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Theorem gex2abl 15143
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 2284 . 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 2283 . . . . . . . . . 10  |-  ( +g  `  G )  =  ( +g  `  G )
91, 8grpass 14496 . . . . . . . . 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 2283 . . . . . . . . . . . 12  |-  (.g `  G
)  =  (.g `  G
)
121, 11, 8mulg2 14576 . . . . . . . . . . 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 2283 . . . . . . . . . . . 12  |-  ( 0g
`  G )  =  ( 0g `  G
)
171, 15, 11, 16gexdvdsi 14894 . . . . . . . . . . 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 2317 . . . . . . . . 9  |-  ( ( ( G  e.  Grp  /\  E  ||  2 )  /\  x  e.  X  /\  y  e.  X
)  ->  ( y
( +g  `  G ) y )  =  ( 0g `  G ) )
2019oveq2d 5874 . . . . . . . 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 14513 . . . . . . . . 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 2319 . . . . . . 7  |-  ( ( ( G  e.  Grp  /\  E  ||  2 )  /\  x  e.  X  /\  y  e.  X
)  ->  ( (
x ( +g  `  G
) y ) ( +g  `  G ) y )  =  x )
2423oveq1d 5873 . . . . . 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 14576 . . . . . . 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 14894 . . . . . . 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 2321 . . . . 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 14495 . . . . . . 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 14894 . . . . . 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 14576 . . . . . 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 2321 . . . 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 14496 . . . . 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 2317 . . 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 14495 . . . . 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 14534 . . . 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 15102 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 1623    e. wcel 1684   class class class wbr 4023   ` cfv 5255  (class class class)co 5858   2c2 9795    || cdivides 12531   Basecbs 13148   +g cplusg 13208   0gc0g 13400   Grpcgrp 14362  .gcmg 14366  gExcgex 14841   Abelcabel 15090
This theorem is referenced by:  lt6abl  15181
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-sup 7194  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-2 9804  df-n0 9966  df-z 10025  df-uz 10231  df-fz 10783  df-seq 11047  df-dvds 12532  df-0g 13404  df-mnd 14367  df-grp 14489  df-minusg 14490  df-mulg 14492  df-gex 14845  df-cmn 15091  df-abl 15092
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