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Theorem relexpcnv 25138
Description: Distributivity of converse and relation exponentiation. (Contributed by Drahflow, 12-Nov-2015.)
Hypotheses
Ref Expression
relexpcnv.1  |-  ( ph  ->  Rel  R )
relexpcnv.2  |-  ( ph  ->  R  e.  _V )
Assertion
Ref Expression
relexpcnv  |-  ( ph  ->  ( N  e.  NN0  ->  `' ( R ^
r N )  =  ( `' R ^
r N ) ) )

Proof of Theorem relexpcnv
Dummy variables  i  n are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eleq1 2498 . . . . . 6  |-  ( i  =  0  ->  (
i  e.  NN0  <->  0  e.  NN0 ) )
21anbi1d 687 . . . . 5  |-  ( i  =  0  ->  (
( i  e.  NN0  /\ 
ph )  <->  ( 0  e.  NN0  /\  ph )
) )
3 oveq2 6092 . . . . . . 7  |-  ( i  =  0  ->  ( R ^ r i )  =  ( R ^
r 0 ) )
43cnveqd 5051 . . . . . 6  |-  ( i  =  0  ->  `' ( R ^ r i )  =  `' ( R ^ r 0 ) )
5 oveq2 6092 . . . . . 6  |-  ( i  =  0  ->  ( `' R ^ r i )  =  ( `' R ^ r 0 ) )
64, 5eqeq12d 2452 . . . . 5  |-  ( i  =  0  ->  ( `' ( R ^
r i )  =  ( `' R ^
r i )  <->  `' ( R ^ r 0 )  =  ( `' R ^ r 0 ) ) )
72, 6imbi12d 313 . . . 4  |-  ( i  =  0  ->  (
( ( i  e. 
NN0  /\  ph )  ->  `' ( R ^
r i )  =  ( `' R ^
r i ) )  <-> 
( ( 0  e. 
NN0  /\  ph )  ->  `' ( R ^
r 0 )  =  ( `' R ^
r 0 ) ) ) )
8 eleq1 2498 . . . . . 6  |-  ( i  =  n  ->  (
i  e.  NN0  <->  n  e.  NN0 ) )
98anbi1d 687 . . . . 5  |-  ( i  =  n  ->  (
( i  e.  NN0  /\ 
ph )  <->  ( n  e.  NN0  /\  ph )
) )
10 oveq2 6092 . . . . . . 7  |-  ( i  =  n  ->  ( R ^ r i )  =  ( R ^
r n ) )
1110cnveqd 5051 . . . . . 6  |-  ( i  =  n  ->  `' ( R ^ r i )  =  `' ( R ^ r n ) )
12 oveq2 6092 . . . . . 6  |-  ( i  =  n  ->  ( `' R ^ r i )  =  ( `' R ^ r n ) )
1311, 12eqeq12d 2452 . . . . 5  |-  ( i  =  n  ->  ( `' ( R ^
r i )  =  ( `' R ^
r i )  <->  `' ( R ^ r n )  =  ( `' R ^ r n ) ) )
149, 13imbi12d 313 . . . 4  |-  ( i  =  n  ->  (
( ( i  e. 
NN0  /\  ph )  ->  `' ( R ^
r i )  =  ( `' R ^
r i ) )  <-> 
( ( n  e. 
NN0  /\  ph )  ->  `' ( R ^
r n )  =  ( `' R ^
r n ) ) ) )
15 eleq1 2498 . . . . . 6  |-  ( i  =  ( n  + 
1 )  ->  (
i  e.  NN0  <->  ( n  +  1 )  e. 
