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Theorem prodmolem2 25214
Description: Lemma for prodmo 25215. (Contributed by Scott Fenton, 4-Dec-2017.)
Hypotheses
Ref Expression
prodmo.1  |-  F  =  ( k  e.  ZZ  |->  if ( k  e.  A ,  B ,  1 ) )
prodmo.2  |-  ( (
ph  /\  k  e.  A )  ->  B  e.  CC )
prodmo.3  |-  G  =  ( j  e.  NN  |->  [_ ( f `  j
)  /  k ]_ B )
Assertion
Ref Expression
prodmolem2  |-  ( (
ph  /\  E. m  e.  ZZ  ( A  C_  ( ZZ>= `  m )  /\  E. n  e.  (
ZZ>= `  m ) E. y ( y  =/=  0  /\  seq  n
(  x.  ,  F
)  ~~>  y )  /\  seq  m (  x.  ,  F )  ~~>  x ) )  ->  ( E. m  e.  NN  E. f
( f : ( 1 ... m ) -1-1-onto-> A  /\  z  =  (  seq  1 (  x.  ,  G ) `  m ) )  ->  x  =  z )
)
Distinct variable groups:    A, k, n    k, F, n    ph, k, n    A, f, j, m    B, j    f, F, j, k, m    ph, f    x, f    z, f    j, G    j, k, m, ph    x, j    k, m, x    ph, m    x, m    z, m
Allowed substitution hints:    ph( x, y, z)    A( x, y, z)    B( x, y, z, f, k, m, n)    F( x, y, z)    G( x, y, z, f, k, m, n)

Proof of Theorem prodmolem2
Dummy variables  g  w are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 3simpb 955 . . 3  |-  ( ( A  C_  ( ZZ>= `  m )  /\  E. n  e.  ( ZZ>= `  m ) E. y
( y  =/=  0  /\  seq  n (  x.  ,  F )  ~~>  y )  /\  seq  m (  x.  ,  F )  ~~>  x )  ->  ( A  C_  ( ZZ>= `  m
)  /\  seq  m (  x.  ,  F )  ~~>  x ) )
21reximi 2773 . 2  |-  ( E. m  e.  ZZ  ( A  C_  ( ZZ>= `  m
)  /\  E. n  e.  ( ZZ>= `  m ) E. y ( y  =/=  0  /\  seq  n
(  x.  ,  F
)  ~~>  y )  /\  seq  m (  x.  ,  F )  ~~>  x )  ->  E. m  e.  ZZ  ( A  C_  ( ZZ>= `  m )  /\  seq  m (  x.  ,  F )  ~~>  x ) )
3 fveq2 5687 . . . . . 6  |-  ( m  =  w  ->  ( ZZ>=
`  m )  =  ( ZZ>= `  w )
)
43sseq2d 3336 . . . . 5  |-  ( m  =  w  ->  ( A  C_  ( ZZ>= `  m
)  <->  A  C_  ( ZZ>= `  w ) ) )
5 seqeq1 11281 . . . . . 6  |-  ( m  =  w  ->  seq  m (  x.  ,  F )  =  seq  w (  x.  ,  F ) )
65breq1d 4182 . . . . 5  |-  ( m  =  w  ->  (  seq  m (  x.  ,  F )  ~~>  x  <->  seq  w (  x.  ,  F )  ~~>  x ) )
74, 6anbi12d 692 . . . 4  |-  ( m  =  w  ->  (
( A  C_  ( ZZ>=
`  m )  /\  seq  m (  x.  ,  F )  ~~>  x )  <-> 
( A  C_  ( ZZ>=
`  w )  /\  seq  w (  x.  ,  F )  ~~>  x ) ) )
87cbvrexv 2893 . . 3  |-  ( E. m  e.  ZZ  ( A  C_  ( ZZ>= `  m
)  /\  seq  m (  x.  ,  F )  ~~>  x )  <->  E. w  e.  ZZ  ( A  C_  ( ZZ>= `  w )  /\  seq  w (  x.  ,  F )  ~~>  x ) )
9 reeanv 2835 . . . . 5  |-  ( E. w  e.  ZZ  E. m  e.  NN  (
( A  C_  ( ZZ>=
`  w )  /\  seq  w (  x.  ,  F )  ~~>  x )  /\  E. f ( f : ( 1 ... m ) -1-1-onto-> A  /\  z  =  (  seq  1 (  x.  ,  G ) `  m
) ) )  <->  ( E. w  e.  ZZ  ( A  C_  ( ZZ>= `  w
)  /\  seq  w (  x.  ,  F )  ~~>  x )  /\  E. m  e.  NN  E. f
( f : ( 1 ... m ) -1-1-onto-> A  /\  z  =  (  seq  1 (  x.  ,  G ) `  m ) ) ) )
10 simprlr 740 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
