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Theorem ntrivcvgmul 13936
Description: The product of two non-trivially converging products converges non-trivially. (Contributed by Scott Fenton, 18-Dec-2017.)
Hypotheses
Ref Expression
ntrivcvgmul.1  |-  Z  =  ( ZZ>= `  M )
ntrivcvgmul.3  |-  ( ph  ->  E. n  e.  Z  E. y ( y  =/=  0  /\  seq n
(  x.  ,  F
)  ~~>  y ) )
ntrivcvgmul.4  |-  ( (
ph  /\  k  e.  Z )  ->  ( F `  k )  e.  CC )
ntrivcvgmul.5  |-  ( ph  ->  E. m  e.  Z  E. z ( z  =/=  0  /\  seq m
(  x.  ,  G
)  ~~>  z ) )
ntrivcvgmul.6  |-  ( (
ph  /\  k  e.  Z )  ->  ( G `  k )  e.  CC )
ntrivcvgmul.7  |-  ( (
ph  /\  k  e.  Z )  ->  ( H `  k )  =  ( ( F `
 k )  x.  ( G `  k
) ) )
Assertion
Ref Expression
ntrivcvgmul  |-  ( ph  ->  E. p  e.  Z  E. w ( w  =/=  0  /\  seq p
(  x.  ,  H
)  ~~>  w ) )
Distinct variable groups:    m, F, z    n, G, y    m, H, n, y, z, p    ph, m    w, m, y, z    n, p    ph, n    w, n, y, z, p    ph, y, z    y, w, z    m, Z, n, y, z    w, F   
w, G    H, p, w    Z, p    k, F   
k, G    k, H, m, n    ph, k, y, z    k, Z
Allowed substitution hints:    ph( w, p)    F( y, n, p)    G( z, m, p)    M( y,
z, w, k, m, n, p)    Z( w)

Proof of Theorem ntrivcvgmul
StepHypRef Expression
1 ntrivcvgmul.3 . . 3  |-  ( ph  ->  E. n  e.  Z  E. y ( y  =/=  0  /\  seq n
(  x.  ,  F
)  ~~>  y ) )
2 ntrivcvgmul.5 . . 3  |-  ( ph  ->  E. m  e.  Z  E. z ( z  =/=  0  /\  seq m
(  x.  ,  G
)  ~~>  z ) )
3 eeanv 2045 . . . . 5  |-  ( E. y E. z ( ( y  =/=  0  /\  seq n (  x.  ,  F )  ~~>  y )  /\  ( z  =/=  0  /\  seq m
(  x.  ,  G
)  ~~>  z ) )  <-> 
( E. y ( y  =/=  0  /\ 
seq n (  x.  ,  F )  ~~>  y )  /\  E. z ( z  =/=  0  /\ 
seq m (  x.  ,  G )  ~~>  z ) ) )
432rexbii 2935 . . . 4  |-  ( E. n  e.  Z  E. m  e.  Z  E. y E. z ( ( y  =/=  0  /\ 
seq n (  x.  ,  F )  ~~>  y )  /\  ( z  =/=  0  /\  seq m
(  x.  ,  G
)  ~~>  z ) )  <->  E. n  e.  Z  E. m  e.  Z  ( E. y ( y  =/=  0  /\  seq n (  x.  ,  F )  ~~>  y )  /\  E. z ( z  =/=  0  /\ 
seq m (  x.  ,  G )  ~~>  z ) ) )
5 reeanv 3003 . . . 4  |-  ( E. n  e.  Z  E. m  e.  Z  ( E. y ( y  =/=  0  /\  seq n
(  x.  ,  F
)  ~~>  y )  /\  E. z ( z  =/=  0  /\  seq m
(  x.  ,  G
)  ~~>  z ) )  <-> 
( E. n  e.  Z  E. y ( y  =/=  0  /\ 
seq n (  x.  ,  F )  ~~>  y )  /\  E. m  e.  Z  E. z ( z  =/=  0  /\ 
seq m (  x.  ,  G )  ~~>  z ) ) )
64, 5bitri 252 . . 3  |-  ( E. n  e.  Z  E. m  e.  Z  E. y E. z ( ( y  =/=  0  /\ 