NN0 ) )
1615anbi1d 687 . . . . 5  |-  ( i  =  ( n  + 
1 )  ->  (
( i  e.  NN0  /\ 
ph )  <->  ( (
n  +  1 )  e.  NN0  /\  ph )
) )
17 oveq2 6092 . . . . . . 7  |-  ( i  =  ( n  + 
1 )  ->  ( R ^ r i )  =  ( R ^
r ( n  + 
1 ) ) )
1817cnveqd 5051 . . . . . 6  |-  ( i  =  ( n  + 
1 )  ->  `' ( R ^ r i )  =  `' ( R ^ r ( n  +  1 ) ) )
19 oveq2 6092 . . . . . 6  |-  ( i  =  ( n  + 
1 )  ->  ( `' R ^ r i )  =  ( `' R ^ r ( n  +  1 ) ) )
2018, 19eqeq12d 2452 . . . . 5  |-  ( i  =  ( n  + 
1 )  ->  ( `' ( R ^
r i )  =  ( `' R ^
r i )  <->  `' ( R ^ r ( n  +  1 ) )  =  ( `' R ^ r ( n  +  1 ) ) ) )
2116, 20imbi12d 313 . . . 4  |-  ( i  =  ( n  + 
1 )  ->  (
( ( i  e. 
NN0  /\  ph )  ->  `' ( R ^
r i )  =  ( `' R ^
r i ) )  <-> 
( ( ( n  +  1 )  e. 
NN0  /\  ph )  ->  `' ( R ^
r ( n  + 
1 ) )  =  ( `' R ^
r ( n  + 
1 ) ) ) ) )
22 eleq1 2498 . . . . . 6  |-  ( i  =  N  ->  (
i  e.  NN0  <->  N  e.  NN0 ) )
2322anbi1d 687 . . . . 5  |-  ( i  =  N  ->  (
( i  e.  NN0  /\ 
ph )  <->  ( N  e.  NN0  /\  ph )
) )
24 oveq2 6092 . . . . . . 7  |-  ( i  =  N  ->  ( R ^ r i )  =  ( R ^
r N ) )
2524cnveqd 5051 . . . . . 6  |-  ( i  =  N  ->  `' ( R ^ r i )  =  `' ( R ^ r N ) )
26 oveq2 6092 . . . . . 6  |-  ( i  =  N  ->  ( `' R ^ r i )  =  ( `' R ^ r N ) )
2725, 26eqeq12d 2452 . . . . 5  |-  ( i  =  N  ->  ( `' ( R ^
r i )  =  ( `' R ^
r i )  <->  `' ( R ^ r N )  =  ( `' R ^ r N ) ) )
2823, 27imbi12d 313 . . . 4  |-  ( i  =  N  ->  (
( ( i  e. 
NN0  /\  ph )  ->  `' ( R ^
r i )  =  ( `' R ^
r i ) )  <-> 
( ( N  e. 
NN0  /\  ph )  ->  `' ( R ^
r N )  =  ( `' R ^
r N ) ) ) )
29 relexpcnv.1 . . . . . . 7  |-  ( ph  ->  Rel  R )
30 relexpcnv.2 . . . . . . 7  |-  ( ph  ->  R  e.  _V )
3129, 30relexp0 25134 . . . . . 6  |-  ( ph  ->  ( R ^ r 0 )  =  (  _I  |`  U. U. R
) )
3231adantl 454 . . . . 5  |-  ( ( 0  e.  NN0  /\  ph )  ->  ( R ^ r 0 )  =  (  _I  |`  U. U. R ) )
33 id 21 . . . . . . 7  |-  ( ( R ^ r 0 )  =  (  _I  |`  U. U. R )  ->  ( R ^
r 0 )  =  (  _I  |`  U. U. R ) )
3433cnveqd 5051 . . . . . 6  |-  ( ( R ^ r 0 )  =  (  _I  |`  U. U. R )  ->  `' ( R ^ r 0 )  =  `' (  _I  |`  U. U. R ) )
35 cnvresid 5526 . . . . . . 7  |-  `' (  _I  |`  U. U. R
)  =  (  _I  |`  U. U. R )
3629adantl 454 . . . . . . . . 9  |-  ( ( 0  e.  NN0  /\  ph )  ->  Rel  R )
37 relcnvfld 5403 . . . . . . . . 9  |-  ( Rel 
R  ->  U. U. R  =  U. U. `' R
)
3836, 37syl 16 . . . . . . . 8  |-  ( ( 0  e.  NN0  /\  ph )  ->  U. U. R  =  U. U. `' R
)
39 reseq2 5144 . . . . . . . . 9  |-  ( U. U. R  =  U. U. `' R  ->  (  _I  |`  U. U. R )  =  (  _I  |`  U. U. `' R ) )
