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Theorem symggrp 14796
Description: The symmetry group on  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 2297 . 2  |-  ( A  e.  V  ->  ( Base `  G )  =  ( Base `  G
) )
2 eqidd 2297 . 2  |-  ( A  e.  V  ->  ( +g  `  G )  =  ( +g  `  G
) )
3 symggrp.1 . . . 4  |-  G  =  ( SymGrp `  A )
4 eqid 2296 . . . 4  |-  ( Base `  G )  =  (
Base `  G )
5 eqid 2296 . . . 4  |-  ( +g  `  G )  =  ( +g  `  G )
63, 4, 5symgcl 14794 . . 3  |-  ( ( x  e.  ( Base `  G )  /\  y  e.  ( Base `  G
) )  ->  (
x ( +g  `  G
) y )  e.  ( Base `  G
) )
763adant1 973 . 2  |-  ( ( A  e.  V  /\  x  e.  ( Base `  G )  /\  y  e.  ( Base `  G
) )  ->  (
x ( +g  `  G
) y )  e.  ( Base `  G
) )
8 coass 5207 . . . 4  |-  ( ( x  o.  y )  o.  z )  =  ( x  o.  (
y  o.  z ) )
9 simpr1 961 . . . . . 6  |-  ( ( A  e.  V  /\  ( x  e.  ( Base `  G )  /\  y  e.  ( Base `  G )  /\  z  e.  ( Base `  G
) ) )  ->  x  e.  ( Base `  G ) )
10 simpr2 962 . . . . . 6  |-  ( ( A  e.  V  /\  ( x  e.  ( Base `  G )  /\  y  e.  ( Base `  G )  /\  z  e.  ( Base `  G
) ) )  -> 
y  e.  ( Base `  G ) )
113, 4, 5symgov 14793 . . . . . 6  |-  ( ( x  e.  ( Base `  G )  /\  y  e.  ( Base `  G
) )  ->  (
x ( +g  `  G
) y )  =  ( x  o.  y
) )
129, 10, 11syl2anc 642 . . . . 5  |-  ( ( A  e.  V  /\  ( x  e.  ( Base `  G )  /\  y  e.  ( Base `  G )  /\  z  e.  ( Base `  G
) ) )  -> 
( x ( +g  `  G ) y )  =  ( x  o.  y ) )
1312coeq1d 4861 . . . 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 963 . . . . . 6  |-  ( ( A  e.  V  /\  ( x  e.  ( Base `  G )  /\  y  e.  ( Base `  G )  /\  z  e.  ( Base `  G
) ) )  -> 
z  e.  ( Base `  G ) )
153, 4, 5symgov 14793 . . . . . 6  |-  ( ( y  e.  ( Base `  G )  /\  z  e.  ( Base `  G
) )  ->  (
y ( +g  `  G
) z )  =  ( y  o.  z
) )
1610, 14, 15syl2anc 642 . . . . 5  |-  ( ( A  e.  V  /\  ( x  e.  ( Base `  G )  /\  y  e.  ( Base `  G )  /\  z  e.  ( Base `  G
) ) )  -> 
( y ( +g  `  G ) z )  =  ( y  o.  z ) )
1716coeq2d 4862 . . . 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 2354 . . 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 642 . . . 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 14793 . . . 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 642 . . 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 14794 . . . . 5  |-  ( ( y  e.  ( Base `  G )  /\  z  e.  ( Base `  G
) )  ->  (
y ( +g  `  G
) z )  e.  ( Base `  G
) )
2310, 14, 22syl2anc 642 . . . 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 14793 . . . 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 642 . . 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 2338 . 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 5527 . . 3  |-  (  _I  |`  A ) : A -1-1-onto-> A
283, 4elsymgbas 14790 . . 3  |-  ( A  e.  V  ->  (
(  _I  |`  A )  e.  ( Base `  G
)  <->  (  _I  |`  A ) : A -1-1-onto-> A ) )
2927, 28mpbiri 224 . 2  |-  ( A  e.  V  ->  (  _I  |`  A )  e.  ( Base `  G
) )
303, 4, 5symgov 14793 . . . 4  |-  ( ( (  _I  |`  A )  e.  ( Base `  G
