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Theorem frgp0 16250
Description: The free group is a group. (Contributed by Mario Carneiro, 1-Oct-2015.) (Revised by Mario Carneiro, 27-Feb-2016.)
Hypotheses
Ref Expression
frgp0.m  |-  G  =  (freeGrp `  I )
frgp0.r  |-  .~  =  ( ~FG  `  I )
Assertion
Ref Expression
frgp0  |-  ( I  e.  V  ->  ( G  e.  Grp  /\  [ (/)
]  .~  =  ( 0g `  G ) ) )

Proof of Theorem frgp0
Dummy variables  a 
b  c  d  x  y  z  n  v  w are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 frgp0.m . . 3  |-  G  =  (freeGrp `  I )
2 eqid 2441 . . 3  |-  (freeMnd `  (
I  X.  2o ) )  =  (freeMnd `  (
I  X.  2o ) )
3 frgp0.r . . 3  |-  .~  =  ( ~FG  `  I )
41, 2, 3frgpval 16248 . 2  |-  ( I  e.  V  ->  G  =  ( (freeMnd `  (
I  X.  2o ) )  /.s 
.~  ) )
5 2on 6924 . . . . 5  |-  2o  e.  On
6 xpexg 6506 . . . . 5  |-  ( ( I  e.  V  /\  2o  e.  On )  -> 
( I  X.  2o )  e.  _V )
75, 6mpan2 666 . . . 4  |-  ( I  e.  V  ->  (
I  X.  2o )  e.  _V )
8 eqid 2441 . . . . 5  |-  ( Base `  (freeMnd `  ( I  X.  2o ) ) )  =  ( Base `  (freeMnd `  ( I  X.  2o ) ) )
92, 8frmdbas 15523 . . . 4  |-  ( ( I  X.  2o )  e.  _V  ->  ( Base `  (freeMnd `  (
I  X.  2o ) ) )  = Word  (
I  X.  2o ) )
107, 9syl 16 . . 3  |-  ( I  e.  V  ->  ( Base `  (freeMnd `  (
I  X.  2o ) ) )  = Word  (
I  X.  2o ) )
1110eqcomd 2446 . 2  |-  ( I  e.  V  -> Word  ( I  X.  2o )  =  ( Base `  (freeMnd `  ( I  X.  2o ) ) ) )
12 eqidd 2442 . 2  |-  ( I  e.  V  ->  ( +g  `  (freeMnd `  (
I  X.  2o ) ) )  =  ( +g  `  (freeMnd `  (
I  X.  2o ) ) ) )
13 eqid 2441 . . . 4  |-  (  _I 
` Word  ( I  X.  2o ) )  =  (  _I  ` Word  ( I  X.  2o ) )
1413, 3efger 16208 . . 3  |-  .~  Er  (  _I  ` Word  ( I  X.  2o ) )
15 wrdexg 12240 . . . . 5  |-  ( ( I  X.  2o )  e.  _V  -> Word  ( I  X.  2o )  e. 
