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Theorem subrgugrp 13372
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 13368 . . 3  |-  ( A  e.  (SubRing `  R
)  ->  V  C_  U
)
5 subrgrcl 13358 . . . 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 13235 . . . . 5  |-  ( R  e.  Ring  ->  R  e. SRing
)
106, 8, 9unitgrpbasd 13295 . . . 4  |-  ( R  e.  Ring  ->  U  =  ( Base `  G
) )
115, 10syl 14 . . 3  |-  ( A  e.  (SubRing `  R
)  ->  U  =  ( Base `  G )
)
124, 11sseqtrd 3195 . 2  |-  ( A  e.  (SubRing `  R
)  ->  V  C_  ( Base `  G ) )
131subrgring 13356 . . 3  |-  ( A  e.  (SubRing `  R
)  ->  S  e.  Ring )
14 eqid 2177 . . . 4  |-  ( 1r
`  S )  =  ( 1r `  S
)
153, 141unit 13287 . . 3  |-  ( S  e.  Ring  ->  ( 1r
`  S )  e.  V )
16 elex2 2755 . . 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 2177 . . . . . . . . . . . 12  |-  ( .r
`  R )  =  ( .r `  R
)
191, 18ressmulrg 12606 . . . . . . . . . . 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 1018 . . . . . . . . 9  |-  ( ( A  e.  (SubRing `  R
)  /\  x  e.  V  /\  y  e.  V
)  ->  ( .r `  R )  =  ( .r `  S ) )
2221oveqd 5895 . . . . . . . 8  |-  ( ( A  e.  (SubRing `  R
)  /\  x  e.  V  /\  y  e.  V
)  ->  ( x
( .r `  R
) y )  =  ( x ( .r
`  S ) y ) )
23 eqid 2177 . . . . . . . . . 10  |-  ( .r
`  S )  =  ( .r `  S
)
243, 23unitmulcl 13293 . . . . . . . . 9  |-  ( ( S  e.  Ring  /\  x  e.  V  /\  y  e.  V )  ->  (
x ( .r `  S ) y )  e.  V )
2513, 24syl3an1 1271 . . . . . . . 8  |-  ( ( A  e.  (SubRing `  R
)  /\  x  e.  V  /\  y  e.  V
)  ->  ( x
( .r `  S
) y )  e.  V )
2622, 25eqeltrd 2254 . . . . . . 7  |-  ( ( A  e.  (SubRing `  R
)  /\  x  e.  V  /\  y  e.  V
)  ->  ( x
( .r `  R
) y )  e.  V )
27263expa 1203 . . . . . 6  |-  ( ( ( A  e.  (SubRing `  R )  /\  x  e.  V )  /\  y  e.  V )  ->  (
x ( .r `  R ) y )  e.  V )
2827ralrimiva 2550 . . . . 5  |-  ( ( A  e.  (SubRing `  R
)  /\  x  e.  V )  ->  A. y  e.  V  ( x
( .r `  R
) y )  e.  V )
29 eqid 2177 . . . . . . 7  |-  ( invr `  R )  =  (
invr `  R )
30 eqid 2177 . . . . . . 7  |-  ( invr `  S )  =  (
invr `  S )
311, 29, 3, 30subrginv 13369 . . . . . 6  |-  ( ( A  e.  (SubRing `  R
)  /\  x  e.  V )  ->  (
( invr `  R ) `  x )  =  ( ( invr `  S
) `  x )
)
323, 30unitinvcl 13303 . . . . . . 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 2254 . . . . 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 2550 . . 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 2177 . . . . . . . . . . 11  |-  (mulGrp `  R )  =  (mulGrp `  R )
3837, 18mgpplusgg 13145 . . . . . . . . . 10  |-  ( R  e.  Ring  ->  ( .r
`  R )  =  ( +g  `  (mulGrp `  R ) ) )
39 basfn 12523 . . . . . . . . . . . 12  |-  Base  Fn  _V
40 elex 2750 . . . . . . . . . . . 12  |-  ( R  e.  Ring  ->  R  e. 
_V )
41 funfvex 5534 . . . . . . . . . . . . 13  |-  ( ( Fun  Base  /\  R  e. 
dom  Base )  ->  ( Base `  R )  e. 
