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Theorem subrgugrp 15889
Description: The units of a subring form a subgroup of the unit group of the original ring. (Contributed by Mario Carneiro, 4-Dec-2014.)
Hypotheses
Ref Expression
subrgugrp.1  |-  S  =  ( Rs  A )
subrgugrp.2  |-  U  =  (Unit `  R )
subrgugrp.3  |-  V  =  (Unit `  S )
subrgugrp.4  |-  G  =  ( (mulGrp `  R
)s 
U )
Assertion
Ref Expression
subrgugrp  |-  ( A  e.  (SubRing `  R
)  ->  V  e.  (SubGrp `  G ) )

Proof of Theorem subrgugrp
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 subrgugrp.1 . . 3  |-  S  =  ( Rs  A )
2 subrgugrp.2 . . 3  |-  U  =  (Unit `  R )
3 subrgugrp.3 . . 3  |-  V  =  (Unit `  S )
41, 2, 3subrguss 15885 . 2  |-  ( A  e.  (SubRing `  R
)  ->  V  C_  U
)
51subrgrng 15873 . . 3  |-  ( A  e.  (SubRing `  R
)  ->  S  e.  Ring )
6 eqid 2438 . . . 4  |-  ( 1r
`  S )  =  ( 1r `  S
)
73, 61unit 15765 . . 3  |-  ( S  e.  Ring  ->  ( 1r
`  S )  e.  V )
8 ne0i 3636 . . 3  |-  ( ( 1r `  S )  e.  V  ->  V  =/=  (/) )
95, 7, 83syl 19 . 2  |-  ( A  e.  (SubRing `  R
)  ->  V  =/=  (/) )
10 eqid 2438 . . . . . . . . . 10  |-  ( .r
`  R )  =  ( .r `  R
)
111, 10ressmulr 13584 . . . . . . . . 9  |-  ( A  e.  (SubRing `  R
)  ->  ( .r `  R )  =  ( .r `  S ) )
12113ad2ant1 979 . . . . . . . 8  |-  ( ( A  e.  (SubRing `  R
)  /\  x  e.  V  /\  y  e.  V
)  ->  ( .r `  R )  =  ( .r `  S ) )
1312oveqd 6100 . . . . . . 7  |-  ( ( A  e.  (SubRing `  R
)  /\  x  e.  V  /\  y  e.  V
)  ->  ( x
( .r `  R
) y )  =  ( x ( .r
`  S ) y ) )
14 eqid 2438 . . . . . . . . 9  |-  ( .r
`  S )  =  ( .r `  S
)
153, 14unitmulcl 15771 . . . . . . . 8  |-  ( ( S  e.  Ring  /\  x  e.  V  /\  y  e.  V )  ->  (
x ( .r `  S ) y )  e.  V )
165, 15syl3an1 1218 . . . . . . 7  |-  ( ( A  e.  (SubRing `  R
)  /\  x  e.  V  /\  y  e.  V
)  ->  ( x
( .r `  S
) y )  e.  V )
1713, 16eqeltrd 2512 . . . . . 6  |-  ( ( A  e.  (SubRing `  R
)  /\  x  e.  V  /\  y  e.  V
)  ->  ( x
( .r `  R
) y )  e.  V )
18173expa 1154 . . . . 5  |-  ( ( ( A  e.  (SubRing `  R )  /\  x  e.  V )  /\  y  e.  V )  ->  (
x ( .r `  R ) y )  e.  V )
1918ralrimiva 2791 . . . 4  |-  ( ( A  e.  (SubRing `  R
)  /\  x  e.  V )  ->  A. y  e.  V  ( x
( .r `  R
) y )  e.  V )
20 eqid 2438 . . . . . 6  |-  ( invr `  R )  =  (
invr `  R )
21 eqid 2438 . . . . . 6  |-  ( invr `  S )  =  (
invr `  S )
221, 20, 3, 21subrginv 15886 . . . . 5  |-  ( ( A  e.  (SubRing `  R
)  /\  x  e.  V )  ->  (
( invr `  R ) `  x )  =  ( ( invr `  S
) `  x )
)
233, 21unitinvcl 15781 . . . . . 6  |-  ( ( S  e.  Ring  /\  x  e.  V )  ->  (
( invr `  S ) `  x )  e.  V
)
245, 23sylan 459 . . . . 5  |-  ( ( A  e.  (SubRing `  R
)  /\  x  e.  V )  ->  (
( invr `  S ) `  x )  e.  V
)
2522, 24eqeltrd 2512 . . . 4  |-  ( ( A  e.  (SubRing `  R
)  /\  x  e.  V )  ->  (
( invr `  R ) `  x )  e.  V
)
2619, 25jca 520 . . 3  |-  ( ( A  e.  (SubRing `  R
)  /\  x  e.  V )  ->  ( A. y  e.  V  ( x ( .r
`  R ) y )  e.  V  /\  ( ( invr `  R
) `  x )  e.  V ) )
2726ralrimiva 2791 . 2  |-  ( A  e.  (SubRing `  R
)  ->  A. x  e.  V  ( A. y  e.  V  (
x ( .r `  R ) y )  e.  V  /\  (
( invr `  R ) `  x )  e.  V
) )
28 subrgrcl 15875 . . 3  |-  ( A  e.  (SubRing `  R
)  ->  R  e.  Ring )
29 subrgugrp.4 . . . 4  |-  G  =  ( (mulGrp `  R
)s 
U )
302, 29unitgrp 15774 . . 3  |-  ( R  e.  Ring  ->  G  e. 