_V )
16 fvi 5745 . . . . 5  |-  (Word  (
I  X.  2o )  e.  _V  ->  (  _I  ` Word  ( I  X.  2o ) )  = Word  (
I  X.  2o ) )
177, 15, 163syl 20 . . . 4  |-  ( I  e.  V  ->  (  _I  ` Word  ( I  X.  2o ) )  = Word  (
I  X.  2o ) )
18 ereq2 7105 . . . 4  |-  ( (  _I  ` Word  ( I  X.  2o ) )  = Word  ( I  X.  2o )  ->  (  .~  Er  (  _I  ` Word  ( I  X.  2o ) )  <->  .~  Er Word  ( I  X.  2o ) ) )
1917, 18syl 16 . . 3  |-  ( I  e.  V  ->  (  .~  Er  (  _I  ` Word  ( I  X.  2o ) )  <->  .~  Er Word  (
I  X.  2o ) ) )
2014, 19mpbii 211 . 2  |-  ( I  e.  V  ->  .~  Er Word  ( I  X.  2o ) )
21 fvex 5698 . . 3  |-  (freeMnd `  (
I  X.  2o ) )  e.  _V
2221a1i 11 . 2  |-  ( I  e.  V  ->  (freeMnd `  ( I  X.  2o ) )  e.  _V )
23 eqid 2441 . . . 4  |-  ( +g  `  (freeMnd `  ( I  X.  2o ) ) )  =  ( +g  `  (freeMnd `  ( I  X.  2o ) ) )
241, 2, 3, 23frgpcpbl 16249 . . 3  |-  ( ( a  .~  b  /\  c  .~  d )  -> 
( a ( +g  `  (freeMnd `  ( I  X.  2o ) ) ) c )  .~  (
b ( +g  `  (freeMnd `  ( I  X.  2o ) ) ) d ) )
2524a1i 11 . 2  |-  ( I  e.  V  ->  (
( a  .~  b  /\  c  .~  d
)  ->  ( a
( +g  `  (freeMnd `  (
I  X.  2o ) ) ) c )  .~  ( b ( +g  `  (freeMnd `  (
I  X.  2o ) ) ) d ) ) )
262frmdmnd 15530 . . . . . 6  |-  ( ( I  X.  2o )  e.  _V  ->  (freeMnd `  ( I  X.  2o ) )  e.  Mnd )
277, 26syl 16 . . . . 5  |-  ( I  e.  V  ->  (freeMnd `  ( I  X.  2o ) )  e.  Mnd )
28273ad2ant1 1004 . . . 4  |-  ( ( I  e.  V  /\  x  e. Word  ( I  X.  2o )  /\  y  e. Word  ( I  X.  2o ) )  ->  (freeMnd `  ( I  X.  2o ) )  e.  Mnd )
29 simp2 984 . . . . 5  |-  ( ( I  e.  V  /\  x  e. Word  ( I  X.  2o )  /\  y  e. Word  ( I  X.  2o ) )  ->  x  e. Word  ( I  X.  2o ) )
30113ad2ant1 1004 . . . . 5  |-  ( ( I  e.  V  /\  x  e. Word  ( I  X.  2o )  /\  y  e. Word  ( I  X.  2o ) )  -> Word  ( I  X.  2o )  =  ( Base `  (freeMnd `  ( I  X.  2o ) ) ) )
3129, 30eleqtrd 2517 . . . 4  |-  ( ( I  e.  V  /\  x  e. Word  ( I  X.  2o )  /\  y  e. Word  ( I  X.  2o ) )  ->  x  e.  ( Base `  (freeMnd `  ( I  X.  2o ) ) ) )
32 simp3 985 . . . . 5  |-  ( ( I  e.  V  /\  x  e. Word  ( I  X.  2o )  /\  y  e. Word  ( I  X.  2o ) )  ->  y  e. Word  ( I  X.  2o ) )
3332, 30eleqtrd 2517 . . . 4  |-  ( ( I  e.  V  /\  x  e. Word  ( I  X.  2o )  /\  y  e. Word  ( I  X.  2o ) )  ->  y  e.  ( Base `  (freeMnd `  ( I  X.  2o ) ) ) )
348, 23mndcl 15416 . . . 4  |-  ( ( (freeMnd `  ( I  X.  2o ) )  e. 
