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Theorem gex2abl 16437
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 2452 . 2  |-  ( ( G  e.  Grp  /\  E  ||  2 )  -> 
( +g  `  G )  =  ( +g  `  G
) )
4 simpl 457 . 2  |-  ( ( G  e.  Grp  /\  E  ||  2 )  ->  G  e.  Grp )
5 simp1l 1012 . . . . . . . . 9  |-  ( ( ( G  e.  Grp  /\  E  ||  2 )  /\  x  e.  X  /\  y  e.  X
)  ->  G  e.  Grp )
6 simp2 989 . . . . . . . . 9  |-  ( ( ( G  e.  Grp  /\  E  ||  2 )  /\  x  e.  X  /\  y  e.  X
)  ->  x  e.  X )
7 simp3 990 . . . . . . . . 9  |-  ( ( ( G  e.  Grp  /\  E  ||  2 )  /\  x  e.  X  /\  y  e.  X
)  ->  y  e.  X )
8 eqid 2451 . . . . . . . . . 10  |-  ( +g  `  G )  =  ( +g  `  G )
91, 8grpass 15654 . . . . . . . . 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 1221 . . . . . . . 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 2451 . . . . . . . . . . . 12  |-  (.g `  G
)  =  (.g `  G
)
121, 11, 8mulg2 15738 . . . . . . . . . . 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 1013 . . . . . . . . . . 11  |-  ( ( ( G  e.  Grp  /\  E  ||  2 )  /\  x  e.  X  /\  y  e.  X
)  ->  E  ||  2
)
15 gexex.2 . . . . . . . . . . . 12  |-  E  =  (gEx `  G )
16 eqid 2451 . . . . . . . . . . . 12  |-  ( 0g
`  G )  =  ( 0g `  G
)
171, 15, 11, 16gexdvdsi 16186 . . . . . . . . . . 11  |-  ( ( G  e.  Grp  /\  y  e.  X  /\  E  ||  2 )  -> 
( 2 (.g `  G
) y )  =  ( 0g `  G
) )
185, 7, 14, 17syl3anc 1219 . . . . . . . . . 10  |-  ( ( ( G  e.  Grp  /\  E  ||  2 )  /\  x  e.  X  /\  y  e.  X
)  ->  ( 2 (.g `  G ) y )  =  ( 0g
`  G ) )
1913, 18eqtr3d 2494 . . . . . . . . 9  |-  ( ( ( G  e.  Grp  /\  E  ||  2 )  /\  x  e.  X  /\  y  e.  X
)  ->  ( y
( +g  `  G ) y )  =  ( 0g `  G ) )
2019oveq2d 6206 . . . . . . . 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 15671 . . . . . . . . 9  |-  ( ( G  e.  Grp  /\  x  e.  X )  ->  ( x ( +g  `  G ) ( 0g
`  G ) )  =  x )
225, 6, 21syl2anc 661 . . . . . . . 8  |-  ( ( ( G  e.  Grp  /\  E  ||  2 )  /\  x  e.  X  /\  y  e.  X
)  ->  ( x
( +g  `  G ) ( 0g `  G
) )  =  x )
2310, 20, 223eqtrd 2496 . . . . . . 7  |-  ( ( ( G  e.  Grp  /\  E  ||  2 )  /\  x  e.  X  /\  y  e.  X
)  ->  ( (
x ( +g  `  G
) y ) ( +g  `  G ) y )  =  x )
2423oveq1d 6205 . . . . . 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 15738 . . . . . . 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 16186 . . . . . . 7  |-  ( ( G  e.  Grp  /\  x  e.  X  /\  E  ||  2 )  -> 
( 2 (.g `  G
) x )  =  ( 0g `  G
) )
285, 6, 14, 27syl3anc 1219 . . . . . 6  |-  ( ( ( G  e.  Grp  /\  E  ||  2 )  /\  x  e.  X  /\  y  e.  X
)  ->  ( 2 (.g `  G ) x )  =  ( 0g
`  G ) )
2924, 26, 283eqtr2d 2498 . . . . 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 15653 . . . . . . 7  |-  ( ( G  e.  Grp  /\  x  e.  X  /\  y  e.  X )  ->  ( x ( +g  `  G ) y )  e.  X )
315, 6, 7, 30syl3anc 1219 . . . . . 6  |-  ( ( ( G  e.  Grp  /\  E  ||  2 )  /\  x  e.  X  /\  y  e.  X
)  ->  ( x
( +g  `  G ) y )  e.  X
)
321, 15, 11, 16gexdvdsi 16186 . . . . . 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 1219 . . . . 5  |-  ( ( ( G  e.  Grp  /\  E  ||  2 )  /\  x  e.  X  /\  y  e.  X