Grp )
312, 29unitgrpbas 15773 . . . 4  |-  U  =  ( Base `  G
)
32 fvex 5744 . . . . . 6  |-  (Unit `  R )  e.  _V
332, 32eqeltri 2508 . . . . 5  |-  U  e. 
_V
34 eqid 2438 . . . . . . 7  |-  (mulGrp `  R )  =  (mulGrp `  R )
3534, 10mgpplusg 15654 . . . . . 6  |-  ( .r
`  R )  =  ( +g  `  (mulGrp `  R ) )
3629, 35ressplusg 13573 . . . . 5  |-  ( U  e.  _V  ->  ( .r `  R )  =  ( +g  `  G
) )
3733, 36ax-mp 8 . . . 4  |-  ( .r
`  R )  =  ( +g  `  G
)
382, 29, 20invrfval 15780 . . . 4  |-  ( invr `  R )  =  ( inv g `  G
)
3931, 37, 38issubg2 14961 . . 3  |-  ( G  e.  Grp  ->  ( V  e.  (SubGrp `  G
)  <->  ( V  C_  U  /\  V  =/=  (/)  /\  A. x  e.  V  ( A. y  e.  V  ( x ( .r
`  R ) y )  e.  V  /\  ( ( invr `  R
) `  x )  e.  V ) ) ) )
4028, 30, 393syl 19 . 2  |-  ( A  e.  (SubRing `  R
)  ->  ( V  e.  (SubGrp `  G )  <->  ( V  C_  U  /\  V  =/=  (/)  /\  A. x  e.  V  ( A. y  e.  V  (
x ( .r `  R ) y )  e.  V  /\  (
( invr `  R ) `  x )  e.  V
) ) ) )
414, 9, 27, 40mpbir3and 1138 1  |-  ( A  e.  (SubRing `  R
)  ->  V  e.  (SubGrp `  G ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726    =/= wne 2601   A.wral 2707   _Vcvv 2958    C_ wss 3322   (/)c0 3630   ` cfv 5456  (class class class)co 6083   ↾s cress 13472   +g cplusg 13531   .rcmulr 13532   Grpcgrp 14687  SubGrpcsubg 14940  mulGrpcmgp 15650   Ringcrg 15662   1rcur 15664  Unitcui 15746   invrcinvr 15778  SubRingcsubrg 15866
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703  ax-cnex 9048  ax-resscn 9049  ax-1cn 9050  ax-icn 9051  ax-addcl 9052  ax-addrcl 9053  ax-mulcl 9054  ax-mulrcl 9055  ax-mulcom 9056  ax-addass 9057  ax-mulass 9058  ax-distr 9059  ax-i2m1 9060  ax-1ne0 9061  ax-1rid 9062  ax-rnegex 9063  ax-rrecex 9064  ax-cnre 9065  ax-pre-lttri 9066  ax-pre-lttrn 9067  ax-pre-ltadd 9068  ax-pre-mulgt0 9069
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-tr 4305  df-eprel 4496  df-id 4500  df-po 4505  df-so 4506  df-fr 4543  df-we 4545  df-ord 4586  df-on 4587  df-lim 4588  df-suc 4589  df-om 4848  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-tpos 6481  df-riota 6551  df-recs 6635  df-rdg 6670  df-er 6907  df-en 7112  df-dom 7113  df-sdom 7114  df-pnf 9124  df-mnf 9125  df-xr 9126  df-ltxr 9127  df-le 9128  df-sub 9295  df-neg 9296  df-nn 10003  df-2 10060  df-3 10061  df-ndx 13474  df-slot 13475  df-base 13476  df-sets 13477  df-ress 13478  df-plusg 13544  df-mulr 13545  df-0g 13729  df-mnd 14692  df-grp 14814  df-minusg 14815  df-subg 14943  df-mgp 15651  df-rng 15665  df-ur 15667  df-oppr 15730  df-dvdsr 15748  df-unit 15749  df-invr 15779  df-subrg 15868
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