Mnd  /\  x  e.  ( Base `  (freeMnd `  (
I  X.  2o ) ) )  /\  y  e.  ( Base `  (freeMnd `  ( I  X.  2o ) ) ) )  ->  ( x ( +g  `  (freeMnd `  (
I  X.  2o ) ) ) y )  e.  ( Base `  (freeMnd `  ( I  X.  2o ) ) ) )
3528, 31, 33, 34syl3anc 1213 . . 3  |-  ( ( I  e.  V  /\  x  e. Word  ( I  X.  2o )  /\  y  e. Word  ( I  X.  2o ) )  ->  (
x ( +g  `  (freeMnd `  ( I  X.  2o ) ) ) y )  e.  ( Base `  (freeMnd `  ( I  X.  2o ) ) ) )
3635, 30eleqtrrd 2518 . 2  |-  ( ( I  e.  V  /\  x  e. Word  ( I  X.  2o )  /\  y  e. Word  ( I  X.  2o ) )  ->  (
x ( +g  `  (freeMnd `  ( I  X.  2o ) ) ) y )  e. Word  ( I  X.  2o ) )
3720adantr 462 . . . 4  |-  ( ( I  e.  V  /\  ( x  e. Word  ( I  X.  2o )  /\  y  e. Word  ( I  X.  2o )  /\  z  e. Word  ( I  X.  2o ) ) )  ->  .~  Er Word  ( I  X.  2o ) )
3827adantr 462 . . . . . 6  |-  ( ( I  e.  V  /\  ( x  e. Word  ( I  X.  2o )  /\  y  e. Word  ( I  X.  2o )  /\  z  e. Word  ( I  X.  2o ) ) )  -> 
(freeMnd `  ( I  X.  2o ) )  e.  Mnd )
39353adant3r3 1193 . . . . . 6  |-  ( ( I  e.  V  /\  ( x  e. Word  ( I  X.  2o )  /\  y  e. Word  ( I  X.  2o )  /\  z  e. Word  ( I  X.  2o ) ) )  -> 
( x ( +g  `  (freeMnd `  ( I  X.  2o ) ) ) y )  e.  (
Base `  (freeMnd `  (
I  X.  2o ) ) ) )
40 simpr3 991 . . . . . . 7  |-  ( ( I  e.  V  /\  ( x  e. Word  ( I  X.  2o )  /\  y  e. Word  ( I  X.  2o )  /\  z  e. Word  ( I  X.  2o ) ) )  -> 
z  e. Word  ( I  X.  2o ) )
4111adantr 462 . . . . . . 7  |-  ( ( I  e.  V  /\  ( x  e. Word  ( I  X.  2o )  /\  y  e. Word  ( I  X.  2o )  /\  z  e. Word  ( I  X.  2o ) ) )  -> Word  ( I  X.  2o )  =  ( Base `  (freeMnd `  ( I  X.  2o ) ) ) )
4240, 41eleqtrd 2517 . . . . . 6  |-  ( ( I  e.  V  /\  ( x  e. Word  ( I  X.  2o )  /\  y  e. Word  ( I  X.  2o )  /\  z  e. Word  ( I  X.  2o ) ) )  -> 
z  e.  ( Base `  (freeMnd `  ( I  X.  2o ) ) ) )
438, 23mndcl 15416 . . . . . 6  |-  ( ( (freeMnd `  ( I  X.  2o ) )  e. 
Mnd  /\  ( x
( +g  `  (freeMnd `  (
I  X.  2o ) ) ) y )  e.  ( Base `  (freeMnd `  ( I  X.  2o ) ) )  /\  z  e.  ( Base `  (freeMnd `  ( I  X.  2o ) ) ) )  ->  ( (
x ( +g  `  (freeMnd `  ( I  X.  2o ) ) ) y ) ( +g  `  (freeMnd `  ( I  X.  2o ) ) ) z )  e.  ( Base `  (freeMnd `  ( I  X.  2o ) ) ) )
4438, 39, 42, 43syl3anc 1213 . . . . 5  |-  ( ( I  e.  V  /\  ( x  e. Word  ( I  X.  2o )  /\  y  e. Word  ( I  X.  2o )  /\  z  e. Word  ( I  X.  2o ) ) )  -> 
