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Theorem subgoablo 21901
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 449 . 2  |-  ( ( G  e.  AbelOp  /\  H  e.  ( SubGrpOp `  G )
)  ->  H  e.  ( SubGrpOp `  G )
)
2 eqid 2438 . . . . . . . . 9  |-  ran  G  =  ran  G
3 eqid 2438 . . . . . . . . 9  |-  ran  H  =  ran  H
42, 3subgornss 21896 . . . . . . . 8  |-  ( H  e.  ( SubGrpOp `  G
)  ->  ran  H  C_  ran  G )
54sseld 3349 . . . . . . 7  |-  ( H  e.  ( SubGrpOp `  G
)  ->  ( x  e.  ran  H  ->  x  e.  ran  G ) )
64sseld 3349 . . . . . . 7  |-  ( H  e.  ( SubGrpOp `  G
)  ->  ( y  e.  ran  H  ->  y  e.  ran  G ) )
75, 6anim12d 548 . . . . . 6  |-  ( H  e.  ( SubGrpOp `  G
)  ->  ( (
x  e.  ran  H  /\  y  e.  ran  H )  ->  ( x  e.  ran  G  /\  y  e.  ran  G ) ) )
82isablo 21873 . . . . . . . 8  |-  ( G  e.  AbelOp 
<->  ( G  e.  GrpOp  /\ 
A. x  e.  ran  G A. y  e.  ran  G ( x G y )  =  ( y G x ) ) )
98simprbi 452 . . . . . . 7  |-  ( G  e.  AbelOp  ->  A. x  e.  ran  G A. y  e.  ran  G ( x G y )  =  ( y G x ) )
10 rsp2 2770 . . . . . . 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 16 . . . . . 6  |-  ( G  e.  AbelOp  ->  ( ( x  e.  ran  G  /\  y  e.  ran  G )  ->  ( x G y )  =  ( y G x ) ) )
127, 11sylan9r 641 . . . . 5  |-  ( ( G  e.  AbelOp  /\  H  e.  ( SubGrpOp `  G )
)  ->  ( (
x  e.  ran  H  /\  y  e.  ran  H )  ->  ( x G y )  =  ( y G x ) ) )
1312imp 420 . . . 4  |-  ( ( ( G  e.  AbelOp  /\  H  e.  ( SubGrpOp `  G ) )  /\  ( x  e.  ran  H  /\  y  e.  ran  H ) )  ->  (
x G y )  =  ( y G x ) )
143subgoov 21895 . . . . 5  |-  ( ( H  e.  ( SubGrpOp `  G )  /\  (
x  e.  ran  H  /\  y  e.  ran  H ) )  ->  (
x H y )  =  ( x G y ) )
1514adantll 696 . . . 4  |-  ( ( ( G  e.  AbelOp  /\  H  e.  ( SubGrpOp `  G ) )  /\  ( x  e.  ran  H  /\  y  e.  ran  H ) )  ->  (
x H y )  =  ( x G y ) )
163subgoov 21895 . . . . . 6  |-  ( ( H  e.  ( SubGrpOp `  G )  /\  (
y  e.  ran  H  /\  x  e.  ran  H ) )  ->  (
y H x )  =  ( y G x ) )
1716ancom2s 779 . . . . 5  |-  ( ( H  e.  ( SubGrpOp `  G )  /\  (
x  e.  ran  H  /\  y  e.  ran  H ) )  ->  (
y H x )  =  ( y G x ) )
1817adantll 696 . . . 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 2480 . . 3  |-  ( ( ( G  e.  AbelOp  /\  H  e.  ( SubGrpOp `  G ) )  /\  ( x  e.  ran  H  /\  y  e.  ran  H ) )  ->  (
x H y )  =  ( y H x ) )
2019ralrimivva 2800 . 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 21893 . . . 4  |-  ( H  e.  ( SubGrpOp `  G
)  <->  ( G  e. 
GrpOp  /\  H  e.  GrpOp  /\  H  C_  G )
)
2221simp2bi 974 . . 3  |-  ( H  e.  ( SubGrpOp `  G
)  ->  H  e.  GrpOp
)
233isablo 21873 . . . 4  |-  ( H  e.  AbelOp 
<->  ( H  e.  GrpOp  /\ 
A. x  e.  ran  H A. y  e.  ran  H ( x H y )  =  ( y H x ) ) )
2423biimpri 199 . . 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 459 . 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 644 1  |-  ( ( G  e.  AbelOp  /\  H  e.  ( SubGrpOp `  G )
)  ->  H  e.  AbelOp )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    = wceq 1653    e. wcel 1726   A.wral 2707    C_ wss 3322   ran crn 4881   ` cfv 5456  (class class class)co 6083   GrpOpcgr 21776   AbelOpcablo 21871   SubGrpOpcsubgo 21891
This theorem is referenced by:  efghgrp  21963
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-fo 5462  df-fv 5464  df-ov 6086  df-grpo 21781  df-ablo 21872  df-subgo 21892
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