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Theorem subrgugrp 17317
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 17313 . 2  |-  ( A  e.  (SubRing `  R
)  ->  V  C_  U
)
51subrgring 17301 . . 3  |-  ( A  e.  (SubRing `  R
)  ->  S  e.  Ring )
6 eqid 2467 . . . 4  |-  ( 1r
`  S )  =  ( 1r `  S
)
73, 61unit 17177 . . 3  |-  ( S  e.  Ring  ->  ( 1r
`  S )  e.  V )
8 ne0i 3796 . . 3  |-  ( ( 1r `  S )  e.  V  ->  V  =/=  (/) )
95, 7, 83syl 20 . 2  |-  ( A  e.  (SubRing `  R
)  ->  V  =/=  (/) )
10 eqid 2467 . . . . . . . . . 10  |-  ( .r
`  R )  =  ( .r `  R
)
111, 10ressmulr 14624 . . . . . . . . 9  |-  ( A  e.  (SubRing `  R
)  ->  ( .r `  R )  =  ( .r `  S ) )
12113ad2ant1 1017 . . . . . . . 8  |-  ( ( A  e.  (SubRing `  R
)  /\  x  e.  V  /\  y  e.  V
)  ->  ( .r `  R )  =  ( .r `  S ) )
1312oveqd 6312 . . . . . . 7  |-  ( ( A  e.  (SubRing `  R
)  /\  x  e.  V  /\  y  e.  V
)  ->  ( x
( .r `  R
) y )  =  ( x ( .r
`  S ) y ) )
14 eqid 2467 . . . . . . . . 9  |-  ( .r
`  S )  =  ( .r `  S
)
153, 14unitmulcl 17183 . . . . . . . 8  |-  ( ( S  e.  Ring  /\  x  e.  V  /\  y  e.  V )  ->  (
x ( .r `  S ) y )  e.  V )
165, 15syl3an1 1261 . . . . . . 7  |-  ( ( A  e.  (SubRing `  R
)  /\  x  e.  V  /\  y  e.  V
)  ->  ( x
( .r `  S
) y )  e.  V )
1713, 16eqeltrd 2555 . . . . . 6  |-  ( ( A  e.  (SubRing `  R
)  /\  x  e.  V  /\  y  e.  V
)  ->  ( x
( .r `  R
) y )  e.  V )
18173expa 1196 . . . . 5  |-  ( ( ( A  e.  (SubRing `  R )  /\  x  e.  V )  /\  y  e.  V )  ->  (
x ( .r `  R ) y )  e.  V )
1918ralrimiva 2881 . . . 4  |-  ( ( A  e.  (SubRing `  R
)  /\  x  e.  V )  ->  A. y  e.  V  ( x
( .r `  R
) y )  e.  V )
20 eqid 2467 . . . . . 6  |-  ( invr `  R )  =  (
invr `  R )
21 eqid 2467 . . . . . 6  |-  ( invr `  S )  =  (
invr `  S )
221, 20, 3, 21subrginv 17314 . . . . 5  |-  ( ( A  e.  (SubRing `  R
)  /\  x  e.  V )  ->  (
( invr `  R ) `  x )  =  ( ( invr `  S
) `  x )
)
233, 21unitinvcl 17193 . . . . . 6  |-  ( ( S  e.  Ring  /\  x  e.  V )  ->  (
( invr `  S ) `  x )  e.  V
)
245, 23sylan 471 . . . . 5  |-  ( ( A  e.  (SubRing `  R
)  /\  x  e.  V )  ->  (
( invr `  S ) `  x )  e.  V
)
2522, 24eqeltrd 2555 . . . 4  |-  ( ( A  e.  (SubRing `  R
)  /\  x  e.  V )  ->  (
( invr `  R ) `  x )  e.  V
)
2619, 25jca 532 . . 3  |-  ( ( A  e.  (SubRing `  R
)  /\  x  e.  V )  ->  ( A. y  e.  V  ( x ( .r
`  R ) y )  e.  V  /\  ( ( invr `  R
) `  x )  e.  V ) )
2726ralrimiva 2881 . 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 17303 . . 3  |-  ( A  e.  (SubRing `  R
)  ->  R  e.  Ring )
29 subrgugrp.4 . . . 4  |-  G  =  ( (mulGrp `  R
)s 
U )
302, 29unitgrp 17186 . . 3  |-  ( R  e.  Ring  ->  G  e. 
Grp )
312, 29unitgrpbas 17185 . . . 4  |-  U  =  ( Base `  G
)
32 fvex 5882 . . . . . 6  |-  (Unit `  R )  e.  _V
332, 32eqeltri 2551 . . . . 5  |-  U  e. 
_V
34 eqid 2467 . . . . . . 7  |-  (mulGrp `  R )  =  (mulGrp `  R )
3534, 10mgpplusg 17015 . . . . . 6  |-  ( .r
`  R )  =  ( +g  `  (mulGrp `  R ) )
3629, 35ressplusg 14613 . . . . 5  |-  ( U  e.  _V  ->  ( .r `  R )  =  ( +g  `  G
) )
3733, 36ax-mp 5 . . . 4  |-  ( .r
`  R )  =  ( +g  `  G
)
382, 29, 20invrfval 17192 . . . 4  |-  ( invr `  R )  =  ( invg `  G
)
3931, 37, 38issubg2 16087 . . 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 20 . 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 1179 1  |-  ( A  e.  (SubRing `  R
)  ->  V  e.  (SubGrp `  G ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767    =/= wne 2662   A.wral 2817   _Vcvv 3118    C_ wss 3481   (/)c0 3790   ` cfv 5594  (class class class)co 6295   ↾s cress 14507   +g cplusg 14571   .rcmulr 14572   Grpcgrp 15924  SubGrpcsubg 16066  mulGrpcmgp 17011   1rcur 17023   Ringcrg 17068  Unitcui 17158   invrcinvr 17190  SubRingcsubrg 17294
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 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581
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 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6696  df-tpos 6967  df-recs 7054  df-rdg 7088  df-er 7323  df-en 7529  df-dom 7530  df-sdom 7531  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-nn 10549  df-2 10606  df-3 10607  df-ndx 14509  df-slot 14510  df-base 14511  df-sets 14512  df-ress 14513  df-plusg 14584  df-mulr 14585  df-0g 14713  df-mgm 15745  df-sgrp 15784  df-mnd 15794  df-grp 15928  df-minusg 15929  df-subg 16069  df-mgp 17012  df-ur 17024  df-ring 17070  df-oppr 17142  df-dvdsr 17160  df-unit 17161  df-invr 17191  df-subrg 17296
This theorem is referenced by: (None)
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