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Theorem gex1 14902
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 731 . . . . . . . . 9  |-  ( ( ( G  e.  Mnd  /\  E  =  1 )  /\  x  e.  X
)  ->  E  = 
1 )
21oveq1d 5873 . . . . . . . 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 2283 . . . . . . . . . 10  |-  (.g `  G
)  =  (.g `  G
)
6 eqid 2283 . . . . . . . . . 10  |-  ( 0g
`  G )  =  ( 0g `  G
)
73, 4, 5, 6gexid 14892 . . . . . . . . 9  |-  ( x  e.  X  ->  ( E (.g `  G ) x )  =  ( 0g
`  G ) )
87adantl 452 . . . . . . . 8  |-  ( ( ( G  e.  Mnd  /\  E  =  1 )  /\  x  e.  X
)  ->  ( E
(.g `  G ) x )  =  ( 0g
`  G ) )
93, 5mulg1 14574 . . . . . . . . 9  |-  ( x  e.  X  ->  (
1 (.g `  G ) x )  =  x )
109adantl 452 . . . . . . . 8  |-  ( ( ( G  e.  Mnd  /\  E  =  1 )  /\  x  e.  X
)  ->  ( 1 (.g `  G ) x )  =  x )
112, 8, 103eqtr3rd 2324 . . . . . . 7  |-  ( ( ( G  e.  Mnd  /\  E  =  1 )  /\  x  e.  X
)  ->  x  =  ( 0g `  G ) )
12 elsn 3655 . . . . . . 7  |-  ( x  e.  { ( 0g
`  G ) }  <-> 
x  =  ( 0g
`  G ) )
1311, 12sylibr 203 . . . . . 6  |-  ( ( ( G  e.  Mnd  /\  E  =  1 )  /\  x  e.  X
)  ->  x  e.  { ( 0g `  G
) } )
1413ex 423 . . . . 5  |-  ( ( G  e.  Mnd  /\  E  =  1 )  ->  ( x  e.  X  ->  x  e.  { ( 0g `  G
) } ) )
1514ssrdv 3185 . . . 4  |-  ( ( G  e.  Mnd  /\  E  =  1 )  ->  X  C_  { ( 0g `  G ) } )
163, 6mndidcl 14391 . . . . . 6  |-  ( G  e.  Mnd  ->  ( 0g `  G )  e.  X )
1716adantr 451 . . . . 5  |-  ( ( G  e.  Mnd  /\  E  =  1 )  ->  ( 0g `  G )  e.  X
)
1817snssd 3760 . . . 4  |-  ( ( G  e.  Mnd  /\  E  =  1 )  ->  { ( 0g
`  G ) } 
C_  X )
1915, 18eqssd 3196 . . 3  |-  ( ( G  e.  Mnd  /\  E  =  1 )  ->  X  =  {
( 0g `  G
) } )
20 fvex 5539 . . . 4  |-  ( 0g
`  G )  e. 
_V
2120ensn1 6925 . . 3  |-  { ( 0g `  G ) }  ~~  1o
2219, 21syl6eqbr 4060 . 2  |-  ( ( G  e.  Mnd  /\  E  =  1 )  ->  X  ~~  1o )
23 simpl 443 . . . 4  |-  ( ( G  e.  Mnd  /\  X  ~~  1o )  ->  G  e.  Mnd )
24 1nn 9757 . . . . 5  |-  1  e.  NN
2524a1i 10 . . . 4  |-  ( ( G  e.  Mnd  /\  X  ~~  1o )  -> 
1  e.  NN )
269adantl 452 . . . . . 6  |-  ( ( ( G  e.  Mnd  /\  X  ~~  1o )  /\  x  e.  X
)  ->  ( 1 (.g `  G ) x )  =  x )
27 en1eqsn 7088 . . . . . . . . . 10  |-  ( ( ( 0g `  G
)  e.  X  /\  X  ~~  1o )  ->  X  =  { ( 0g `  G ) } )
2816, 27sylan 457 . . . . . . . . 9  |-  ( ( G  e.  Mnd  /\  X  ~~  1o )  ->  X  =  { ( 0g `  G ) } )
2928eleq2d 2350 . . . . . . . 8  |-  ( ( G  e.  Mnd  /\  X  ~~  1o )  -> 
( x  e.  X  <->  x  e.  { ( 0g
`  G ) } ) )
3029biimpa 470 . . . . . . 7  |-  ( ( ( G  e.  Mnd  /\  X  ~~  1o )  /\  x  e.  X
)  ->  x  e.  { ( 0g `  G
) } )
3130, 12sylib 188 . . . . . 6  |-  ( ( ( G  e.  Mnd  /\  X  ~~  1o )  /\  x  e.  X
)  ->  x  =  ( 0g `  G ) )
3226, 31eqtrd 2315 . . . . 5  |-  ( ( ( G  e.  Mnd  /\  X  ~~  1o )  /\  x  e.  X
)  ->  ( 1 (.g `  G ) x )  =  ( 0g
`  G ) )
3332ralrimiva 2626 . . . 4  |-  ( ( G  e.  Mnd  /\  X  ~~  1o )  ->  A. x  e.  X  ( 1 (.g `  G
) x )  =  ( 0g `  G
) )
343, 4, 5, 6gexlem2 14893 . . . 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 1182 . . 3  |-  ( ( G  e.  Mnd  /\  X  ~~  1o )  ->  E  e.  ( 1 ... 1 ) )
36 elfz1eq 10807 . . 3  |-  ( E  e.  ( 1 ... 1 )  ->  E  =  1 )
3735, 36syl 15 . 2  |-  ( ( G  e.  Mnd  /\  X  ~~  1o )  ->  E  =  1 )
3822, 37impbida 805 1  |-  ( G  e.  Mnd  ->  ( E  =  1  <->  X  ~~  1o ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543   {csn 3640   class class class wbr 4023   ` cfv 5255  (class class class)co 5858   1oc1o 6472    ~~ cen 6860   1c1 8738   NNcn 9746   ...cfz 10782   Basecbs 13148   0gc0g 13400   Mndcmnd 14361  .gcmg 14366  gExcgex 14841
This theorem is referenced by:  pgpfac1lem3a  15311  pgpfaclem3  15318
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-1o 6479  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  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-n0 9966  df-z 10025  df-uz 10231  df-fz 10783  df-seq 11047  df-0g 13404  df-mnd 14367  df-mulg 14492  df-gex 14845
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