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Theorem gex2abl 15192
Description: A group with exponent 2 (or 1) is abelian. (Contributed by Mario Carneiro, 24-Apr-2016.)
Hypotheses
Ref Expression
gexex.1  |-  X  =  ( Base `  G
)
gexex.2  |-  E  =  (gEx `  G )
Assertion
Ref Expression
gex2abl  |-  ( ( G  e.  Grp  /\  E  ||  2 )  ->  G  e.  Abel )

Proof of Theorem gex2abl
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 gexex.1 . . 3  |-  X  =  ( Base `  G
)
21a1i 10 . 2  |-  ( ( G  e.  Grp  /\  E  ||  2 )  ->  X  =  ( Base `  G ) )
3 eqidd 2317 . 2  |-  ( ( G  e.  Grp  /\  E  ||  2 )  -> 
( +g  `  G )  =  ( +g  `  G
) )
4 simpl 443 . 2  |-  ( ( G  e.  Grp  /\  E  ||  2 )  ->  G  e.  Grp )
5 simp1l 979 . . . . . . . . 9  |-  ( ( ( G  e.  Grp  /\  E  ||  2 )  /\  x  e.  X  /\  y  e.  X
)  ->  G  e.  Grp )
6 simp2 956 . . . . . . . . 9  |-  ( ( ( G  e.  Grp  /\  E  ||  2 )  /\  x  e.  X  /\  y  e.  X
)  ->  x  e.  X )
7 simp3 957 . . . . . . . . 9  |-  ( ( ( G  e.  Grp  /\  E  ||  2 )  /\  x  e.  X  /\  y  e.  X
)  ->  y  e.  X )
8 eqid 2316 . . . . . . . . . 10  |-  ( +g  `  G )  =  ( +g  `  G )
91, 8grpass 14545 . . . . . . . . 9  |-  ( ( G  e.  Grp  /\  ( x  e.  X  /\  y  e.  X  /\  y  e.  X
) )  ->  (
( x ( +g  `  G ) y ) ( +g  `  G
) y )  =  ( x ( +g  `  G ) ( y ( +g  `  G
) y ) ) )
105, 6, 7, 7, 9syl13anc 1184 . . . . . . . 8  |-  ( ( ( G  e.  Grp  /\  E  ||  2 )  /\  x  e.  X  /\  y  e.  X
)  ->  ( (
x ( +g  `  G
) y ) ( +g  `  G ) y )  =  ( x ( +g  `  G
) ( y ( +g  `  G ) y ) ) )
11 eqid 2316 . . . . . . . . . . . 12  |-  (.g `  G
)  =  (.g `  G
)
121, 11, 8mulg2 14625 . . . . . . . . . . 11  |-  ( y  e.  X  ->  (
2 (.g `  G ) y )  =  ( y ( +g  `  G
) y ) )
137, 12syl 15 . . . . . . . . . 10  |-  ( ( ( G  e.  Grp  /\  E  ||  2 )  /\  x  e.  X  /\  y  e.  X
)  ->  ( 2 (.g `  G ) y )  =  ( y ( +g  `  G
) y ) )
14 simp1r 980 . . . . . . . . . . 11  |-  ( ( ( G  e.  Grp  /\  E  ||  2 )  /\  x  e.  X  /\  y  e.  X
)  ->  E  ||  2
)
15 gexex.2 . . . . . . . . . . . 12  |-  E  =  (gEx `  G )
16 eqid 2316 . . . . . . . . . . . 12  |-  ( 0g
`  G )  =  ( 0g `  G
)
171, 15, 11, 16gexdvdsi 14943 . . . . . . . . . . 11  |-  ( ( G  e.  Grp  /\  y  e.  X  /\  E  ||  2 )  -> 
( 2 (.g `  G
) y )  =  ( 0g `  G
) )
