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Theorem subrgugrp 13455
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 13451 . . 3  |-  ( A  e.  (SubRing `  R
)  ->  V  C_  U
)
5 subrgrcl 13441 . . . 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 13292 . . . . 5  |-  ( R  e.  Ring  ->  R  e. SRing
)
106, 8, 9unitgrpbasd 13358 . . . 4  |-  ( R  e.  Ring  ->  U  =  ( Base `  G
) )
115, 10syl 14 . . 3  |-  ( A  e.  (SubRing `  R
)  ->  U  =  ( Base `  G )
)
124, 11sseqtrd 3205 . 2  |-  ( A  e.  (SubRing `  R
)  ->  V  C_  ( Base `  G ) )
131subrgring 13439 . . 3  |-  ( A  e.  (SubRing `  R
)  ->  S  e.  Ring )
14 eqid 2187 . . . 4  |-  ( 1r
`  S )  =  ( 1r `  S
)
153, 141unit 13350 . . 3  |-  ( S  e.  Ring  ->  ( 1r
`  S )  e.  V )
16 elex2 2765 . . 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 2187 . . . . . . . . . . . 12  |-  ( .r
`  R )  =  ( .r `  R
)
191, 18ressmulrg 12617 . . . . . . . . . . 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 1019 . . . . . . . . 9  |-  ( ( A  e.  (SubRing `  R
)  /\  x  e.  V  /\  y  e.  V
)  ->  ( .r `  R )  =  ( .r `  S ) )
2221oveqd 5905 . . . . . . . 8  |-  ( ( A  e.  (SubRing `  R
)  /\  x  e.  V  /\  y  e.  V
)  ->  ( x
( .r `  R
) y )  =  ( x ( .r
`  S ) y ) )
23 eqid 2187 . . . . . . . . . 10  |-  ( .r
`  S )  =  ( .r `  S
)
243, 23unitmulcl 13356 . . . . . . . . 9  |-  ( ( S  e.  Ring  /\  x  e.  V  /\  y  e.  V )  ->  (
x ( .r `  S ) y )  e.  V )
2513, 24syl3an1 1281 . . . . . . . 8  |-  ( ( A  e.  (SubRing `  R
)  /\  x  e.  V  /\  y  e.  V
)  ->  ( x
( .r `  S
) y )  e.  V )
2622, 25eqeltrd 2264 . . . . . . 7  |-  ( ( A  e.  (SubRing `  R
)  /\  x  e.  V  /\  y  e.  V
)  ->  ( x
( .r `  R
) y )  e.  V )
27263expa 1204 . . . . . 6  |-  ( ( ( A  e.  (SubRing `  R )  /\  x  e.  V )  /\  y  e.  V )  ->  (
x ( .r `  R ) y )  e.  V )
2827ralrimiva 2560 . . . . 5  |-  ( ( A  e.  (SubRing `  R
)  /\  x  e.  V )  ->  A. y  e.  V  ( x
( .r `  R
) y )  e.  V )
29 eqid 2187 . . . . . . 7  |-  ( invr `  R )  =  (
invr `  R )
30 eqid 2187 . . . . . . 7  |-  ( invr `  S )  =  (
invr `  S )
311, 29, 3, 30subrginv 13452 . . . . . 6  |-  ( ( A  e.  (SubRing `  R
)  /\  x  e.  V )  ->  (
( invr `  R ) `  x )  =  ( ( invr `  S
) `  x )
)
323, 30unitinvcl 13366 . . . . . . 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 2264 . . . . 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 2560 . . 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 2187 . . . . . . . . . . 11  |-  (mulGrp `  R )  =  (mulGrp `  R )
3837, 18mgpplusgg 13174 . . . . . . . . . 10  |-  ( R  e.  Ring  ->  ( .r
`  R )  =  ( +g  `  (mulGrp `  R ) ) )
39 basfn 12533 . . . . . . . . . . . 12  |-  Base  Fn  _V
40 elex 2760 . . . . . . . . . . . 12  |-  ( R  e.  Ring  ->  R  e. 
_V )
41 funfvex 5544 . . . . . . . . . . . . 13  |-  ( ( Fun  Base  /\  R  e. 
dom  Base )  ->  ( Base `  R )  e. 
_V )
4241funfni 5328 . . . . . . . . . . . 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 2188 . . . . . . . . . . . 12  |-  ( R  e.  Ring  ->  ( Base `  R )  =  (
Base `  R )
)
4544, 6, 9unitssd 13352 . . . . . . . . . . 11  |-  ( R  e.  Ring  ->  U  C_  ( Base `  R )
)
4643, 45ssexd 4155 . . . . . . . . . 10  |-  ( R  e.  Ring  ->  U  e. 
