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Theorem oppgsubg 14929
Description: Being a subgroup is a symmetric property. (Contributed by Mario Carneiro, 17-Sep-2015.)
Hypothesis
Ref Expression
oppggic.o  |-  O  =  (oppg
`  G )
Assertion
Ref Expression
oppgsubg  |-  (SubGrp `  G )  =  (SubGrp `  O )

Proof of Theorem oppgsubg
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 subgrcl 14719 . . 3  |-  ( x  e.  (SubGrp `  G
)  ->  G  e.  Grp )
2 subgrcl 14719 . . . 4  |-  ( x  e.  (SubGrp `  O
)  ->  O  e.  Grp )
3 oppggic.o . . . . 5  |-  O  =  (oppg
`  G )
43oppggrpb 14924 . . . 4  |-  ( G  e.  Grp  <->  O  e.  Grp )
52, 4sylibr 203 . . 3  |-  ( x  e.  (SubGrp `  O
)  ->  G  e.  Grp )
63oppgsubm 14928 . . . . . . 7  |-  (SubMnd `  G )  =  (SubMnd `  O )
76eleq2i 2422 . . . . . 6  |-  ( x  e.  (SubMnd `  G
)  <->  x  e.  (SubMnd `  O ) )
87a1i 10 . . . . 5  |-  ( G  e.  Grp  ->  (
x  e.  (SubMnd `  G )  <->  x  e.  (SubMnd `  O ) ) )
9 eqid 2358 . . . . . . . . 9  |-  ( inv g `  G )  =  ( inv g `  G )
103, 9oppginv 14925 . . . . . . . 8  |-  ( G  e.  Grp  ->  ( inv g `  G )  =  ( inv g `  O ) )
1110fveq1d 5607 . . . . . . 7  |-  ( G  e.  Grp  ->  (
( inv g `  G ) `  y
)  =  ( ( inv g `  O
) `  y )
)
1211eleq1d 2424 . . . . . 6  |-  ( G  e.  Grp  ->  (
( ( inv g `  G ) `  y
)  e.  x  <->  ( ( inv g `  O ) `
 y )  e.  x ) )
1312ralbidv 2639 . . . . 5  |-  ( G  e.  Grp  ->  ( A. y  e.  x  ( ( inv g `  G ) `  y
)  e.  x  <->  A. y  e.  x  ( ( inv g `  O ) `
 y )  e.  x ) )
148, 13anbi12d 691 . . . 4  |-  ( G  e.  Grp  ->  (
( x  e.  (SubMnd `  G )  /\  A. y  e.  x  (
( inv g `  G ) `  y
)  e.  x )  <-> 
( x  e.  (SubMnd `  O )  /\  A. y  e.  x  (
( inv g `  O ) `  y
)  e.  x ) ) )
159issubg3 14730 . . . 4  |-  ( G  e.  Grp  ->  (
x  e.  (SubGrp `  G )  <->  ( x  e.  (SubMnd `  G )  /\  A. y  e.  x  ( ( inv g `  G ) `  y
)  e.  x ) ) )
16 eqid 2358 . . . . . 6  |-  ( inv g `  O )  =  ( inv g `  O )
1716issubg3 14730 . . . . 5  |-  ( O  e.  Grp  ->  (
x  e.  (SubGrp `  O )  <->  ( x  e.  (SubMnd `  O )  /\  A. y  e.  x  ( ( inv g `  O ) `  y
)  e.  x ) ) )
184, 17sylbi 187 . . . 4  |-  ( G  e.  Grp  ->  (
x  e.  (SubGrp `  O )  <->  ( x  e.  (SubMnd `  O )  /\  A. y  e.  x  ( ( inv g `  O ) `  y
)  e.  x ) ) )
1914, 15, 183bitr4d 276 . . 3  |-  ( G  e.  Grp  ->  (
x  e.  (SubGrp `  G )  <->  x  e.  (SubGrp `  O ) ) )
201, 5, 19pm5.21nii 342 . 2  |-  ( x  e.  (SubGrp `  G
)  <->  x  e.  (SubGrp `  O ) )
2120eqriv 2355 1  |-  (SubGrp `  G )  =  (SubGrp `  O )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358    = wceq 1642    e. wcel 1710   A.wral 2619   ` cfv 5334   Grpcgrp 14455   inv gcminusg 14456  SubMndcsubmnd 14507  SubGrpcsubg 14708  oppgcoppg 14911
This theorem is referenced by:  lsmmod2  15078  lsmdisj2r  15087
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-rep 4210  ax-sep 4220  ax-nul 4228  ax-pow 4267  ax-pr 4293  ax-un 4591  ax-cnex 8880  ax-resscn 8881  ax-1cn 8882  ax-icn 8883  ax-addcl 8884  ax-addrcl 8885  ax-mulcl 8886  ax-mulrcl 8887  ax-mulcom 8888  ax-addass 8889  ax-mulass 8890  ax-distr 8891  ax-i2m1 8892  ax-1ne0 8893  ax-1rid 8894  ax-rnegex 8895  ax-rrecex 8896  ax-cnre 8897  ax-pre-lttri 8898  ax-pre-lttrn 8899  ax-pre-ltadd 8900  ax-pre-mulgt0 8901
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-nel 2524  df-ral 2624  df-rex 2625  df-reu 2626  df-rmo 2627  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-pss 3244  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-tp 3724  df-op 3725  df-uni 3907  df-iun 3986  df-br 4103  df-opab 4157  df-mpt 4158  df-tr 4193  df-eprel 4384  df-id 4388  df-po 4393  df-so 4394  df-fr 4431  df-we 4433  df-ord 4474  df-on 4475  df-lim 4476  df-suc 4477  df-om 4736  df-xp 4774  df-rel 4775  df-cnv 4776  df-co 4777  df-dm 4778  df-rn 4779  df-res 4780  df-ima 4781  df-iota 5298  df-fun 5336  df-fn 5337  df-f 5338  df-f1 5339  df-fo 5340  df-f1o 5341  df-fv 5342  df-ov 5945  df-oprab 5946  df-mpt2 5947  df-tpos 6318  df-riota 6388  df-recs 6472  df-rdg 6507  df-er 6744  df-en 6949  df-dom 6950  df-sdom 6951  df-pnf 8956  df-mnf 8957  df-xr 8958  df-ltxr 8959  df-le 8960  df-sub 9126  df-neg 9127  df-nn 9834  df-2 9891  df-ndx 13242  df-slot 13243  df-base 13244  df-sets 13245  df-ress 13246  df-plusg 13312  df-0g 13497  df-mnd 14460  df-submnd 14509  df-grp 14582  df-minusg 14583  df-subg 14711  df-oppg 14912
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