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Theorem frgp0 16757
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 2443 . . 3  |-  (freeMnd `  (
I  X.  2o ) )  =  (freeMnd `  (
I  X.  2o ) )
3 frgp0.r . . 3  |-  .~  =  ( ~FG  `  I )
41, 2, 3frgpval 16755 . 2  |-  ( I  e.  V  ->  G  =  ( (freeMnd `  (
I  X.  2o ) )  /.s 
.~  ) )
5 2on 7140 . . . . 5  |-  2o  e.  On
6 xpexg 6587 . . . . 5  |-  ( ( I  e.  V  /\  2o  e.  On )  -> 
( I  X.  2o )  e.  _V )
75, 6mpan2 671 . . . 4  |-  ( I  e.  V  ->  (
I  X.  2o )  e.  _V )
8 eqid 2443 . . . . 5  |-  ( Base `  (freeMnd `  ( I  X.  2o ) ) )  =  ( Base `  (freeMnd `  ( I  X.  2o ) ) )
92, 8frmdbas 15999 . . . 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 2451 . 2  |-  ( I  e.  V  -> Word  ( I  X.  2o )  =  ( Base `  (freeMnd `  ( I  X.  2o ) ) ) )
12 eqidd 2444 . 2  |-  ( I  e.  V  ->  ( +g  `  (freeMnd `  (
I  X.  2o ) ) )  =  ( +g  `  (freeMnd `  (
I  X.  2o ) ) ) )
13 eqid 2443 . . . 4  |-  (  _I 
` Word  ( I  X.  2o ) )  =  (  _I  ` Word  ( I  X.  2o ) )
1413, 3efger 16715 . . 3  |-  .~  Er  (  _I  ` Word  ( I  X.  2o ) )
15 wrdexg 12539 . . . . 5  |-  ( ( I  X.  2o )  e.  _V  -> Word  ( I  X.  2o )  e. 
_V )
16 fvi 5915 . . . . 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 7321 . . . 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 5866 . . 3  |-  (freeMnd `  (
I  X.  2o ) )  e.  _V
2221a1i 11 . 2  |-  ( I  e.  V  ->  (freeMnd `  ( I  X.  2o ) )  e.  _V )
23 eqid 2443 . . . 4  |-  ( +g  `  (freeMnd `  ( I  X.  2o ) ) )  =  ( +g  `  (freeMnd `  ( I  X.  2o ) ) )
241, 2, 3, 23frgpcpbl 16756 . . 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 16006 . . . . . 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 1018 . . . 4  |-  ( ( I  e.  V  /\  x  e. Word  ( I  X.  2o )  /\  y  e. Word  ( I  X.  2o ) )  ->  (freeMnd `  ( I  X.  2o ) )  e.  Mnd )
29 simp2 998 . . . . 5  |-  ( ( I  e.  V  /\  x  e. Word  ( I  X.  2o )  /\  y  e. Word  ( I  X.  2o ) )  ->  x  e. Word  ( I  X.  2o ) )
30113ad2ant1 1018 . . . . 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 2533 . . . 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 999 . . . . 5  |-  ( ( I  e.  V  /\  x  e. Word  ( I  X.  2o )  /\  y  e. Word  ( I  X.  2o ) )  ->  y  e. Word  ( I  X.  2o ) )
3332, 30eleqtrd 2533 . . . 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 15908 . . . 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 1229 . . 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 2534 . 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 465 . . . 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 465 . . . . . 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 1208 . . . . . 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 1005 . . . . . . 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 465 . . . . . . 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 2533 . . . . . 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 15908 . . . . . 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 1229 . . . . 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 2534 . . . 