ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  subrgugrp Unicode version

Theorem subrgugrp 13796
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 13792 . . 3  |-  ( A  e.  (SubRing `  R
)  ->  V  C_  U
)
5 subrgrcl 13782 . . . 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 13603 . . . . 5  |-  ( R  e.  Ring  ->  R  e. SRing
)
106, 8, 9unitgrpbasd 13671 . . . 4  |-  ( R  e.  Ring  ->  U  =  ( Base `  G
) )
115, 10syl 14 . . 3  |-  ( A  e.  (SubRing `  R
)  ->  U  =  ( Base `  G )
)
124, 11sseqtrd 3221 . 2  |-  ( A  e.  (SubRing `  R
)  ->  V  C_  ( Base `  G ) )
131subrgring 13780 . . 3  |-  ( A  e.  (SubRing `  R
)  ->  S  e.  Ring )
14 eqid 2196 . . . 4  |-  ( 1r
`  S )  =  ( 1r `  S
)
153, 141unit 13663 . . 3  |-  ( S  e.  Ring  ->  ( 1r
`  S )  e.  V )
16 elex2 2779 . . 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 2196 . . . . . . . . . . . 12  |-  ( .r
`  R )  =  ( .r `  R
)
191, 18ressmulrg 12822 . . . . . . . . . . 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 5939 . . . . . . . 8  |-  ( ( A  e.  (SubRing `  R
)  /\  x  e.  V  /\  y  e.  V
)  ->  ( x
( .r `  R
) y )  =  ( x ( .r
`  S ) y ) )
23 eqid 2196 . . . . . . . . . 10  |-  ( .r
`  S )  =  ( .r `  S
)
243, 23unitmulcl 13669 . . . . . . . . 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 2273 . . . . . . 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 2570 . . . . 5  |-  ( ( A  e.  (SubRing `  R
)  /\  x  e.  V )  ->  A. y  e.  V  ( x
( .r `  R
) y )  e.  V )
29 eqid 2196 . . . . . . 7  |-  ( invr `  R )  =  (
invr `  R )
30 eqid 2196 . . . . . . 7  |-  ( invr `  S )  =  (
invr `  S )
311, 29, 3, 30subrginv 13793 . . . . . 6  |-  ( ( A  e.  (SubRing `  R
)  /\  x  e.  V )  ->  (
( invr `  R ) `  x )  =  ( ( invr `  S
) `  x )
)
323, 30unitinvcl 13679 . . . . . . 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 2273 . . . . 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 2570 . . 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 2196 . . . . . . . . . . 11  |-  (mulGrp `  R )  =  (mulGrp `  R )
3837, 18mgpplusgg 13480 . . . . . . . . . 10  |-  ( R  e.  Ring  ->  ( .r
`  R )  =  ( +g  `  (mulGrp `  R ) ) )
39 basfn 12736 . . . . . . . . . . . 12  |-  Base  Fn  _V
40 elex 2774 . . . . . . . . . . . 12  |-  ( R  e.  Ring  ->  R  e. 
_V )
41 funfvex 5575 . . . . . . . . . . . . 13  |-  ( ( Fun  Base  /\  R  e. 
dom  Base )  ->  ( Base `  R )  e. 
_V )
4241funfni 5358 . . . . . . . . . . . 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 2197 . . . . . . . . . . . 12  |-  ( R  e.  Ring  ->  ( Base `  R )  =  (
Base `  R )
)
4544, 6, 9unitssd 13665 . . . . . . . . . . 11  |-  ( R  e.  Ring  ->  U  C_  ( Base `  R )
)
4643, 45ssexd 4173 . . . . . . . . . 10  |-  ( R  e.  Ring  ->  U  e. 
