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Theorem opprsubg 17098
Description: Being a subgroup is a symmetric property. (Contributed by Mario Carneiro, 6-Dec-2014.)
Hypothesis
Ref Expression
opprbas.1  |-  O  =  (oppr
`  R )
Assertion
Ref Expression
opprsubg  |-  (SubGrp `  R )  =  (SubGrp `  O )

Proof of Theorem opprsubg
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 opprbas.1 . . . . . 6  |-  O  =  (oppr
`  R )
2 eqid 2467 . . . . . 6  |-  ( Base `  R )  =  (
Base `  R )
31, 2opprbas 17091 . . . . 5  |-  ( Base `  R )  =  (
Base `  O )
4 eqid 2467 . . . . . 6  |-  ( +g  `  R )  =  ( +g  `  R )
51, 4oppradd 17092 . . . . 5  |-  ( +g  `  R )  =  ( +g  `  O )
63, 5grpprop 15883 . . . 4  |-  ( R  e.  Grp  <->  O  e.  Grp )
7 biid 236 . . . 4  |-  ( x 
C_  ( Base `  R
)  <->  x  C_  ( Base `  R ) )
8 vex 3116 . . . . . . 7  |-  x  e. 
_V
9 eqid 2467 . . . . . . . 8  |-  ( Rs  x )  =  ( Rs  x )
109, 2ressbas 14548 . . . . . . 7  |-  ( x  e.  _V  ->  (
x  i^i  ( Base `  R ) )  =  ( Base `  ( Rs  x ) ) )
118, 10ax-mp 5 . . . . . 6  |-  ( x  i^i  ( Base `  R
) )  =  (
Base `  ( Rs  x
) )
12 eqid 2467 . . . . . . . 8  |-  ( Os  x )  =  ( Os  x )
1312, 3ressbas 14548 . . . . . . 7  |-  ( x  e.  _V  ->  (
x  i^i  ( Base `  R ) )  =  ( Base `  ( Os  x ) ) )
148, 13ax-mp 5 . . . . . 6  |-  ( x  i^i  ( Base `  R
) )  =  (
Base `  ( Os  x
) )
1511, 14eqtr3i 2498 . . . . 5  |-  ( Base `  ( Rs  x ) )  =  ( Base `  ( Os  x ) )
169, 4ressplusg 14600 . . . . . . 7  |-  ( x  e.  _V  ->  ( +g  `  R )  =  ( +g  `  ( Rs  x ) ) )
1712, 5ressplusg 14600 . . . . . . 7  |-  ( x  e.  _V  ->  ( +g  `  R )  =  ( +g  `  ( Os  x ) ) )
1816, 17eqtr3d 2510 . . . . . 6  |-  ( x  e.  _V  ->  ( +g  `  ( Rs  x ) )  =  ( +g  `  ( Os  x ) ) )
198, 18ax-mp 5 . . . . 5  |-  ( +g  `  ( Rs  x ) )  =  ( +g  `  ( Os  x ) )
2015, 19grpprop 15883 . . . 4  |-  ( ( Rs  x )  e.  Grp  <->  ( Os  x )  e.  Grp )
216, 7, 203anbi123i 1185 . . 3  |-  ( ( R  e.  Grp  /\  x  C_  ( Base `  R
)  /\  ( Rs  x
)  e.  Grp )  <->  ( O  e.  Grp  /\  x  C_  ( Base `  R
)  /\  ( Os  x
)  e.  Grp )
)
222issubg 16015 . . 3  |-  ( x  e.  (SubGrp `  R
)  <->  ( R  e. 
Grp  /\  x  C_  ( Base `  R )  /\  ( Rs  x )  e.  Grp ) )
233issubg 16015 . . 3  |-  ( x  e.  (SubGrp `  O
)  <->  ( O  e. 
Grp  /\  x  C_  ( Base `  R )  /\  ( Os  x )  e.  Grp ) )
2421, 22, 233bitr4i 277 . 2  |-  ( x  e.  (SubGrp `  R
)  <->  x  e.  (SubGrp `  O ) )
2524eqriv 2463 1  |-  (SubGrp `  R )  =  (SubGrp `  O )
Colors of variables: wff setvar class
Syntax hints:    /\ w3a 973    = wceq 1379    e. wcel 1767   _Vcvv 3113    i^i cin 3475    C_ wss 3476   ` cfv 5588  (class class class)co 6285   Basecbs 14493   ↾s cress 14494   +g cplusg 14558   Grpcgrp 15730  SubGrpcsubg 16009  opprcoppr 17084
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6577  ax-cnex 9549  ax-resscn 9550  ax-1cn 9551  ax-icn 9552  ax-addcl 9553  ax-addrcl 9554  ax-mulcl 9555  ax-mulrcl 9556  ax-mulcom 9557  ax-addass 9558  ax-mulass 9559  ax-distr 9560  ax-i2m1 9561  ax-1ne0 9562  ax-1rid 9563  ax-rnegex 9564  ax-rrecex 9565  ax-cnre 9566  ax-pre-lttri 9567  ax-pre-lttrn 9568  ax-pre-ltadd 9569  ax-pre-mulgt0 9570
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-riota 6246  df-ov 6288  df-oprab 6289  df-mpt2 6290  df-om 6686  df-tpos 6956  df-recs 7043  df-rdg 7077  df-er 7312  df-en 7518  df-dom 7519  df-sdom 7520  df-pnf 9631  df-mnf 9632  df-xr 9633  df-ltxr 9634  df-le 9635  df-sub 9808  df-neg 9809  df-nn 10538  df-2 10595  df-3 10596  df-ndx 14496  df-slot 14497  df-base 14498  df-sets 14499  df-ress 14500  df-plusg 14571  df-mulr 14572  df-0g 14700  df-mnd 15735  df-grp 15871  df-subg 16012  df-oppr 17085
This theorem is referenced by:  opprsubrg  17262
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