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Theorem frgp0 15394
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 2438 . . 3  |-  (freeMnd `  (
I  X.  2o ) )  =  (freeMnd `  (
I  X.  2o ) )
3 frgp0.r . . 3  |-  .~  =  ( ~FG  `  I )
41, 2, 3frgpval 15392 . 2  |-  ( I  e.  V  ->  G  =  ( (freeMnd `  (
I  X.  2o ) )  /.s 
.~  ) )
5 2on 6734 . . . . 5  |-  2o  e.  On
6 xpexg 4991 . . . . 5  |-  ( ( I  e.  V  /\  2o  e.  On )  -> 
( I  X.  2o )  e.  _V )
75, 6mpan2 654 . . . 4  |-  ( I  e.  V  ->  (
I  X.  2o )  e.  _V )
8 eqid 2438 . . . . 5  |-  ( Base `  (freeMnd `  ( I  X.  2o ) ) )  =  ( Base `  (freeMnd `  ( I  X.  2o ) ) )
92, 8frmdbas 14799 . . . 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 2443 . 2  |-  ( I  e.  V  -> Word  ( I  X.  2o )  =  ( Base `  (freeMnd `  ( I  X.  2o ) ) ) )
12 eqidd 2439 . 2  |-  ( I  e.  V  ->  ( +g  `  (freeMnd `  (
I  X.  2o ) ) )  =  ( +g  `  (freeMnd `  (
I  X.  2o ) ) ) )
13 eqid 2438 . . . 4  |-  (  _I 
` Word  ( I  X.  2o ) )  =  (  _I  ` Word  ( I  X.  2o ) )
1413, 3efger 15352 . . 3  |-  .~  Er  (  _I  ` Word  ( I  X.  2o ) )
15 wrdexg 11741 . . . . 5  |-  ( ( I  X.  2o )  e.  _V  -> Word  ( I  X.  2o )  e. 
_V )
16 fvi 5785 . . . . 5  |-  (Word  (
I  X.  2o )  e.  _V  ->  (  _I  ` Word  ( I  X.  2o ) )  = Word  (
I  X.  2o ) )
177, 15, 163syl 19 . . . 4  |-  ( I  e.  V  ->  (  _I  ` Word  ( I  X.  2o ) )  = Word  (
I  X.  2o ) )
18 ereq2 6915 . . . 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 204 . 2  |-  ( I  e.  V  ->  .~  Er Word  ( I  X.  2o ) )
21 fvex 5744 . . 3  |-  (freeMnd `  (
I  X.  2o ) )  e.  _V
2221a1i 11 . 2  |-  ( I  e.  V  ->  (freeMnd `  ( I  X.  2o ) )  e.  _V )
23 eqid 2438 . . . 4  |-  ( +g  `  (freeMnd `  ( I  X.  2o ) ) )  =  ( +g  `  (freeMnd `  ( I  X.  2o ) ) )
241, 2, 3, 23frgpcpbl 15393 . . 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 14806 . . . . . 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 979 . . . 4  |-  ( ( I  e.  V  /\  x  e. Word  ( I  X.  2o )  /\  y  e. Word  ( I  X.  2o ) )  ->  (freeMnd `  ( I  X.  2o ) )  e.  Mnd )
29 simp2 959 . . . . 5  |-  ( ( I  e.  V  /\  x  e. Word  ( I  X.  2o )  /\  y  e. Word  ( I  X.  2o ) )  ->  x  e. Word  ( I  X.  2o ) )
30113ad2ant1 979 . . . . 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 2514 . . . 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 960 . . . . 5  |-  ( ( I  e.  V  /\  x  e. Word  ( I  X.  2o )  /\  y  e. Word  ( I  X.  2o ) )  ->  y  e. Word  ( I  X.  2o ) )
3332, 30eleqtrd 2514 . . . 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 14697 . . . 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 1185 . . 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 2515 . 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 453 . . . 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 453 . . . . . 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 1165 . . . . . 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 966 . . . . . . 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 453 . . . . . . 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 2514 . . . . . 