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Theorem subrgugrp 16896
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 16892 . 2  |-  ( A  e.  (SubRing `  R
)  ->  V  C_  U
)
51subrgrng 16880 . . 3  |-  ( A  e.  (SubRing `  R
)  ->  S  e.  Ring )
6 eqid 2443 . . . 4  |-  ( 1r
`  S )  =  ( 1r `  S
)
73, 61unit 16762 . . 3  |-  ( S  e.  Ring  ->  ( 1r
`  S )  e.  V )
8 ne0i 3655 . . 3  |-  ( ( 1r `  S )  e.  V  ->  V  =/=  (/) )
95, 7, 83syl 20 . 2  |-  ( A  e.  (SubRing `  R
)  ->  V  =/=  (/) )
10 eqid 2443 . . . . . . . . . 10  |-  ( .r
`  R )  =  ( .r `  R
)
111, 10ressmulr 14303 . . . . . . . . 9  |-  ( A  e.  (SubRing `  R
)  ->  ( .r `  R )  =  ( .r `  S ) )
12113ad2ant1 1009 . . . . . . . 8  |-  ( ( A  e.  (SubRing `  R
)  /\  x  e.  V  /\  y  e.  V
)  ->  ( .r `  R )  =  ( .r `  S ) )
1312oveqd 6120 . . . . . . 7  |-  ( ( A  e.  (SubRing `  R
)  /\  x  e.  V  /\  y  e.  V
)  ->  ( x
( .r `  R
) y )  =  ( x ( .r
`  S ) y ) )
14 eqid 2443 . . . . . . . . 9  |-  ( .r
`  S )  =  ( .r `  S
)
153, 14unitmulcl 16768 . . . . . . . 8  |-  ( ( S  e.  Ring  /\  x  e.  V  /\  y  e.  V )  ->  (
x ( .r `  S ) y )  e.  V )
165, 15syl3an1 1251 . . . . . . 7  |-  ( ( A  e.  (SubRing `  R
)  /\  x  e.  V  /\  y  e.  V
)  ->  ( x
( .r `  S
) y )  e.  V )
1713, 16eqeltrd 2517 . . . . . 6  |-  ( ( A  e.  (SubRing `  R
)  /\  x  e.  V  /\  y  e.  V
)  ->  ( x
( .r `  R
) y )  e.  V )
18173expa 1187 . . . . 5  |-  ( ( ( A  e.  (SubRing `  R )  /\  x  e.  V )  /\  y  e.  V )  ->  (
x ( .r `  R ) y )  e.  V )
1918ralrimiva 2811 . . . 4  |-  ( ( A  e.  (SubRing `  R
)  /\  x  e.  V )  ->  A. y  e.  V  ( x
( .r `  R
) y )  e.  V )
20 eqid 2443 . . . . . 6  |-  ( invr `  R )  =  (
invr `  R )
21 eqid 2443 . . . . . 6  |-  ( invr `  S )  =  (
invr `  S )
221, 20, 3, 21subrginv 16893 . . . . 5  |-  ( ( A  e.  (SubRing `  R
)  /\  x  e.  V )  ->  (
( invr `  R ) `  x )  =  ( ( invr `  S
) `  x )
)
233, 21unitinvcl 16778 . . . . . 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 2517 . . . 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 2811 . 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 16882 . . 3  |-  ( A  e.  (SubRing `  R
)  ->  R  e.  Ring )
29 subrgugrp.4 . . . 4  |-  G  =  ( (mulGrp `  R
)s 
U )
302, 29unitgrp 16771 . . 3  |-  ( R  e.  Ring  ->  G  e. 
Grp )
312, 29unitgrpbas 16770 . . . 4  |-  U  =  ( Base `  G
)
32 fvex 5713 . . . . . 6  |-  (Unit `  R )  e.  _V
332, 32eqeltri 2513 . . . . 5  |-  U  e. 
_V
34 eqid 2443 . . . . . . 7  |-  (mulGrp `  R )  =  (mulGrp `  R )
3534, 10mgpplusg 16607 . . . . . 6  |-  ( .r
`  R )  =  ( +g  `  (mulGrp `  R ) )
3629, 35ressplusg 14292 . . . . 5  |-  ( U  e.  _V  ->  ( .r `  R )  =  ( +g  `  G
) )
3733, 36ax-mp 5 . . . 4  |-  ( .r
`  R )  =  ( +g  `  G
)
382, 29, 20invrfval 16777 . . . 4  |-  ( invr `  R )  =  ( invg `  G
)
3931, 37, 38issubg2 15708 . . 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 1171 1  |-  ( A  e.  (SubRing `  R
)  ->  V  e.  (SubGrp `  G ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756    =/= wne 2618   A.wral 2727   _Vcvv 2984    C_ wss 3340   (/)c0 3649   ` cfv 5430  (class class class)co 6103   ↾s cress 14187   +g cplusg 14250   .rcmulr 14251   Grpcgrp 15422  SubGrpcsubg 15687  mulGrpcmgp 16603   1rcur 16615   Ringcrg 16657  Unitcui 16743   invrcinvr 16775  SubRingcsubrg 16873
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4415  ax-sep 4425  ax-nul 4433  ax-pow 4482  ax-pr 4543  ax-un 6384  ax-cnex 9350  ax-resscn 9351  ax-1cn 9352  ax-icn 9353  ax-addcl 9354  ax-addrcl 9355  ax-mulcl 9356  ax-mulrcl 9357  ax-mulcom 9358  ax-addass 9359  ax-mulass 9360  ax-distr 9361  ax-i2m1 9362  ax-1ne0 9363  ax-1rid 9364  ax-rnegex 9365  ax-rrecex 9366  ax-cnre 9367  ax-pre-lttri 9368  ax-pre-lttrn 9369  ax-pre-ltadd 9370  ax-pre-mulgt0 9371
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2620  df-nel 2621  df-ral 2732  df-rex 2733  df-reu 2734  df-rmo 2735  df-rab 2736  df-v 2986  df-sbc 3199  df-csb 3301  df-dif 3343  df-un 3345  df-in 3347  df-ss 3354  df-pss 3356  df-nul 3650  df-if 3804  df-pw 3874  df-sn 3890  df-pr 3892  df-tp 3894  df-op 3896  df-uni 4104  df-iun 4185  df-br 4305  df-opab 4363  df-mpt 4364  df-tr 4398  df-eprel 4644  df-id 4648  df-po 4653  df-so 4654  df-fr 4691  df-we 4693  df-ord 4734  df-on 4735  df-lim 4736  df-suc 4737  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-iota 5393  df-fun 5432  df-fn 5433  df-f 5434  df-f1 5435  df-fo 5436  df-f1o 5437  df-fv 5438  df-riota 6064  df-ov 6106  df-oprab 6107  df-mpt2 6108  df-om 6489  df-tpos 6757  df-recs 6844  df-rdg 6878  df-er 7113  df-en 7323  df-dom 7324  df-sdom 7325  df-pnf 9432  df-mnf 9433  df-xr 9434  df-ltxr 9435  df-le 9436  df-sub 9609  df-neg 9610  df-nn 10335  df-2 10392  df-3 10393  df-ndx 14189  df-slot 14190  df-base 14191  df-sets 14192  df-ress 14193  df-plusg 14263  df-mulr 14264  df-0g 14392  df-mnd 15427  df-grp 15557  df-minusg 15558  df-subg 15690  df-mgp 16604  df-ur 16616  df-rng 16659  df-oppr 16727  df-dvdsr 16745  df-unit 16746  df-invr 16776  df-subrg 16875
This theorem is referenced by: (None)
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