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Theorem subgoablo 20978
Description: A subgroup of an Abelian group is Abelian. (Contributed by Paul Chapman, 25-Apr-2008.) (New usage is discouraged.)
Assertion
Ref Expression
subgoablo  |-  ( ( G  e.  AbelOp  /\  H  e.  ( SubGrpOp `  G )
)  ->  H  e.  AbelOp )

Proof of Theorem subgoablo
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpr 447 . 2  |-  ( ( G  e.  AbelOp  /\  H  e.  ( SubGrpOp `  G )
)  ->  H  e.  ( SubGrpOp `  G )
)
2 eqid 2283 . . . . . . . . 9  |-  ran  G  =  ran  G
3 eqid 2283 . . . . . . . . 9  |-  ran  H  =  ran  H
42, 3subgornss 20973 . . . . . . . 8  |-  ( H  e.  ( SubGrpOp `  G
)  ->  ran  H  C_  ran  G )
54sseld 3179 . . . . . . 7  |-  ( H  e.  ( SubGrpOp `  G
)  ->  ( x  e.  ran  H  ->  x  e.  ran  G ) )
64sseld 3179 . . . . . . 7  |-  ( H  e.  ( SubGrpOp `  G
)  ->  ( y  e.  ran  H  ->  y  e.  ran  G ) )
75, 6anim12d 546 . . . . . 6  |-  ( H  e.  ( SubGrpOp `  G
)  ->  ( (
x  e.  ran  H  /\  y  e.  ran  H )  ->  ( x  e.  ran  G  /\  y  e.  ran  G ) ) )
82isablo 20950 . . . . . . . 8  |-  ( G  e.  AbelOp 
<->  ( G  e.  GrpOp  /\ 
A. x  e.  ran  G A. y  e.  ran  G ( x G y )  =  ( y G x ) ) )
98simprbi 450 . . . . . . 7  |-  ( G  e.  AbelOp  ->  A. x  e.  ran  G A. y  e.  ran  G ( x G y )  =  ( y G x ) )
10 rsp2 2605 . . . . . . 7  |-  ( A. x  e.  ran  G A. y  e.  ran  G ( x G y )  =  ( y G x )  ->  (
( x  e.  ran  G  /\  y  e.  ran  G )  ->  ( x G y )  =  ( y G x ) ) )
119, 10syl 15 . . . . . 6  |-  ( G  e.  AbelOp  ->  ( ( x  e.  ran  G  /\  y  e.  ran  G )  ->  ( x G y )  =  ( y G x ) ) )
127, 11sylan9r 639 . . . . 5  |-  ( ( G  e.  AbelOp  /\  H  e.  ( SubGrpOp `  G )
)  ->  ( (
x  e.  ran  H  /\  y  e.  ran  H )  ->  ( x G y )  =  ( y G x ) ) )
1312imp 418 . . . 4  |-  ( ( ( G  e.  AbelOp  /\  H  e.  ( SubGrpOp `  G ) )  /\  ( x  e.  ran  H  /\  y  e.  ran  H ) )  ->  (
x G y )  =  ( y G x ) )
143subgoov 20972 . . . . 5  |-  ( ( H  e.  ( SubGrpOp `  G )  /\  (
x  e.  ran  H  /\  y  e.  ran  H ) )  ->  (
x H y )  =  ( x G y ) )
1514adantll 694 . . . 4  |-  ( ( ( G  e.  AbelOp  /\  H  e.  ( SubGrpOp `  G ) )  /\  ( x  e.  ran  H  /\  y  e.  ran  H ) )  ->  (
x H y )  =  ( x G y ) )
163subgoov 20972 . . . . . 6  |-  ( ( H  e.  ( SubGrpOp `  G )  /\  (
y  e.  ran  H  /\  x  e.  ran  H ) )  ->  (
y H x )  =  ( y G x ) )
1716ancom2s 777 . . . . 5  |-  ( ( H  e.  ( SubGrpOp `  G )  /\  (
x  e.  ran  H  /\  y  e.  ran  H ) )  ->  (
y H x )  =  ( y G x ) )
1817adantll 694 . . . 4  |-  ( ( ( G  e.  AbelOp  /\  H  e.  ( SubGrpOp `  G ) )  /\  ( x  e.  ran  H  /\  y  e.  ran  H ) )  ->  (
y H x )  =  ( y G x ) )
1913, 15, 183eqtr4d 2325 . . 3  |-  ( ( ( G  e.  AbelOp  /\  H  e.  ( SubGrpOp `  G ) )  /\  ( x  e.  ran  H  /\  y  e.  ran  H ) )  ->  (
x H y )  =  ( y H x ) )
2019ralrimivva 2635 . 2  |-  ( ( G  e.  AbelOp  /\  H  e.  ( SubGrpOp `  G )
)  ->  A. x  e.  ran  H A. y  e.  ran  H ( x H y )  =  ( y H x ) )
21 issubgo 20970 . . . 4  |-  ( H  e.  ( SubGrpOp `  G
)  <->  ( G  e. 
GrpOp  /\  H  e.  GrpOp  /\  H  C_  G )
)
2221simp2bi 971 . . 3  |-  ( H  e.  ( SubGrpOp `  G
)  ->  H  e.  GrpOp
)
233isablo 20950 . . . 4  |-  ( H  e.  AbelOp 
<->  ( H  e.  GrpOp  /\ 
A. x  e.  ran  H A. y  e.  ran  H ( x H y )  =  ( y H x ) ) )
2423biimpri 197 . . 3  |-  ( ( H  e.  GrpOp  /\  A. x  e.  ran  H A. y  e.  ran  H ( x H y )  =  ( y H x ) )  ->  H  e.  AbelOp )
2522, 24sylan 457 . 2  |-  ( ( H  e.  ( SubGrpOp `  G )  /\  A. x  e.  ran  H A. y  e.  ran  H ( x H y )  =  ( y H x ) )  ->  H  e.  AbelOp )
261, 20, 25syl2anc 642 1  |-  ( ( G  e.  AbelOp  /\  H  e.  ( SubGrpOp `  G )
)  ->  H  e.  AbelOp )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543    C_ wss 3152   ran crn 4690   ` cfv 5255  (class class class)co 5858   GrpOpcgr 20853   AbelOpcablo 20948   SubGrpOpcsubgo 20968
This theorem is referenced by:  efghgrp  21040
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
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  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-ral 2548  df-rex 2549  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-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  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-fo 5261  df-fv 5263  df-ov 5861  df-grpo 20858  df-ablo 20949  df-subgo 20969
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