( ( x ( +g  `  (freeMnd `  (
I  X.  2o ) ) ) y ) ( +g  `  (freeMnd `  ( I  X.  2o ) ) ) z )  e.  ( Base `  (freeMnd `  ( I  X.  2o ) ) ) )
4544, 41eleqtrrd 2518 . . . 4  |-  ( ( I  e.  V  /\  ( x  e. Word  ( I  X.  2o )  /\  y  e. Word  ( I  X.  2o )  /\  z  e. Word  ( I  X.  2o ) ) )  -> 
( ( x ( +g  `  (freeMnd `  (
I  X.  2o ) ) ) y ) ( +g  `  (freeMnd `  ( I  X.  2o ) ) ) z )  e. Word  ( I  X.  2o ) )
4637, 45erref 7117 . . 3  |-  ( ( I  e.  V  /\  ( x  e. Word  ( I  X.  2o )  /\  y  e. Word  ( I  X.  2o )  /\  z  e. Word  ( I  X.  2o ) ) )  -> 
( ( x ( +g  `  (freeMnd `  (
I  X.  2o ) ) ) y ) ( +g  `  (freeMnd `  ( I  X.  2o ) ) ) z )  .~  ( ( x ( +g  `  (freeMnd `  ( I  X.  2o ) ) ) y ) ( +g  `  (freeMnd `  ( I  X.  2o ) ) ) z ) )
47313adant3r3 1193 . . . 4  |-  ( ( I  e.  V  /\  ( x  e. Word  ( I  X.  2o )  /\  y  e. Word  ( I  X.  2o )  /\  z  e. Word  ( I  X.  2o ) ) )  ->  x  e.  ( Base `  (freeMnd `  ( I  X.  2o ) ) ) )
48333adant3r3 1193 . . . 4  |-  ( ( I  e.  V  /\  ( x  e. Word  ( I  X.  2o )  /\  y  e. Word  ( I  X.  2o )  /\  z  e. Word  ( I  X.  2o ) ) )  -> 
y  e.  ( Base `  (freeMnd `  ( I  X.  2o ) ) ) )
498, 23mndass 15417 . . . 4  |-  ( ( (freeMnd `  ( I  X.  2o ) )  e. 
Mnd  /\  ( x  e.  ( Base `  (freeMnd `  ( I  X.  2o ) ) )  /\  y  e.  ( Base `  (freeMnd `  ( I  X.  2o ) ) )  /\  z  e.  (
Base `  (freeMnd `  (
I  X.  2o ) ) ) ) )  ->  ( ( x ( +g  `  (freeMnd `  ( I  X.  2o ) ) ) y ) ( +g  `  (freeMnd `  ( I  X.  2o ) ) ) z )  =  ( x ( +g  `  (freeMnd `  ( I  X.  2o ) ) ) ( y ( +g  `  (freeMnd `  ( I  X.  2o ) ) ) z ) ) )
5038, 47, 48, 42, 49syl13anc 1215 . . 3  |-  ( ( I  e.  V  /\  ( x  e. Word  ( I  X.  2o )  /\  y  e. Word  ( I  X.  2o )  /\  z  e. Word  ( I  X.  2o ) ) )  -> 
( ( x ( +g  `  (freeMnd `  (
I  X.  2o ) ) ) y ) ( +g  `  (freeMnd `  ( I  X.  2o ) ) ) z )  =  ( x ( +g  `  (freeMnd `  ( I  X.  2o ) ) ) ( y ( +g  `  (freeMnd `  ( I  X.  2o ) ) ) z ) ) )
5146, 50breqtrd 4313 . 2  |-  ( ( I  e.  V  /\  ( x  e. Word  ( I  X.  2o )  /\  y  e. Word  ( I  X.  2o )  /\  z  e. Word  ( I  X.  2o ) ) )  -> 
( ( x ( +g  `  (freeMnd `  (
I  X.  2o ) ) ) y ) ( +g  `  (freeMnd `  ( I  X.  2o ) ) ) z )  .~  ( x ( +g  `  (freeMnd `  ( I  X.  2o ) ) ) ( y ( +g  `  (freeMnd `  ( I  X.  2o ) ) ) z ) ) )
52 wrd0 12248 . . 3  |-  (/)  e. Word  (
I  X.  2o )