40 simpr 449 . . . . . . . . . . 11  |-  ( ( 0  e.  NN0  /\  ph )  ->  ph )
41 relcnv 5245 . . . . . . . . . . 11  |-  Rel  `' R
42 simpr 449 . . . . . . . . . . . 12  |-  ( (
ph  /\  Rel  `' R
)  ->  Rel  `' R
)
43 simpl 445 . . . . . . . . . . . . 13  |-  ( (
ph  /\  Rel  `' R
)  ->  ph )
44 cnvexg 5408 . . . . . . . . . . . . 13  |-  ( R  e.  _V  ->  `' R  e.  _V )
4543, 30, 443syl 19 . . . . . . . . . . . 12  |-  ( (
ph  /\  Rel  `' R
)  ->  `' R  e.  _V )
4642, 45relexp0 25134 . . . . . . . . . . 11  |-  ( (
ph  /\  Rel  `' R
)  ->  ( `' R ^ r 0 )  =  (  _I  |`  U. U. `' R ) )
4740, 41, 46sylancl 645 . . . . . . . . . 10  |-  ( ( 0  e.  NN0  /\  ph )  ->  ( `' R ^ r 0 )  =  (  _I  |`  U. U. `' R ) )
4847eqcomd 2443 . . . . . . . . 9  |-  ( ( 0  e.  NN0  /\  ph )  ->  (  _I  |` 
U. U. `' R )  =  ( `' R ^ r 0 ) )
4939, 48sylan9eq 2490 . . . . . . . 8  |-  ( ( U. U. R  = 
U. U. `' R  /\  ( 0  e.  NN0  /\ 
ph ) )  -> 
(  _I  |`  U. U. R )  =  ( `' R ^ r 0 ) )
5038, 49mpancom 652 . . . . . . 7  |-  ( ( 0  e.  NN0  /\  ph )  ->  (  _I  |` 
U. U. R )  =  ( `' R ^
r 0 ) )
5135, 50syl5eq 2482 . . . . . 6  |-  ( ( 0  e.  NN0  /\  ph )  ->  `' (  _I  |`  U. U. R
)  =  ( `' R ^ r 0 ) )
5234, 51sylan9eq 2490 . . . . 5  |-  ( ( ( R ^ r 0 )  =  (  _I  |`  U. U. R
)  /\  ( 0  e.  NN0  /\  ph )
)  ->  `' ( R ^ r 0 )  =  ( `' R ^ r 0 ) )
5332, 52mpancom 652 . . . 4  |-  ( ( 0  e.  NN0  /\  ph )  ->  `' ( R ^ r 0 )  =  ( `' R ^ r 0 ) )
54 simprrr 743 . . . . . . . . . 10  |-  ( ( ( n  +  1 )  e.  NN0  /\  ( ph  /\  ( ( ( n  e.  NN0  /\ 
ph )  ->  `' ( R ^ r n )  =  ( `' R ^ r n ) )  /\  n  e.  NN0 ) ) )  ->  n  e.  NN0 )
55 simpl 445 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( (
( n  e.  NN0  /\ 
ph )  ->  `' ( R ^ r n )  =  ( `' R ^ r n ) )  /\  n  e.  NN0 ) )  ->  ph )
5655adantl 454 . . . . . . . . . . . 12  |-  ( ( ( n  +  1 )  e.  NN0  /\  ( ph  /\  ( ( ( n  e.  NN0  /\ 
ph )  ->  `' ( R ^ r n )  =  ( `' R ^ r n ) )  /\  n  e.  NN0 ) ) )  ->  ph )
5756, 29syl 16 . . . . . . . . . . 11  |-  ( ( ( n  +  1 )  e.  NN0  /\  ( ph  /\  ( ( ( n  e.  NN0  /\ 
ph )  ->  `' ( R ^ r n )  =  ( `' R ^ r n ) )  /\  n  e.  NN0 ) ) )  ->  Rel  R )
5856, 30syl 16 . . . . . . . . . . 11  |-  ( ( ( n  +  1 )  e.  NN0  /\  ( ph  /\  ( ( ( n  e.  NN0  /\ 
ph )  ->  `' ( R ^ r n )  =  ( `' R ^ r n ) )  /\  n  e.  NN0 ) ) )  ->  R  e.  _V )
5957, 58relexpsucl 25137 . . . . . . . . . 10  |-  ( ( ( n  +  1 )  e.  NN0  /\  ( ph  /\  ( ( ( n  e.  NN0  /\ 
ph )  ->  `' ( R ^ r n )  =  ( `' R ^ r n ) )  /\  n  e.  NN0 ) ) )  ->  ( n  e. 
