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Theorem gex1 15153
Description: A group or monoid has exponent 1 iff it is trivial. (Contributed by Mario Carneiro, 24-Apr-2016.)
Hypotheses
Ref Expression
gexcl2.1  |-  X  =  ( Base `  G
)
gexcl2.2  |-  E  =  (gEx `  G )
Assertion
Ref Expression
gex1  |-  ( G  e.  Mnd  ->  ( E  =  1  <->  X  ~~  1o ) )

Proof of Theorem gex1
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 simplr 732 . . . . . . . . 9  |-  ( ( ( G  e.  Mnd  /\  E  =  1 )  /\  x  e.  X
)  ->  E  = 
1 )
21oveq1d 6036 . . . . . . . 8  |-  ( ( ( G  e.  Mnd  /\  E  =  1 )  /\  x  e.  X
)  ->  ( E
(.g `  G ) x )  =  ( 1 (.g `  G ) x ) )
3 gexcl2.1 . . . . . . . . . 10  |-  X  =  ( Base `  G
)
4 gexcl2.2 . . . . . . . . . 10  |-  E  =  (gEx `  G )
5 eqid 2388 . . . . . . . . . 10  |-  (.g `  G
)  =  (.g `  G
)
6 eqid 2388 . . . . . . . . . 10  |-  ( 0g
`  G )  =  ( 0g `  G
)
73, 4, 5, 6gexid 15143 . . . . . . . . 9  |-  ( x  e.  X  ->  ( E (.g `  G ) x )  =  ( 0g
`  G ) )
87adantl 453 . . . . . . . 8  |-  ( ( ( G  e.  Mnd  /\  E  =  1 )  /\  x  e.  X
)  ->  ( E
(.g `  G ) x )  =  ( 0g
`  G ) )
93, 5mulg1 14825 . . . . . . . . 9  |-  ( x  e.  X  ->  (
1 (.g `  G ) x )  =  x )
109adantl 453 . . . . . . . 8  |-  ( ( ( G  e.  Mnd  /\  E  =  1 )  /\  x  e.  X
)  ->  ( 1 (.g `  G ) x )  =  x )
112, 8, 103eqtr3rd 2429 . . . . . . 7  |-  ( ( ( G  e.  Mnd  /\  E  =  1 )  /\  x  e.  X
)  ->  x  =  ( 0g `  G ) )
12 elsn 3773 . . . . . . 7  |-  ( x  e.  { ( 0g
`  G ) }  <-> 
x  =  ( 0g
`  G ) )
1311, 12sylibr 204 . . . . . 6  |-  ( ( ( G  e.  Mnd  /\  E  =  1 )  /\  x  e.  X
)  ->  x  e.  { ( 0g `  G
) } )
1413ex 424 . . . . 5  |-  ( ( G  e.  Mnd  /\  E  =  1 )  ->  ( x  e.  X  ->  x  e.  { ( 0g `  G
) } ) )
1514ssrdv 3298 . . . 4  |-  ( ( G  e.  Mnd  /\  E  =  1 )  ->  X  C_  { ( 0g `  G ) } )
163, 6mndidcl 14642 . . . . . 6  |-  ( G  e.  Mnd  ->  ( 0g `  G )  e.  X )
1716adantr 452 . . . . 5  |-  ( ( G  e.  Mnd  /\  E  =  1 )  ->  ( 0g `  G )  e.  X
)
1817snssd 3887 . . . 4  |-  ( ( G  e.  Mnd  /\  E  =  1 )  ->  { ( 0g
`  G ) } 
C_  X )
1915, 18eqssd 3309 . . 3  |-  ( ( G  e.  Mnd  /\  E  =  1 )  ->  X  =  {
( 0g `  G
) } )
20 fvex 5683 . . . 4  |-  ( 0g
`  G )  e. 