5352a1i 11 . 2  |-  ( I  e.  V  ->  (/)  e. Word  (
I  X.  2o ) )
5452, 11syl5eleq 2527 . . . . . 6  |-  ( I  e.  V  ->  (/)  e.  (
Base `  (freeMnd `  (
I  X.  2o ) ) ) )
5554adantr 462 . . . . 5  |-  ( ( I  e.  V  /\  x  e. Word  ( I  X.  2o ) )  ->  (/) 
e.  ( Base `  (freeMnd `  ( I  X.  2o ) ) ) )
5611eleq2d 2508 . . . . . 6  |-  ( I  e.  V  ->  (
x  e. Word  ( I  X.  2o )  <->  x  e.  ( Base `  (freeMnd `  (
I  X.  2o ) ) ) ) )
5756biimpa 481 . . . . 5  |-  ( ( I  e.  V  /\  x  e. Word  ( I  X.  2o ) )  ->  x  e.  ( Base `  (freeMnd `  ( I  X.  2o ) ) ) )
582, 8, 23frmdadd 15526 . . . . 5  |-  ( (
(/)  e.  ( Base `  (freeMnd `  ( I  X.  2o ) ) )  /\  x  e.  (
Base `  (freeMnd `  (
I  X.  2o ) ) ) )  -> 
( (/) ( +g  `  (freeMnd `  ( I  X.  2o ) ) ) x )  =  ( (/) concat  x ) )
5955, 57, 58syl2anc 656 . . . 4  |-  ( ( I  e.  V  /\  x  e. Word  ( I  X.  2o ) )  -> 
( (/) ( +g  `  (freeMnd `  ( I  X.  2o ) ) ) x )  =  ( (/) concat  x ) )
60 ccatlid 12280 . . . . 5  |-  ( x  e. Word  ( I  X.  2o )  ->  ( (/) concat  x )  =  x )
6160adantl 463 . . . 4  |-  ( ( I  e.  V  /\  x  e. Word  ( I  X.  2o ) )  -> 
( (/) concat  x )  =  x )
6259, 61eqtrd 2473 . . 3  |-  ( ( I  e.  V  /\  x  e. Word  ( I  X.  2o ) )  -> 
( (/) ( +g  `  (freeMnd `  ( I  X.  2o ) ) ) x )  =  x )
6320adantr 462 . . . 4  |-  ( ( I  e.  V  /\  x  e. Word  ( I  X.  2o ) )  ->  .~  Er Word  ( I  X.  2o ) )
64 simpr 458 . . . 4  |-  ( ( I  e.  V  /\  x  e. Word  ( I  X.  2o ) )  ->  x  e. Word  ( I  X.  2o ) )
6563, 64erref 7117 . . 3  |-  ( ( I  e.  V  /\  x  e. Word  ( I  X.  2o ) )  ->  x  .~  x )
6662, 65eqbrtrd 4309 . 2  |-  ( ( I  e.  V  /\  x  e. Word  ( I  X.  2o ) )  -> 
( (/) ( +g  `  (freeMnd `  ( I  X.  2o ) ) ) x )  .~  x )
67 revcl 12397 . . . 4  |-  ( x  e. Word  ( I  X.  2o )  ->  (reverse `  x
)  e. Word  ( I  X.  2o ) )
6867adantl 463 . . 3  |-  ( ( I  e.  V  /\  x  e. Word  ( I  X.  2o ) )  -> 
(reverse `  x )  e. Word 
( I  X.  2o ) )
69 eqid 2441 . . . . 5  |-  ( y  e.  I ,  z  e.  2o  |->  <. y ,  ( 1o  \ 
z ) >. )  =  ( y  e.  I ,  z  e.  2o  |->  <. y ,  ( 1o  \  z )
>. )
7069efgmf 16203 . . . 4  |-  ( y  e.  I ,  z  e.  2o  |->  <. y ,  ( 1o  \ 
z ) >. ) : ( I  X.  2o ) --> ( I  X.  2o )
7170a1i 11 . . 3  |-  ( ( I  e.  V  /\  x  e. Word  ( I  X.  2o ) )  -> 
( y  e.  I ,  z  e.  2o  |->  <. y ,  ( 1o 
\  z ) >.