185, 7, 14, 17syl3anc 1182 . . . . . . . . . 10  |-  ( ( ( G  e.  Grp  /\  E  ||  2 )  /\  x  e.  X  /\  y  e.  X
)  ->  ( 2 (.g `  G ) y )  =  ( 0g
`  G ) )
1913, 18eqtr3d 2350 . . . . . . . . 9  |-  ( ( ( G  e.  Grp  /\  E  ||  2 )  /\  x  e.  X  /\  y  e.  X
)  ->  ( y
( +g  `  G ) y )  =  ( 0g `  G ) )
2019oveq2d 5916 . . . . . . . 8  |-  ( ( ( G  e.  Grp  /\  E  ||  2 )  /\  x  e.  X  /\  y  e.  X
)  ->  ( x
( +g  `  G ) ( y ( +g  `  G ) y ) )  =  ( x ( +g  `  G
) ( 0g `  G ) ) )
211, 8, 16grprid 14562 . . . . . . . . 9  |-  ( ( G  e.  Grp  /\  x  e.  X )  ->  ( x ( +g  `  G ) ( 0g
`  G ) )  =  x )
225, 6, 21syl2anc 642 . . . . . . . 8  |-  ( ( ( G  e.  Grp  /\  E  ||  2 )  /\  x  e.  X  /\  y  e.  X
)  ->  ( x
( +g  `  G ) ( 0g `  G
) )  =  x )
2310, 20, 223eqtrd 2352 . . . . . . 7  |-  ( ( ( G  e.  Grp  /\  E  ||  2 )  /\  x  e.  X  /\  y  e.  X
)  ->  ( (
x ( +g  `  G
) y ) ( +g  `  G ) y )  =  x )
2423oveq1d 5915 . . . . . 6  |-  ( ( ( G  e.  Grp  /\  E  ||  2 )  /\  x  e.  X  /\  y  e.  X
)  ->  ( (
( x ( +g  `  G ) y ) ( +g  `  G
) y ) ( +g  `  G ) x )  =  ( x ( +g  `  G
) x ) )
251, 11, 8mulg2 14625 . . . . . . 7  |-  ( x  e.  X  ->  (
2 (.g `  G ) x )  =  ( x ( +g  `  G
) x ) )
266, 25syl 15 . . . . . 6  |-  ( ( ( G  e.  Grp  /\  E  ||  2 )  /\  x  e.  X  /\  y  e.  X
)  ->  ( 2 (.g `  G ) x )  =  ( x ( +g  `  G
) x ) )
271, 15, 11, 16gexdvdsi 14943 . . . . . . 7  |-  ( ( G  e.  Grp  /\  x  e.  X  /\  E  ||  2 )  -> 
( 2 (.g `  G
) x )  =  ( 0g `  G
) )
285, 6, 14, 27syl3anc 1182 . . . . . 6  |-  ( ( ( G  e.  Grp  /\  E  ||  2 )  /\  x  e.  X  /\  y  e.  X
)  ->  ( 2 (.g `  G ) x )  =  ( 0g
`  G ) )
2924, 26, 283eqtr2d 2354 . . . . 5  |-  ( ( ( G  e.  Grp  /\  E  ||  2 )  /\  x  e.  X  /\  y  e.  X
)  ->  ( (
( x ( +g  `  G ) y ) ( +g  `  G
) y ) ( +g  `  G ) x )  =  ( 0g `  G ) )
301, 8grpcl 14544 . . . . . . 7  |-  ( ( G  e.  Grp  /\  x  e.  X  /\  y  e.  X )  ->  ( x ( +g  `  G ) y )  e.  X )
315, 6, 7, 30syl3anc 1182 . . . . . 6  |-  ( ( ( G  e.  Grp  /\  E  ||  2 )  /\  x  e.  X  /\  y  e.  X
)  ->  ( x
( +g  `  G ) y )  e.  X
)
321, 15, 11, 16gexdvdsi 14943 . . . . . 6  |-  ( ( G  e.  Grp  /\  ( x ( +g  `  G ) y )  e.  X  /\  E  ||  2 )  ->  (
2 (.g `  G ) ( x ( +g  `  G
) y ) )  =  ( 0g `  G ) )
335, 31, 14, 32syl3anc 1182 . . . . 5  |-  ( ( ( G  e.  Grp  /\  E  ||  2 )  /\  x  e.  X  /\  y  e.  X
