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Theorem opprsubg 15741
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 2436 . . . . . 6  |-  ( Base `  R )  =  (
Base `  R )
31, 2opprbas 15734 . . . . 5  |-  ( Base `  R )  =  (
Base `  O )
4 eqid 2436 . . . . . 6  |-  ( +g  `  R )  =  ( +g  `  R )
51, 4oppradd 15735 . . . . 5  |-  ( +g  `  R )  =  ( +g  `  O )
63, 5grpprop 14824 . . . 4  |-  ( R  e.  Grp  <->  O  e.  Grp )
7 biid 228 . . . 4  |-  ( x 
C_  ( Base `  R
)  <->  x  C_  ( Base `  R ) )
8 vex 2959 . . . . . . 7  |-  x  e. 
_V
9 eqid 2436 . . . . . . . 8  |-  ( Rs  x )  =  ( Rs  x )
109, 2ressbas 13519 . . . . . . 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 2436 . . . . . . . 8  |-  ( Os  x )  =  ( Os  x )
1312, 3ressbas 13519 . . . . . . 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 2458 . . . . 5  |-  ( Base `  ( Rs  x ) )  =  ( Base `  ( Os  x ) )
169, 4ressplusg 13571 . . . . . . 7  |-  ( x  e.  _V  ->  ( +g  `  R )  =  ( +g  `  ( Rs  x ) ) )
1712, 5ressplusg 13571 . . . . . . 7  |-  ( x  e.  _V  ->  ( +g  `  R )  =  ( +g  `  ( Os  x ) ) )
1816, 17eqtr3d 2470 . . . . . 6  |-  ( x  e.  _V  ->  ( +g  `  ( Rs  x ) )  =  ( +g  `  ( Os  x ) ) )
198, 18ax-mp 8 . . . . 5  |-  ( +g  `  ( Rs  x ) )  =  ( +g  `  ( Os  x ) )
2015, 19grpprop 14824 . . . 4  |-  ( ( Rs  x )  e.  Grp  <->  ( Os  x )  e.  Grp )
216, 7, 203anbi123i 1142 . . 3  |-  ( ( R  e.  Grp  /\  x  C_  ( Base `  R
)  /\  ( Rs  x
)  e.  Grp )  <->  ( O  e.  Grp  /\  x  C_  ( Base `  R
)  /\  ( Os  x
)  e.  Grp )
)
222issubg 14944 . . 3  |-  ( x  e.  (SubGrp `  R
)  <->  ( R  e. 
Grp  /\  x  C_  ( Base `  R )  /\  ( Rs  x )  e.  Grp ) )
233issubg 14944 . . 3  |-  ( x  e.  (SubGrp `  O
)  <->  ( O  e. 
Grp  /\  x  C_  ( Base `  R )  /\  ( Os  x )  e.  Grp ) )
2421, 22, 233bitr4i 269 . 2  |-  ( x  e.  (SubGrp `  R
)  <->  x  e.  (SubGrp `  O ) )
2524eqriv 2433 1  |-  (SubGrp `  R )  =  (SubGrp `  O )
Colors of variables: wff set class
Syntax hints:    /\ w3a 936    = wceq 1652    e. wcel 1725   _Vcvv 2956    i^i cin 3319    C_ wss 3320   ` cfv 5454  (class class class)co 6081   Basecbs 13469   ↾s cress 13470   +g cplusg 13529   Grpcgrp 14685  SubGrpcsubg 14938  opprcoppr 15727
This theorem is referenced by:  opprsubrg  15889
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-tpos 6479  df-riota 6549  df-recs 6633  df-rdg 6668  df-er 6905  df-en 7110  df-dom 7111  df-sdom 7112  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-nn 10001  df-2 10058  df-3 10059  df-ndx 13472  df-slot 13473  df-base 13474  df-sets 13475  df-ress 13476  df-plusg 13542  df-mulr 13543  df-0g 13727  df-mnd 14690  df-grp 14812  df-subg 14941  df-oppr 15728
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