)  ->  ( 2 (.g `  G ) ( x ( +g  `  G
) y ) )  =  ( 0g `  G ) )
341, 11, 8mulg2 15738 . . . . . 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 2498 . . . 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 15654 . . . . 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 1221 . . . 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 2494 . . 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 15653 . . . . 5  |-  ( ( G  e.  Grp  /\  y  e.  X  /\  x  e.  X )  ->  ( y ( +g  `  G ) x )  e.  X )
415, 7, 6, 40syl3anc 1219 . . . 4  |-  ( ( ( G  e.  Grp  /\  E  ||  2 )  /\  x  e.  X  /\  y  e.  X
)  ->  ( y
( +g  `  G ) x )  e.  X
)
421, 8grplcan 15692 . . . 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 1221 . . 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 210 . 2  |-  ( ( ( G  e.  Grp  /\  E  ||  2 )  /\  x  e.  X  /\  y  e.  X
)  ->  ( x
( +g  `  G ) y )  =  ( y ( +g  `  G
) x ) )
452, 3, 4, 44isabld 16394 1  |-  ( ( G  e.  Grp  /\  E  ||  2 )  ->  G  e.  Abel )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758   class class class wbr 4390   ` cfv 5516  (class class class)co 6190   2c2 10472    || cdivides 13637   Basecbs 14276   +g cplusg 14340   0gc0g 14480   Grpcgrp 15512  .gcmg 15516  gExcgex 16133   Abelcabel 16382
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4501  ax-sep 4511  ax-nul 4519  ax-pow 4568  ax-pr 4629  ax-un 6472  ax-inf2 7948  ax-cnex 9439  ax-resscn 9440  ax-1cn 9441  ax-icn 9442  ax-addcl 9443  ax-addrcl 9444  ax-mulcl 9445  ax-mulrcl 9446  ax-mulcom 9447  ax-addass 9448  ax-mulass 9449  ax-distr 9450  ax-i2m1 9451  ax-1ne0 9452  ax-1rid 9453  ax-rnegex 9454  ax-rrecex 9455  ax-cnre 9456  ax-pre-lttri 9457  ax-pre-lttrn 9458  ax-pre-ltadd 9459  ax-pre-mulgt0 9460
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rmo 2803  df-rab 2804  df-v 3070  df-sbc 3285  df-csb 3387  df-dif 3429  df-un 3431  df-in 3433  df-ss 3440  df-pss 3442  df-nul 3736  df-if 3890  df-pw 3960  df-sn 3976  df-pr 3978  df-tp 3980  df-op 3982  df-uni 4190  df-iun 4271  df-br 4391  df-opab 4449  df-mpt 4450  df-tr 4484  df-eprel 4730  df-id 4734  df-po 4739  df-so 4740  df-fr 4777  df-we 4779  df-ord 4820  df-on 4821  df-lim 4822  df-suc 4823  df-xp 4944  df-rel 4945  df-cnv 4946  df-co 4947  df-dm 4948  df-rn 4949  df-res 4950  df-ima 4951  df-iota 5479  df-fun 5518  df-fn 5519  df-f 5520  df-f1 5521  df-fo 5522  df-f1o 5523  df-fv 5524  df-riota 6151  df-ov 6193  df-oprab 6194  df-mpt2 6195  df-om 6577  df-1st 6677  df-2nd 6678  df-recs 6932  df-rdg 6966  df-er 7201  df-en 7411  df-dom 7412  df-sdom 7413  df-sup 7792  df-pnf 9521  df-mnf 9522  df-xr 9523  df-ltxr 9524  df-le 9525  df-sub 9698  df-neg 9699  df-nn 10424  df-2 10481  df-n0 10681  df-z 10748  df-uz 10963  df-fz 11539  df-seq 11908  df-dvds 13638  df-0g 14482  df-mnd 15517  df-grp 15647  df-minusg 15648  df-mulg 15650  df-gex 16137  df-cmn 16383  df-abl 16384
This theorem is referenced by:  lt6abl  16475
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