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Theorem subgoablo 20994
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 2296 . . . . . . . . 9  |-  ran  G  =  ran  G
3 eqid 2296 . . . . . . . . 9  |-  ran  H  =  ran  H
42, 3subgornss 20989 . . . . . . . 8  |-  ( H  e.  ( SubGrpOp `  G
)  ->  ran  H  C_  ran  G )
54sseld 3192 . . . . . . 7  |-  ( H  e.  ( SubGrpOp `  G
)  ->  ( x  e.  ran  H  ->  x  e.  ran  G ) )
64sseld 3192 . . . . . . 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 20966 . . . . . . . 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 2618 . . . . . . 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 20988 . . . . 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 20988 . . . . . 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 2338 . . 3  |-  ( ( ( G  e.  AbelOp  /\  H  e.  ( SubGrpOp `  G ) )  /\  ( x  e.  ran  H  /\  y  e.  ran  H ) )  ->  (
x H y )  =  ( y H x ) )
2019ralrimivva 2648 . 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 20986 . . . 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 20966 . . . 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 1632    e. wcel 1696   A.wral 2556    C_ wss 3165   ran crn 4706   ` cfv 5271  (class class class)co 5874   GrpOpcgr 20869   AbelOpcablo 20964   SubGrpOpcsubgo 20984
This theorem is referenced by:  efghgrp  21056
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-fo 5277  df-fv 5279  df-ov 5877  df-grpo 20874  df-ablo 20965  df-subgo 20985
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