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 14697 . . . . . 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 1185 . . . . 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 2515 . . . 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 6927 . . 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 1165 . . . 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 1165 . . . 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 14698 . . . 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 1187 . . 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 4238 . 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 11734 . . 3  |-  (/)  e. Word  (
I  X.  2o )
5352a1i 11 . 2  |-  ( I  e.  V  ->  (/)  e. Word  (
I  X.  2o ) )
5452, 11syl5eleq 2524 . . . . . 6  |-  ( I  e.  V  ->  (/)  e.  (
Base `  (freeMnd `  (
I  X.  2o ) ) ) )
5554adantr 453 . . . . 5  |-  ( ( I  e.  V  /\  x  e. Word  ( I  X.  2o ) )  ->  (/) 
e.  ( Base `  (freeMnd `  ( I  X.  2o ) ) ) )
5611eleq2d 2505 . . . . . 6  |-  ( I  e.  V  ->  (
x  e. Word  ( I  X.  2o )  <->  x  e.  ( Base `  (freeMnd `  (
I  X.  2o ) ) ) ) )
5756biimpa 472 . . . . 5  |-  ( ( I  e.  V  /\  x  e. Word  ( I  X.  2o ) )  ->  x  e.  ( Base `  (freeMnd `  ( I  X.  2o ) ) ) )
582, 8, 23frmdadd 14802 . . . . 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 644 . . . 4  |-  ( ( I  e.  V  /\  x  e. Word  ( I  X.  2o ) )  -> 
( (/) ( +g  `  (freeMnd `  ( I  X.  2o ) ) ) x )  =  ( (/) concat  x ) )
60 ccatlid 11750 . . . . 5  |-  ( x  e. Word  ( I  X.  2o )  ->  ( (/) concat  x )  =  x )
6160adantl 454 . . . 4  |-  ( ( I  e.  V  /\  x  e. Word  ( I  X.  2o ) )  -> 
( (/) concat  x )  =  x )
6259, 61eqtrd 2470 . . 3  |-  ( ( I  e.  V  /\  x  e. Word  ( I  X.  2o ) )  -> 
( (/) ( +g  `  (freeMnd `  ( I  X.  2o ) ) ) x )  =  x )
6320adantr 453 . . . 4  |-  ( ( I  e.  V  /\  x  e. Word  ( I  X.  2o ) )  ->  .~  Er Word  ( I  X.  2o ) )
64 simpr 449 . . . 4  |-  ( ( I  e.  V  /\  x  e. Word  ( I  X.  2o ) )  ->  x  e. Word  ( I  X.  2o ) )
6563, 64erref 6927 . . 3  |-  ( ( I  e.  V  /\  x  e. Word  ( I  X.  2o ) )  ->  x  .~  x )
6662, 65eqbrtrd 4234 . 2  |-  ( ( I  e.  V  /\  x  e. Word  ( I  X.  2o ) )  -> 
( (/) ( +g  `  (freeMnd `  ( I  X.  2o ) ) ) x )  .~  x )
67 revcl 11795 . . . 4  |-  ( x  e. Word  ( I  X.  2o )  ->  (reverse `  x
)  e. Word  ( I  X.  2o ) )
6867adantl 454 . . 3  |-  ( ( I  e.  V  /\  x  e. Word  ( I  X.  2o ) )  -> 
(reverse `  x )  e. Word 
( I  X.  2o ) )
69 eqid 2438 . . . . 5  |-  ( y  e.  I ,  z  e.  2o  |->  <. y ,  ( 1o  \ 
z ) >. )  =  ( y  e.  I ,  z  e.  2o  |->  <. y ,  ( 1o  \  z )
>. )
7069efgmf 15347 . . . 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 11802 . . 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 644 . 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 453 . . . . 5  |-  ( ( I  e.  V  /\  x  e. Word  ( I  X.  2o ) )  -> Word  ( I  X.  2o )  =  ( Base `  (freeMnd `  ( I  X.  2o ) ) ) )
