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Theorem rezh 26538
Description: The  ZZ-module of  RR is a normed module. (Contributed by Thierry Arnoux, 14-Feb-2018.)
Assertion
Ref Expression
rezh  |-  ( ZMod
` RRfld )  e. NrmMod

Proof of Theorem rezh
Dummy variables  x  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cnnrg 20485 . . . . . 6  |-fld  e. NrmRing
2 resubdrg 18156 . . . . . . 7  |-  ( RR  e.  (SubRing ` fld )  /\ RRfld  e.  DivRing )
32simpli 458 . . . . . 6  |-  RR  e.  (SubRing ` fld )
4 df-refld 18153 . . . . . . 7  |- RRfld  =  (flds  RR )
54subrgnrg 20379 . . . . . 6  |-  ( (fld  e. NrmRing  /\  RR  e.  (SubRing ` fld ) )  -> RRfld  e. NrmRing )
61, 3, 5mp2an 672 . . . . 5  |- RRfld  e. NrmRing
7 eqid 2451 . . . . . 6  |-  ( ZMod
` RRfld )  =  ( ZMod ` RRfld )
87zhmnrg 26534 . . . . 5  |-  (RRfld  e. NrmRing  -> 
( ZMod ` RRfld )  e. NrmRing )
9 nrgngp 20368 . . . . 5  |-  ( ( ZMod ` RRfld )  e. NrmRing  ->  ( ZMod ` RRfld )  e. NrmGrp )
106, 8, 9mp2b 10 . . . 4  |-  ( ZMod
` RRfld )  e. NrmGrp
11 nrgrng 20369 . . . . . 6  |-  (RRfld  e. NrmRing  -> RRfld  e. 
Ring )
12 rngabl 16789 . . . . . 6  |-  (RRfld  e.  Ring 
-> RRfld  e.  Abel )
136, 11, 12mp2b 10 . . . . 5  |- RRfld  e.  Abel
147zlmlmod 18072 . . . . 5  |-  (RRfld  e.  Abel  <->  ( ZMod ` RRfld )  e.  LMod )
1513, 14mpbi 208 . . . 4  |-  ( ZMod
` RRfld )  e.  LMod
16 zringnrg 20490 . . . 4  |-ring  e. NrmRing
1710, 15, 163pm3.2i 1166 . . 3  |-  ( ( ZMod ` RRfld )  e. NrmGrp  /\  ( ZMod ` RRfld )  e.  LMod  /\ring  e. NrmRing )
18 zsscn 10758 . . . . . . . 8  |-  ZZ  C_  CC
19 simpl 457 . . . . . . . 8  |-  ( ( z  e.  ZZ  /\  x  e.  RR )  ->  z  e.  ZZ )
2018, 19sseldi 3455 . . . . . . 7  |-  ( ( z  e.  ZZ  /\  x  e.  RR )  ->  z  e.  CC )
21 ax-resscn 9443 . . . . . . . 8  |-  RR  C_  CC
22 simpr 461 . . . . . . . 8  |-  ( ( z  e.  ZZ  /\  x  e.  RR )  ->  x  e.  RR )
2321, 22sseldi 3455 . . . . . . 7  |-  ( ( z  e.  ZZ  /\  x  e.  RR )  ->  x  e.  CC )
24 absmul 12894 . . . . . . 7  |-  ( ( z  e.  CC  /\  x  e.  CC )  ->  ( abs `  (
z  x.  x ) )  =  ( ( abs `  z )  x.  ( abs `  x
) ) )
2520, 23, 24syl2anc 661 . . . . . 6  |-  ( ( z  e.  ZZ  /\  x  e.  RR )  ->  ( abs `  (
z  x.  x ) )  =  ( ( abs `  z )  x.  ( abs `  x
) ) )
26 subrgsubg 16986 . . . . . . . . . . 11  |-  ( RR  e.  (SubRing ` fld )  ->  RR  e.  (SubGrp ` fld ) )
273, 26ax-mp 5 . . . . . . . . . 10  |-  RR  e.  (SubGrp ` fld )
28 eqid 2451 . . . . . . . . . . 11  |-  (.g ` fld )  =  (.g ` fld )
29 eqid 2451 . . . . . . . . . . . . 13  |-  (.g ` RRfld )  =  (.g ` RRfld )
307, 29zlmvsca 18071 . . . . . . . . . . . 12  |-  (.g ` RRfld )  =  ( .s `  ( ZMod ` RRfld ) )