seq n (  x.  ,  F )  ~~>  y )  /\  ( z  =/=  0  /\  seq m
(  x.  ,  G
)  ~~>  z ) )  <-> 
( E. n  e.  Z  E. y ( y  =/=  0  /\ 
seq n (  x.  ,  F )  ~~>  y )  /\  E. m  e.  Z  E. z ( z  =/=  0  /\ 
seq m (  x.  ,  G )  ~~>  z ) ) )
71, 2, 6sylanbrc 668 . 2  |-  ( ph  ->  E. n  e.  Z  E. m  e.  Z  E. y E. z ( ( y  =/=  0  /\  seq n (  x.  ,  F )  ~~>  y )  /\  ( z  =/=  0  /\  seq m
(  x.  ,  G
)  ~~>  z ) ) )
8 ntrivcvgmul.1 . . . . . . . . 9  |-  Z  =  ( ZZ>= `  M )
9 uzssz 11178 . . . . . . . . 9  |-  ( ZZ>= `  M )  C_  ZZ
108, 9eqsstri 3500 . . . . . . . 8  |-  Z  C_  ZZ
11 simp2l 1031 . . . . . . . 8  |-  ( (
ph  /\  ( n  e.  Z  /\  m  e.  Z )  /\  (
( y  =/=  0  /\  seq n (  x.  ,  F )  ~~>  y )  /\  ( z  =/=  0  /\  seq m
(  x.  ,  G
)  ~~>  z ) ) )  ->  n  e.  Z )
1210, 11sseldi 3468 . . . . . . 7  |-  ( (
ph  /\  ( n  e.  Z  /\  m  e.  Z )  /\  (
( y  =/=  0  /\  seq n (  x.  ,  F )  ~~>  y )  /\  ( z  =/=  0  /\  seq m
(  x.  ,  G
)  ~~>  z ) ) )  ->  n  e.  ZZ )
1312zred 11040 . . . . . 6  |-  ( (
ph  /\  ( n  e.  Z  /\  m  e.  Z )  /\  (
( y  =/=  0  /\  seq n (  x.  ,  F )  ~~>  y )  /\  ( z  =/=  0  /\  seq m
(  x.  ,  G
)  ~~>  z ) ) )  ->  n  e.  RR )
14 simp2r 1032 . . . . . . . 8  |-  ( (
ph  /\  ( n  e.  Z  /\  m  e.  Z )  /\  (
( y  =/=  0  /\  seq n (  x.  ,  F )  ~~>  y )  /\  ( z  =/=  0  /\  seq m
(  x.  ,  G
)  ~~>  z ) ) )  ->  m  e.  Z )
1510, 14sseldi 3468 . . . . . . 7  |-  ( (
ph  /\  ( n  e.  Z  /\  m  e.  Z )  /\  (
( y  =/=  0  /\  seq n (  x.  ,  F )  ~~>  y )  /\  ( z  =/=  0  /\  seq m
(  x.  ,  G
)  ~~>  z ) ) )  ->  m  e.  ZZ )
1615zred 11040 . . . . . 6  |-  ( (
ph  /\  ( n  e.  Z  /\  m  e.  Z )  /\  (
( y  =/=  0  /\  seq n (  x.  ,  F )  ~~>  y )  /\  ( z  =/=  0  /\  seq m
(  x.  ,  G
)  ~~>  z ) ) )  ->  m  e.  RR )
17 simpl2l 1058 . . . . . . 7  |-  ( ( ( ph  /\  (
n  e.  Z  /\  m  e.  Z )  /\  ( ( y  =/=  0  /\  seq n
(  x.  ,  F
)  ~~>  y )  /\  ( z  =/=  0  /\  seq m (  x.  ,  G )  ~~>  z ) ) )  /\  n  <_  m )  ->  n  e.  Z )
18 simpl2r 1059 . . . . . . 7  |-  ( ( ( ph  /\  (
n  e.  Z  /\  m  e.  Z )  /\  ( ( y  =/=  0  /\  seq n
(  x.  ,  F
)  ~~>  y )  /\  ( z  =/=  0  /\  seq m (  x.  ,  G )  ~~>  z ) ) )  /\  n  <_  m )  ->  m  e.  Z )
19 simp3ll 1076 . . . . . . . 8  |-  ( (
ph  /\  ( n  e.  Z  /\  m  e.  Z )  /\  (
( y  =/=  0  /\  seq n (  x.  ,  F )  ~~>  y )  /\  ( z  =/=  0  /\  seq m
