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Theorem prodmolem2 25266
Description: Lemma for prodmo 25267. (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 956 . . 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 2815 . 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 5731 . . . . . 6  |-  ( m  =  w  ->  ( ZZ>=
`  m )  =  ( ZZ>= `  w )
)
43sseq2d 3378 . . . . 5  |-  ( m  =  w  ->  ( A  C_  ( ZZ>= `  m
)  <->  A  C_  ( ZZ>= `  w ) ) )
5 seqeq1 11331 . . . . . 6  |-  ( m  =  w  ->  seq  m (  x.  ,  F )  =  seq  w (  x.  ,  F ) )
65breq1d 4225 . . . . 5  |-  ( m  =  w  ->  (  seq  m (  x.  ,  F )  ~~>  x  <->  seq  w (  x.  ,  F )  ~~>  x ) )
74, 6anbi12d 693 . . . 4  |-  ( m  =  w  ->  (
( A  C_  ( ZZ>=
`  m )  /\  seq  m (  x.  ,  F )  ~~>  x )  <-> 
( A  C_  ( ZZ>=
`  w )  /\  seq  w (  x.  ,  F )  ~~>  x ) ) )
87cbvrexv 2935 . . 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 2877 . . . . 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 741 . . . . . . . . . . . 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 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 ) )  ->  A  C_  ( ZZ>= `  w
) )
12 uzssz 10510 . . . . . . . . . . . . . . . . 17  |-  ( ZZ>= `  w )  C_  ZZ
13 zssre 10294 . . . . . . . . . . . . . . . . 17  |-  ZZ  C_  RR
1412, 13sstri 3359 . . . . . . . . . . . . . . . 16  |-  ( ZZ>= `  w )  C_  RR
1511, 14syl6ss 3362 . . . . . . . . . . . . . . 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 9161 . . . . . . . . . . . . . . 15  |-  <  Or  RR
17 soss 4524 . . . . . . . . . . . . . . 15  |-  ( A 
C_  RR  ->  (  < 
Or  RR  ->  <  Or  A ) )
1815, 16, 17ee10 1386 . . . . . . . . . . . . . 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 11316 . . . . . . . . . . . . . . 15  |-  ( 1 ... m )  e. 
Fin
20 ovex 6109 . . . . . . . . . . . . . . . . . 18  |-  ( 1 ... m )  e. 
_V
2120f1oen 7131 . . . . . . . . . . . . . . . . 17  |-  ( f : ( 1 ... m ) -1-1-onto-> A  ->  ( 1 ... m )  ~~  A )
2221ad2antll 711 . . . . . . . . . . . . . . . 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 7161 . . . . . . . . . . . . . . 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 7329 . . . . . . . . . . . . . . 15  |-  ( ( ( 1 ... m
)  e.  Fin  /\  A  ~~  ( 1 ... m ) )  ->  A  e.  Fin )
2519, 23, 24sylancr 646 . . . . . . . . . . . . . 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 11716 . . . . . . . . . . . . . 14  |-  ( (  <  Or  A  /\  A  e.  Fin )  ->  E. g  g  Isom  <  ,  <  ( ( 1 ... ( # `  A
) ) ,  A
) )
2718, 25, 26syl2anc 644 . . . . . . . . . . . . 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 732 . . . . . . . . . . . . . . . . 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 459 . . . . . . . . . . . . . . . 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 2438 . . . . . . . . . . . . . . . 16  |-  ( j  e.  NN  |->  [_ (
g `  j )  /  k ]_ B
)  =  ( j  e.  NN  |->  [_ (
g `  j )  /  k ]_ B
)
34 simplrr 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 )  /\  g  Isom  <  ,  <  ( ( 1 ... ( # `  A
) ) ,  A
) ) )  ->  m  e.  NN )
35 simplrl 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
) ) )  ->  w  e.  ZZ )
36 simplll 736 . . . . . . . . . . . . . . . . 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 454 . . . . . . . . . . . . . . . 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 741 . . . . . . . . . . . . . . . 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 735 . . . . . . . . . . . . . . . 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 25265 . . . . . . . . . . . . . . 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 600 . . . . . . . . . . . . . 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 1647 . . . . . . . . . . . . 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 12351 . . . . . . . . . . . 12  |-  ( (  seq  w (  x.  ,  F )  ~~>  x  /\  seq  w (  x.  ,  F )  ~~>  (  seq  1 (  x.  ,  G ) `  m
) )  ->  x  =  (  seq  1
(  x.  ,  G
) `  m )
)
4510, 43, 44syl2anc 644 . . . . . . . . . . 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 2447 . . . . . . . . . . 11  |-  ( z  =  (  seq  1
(  x.  ,  G
) `  m )  ->  ( x  =  z  <-> 
x  =  (  seq  1 (  x.  ,  G ) `  m
) ) )
4745, 46syl5ibrcom 215 . . . . . . . . . 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 600 . . . . . . . . 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 422 . . . . . . . 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 1647 . . . . . . 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 588 . . . . . 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 2839 . . . . 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 211 . . . 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 428 . . 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 463 . 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 462 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 360    /\ w3a 937   E.wex 1551    = wceq 1653    e. wcel 1726    =/= wne 2601   E.wrex 2708   [_csb 3253    C_ wss 3322   ifcif 3741   class class class wbr 4215    e. cmpt 4269    Or wor 4505   -1-1-onto->wf1o 5456   ` cfv 5457    Isom wiso 5458  (class class class)co 6084    ~~ cen 7109   Fincfn 7112   CCcc 8993   RRcr 8994   0cc0 8995   1c1 8996    x. cmul 9000    < clt 9125   NNcn 10005   ZZcz 10287   ZZ>=cuz 10493   ...cfz 11048    seq cseq 11328   #chash 11623    ~~> cli 12283
This theorem is referenced by:  prodmo  25267
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-rep 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704  ax-inf2 7599  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  ax-pre-sup 9073
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-uni 4018  df-int 4053  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-se 4545  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-isom 5466  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-1st 6352  df-2nd 6353  df-riota 6552  df-recs 6636  df-rdg 6671  df-1o 6727  df-oadd 6731  df-er 6908  df-en 7113  df-dom 7114  df-sdom 7115  df-fin 7116  df-sup 7449  df-oi 7482  df-card 7831  df-pnf 9127  df-mnf 9128  df-xr 9129  df-ltxr 9130  df-le 9131  df-sub 9298  df-neg 9299  df-div 9683  df-nn 10006  df-2 10063  df-3 10064  df-n0 10227  df-z 10288  df-uz 10494  df-rp 10618  df-fz 11049  df-fzo 11141  df-seq 11329  df-exp 11388  df-hash 11624  df-cj 11909  df-re 11910  df-im 11911  df-sqr 12045  df-abs 12046  df-clim 12287
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