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Theorem zhmnrg 27699
Description: The  ZZ-module built from a normed ring is also a normed ring. (Contributed by Thierry Arnoux, 8-Nov-2017.)
Hypothesis
Ref Expression
zlmlem2.1  |-  W  =  ( ZMod `  G
)
Assertion
Ref Expression
zhmnrg  |-  ( G  e. NrmRing  ->  W  e. NrmRing )

Proof of Theorem zhmnrg
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2467 . . . . . . . 8  |-  ( Base `  G )  =  (
Base `  G )
21a1i 11 . . . . . . 7  |-  ( G  e. NrmRing  ->  ( Base `  G
)  =  ( Base `  G ) )
3 zlmlem2.1 . . . . . . . . 9  |-  W  =  ( ZMod `  G
)
43, 1zlmbas 18362 . . . . . . . 8  |-  ( Base `  G )  =  (
Base `  W )
54a1i 11 . . . . . . 7  |-  ( G  e. NrmRing  ->  ( Base `  G
)  =  ( Base `  W ) )
6 eqid 2467 . . . . . . . . . 10  |-  ( +g  `  G )  =  ( +g  `  G )
73, 6zlmplusg 18363 . . . . . . . . 9  |-  ( +g  `  G )  =  ( +g  `  W )
87a1i 11 . . . . . . . 8  |-  ( G  e. NrmRing  ->  ( +g  `  G
)  =  ( +g  `  W ) )
98proplem3 14949 . . . . . . 7  |-  ( ( G  e. NrmRing  /\  (
x  e.  ( Base `  G )  /\  y  e.  ( Base `  G
) ) )  -> 
( x ( +g  `  G ) y )  =  ( x ( +g  `  W ) y ) )
102, 5, 9grppropd 15882 . . . . . 6  |-  ( G  e. NrmRing  ->  ( G  e. 
Grp 
<->  W  e.  Grp )
)
11 eqid 2467 . . . . . . . . 9  |-  ( dist `  G )  =  (
dist `  G )
123, 11zlmds 27696 . . . . . . . 8  |-  ( G  e. NrmRing  ->  ( dist `  G
)  =  ( dist `  W ) )
1312reseq1d 5272 . . . . . . 7  |-  ( G  e. NrmRing  ->  ( ( dist `  G )  |`  (
( Base `  G )  X.  ( Base `  G
) ) )  =  ( ( dist `  W
)  |`  ( ( Base `  G )  X.  ( Base `  G ) ) ) )
14 eqid 2467 . . . . . . . . 9  |-  (TopSet `  G )  =  (TopSet `  G )
153, 14zlmtset 27697 . . . . . . . 8  |-  ( G  e. NrmRing  ->  (TopSet `  G )  =  (TopSet `  W )
)
165, 15topnpropd 14695 . . . . . . 7  |-  ( G  e. NrmRing  ->  ( TopOpen `  G
)  =  ( TopOpen `  W ) )
172, 5, 13, 16mspropd 20804 . . . . . 6  |-  ( G  e. NrmRing  ->  ( G  e. 
MetSp 
<->  W  e.  MetSp ) )
18 eqid 2467 . . . . . . . . 9  |-  ( norm `  G )  =  (
norm `  G )
193, 18zlmnm 27698 . . . . . . . 8  |-  ( G  e. NrmRing  ->  ( norm `  G
)  =  ( norm `  W ) )
205, 8grpsubpropd 15954 . . . . . . . 8  |-  ( G  e. NrmRing  ->  ( -g `  G
)  =  ( -g `  W ) )
2119, 20coeq12d 5167 . . . . . . 7  |-  ( G  e. NrmRing  ->  ( ( norm `  G )  o.  ( -g `  G ) )  =  ( ( norm `  W )  o.  ( -g `  W ) ) )
2221, 12sseq12d 3533 . . . . . 6  |-  ( G  e. NrmRing  ->  ( ( (
norm `  G )  o.  ( -g `  G
) )  C_  ( dist `  G )  <->  ( ( norm `  W )  o.  ( -g `  W
) )  C_  ( dist `  W ) ) )
2310, 17, 223anbi123d 1299 . . . . 5  |-  ( G  e. NrmRing  ->  ( ( G  e.  Grp  /\  G  e.  MetSp  /\  ( ( norm `  G )  o.  ( -g `  G
) )  C_  ( dist `  G ) )  <-> 
( W  e.  Grp  /\  W  e.  MetSp  /\  (
( norm `  W )  o.  ( -g `  W
) )  C_  ( dist `  W ) ) ) )
24 eqid 2467 . . . . . 6  |-  ( -g `  G )  =  (
-g `  G )
2518, 24, 11isngp 20943 . . . . 5  |-  ( G  e. NrmGrp 
<->  ( G  e.  Grp  /\  G  e.  MetSp  /\  (
( norm `  G )  o.  ( -g `  G
) )  C_  ( dist `  G ) ) )
26 eqid 2467 . . . . . 6  |-  ( norm `  W )  =  (
norm `  W )
27 eqid 2467 . . . . . 6  |-  ( -g `  W )  =  (
-g `  W )
28 eqid 2467 . . . . . 6  |-  ( dist `  W )  =  (
