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Theorem symggrp 16296
Description: The symmetric group on a set  A is a group. (Contributed by Paul Chapman, 25-Feb-2008.) (Revised by Mario Carneiro, 13-Jan-2015.)
Hypothesis
Ref Expression
symggrp.1  |-  G  =  ( SymGrp `  A )
Assertion
Ref Expression
symggrp  |-  ( A  e.  V  ->  G  e.  Grp )

Proof of Theorem symggrp
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqidd 2442 . 2  |-  ( A  e.  V  ->  ( Base `  G )  =  ( Base `  G
) )
2 eqidd 2442 . 2  |-  ( A  e.  V  ->  ( +g  `  G )  =  ( +g  `  G
) )
3 symggrp.1 . . . 4  |-  G  =  ( SymGrp `  A )
4 eqid 2441 . . . 4  |-  ( Base `  G )  =  (
Base `  G )
5 eqid 2441 . . . 4  |-  ( +g  `  G )  =  ( +g  `  G )
63, 4, 5symgcl 16287 . . 3  |-  ( ( x  e.  ( Base `  G )  /\  y  e.  ( Base `  G
) )  ->  (
x ( +g  `  G
) y )  e.  ( Base `  G
) )
763adant1 1013 . 2  |-  ( ( A  e.  V  /\  x  e.  ( Base `  G )  /\  y  e.  ( Base `  G
) )  ->  (
x ( +g  `  G
) y )  e.  ( Base `  G
) )
8 coass 5513 . . . 4  |-  ( ( x  o.  y )  o.  z )  =  ( x  o.  (
y  o.  z ) )
9 simpr1 1001 . . . . . 6  |-  ( ( A  e.  V  /\  ( x  e.  ( Base `  G )  /\  y  e.  ( Base `  G )  /\  z  e.  ( Base `  G
) ) )  ->  x  e.  ( Base `  G ) )
10 simpr2 1002 . . . . . 6  |-  ( ( A  e.  V  /\  ( x  e.  ( Base `  G )  /\  y  e.  ( Base `  G )  /\  z  e.  ( Base `  G
) ) )  -> 
y  e.  ( Base `  G ) )
113, 4, 5symgov 16286 . . . . . 6  |-  ( ( x  e.  ( Base `  G )  /\  y  e.  ( Base `  G
) )  ->  (
x ( +g  `  G
) y )  =  ( x  o.  y
) )
129, 10, 11syl2anc 661 . . . . 5  |-  ( ( A  e.  V  /\  ( x  e.  ( Base `  G )  /\  y  e.  ( Base `  G )  /\  z  e.  ( Base `  G
) ) )  -> 
( x ( +g  `  G ) y )  =  ( x  o.  y ) )
1312coeq1d 5151 . . . 4  |-  ( ( A  e.  V  /\  ( x  e.  ( Base `  G )  /\  y  e.  ( Base `  G )  /\  z  e.  ( Base `  G
) ) )  -> 
( ( x ( +g  `  G ) y )  o.  z
)  =  ( ( x  o.  y )  o.  z ) )
14 simpr3 1003 . . . . . 6  |-  ( ( A  e.  V  /\  ( x  e.  ( Base `  G )  /\  y  e.  ( Base `  G )  /\  z  e.  ( Base `  G
) ) )  -> 
z  e.  ( Base `  G ) )
153, 4, 5symgov 16286 . . . . . 6  |-  ( ( y  e.  ( Base `  G )  /\  z  e.  ( Base `  G
) )  ->  (
y ( +g  `  G
) z )  =  ( y  o.  z
) )
1610, 14, 15syl2anc 661 . . . . 5  |-  ( ( A  e.  V  /\  ( x  e.  ( Base `  G )  /\  y  e.  ( Base `  G )  /\  z  e.  ( Base `  G
) ) )  -> 
( y ( +g  `  G ) z )  =  ( y  o.  z ) )
1716coeq2d 5152 . . . 4  |-  ( ( A  e.  V  /\  ( x  e.  ( Base `  G )  /\  y  e.  ( Base `  G )  /\  z  e.  ( Base `  G
) ) )  -> 
( x  o.  (
y ( +g  `  G
) z ) )  =  ( x  o.  ( y  o.  z
) ) )
188, 13, 173eqtr4a 2508 . . 3  |-  ( ( A  e.  V  /\  ( x  e.  ( Base `  G )  /\  y  e.  ( Base `  G )  /\  z  e.  ( Base `  G
) ) )  -> 
( ( x ( +g  `  G ) y )  o.  z
)  =  ( x  o.  ( y ( +g  `  G ) z ) ) )
199, 10, 6syl2anc 661 . . . 4  |-  ( ( A  e.  V  /\  ( x  e.  ( Base `  G )  /\  y  e.  ( Base `  G )  /\  z  e.  ( Base `  G
) ) )  -> 
( x ( +g  `  G ) y )  e.  ( Base `  G
) )
203, 4, 5symgov 16286 . . . 4  |-  ( ( ( x ( +g  `  G ) y )  e.  ( Base `  G
)  /\  z  e.  ( Base `  G )
)  ->  ( (
x ( +g  `  G
) y ) ( +g  `  G ) z )  =  ( ( x ( +g  `  G ) y )  o.  z ) )
