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Theorem zhmnrg 28636
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 2420 . . . . . . . 8  |-  ( Base `  G )  =  (
Base `  G )
21a1i 11 . . . . . . 7  |-  ( G  e. NrmRing  ->  ( Base `  G
)  =  ( Base `  G ) )
3 zlmlem2.1 . . . . . . . . 9  |-  W  =  ( ZMod `  G
)
43, 1zlmbas 19013 . . . . . . . 8  |-  ( Base `  G )  =  (
Base `  W )
54a1i 11 . . . . . . 7  |-  ( G  e. NrmRing  ->  ( Base `  G
)  =  ( Base `  W ) )
6 eqid 2420 . . . . . . . . . 10  |-  ( +g  `  G )  =  ( +g  `  G )
73, 6zlmplusg 19014 . . . . . . . . 9  |-  ( +g  `  G )  =  ( +g  `  W )
87a1i 11 . . . . . . . 8  |-  ( G  e. NrmRing  ->  ( +g  `  G
)  =  ( +g  `  W ) )
98oveqdr 6320 . . . . . . 7  |-  ( ( G  e. NrmRing  /\  (
x  e.  ( Base `  G )  /\  y  e.  ( Base `  G
) ) )  -> 
( x ( +g  `  G ) y )  =  ( x ( +g  `  W ) y ) )
102, 5, 9grppropd 16628 . . . . . 6  |-  ( G  e. NrmRing  ->  ( G  e. 
Grp 
<->  W  e.  Grp )
)
11 eqid 2420 . . . . . . . . 9  |-  ( dist `  G )  =  (
dist `  G )
123, 11zlmds 28633 . . . . . . . 8  |-  ( G  e. NrmRing  ->  ( dist `  G
)  =  ( dist `  W ) )
1312reseq1d 5115 . . . . . . 7  |-  ( G  e. NrmRing  ->  ( ( dist `  G )  |`  (
( Base `  G )  X.  ( Base `  G
) ) )  =  ( ( dist `  W
)  |`  ( ( Base `  G )  X.  ( Base `  G ) ) ) )
14 eqid 2420 . . . . . . . . 9  |-  (TopSet `  G )  =  (TopSet `  G )
153, 14zlmtset 28634 . . . . . . . 8  |-  ( G  e. NrmRing  ->  (TopSet `  G )  =  (TopSet `  W )
)
165, 15topnpropd 15287 . . . . . . 7  |-  ( G  e. NrmRing  ->  ( TopOpen `  G
)  =  ( TopOpen `  W ) )
172, 5, 13, 16mspropd 21413 . . . . . 6  |-  ( G  e. NrmRing  ->  ( G  e. 
MetSp 
<->  W  e.  MetSp ) )
18 eqid 2420 . . . . . . . . 9  |-  ( norm `  G )  =  (
norm `  G )
193, 18zlmnm 28635 . . . . . . . 8  |-  ( G  e. NrmRing  ->  ( norm `  G
)  =  ( norm `  W ) )
205, 8grpsubpropd 16700 . . . . . . . 8  |-  ( G  e. NrmRing  ->  ( -g `  G
)  =  ( -g `  W ) )
2119, 20coeq12d 5010 . . . . . . 7  |-  ( G  e. NrmRing  ->  ( ( norm `  G )  o.  ( -g `  G ) )  =  ( ( norm `  W )  o.  ( -g `  W ) ) )
2221, 12sseq12d 3490 . . . . . 6  |-  ( G  e. NrmRing  ->  ( ( (
norm `  G )  o.  ( -g `  G
) )  C_  ( dist `  G )  <->  ( ( norm `  W )  o.  ( -g `  W
) )  C_  ( dist `  W ) ) )
2310, 17, 223anbi123d 1335 . . . . 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 2420 . . . . . 6  |-  ( -g `  G )  =  (
-g `  G )
2518, 24, 11isngp 21534 . . . . 5  |-  ( G  e. NrmGrp 
<->  ( G  e.  Grp  /\  G  e.  MetSp  /\  (
( norm `  G )  o.  ( -g `  G
) )  C_  ( dist `  G ) ) )
26 eqid 2420 . . . . . 6  |-  ( norm `  W )  =  (
norm `  W )
27 eqid 2420 . . . . . 6  |-  ( -g `  W )  =  (
-g `  W )
28 eqid 2420 . . . . . 6  |-  ( dist `  W )  =  (
dist `  W )
2926, 27, 28isngp 21534 . . . . 5  |-  ( W  e. NrmGrp 
<->  ( W  e.  Grp  /\  W  e.  MetSp  /\  (
( norm `  W )  o.  ( -g `  W
) )  C_  ( dist `  W ) ) )
3023, 25, 293bitr4g 291 . . . 4  |-  ( G  e. NrmRing  ->  ( G  e. NrmGrp  <->  W  e. NrmGrp ) )
31 eqid 2420 . . . . . . . 8  |-  ( .r
`  G )  =  ( .r `  G
)
323, 31zlmmulr 19015 . . . . . . 7  |-  ( .r
`  G )  =  ( .r `  W
)
3332a1i 11 . . . . . 6  |-  ( G  e. NrmRing  ->  ( .r `  G )  =  ( .r `  W ) )
345, 8, 33abvpropd2 28277 . . . . 5  |-  ( G  e. NrmRing  ->  (AbsVal `  G )  =  (AbsVal `  W )
)