) : ( I  X.  2o ) --> ( I  X.  2o ) )
72 wrdco 12455 . . 3  |-  ( ( (reverse `  x )  e. Word  ( I  X.  2o )  /\  ( y  e.  I ,  z  e.  2o  |->  <. y ,  ( 1o  \  z )
>. ) : ( I  X.  2o ) --> ( I  X.  2o ) )  ->  ( (
y  e.  I ,  z  e.  2o  |->  <.
y ,  ( 1o 
\  z ) >.
)  o.  (reverse `  x
) )  e. Word  (
I  X.  2o ) )
7368, 71, 72syl2anc 656 . 2  |-  ( ( I  e.  V  /\  x  e. Word  ( I  X.  2o ) )  -> 
( ( y  e.  I ,  z  e.  2o  |->  <. y ,  ( 1o  \  z )
>. )  o.  (reverse `  x ) )  e. Word 
( I  X.  2o ) )
7411adantr 462 . . . . 5  |-  ( ( I  e.  V  /\  x  e. Word  ( I  X.  2o ) )  -> Word  ( I  X.  2o )  =  ( Base `  (freeMnd `  ( I  X.  2o ) ) ) )
7573, 74eleqtrd 2517 . . . 4  |-  ( ( I  e.  V  /\  x  e. Word  ( I  X.  2o ) )  -> 
( ( y  e.  I ,  z  e.  2o  |->  <. y ,  ( 1o  \  z )
>. )  o.  (reverse `  x ) )  e.  ( Base `  (freeMnd `  ( I  X.  2o ) ) ) )
762, 8, 23frmdadd 15526 . . . 4  |-  ( ( ( ( y  e.  I ,  z  e.  2o  |->  <. y ,  ( 1o  \  z )
>. )  o.  (reverse `  x ) )  e.  ( Base `  (freeMnd `  ( I  X.  2o ) ) )  /\  x  e.  ( Base `  (freeMnd `  ( I  X.  2o ) ) ) )  ->  ( (
( y  e.  I ,  z  e.  2o  |->  <. y ,  ( 1o 
\  z ) >.
)  o.  (reverse `  x
) ) ( +g  `  (freeMnd `  ( I  X.  2o ) ) ) x )  =  ( ( ( y  e.  I ,  z  e.  2o  |->  <. y ,  ( 1o  \  z )
>. )  o.  (reverse `  x ) ) concat  x
) )
7775, 57, 76syl2anc 656 . . 3  |-  ( ( I  e.  V  /\  x  e. Word  ( I  X.  2o ) )  -> 
( ( ( y  e.  I ,  z  e.  2o  |->  <. y ,  ( 1o  \ 
z ) >. )  o.  (reverse `  x )
) ( +g  `  (freeMnd `  ( I  X.  2o ) ) ) x )  =  ( ( ( y  e.  I ,  z  e.  2o  |->  <. y ,  ( 1o 
\  z ) >.
)  o.  (reverse `  x
) ) concat  x )
)
7817eleq2d 2508 . . . . 5  |-  ( I  e.  V  ->  (
x  e.  (  _I 
` Word  ( I  X.  2o ) )  <->  x  e. Word  ( I  X.  2o ) ) )
7978biimpar 482 . . . 4  |-  ( ( I  e.  V  /\  x  e. Word  ( I  X.  2o ) )  ->  x  e.  (  _I  ` Word 
( I  X.  2o ) ) )
80 eqid 2441 . . . . 5  |-  ( v  e.  (  _I  ` Word  ( I  X.  2o ) )  |->  ( n  e.  ( 0 ... ( # `  v
) ) ,  w  e.  ( I  X.  2o )  |->  ( v splice  <. n ,  n ,  <" w ( ( y  e.  I ,  z  e.  2o  |->  <. y ,  ( 1o  \ 
z ) >. ) `  w ) "> >.