NN0  ->  ( R ^
r ( n  + 
1 ) )  =  ( R  o.  ( R ^ r n ) ) ) )
6054, 59mpd 15 . . . . . . . . 9  |-  ( ( ( n  +  1 )  e.  NN0  /\  ( ph  /\  ( ( ( n  e.  NN0  /\ 
ph )  ->  `' ( R ^ r n )  =  ( `' R ^ r n ) )  /\  n  e.  NN0 ) ) )  ->  ( R ^
r ( n  + 
1 ) )  =  ( R  o.  ( R ^ r n ) ) )
6160cnveqd 5051 . . . . . . . 8  |-  ( ( ( n  +  1 )  e.  NN0  /\  ( ph  /\  ( ( ( n  e.  NN0  /\ 
ph )  ->  `' ( R ^ r n )  =  ( `' R ^ r n ) )  /\  n  e.  NN0 ) ) )  ->  `' ( R ^ r ( n  +  1 ) )  =  `' ( R  o.  ( R ^
r n ) ) )
62 id 21 . . . . . . . . . 10  |-  ( `' ( R ^ r ( n  +  1 ) )  =  `' ( R  o.  ( R ^ r n ) )  ->  `' ( R ^ r ( n  +  1 ) )  =  `' ( R  o.  ( R ^
r n ) ) )
63 cnvco 5059 . . . . . . . . . . 11  |-  `' ( R  o.  ( R ^ r n ) )  =  ( `' ( R ^ r n )  o.  `' R )
64 simprrl 742 . . . . . . . . . . . . 13  |-  ( ( ( n  +  1 )  e.  NN0  /\  ( ph  /\  ( ( ( n  e.  NN0  /\ 
ph )  ->  `' ( R ^ r n )  =  ( `' R ^ r n ) )  /\  n  e.  NN0 ) ) )  ->  ( ( n  e.  NN0  /\  ph )  ->  `' ( R ^
r n )  =  ( `' R ^
r n ) ) )
6554, 56, 64mp2and 662 . . . . . . . . . . . 12  |-  ( ( ( n  +  1 )  e.  NN0  /\  ( ph  /\  ( ( ( n  e.  NN0  /\ 
ph )  ->  `' ( R ^ r n )  =  ( `' R ^ r n ) )  /\  n  e.  NN0 ) ) )  ->  `' ( R ^ r n )  =  ( `' R ^ r n ) )
6665coeq1d 5037 . . . . . . . . . . 11  |-  ( ( ( n  +  1 )  e.  NN0  /\  ( ph  /\  ( ( ( n  e.  NN0  /\ 
ph )  ->  `' ( R ^ r n )  =  ( `' R ^ r n ) )  /\  n  e.  NN0 ) ) )  ->  ( `' ( R ^ r n )  o.  `' R
)  =  ( ( `' R ^ r n )  o.  `' R
) )
6763, 66syl5eq 2482 . . . . . . . . . 10  |-  ( ( ( n  +  1 )  e.  NN0  /\  ( ph  /\  ( ( ( n  e.  NN0  /\ 
ph )  ->  `' ( R ^ r n )  =  ( `' R ^ r n ) )  /\  n  e.  NN0 ) ) )  ->  `' ( R  o.  ( R ^
r n ) )  =  ( ( `' R ^ r n )  o.  `' R
) )
6862, 67sylan9eq 2490 . . . . . . . . 9  |-  ( ( `' ( R ^
r ( n  + 
1 ) )  =  `' ( R  o.  ( R ^ r n ) )  /\  (
( n  +  1 )  e.  NN0  /\  ( ph  /\  ( ( ( n  e.  NN0  /\ 
ph )  ->  `' ( R ^ r n )  =  ( `' R ^ r n ) )  /\  n  e.  NN0 ) ) ) )  ->  `' ( R ^ r ( n  +  1 ) )  =  ( ( `' R ^ r n )  o.  `' R
) )
6954adantl 454 . . . . . . . . . 10  |-  ( ( `' ( R ^
r ( n  + 
1 ) )  =  `' ( R  o.  ( R ^ r n ) )  /\  (