)  ->  ( 2 (.g `  G ) ( x ( +g  `  G
) y ) )  =  ( 0g `  G ) )
341, 11, 8mulg2 14625 . . . . . 6  |-  ( ( x ( +g  `  G
) y )  e.  X  ->  ( 2 (.g `  G ) ( x ( +g  `  G
) y ) )  =  ( ( x ( +g  `  G
) y ) ( +g  `  G ) ( x ( +g  `  G ) y ) ) )
3531, 34syl 15 . . . . 5  |-  ( ( ( G  e.  Grp  /\  E  ||  2 )  /\  x  e.  X  /\  y  e.  X
)  ->  ( 2 (.g `  G ) ( x ( +g  `  G
) y ) )  =  ( ( x ( +g  `  G
) y ) ( +g  `  G ) ( x ( +g  `  G ) y ) ) )
3629, 33, 353eqtr2d 2354 . . . 4  |-  ( ( ( G  e.  Grp  /\  E  ||  2 )  /\  x  e.  X  /\  y  e.  X
)  ->  ( (
( x ( +g  `  G ) y ) ( +g  `  G
) y ) ( +g  `  G ) x )  =  ( ( x ( +g  `  G ) y ) ( +g  `  G
) ( x ( +g  `  G ) y ) ) )
371, 8grpass 14545 . . . . 5  |-  ( ( G  e.  Grp  /\  ( ( x ( +g  `  G ) y )  e.  X  /\  y  e.  X  /\  x  e.  X
) )  ->  (
( ( x ( +g  `  G ) y ) ( +g  `  G ) y ) ( +g  `  G
) x )  =  ( ( x ( +g  `  G ) y ) ( +g  `  G ) ( y ( +g  `  G
) x ) ) )
385, 31, 7, 6, 37syl13anc 1184 . . . 4  |-  ( ( ( G  e.  Grp  /\  E  ||  2 )  /\  x  e.  X  /\  y  e.  X
)  ->  ( (
( x ( +g  `  G ) y ) ( +g  `  G
) y ) ( +g  `  G ) x )  =  ( ( x ( +g  `  G ) y ) ( +g  `  G
) ( y ( +g  `  G ) x ) ) )
3936, 38eqtr3d 2350 . . 3  |-  ( ( ( G  e.  Grp  /\  E  ||  2 )  /\  x  e.  X  /\  y  e.  X
)  ->  ( (
x ( +g  `  G
) y ) ( +g  `  G ) ( x ( +g  `  G ) y ) )  =  ( ( x ( +g  `  G
) y ) ( +g  `  G ) ( y ( +g  `  G ) x ) ) )
401, 8grpcl 14544 . . . . 5  |-  ( ( G  e.  Grp  /\  y  e.  X  /\  x  e.  X )  ->  ( y ( +g  `  G ) x )  e.  X )
415, 7, 6, 40syl3anc 1182 . . . 4  |-  ( ( ( G  e.  Grp  /\  E  ||  2 )  /\  x  e.  X  /\  y  e.  X
)  ->  ( y
( +g  `  G ) x )  e.  X
)
421, 8grplcan 14583 . . . 4  |-  ( ( G  e.  Grp  /\  ( ( x ( +g  `  G ) y )  e.  X  /\  ( y ( +g  `  G ) x )  e.  X  /\  (
x ( +g  `  G
) y )  e.  X ) )  -> 
( ( ( x ( +g  `  G
) y ) ( +g  `  G ) ( x ( +g  `  G ) y ) )  =  ( ( x ( +g  `  G
) y ) ( +g  `  G ) ( y ( +g  `  G ) x ) )  <->  ( x ( +g  `  G ) y )  =  ( y ( +g  `  G
) x ) ) )
435, 31, 41, 31, 42syl13anc 1184 . . 3  |-  ( ( ( G  e.  Grp  /\  E  ||  2 )  /\  x  e.  X  /\  y  e.  X
)  ->  ( (
( x ( +g  `  G ) y ) ( +g  `  G
) ( x ( +g  `  G ) y ) )  =  ( ( x ( +g  `  G ) y ) ( +g  `  G ) ( y ( +g  `  G
) x ) )  <-> 
( x ( +g  `  G ) y )  =  ( y ( +g  `  G ) x ) ) )