) ) )  =  ( v  e.  (  _I  ` Word  ( I  X.  2o ) )  |->  ( n  e.  ( 0 ... ( # `  v
) ) ,  w  e.  ( I  X.  2o )  |->  ( v splice  <. n ,  n ,  <" w ( ( y  e.  I ,  z  e.  2o  |->  <. y ,  ( 1o  \ 
z ) >. ) `  w ) "> >.
) ) )
8113, 3, 69, 80efginvrel1 16218 . . . 4  |-  ( x  e.  (  _I  ` Word  ( I  X.  2o ) )  ->  (
( ( y  e.  I ,  z  e.  2o  |->  <. y ,  ( 1o  \  z )
>. )  o.  (reverse `  x ) ) concat  x
)  .~  (/) )
8279, 81syl 16 . . 3  |-  ( ( I  e.  V  /\  x  e. Word  ( I  X.  2o ) )  -> 
( ( ( y  e.  I ,  z  e.  2o  |->  <. y ,  ( 1o  \ 
z ) >. )  o.  (reverse `  x )
) concat  x )  .~  (/) )
8377, 82eqbrtrd 4309 . 2  |-  ( ( I  e.  V  /\  x  e. Word  ( I  X.  2o ) )  -> 
( ( ( y  e.  I ,  z  e.  2o  |->  <. y ,  ( 1o  \ 
z ) >. )  o.  (reverse `  x )
) ( +g  `  (freeMnd `  ( I  X.  2o ) ) ) x )  .~  (/) )
844, 11, 12, 20, 22, 25, 36, 51, 53, 66, 73, 83divsgrp2 15666 1  |-  ( I  e.  V  ->  ( G  e.  Grp  /\  [ (/)
]  .~  =  ( 0g `  G ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 960    = wceq 1364    e. wcel 1761   _Vcvv 2970    \ cdif 3322   (/)c0 3634   <.cop 3880   <.cotp 3882   class class class wbr 4289    e. cmpt 4347    _I cid 4627   Oncon0 4715    X. cxp 4834    o. ccom 4840   -->wf 5411   ` cfv 5415  (class class class)co 6090    e. cmpt2 6092   1oc1o 6909   2oc2o 6910    Er wer 7094   [cec 7095   0cc0 9278   ...cfz 11433   #chash 12099  Word cword 12217   concat cconcat 12219   splice csplice 12222  reversecreverse 12223   <"cs2 12464   Basecbs 14170   +g cplusg 14234   0gc0g 14374   Mndcmnd 15405   Grpcgrp 15406  freeMndcfrmd 15518   ~FG cefg 16196  freeGrpcfrgp 16197
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2263  df-mo 2264  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-ot 3883  df-uni 4089  df-int 4126  df-iun 4170  df-iin 4171  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-1st 6576  df-2nd 6577  df-recs 6828  df-rdg 6862  df-1o 6916  df-2o 6917  df-oadd 6920  df-er 7097  df-ec 7099  df-qs 7103  df-map 7212  df-pm 7213  df-en 7307  df-dom 7308  df-sdom 7309  df-fin 7310  df-sup 7687  df-card 8105  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-nn 10319  df-2 10376  df-3 10377  df-4 10378  df-5 10379  df-6 10380  df-7 10381  df-8 10382  df-9 10383  df-10 10384  df-n0 10576  df-z 10643  df-dec 10752  df-uz 10858  df-fz 11434  df-fzo 11545  df-hash 12100  df-word 12225  df-concat 12227  df-s1 12228  df-substr 12229  df-splice 12230  df-reverse 12231  df-s2 12471  df-struct 14172  df-ndx 14173  df-slot 14174  df-base 14175  df-plusg 14247  df-mulr 14248  df-sca 14250  df-vsca 14251  df-ip 14252  df-tset 14253  df-ple 14254  df-ds 14256  df-0g 14376  df-imas 14442  df-divs 14443  df-mnd 15411  df-frmd 15520  df-grp 15538  df-efg 16199  df-frgp 16200
This theorem is referenced by:  frgpgrp  16252  frgpinv  16254  frgpmhm  16255
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