3130eqcomi 2464 . . . . . . . . . . 11  |-  ( .s
`  ( ZMod ` RRfld ) )  =  (.g ` RRfld
)
3228, 4, 31subgmulg 15806 . . . . . . . . . 10  |-  ( ( RR  e.  (SubGrp ` fld )  /\  z  e.  ZZ  /\  x  e.  RR )  ->  ( z (.g ` fld ) x )  =  ( z ( .s `  ( ZMod ` RRfld ) )
x ) )
3327, 32mp3an1 1302 . . . . . . . . 9  |-  ( ( z  e.  ZZ  /\  x  e.  RR )  ->  ( z (.g ` fld ) x )  =  ( z ( .s
`  ( ZMod ` RRfld ) ) x ) )
34 cnfldmulg 17966 . . . . . . . . . 10  |-  ( ( z  e.  ZZ  /\  x  e.  CC )  ->  ( z (.g ` fld ) x )  =  ( z  x.  x
) )
3523, 34syldan 470 . . . . . . . . 9  |-  ( ( z  e.  ZZ  /\  x  e.  RR )  ->  ( z (.g ` fld ) x )  =  ( z  x.  x
) )
3633, 35eqtr3d 2494 . . . . . . . 8  |-  ( ( z  e.  ZZ  /\  x  e.  RR )  ->  ( z ( .s
`  ( ZMod ` RRfld ) ) x )  =  ( z  x.  x ) )
3736fveq2d 5796 . . . . . . 7  |-  ( ( z  e.  ZZ  /\  x  e.  RR )  ->  ( ( abs  |`  RR ) `
 ( z ( .s `  ( ZMod
` RRfld ) ) x ) )  =  ( ( abs  |`  RR ) `
 ( z  x.  x ) ) )
38 zre 10754 . . . . . . . 8  |-  ( z  e.  ZZ  ->  z  e.  RR )
39 remulcl 9471 . . . . . . . . 9  |-  ( ( z  e.  RR  /\  x  e.  RR )  ->  ( z  x.  x
)  e.  RR )
40 fvres 5806 . . . . . . . . 9  |-  ( ( z  x.  x )  e.  RR  ->  (
( abs  |`  RR ) `
 ( z  x.  x ) )  =  ( abs `  (
z  x.  x ) ) )
4139, 40syl 16 . . . . . . . 8  |-  ( ( z  e.  RR  /\  x  e.  RR )  ->  ( ( abs  |`  RR ) `
 ( z  x.  x ) )  =  ( abs `  (
z  x.  x ) ) )
4238, 41sylan 471 . . . . . . 7  |-  ( ( z  e.  ZZ  /\  x  e.  RR )  ->  ( ( abs  |`  RR ) `
 ( z  x.  x ) )  =  ( abs `  (
z  x.  x ) ) )
4337, 42eqtrd 2492 . . . . . 6  |-  ( ( z  e.  ZZ  /\  x  e.  RR )  ->  ( ( abs  |`  RR ) `
 ( z ( .s `  ( ZMod
` RRfld ) ) x ) )  =  ( abs `  ( z  x.  x ) ) )
44 fvres 5806 . . . . . . 7  |-  ( z  e.  ZZ  ->  (
( abs  |`  ZZ ) `
 z )  =  ( abs `  z
) )
45 fvres 5806 . . . . . . 7  |-  ( x  e.  RR  ->  (
( abs  |`  RR ) `
 x )  =  ( abs `  x
) )
4644, 45oveqan12d 6212 . . . . . 6  |-  ( ( z  e.  ZZ  /\  x  e.  RR )  ->  ( ( ( abs  |`  ZZ ) `  z
)  x.  ( ( abs  |`  RR ) `  x ) )  =  ( ( abs `  z
)  x.  ( abs `  x ) ) )
4725, 43, 463eqtr4d 2502 . . . . 5  |-  ( ( z  e.  ZZ  /\  x  e.  RR )  ->  ( ( abs  |`  RR ) `
 ( z ( .s `  ( ZMod
` RRfld ) ) x ) )  =  ( ( ( abs  |`  ZZ ) `
 z )  x.  ( ( abs  |`  RR ) `
 x ) ) )
4847ralrimiva 2825 . . . 4  |-  ( z  e.  ZZ  ->  A. x  e.  RR  ( ( abs  |`  RR ) `  (
z ( .s `  ( ZMod ` RRfld ) )
x ) )  =  ( ( ( abs  |`  ZZ ) `  z
)  x.  ( ( abs  |`  RR ) `  x ) ) )
4948rgen 2892 . . 3  |-  A. z  e.  ZZ  A. x  e.  RR  ( ( abs  |`  RR ) `  (
z ( .s `  ( ZMod ` RRfld ) )
x ) )  =  ( ( ( abs  |`  ZZ ) `  z
)  x.  ( ( abs  |`  RR ) `  x ) )
5017, 49pm3.2i 455 . 2  |-  ( ( ( ZMod ` RRfld )  e. NrmGrp  /\  ( ZMod ` RRfld )  e. 
