e. NrmRing  /\  S  e.  (SubRing `  W )
)  ->  ( Base `  ( Ws  S ) )  =  ( Base `  (Scalar `  A ) ) )
3331, 32eqtrd 2501 . . . . . . . . 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 2551 . . . . . . 7  |-  ( ( ( W  e. NrmRing  /\  S  e.  (SubRing `  W )
)  /\  ( x  e.  ( Base `  (Scalar `  A ) )  /\  y  e.  ( Base `  A ) ) )  ->  x  e.  S
)
3628, 35sseldd 3498 . . . . . 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 2551 . . . . . 6  |-  ( ( ( W  e. NrmRing  /\  S  e.  (SubRing `  W )
)  /\  ( x  e.  ( Base `  (Scalar `  A ) )  /\  y  e.  ( Base `  A ) ) )  ->  y  e.  (
Base `  W )
)
40 eqid 2460 . . . . . . 7  |-  ( .r
`  W )  =  ( .r `  W
)
4125, 6, 40abvmul 17254 . . . . . 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 1223 . . . . 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 20842 . . . . . . 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 17604 . . . . . . 7  |-  ( ( W  e. NrmRing  /\  S  e.  (SubRing `  W )
)  ->  ( .r `  W )  =  ( .s `  A ) )
4645proplem3 14935 . . . . . 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 5863 . . . . 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 2503 . . . 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 17211 . . . . . . . 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 2460 . . . . . . . 8  |-  ( norm `  ( Ws  S ) )  =  ( norm `  ( Ws  S ) )
5220, 24, 51subgnm2 20876 . . . . . . 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 5861 . . . . . . 7  |-  ( ( ( W  e. NrmRing  /\  S  e.  (SubRing `  W )
)  /\  ( x  e.  ( Base `  (Scalar `  A ) )  /\  y  e.  ( Base `  A ) ) )  ->  ( norm `  ( Ws  S ) )  =  ( norm `  (Scalar `  A ) ) )
5655fveq1d 5859 . . . . . 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 2503 . . . . 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 5859 . . . . 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 6293 . . . 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 2503 . . 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 2878 . 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 2460 . . 3  |-  ( Base `  A )  =  (
Base `  A )
63 eqid 2460 . . 3  |-  ( norm `  A )  =  (
norm `  A )
64 eqid 2460 . . 3  |-  ( .s
`  A )  =  ( .s `  A
)
65 eqid 2460 . . 3  |-  (Scalar `  A )  =  (Scalar `  A )
66 eqid 2460 . . 3  |-  ( Base `  (Scalar `  A )
)  =  ( Base `  (Scalar `  A )
)
67 eqid 2460 . . 3  |-  ( norm `  (Scalar `  A )
)  =  ( norm `  (Scalar `  A )
)
6862, 63, 64, 65, 66, 67isnlm 20912 . 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 968    = wceq 1374    e. wcel 1762   A.wral 2807    C_ wss 3469    X. cxp 4990   ` cfv 5579  (class class class)co 6275    x. cmul 9486   Basecbs 14479   ↾s cress 14480   +g cplusg 14544   .rcmulr 14545  Scalarcsca 14547   .scvsca 14548   distcds 14553  SubGrpcsubg 15983  SubRingcsubrg 17201  AbsValcabv 17241   LModclmod 17288  subringAlg csra 17590   normcnm 20825  NrmGrpcngp 20826  NrmRingcnrg 20828  NrmModcnlm 20829
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558  ax-pre-sup 9559
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-nel 2658  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-uni 4239  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-tr 4534  df-eprel 4784  df-id 4788  df-po 4793  df-so 4794  df-fr 4831  df-we 4833  df-ord 4874  df-on 4875  df-lim 4876  df-suc 4877  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-riota 6236  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-om 6672  df-1st 6774  df-2nd 6775  df-recs 7032  df-rdg 7066  df-er 7301  df-map 7412  df-en 7507  df-dom 7508  df-sdom 7509  df-sup 7890  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9796  df-neg 9797  df-div 10196  df-nn 10526  df-2 10583  df-3 10584  df-4 10585  df-5 10586  df-6 10587  df-7 10588  df-8 10589  df-9 10590  df-10 10591  df-n0 10785  df-z 10854  df-dec 10966  df-uz 11072  df-q 11172  df-rp 11210  df-xneg 11307  df-xadd 11308  df-xmul 11309  df-ico 11524  df-ndx 14482  df-slot 14483  df-base 14484  df-sets 14485  df-ress 14486  df-plusg 14557  df-mulr 14558  df-sca 14560  df-vsca 14561  df-ip 14562  df-tset 14563  df-ds 14566  df-rest 14667  df-topn 14668  df-0g 14686  df-topgen 14688  df-mnd 15721  df-grp 15851  df-minusg 15852  df-sbg 15853  df-subg 15986  df-mgp 16925  df-ur 16937  df-rng 16981  df-subrg 17203  df-abv 17242  df-lmod 17290  df-sra 17594  df-psmet 18175  df-xmet 18176  df-met 18177  df-bl 18178  df-mopn 18179  df-top 19159  df-bases 19161  df-topon 19162  df-topsp 19163  df-xms 20551  df-ms 20552  df-nm 20831  df-ngp 20832  df-nrg 20834  df-nlm 20835
This theorem is referenced by:  rlmnlm  20925  srabn  21528
  Copyright terms: Public domain W3C validator