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 7333 . . 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 1208 . . . 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 1208 . . . 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 15909 . . . 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 1231 . . 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 4461 . 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 12547 . . 3  |-  (/)  e. Word  (
I  X.  2o )
5352a1i 11 . 2  |-  ( I  e.  V  ->  (/)  e. Word  (
I  X.  2o ) )
5452, 11syl5eleq 2537 . . . . . 6  |-  ( I  e.  V  ->  (/)  e.  (
Base `  (freeMnd `  (
I  X.  2o ) ) ) )
5554adantr 465 . . . . 5  |-  ( ( I  e.  V  /\  x  e. Word  ( I  X.  2o ) )  ->  (/) 
e.  ( Base `  (freeMnd `  ( I  X.  2o ) ) ) )
5611eleq2d 2513 . . . . . 6  |-  ( I  e.  V  ->  (
x  e. Word  ( I  X.  2o )  <->  x  e.  ( Base `  (freeMnd `  (
I  X.  2o ) ) ) ) )
5756biimpa 484 . . . . 5  |-  ( ( I  e.  V  /\  x  e. Word  ( I  X.  2o ) )  ->  x  e.  ( Base `  (freeMnd `  ( I  X.  2o ) ) ) )
582, 8, 23frmdadd 16002 . . . . 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 661 . . . 4  |-  ( ( I  e.  V  /\  x  e. Word  ( I  X.  2o ) )  -> 
( (/) ( +g  `  (freeMnd `  ( I  X.  2o ) ) ) x )  =  ( (/) concat  x ) )
60 ccatlid 12585 . . . . 5  |-  ( x  e. Word  ( I  X.  2o )  ->  ( (/) concat  x )  =  x )
6160adantl 466 . . . 4  |-  ( ( I  e.  V  /\  x  e. Word  ( I  X.  2o ) )  -> 
( (/) concat  x )  =  x )
6259, 61eqtrd 2484 . . 3  |-  ( ( I  e.  V  /\  x  e. Word  ( I  X.  2o ) )  -> 
( (/) ( +g  `  (freeMnd `  ( I  X.  2o ) ) ) x )  =  x )
6320adantr 465 . . . 4  |-  ( ( I  e.  V  /\  x  e. Word  ( I  X.  2o ) )  ->  .~  Er Word  ( I  X.  2o ) )
64 simpr 461 . . . 4  |-  ( ( I  e.  V  /\  x  e. Word  ( I  X.  2o ) )  ->  x  e. Word  ( I  X.  2o ) )
6563, 64erref 7333 . . 3  |-  ( ( I  e.  V  /\  x  e. Word  ( I  X.  2o ) )  ->  x  .~  x )
6662, 65eqbrtrd 4457 . 2  |-  ( ( I  e.  V  /\  x  e. Word  ( I  X.  2o ) )  -> 
( (/) ( +g  `  (freeMnd `  ( I  X.  2o ) ) ) x )  .~  x )
67 revcl 12717 . . . 4  |-  ( x  e. Word  ( I  X.  2o )  ->  (reverse `  x
)  e. Word  ( I  X.  2o ) )
6867adantl 466 . . 3  |-  ( ( I  e.  V  /\  x  e. Word  ( I  X.  2o ) )  -> 
(reverse `  x )  e. Word 
( I  X.  2o ) )
69 eqid 2443 . . . . 5  |-  ( y  e.  I ,  z  e.  2o  |->  <. y ,  ( 1o  \ 
z ) >. )  =  ( y  e.  I ,  z  e.  2o  |->  <. y ,  ( 1o  \  z )
>. )
7069efgmf 16710 . . . 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 12779 . . 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 661 . 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 465 . . . . 5  |-  ( ( I  e.  V  /\  x  e. Word  ( I  X.  2o ) )  -> Word  ( I  X.  2o )  =  ( Base `  (freeMnd `  ( I  X.  2o ) ) ) )
7573, 74eleqtrd 2533 . . . 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 16002 . . . 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 661 . . 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 2513 . . . . 5  |-  ( I  e.  V  ->  (
x  e.  (  _I 
` Word  ( I  X.  2o ) )  <->  x  e. Word  ( I  X.  2o ) ) )
