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Theorem subrgugrp 13554
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  w are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 subrgugrp.1 . . . 4  |-  S  =  ( Rs  A )
2 subrgugrp.2 . . . 4  |-  U  =  (Unit `  R )
3 subrgugrp.3 . . . 4  |-  V  =  (Unit `  S )
41, 2, 3subrguss 13550 . . 3  |-  ( A  e.  (SubRing `  R
)  ->  V  C_  U
)
5 subrgrcl 13540 . . . 4  |-  ( A  e.  (SubRing `  R
)  ->  R  e.  Ring )
62a1i 9 . . . . 5  |-  ( R  e.  Ring  ->  U  =  (Unit `  R )
)
7 subrgugrp.4 . . . . . 6  |-  G  =  ( (mulGrp `  R
)s 
U )
87a1i 9 . . . . 5  |-  ( R  e.  Ring  ->  G  =  ( (mulGrp `  R
)s 
U ) )
9 ringsrg 13366 . . . . 5  |-  ( R  e.  Ring  ->  R  e. SRing
)
106, 8, 9unitgrpbasd 13432 . . . 4  |-  ( R  e.  Ring  ->  U  =  ( Base `  G
) )
115, 10syl 14 . . 3  |-  ( A  e.  (SubRing `  R
)  ->  U  =  ( Base `  G )
)
124, 11sseqtrd 3208 . 2  |-  ( A  e.  (SubRing `  R
)  ->  V  C_  ( Base `  G ) )
131subrgring 13538 . . 3  |-  ( A  e.  (SubRing `  R
)  ->  S  e.  Ring )
14 eqid 2189 . . . 4  |-  ( 1r
`  S )  =  ( 1r `  S
)
153, 141unit 13424 . . 3  |-  ( S  e.  Ring  ->  ( 1r
`  S )  e.  V )
16 elex2 2768 . . 3  |-  ( ( 1r `  S )  e.  V  ->  E. w  w  e.  V )
1713, 15, 163syl 17 . 2  |-  ( A  e.  (SubRing `  R
)  ->  E. w  w  e.  V )
18 eqid 2189 . . . . . . . . . . . 12  |-  ( .r
`  R )  =  ( .r `  R
)
191, 18ressmulrg 12628 . . . . . . . . . . 11  |-  ( ( A  e.  (SubRing `  R
)  /\  R  e.  Ring )  ->  ( .r `  R )  =  ( .r `  S ) )
205, 19mpdan 421 . . . . . . . . . 10  |-  ( A  e.  (SubRing `  R
)  ->  ( .r `  R )  =  ( .r `  S ) )
21203ad2ant1 1020 . . . . . . . . 9  |-  ( ( A  e.  (SubRing `  R
)  /\  x  e.  V  /\  y  e.  V
)  ->  ( .r `  R )  =  ( .r `  S ) )
2221oveqd 5908 . . . . . . . 8  |-  ( ( A  e.  (SubRing `  R
)  /\  x  e.  V  /\  y  e.  V
)  ->  ( x
( .r `  R
) y )  =  ( x ( .r
`  S ) y ) )
23 eqid 2189 . . . . . . . . . 10  |-  ( .r
`  S )  =  ( .r `  S
)
243, 23unitmulcl 13430 . . . . . . . . 9  |-  ( ( S  e.  Ring  /\  x  e.  V  /\  y  e.  V )  ->  (
x ( .r `  S ) y )  e.  V )
2513, 24syl3an1 1282 . . . . . . . 8  |-  ( ( A  e.  (SubRing `  R
)  /\  x  e.  V  /\  y  e.  V
)  ->  ( x
( .r `  S
) y )  e.  V )
2622, 25eqeltrd 2266 . . . . . . 7  |-  ( ( A  e.  (SubRing `  R
)  /\  x  e.  V  /\  y  e.  V
)  ->  ( x
( .r `  R
) y )  e.  V )
27263expa 1205 . . . . . 6  |-  ( ( ( A  e.  (SubRing `  R )  /\  x  e.  V )  /\  y  e.  V )  ->  (
x ( .r `  R ) y )  e.  V )
2827ralrimiva 2563 . . . . 5  |-  ( ( A  e.  (SubRing `  R
)  /\  x  e.  V )  ->  A. y  e.  V  ( x
( .r `  R
) y )  e.  V )
29 eqid 2189 . . . . . . 7  |-  ( invr `  R )  =  (
invr `  R )
30 eqid 2189 . . . . . . 7  |-  ( invr `  S )  =  (
invr `  S )
311, 29, 3, 30subrginv 13551 . . . . . 6  |-  ( ( A  e.  (SubRing `  R
)  /\  x  e.  V )  ->  (
( invr `  R ) `  x )  =  ( ( invr `  S
) `  x )
)
323, 30unitinvcl 13440 . . . . . . 7  |-  ( ( S  e.  Ring  /\  x  e.  V )  ->  (
( invr `  S ) `  x )  e.  V
)
3313, 32sylan 283 . . . . . 6  |-  ( ( A  e.  (SubRing `  R
)  /\  x  e.  V )  ->  (
( invr `  S ) `  x )  e.  V
)
3431, 33eqeltrd 2266 . . . . 5  |-  ( ( A  e.  (SubRing `  R
)  /\  x  e.  V )  ->  (
( invr `  R ) `  x )  e.  V
)