LMod  /\ring 
e. NrmRing )  /\  A. z  e.  ZZ  A. x  e.  RR  ( ( abs  |`  RR ) `  (
z ( .s `  ( ZMod ` RRfld ) )
x ) )  =  ( ( ( abs  |`  ZZ ) `  z
)  x.  ( ( abs  |`  RR ) `  x ) ) )
51 rebase 18154 . . . 4  |-  RR  =  ( Base ` RRfld )
527, 51zlmbas 18067 . . 3  |-  RR  =  ( Base `  ( ZMod ` RRfld
) )
53 recusp 21011 . . . . 5  |- RRfld  e. CUnifSp
5453elexi 3081 . . . 4  |- RRfld  e.  _V
55 cnrng 17956 . . . . . . 7  |-fld  e.  Ring
56 rngmnd 16769 . . . . . . 7  |-  (fld  e.  Ring  ->fld  e.  Mnd )
5755, 56ax-mp 5 . . . . . 6  |-fld  e.  Mnd
58 0re 9490 . . . . . 6  |-  0  e.  RR
59 cnfldbas 17940 . . . . . . 7  |-  CC  =  ( Base ` fld )
60 cnfld0 17958 . . . . . . 7  |-  0  =  ( 0g ` fld )
61 cnfldnm 20483 . . . . . . 7  |-  abs  =  ( norm ` fld )
624, 59, 60, 61ressnm 26250 . . . . . 6  |-  ( (fld  e. 
Mnd  /\  0  e.  RR  /\  RR  C_  CC )  ->  ( abs  |`  RR )  =  ( norm ` RRfld ) )
6357, 58, 21, 62mp3an 1315 . . . . 5  |-  ( abs  |`  RR )  =  (
norm ` RRfld )
647, 63zlmnm 26533 . . . 4  |-  (RRfld  e.  _V  ->  ( abs  |`  RR )  =  ( norm `  ( ZMod ` RRfld ) ) )
6554, 64ax-mp 5 . . 3  |-  ( abs  |`  RR )  =  (
norm `  ( ZMod ` RRfld
) )
66 eqid 2451 . . 3  |-  ( .s
`  ( ZMod ` RRfld ) )  =  ( .s `  ( ZMod
` RRfld ) )
677zlmsca 18070 . . . 4  |-  (RRfld  e.  _V  ->ring  =  (Scalar `  ( ZMod ` RRfld
) ) )
6854, 67ax-mp 5 . . 3  |-ring  =  (Scalar `  ( ZMod ` RRfld ) )
69 zringbas 18007 . . 3  |-  ZZ  =  ( Base ` ring )
70 zringnm 26526 . . . 4  |-  ( norm ` ring )  =  ( abs  |`  ZZ )
7170eqcomi 2464 . . 3  |-  ( abs  |`  ZZ )  =  (
norm ` ring )
7252, 65, 66, 68, 69, 71isnlm 20381 . 2  |-  ( ( ZMod ` RRfld )  e. NrmMod  <->  ( (
( ZMod ` RRfld )  e. NrmGrp  /\  ( ZMod ` RRfld )  e. 
LMod  /\ring 
e. NrmRing )  /\  A. z  e.  ZZ  A. x  e.  RR  ( ( abs  |`  RR ) `  (
z ( .s `  ( ZMod ` RRfld ) )
x ) )  =  ( ( ( abs  |`  ZZ ) `  z
)  x.  ( ( abs  |`  RR ) `  x ) ) ) )
7350, 72mpbir 209 1  |-  ( ZMod
` RRfld )  e. NrmMod
Colors of variables: wff setvar class
Syntax hints:    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758   A.wral 2795   _Vcvv 3071    C_ wss 3429    |` cres 4943   ` cfv 5519  (class class class)co 6193   CCcc 9384   RRcr 9385   0cc0 9386    x. cmul 9391   ZZcz 10750   abscabs 12834  Scalarcsca 14352   .scvsca 14353   Mndcmnd 15520  .gcmg 15525  SubGrpcsubg 15786   Abelcabel 16391   Ringcrg 16760   DivRingcdr 16947  SubRingcsubrg 16976   LModclmod 17063  ℂfldccnfld 17936  ℤringzring 18001   ZModczlm 18050  RRfldcrefld 18152  CUnifSpccusp 19997   normcnm 20294  NrmGrpcngp 20295  NrmRingcnrg 20297  NrmModcnlm 20298
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4504  ax-sep 4514  ax-nul 4522  ax-pow 4571  ax-pr 4632  ax-un 6475  ax-inf2 7951  ax-cnex 9442  ax-resscn 9443  ax-1cn 9444  ax-icn 9445  ax-addcl 9446  ax-addrcl 9447  ax-mulcl 9448  ax-mulrcl 9449  ax-mulcom 9450  ax-addass 9451  ax-mulass 9452  ax-distr 9453  ax-i2m1 9454  ax-1ne0 9455  ax-1rid 9456  ax-rnegex 9457  ax-rrecex 9458  ax-cnre 9459  ax-pre-lttri 9460  ax-pre-lttrn 9461  ax-pre-ltadd 9462  ax-pre-mulgt0 9463  ax-pre-sup 9464  ax-addf 9465  ax-mulf 9466