w  e.  ZZ  /\  m  e.  NN )
)  /\  ( ( A  C_  ( ZZ>= `  w
)  /\  seq  w (  x.  ,  F )  ~~>  x )  /\  f : ( 1 ... m ) -1-1-onto-> A ) )  ->  seq  w (  x.  ,  F )  ~~>  x )
11 simprll 739 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  (
w  e.  ZZ  /\  m  e.  NN )
)  /\  ( ( A  C_  ( ZZ>= `  w
)  /\  seq  w (  x.  ,  F )  ~~>  x )  /\  f : ( 1 ... m ) -1-1-onto-> A ) )  ->  A  C_  ( ZZ>= `  w
) )
12 uzssz 10461 . . . . . . . . . . . . . . . . 17  |-  ( ZZ>= `  w )  C_  ZZ
13 zssre 10245 . . . . . . . . . . . . . . . . 17  |-  ZZ  C_  RR
1412, 13sstri 3317 . . . . . . . . . . . . . . . 16  |-  ( ZZ>= `  w )  C_  RR
1511, 14syl6ss 3320 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  (
w  e.  ZZ  /\  m  e.  NN )
)  /\  ( ( A  C_  ( ZZ>= `  w
)  /\  seq  w (  x.  ,  F )  ~~>  x )  /\  f : ( 1 ... m ) -1-1-onto-> A ) )  ->  A  C_  RR )
16 ltso 9112 . . . . . . . . . . . . . . 15  |-  <  Or  RR
17 soss 4481 . . . . . . . . . . . . . . 15  |-  ( A 
C_  RR  ->  (  < 
Or  RR  ->  <  Or  A ) )
1815, 16, 17ee10 1382 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  (
w  e.  ZZ  /\  m  e.  NN )
)  /\  ( ( A  C_  ( ZZ>= `  w
)  /\  seq  w (  x.  ,  F )  ~~>  x )  /\  f : ( 1 ... m ) -1-1-onto-> A ) )  ->  <  Or  A )
19 fzfi 11266 . . . . . . . . . . . . . . 15  |-  ( 1 ... m )  e. 
Fin
20 ovex 6065 . . . . . . . . . . . . . . . . . 18  |-  ( 1 ... m )  e. 
_V
2120f1oen 7087 . . . . . . . . . . . . . . . . 17  |-  ( f : ( 1 ... m ) -1-1-onto-> A  ->  ( 1 ... m )  ~~  A )
2221ad2antll 710 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  (
w  e.  ZZ  /\  m  e.  NN )
)  /\  ( ( A  C_  ( ZZ>= `  w
)  /\  seq  w (  x.  ,  F )  ~~>  x )  /\  f : ( 1 ... m ) -1-1-onto-> A ) )  -> 
( 1 ... m
)  ~~  A )
2322ensymd 7117 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  (
w  e.  ZZ  /\  m  e.  NN )
)  /\  ( ( A  C_  ( ZZ>= `  w
)  /\  seq  w (  x.  ,  F )  ~~>  x )  /\  f : ( 1 ... m ) -1-1-onto-> A ) )  ->  A  ~~  ( 1 ... m ) )
24 enfii 7285 . . . . . . . . . . . . . . 15  |-  ( ( ( 1 ... m
)  e.  Fin  /\  A  ~~  ( 1 ... m ) )  ->  A  e.  Fin )
2519, 23, 24sylancr 645 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  (
w  e.  ZZ  /\  m  e.  NN )
)  /\  ( ( A  C_  ( ZZ>= `  w
)  /\  seq  w (  x.  ,  F )  ~~>  x )  /\  f : ( 1 ... m ) -1-1-onto-> A ) )  ->  A  e.  Fin )
26 fz1iso 11666 . . . . . . . . . . . . . 14  |-  ( (  <  Or  A  /\  A  e.  Fin )  ->  E. g  g  Isom  <  ,  <  ( ( 1 ... ( # `  A
) ) ,  A
) )
2718, 25, 26syl2anc 643 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  (
w  e.  ZZ  /\  m  e.  NN )
)  /\  ( ( A  C_  ( ZZ>= `  w
)  /\  seq  w (  x.  ,  F )  ~~>  x )  /\  f : ( 1 ... m ) -1-1-onto-> A ) )  ->  E. g  g  Isom  <  ,  <  ( ( 1 ... ( # `  A
) ) ,  A
) )