7978biimpar 485 . . . 4  |-  ( ( I  e.  V  /\  x  e. Word  ( I  X.  2o ) )  ->  x  e.  (  _I  ` Word 
( I  X.  2o ) ) )
80 eqid 2443 . . . . 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 16725 . . . 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 4457 . 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, 83qusgrp2 16167 1  |-  ( I  e.  V  ->  ( G  e.  Grp  /\  [ (/)
]  .~  =  ( 0g `  G ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 974    = wceq 1383    e. wcel 1804   _Vcvv 3095    \ cdif 3458   (/)c0 3770   <.cop 4020   <.cotp 4022   class class class wbr 4437    |-> cmpt 4495    _I cid 4780   Oncon0 4868    X. cxp 4987    o. ccom 4993   -->wf 5574   ` cfv 5578  (class class class)co 6281    |-> cmpt2 6283   1oc1o 7125   2oc2o 7126    Er wer 7310   [cec 7311   0cc0 9495   ...cfz 11683   #chash 12387  Word cword 12516   concat cconcat 12518   splice csplice 12521  reversecreverse 12522   <"cs2 12788   Basecbs 14614   +g cplusg 14679   0gc0g 14819   Mndcmnd 15898  freeMndcfrmd 15994   Grpcgrp 16032   ~FG cefg 16703  freeGrpcfrgp 16704
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577  ax-cnex 9551  ax-resscn 9552  ax-1cn 9553  ax-icn 9554  ax-addcl 9555  ax-addrcl 9556  ax-mulcl 9557  ax-mulrcl 9558  ax-mulcom 9559  ax-addass 9560  ax-mulass 9561  ax-distr 9562  ax-i2m1 9563  ax-1ne0 9564  ax-1rid 9565  ax-rnegex 9566  ax-rrecex 9567  ax-cnre 9568  ax-pre-lttri 9569  ax-pre-lttrn 9570  ax-pre-ltadd 9571  ax-pre-mulgt0 9572
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-nel 2641  df-ral 2798  df-rex 2799  df-reu 2800  df-rmo 2801  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-ot 4023  df-uni 4235  df-int 4272  df-iun 4317  df-iin 4318  df-br 4438  df-opab 4496  df-mpt 4497  df-tr 4531  df-eprel 4781  df-id 4785  df-po 4790  df-so 4791  df-fr 4828  df-we 4830  df-ord 4871  df-on 4872  df-lim 4873  df-suc 4874  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-riota 6242  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-om 6686  df-1st 6785  df-2nd 6786  df-recs 7044  df-rdg 7078  df-1o 7132  df-2o 7133  df-oadd 7136  df-er 7313  df-ec 7315  df-qs 7319  df-map 7424  df-pm 7425  df-en 7519  df-dom 7520  df-sdom 7521  df-fin 7522  df-sup 7903  df-card 8323  df-cda 8551  df-pnf 9633  df-mnf 9634  df-xr 9635  df-ltxr 9636  df-le 9637  df-sub 9812  df-neg 9813  df-nn 10544  df-2 10601  df-3 10602  df-4 10603  df-5 10604  df-6 10605  df-7 10606  df-8 10607  df-9 10608  df-10 10609  df-n0 10803  df-z 10872  df-dec 10987  df-uz 11093  df-fz 11684  df-fzo 11807  df-hash 12388  df-word 12524  df-concat 12526  df-s1 12527  df-substr 12528  df-splice 12529  df-reverse 12530  df-s2 12795  df-struct 14616  df-ndx 14617  df-slot 14618  df-base 14619  df-plusg 14692  df-mulr 14693  df-sca 14695  df-vsca 14696  df-ip 14697  df-tset 14698  df-ple 14699  df-ds 14701  df-0g 14821  df-imas 14887  df-qus 14888  df-mgm 15851  df-sgrp 15890  df-mnd 15900  df-frmd 15996  df-grp 16036  df-efg 16706  df-frgp 16707
This theorem is referenced by:  frgpgrp  16759  frgpinv  16761  frgpmhm  16762
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