3528, 34jca 306 . . . 4  |-  ( ( A  e.  (SubRing `  R
)  /\  x  e.  V )  ->  ( A. y  e.  V  ( x ( .r
`  R ) y )  e.  V  /\  ( ( invr `  R
) `  x )  e.  V ) )
3635ralrimiva 2563 . . 3  |-  ( A  e.  (SubRing `  R
)  ->  A. x  e.  V  ( A. y  e.  V  (
x ( .r `  R ) y )  e.  V  /\  (
( invr `  R ) `  x )  e.  V
) )
37 eqid 2189 . . . . . . . . . . 11  |-  (mulGrp `  R )  =  (mulGrp `  R )
3837, 18mgpplusgg 13245 . . . . . . . . . 10  |-  ( R  e.  Ring  ->  ( .r
`  R )  =  ( +g  `  (mulGrp `  R ) ) )
39 basfn 12544 . . . . . . . . . . . 12  |-  Base  Fn  _V
40 elex 2763 . . . . . . . . . . . 12  |-  ( R  e.  Ring  ->  R  e. 
_V )
41 funfvex 5547 . . . . . . . . . . . . 13  |-  ( ( Fun  Base  /\  R  e. 
dom  Base )  ->  ( Base `  R )  e. 
_V )
4241funfni 5331 . . . . . . . . . . . 12  |-  ( (
Base  Fn  _V  /\  R  e.  _V )  ->  ( Base `  R )  e. 
_V )
4339, 40, 42sylancr 414 . . . . . . . . . . 11  |-  ( R  e.  Ring  ->  ( Base `  R )  e.  _V )
44 eqidd 2190 . . . . . . . . . . . 12  |-  ( R  e.  Ring  ->  ( Base `  R )  =  (
Base `  R )
)
4544, 6, 9unitssd 13426 . . . . . . . . . . 11  |-  ( R  e.  Ring  ->  U  C_  ( Base `  R )
)
4643, 45ssexd 4158 . . . . . . . . . 10  |-  ( R  e.  Ring  ->  U  e. 
_V )
4737ringmgp 13323 . . . . . . . . . 10  |-  ( R  e.  Ring  ->  (mulGrp `  R )  e.  Mnd )
488, 38, 46, 47ressplusgd 12612 . . . . . . . . 9  |-  ( R  e.  Ring  ->  ( .r
`  R )  =  ( +g  `  G
) )
495, 48syl 14 . . . . . . . 8  |-  ( A  e.  (SubRing `  R
)  ->  ( .r `  R )  =  ( +g  `  G ) )
5049oveqd 5908 . . . . . . 7  |-  ( A  e.  (SubRing `  R
)  ->  ( x
( .r `  R
) y )  =  ( x ( +g  `  G ) y ) )
5150eleq1d 2258 . . . . . 6  |-  ( A  e.  (SubRing `  R
)  ->  ( (
x ( .r `  R ) y )  e.  V  <->  ( x
( +g  `  G ) y )  e.  V
) )
5251ralbidv 2490 . . . . 5  |-  ( A  e.  (SubRing `  R
)  ->  ( A. y  e.  V  (
x ( .r `  R ) y )  e.  V  <->  A. y  e.  V  ( x
( +g  `  G ) y )  e.  V
) )
532a1i 9 . . . . . . . 8  |-  ( A  e.  (SubRing `  R
)  ->  U  =  (Unit `  R ) )
547a1i 9 . . . . . . . 8  |-  ( A  e.  (SubRing `  R
)  ->  G  =  ( (mulGrp `  R )s  U
) )
55 eqidd 2190 . . . . . . . 8  |-  ( A  e.  (SubRing `  R
)  ->  ( invr `  R )  =  (
invr `  R )
)
5653, 54, 55, 5invrfvald 13439 . . . . . . 7  |-  ( A  e.  (SubRing `  R
)  ->  ( invr `  R )  =  ( invg `  G
) )
5756fveq1d 5532 . . . . . 6  |-  ( A  e.  (SubRing `  R
)  ->  ( ( invr `  R ) `  x )  =  ( ( invg `  G ) `  x
) )
5857eleq1d 2258 . . . . 5  |-  ( A  e.  (SubRing `  R
)  ->  ( (
( invr `  R ) `  x )  e.  V  <->  ( ( invg `  G ) `  x
)  e.  V ) )
5952, 58anbi12d 473 . . . 4  |-  ( A  e.  (SubRing `  R
)  ->  ( ( A. y  e.  V  ( x ( .r
`  R ) y )  e.  V  /\  ( ( invr `  R
) `  x )  e.  V )  <->  ( A. y  e.  V  (
x ( +g  `  G
) y )  e.  V  /\  ( ( invg `  G
) `  x )  e.  V ) ) )
6059ralbidv 2490 . . 3  |-  ( A  e.  (SubRing `  R
)  ->  ( A. x  e.  V  ( A. y  e.  V  ( x ( .r
`  R ) y )  e.  V  /\  ( ( invr `  R
) `  x )  e.  V )  <->  A. x  e.  V  ( A. y  e.  V  (
x ( +g  `  G
) y )  e.  V  /\  ( ( invg `  G
) `  x )  e.  V ) ) )
6136, 60mpbid 147 . 2  |-  ( A  e.  (SubRing `  R
)  ->  A. x  e.  V  ( A. y  e.  V  (
x ( +g  `  G
) y )  e.  V  /\  ( ( invg `  G
) `  x )  e.  V ) )
622, 7unitgrp 13433 . . 3  |-  ( R  e.  Ring  ->  G  e. 