_V )
4241funfni 5318 . . . . . . . . . . . 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 2178 . . . . . . . . . . . 12  |-  ( R  e.  Ring  ->  ( Base `  R )  =  (
Base `  R )
)
4544, 6, 9unitssd 13289 . . . . . . . . . . 11  |-  ( R  e.  Ring  ->  U  C_  ( Base `  R )
)
4643, 45ssexd 4145 . . . . . . . . . 10  |-  ( R  e.  Ring  ->  U  e. 
_V )
4737ringmgp 13196 . . . . . . . . . 10  |-  ( R  e.  Ring  ->  (mulGrp `  R )  e.  Mnd )
488, 38, 46, 47ressplusgd 12590 . . . . . . . . 9  |-  ( R  e.  Ring  ->  ( .r
`  R )  =  ( +g  `  G
) )
495, 48syl 14 . . . . . . . 8  |-  ( A  e.  (SubRing `  R
)  ->  ( .r `  R )  =  ( +g  `  G ) )
5049oveqd 5895 . . . . . . 7  |-  ( A  e.  (SubRing `  R
)  ->  ( x
( .r `  R
) y )  =  ( x ( +g  `  G ) y ) )
5150eleq1d 2246 . . . . . 6  |-  ( A  e.  (SubRing `  R
)  ->  ( (
x ( .r `  R ) y )  e.  V  <->  ( x
( +g  `  G ) y )  e.  V
) )
5251ralbidv 2477 . . . . 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 2178 . . . . . . . 8  |-  ( A  e.  (SubRing `  R
)  ->  ( invr `  R )  =  (
invr `  R )
)
5653, 54, 55, 5invrfvald 13302 . . . . . . 7  |-  ( A  e.  (SubRing `  R
)  ->  ( invr `  R )  =  ( invg `  G
) )
5756fveq1d 5519 . . . . . 6  |-  ( A  e.  (SubRing `  R
)  ->  ( ( invr `  R ) `  x )  =  ( ( invg `  G ) `  x
) )
5857eleq1d 2246 . . . . 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 2477 . . 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 13296 . . 3  |-  ( R  e.  Ring  ->  G  e. 
Grp )
63 eqid 2177 . . . 4  |-  ( Base `  G )  =  (
Base `  G )
64 eqid 2177 . . . 4  |-  ( +g  `  G )  =  ( +g  `  G )
65 eqid 2177 . . . 4  |-  ( invg `  G )  =  ( invg `  G )
6663, 64, 65issubg2m 13059 . . 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 1180 1  |-  ( A  e.  (SubRing `  R
)  ->  V  e.  (SubGrp `  G ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    <-> wb 105    /\ w3a 978    = wceq 1353   E.wex 1492    e. wcel 2148   A.wral 2455   _Vcvv 2739    C_ wss 3131    Fn wfn 5213   ` cfv 5218  (class class class)co 5878   Basecbs 12465   ↾s cress 12466   +g cplusg 12539   .rcmulr 12540   Mndcmnd 12824   Grpcgrp 12884   invgcminusg 12885  SubGrpcsubg 13037  mulGrpcmgp 13141   1rcur 13153   Ringcrg 13190  Unitcui 13267   invrcinvr 13300  SubRingcsubrg 13349
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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4120  ax-sep 4123  ax-nul 4131  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-cnex 7905  ax-resscn 7906  ax-1cn 7907  ax-1re 7908  ax-icn 7909  ax-addcl 7910  ax-addrcl 7911  ax-mulcl 7912  ax-addcom 7914  ax-addass 7916  ax-i2m1 7919  ax-0lt1 7920  ax-0id 7922  ax-rnegex 7923  ax-pre-ltirr 7926  ax-pre-lttrn 7928  ax-pre-ltadd 7930
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-iun 3890  df-br 4006  df-opab 4067  df-mpt 4068  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225  df-fv 5226  df-riota 5834  df-ov 5881  df-oprab 5882  df-mpo 5883  df-tpos 6249  df-pnf 7997  df-mnf 7998  df-ltxr 8000  df-inn 8923  df-2 8981  df-3 8982  df-ndx 12468  df-slot 12469  df-base 12471  df-sets 12472  df-iress 12473  df-plusg 12552  df-mulr 12553  df-0g 12713  df-mgm 12782  df-sgrp 12815  df-mnd 12825  df-grp 12887  df-minusg 12888  df-subg 13040  df-cmn 13101  df-abl 13102  df-mgp 13142  df-ur 13154  df-srg 13158  df-ring 13192  df-oppr 13251  df-dvdsr 13269  df-unit 13270  df-invr 13301  df-subrg 13351
This theorem is referenced by: (None)
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