)  /\  x  e.  ( Base `  G )
)  ->  ( (  _I  |`  A ) ( +g  `  G ) x )  =  ( (  _I  |`  A )  o.  x ) )
3129, 30sylan 457 . . 3  |-  ( ( A  e.  V  /\  x  e.  ( Base `  G ) )  -> 
( (  _I  |`  A ) ( +g  `  G
) x )  =  ( (  _I  |`  A )  o.  x ) )
323, 4elsymgbas 14790 . . . . 5  |-  ( A  e.  V  ->  (
x  e.  ( Base `  G )  <->  x : A
-1-1-onto-> A ) )
3332biimpa 470 . . . 4  |-  ( ( A  e.  V  /\  x  e.  ( Base `  G ) )  ->  x : A -1-1-onto-> A )
34 f1of 5488 . . . 4  |-  ( x : A -1-1-onto-> A  ->  x : A
--> A )
35 fcoi2 5432 . . . 4  |-  ( x : A --> A  -> 
( (  _I  |`  A )  o.  x )  =  x )
3633, 34, 353syl 18 . . 3  |-  ( ( A  e.  V  /\  x  e.  ( Base `  G ) )  -> 
( (  _I  |`  A )  o.  x )  =  x )
3731, 36eqtrd 2328 . 2  |-  ( ( A  e.  V  /\  x  e.  ( Base `  G ) )  -> 
( (  _I  |`  A ) ( +g  `  G
) x )  =  x )
38 f1ocnv 5501 . . . . 5  |-  ( x : A -1-1-onto-> A  ->  `' x : A -1-1-onto-> A )
3938a1i 10 . . . 4  |-  ( A  e.  V  ->  (
x : A -1-1-onto-> A  ->  `' x : A -1-1-onto-> A ) )
403, 4elsymgbas 14790 . . . 4  |-  ( A  e.  V  ->  ( `' x  e.  ( Base `  G )  <->  `' x : A -1-1-onto-> A ) )
4139, 32, 403imtr4d 259 . . 3  |-  ( A  e.  V  ->  (
x  e.  ( Base `  G )  ->  `' x  e.  ( Base `  G ) ) )
4241imp 418 . 2  |-  ( ( A  e.  V  /\  x  e.  ( Base `  G ) )  ->  `' x  e.  ( Base `  G ) )
433, 4, 5symgov 14793 . . . 4  |-  ( ( `' x  e.  ( Base `  G )  /\  x  e.  ( Base `  G ) )  -> 
( `' x ( +g  `  G ) x )  =  ( `' x  o.  x
) )
4442, 43sylancom 648 . . 3  |-  ( ( A  e.  V  /\  x  e.  ( Base `  G ) )  -> 
( `' x ( +g  `  G ) x )  =  ( `' x  o.  x
) )
45 f1ococnv1 5518 . . . 4  |-  ( x : A -1-1-onto-> A  ->  ( `' x  o.  x )  =  (  _I  |`  A ) )
4633, 45syl 15 . . 3  |-  ( ( A  e.  V  /\  x  e.  ( Base `  G ) )  -> 
( `' x  o.  x )  =  (  _I  |`  A )
)
4744, 46eqtrd 2328 . 2  |-  ( ( A  e.  V  /\  x  e.  ( Base `  G ) )  -> 
( `' x ( +g  `  G ) x )  =  (  _I  |`  A )
)
481, 2, 7, 26, 29, 37, 42, 47isgrpd 14523 1  |-  ( A  e.  V  ->  G  e.  Grp )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696    _I cid 4320   `'ccnv 4704    |` cres 4707    o. ccom 4709   -->wf 5267   -1-1-onto->wf1o 5270   ` cfv 5271  (class class class)co 5874   Basecbs 13164   +g cplusg 13224   Grpcgrp 14378   SymGrpcsymg 14785
This theorem is referenced by:  symgid  14797  symginv  14798  galactghm  14799  symgga  14802  symgtgp  17800  symgfo  25471  symgsssg  27511  symgfisg  27512  symggen  27514  symgtrinv  27516  psgnunilem5  27520  psgnunilem2  27521  psgnuni  27525  psgneldm2  27530  psgnghm  27540
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-oadd 6499  df-er 6676  df-map 6790  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-nn 9763  df-2 9820  df-3 9821  df-4 9822  df-5 9823  df-6 9824  df-7 9825  df-8 9826  df-9 9827  df-n0 9982  df-z 10041  df-uz 10247  df-fz 10799  df-struct 13166  df-ndx 13167  df-slot 13168  df-base 13169  df-plusg 13237  df-tset 13243  df-0g 13420  df-mnd 14383  df-grp 14505  df-symg 14786
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