28 prodmo.1 . . . . . . . . . . . . . . . 16  |-  F  =  ( k  e.  ZZ  |->  if ( k  e.  A ,  B ,  1 ) )
29 simpll 731 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  (
w  e.  ZZ  /\  m  e.  NN )
)  /\  ( (
( A  C_  ( ZZ>=
`  w )  /\  seq  w (  x.  ,  F )  ~~>  x )  /\  f : ( 1 ... m ) -1-1-onto-> A )  /\  g  Isom  <  ,  <  ( ( 1 ... ( # `  A
) ) ,  A
) ) )  ->  ph )
30 prodmo.2 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  k  e.  A )  ->  B  e.  CC )
3129, 30sylan 458 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ph  /\  ( w  e.  ZZ  /\  m  e.  NN ) )  /\  ( ( ( A  C_  ( ZZ>=
`  w )  /\  seq  w (  x.  ,  F )  ~~>  x )  /\  f : ( 1 ... m ) -1-1-onto-> A )  /\  g  Isom  <  ,  <  ( ( 1 ... ( # `  A
) ) ,  A
) ) )  /\  k  e.  A )  ->  B  e.  CC )
32 prodmo.3 . . . . . . . . . . . . . . . 16  |-  G  =  ( j  e.  NN  |->  [_ ( f `  j
)  /  k ]_ B )
33 eqid 2404 . . . . . . . . . . . . . . . 16  |-  ( j  e.  NN  |->  [_ (
g `  j )  /  k ]_ B
)  =  ( j  e.  NN  |->  [_ (
g `  j )  /  k ]_ B
)
34 simplrr 738 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  (
w  e.  ZZ  /\  m  e.  NN )
)  /\  ( (
( A  C_  ( ZZ>=
`  w )  /\  seq  w (  x.  ,  F )  ~~>  x )  /\  f : ( 1 ... m ) -1-1-onto-> A )  /\  g  Isom  <  ,  <  ( ( 1 ... ( # `  A
) ) ,  A
) ) )  ->  m  e.  NN )
35 simplrl 737 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  (
w  e.  ZZ  /\  m  e.  NN )
)  /\  ( (
( A  C_  ( ZZ>=
`  w )  /\  seq  w (  x.  ,  F )  ~~>  x )  /\  f : ( 1 ... m ) -1-1-onto-> A )  /\  g  Isom  <  ,  <  ( ( 1 ... ( # `  A
) ) ,  A
) ) )  ->  w  e.  ZZ )
36 simplll 735 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( A  C_  ( ZZ>= `  w )  /\  seq  w (  x.  ,  F )  ~~>  x )  /\  f : ( 1 ... m ) -1-1-onto-> A )  /\  g  Isom  <  ,  <  ( ( 1 ... ( # `  A
) ) ,  A
) )  ->  A  C_  ( ZZ>= `  w )
)
3736adantl 453 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  (
w  e.  ZZ  /\  m  e.  NN )
)  /\  ( (
( A  C_  ( ZZ>=
`  w )  /\  seq  w (  x.  ,  F )  ~~>  x )  /\  f : ( 1 ... m ) -1-1-onto-> A )  /\  g  Isom  <  ,  <  ( ( 1 ... ( # `  A
) ) ,  A
) ) )  ->  A  C_  ( ZZ>= `  w
) )
38 simprlr 740 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  (
w  e.  ZZ  /\  m  e.  NN )
)  /\  ( (
( A  C_  ( ZZ>=
`  w )  /\  seq  w (  x.  ,  F )  ~~>  x )  /\  f : ( 1 ... m ) -1-1-onto-> A )  /\  g  Isom  <  ,  <  ( ( 1 ... ( # `  A
) ) ,  A
) ) )  -> 
f : ( 1 ... m ) -1-1-onto-> A )
39 simprr 734 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  (
w  e.  ZZ  /\  m  e.  NN )
)  /\  ( (
( A  C_  ( ZZ>=
`  w )  /\  seq  w (  x.  ,  F )  ~~>  x )  /\  f : ( 1 ... m ) -1-1-onto-> A )  /\  g  Isom  <  ,  <  ( ( 1 ... ( # `  A
) ) ,  A
) ) )  -> 
g  Isom  <  ,  <  ( ( 1 ... ( # `
 A ) ) ,  A ) )
4028, 31, 32, 33, 34, 35, 37, 38, 39prodmolem2a 25213 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  (