(  x.  ,  G
)  ~~>  z ) ) )  ->  y  =/=  0 )
2019adantr 466 . . . . . . 7  |-  ( ( ( ph  /\  (
n  e.  Z  /\  m  e.  Z )  /\  ( ( y  =/=  0  /\  seq n
(  x.  ,  F
)  ~~>  y )  /\  ( z  =/=  0  /\  seq m (  x.  ,  G )  ~~>  z ) ) )  /\  n  <_  m )  ->  y  =/=  0 )
21 simp3rl 1078 . . . . . . . 8  |-  ( (
ph  /\  ( n  e.  Z  /\  m  e.  Z )  /\  (
( y  =/=  0  /\  seq n (  x.  ,  F )  ~~>  y )  /\  ( z  =/=  0  /\  seq m
(  x.  ,  G
)  ~~>  z ) ) )  ->  z  =/=  0 )
2221adantr 466 . . . . . . 7  |-  ( ( ( ph  /\  (
n  e.  Z  /\  m  e.  Z )  /\  ( ( y  =/=  0  /\  seq n
(  x.  ,  F
)  ~~>  y )  /\  ( z  =/=  0  /\  seq m (  x.  ,  G )  ~~>  z ) ) )  /\  n  <_  m )  ->  z  =/=  0 )
23 simp3lr 1077 . . . . . . . 8  |-  ( (
ph  /\  ( n  e.  Z  /\  m  e.  Z )  /\  (
( y  =/=  0  /\  seq n (  x.  ,  F )  ~~>  y )  /\  ( z  =/=  0  /\  seq m
(  x.  ,  G
)  ~~>  z ) ) )  ->  seq n
(  x.  ,  F
)  ~~>  y )
2423adantr 466 . . . . . . 7  |-  ( ( ( ph  /\  (
n  e.  Z  /\  m  e.  Z )  /\  ( ( y  =/=  0  /\  seq n
(  x.  ,  F
)  ~~>  y )  /\  ( z  =/=  0  /\  seq m (  x.  ,  G )  ~~>  z ) ) )  /\  n  <_  m )  ->  seq n (  x.  ,  F )  ~~>  y )
25 simp3rr 1079 . . . . . . . 8  |-  ( (
ph  /\  ( n  e.  Z  /\  m  e.  Z )  /\  (
( y  =/=  0  /\  seq n (  x.  ,  F )  ~~>  y )  /\  ( z  =/=  0  /\  seq m
(  x.  ,  G
)  ~~>  z ) ) )  ->  seq m
(  x.  ,  G
)  ~~>  z )
2625adantr 466 . . . . . . 7  |-  ( ( ( ph  /\  (
n  e.  Z  /\  m  e.  Z )  /\  ( ( y  =/=  0  /\  seq n
(  x.  ,  F
)  ~~>  y )  /\  ( z  =/=  0  /\  seq m (  x.  ,  G )  ~~>  z ) ) )  /\  n  <_  m )  ->  seq m (  x.  ,  G )  ~~>  z )
27 simpl1 1008 . . . . . . . 8  |-  ( ( ( ph  /\  (
n  e.  Z  /\  m  e.  Z )  /\  ( ( y  =/=  0  /\  seq n
(  x.  ,  F
)  ~~>  y )  /\  ( z  =/=  0  /\  seq m (  x.  ,  G )  ~~>  z ) ) )  /\  n  <_  m )  ->  ph )
28 ntrivcvgmul.4 . . . . . . . 8  |-  ( (
ph  /\  k  e.  Z )  ->  ( F `  k )  e.  CC )
2927, 28sylan 473 . . . . . . 7  |-  ( ( ( ( ph  /\  ( n  e.  Z  /\  m  e.  Z
)  /\  ( (
y  =/=  0  /\ 
seq n (  x.  ,  F )  ~~>  y )  /\  ( z  =/=  0  /\  seq m
(  x.  ,  G
)  ~~>  z ) ) )  /\  n  <_  m )  /\  k  e.  Z )  ->  ( F `  k )  e.  CC )
30 ntrivcvgmul.6 . . . . . . . 8  |-  ( (
ph  /\  k  e.  Z )  ->  ( G `  k )  e.  CC )
3127, 30sylan 473 . . . . . . 7  |-  ( ( ( ( ph  /\  ( n  e.  Z  /\  m  e.  Z
)  /\  ( (
y  =/=  0  /\ 
seq n (  x.  ,  F )  ~~>  y )  /\  ( z  =/=  0  /\  seq m