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rmo 2803  df-rab 2804  df-v 3073  df-sbc 3288  df-csb 3390  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-pss 3445  df-nul 3739  df-if 3893  df-pw 3963  df-sn 3979  df-pr 3981  df-tp 3983  df-op 3985  df-uni 4193  df-int 4230  df-iun 4274  df-iin 4275  df-br 4394  df-opab 4452  df-mpt 4453  df-tr 4487  df-eprel 4733  df-id 4737  df-po 4742  df-so 4743  df-fr 4780  df-se 4781  df-we 4782  df-ord 4823  df-on 4824  df-lim 4825  df-suc 4826  df-xp 4947  df-rel 4948  df-cnv 4949  df-co 4950  df-dm 4951  df-rn 4952  df-res 4953  df-ima 4954  df-iota 5482  df-fun 5521  df-fn 5522  df-f 5523  df-f1 5524  df-fo 5525  df-f1o 5526  df-fv 5527  df-isom 5528  df-riota 6154  df-ov 6196  df-oprab 6197  df-mpt2 6198  df-of 6423  df-om 6580  df-1st 6680  df-2nd 6681  df-supp 6794  df-tpos 6848  df-recs 6935  df-rdg 6969  df-1o 7023  df-2o 7024  df-oadd 7027  df-er 7204  df-map 7319  df-ixp 7367  df-en 7414  df-dom 7415  df-sdom 7416  df-fin 7417  df-fsupp 7725  df-fi 7765  df-sup 7795  df-oi 7828  df-card 8213  df-cda 8441  df-pnf 9524  df-mnf 9525  df-xr 9526  df-ltxr 9527  df-le 9528  df-sub 9701  df-neg 9702  df-div 10098  df-nn 10427  df-2 10484  df-3 10485  df-4 10486  df-5 10487  df-6 10488  df-7 10489  df-8 10490  df-9 10491  df-10 10492  df-n0 10684  df-z 10751  df-dec 10860  df-uz 10966  df-q 11058  df-rp 11096  df-xneg 11193  df-xadd 11194  df-xmul 11195  df-ioo 11408  df-ico 11410  df-icc 11411  df-fz 11548  df-fzo 11659  df-seq 11917  df-exp 11976  df-hash 12214  df-cj 12699  df-re 12700  df-im 12701  df-sqr 12835  df-abs 12836  df-struct 14287  df-ndx 14288  df-slot 14289  df-base 14290  df-sets 14291  df-ress 14292  df-plusg 14362  df-mulr 14363  df-starv 14364  df-sca 14365  df-vsca 14366  df-ip 14367  df-tset 14368  df-ple 14369  df-ds 14371  df-unif 14372  df-hom 14373  df-cco 14374  df-rest 14472  df-topn 14473  df-0g 14491  df-gsum 14492  df-topgen 14493  df-pt 14494  df-prds 14497  df-xrs 14551  df-qtop 14556  df-imas 14557  df-xps 14559  df-mre 14635  df-mrc 14636  df-acs 14638  df-mnd 15526  df-submnd 15576  df-grp 15656  df-minusg 15657  df-sbg 15658  df-mulg 15659  df-subg 15789  df-cntz 15946  df-cmn 16392  df-abl 16393  df-mgp 16706  df-ur 16718  df-rng 16762  df-cring 16763  df-oppr 16830  df-dvdsr 16848  df-unit 16849  df-invr 16879  df-dvr 16890  df-drng 16949  df-subrg 16978  df-abv 17017  df-lmod 17065  df-psmet 17927  df-xmet 17928  df-met 17929  df-bl 17930  df-mopn 17931  df-fbas 17932  df-fg 17933  df-metu 17935  df-cnfld 17937  df-zring 18002  df-zlm 18054  df-refld 18153  df-top 18628  df-bases 18630  df-topon 18631  df-topsp 18632  df-cld 18748  df-ntr 18749  df-cls 18750  df-nei 18827  df-cn 18956  df-cnp 18957  df-haus 19044  df-cmp 19115  df-tx 19260  df-hmeo 19453  df-fil 19544  df-flim 19637  df-fcls 19639  df-ust 19900  df-utop 19931  df-uss 19956  df-usp 19957  df-cfilu 19987  df-cusp 19998  df-xms 20020  df-ms 20021  df-tms 20022  df-nm 20300  df-ngp 20301  df-nrg 20303  df-nlm 20304  df-cncf 20579  df-cfil 20891  df-cmet 20893  df-cms 20971
This theorem is referenced by:  rerrext  26576
  Copyright terms: Public domain W3C validator