w  e.  ZZ  /\  m  e.  NN )
)  /\  ( (
( A  C_  ( ZZ>=
`  w )  /\  seq  w (  x.  ,  F )  ~~>  x )  /\  f : ( 1 ... m ) -1-1-onto-> A )  /\  g  Isom  <  ,  <  ( ( 1 ... ( # `  A
) ) ,  A
) ) )  ->  seq  w (  x.  ,  F )  ~~>  (  seq  1 (  x.  ,  G ) `  m
) )
4140expr 599 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  (
w  e.  ZZ  /\  m  e.  NN )
)  /\  ( ( A  C_  ( ZZ>= `  w
)  /\  seq  w (  x.  ,  F )  ~~>  x )  /\  f : ( 1 ... m ) -1-1-onto-> A ) )  -> 
( g  Isom  <  ,  <  ( ( 1 ... ( # `  A
) ) ,  A
)  ->  seq  w (  x.  ,  F )  ~~>  (  seq  1 (  x.  ,  G ) `
 m ) ) )
4241exlimdv 1643 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  (
w  e.  ZZ  /\  m  e.  NN )
)  /\  ( ( A  C_  ( ZZ>= `  w
)  /\  seq  w (  x.  ,  F )  ~~>  x )  /\  f : ( 1 ... m ) -1-1-onto-> A ) )  -> 
( E. g  g 
Isom  <  ,  <  (
( 1 ... ( # `
 A ) ) ,  A )  ->  seq  w (  x.  ,  F )  ~~>  (  seq  1 (  x.  ,  G ) `  m
) ) )
4327, 42mpd 15 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
w  e.  ZZ  /\  m  e.  NN )
)  /\  ( ( A  C_  ( ZZ>= `  w
)  /\  seq  w (  x.  ,  F )  ~~>  x )  /\  f : ( 1 ... m ) -1-1-onto-> A ) )  ->  seq  w (  x.  ,  F )  ~~>  (  seq  1 (  x.  ,  G ) `  m
) )
44 climuni 12301 . . . . . . . . . . . 12  |-  ( (  seq  w (  x.  ,  F )  ~~>  x  /\  seq  w (  x.  ,  F )  ~~>  (  seq  1 (  x.  ,  G ) `  m
) )  ->  x  =  (  seq  1
(  x.  ,  G
) `  m )
)
4510, 43, 44syl2anc 643 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
w  e.  ZZ  /\  m  e.  NN )
)  /\  ( ( A  C_  ( ZZ>= `  w
)  /\  seq  w (  x.  ,  F )  ~~>  x )  /\  f : ( 1 ... m ) -1-1-onto-> A ) )  ->  x  =  (  seq  1 (  x.  ,  G ) `  m
) )
46 eqeq2 2413 . . . . . . . . . . 11  |-  ( z  =  (  seq  1
(  x.  ,  G
) `  m )  ->  ( x  =  z  <-> 
x  =  (  seq  1 (  x.  ,  G ) `  m
) ) )
4745, 46syl5ibrcom 214 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
w  e.  ZZ  /\  m  e.  NN )
)  /\  ( ( A  C_  ( ZZ>= `  w
)  /\  seq  w (  x.  ,  F )  ~~>  x )  /\  f : ( 1 ... m ) -1-1-onto-> A ) )  -> 
( z  =  (  seq  1 (  x.  ,  G ) `  m )  ->  x  =  z ) )
4847expr 599 . . . . . . . . 9  |-  ( ( ( ph  /\  (
w  e.  ZZ  /\  m  e.  NN )
)  /\  ( A  C_  ( ZZ>= `  w )  /\  seq  w (  x.  ,  F )  ~~>  x ) )  ->  ( f : ( 1 ... m ) -1-1-onto-> A  ->  ( z  =  (  seq  1
(  x.  ,  G
) `  m )  ->  x  =  z ) ) )
4948imp3a 421 . . . . . . . 8  |-  ( ( ( ph  /\  (
w  e.  ZZ  /\  m  e.  NN )
)  /\  ( A  C_  ( ZZ>= `  w )  /\  seq  w (  x.  ,  F )  ~~>  x ) )  ->  ( (
f : ( 1 ... m ) -1-1-onto-> A  /\  z  =  (  seq  1 (  x.  ,  G ) `  m
) )  ->  x  =  z ) )
5049exlimdv 1643 . . . . . . 7  |-  ( ( ( ph  /\  (
w  e.  ZZ  /\  m  e.  NN )
)  /\  ( A  C_  ( ZZ>= `  w )  /\  seq  w (  x.  ,  F )  ~~>  x ) )  ->  ( E. f ( f : ( 1 ... m
)
-1-1-onto-> A  /\  z  =  (  seq  1 (  x.  ,  G ) `  m ) )  ->  x  =  z )
