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Theorem frgp0 15085
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 2296 . . 3  |-  (freeMnd `  (
I  X.  2o ) )  =  (freeMnd `  (
I  X.  2o ) )
3 frgp0.r . . 3  |-  .~  =  ( ~FG  `  I )
41, 2, 3frgpval 15083 . 2  |-  ( I  e.  V  ->  G  =  ( (freeMnd `  (
I  X.  2o ) )  /.s 
.~  ) )
5 2on 6503 . . . . 5  |-  2o  e.  On
6 xpexg 4816 . . . . 5  |-  ( ( I  e.  V  /\  2o  e.  On )  -> 
( I  X.  2o )  e.  _V )
75, 6mpan2 652 . . . 4  |-  ( I  e.  V  ->  (
I  X.  2o )  e.  _V )
8 eqid 2296 . . . . 5  |-  ( Base `  (freeMnd `  ( I  X.  2o ) ) )  =  ( Base `  (freeMnd `  ( I  X.  2o ) ) )
92, 8frmdbas 14490 . . . 4  |-  ( ( I  X.  2o )  e.  _V  ->  ( Base `  (freeMnd `  (
I  X.  2o ) ) )  = Word  (
I  X.  2o ) )
107, 9syl 15 . . 3  |-  ( I  e.  V  ->  ( Base `  (freeMnd `  (
I  X.  2o ) ) )  = Word  (
I  X.  2o ) )
1110eqcomd 2301 . 2  |-  ( I  e.  V  -> Word  ( I  X.  2o )  =  ( Base `  (freeMnd `  ( I  X.  2o ) ) ) )
12 eqidd 2297 . 2  |-  ( I  e.  V  ->  ( +g  `  (freeMnd `  (
I  X.  2o ) ) )  =  ( +g  `  (freeMnd `  (
I  X.  2o ) ) ) )
13 eqid 2296 . . . 4  |-  (  _I 
` Word  ( I  X.  2o ) )  =  (  _I  ` Word  ( I  X.  2o ) )
1413, 3efger 15043 . . 3  |-  .~  Er  (  _I  ` Word  ( I  X.  2o ) )
15 wrdexg 11441 . . . . 5  |-  ( ( I  X.  2o )  e.  _V  -> Word  ( I  X.  2o )  e. 
_V )
16 fvi 5595 . . . . 5  |-  (Word  (
I  X.  2o )  e.  _V  ->  (  _I  ` Word  ( I  X.  2o ) )  = Word  (
I  X.  2o ) )
177, 15, 163syl 18 . . . 4  |-  ( I  e.  V  ->  (  _I  ` Word  ( I  X.  2o ) )  = Word  (
I  X.  2o ) )
18 ereq2 6684 . . . 4  |-  ( (  _I  ` Word  ( I  X.  2o ) )  = Word  ( I  X.  2o )  ->  (  .~  Er  (  _I  ` Word  ( I  X.  2o ) )  <->  .~  Er Word  ( I  X.  2o ) ) )
1917, 18syl 15 . . 3  |-  ( I  e.  V  ->  (  .~  Er  (  _I  ` Word  ( I  X.  2o ) )  <->  .~  Er Word  (
I  X.  2o ) ) )
2014, 19mpbii 202 . 2  |-  ( I  e.  V  ->  .~  Er Word  ( I  X.  2o ) )
21 fvex 5555 . . 3  |-  (freeMnd `  (
I  X.  2o ) )  e.  _V
2221a1i 10 . 2  |-  ( I  e.  V  ->  (freeMnd `  ( I  X.  2o ) )  e.  _V )
23 eqid 2296 . . . 4  |-  ( +g  `  (freeMnd `  ( I  X.  2o ) ) )  =  ( +g  `  (freeMnd `  ( I  X.  2o ) ) )
241, 2, 3, 23frgpcpbl 15084 . . 3  |-  ( ( a  .~  b  /\  c  .~  d )  -> 
( a ( +g  `  (freeMnd `  ( I  X.  2o ) ) ) c )  .~  (
b ( +g  `  (freeMnd `  ( I  X.  2o ) ) ) d ) )
2524a1i 10 . 2  |-  ( I  e.  V  ->  (
( a  .~  b  /\  c  .~  d
)  ->  ( a
( +g  `  (freeMnd `  (
I  X.  2o ) ) ) c )  .~  ( b ( +g  `  (freeMnd `  (
I  X.  2o ) ) ) d ) ) )
262frmdmnd 14497 . . . . . 6  |-  ( ( I  X.  2o )  e.  _V  ->  (freeMnd `  ( I  X.  2o ) )  e.  Mnd )
277, 26syl 15 . . . . 5  |-  ( I  e.  V  ->  (freeMnd `  ( I  X.  2o ) )  e.  Mnd )
28273ad2ant1 976 . . . 4  |-  ( ( I  e.  V  /\  x  e. Word  ( I  X.  2o )  /\  y  e. Word  ( I  X.  2o ) )  ->  (freeMnd `  ( I  X.  2o ) )  e.  Mnd )
29 simp2 956 . . . . 5  |-  ( ( I  e.  V  /\  x  e. Word  ( I  X.  2o )  /\  y  e. Word  ( I  X.  2o ) )  ->  x  e. Word  ( I  X.  2o ) )
30113ad2ant1 976 . . . . 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 2372 . . . 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 957 . . . . 5  |-  ( ( I  e.  V  /\  x  e. Word  ( I  X.  2o )  /\  y  e. Word  ( I  X.  2o ) )  ->  y  e. Word  ( I  X.  2o ) )