7573, 74eleqtrd 2514 . . . 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 14802 . . . 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 644 . . 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 2505 . . . . 5  |-  ( I  e.  V  ->  (
x  e.  (  _I 
` Word  ( I  X.  2o ) )  <->  x  e. Word  ( I  X.  2o ) ) )
7978biimpar 473 . . . 4  |-  ( ( I  e.  V  /\  x  e. Word  ( I  X.  2o ) )  ->  x  e.  (  _I  ` Word 
( I  X.  2o ) ) )
80 eqid 2438 . . . . 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 15362 . . . 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 4234 . 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 14938 1  |-  ( I  e.  V  ->  ( G  e.  Grp  /\  [ (/)
]  .~  =  ( 0g `  G ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726   _Vcvv 2958    \ cdif 3319   (/)c0 3630   <.cop 3819   <.cotp 3820   class class class wbr 4214    e. cmpt 4268    _I cid 4495   Oncon0 4583    X. cxp 4878    o. ccom 4884   -->wf 5452   ` cfv 5456  (class class class)co 6083    e. cmpt2 6085   1oc1o 6719   2oc2o 6720    Er wer 6904   [cec 6905   0cc0 8992   ...cfz 11045   #chash 11620  Word cword 11719   concat cconcat 11720   splice csplice 11723  reversecreverse 11724   <"cs2 11807   Basecbs 13471   +g cplusg 13531   0gc0g 13725   Mndcmnd 14686   Grpcgrp 14687  freeMndcfrmd 14794   ~FG cefg 15340  freeGrpcfrgp 15341
This theorem is referenced by:  frgpgrp  15396  frgpinv  15398  frgpmhm  15399
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703  ax-cnex 9048  ax-resscn 9049  ax-1cn 9050  ax-icn 9051  ax-addcl 9052  ax-addrcl 9053  ax-mulcl 9054  ax-mulrcl 9055  ax-mulcom 9056  ax-addass 9057  ax-mulass 9058  ax-distr 9059  ax-i2m1 9060  ax-1ne0 9061  ax-1rid 9062  ax-rnegex 9063  ax-rrecex 9064  ax-cnre 9065  ax-pre-lttri 9066  ax-pre-lttrn 9067  ax-pre-ltadd 9068  ax-pre-mulgt0 9069
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-ot 3826  df-uni 4018  df-int 4053  df-iun 4097  df-iin 4098  df-br 4215  df-opab 4269  df-mpt 4270  df-tr 4305  df-eprel 4496  df-id 4500  df-po 4505  df-so 4506  df-fr 4543  df-we 4545  df-ord 4586  df-on 4587  df-lim 4588  df-suc 4589  df-om 4848  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-1st 6351  df-2nd 6352  df-riota 6551  df-recs 6635  df-rdg 6670  df-1o 6726  df-2o 6727  df-oadd 6730  df-er 6907  df-ec 6909  df-qs 6913  df-map 7022  df-pm 7023  df-en 7112  df-dom 7113  df-sdom 7114  df-fin 7115  df-sup 7448  df-card 7828  df-pnf 9124  df-mnf 9125  df-xr 9126  df-ltxr 9127  df-le 9128  df-sub 9295  df-neg 9296  df-nn 10003  df-2 10060  df-3 10061  df-4 10062  df-5 10063  df-6 10064  df-7 10065  df-8 10066  df-9 10067  df-10 10068  df-n0 10224  df-z 10285  df-dec 10385  df-uz 10491  df-fz 11046  df-fzo 11138  df-hash 11621  df-word 11725  df-concat 11726  df-s1 11727  df-substr 11728  df-splice 11729  df-reverse 11730  df-s2 11814  df-struct 13473  df-ndx 13474  df-slot 13475  df-base 13476  df-plusg 13544  df-mulr 13545  df-sca 13547  df-vsca 13548  df-tset 13550  df-ple 13551  df-ds 13553  df-0g 13729  df-imas 13736  df-divs 13737  df-mnd 14692  df-frmd 14796  df-grp 14814  df-efg 15343  df-frgp 15344
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