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Theorem gex2abl 15467
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 11 . 2  |-  ( ( G  e.  Grp  /\  E  ||  2 )  ->  X  =  ( Base `  G ) )
3 eqidd 2438 . 2  |-  ( ( G  e.  Grp  /\  E  ||  2 )  -> 
( +g  `  G )  =  ( +g  `  G
) )
4 simpl 445 . 2  |-  ( ( G  e.  Grp  /\  E  ||  2 )  ->  G  e.  Grp )
5 simp1l 982 . . . . . . . . 9  |-  ( ( ( G  e.  Grp  /\  E  ||  2 )  /\  x  e.  X  /\  y  e.  X
)  ->  G  e.  Grp )
6 simp2 959 . . . . . . . . 9  |-  ( ( ( G  e.  Grp  /\  E  ||  2 )  /\  x  e.  X  /\  y  e.  X
)  ->  x  e.  X )
7 simp3 960 . . . . . . . . 9  |-  ( ( ( G  e.  Grp  /\  E  ||  2 )  /\  x  e.  X  /\  y  e.  X
)  ->  y  e.  X )
8 eqid 2437 . . . . . . . . . 10  |-  ( +g  `  G )  =  ( +g  `  G )
91, 8grpass 14820 . . . . . . . . 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 1187 . . . . . . . 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 2437 . . . . . . . . . . . 12  |-  (.g `  G
)  =  (.g `  G
)
121, 11, 8mulg2 14900 . . . . . . . . . . 11  |-  ( y  e.  X  ->  (
2 (.g `  G ) y )  =  ( y ( +g  `  G
) y ) )
137, 12syl 16 . . . . . . . . . 10  |-  ( ( ( G  e.  Grp  /\  E  ||  2 )  /\  x  e.  X  /\  y  e.  X
)  ->  ( 2 (.g `  G ) y )  =  ( y ( +g  `  G
) y ) )
14 simp1r 983 . . . . . . . . . . 11  |-  ( ( ( G  e.  Grp  /\  E  ||  2 )  /\  x  e.  X  /\  y  e.  X
)  ->  E  ||  2
)
15 gexex.2 . . . . . . . . . . . 12  |-  E  =  (gEx `  G )
16 eqid 2437 . . . . . . . . . . . 12  |-  ( 0g
`  G )  =  ( 0g `  G
)
171, 15, 11, 16gexdvdsi 15218 . . . . . . . . . . 11  |-  ( ( G  e.  Grp  /\  y  e.  X  /\  E  ||  2 )  -> 
( 2 (.g `  G
) y )  =  ( 0g `  G
) )
185, 7, 14, 17syl3anc 1185 . . . . . . . . . 10  |-  ( ( ( G  e.  Grp  /\  E  ||  2 )  /\  x  e.  X  /\  y  e.  X
)  ->  ( 2 (.g `  G ) y )  =  ( 0g
`  G ) )
1913, 18eqtr3d 2471 . . . . . . . . 9  |-  ( ( ( G  e.  Grp  /\  E  ||  2 )  /\  x  e.  X  /\  y  e.  X
)  ->  ( y
( +g  `  G ) y )  =  ( 0g `  G ) )
2019oveq2d 6098 . . . . . . . 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 14837 . . . . . . . . 9  |-  ( ( G  e.  Grp  /\  x  e.  X )  ->  ( x ( +g  `  G ) ( 0g
`  G ) )  =  x )
225, 6, 21syl2anc 644 . . . . . . . 8  |-  ( ( ( G  e.  Grp  /\  E  ||  2 )  /\  x  e.  X  /\  y  e.  X
)  ->  ( x
( +g  `  G ) ( 0g `  G
) )  =  x )
2310, 20, 223eqtrd 2473 . . . . . . 7  |-  ( ( ( G  e.  Grp  /\  E  ||  2 )  /\  x  e.  X  /\  y  e.  X
)  ->  ( (
x ( +g  `  G
) y ) ( +g  `  G ) y )  =  x )
2423oveq1d 6097 . . . . . 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 14900 . . . . . . 7  |-  ( x  e.  X  ->  (
2 (.g `  G ) x )  =  ( x ( +g  `  G
) x ) )
266, 25syl 16 . . . . . 6  |-  ( ( ( G  e.  Grp  /\  E  ||  2 )  /\  x  e.  X  /\  y  e.  X
)  ->  ( 2 (.g `  G ) x )  =  ( x ( +g  `  G
) x ) )
271, 15, 11, 16gexdvdsi 15218 . . . . . . 7  |-  ( ( G  e.  Grp  /\  x  e.  X  /\  E  ||  2 )  -> 
( 2 (.g `  G
) x )  =  ( 0g `  G
) )
285, 6, 14, 27syl3anc 1185 . . . . . 6  |-  ( ( ( G  e.  Grp  /\  E  ||  2 )  /\  x  e.  X  /\  y  e.  X
)  ->  ( 2 (.g `  G ) x )  =  ( 0g
`  G ) )
2924, 26, 283eqtr2d 2475 . . . . 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 14819 . . . . . . 7  |-  ( ( G  e.  Grp  /\  x  e.  X  /\  y  e.  X )  ->  ( x ( +g  `  G ) y )  e.  X )
315, 6, 7, 30syl3anc 1185 . . . . . 6  |-  ( ( ( G  e.  Grp  /\  E  ||  2 )  /\  x  e.  X  /\  y  e.  X