3332, 30eleqtrd 2372 . . . 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 14388 . . . 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 1182 . . 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 2373 . 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 451 . . . 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 451 . . . . . 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 1162 . . . . . 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 963 . . . . . . 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 451 . . . . . . 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 2372 . . . . . 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 14388 . . . . . 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 1182 . . . . 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 2373 . . . 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 6696 . . 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 1162 . . . 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 1162 . . . 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 14389 . . . 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 1184 . . 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 4063 . 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 11434 . . 3  |-  (/)  e. Word  (
I  X.  2o )
5352a1i 10 . 2  |-  ( I  e.  V  ->  (/)  e. Word  (
I  X.  2o ) )
5452, 11syl5eleq 2382 . . . . . 6  |-  ( I  e.  V  ->  (/)  e.  (
Base `  (freeMnd `  (
I  X.  2o ) ) ) )
5554adantr 451 . . . . 5  |-  ( ( I  e.  V  /\  x  e. Word  ( I  X.  2o ) )  ->  (/) 
e.  ( Base `  (freeMnd `  ( I  X.  2o ) ) ) )
5611eleq2d 2363 . . . . . 6  |-  ( I  e.  V  ->  (
x  e. Word  ( I  X.  2o )  <->  x  e.  ( Base `  (freeMnd `  (
I  X.  2o ) ) ) ) )
5756biimpa 470 . . . . 5  |-  ( ( I  e.  V  /\  x  e. Word  ( I  X.  2o ) )  ->  x  e.  ( Base `  (freeMnd `  ( I  X.  2o ) ) ) )
582, 8, 23frmdadd 14493 . . . . 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 642 . . . 4  |-  ( ( I  e.  V  /\  x  e. Word  ( I  X.  2o ) )  -> 
( (/) ( +g  `  (freeMnd `  ( I  X.  2o ) ) ) x )  =  ( (/) concat  x ) )
60 ccatlid 11450 . . . . 5  |-  ( x  e. Word  ( I  X.  2o )  ->  ( (/) concat  x )  =  x )
6160adantl 452 . . . 4  |-  ( ( I  e.  V  /\  x  e. Word  ( I  X.  2o ) )  -> 
( (/) concat  x )  =  x )
6259, 61eqtrd 2328 . . 3  |-  ( ( I  e.  V  /\  x  e. Word  ( I  X.  2o ) )  -> 
( (/) ( +g  `  (freeMnd `  ( I  X.  2o ) ) ) x )  =  x )
6320adantr 451 . . . 4  |-  ( ( I  e.  V  /\  x  e. Word  ( I  X.  2o ) )  ->  .~  Er Word  ( I  X.  2o ) )
64 simpr 447 . . . 4  |-  ( ( I  e.  V  /\  x  e. Word  ( I  X.  2o ) )  ->  x  e. Word  ( I  X.  2o ) )
6563, 64erref 6696 . . 3  |-  ( ( I  e.  V  /\  x  e. Word  ( I  X.  2o ) )  ->  x  .~  x )
6662, 65eqbrtrd 4059 . 2  |-  ( ( I  e.  V  /\  x  e. Word  ( I  X.  2o ) )  -> 
( (/) ( +g  `  (freeMnd `  ( I  X.  2o ) ) ) x )  .~  x )
67 revcl 11495 . . . 4  |-  ( x  e. Word  ( I  X.  2o )  ->  (reverse `  x
)  e. Word  ( I  X.  2o ) )
6867adantl 452 . . 3  |-  ( ( I  e.  V  /\  x  e. Word  ( I  X.  2o ) )  -> 
