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Theorem gex2abl 17059
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 2455 . 2  |-  ( ( G  e.  Grp  /\  E  ||  2 )  -> 
( +g  `  G )  =  ( +g  `  G
) )
4 simpl 455 . 2  |-  ( ( G  e.  Grp  /\  E  ||  2 )  ->  G  e.  Grp )
5 simp1l 1018 . . . . . . . . 9  |-  ( ( ( G  e.  Grp  /\  E  ||  2 )  /\  x  e.  X  /\  y  e.  X
)  ->  G  e.  Grp )
6 simp2 995 . . . . . . . . 9  |-  ( ( ( G  e.  Grp  /\  E  ||  2 )  /\  x  e.  X  /\  y  e.  X
)  ->  x  e.  X )
7 simp3 996 . . . . . . . . 9  |-  ( ( ( G  e.  Grp  /\  E  ||  2 )  /\  x  e.  X  /\  y  e.  X
)  ->  y  e.  X )
8 eqid 2454 . . . . . . . . . 10  |-  ( +g  `  G )  =  ( +g  `  G )
91, 8grpass 16266 . . . . . . . . 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 1228 . . . . . . . 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 2454 . . . . . . . . . . . 12  |-  (.g `  G
)  =  (.g `  G
)
121, 11, 8mulg2 16353 . . . . . . . . . . 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 1019 . . . . . . . . . . 11  |-  ( ( ( G  e.  Grp  /\  E  ||  2 )  /\  x  e.  X  /\  y  e.  X
)  ->  E  ||  2
)
15 gexex.2 . . . . . . . . . . . 12  |-  E  =  (gEx `  G )
16 eqid 2454 . . . . . . . . . . . 12  |-  ( 0g
`  G )  =  ( 0g `  G
)
171, 15, 11, 16gexdvdsi 16805 . . . . . . . . . . 11  |-  ( ( G  e.  Grp  /\  y  e.  X  /\  E  ||  2 )  -> 
( 2 (.g `  G
) y )  =  ( 0g `  G
) )
185, 7, 14, 17syl3anc 1226 . . . . . . . . . 10  |-  ( ( ( G  e.  Grp  /\  E  ||  2 )  /\  x  e.  X  /\  y  e.  X
)  ->  ( 2 (.g `  G ) y )  =  ( 0g
`  G ) )
1913, 18eqtr3d 2497 . . . . . . . . 9  |-  ( ( ( G  e.  Grp  /\  E  ||  2 )  /\  x  e.  X  /\  y  e.  X
)  ->  ( y
( +g  `  G ) y )  =  ( 0g `  G ) )
2019oveq2d 6286 . . . . . . . 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 16283 . . . . . . . . 9  |-  ( ( G  e.  Grp  /\  x  e.  X )  ->  ( x ( +g  `  G ) ( 0g
`  G ) )  =  x )
225, 6, 21syl2anc 659 . . . . . . . 8  |-  ( ( ( G  e.  Grp  /\  E  ||  2 )  /\  x  e.  X  /\  y  e.  X
)  ->  ( x
( +g  `  G ) ( 0g `  G
) )  =  x )
2310, 20, 223eqtrd 2499 . . . . . . 7  |-  ( ( ( G  e.  Grp  /\  E  ||  2 )  /\  x  e.  X  /\  y  e.  X
)  ->  ( (
x ( +g  `  G
) y ) ( +g  `  G ) y )  =  x )
2423oveq1d 6285 . . . . . 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 16353 . . . . . . 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 16805 . . . . . . 7  |-  ( ( G  e.  Grp  /\  x  e.  X  /\  E  ||  2 )  -> 
( 2 (.g `  G
) x )  =  ( 0g `  G
) )
285, 6, 14, 27syl3anc 1226 . . . . . 6  |-  ( ( ( G  e.  Grp  /\  E  ||  2 )  /\  x  e.  X  /\  y  e.  X
)  ->  ( 2 (.g `  G ) x )  =  ( 0g
`  G ) )
2924, 26, 283eqtr2d 2501 . . . . 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 16265 . . . . . . 7  |-  ( ( G  e.  Grp  /\  x  e.  X  /\  y  e.  X )  ->  ( x ( +g  `  G ) y )  e.  X )
315, 6, 7, 30syl3anc 1226 . . . . . 6  |-  ( ( ( G  e.  Grp  /\  E  ||  2 )  /\  x  e.  X  /\  y  e.  X
)  ->  ( x
( +g  `  G ) y )  e.  X
)
321, 15, 11, 16gexdvdsi 16805 . . . . . 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 1226 . . . . 5  |-  ( ( ( G  e.  Grp  /\  E  ||  2 )  /\  x  e.  X  /\  y  e.  X
)  ->  ( 2 (.g `  G ) ( x ( +g  `  G
) y ) )  =  ( 0g `  G ) )
341, 11, 8mulg2 16353 . . . . . 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 2501 . . . 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 16266 . . . . 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 1228 . . . 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 2497 . . 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 16265 . . . . 5  |-  ( ( G  e.  Grp  /\  y  e.  X  /\  x  e.  X )  ->  ( y ( +g  `  G ) x )  e.  X )
415, 7, 6, 40syl3anc 1226 . . . 4  |-  ( ( ( G  e.  Grp  /\  E  ||  2 )  /\  x  e.  X  /\  y  e.  X
)  ->  ( y
( +g  `  G ) x )  e.  X
)
421, 8grplcan 16304 . . . 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 1228 . . 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 17013 1  |-  ( ( G  e.  Grp  /\  E  ||  2 )  ->  G  e.  Abel )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    /\ w3a 971    = wceq 1398    e. wcel 1823   class class class wbr 4439   ` cfv 5570  (class class class)co 6270   2c2 10581    || cdvds 14073   Basecbs 14719   +g cplusg 14787   0gc0g 14932   Grpcgrp 16255  .gcmg 16258  gExcgex 16752   Abelcabl 17001
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-inf2 8049  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-1st 6773  df-2nd 6774  df-recs 7034  df-rdg 7068  df-er 7303  df-en 7510  df-dom 7511  df-sdom 7512  df-sup 7893  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-nn 10532  df-2 10590  df-n0 10792  df-z 10861  df-uz 11083  df-fz 11676  df-seq 12093  df-dvds 14074  df-0g 14934  df-mgm 16074  df-sgrp 16113  df-mnd 16123  df-grp 16259  df-minusg 16260  df-mulg 16262  df-gex 16756  df-cmn 17002  df-abl 17003
This theorem is referenced by:  lt6abl  17099
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