_V )
4737ringmgp 13558 . . . . . . . . . 10  |-  ( R  e.  Ring  ->  (mulGrp `  R )  e.  Mnd )
488, 38, 46, 47ressplusgd 12806 . . . . . . . . 9  |-  ( R  e.  Ring  ->  ( .r
`  R )  =  ( +g  `  G
) )
495, 48syl 14 . . . . . . . 8  |-  ( A  e.  (SubRing `  R
)  ->  ( .r `  R )  =  ( +g  `  G ) )
5049oveqd 5939 . . . . . . 7  |-  ( A  e.  (SubRing `  R
)  ->  ( x
( .r `  R
) y )  =  ( x ( +g  `  G ) y ) )
5150eleq1d 2265 . . . . . 6  |-  ( A  e.  (SubRing `  R
)  ->  ( (
x ( .r `  R ) y )  e.  V  <->  ( x
( +g  `  G ) y )  e.  V
) )
5251ralbidv 2497 . . . . 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 2197 . . . . . . . 8  |-  ( A  e.  (SubRing `  R
)  ->  ( invr `  R )  =  (
invr `  R )
)
5653, 54, 55, 5invrfvald 13678 . . . . . . 7  |-  ( A  e.  (SubRing `  R
)  ->  ( invr `  R )  =  ( invg `  G
) )
5756fveq1d 5560 . . . . . 6  |-  ( A  e.  (SubRing `  R
)  ->  ( ( invr `  R ) `  x )  =  ( ( invg `  G ) `  x
) )
5857eleq1d 2265 . . . . 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 2497 . . 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 13672 . . 3  |-  ( R  e.  Ring  ->  G  e. 
Grp )
63 eqid 2196 . . . 4  |-  ( Base `  G )  =  (
Base `  G )
64 eqid 2196 . . . 4  |-  ( +g  `  G )  =  ( +g  `  G )
65 eqid 2196 . . . 4  |-  ( invg `  G )  =  ( invg `  G )
6663, 64, 65issubg2m 13319 . . 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 1506    e. wcel 2167   A.wral 2475   _Vcvv 2763    C_ wss 3157    Fn wfn 5253   ` cfv 5258  (class class class)co 5922   Basecbs 12678   ↾s cress 12679   +g cplusg 12755   .rcmulr 12756   Mndcmnd 13057   Grpcgrp 13132   invgcminusg 13133  SubGrpcsubg 13297  mulGrpcmgp 13476   1rcur 13515   Ringcrg 13552  Unitcui 13643   invrcinvr 13676  SubRingcsubrg 13773
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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4148  ax-sep 4151  ax-nul 4159  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573  ax-cnex 7970  ax-resscn 7971  ax-1cn 7972  ax-1re 7973  ax-icn 7974  ax-addcl 7975  ax-addrcl 7976  ax-mulcl 7977  ax-addcom 7979  ax-addass 7981  ax-i2m1 7984  ax-0lt1 7985  ax-0id 7987  ax-rnegex 7988  ax-pre-ltirr 7991  ax-pre-lttrn 7993  ax-pre-ltadd 7995
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rmo 2483  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3451  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-int 3875  df-iun 3918  df-br 4034  df-opab 4095  df-mpt 4096  df-id 4328  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-f1 5263  df-fo 5264  df-f1o 5265  df-fv 5266  df-riota 5877  df-ov 5925  df-oprab 5926  df-mpo 5927  df-tpos 6303  df-pnf 8063  df-mnf 8064  df-ltxr 8066  df-inn 8991  df-2 9049  df-3 9050  df-ndx 12681  df-slot 12682  df-base 12684  df-sets 12685  df-iress 12686  df-plusg 12768  df-mulr 12769  df-0g 12929  df-mgm 12999  df-sgrp 13045  df-mnd 13058  df-grp 13135  df-minusg 13136  df-subg 13300  df-cmn 13416  df-abl 13417  df-mgp 13477  df-ur 13516  df-srg 13520  df-ring 13554  df-oppr 13624  df-dvdsr 13645  df-unit 13646  df-invr 13677  df-subrg 13775
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator