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Theorem subrgugrp 15580
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 15576 . 2  |-  ( A  e.  (SubRing `  R
)  ->  V  C_  U
)
51subrgrng 15564 . . 3  |-  ( A  e.  (SubRing `  R
)  ->  S  e.  Ring )
6 eqid 2296 . . . 4  |-  ( 1r
`  S )  =  ( 1r `  S
)
73, 61unit 15456 . . 3  |-  ( S  e.  Ring  ->  ( 1r
`  S )  e.  V )
8 ne0i 3474 . . 3  |-  ( ( 1r `  S )  e.  V  ->  V  =/=  (/) )
95, 7, 83syl 18 . 2  |-  ( A  e.  (SubRing `  R
)  ->  V  =/=  (/) )
10 eqid 2296 . . . . . . . . . 10  |-  ( .r
`  R )  =  ( .r `  R
)
111, 10ressmulr 13277 . . . . . . . . 9  |-  ( A  e.  (SubRing `  R
)  ->  ( .r `  R )  =  ( .r `  S ) )
12113ad2ant1 976 . . . . . . . 8  |-  ( ( A  e.  (SubRing `  R
)  /\  x  e.  V  /\  y  e.  V
)  ->  ( .r `  R )  =  ( .r `  S ) )
1312oveqd 5891 . . . . . . 7  |-  ( ( A  e.  (SubRing `  R
)  /\  x  e.  V  /\  y  e.  V
)  ->  ( x
( .r `  R
) y )  =  ( x ( .r
`  S ) y ) )
14 eqid 2296 . . . . . . . . 9  |-  ( .r
`  S )  =  ( .r `  S
)
153, 14unitmulcl 15462 . . . . . . . 8  |-  ( ( S  e.  Ring  /\  x  e.  V  /\  y  e.  V )  ->  (
x ( .r `  S ) y )  e.  V )
165, 15syl3an1 1215 . . . . . . 7  |-  ( ( A  e.  (SubRing `  R
)  /\  x  e.  V  /\  y  e.  V
)  ->  ( x
( .r `  S
) y )  e.  V )
1713, 16eqeltrd 2370 . . . . . 6  |-  ( ( A  e.  (SubRing `  R
)  /\  x  e.  V  /\  y  e.  V
)  ->  ( x
( .r `  R
) y )  e.  V )
18173expa 1151 . . . . 5  |-  ( ( ( A  e.  (SubRing `  R )  /\  x  e.  V )  /\  y  e.  V )  ->  (
x ( .r `  R ) y )  e.  V )
1918ralrimiva 2639 . . . 4  |-  ( ( A  e.  (SubRing `  R
)  /\  x  e.  V )  ->  A. y  e.  V  ( x
( .r `  R
) y )  e.  V )
20 eqid 2296 . . . . . 6  |-  ( invr `  R )  =  (
invr `  R )
21 eqid 2296 . . . . . 6  |-  ( invr `  S )  =  (
invr `  S )
221, 20, 3, 21subrginv 15577 . . . . 5  |-  ( ( A  e.  (SubRing `  R
)  /\  x  e.  V )  ->  (
( invr `  R ) `  x )  =  ( ( invr `  S
) `  x )
)
233, 21unitinvcl 15472 . . . . . 6  |-  ( ( S  e.  Ring  /\  x  e.  V )  ->  (
( invr `  S ) `  x )  e.  V
)
245, 23sylan 457 . . . . 5  |-  ( ( A  e.  (SubRing `  R
)  /\  x  e.  V )  ->  (
( invr `  S ) `  x )  e.  V
)
2522, 24eqeltrd 2370 . . . 4  |-  ( ( A  e.  (SubRing `  R
)  /\  x  e.  V )  ->  (
( invr `  R ) `  x )  e.  V
)
2619, 25jca 518 . . 3  |-  ( ( A  e.  (SubRing `  R
)  /\  x  e.  V )  ->  ( A. y  e.  V  ( x ( .r
`  R ) y )  e.  V  /\  ( ( invr `  R
) `  x )  e.  V ) )
2726ralrimiva 2639 . 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 15566 . . 3  |-  ( A  e.  (SubRing `  R
)  ->  R  e.  Ring )
29 subrgugrp.4 . . . 4  |-  G  =  ( (mulGrp `  R
)s 
U )
302, 29unitgrp 15465 . . 3  |-  ( R  e.  Ring  ->  G  e. 
Grp )
312, 29unitgrpbas 15464 . . . 4  |-  U  =  ( Base `  G
)
32 fvex 5555 . . . . . 6  |-  (Unit `  R )  e.  _V
332, 32eqeltri 2366 . . . . 5  |-  U  e. 
_V
34 eqid 2296 . . . . . . 7  |-  (mulGrp `  R )  =  (mulGrp `  R )
3534, 10mgpplusg 15345 . . . . . 6  |-  ( .r
`  R )  =  ( +g  `  (mulGrp `  R ) )
3629, 35ressplusg 13266 . . . . 5  |-  ( U  e.  _V  ->  ( .r `  R )  =  ( +g  `  G
) )
3733, 36ax-mp 8 . . . 4  |-  ( .r
`  R )  =  ( +g  `  G
)
382, 29, 20invrfval 15471 . . . 4  |-  ( invr `  R )  =  ( inv g `  G
)
3931, 37, 38issubg2 14652 . . 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 18 . 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 1135 1  |-  ( A  e.  (SubRing `  R
)  ->  V  e.  (SubGrp `  G ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696    =/= wne 2459   A.wral 2556   _Vcvv 2801    C_ wss 3165   (/)c0 3468   ` cfv 5271  (class class class)co 5874   ↾s cress 13165   +g cplusg 13224   .rcmulr 13225   Grpcgrp 14378  SubGrpcsubg 14631  mulGrpcmgp 15341   Ringcrg 15353   1rcur 15355  Unitcui 15437   invrcinvr 15469  SubRingcsubrg 15557
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-tpos 6250  df-riota 6320  df-recs 6404  df-rdg 6439  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-nn 9763  df-2 9820  df-3 9821  df-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-ress 13171  df-plusg 13237  df-mulr 13238  df-0g 13420  df-mnd 14383  df-grp 14505  df-minusg 14506  df-subg 14634  df-mgp 15342  df-rng 15356  df-ur 15358  df-oppr 15421  df-dvdsr 15439  df-unit 15440  df-invr 15470  df-subrg 15559
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