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Theorem frgp0 15069
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 2283 . . 3  |-  (freeMnd `  (
I  X.  2o ) )  =  (freeMnd `  (
I  X.  2o ) )
3 frgp0.r . . 3  |-  .~  =  ( ~FG  `  I )
41, 2, 3frgpval 15067 . 2  |-  ( I  e.  V  ->  G  =  ( (freeMnd `  (
I  X.  2o ) )  /.s 
.~  ) )
5 2on 6487 . . . . 5  |-  2o  e.  On
6 xpexg 4800 . . . . 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 2283 . . . . 5  |-  ( Base `  (freeMnd `  ( I  X.  2o ) ) )  =  ( Base `  (freeMnd `  ( I  X.  2o ) ) )
92, 8frmdbas 14474 . . . 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 2288 . 2  |-  ( I  e.  V  -> Word  ( I  X.  2o )  =  ( Base `  (freeMnd `  ( I  X.  2o ) ) ) )
12 eqidd 2284 . 2  |-  ( I  e.  V  ->  ( +g  `  (freeMnd `  (
I  X.  2o ) ) )  =  ( +g  `  (freeMnd `  (
I  X.  2o ) ) ) )
13 eqid 2283 . . . 4  |-  (  _I 
` Word  ( I  X.  2o ) )  =  (  _I  ` Word  ( I  X.  2o ) )
1413, 3efger 15027 . . 3  |-  .~  Er  (  _I  ` Word  ( I  X.  2o ) )
15 wrdexg 11425 . . . . 5  |-  ( ( I  X.  2o )  e.  _V  -> Word  ( I  X.  2o )  e. 
_V )
16 fvi 5579 . . . . 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 6668 . . . 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 5539 . . 3  |-  (freeMnd `  (
I  X.  2o ) )  e.  _V
2221a1i 10 . 2  |-  ( I  e.  V  ->  (freeMnd `  ( I  X.  2o ) )  e.  _V )
23 eqid 2283 . . . 4  |-  ( +g  `  (freeMnd `  ( I  X.  2o ) ) )  =  ( +g  `  (freeMnd `  ( I  X.  2o ) ) )
241, 2, 3, 23frgpcpbl 15068 . . 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 14481 . . . . . 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 2359 . . . 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 2359 . . . 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 14372 . . . 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 2360 . 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 2359 . . . . . 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 14372 . . . . . 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 2360 . . . 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 6680 . . 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 14373 . . . 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 4047 . 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 11418 . . 3  |-  (/)  e. Word  (
I  X.  2o )
5352a1i 10 . 2  |-  ( I  e.  V  ->  (/)  e. Word  (
I  X.  2o ) )
5452, 11syl5eleq 2369 . . . . . 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 2350 . . . . . 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 14477 . . . . 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 11434 . . . . 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 2315 . . 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 6680 . . 3  |-  ( ( I  e.  V  /\  x  e. Word  ( I  X.  2o ) )  ->  x  .~  x )
6662, 65eqbrtrd 4043 . 2  |-  ( ( I  e.  V  /\  x  e. Word  ( I  X.  2o ) )  -> 
( (/) ( +g  `  (freeMnd `  ( I  X.  2o ) ) ) x )  .~  x )
67 revcl 11479 . . . 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 2283 . . . . 5  |-  ( y  e.  I ,  z  e.  2o  |->  <. y ,  ( 1o  \ 
z ) >. )  =  ( y  e.  I ,  z  e.  2o  |->  <. y ,  ( 1o  \  z )
>. )
7069efgmf 15022 . . . 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 11486 . . 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 2359 . . . 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 14477 . . . 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 2350 . . . . 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 2283 . . . . 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 15037 . . . 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 4043 . 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 14613 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 1623    e. wcel 1684   _Vcvv 2788    \ cdif 3149   (/)c0 3455   <.cop 3643   <.cotp 3644   class class class wbr 4023    e. cmpt 4077    _I cid 4304   Oncon0 4392    X. cxp 4687    o. ccom 4693   -->wf 5251   ` cfv 5255  (class class class)co 5858    e. cmpt2 5860   1oc1o 6472   2oc2o 6473    Er wer 6657   [cec 6658   0cc0 8737   ...cfz 10782   #chash 11337  Word cword 11403   concat cconcat 11404   splice csplice 11407  reversecreverse 11408   <"cs2 11491   Basecbs 13148   +g cplusg 13208   0gc0g 13400   Mndcmnd 14361   Grpcgrp 14362  freeMndcfrmd 14469   ~FG cefg 15015  freeGrpcfrgp 15016
This theorem is referenced by:  frgpgrp  15071  frgpinv  15073  frgpmhm  15074
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-ot 3650  df-uni 3828  df-int 3863  df-iun 3907  df-iin 3908  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-2o 6480  df-oadd 6483  df-er 6660  df-ec 6662  df-qs 6666  df-map 6774  df-pm 6775  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-sup 7194  df-card 7572  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-7 9809  df-8 9810  df-9 9811  df-10 9812  df-n0 9966  df-z 10025  df-dec 10125  df-uz 10231  df-fz 10783  df-fzo 10871  df-hash 11338  df-word 11409  df-concat 11410  df-s1 11411  df-substr 11412  df-splice 11413  df-reverse 11414  df-s2 11498  df-struct 13150  df-ndx 13151  df-slot 13152  df-base 13153  df-plusg 13221  df-mulr 13222  df-sca 13224  df-vsca 13225  df-tset 13227  df-ple 13228  df-ds 13230  df-0g 13404  df-imas 13411  df-divs 13412  df-mnd 14367  df-frmd 14471  df-grp 14489  df-efg 15018  df-frgp 15019
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