3519, 34eleq12d 2502 . . . 4  |-  ( G  e. NrmRing  ->  ( ( norm `  G )  e.  (AbsVal `  G )  <->  ( norm `  W )  e.  (AbsVal `  W ) ) )
3630, 35anbi12d 715 . . 3  |-  ( G  e. NrmRing  ->  ( ( G  e. NrmGrp  /\  ( norm `  G
)  e.  (AbsVal `  G ) )  <->  ( W  e. NrmGrp  /\  ( norm `  W
)  e.  (AbsVal `  W ) ) ) )
37 eqid 2420 . . . 4  |-  (AbsVal `  G )  =  (AbsVal `  G )
3818, 37isnrg 21587 . . 3  |-  ( G  e. NrmRing 
<->  ( G  e. NrmGrp  /\  ( norm `  G )  e.  (AbsVal `  G )
) )
39 eqid 2420 . . . 4  |-  (AbsVal `  W )  =  (AbsVal `  W )
4026, 39isnrg 21587 . . 3  |-  ( W  e. NrmRing 
<->  ( W  e. NrmGrp  /\  ( norm `  W )  e.  (AbsVal `  W )
) )
4136, 38, 403bitr4g 291 . 2  |-  ( G  e. NrmRing  ->  ( G  e. NrmRing  <->  W  e. NrmRing ) )
4241ibi 244 1  |-  ( G  e. NrmRing  ->  W  e. NrmRing )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1867    C_ wss 3433    X. cxp 4843    o. ccom 4849   ` cfv 5592   Basecbs 15073   +g cplusg 15142   .rcmulr 15143  TopSetcts 15148   distcds 15151   Grpcgrp 16613   -gcsg 16615  AbsValcabv 17972   ZModczlm 18996   MetSpcmt 21257   normcnm 21515  NrmGrpcngp 21516  NrmRingcnrg 21518
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-8 1869  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-rep 4529  ax-sep 4539  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6588  ax-cnex 9584  ax-resscn 9585  ax-1cn 9586  ax-icn 9587  ax-addcl 9588  ax-addrcl 9589  ax-mulcl 9590  ax-mulrcl 9591  ax-mulcom 9592  ax-addass 9593  ax-mulass 9594  ax-distr 9595  ax-i2m1 9596  ax-1ne0 9597  ax-1rid 9598  ax-rnegex 9599  ax-rrecex 9600  ax-cnre 9601  ax-pre-lttri 9602  ax-pre-lttrn 9603  ax-pre-ltadd 9604  ax-pre-mulgt0 9605
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-nel 2619  df-ral 2778  df-rex 2779  df-reu 2780  df-rab 2782  df-v 3080  df-sbc 3297  df-csb 3393  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-pss 3449  df-nul 3759  df-if 3907  df-pw 3978  df-sn 3994  df-pr 3996  df-tp 3998  df-op 4000  df-uni 4214  df-iun 4295  df-br 4418  df-opab 4476  df-mpt 4477  df-tr 4512  df-eprel 4756  df-id 4760  df-po 4766  df-so 4767  df-fr 4804  df-we 4806  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-pred 5390  df-ord 5436  df-on 5437  df-lim 5438  df-suc 5439  df-iota 5556  df-fun 5594  df-fn 5595  df-f 5596  df-f1 5597  df-fo 5598  df-f1o 5599  df-fv 5600  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6698  df-1st 6798  df-2nd 6799  df-wrecs 7027  df-recs 7089  df-rdg 7127  df-er 7362  df-map 7473  df-en 7569  df-dom 7570  df-sdom 7571  df-pnf 9666  df-mnf 9667  df-xr 9668  df-ltxr 9669  df-le 9670  df-sub 9851  df-neg 9852  df-nn 10599  df-2 10657  df-3 10658  df-4 10659  df-5 10660  df-6 10661  df-7 10662  df-8 10663  df-9 10664  df-10 10665  df-n0 10859  df-z 10927  df-dec 11041  df-ndx 15076  df-slot 15077  df-base 15078  df-sets 15079  df-plusg 15155  df-mulr 15156  df-sca 15158  df-vsca 15159  df-tset 15161  df-ds 15164  df-rest 15273  df-topn 15274  df-0g 15292  df-mgm 16432  df-sgrp 16471  df-mnd 16481  df-grp 16617  df-minusg 16618  df-sbg 16619  df-mgp 17652  df-ring 17710  df-abv 17973  df-zlm 19000  df-top 19845  df-topon 19847  df-topsp 19848  df-xms 21259  df-ms 21260  df-nm 21521  df-ngp 21522  df-nrg 21524
This theorem is referenced by:  cnzh  28639  rezh  28640  qqhnm  28659
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