( n  +  1 )  e.  NN0  /\  ( ph  /\  ( ( ( n  e.  NN0  /\ 
ph )  ->  `' ( R ^ r n )  =  ( `' R ^ r n ) )  /\  n  e.  NN0 ) ) ) )  ->  n  e.  NN0 )
7041a1i 11 . . . . . . . . . . 11  |-  ( ( `' ( R ^
r ( n  + 
1 ) )  =  `' ( R  o.  ( R ^ r n ) )  /\  (
( n  +  1 )  e.  NN0  /\  ( ph  /\  ( ( ( n  e.  NN0  /\ 
ph )  ->  `' ( R ^ r n )  =  ( `' R ^ r n ) )  /\  n  e.  NN0 ) ) ) )  ->  Rel  `' R
)
71 simprrl 742 . . . . . . . . . . . 12  |-  ( ( `' ( R ^
r ( n  + 
1 ) )  =  `' ( R  o.  ( R ^ r n ) )  /\  (
( n  +  1 )  e.  NN0  /\  ( ph  /\  ( ( ( n  e.  NN0  /\ 
ph )  ->  `' ( R ^ r n )  =  ( `' R ^ r n ) )  /\  n  e.  NN0 ) ) ) )  ->  ph )
7271, 30, 443syl 19 . . . . . . . . . . 11  |-  ( ( `' ( R ^
r ( n  + 
1 ) )  =  `' ( R  o.  ( R ^ r n ) )  /\  (
( n  +  1 )  e.  NN0  /\  ( ph  /\  ( ( ( n  e.  NN0  /\ 
ph )  ->  `' ( R ^ r n )  =  ( `' R ^ r n ) )  /\  n  e.  NN0 ) ) ) )  ->  `' R  e.  _V )
7370, 72relexpsucr 25135 . . . . . . . . . 10  |-  ( ( `' ( R ^
r ( n  + 
1 ) )  =  `' ( R  o.  ( R ^ r n ) )  /\  (
( n  +  1 )  e.  NN0  /\  ( ph  /\  ( ( ( n  e.  NN0  /\ 
ph )  ->  `' ( R ^ r n )  =  ( `' R ^ r n ) )  /\  n  e.  NN0 ) ) ) )  ->  ( n  e.  NN0  ->  ( `' R ^ r ( n  +  1 ) )  =  ( ( `' R ^ r n )  o.  `' R
) ) )
7469, 73mpd 15 . . . . . . . . 9  |-  ( ( `' ( R ^
r ( n  + 
1 ) )  =  `' ( R  o.  ( R ^ r n ) )  /\  (
( n  +  1 )  e.  NN0  /\  ( ph  /\  ( ( ( n  e.  NN0  /\ 
ph )  ->  `' ( R ^ r n )  =  ( `' R ^ r n ) )  /\  n  e.  NN0 ) ) ) )  ->  ( `' R ^ r ( n  +  1 ) )  =  ( ( `' R ^ r n )  o.  `' R
) )
7568, 74eqtr4d 2473 . . . . . . . 8  |-  ( ( `' ( R ^
r ( n  + 
1 ) )  =  `' ( R  o.  ( R ^ r n ) )  /\  (
( n  +  1 )  e.  NN0  /\  ( ph  /\  ( ( ( n  e.  NN0  /\ 
ph )  ->  `' ( R ^ r n )  =  ( `' R ^ r n ) )  /\  n  e.  NN0 ) ) ) )  ->  `' ( R ^ r ( n  +  1 ) )  =  ( `' R ^ r ( n  +  1 ) ) )
7661, 75mpancom 652 . . . . . . 7  |-  ( ( ( n  +  1 )  e.  NN0  /\  ( ph  /\  ( ( ( n  e.  NN0  /\ 
ph )  ->  `' ( R ^ r n )  =  ( `' R ^ r n ) )  /\  n  e.  NN0 ) ) )  ->  `' ( R ^ r ( n  +  1 ) )  =  ( `' R ^ r ( n  +  1 ) ) )
7776anassrs 631 . . . . . 6  |-  ( ( ( ( n  + 
1 )  e.  NN0  /\ 
ph )  /\  (
( ( n  e. 