dist `  W )
2926, 27, 28isngp 20943 . . . . 5  |-  ( W  e. NrmGrp 
<->  ( W  e.  Grp  /\  W  e.  MetSp  /\  (
( norm `  W )  o.  ( -g `  W
) )  C_  ( dist `  W ) ) )
3023, 25, 293bitr4g 288 . . . 4  |-  ( G  e. NrmRing  ->  ( G  e. NrmGrp  <->  W  e. NrmGrp ) )
31 eqid 2467 . . . . . . . 8  |-  ( .r
`  G )  =  ( .r `  G
)
323, 31zlmmulr 18364 . . . . . . 7  |-  ( .r
`  G )  =  ( .r `  W
)
3332a1i 11 . . . . . 6  |-  ( G  e. NrmRing  ->  ( .r `  G )  =  ( .r `  W ) )
345, 8, 33abvpropd2 27399 . . . . 5  |-  ( G  e. NrmRing  ->  (AbsVal `  G )  =  (AbsVal `  W )
)
3519, 34eleq12d 2549 . . . 4  |-  ( G  e. NrmRing  ->  ( ( norm `  G )  e.  (AbsVal `  G )  <->  ( norm `  W )  e.  (AbsVal `  W ) ) )
3630, 35anbi12d 710 . . 3  |-  ( G  e. NrmRing  ->  ( ( G  e. NrmGrp  /\  ( norm `  G
)  e.  (AbsVal `  G ) )  <->  ( W  e. NrmGrp  /\  ( norm `  W
)  e.  (AbsVal `  W ) ) ) )
37 eqid 2467 . . . 4  |-  (AbsVal `  G )  =  (AbsVal `  G )
3818, 37isnrg 20996 . . 3  |-  ( G  e. NrmRing 
<->  ( G  e. NrmGrp  /\  ( norm `  G )  e.  (AbsVal `  G )
) )
39 eqid 2467 . . . 4  |-  (AbsVal `  W )  =  (AbsVal `  W )
4026, 39isnrg 20996 . . 3  |-  ( W  e. NrmRing 
<->  ( W  e. NrmGrp  /\  ( norm `  W )  e.  (AbsVal `  W )
) )
4136, 38, 403bitr4g 288 . 2  |-  ( G  e. NrmRing  ->  ( G  e. NrmRing  <->  W  e. NrmRing ) )
4241ibi 241 1  |-  ( G  e. NrmRing  ->  W  e. NrmRing )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767    C_ wss 3476    X. cxp 4997    o. ccom 5003   ` cfv 5588   Basecbs 14493   +g cplusg 14558   .rcmulr 14559  TopSetcts 14564   distcds 14567   Grpcgrp 15730   -gcsg 15733  AbsValcabv 17277   ZModczlm 18345   MetSpcmt 20648   normcnm 20924  NrmGrpcngp 20925  NrmRingcnrg 20927
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6577  ax-cnex 9549  ax-resscn 9550  ax-1cn 9551  ax-icn 9552  ax-addcl 9553  ax-addrcl 9554  ax-mulcl 9555  ax-mulrcl 9556  ax-mulcom 9557  ax-addass 9558  ax-mulass 9559  ax-distr 9560  ax-i2m1 9561  ax-1ne0 9562  ax-1rid 9563  ax-rnegex 9564  ax-rrecex 9565  ax-cnre 9566  ax-pre-lttri 9567  ax-pre-lttrn 9568  ax-pre-ltadd 9569  ax-pre-mulgt0 9570
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-riota 6246  df-ov 6288  df-oprab 6289  df-mpt2 6290  df-om 6686  df-1st 6785  df-2nd 6786  df-recs 7043  df-rdg 7077  df-er 7312  df-map 7423  df-en 7518  df-dom 7519  df-sdom 7520  df-pnf 9631  df-mnf 9632  df-xr 9633  df-ltxr 9634  df-le 9635  df-sub 9808  df-neg 9809  df-nn 10538  df-2 10595  df-3 10596  df-4 10597  df-5 10598  df-6 10599  df-7 10600  df-8 10601  df-9 10602  df-10 10603  df-n0 10797  df-z 10866  df-dec 10978  df-ndx 14496  df-slot 14497  df-base 14498  df-sets 14499  df-plusg 14571  df-mulr 14572  df-sca 14574  df-vsca 14575  df-tset 14577  df-ds 14580  df-rest 14681  df-topn 14682  df-0g 14700  df-mnd 15735  df-grp 15871  df-minusg 15872  df-sbg 15873  df-mgp 16956  df-rng 17014  df-abv 17278  df-zlm 18349  df-top 19206  df-topon 19209  df-topsp 19210  df-xms 20650  df-ms 20651  df-nm 20930  df-ngp 20931  df-nrg 20933
This theorem is referenced by:  cnzh  27702  rezh  27703  qqhnm  27722
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