2119, 14, 20syl2anc 661 . . 3  |-  ( ( A  e.  V  /\  ( x  e.  ( Base `  G )  /\  y  e.  ( Base `  G )  /\  z  e.  ( Base `  G
) ) )  -> 
( ( x ( +g  `  G ) y ) ( +g  `  G ) z )  =  ( ( x ( +g  `  G
) y )  o.  z ) )
223, 4, 5symgcl 16287 . . . . 5  |-  ( ( y  e.  ( Base `  G )  /\  z  e.  ( Base `  G
) )  ->  (
y ( +g  `  G
) z )  e.  ( Base `  G
) )
2310, 14, 22syl2anc 661 . . . 4  |-  ( ( A  e.  V  /\  ( x  e.  ( Base `  G )  /\  y  e.  ( Base `  G )  /\  z  e.  ( Base `  G
) ) )  -> 
( y ( +g  `  G ) z )  e.  ( Base `  G
) )
243, 4, 5symgov 16286 . . . 4  |-  ( ( x  e.  ( Base `  G )  /\  (
y ( +g  `  G
) z )  e.  ( Base `  G
) )  ->  (
x ( +g  `  G
) ( y ( +g  `  G ) z ) )  =  ( x  o.  (
y ( +g  `  G
) z ) ) )
259, 23, 24syl2anc 661 . . 3  |-  ( ( A  e.  V  /\  ( x  e.  ( Base `  G )  /\  y  e.  ( Base `  G )  /\  z  e.  ( Base `  G
) ) )  -> 
( x ( +g  `  G ) ( y ( +g  `  G
) z ) )  =  ( x  o.  ( y ( +g  `  G ) z ) ) )
2618, 21, 253eqtr4d 2492 . 2  |-  ( ( A  e.  V  /\  ( x  e.  ( Base `  G )  /\  y  e.  ( Base `  G )  /\  z  e.  ( Base `  G
) ) )  -> 
( ( x ( +g  `  G ) y ) ( +g  `  G ) z )  =  ( x ( +g  `  G ) ( y ( +g  `  G ) z ) ) )
27 f1oi 5838 . . 3  |-  (  _I  |`  A ) : A -1-1-onto-> A
283, 4elsymgbas 16278 . . 3  |-  ( A  e.  V  ->  (
(  _I  |`  A )  e.  ( Base `  G
)  <->  (  _I  |`  A ) : A -1-1-onto-> A ) )
2927, 28mpbiri 233 . 2  |-  ( A  e.  V  ->  (  _I  |`  A )  e.  ( Base `  G
) )
303, 4, 5symgov 16286 . . . 4  |-  ( ( (  _I  |`  A )  e.  ( Base `  G
)  /\  x  e.  ( Base `  G )
)  ->  ( (  _I  |`  A ) ( +g  `  G ) x )  =  ( (  _I  |`  A )  o.  x ) )
3129, 30sylan 471 . . 3  |-  ( ( A  e.  V  /\  x  e.  ( Base `  G ) )  -> 
( (  _I  |`  A ) ( +g  `  G
) x )  =  ( (  _I  |`  A )  o.  x ) )
323, 4elsymgbas 16278 . . . . 5  |-  ( A  e.  V  ->  (
x  e.  ( Base `  G )  <->  x : A
-1-1-onto-> A ) )
3332biimpa 484 . . . 4  |-  ( ( A  e.  V  /\  x  e.  ( Base `  G ) )  ->  x : A -1-1-onto-> A )
34 f1of 5803 . . . 4  |-  ( x : A -1-1-onto-> A  ->  x : A
--> A )
35 fcoi2 5747 . . . 4  |-  ( x : A --> A  -> 
( (  _I  |`  A )  o.  x )  =  x )
3633, 34, 353syl 20 . . 3  |-  ( ( A  e.  V  /\  x  e.  ( Base `  G ) )  -> 
( (  _I  |`  A )  o.  x )  =  x )
3731, 36eqtrd 2482 . 2  |-  ( ( A  e.  V  /\  x  e.  ( Base `  G ) )  -> 
( (  _I  |`  A ) ( +g  `  G
) x )  =  x )
38 f1ocnv 5815 . . . . 5  |-  ( x : A -1-1-onto-> A  ->  `' x : A -1-1-onto-> A )
3938a1i 11 . . . 4  |-  ( A  e.  V  ->  (
x : A -1-1-onto-> A  ->  `' x : A -1-1-onto-> A ) )
403, 4elsymgbas 16278 . . . 4  |-  ( A  e.  V  ->  ( `' x  e.  ( Base `  G )  <->  `' x : A -1-1-onto-> A ) )
4139, 32, 403imtr4d 268 . . 3  |-  ( A  e.  V  ->  (
x  e.  ( Base `  G )  ->  `' x  e.  ( Base `  G ) ) )
4241imp 429 . 2  |-  ( ( A  e.  V  /\  x  e.  ( Base `  G ) )  ->  `' x  e.  ( Base `  G ) )
433, 4, 5symgov 16286 . . . 4  |-  ( ( `' x  e.  ( Base `  G )  /\  x  e.  ( Base `  G ) )  -> 
( `' x ( +g  `  G ) x )  =  ( `' x  o.  x
) )
4442, 43sylancom 667 . . 3  |-  ( ( A  e.  V  /\  x  e.  ( Base `  G ) )  -> 
( `' x ( +g  `  G ) x )  =  ( `' x  o.  x
) )
45 f1ococnv1 5831 . . . 4  |-  ( x : A -1-1-onto-> A  ->  ( `' x  o.  x )  =  (  _I  |`  A ) )