)
5150expimpd 587 . . . . . 6  |-  ( (
ph  /\  ( w  e.  ZZ  /\  m  e.  NN ) )  -> 
( ( ( A 
C_  ( ZZ>= `  w
)  /\  seq  w (  x.  ,  F )  ~~>  x )  /\  E. f ( f : ( 1 ... m
)
-1-1-onto-> A  /\  z  =  (  seq  1 (  x.  ,  G ) `  m ) ) )  ->  x  =  z ) )
5251rexlimdvva 2797 . . . . 5  |-  ( ph  ->  ( E. w  e.  ZZ  E. m  e.  NN  ( ( A 
C_  ( ZZ>= `  w
)  /\  seq  w (  x.  ,  F )  ~~>  x )  /\  E. f ( f : ( 1 ... m
)
-1-1-onto-> A  /\  z  =  (  seq  1 (  x.  ,  G ) `  m ) ) )  ->  x  =  z ) )
539, 52syl5bir 210 . . . 4  |-  ( ph  ->  ( ( E. w  e.  ZZ  ( A  C_  ( ZZ>= `  w )  /\  seq  w (  x.  ,  F )  ~~>  x )  /\  E. m  e.  NN  E. f ( f : ( 1 ... m ) -1-1-onto-> A  /\  z  =  (  seq  1 (  x.  ,  G ) `  m
) ) )  ->  x  =  z )
)
5453expdimp 427 . . 3  |-  ( (
ph  /\  E. w  e.  ZZ  ( A  C_  ( ZZ>= `  w )  /\  seq  w (  x.  ,  F )  ~~>  x ) )  ->  ( E. m  e.  NN  E. f
( f : ( 1 ... m ) -1-1-onto-> A  /\  z  =  (  seq  1 (  x.  ,  G ) `  m ) )  ->  x  =  z )
)
558, 54sylan2b 462 . 2  |-  ( (
ph  /\  E. m  e.  ZZ  ( A  C_  ( ZZ>= `  m )  /\  seq  m (  x.  ,  F )  ~~>  x ) )  ->  ( E. m  e.  NN  E. f
( f : ( 1 ... m ) -1-1-onto-> A  /\  z  =  (  seq  1 (  x.  ,  G ) `  m ) )  ->  x  =  z )
)
562, 55sylan2 461 1  |-  ( (
ph  /\  E. m  e.  ZZ  ( A  C_  ( ZZ>= `  m )  /\  E. n  e.  (
ZZ>= `  m ) E. y ( y  =/=  0  /\  seq  n
(  x.  ,  F
)  ~~>  y )  /\  seq  m (  x.  ,  F )  ~~>  x ) )  ->  ( E. m  e.  NN  E. f
( f : ( 1 ... m ) -1-1-onto-> A  /\  z  =  (  seq  1 (  x.  ,  G ) `  m ) )  ->  x  =  z )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936   E.wex 1547    = wceq 1649    e. wcel 1721    =/= wne 2567   E.wrex 2667   [_csb 3211    C_ wss 3280   ifcif 3699   class class class wbr 4172    e. cmpt 4226    Or wor 4462   -1-1-onto->wf1o 5412   ` cfv 5413    Isom wiso 5414  (class class class)co 6040    ~~ cen 7065   Fincfn 7068   CCcc 8944   RRcr 8945   0cc0 8946   1c1 8947    x. cmul 8951    < clt 9076   NNcn 9956   ZZcz 10238   ZZ>=cuz 10444   ...cfz 10999    seq cseq 11278   #chash 11573    ~~> cli 12233
This theorem is referenced by:  prodmo  25215
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-inf2 7552  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023  ax-pre-sup 9024
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-se 4502  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-oadd 6687  df-er 6864  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-sup 7404  df-oi 7435  df-card 7782  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-nn 9957  df-2 10014  df-3 10015  df-n0 10178  df-z 10239  df-uz 10445  df-rp 10569  df-fz 11000  df-fzo 11091  df-seq 11279  df-exp 11338  df-hash 11574  df-cj 11859  df-re 11860  df-im 11861  df-sqr 11995  df-abs 11996  df-clim 12237
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