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Theorem gex1 15217
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 6088 . . . . . . . 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 2435 . . . . . . . . . 10  |-  (.g `  G
)  =  (.g `  G
)
6 eqid 2435 . . . . . . . . . 10  |-  ( 0g
`  G )  =  ( 0g `  G
)
73, 4, 5, 6gexid 15207 . . . . . . . . 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 14889 . . . . . . . . 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 2476 . . . . . . 7  |-  ( ( ( G  e.  Mnd  /\  E  =  1 )  /\  x  e.  X
)  ->  x  =  ( 0g `  G ) )
12 elsn 3821 . . . . . . 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 3346 . . . 4  |-  ( ( G  e.  Mnd  /\  E  =  1 )  ->  X  C_  { ( 0g `  G ) } )
163, 6mndidcl 14706 . . . . . 6  |-  ( G  e.  Mnd  ->  ( 0g `  G )  e.  X )
1716adantr 452 . . . . 5  |-  ( ( G  e.  Mnd  /\  E  =  1 )  ->  ( 0g `  G )  e.  X
)
1817snssd 3935 . . . 4  |-  ( ( G  e.  Mnd  /\  E  =  1 )  ->  { ( 0g
`  G ) } 
C_  X )
1915, 18eqssd 3357 . . 3  |-  ( ( G  e.  Mnd  /\  E  =  1 )  ->  X  =  {
( 0g `  G
) } )
20 fvex 5734 . . . 4  |-  ( 0g
`  G )  e. 
_V
2120ensn1 7163 . . 3  |-  { ( 0g `  G ) }  ~~  1o
2219, 21syl6eqbr 4241 . 2  |-  ( ( G  e.  Mnd  /\  E  =  1 )  ->  X  ~~  1o )
23 simpl 444 . . . 4  |-  ( ( G  e.  Mnd  /\  X  ~~  1o )  ->  G  e.  Mnd )
24 1nn 10003 . . . . 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 7330 . . . . . . . . . 10  |-  ( ( ( 0g `  G
)  e.  X  /\  X  ~~  1o )  ->  X  =  { ( 0g `  G ) } )
2816, 27sylan 458 . . . . . . . . 9  |-  ( ( G  e.  Mnd  /\  X  ~~  1o )  ->  X  =  { ( 0g `  G ) } )
2928eleq2d 2502 . . . . . . . 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 2467 . . . . 5  |-  ( ( ( G  e.  Mnd  /\  X  ~~  1o )  /\  x  e.  X
)  ->  ( 1 (.g `  G ) x )  =  ( 0g
`  G ) )
3332ralrimiva 2781 . . . 4  |-  ( ( G  e.  Mnd  /\  X  ~~  1o )  ->  A. x  e.  X  ( 1 (.g `  G
) x )  =  ( 0g `  G
) )
343, 4, 5, 6gexlem2 15208 . . . 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 11060 . . 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 1652    e. wcel 1725   A.wral 2697   {csn 3806   class class class wbr 4204   ` cfv 5446  (class class class)co 6073   1oc1o 6709    ~~ cen 7098   1c1 8983   NNcn 9992   ...cfz 11035   Basecbs 13461   0gc0g 13715   Mndcmnd 14676  .gcmg 14681  gExcgex 15156
This theorem is referenced by:  pgpfac1lem3a  15626  pgpfaclem3  15633
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-inf2 7588  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-1o 6716  df-er 6897  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-sup 7438  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-nn 9993  df-n0 10214  df-z 10275  df-uz 10481  df-fz 11036  df-seq 11316  df-0g 13719  df-mnd 14682  df-mulg 14807  df-gex 15160
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