4633, 45syl 16 . . 3  |-  ( ( A  e.  V  /\  x  e.  ( Base `  G ) )  -> 
( `' x  o.  x )  =  (  _I  |`  A )
)
4744, 46eqtrd 2482 . 2  |-  ( ( A  e.  V  /\  x  e.  ( Base `  G ) )  -> 
( `' x ( +g  `  G ) x )  =  (  _I  |`  A )
)
481, 2, 7, 26, 29, 37, 42, 47isgrpd 15946 1  |-  ( A  e.  V  ->  G  e.  Grp )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 972    = wceq 1381    e. wcel 1802    _I cid 4777   `'ccnv 4985    |` cres 4988    o. ccom 4990   -->wf 5571   -1-1-onto->wf1o 5574   ` cfv 5575  (class class class)co 6278   Basecbs 14506   +g cplusg 14571   Grpcgrp 15924   SymGrpcsymg 16273
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-8 1804  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-rep 4545  ax-sep 4555  ax-nul 4563  ax-pow 4612  ax-pr 4673  ax-un 6574  ax-cnex 9548  ax-resscn 9549  ax-1cn 9550  ax-icn 9551  ax-addcl 9552  ax-addrcl 9553  ax-mulcl 9554  ax-mulrcl 9555  ax-mulcom 9556  ax-addass 9557  ax-mulass 9558  ax-distr 9559  ax-i2m1 9560  ax-1ne0 9561  ax-1rid 9562  ax-rnegex 9563  ax-rrecex 9564  ax-cnre 9565  ax-pre-lttri 9566  ax-pre-lttrn 9567  ax-pre-ltadd 9568  ax-pre-mulgt0 9569
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 973  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-nel 2639  df-ral 2796  df-rex 2797  df-reu 2798  df-rmo 2799  df-rab 2800  df-v 3095  df-sbc 3312  df-csb 3419  df-dif 3462  df-un 3464  df-in 3466  df-ss 3473  df-pss 3475  df-nul 3769  df-if 3924  df-pw 3996  df-sn 4012  df-pr 4014  df-tp 4016  df-op 4018  df-uni 4232  df-int 4269  df-iun 4314  df-br 4435  df-opab 4493  df-mpt 4494  df-tr 4528  df-eprel 4778  df-id 4782  df-po 4787  df-so 4788  df-fr 4825  df-we 4827  df-ord 4868  df-on 4869  df-lim 4870  df-suc 4871  df-xp 4992  df-rel 4993  df-cnv 4994  df-co 4995  df-dm 4996  df-rn 4997  df-res 4998  df-ima 4999  df-iota 5538  df-fun 5577  df-fn 5578  df-f 5579  df-f1 5580  df-fo 5581  df-f1o 5582  df-fv 5583  df-riota 6239  df-ov 6281  df-oprab 6282  df-mpt2 6283  df-om 6683  df-1st 6782  df-2nd 6783  df-recs 7041  df-rdg 7075  df-1o 7129  df-oadd 7133  df-er 7310  df-map 7421  df-en 7516  df-dom 7517  df-sdom 7518  df-fin 7519  df-pnf 9630  df-mnf 9631  df-xr 9632  df-ltxr 9633  df-le 9634  df-sub 9809  df-neg 9810  df-nn 10540  df-2 10597  df-3 10598  df-4 10599  df-5 10600  df-6 10601  df-7 10602  df-8 10603  df-9 10604  df-n0 10799  df-z 10868  df-uz 11088  df-fz 11679  df-struct 14508  df-ndx 14509  df-slot 14510  df-base 14511  df-plusg 14584  df-tset 14590  df-0g 14713  df-mgm 15743  df-sgrp 15782  df-mnd 15792  df-grp 15928  df-symg 16274
This theorem is referenced by:  symgid  16297  symginv  16298  galactghm  16299  symgga  16302  pgrpsubgsymgbi  16303  pgrpsubgsymg  16304  idressubgsymg  16306  gsumccatsymgsn  16322  symgsssg  16363  symgfisg  16364  symggen  16366  symgtrinv  16368  psgnunilem5  16390  psgnunilem2  16391  psgnuni  16395  psgneldm2  16400  psgnfitr  16413  psgnghm  18486  zrhpsgninv  18491  evpmodpmf1o  18502  mdetleib2  18960  mdetdiag  18971  mdetralt  18980  mdetunilem7  18990  symgtgp  20470  pgrple2abl  32686
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