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Theorem subrgugrp 17766
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 17762 . 2  |-  ( A  e.  (SubRing `  R
)  ->  V  C_  U
)
51subrgring 17750 . . 3  |-  ( A  e.  (SubRing `  R
)  ->  S  e.  Ring )
6 eqid 2402 . . . 4  |-  ( 1r
`  S )  =  ( 1r `  S
)
73, 61unit 17625 . . 3  |-  ( S  e.  Ring  ->  ( 1r
`  S )  e.  V )
8 ne0i 3743 . . 3  |-  ( ( 1r `  S )  e.  V  ->  V  =/=  (/) )
95, 7, 83syl 20 . 2  |-  ( A  e.  (SubRing `  R
)  ->  V  =/=  (/) )
10 eqid 2402 . . . . . . . . . 10  |-  ( .r
`  R )  =  ( .r `  R
)
111, 10ressmulr 14964 . . . . . . . . 9  |-  ( A  e.  (SubRing `  R
)  ->  ( .r `  R )  =  ( .r `  S ) )
12113ad2ant1 1018 . . . . . . . 8  |-  ( ( A  e.  (SubRing `  R
)  /\  x  e.  V  /\  y  e.  V
)  ->  ( .r `  R )  =  ( .r `  S ) )
1312oveqd 6294 . . . . . . 7  |-  ( ( A  e.  (SubRing `  R
)  /\  x  e.  V  /\  y  e.  V
)  ->  ( x
( .r `  R
) y )  =  ( x ( .r
`  S ) y ) )
14 eqid 2402 . . . . . . . . 9  |-  ( .r
`  S )  =  ( .r `  S
)
153, 14unitmulcl 17631 . . . . . . . 8  |-  ( ( S  e.  Ring  /\  x  e.  V  /\  y  e.  V )  ->  (
x ( .r `  S ) y )  e.  V )
165, 15syl3an1 1263 . . . . . . 7  |-  ( ( A  e.  (SubRing `  R
)  /\  x  e.  V  /\  y  e.  V
)  ->  ( x
( .r `  S
) y )  e.  V )
1713, 16eqeltrd 2490 . . . . . 6  |-  ( ( A  e.  (SubRing `  R
)  /\  x  e.  V  /\  y  e.  V
)  ->  ( x
( .r `  R
) y )  e.  V )
18173expa 1197 . . . . 5  |-  ( ( ( A  e.  (SubRing `  R )  /\  x  e.  V )  /\  y  e.  V )  ->  (
x ( .r `  R ) y )  e.  V )
1918ralrimiva 2817 . . . 4  |-  ( ( A  e.  (SubRing `  R
)  /\  x  e.  V )  ->  A. y  e.  V  ( x
( .r `  R
) y )  e.  V )
20 eqid 2402 . . . . . 6  |-  ( invr `  R )  =  (
invr `  R )
21 eqid 2402 . . . . . 6  |-  ( invr `  S )  =  (
invr `  S )
221, 20, 3, 21subrginv 17763 . . . . 5  |-  ( ( A  e.  (SubRing `  R
)  /\  x  e.  V )  ->  (
( invr `  R ) `  x )  =  ( ( invr `  S
) `  x )
)
233, 21unitinvcl 17641 . . . . . 6  |-  ( ( S  e.  Ring  /\  x  e.  V )  ->  (
( invr `  S ) `  x )  e.  V
)
245, 23sylan 469 . . . . 5  |-  ( ( A  e.  (SubRing `  R
)  /\  x  e.  V )  ->  (
( invr `  S ) `  x )  e.  V
)
2522, 24eqeltrd 2490 . . . 4  |-  ( ( A  e.  (SubRing `  R
)  /\  x  e.  V )  ->  (
( invr `  R ) `  x )  e.  V
)
2619, 25jca 530 . . 3  |-  ( ( A  e.  (SubRing `  R
)  /\  x  e.  V )  ->  ( A. y  e.  V  ( x ( .r
`  R ) y )  e.  V  /\  ( ( invr `  R
) `  x )  e.  V ) )
2726ralrimiva 2817 . 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 17752 . . 3  |-  ( A  e.  (SubRing `  R
)  ->  R  e.  Ring )
29 subrgugrp.4 . . . 4  |-  G  =  ( (mulGrp `  R
)s 
U )
302, 29unitgrp 17634 . . 3  |-  ( R  e.  Ring  ->  G  e. 