4439, 43mpbid 201 . 2  |-  ( ( ( G  e.  Grp  /\  E  ||  2 )  /\  x  e.  X  /\  y  e.  X
)  ->  ( x
( +g  `  G ) y )  =  ( y ( +g  `  G
) x ) )
452, 3, 4, 44isabld 15151 1  |-  ( ( G  e.  Grp  /\  E  ||  2 )  ->  G  e.  Abel )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1633    e. wcel 1701   class class class wbr 4060   ` cfv 5292  (class class class)co 5900   2c2 9840    || cdivides 12578   Basecbs 13195   +g cplusg 13255   0gc0g 13449   Grpcgrp 14411  .gcmg 14415  gExcgex 14890   Abelcabel 15139
This theorem is referenced by:  lt6abl  15230
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-rep 4168  ax-sep 4178  ax-nul 4186  ax-pow 4225  ax-pr 4251  ax-un 4549  ax-inf2 7387  ax-cnex 8838  ax-resscn 8839  ax-1cn 8840  ax-icn 8841  ax-addcl 8842  ax-addrcl 8843  ax-mulcl 8844  ax-mulrcl 8845  ax-mulcom 8846  ax-addass 8847  ax-mulass 8848  ax-distr 8849  ax-i2m1 8850  ax-1ne0 8851  ax-1rid 8852  ax-rnegex 8853  ax-rrecex 8854  ax-cnre 8855  ax-pre-lttri 8856  ax-pre-lttrn 8857  ax-pre-ltadd 8858  ax-pre-mulgt0 8859
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-nel 2482  df-ral 2582  df-rex 2583  df-reu 2584  df-rmo 2585  df-rab 2586  df-v 2824  df-sbc 3026  df-csb 3116  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-pss 3202  df-nul 3490  df-if 3600  df-pw 3661  df-sn 3680  df-pr 3681  df-tp 3682  df-op 3683  df-uni 3865  df-iun 3944  df-br 4061  df-opab 4115  df-mpt 4116  df-tr 4151  df-eprel 4342  df-id 4346  df-po 4351  df-so 4352  df-fr 4389  df-we 4391  df-ord 4432  df-on 4433  df-lim 4434  df-suc 4435  df-om 4694  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-res 4738  df-ima 4739  df-iota 5256  df-fun 5294  df-fn 5295  df-f 5296  df-f1 5297  df-fo 5298  df-f1o 5299  df-fv 5300  df-ov 5903  df-oprab 5904  df-mpt2 5905  df-1st 6164  df-2nd 6165  df-riota 6346  df-recs 6430  df-rdg 6465  df-er 6702  df-en 6907  df-dom 6908  df-sdom 6909  df-sup 7239  df-pnf 8914  df-mnf 8915  df-xr 8916  df-ltxr 8917  df-le 8918  df-sub 9084  df-neg 9085  df-nn 9792  df-2 9849  df-n0 10013  df-z 10072  df-uz 10278  df-fz 10830  df-seq 11094  df-dvds 12579  df-0g 13453  df-mnd 14416  df-grp 14538  df-minusg 14539  df-mulg 14541  df-gex 14894  df-cmn 15140  df-abl 15141
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