_V
2120ensn1 7108 . . 3  |-  { ( 0g `  G ) }  ~~  1o
2219, 21syl6eqbr 4191 . 2  |-  ( ( G  e.  Mnd  /\  E  =  1 )  ->  X  ~~  1o )
23 simpl 444 . . . 4  |-  ( ( G  e.  Mnd  /\  X  ~~  1o )  ->  G  e.  Mnd )
24 1nn 9944 . . . . 5  |-  1  e.  NN
2524a1i 11 . . . 4  |-  ( ( G  e.  Mnd  /\  X  ~~  1o )  -> 
1  e.  NN )
269adantl 453 . . . . . 6  |-  ( ( ( G  e.  Mnd  /\  X  ~~  1o )  /\  x  e.  X
)  ->  ( 1 (.g `  G ) x )  =  x )
27 en1eqsn 7275 . . . . . . . . . 10  |-  ( ( ( 0g `  G
)  e.  X  /\  X  ~~  1o )  ->  X  =  { ( 0g `  G ) } )
2816, 27sylan 458 . . . . . . . . 9  |-  ( ( G  e.  Mnd  /\  X  ~~  1o )  ->  X  =  { ( 0g `  G ) } )
2928eleq2d 2455 . . . . . . . 8  |-  ( ( G  e.  Mnd  /\  X  ~~  1o )  -> 
( x  e.  X  <->  x  e.  { ( 0g
`  G ) } ) )
3029biimpa 471 . . . . . . 7  |-  ( ( ( G  e.  Mnd  /\  X  ~~  1o )  /\  x  e.  X
)  ->  x  e.  { ( 0g `  G
) } )
3130, 12sylib 189 . . . . . 6  |-  ( ( ( G  e.  Mnd  /\  X  ~~  1o )  /\  x  e.  X
)  ->  x  =  ( 0g `  G ) )
3226, 31eqtrd 2420 . . . . 5  |-  ( ( ( G  e.  Mnd  /\  X  ~~  1o )  /\  x  e.  X
)  ->  ( 1 (.g `  G ) x )  =  ( 0g
`  G ) )
3332ralrimiva 2733 . . . 4  |-  ( ( G  e.  Mnd  /\  X  ~~  1o )  ->  A. x  e.  X  ( 1 (.g `  G
) x )  =  ( 0g `  G
) )
343, 4, 5, 6gexlem2 15144 . . . 4  |-  ( ( G  e.  Mnd  /\  1  e.  NN  /\  A. x  e.  X  (
1 (.g `  G ) x )  =  ( 0g
`  G ) )  ->  E  e.  ( 1 ... 1 ) )
3523, 25, 33, 34syl3anc 1184 . . 3  |-  ( ( G  e.  Mnd  /\  X  ~~  1o )  ->  E  e.  ( 1 ... 1 ) )
36 elfz1eq 11001 . . 3  |-  ( E  e.  ( 1 ... 1 )  ->  E  =  1 )
3735, 36syl 16 . 2  |-  ( ( G  e.  Mnd  /\  X  ~~  1o )  ->  E  =  1 )
3822, 37impbida 806 1  |-  ( G  e.  Mnd  ->  ( E  =  1  <->  X  ~~  1o ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1717   A.wral 2650   {csn 3758   class class class wbr 4154   ` cfv 5395  (class class class)co 6021   1oc1o 6654    ~~ cen 7043   1c1 8925   NNcn 9933   ...cfz 10976   Basecbs 13397   0gc0g 13651   Mndcmnd 14612  .gcmg 14617  gExcgex 15092
This theorem is referenced by:  pgpfac1lem3a  15562  pgpfaclem3  15569
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-rep 4262  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345  ax-un 4642  ax-inf2 7530  ax-cnex 8980  ax-resscn 8981  ax-1cn 8982  ax-icn 8983  ax-addcl 8984  ax-addrcl 8985  ax-mulcl 8986  ax-mulrcl 8987  ax-mulcom 8988  ax-addass 8989  ax-mulass 8990  ax-distr 8991  ax-i2m1 8992  ax-1ne0 8993  ax-1rid 8994  ax-rnegex 8995  ax-rrecex 8996  ax-cnre 8997  ax-pre-lttri 8998  ax-pre-lttrn 8999  ax-pre-ltadd 9000  ax-pre-mulgt0 9001
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-nel 2554  df-ral 2655  df-rex 2656  df-reu 2657  df-rmo 2658  df-rab 2659  df-v 2902  df-sbc 3106  df-csb 3196  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-pss 3280  df-nul 3573  df-if 3684  df-pw 3745  df-sn 3764  df-pr 3765  df-tp 3766  df-op 3767  df-uni 3959  df-iun 4038  df-br 4155  df-opab 4209  df-mpt 4210  df-tr 4245  df-eprel 4436  df-id 4440  df-po 4445  df-so 4446  df-fr 4483  df-we 4485  df-ord 4526  df-on 4527  df-lim 4528  df-suc 4529  df-om 4787  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-f1 5400  df-fo 5401  df-f1o 5402  df-fv 5403  df-ov 6024  df-oprab 6025  df-mpt2 6026  df-1st 6289  df-2nd 6290  df-riota 6486  df-recs 6570  df-rdg 6605  df-1o 6661  df-er 6842  df-en 7047  df-dom 7048  df-sdom 7049  df-fin 7050  df-sup 7382  df-pnf 9056  df-mnf 9057  df-xr 9058  df-ltxr 9059  df-le 9060  df-sub 9226  df-neg 9227  df-nn 9934  df-n0 10155  df-z 10216  df-uz 10422  df-fz 10977  df-seq 11252  df-0g 13655  df-mnd 14618  df-mulg 14743  df-gex 15096
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