Grp )
312, 29unitgrpbas 17633 . . . 4  |-  U  =  ( Base `  G
)
32 fvex 5858 . . . . . 6  |-  (Unit `  R )  e.  _V
332, 32eqeltri 2486 . . . . 5  |-  U  e. 
_V
34 eqid 2402 . . . . . . 7  |-  (mulGrp `  R )  =  (mulGrp `  R )
3534, 10mgpplusg 17463 . . . . . 6  |-  ( .r
`  R )  =  ( +g  `  (mulGrp `  R ) )
3629, 35ressplusg 14953 . . . . 5  |-  ( U  e.  _V  ->  ( .r `  R )  =  ( +g  `  G
) )
3733, 36ax-mp 5 . . . 4  |-  ( .r
`  R )  =  ( +g  `  G
)
382, 29, 20invrfval 17640 . . . 4  |-  ( invr `  R )  =  ( invg `  G
)
3931, 37, 38issubg2 16538 . . 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 1180 1  |-  ( A  e.  (SubRing `  R
)  ->  V  e.  (SubGrp `  G ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    /\ w3a 974    = wceq 1405    e. wcel 1842    =/= wne 2598   A.wral 2753   _Vcvv 3058    C_ wss 3413   (/)c0 3737   ` cfv 5568  (class class class)co 6277   ↾s cress 14840   +g cplusg 14907   .rcmulr 14908   Grpcgrp 16375  SubGrpcsubg 16517  mulGrpcmgp 17459   1rcur 17471   Ringcrg 17516  Unitcui 17606   invrcinvr 17638  SubRingcsubrg 17743
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6573  ax-cnex 9577  ax-resscn 9578  ax-1cn 9579  ax-icn 9580  ax-addcl 9581  ax-addrcl 9582  ax-mulcl 9583  ax-mulrcl 9584  ax-mulcom 9585  ax-addass 9586  ax-mulass 9587  ax-distr 9588  ax-i2m1 9589  ax-1ne0 9590  ax-1rid 9591  ax-rnegex 9592  ax-rrecex 9593  ax-cnre 9594  ax-pre-lttri 9595  ax-pre-lttrn 9596  ax-pre-ltadd 9597  ax-pre-mulgt0 9598
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2758  df-rex 2759  df-reu 2760  df-rmo 2761  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-pss 3429  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-tp 3976  df-op 3978  df-uni 4191  df-iun 4272  df-br 4395  df-opab 4453  df-mpt 4454  df-tr 4489  df-eprel 4733  df-id 4737  df-po 4743  df-so 4744  df-fr 4781  df-we 4783  df-xp 4828  df-rel 4829  df-cnv 4830  df-co 4831  df-dm 4832  df-rn 4833  df-res 4834  df-ima 4835  df-pred 5366  df-ord 5412  df-on 5413  df-lim 5414  df-suc 5415  df-iota 5532  df-fun 5570  df-fn 5571  df-f 5572  df-f1 5573  df-fo 5574  df-f1o 5575  df-fv 5576  df-riota 6239  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-om 6683  df-tpos 6957  df-wrecs 7012  df-recs 7074  df-rdg 7112  df-er 7347  df-en 7554  df-dom 7555  df-sdom 7556  df-pnf 9659  df-mnf 9660  df-xr 9661  df-ltxr 9662  df-le 9663  df-sub 9842  df-neg 9843  df-nn 10576  df-2 10634  df-3 10635  df-ndx 14842  df-slot 14843  df-base 14844  df-sets 14845  df-ress 14846  df-plusg 14920  df-mulr 14921  df-0g 15054  df-mgm 16194  df-sgrp 16233  df-mnd 16243  df-grp 16379  df-minusg 16380  df-subg 16520  df-mgp 17460  df-ur 17472  df-ring 17518  df-oppr 17590  df-dvdsr 17608  df-unit 17609  df-invr 17639  df-subrg 17745
This theorem is referenced by: (None)
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