NN0  /\  ph )  ->  `' ( R ^
r n )  =  ( `' R ^
r n ) )  /\  n  e.  NN0 ) )  ->  `' ( R ^ r ( n  +  1 ) )  =  ( `' R ^ r ( n  +  1 ) ) )
7877expcom 426 . . . . 5  |-  ( ( ( ( n  e. 
NN0  /\  ph )  ->  `' ( R ^
r n )  =  ( `' R ^
r n ) )  /\  n  e.  NN0 )  ->  ( ( ( n  +  1 )  e.  NN0  /\  ph )  ->  `' ( R ^
r ( n  + 
1 ) )  =  ( `' R ^
r ( n  + 
1 ) ) ) )
7978expcom 426 . . . 4  |-  ( n  e.  NN0  ->  ( ( ( n  e.  NN0  /\ 
ph )  ->  `' ( R ^ r n )  =  ( `' R ^ r n ) )  ->  (
( ( n  + 
1 )  e.  NN0  /\ 
ph )  ->  `' ( R ^ r ( n  +  1 ) )  =  ( `' R ^ r ( n  +  1 ) ) ) ) )
807, 14, 21, 28, 53, 79nn0ind 10371 . . 3  |-  ( N  e.  NN0  ->  ( ( N  e.  NN0  /\  ph )  ->  `' ( R ^ r N )  =  ( `' R ^ r N ) ) )
8180anabsi5 792 . 2  |-  ( ( N  e.  NN0  /\  ph )  ->  `' ( R ^ r N )  =  ( `' R ^ r N ) )
8281expcom 426 1  |-  ( ph  ->  ( N  e.  NN0  ->  `' ( R ^
r N )  =  ( `' R ^
r N ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    = wceq 1653    e. wcel 1726   _Vcvv 2958   U.cuni 4017    _I cid 4496   `'ccnv 4880    |` cres 4883    o. ccom 4885   Rel wrel 4886  (class class class)co 6084   0cc0 8995   1c1 8996    + caddc 8998   NN0cn0 10226   ^ rcrelexp 25132
This theorem is referenced by:  relexprel  25139
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 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-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704  ax-cnex 9051  ax-resscn 9052  ax-1cn 9053  ax-icn 9054  ax-addcl 9055  ax-addrcl 9056  ax-mulcl 9057  ax-mulrcl 9058  ax-mulcom 9059  ax-addass 9060  ax-mulass 9061  ax-distr 9062  ax-i2m1 9063  ax-1ne0 9064  ax-1rid 9065  ax-rnegex 9066  ax-rrecex 9067  ax-cnre 9068  ax-pre-lttri 9069  ax-pre-lttrn 9070  ax-pre-ltadd 9071  ax-pre-mulgt0 9072
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-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-uni 4018  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-tr 4306  df-eprel 4497  df-id 4501  df-po 4506  df-so 4507  df-fr 4544  df-we 4546  df-ord 4587  df-on 4588  df-lim 4589  df-suc 4590  df-om 4849  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-2nd 6353  df-riota 6552  df-recs 6636  df-rdg 6671  df-er 6908  df-en 7113  df-dom 7114  df-sdom 7115  df-pnf 9127  df-mnf 9128  df-xr 9129  df-ltxr 9130  df-le 9131  df-sub 9298  df-neg 9299  df-nn 10006  df-n0 10227  df-z 10288  df-uz 10494  df-seq 11329  df-relexp 25133
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