_V )
4737ringmgp 13249 . . . . . . . . . 10  |-  ( R  e.  Ring  ->  (mulGrp `  R )  e.  Mnd )
488, 38, 46, 47ressplusgd 12601 . . . . . . . . 9  |-  ( R  e.  Ring  ->  ( .r
`  R )  =  ( +g  `  G
) )
495, 48syl 14 . . . . . . . 8  |-  ( A  e.  (SubRing `  R
)  ->  ( .r `  R )  =  ( +g  `  G ) )
5049oveqd 5905 . . . . . . 7  |-  ( A  e.  (SubRing `  R
)  ->  ( x
( .r `  R
) y )  =  ( x ( +g  `  G ) y ) )
5150eleq1d 2256 . . . . . 6  |-  ( A  e.  (SubRing `  R
)  ->  ( (
x ( .r `  R ) y )  e.  V  <->  ( x
( +g  `  G ) y )  e.  V
) )
5251ralbidv 2487 . . . . 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 2188 . . . . . . . 8  |-  ( A  e.  (SubRing `  R
)  ->  ( invr `  R )  =  (
invr `  R )
)
5653, 54, 55, 5invrfvald 13365 . . . . . . 7  |-  ( A  e.  (SubRing `  R
)  ->  ( invr `  R )  =  ( invg `  G
) )
5756fveq1d 5529 . . . . . 6  |-  ( A  e.  (SubRing `  R
)  ->  ( ( invr `  R ) `  x )  =  ( ( invg `  G ) `  x
) )
5857eleq1d 2256 . . . . 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 2487 . . 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 13359 . . 3  |-  ( R  e.  Ring  ->  G  e. 
Grp )
63 eqid 2187 . . . 4  |-  ( Base `  G )  =  (
Base `  G )
64 eqid 2187 . . . 4  |-  ( +g  `  G )  =  ( +g  `  G )
65 eqid 2187 . . . 4  |-  ( invg `  G )  =  ( invg `  G )
6663, 64, 65issubg2m 13080 . . 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 1181 1  |-  ( A  e.  (SubRing `  R
)  ->  V  e.  (SubGrp `  G ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    <-> wb 105    /\ w3a 979    = wceq 1363   E.wex 1502    e. wcel 2158   A.wral 2465   _Vcvv 2749    C_ wss 3141    Fn wfn 5223   ` cfv 5228  (class class class)co 5888   Basecbs 12475   ↾s cress 12476   +g cplusg 12550   .rcmulr 12551   Mndcmnd 12838   Grpcgrp 12898   invgcminusg 12899  SubGrpcsubg 13058  mulGrpcmgp 13170   1rcur 13206   Ringcrg 13243  Unitcui 13330   invrcinvr 13363  SubRingcsubrg 13432
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 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-coll 4130  ax-sep 4133  ax-nul 4141  ax-pow 4186  ax-pr 4221  ax-un 4445  ax-setind 4548  ax-cnex 7915  ax-resscn 7916  ax-1cn 7917  ax-1re 7918  ax-icn 7919  ax-addcl 7920  ax-addrcl 7921  ax-mulcl 7922  ax-addcom 7924  ax-addass 7926  ax-i2m1 7929  ax-0lt1 7930  ax-0id 7932  ax-rnegex 7933  ax-pre-ltirr 7936  ax-pre-lttrn 7938  ax-pre-ltadd 7940
This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ne 2358  df-nel 2453  df-ral 2470  df-rex 2471  df-reu 2472  df-rmo 2473  df-rab 2474  df-v 2751  df-sbc 2975  df-csb 3070  df-dif 3143  df-un 3145  df-in 3147  df-ss 3154  df-nul 3435  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-int 3857  df-iun 3900  df-br 4016  df-opab 4077  df-mpt 4078  df-id 4305  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-rn 4649  df-res 4650  df-ima 4651  df-iota 5190  df-fun 5230  df-fn 5231  df-f 5232  df-f1 5233  df-fo 5234  df-f1o 5235  df-fv 5236  df-riota 5844  df-ov 5891  df-oprab 5892  df-mpo 5893  df-tpos 6259  df-pnf 8007  df-mnf 8008  df-ltxr 8010  df-inn 8933  df-2 8991  df-3 8992  df-ndx 12478  df-slot 12479  df-base 12481  df-sets 12482  df-iress 12483  df-plusg 12563  df-mulr 12564  df-0g 12724  df-mgm 12793  df-sgrp 12826  df-mnd 12839  df-grp 12901  df-minusg 12902  df-subg 13061  df-cmn 13122  df-abl 13123  df-mgp 13171  df-ur 13207  df-srg 13211  df-ring 13245  df-oppr 13311  df-dvdsr 13332  df-unit 13333  df-invr 13364  df-subrg 13434
This theorem is referenced by: (None)
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