)  ->  ( x
( +g  `  G ) y )  e.  X
)
321, 15, 11, 16gexdvdsi 15218 . . . . . 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 1185 . . . . 5  |-  ( ( ( G  e.  Grp  /\  E  ||  2 )  /\  x  e.  X  /\  y  e.  X
)  ->  ( 2 (.g `  G ) ( x ( +g  `  G
) y ) )  =  ( 0g `  G ) )
341, 11, 8mulg2 14900 . . . . . 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 16 . . . . 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 2475 . . . 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 14820 . . . . 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 1187 . . . 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 2471 . . 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 14819 . . . . 5  |-  ( ( G  e.  Grp  /\  y  e.  X  /\  x  e.  X )  ->  ( y ( +g  `  G ) x )  e.  X )
415, 7, 6, 40syl3anc 1185 . . . 4  |-  ( ( ( G  e.  Grp  /\  E  ||  2 )  /\  x  e.  X  /\  y  e.  X
)  ->  ( y
( +g  `  G ) x )  e.  X
)
421, 8grplcan 14858 . . . 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 1187 . . 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 203 . 2  |-  ( ( ( G  e.  Grp  /\  E  ||  2 )  /\  x  e.  X  /\  y  e.  X
)  ->  ( x
( +g  `  G ) y )  =  ( y ( +g  `  G
) x ) )
452, 3, 4, 44isabld 15426 1  |-  ( ( G  e.  Grp  /\  E  ||  2 )  ->  G  e.  Abel )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726   class class class wbr 4213   ` cfv 5455  (class class class)co 6082   2c2 10050    || cdivides 12853   Basecbs 13470   +g cplusg 13530   0gc0g 13724   Grpcgrp 14686  .gcmg 14690  gExcgex 15165   Abelcabel 15414
This theorem is referenced by:  lt6abl  15505
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 2418  ax-rep 4321  ax-sep 4331  ax-nul 4339  ax-pow 4378  ax-pr 4404  ax-un 4702  ax-inf2 7597  ax-cnex 9047  ax-resscn 9048  ax-1cn 9049  ax-icn 9050  ax-addcl 9051  ax-addrcl 9052  ax-mulcl 9053  ax-mulrcl 9054  ax-mulcom 9055  ax-addass 9056  ax-mulass 9057  ax-distr 9058  ax-i2m1 9059  ax-1ne0 9060  ax-1rid 9061  ax-rnegex 9062  ax-rrecex 9063  ax-cnre 9064  ax-pre-lttri 9065  ax-pre-lttrn 9066  ax-pre-ltadd 9067  ax-pre-mulgt0 9068
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-nel 2603  df-ral 2711  df-rex 2712  df-reu 2713  df-rmo 2714  df-rab 2715  df-v 2959  df-sbc 3163  df-csb 3253  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-pss 3337  df-nul 3630  df-if 3741  df-pw 3802  df-sn 3821  df-pr 3822  df-tp 3823  df-op 3824  df-uni 4017  df-iun 4096  df-br 4214  df-opab 4268  df-mpt 4269  df-tr 4304  df-eprel 4495  df-id 4499  df-po 4504  df-so 4505  df-fr 4542  df-we 4544  df-ord 4585  df-on 4586  df-lim 4587  df-suc 4588  df-om 4847  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-rn 4890  df-res 4891  df-ima 4892  df-iota 5419  df-fun 5457  df-fn 5458  df-f 5459  df-f1 5460  df-fo 5461  df-f1o 5462  df-fv 5463  df-ov 6085  df-oprab 6086  df-mpt2 6087  df-1st 6350  df-2nd 6351  df-riota 6550  df-recs 6634  df-rdg 6669  df-er 6906  df-en 7111  df-dom 7112  df-sdom 7113  df-sup 7447  df-pnf 9123  df-mnf 9124  df-xr 9125  df-ltxr 9126  df-le 9127  df-sub 9294  df-neg 9295  df-nn 10002  df-2 10059  df-n0 10223  df-z 10284  df-uz 10490  df-fz 11045  df-seq 11325  df-dvds 12854  df-0g 13728  df-mnd 14691  df-grp 14813  df-minusg 14814  df-mulg 14816  df-gex 15169  df-cmn 15415  df-abl 15416
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