Grp )
63 eqid 2189 . . . 4  |-  ( Base `  G )  =  (
Base `  G )
64 eqid 2189 . . . 4  |-  ( +g  `  G )  =  ( +g  `  G )
65 eqid 2189 . . . 4  |-  ( invg `  G )  =  ( invg `  G )
6663, 64, 65issubg2m 13100 . . 3  |-  ( G  e.  Grp  ->  ( V  e.  (SubGrp `  G
)  <->  ( V  C_  ( Base `  G )  /\  E. w  w  e.  V  /\  A. x  e.  V  ( A. y  e.  V  (
x ( +g  `  G
) y )  e.  V  /\  ( ( invg `  G
) `  x )  e.  V ) ) ) )
675, 62, 663syl 17 . 2  |-  ( A  e.  (SubRing `  R
)  ->  ( V  e.  (SubGrp `  G )  <->  ( V  C_  ( Base `  G )  /\  E. w  w  e.  V  /\  A. x  e.  V  ( A. y  e.  V  ( x ( +g  `  G ) y )  e.  V  /\  (
( invg `  G ) `  x
)  e.  V ) ) ) )
6812, 17, 61, 67mpbir3and 1182 1  |-  ( A  e.  (SubRing `  R
)  ->  V  e.  (SubGrp `  G ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    <-> wb 105    /\ w3a 980    = wceq 1364   E.wex 1503    e. wcel 2160   A.wral 2468   _Vcvv 2752    C_ wss 3144    Fn wfn 5226   ` cfv 5231  (class class class)co 5891   Basecbs 12486   ↾s cress 12487   +g cplusg 12561   .rcmulr 12562   Mndcmnd 12849   Grpcgrp 12917   invgcminusg 12918  SubGrpcsubg 13078  mulGrpcmgp 13241   1rcur 13280   Ringcrg 13317  Unitcui 13404   invrcinvr 13437  SubRingcsubrg 13531
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-coll 4133  ax-sep 4136  ax-nul 4144  ax-pow 4189  ax-pr 4224  ax-un 4448  ax-setind 4551  ax-cnex 7921  ax-resscn 7922  ax-1cn 7923  ax-1re 7924  ax-icn 7925  ax-addcl 7926  ax-addrcl 7927  ax-mulcl 7928  ax-addcom 7930  ax-addass 7932  ax-i2m1 7935  ax-0lt1 7936  ax-0id 7938  ax-rnegex 7939  ax-pre-ltirr 7942  ax-pre-lttrn 7944  ax-pre-ltadd 7946
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-nel 2456  df-ral 2473  df-rex 2474  df-reu 2475  df-rmo 2476  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-id 4308  df-xp 4647  df-rel 4648  df-cnv 4649  df-co 4650  df-dm 4651  df-rn 4652  df-res 4653  df-ima 4654  df-iota 5193  df-fun 5233  df-fn 5234  df-f 5235  df-f1 5236  df-fo 5237  df-f1o 5238  df-fv 5239  df-riota 5847  df-ov 5894  df-oprab 5895  df-mpo 5896  df-tpos 6264  df-pnf 8013  df-mnf 8014  df-ltxr 8016  df-inn 8939  df-2 8997  df-3 8998  df-ndx 12489  df-slot 12490  df-base 12492  df-sets 12493  df-iress 12494  df-plusg 12574  df-mulr 12575  df-0g 12735  df-mgm 12804  df-sgrp 12837  df-mnd 12850  df-grp 12920  df-minusg 12921  df-subg 13081  df-cmn 13192  df-abl 13193  df-mgp 13242  df-ur 13281  df-srg 13285  df-ring 13319  df-oppr 13385  df-dvdsr 13406  df-unit 13407  df-invr 13438  df-subrg 13533
This theorem is referenced by: (None)
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