(reverse `  x )  e. Word 
( I  X.  2o ) )
69 eqid 2296 . . . . 5  |-  ( y  e.  I ,  z  e.  2o  |->  <. y ,  ( 1o  \ 
z ) >. )  =  ( y  e.  I ,  z  e.  2o  |->  <. y ,  ( 1o  \  z )
>. )
7069efgmf 15038 . . . 4  |-  ( y  e.  I ,  z  e.  2o  |->  <. y ,  ( 1o  \ 
z ) >. ) : ( I  X.  2o ) --> ( I  X.  2o )
7170a1i 10 . . 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 11502 . . 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 642 . 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 451 . . . . 5  |-  ( ( I  e.  V  /\  x  e. Word  ( I  X.  2o ) )  -> Word  ( I  X.  2o )  =  ( Base `  (freeMnd `  ( I  X.  2o ) ) ) )
7573, 74eleqtrd 2372 . . . 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 14493 . . . 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 642 . . 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 2363 . . . . 5  |-  ( I  e.  V  ->  (
x  e.  (  _I 
` Word  ( I  X.  2o ) )  <->  x  e. Word  ( I  X.  2o ) ) )
7978biimpar 471 . . . 4  |-  ( ( I  e.  V  /\  x  e. Word  ( I  X.  2o ) )  ->  x  e.  (  _I  ` Word 
( I  X.  2o ) ) )
80 eqid 2296 . . . . 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 15053 . . . 4  |-  ( x  e.  (  _I  ` Word  ( I  X.  2o ) )  ->  (
( ( y  e.  I ,  z  e.  2o  |->  <. y ,  ( 1o  \  z )
>. )  o.  (reverse `  x ) ) concat  x
)  .~  (/) )
8279, 81syl 15 . . 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 4059 . 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 14629 1  |-  ( I  e.  V  ->  ( G  e.  Grp  /\  [ (/)
]  .~  =  ( 0g `  G ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696   _Vcvv 2801    \ cdif 3162   (/)c0 3468   <.cop 3656   <.cotp 3657   class class class wbr 4039    e. cmpt 4093    _I cid 4320   Oncon0 4408    X. cxp 4703    o. ccom 4709   -->wf 5267   ` cfv 5271  (class class class)co 5874    e. cmpt2 5876   1oc1o 6488   2oc2o 6489    Er wer 6673   [cec 6674   0cc0 8753   ...cfz 10798   #chash 11353  Word cword 11419   concat cconcat 11420   splice csplice 11423  reversecreverse 11424   <"cs2 11507   Basecbs 13164   +g cplusg 13224   0gc0g 13416   Mndcmnd 14377   Grpcgrp 14378  freeMndcfrmd 14485   ~FG cefg 15031  freeGrpcfrgp 15032
This theorem is referenced by:  frgpgrp  15087  frgpinv  15089  frgpmhm  15090
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-ot 3663  df-uni 3844  df-int 3879  df-iun 3923  df-iin 3924  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-2o 6496  df-oadd 6499  df-er 6676  df-ec 6678  df-qs 6682  df-map 6790  df-pm 6791  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-sup 7210  df-card 7588  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-10 9828  df-n0 9982  df-z 10041  df-dec 10141  df-uz 10247  df-fz 10799  df-fzo 10887  df-hash 11354  df-word 11425  df-concat 11426  df-s1 11427  df-substr 11428  df-splice 11429  df-reverse 11430  df-s2 11514  df-struct 13166  df-ndx 13167  df-slot 13168  df-base 13169  df-plusg 13237  df-mulr 13238  df-sca 13240  df-vsca 13241  df-tset 13243  df-ple 13244  df-ds 13246  df-0g 13420  df-imas 13427  df-divs 13428  df-mnd 14383  df-frmd 14487  df-grp 14505  df-efg 15034  df-frgp 15035
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