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Theorem opprsubg 15418
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 2283 . . . . . 6  |-  ( Base `  R )  =  (
Base `  R )
31, 2opprbas 15411 . . . . 5  |-  ( Base `  R )  =  (
Base `  O )
4 eqid 2283 . . . . . 6  |-  ( +g  `  R )  =  ( +g  `  R )
51, 4oppradd 15412 . . . . 5  |-  ( +g  `  R )  =  ( +g  `  O )
63, 5grpprop 14501 . . . 4  |-  ( R  e.  Grp  <->  O  e.  Grp )
7 biid 227 . . . 4  |-  ( x 
C_  ( Base `  R
)  <->  x  C_  ( Base `  R ) )
8 vex 2791 . . . . . . 7  |-  x  e. 
_V
9 eqid 2283 . . . . . . . 8  |-  ( Rs  x )  =  ( Rs  x )
109, 2ressbas 13198 . . . . . . 7  |-  ( x  e.  _V  ->  (
x  i^i  ( Base `  R ) )  =  ( Base `  ( Rs  x ) ) )
118, 10ax-mp 8 . . . . . 6  |-  ( x  i^i  ( Base `  R
) )  =  (
Base `  ( Rs  x
) )
12 eqid 2283 . . . . . . . 8  |-  ( Os  x )  =  ( Os  x )
1312, 3ressbas 13198 . . . . . . 7  |-  ( x  e.  _V  ->  (
x  i^i  ( Base `  R ) )  =  ( Base `  ( Os  x ) ) )
148, 13ax-mp 8 . . . . . 6  |-  ( x  i^i  ( Base `  R
) )  =  (
Base `  ( Os  x
) )
1511, 14eqtr3i 2305 . . . . 5  |-  ( Base `  ( Rs  x ) )  =  ( Base `  ( Os  x ) )
169, 4ressplusg 13250 . . . . . . 7  |-  ( x  e.  _V  ->  ( +g  `  R )  =  ( +g  `  ( Rs  x ) ) )
1712, 5ressplusg 13250 . . . . . . 7  |-  ( x  e.  _V  ->  ( +g  `  R )  =  ( +g  `  ( Os  x ) ) )
1816, 17eqtr3d 2317 . . . . . 6  |-  ( x  e.  _V  ->  ( +g  `  ( Rs  x ) )  =  ( +g  `  ( Os  x ) ) )
198, 18ax-mp 8 . . . . 5  |-  ( +g  `  ( Rs  x ) )  =  ( +g  `  ( Os  x ) )
2015, 19grpprop 14501 . . . 4  |-  ( ( Rs  x )  e.  Grp  <->  ( Os  x )  e.  Grp )
216, 7, 203anbi123i 1140 . . 3  |-  ( ( R  e.  Grp  /\  x  C_  ( Base `  R
)  /\  ( Rs  x
)  e.  Grp )  <->  ( O  e.  Grp  /\  x  C_  ( Base `  R
)  /\  ( Os  x
)  e.  Grp )
)
222issubg 14621 . . 3  |-  ( x  e.  (SubGrp `  R
)  <->  ( R  e. 
Grp  /\  x  C_  ( Base `  R )  /\  ( Rs  x )  e.  Grp ) )
233issubg 14621 . . 3  |-  ( x  e.  (SubGrp `  O
)  <->  ( O  e. 
Grp  /\  x  C_  ( Base `  R )  /\  ( Os  x )  e.  Grp ) )
2421, 22, 233bitr4i 268 . 2  |-  ( x  e.  (SubGrp `  R
)  <->  x  e.  (SubGrp `  O ) )
2524eqriv 2280 1  |-  (SubGrp `  R )  =  (SubGrp `  O )
Colors of variables: wff set class
Syntax hints:    /\ w3a 934    = wceq 1623    e. wcel 1684   _Vcvv 2788    i^i cin 3151    C_ wss 3152   ` cfv 5255  (class class class)co 5858   Basecbs 13148   ↾s cress 13149   +g cplusg 13208   Grpcgrp 14362  SubGrpcsubg 14615  opprcoppr 15404
This theorem is referenced by:  opprsubrg  15566
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-tpos 6234  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-2 9804  df-3 9805  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-mulr 13222  df-0g 13404  df-mnd 14367  df-grp 14489  df-subg 14618  df-oppr 15405
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