(  x.  ,  G
)  ~~>  z ) ) )  /\  n  <_  m )  /\  k  e.  Z )  ->  ( G `  k )  e.  CC )
32 simpr 462 . . . . . . 7  |-  ( ( ( ph  /\  (
n  e.  Z  /\  m  e.  Z )  /\  ( ( y  =/=  0  /\  seq n
(  x.  ,  F
)  ~~>  y )  /\  ( z  =/=  0  /\  seq m (  x.  ,  G )  ~~>  z ) ) )  /\  n  <_  m )  ->  n  <_  m )
33 ntrivcvgmul.7 . . . . . . . 8  |-  ( (
ph  /\  k  e.  Z )  ->  ( H `  k )  =  ( ( F `
 k )  x.  ( G `  k
) ) )
3427, 33sylan 473 . . . . . . 7  |-  ( ( ( ( ph  /\  ( n  e.  Z  /\  m  e.  Z
)  /\  ( (
y  =/=  0  /\ 
seq n (  x.  ,  F )  ~~>  y )  /\  ( z  =/=  0  /\  seq m
(  x.  ,  G
)  ~~>  z ) ) )  /\  n  <_  m )  /\  k  e.  Z )  ->  ( H `  k )  =  ( ( F `
 k )  x.  ( G `  k
) ) )
358, 17, 18, 20, 22, 24, 26, 29, 31, 32, 34ntrivcvgmullem 13935 . . . . . 6  |-  ( ( ( ph  /\  (
n  e.  Z  /\  m  e.  Z )  /\  ( ( y  =/=  0  /\  seq n
(  x.  ,  F
)  ~~>  y )  /\  ( z  =/=  0  /\  seq m (  x.  ,  G )  ~~>  z ) ) )  /\  n  <_  m )  ->  E. p  e.  Z  E. w
( w  =/=  0  /\  seq p (  x.  ,  H )  ~~>  w ) )
36 simpl2r 1059 . . . . . . 7  |-  ( ( ( ph  /\  (
n  e.  Z  /\  m  e.  Z )  /\  ( ( y  =/=  0  /\  seq n
(  x.  ,  F
)  ~~>  y )  /\  ( z  =/=  0  /\  seq m (  x.  ,  G )  ~~>  z ) ) )  /\  m  <_  n )  ->  m  e.  Z )
37 simpl2l 1058 . . . . . . 7  |-  ( ( ( ph  /\  (
n  e.  Z  /\  m  e.  Z )  /\  ( ( y  =/=  0  /\  seq n
(  x.  ,  F
)  ~~>  y )  /\  ( z  =/=  0  /\  seq m (  x.  ,  G )  ~~>  z ) ) )  /\  m  <_  n )  ->  n  e.  Z )
3821adantr 466 . . . . . . 7  |-  ( ( ( ph  /\  (
n  e.  Z  /\  m  e.  Z )  /\  ( ( y  =/=  0  /\  seq n
(  x.  ,  F
)  ~~>  y )  /\  ( z  =/=  0  /\  seq m (  x.  ,  G )  ~~>  z ) ) )  /\  m  <_  n )  ->  z  =/=  0 )
3919adantr 466 . . . . . . 7  |-  ( ( ( ph  /\  (
n  e.  Z  /\  m  e.  Z )  /\  ( ( y  =/=  0  /\  seq n
(  x.  ,  F
)  ~~>  y )  /\  ( z  =/=  0  /\  seq m (  x.  ,  G )  ~~>  z ) ) )  /\  m  <_  n )  ->  y  =/=  0 )
4025adantr 466 . . . . . . 7  |-  ( ( ( ph  /\  (
n  e.  Z  /\  m  e.  Z )  /\  ( ( y  =/=  0  /\  seq n
(  x.  ,  F
)  ~~>  y )  /\  ( z  =/=  0  /\  seq m (  x.  ,  G )  ~~>  z ) ) )  /\  m  <_  n )  ->  seq m (  x.  ,  G )  ~~>  z )
4123adantr 466 . . . . . . 7  |-  ( ( ( ph  /\  (
n  e.  Z  /\  m  e.  Z )  /\  ( ( y  =/=  0  /\  seq n
(  x.  ,  F
)  ~~>  y )  /\  ( z  =/=  0  /\  seq m (  x.  ,  G )  ~~>  z ) ) )  /\  m  <_  n )  ->  seq n (  x.  ,  F )  ~~>  y )
42 simpl1 1008 . . . . . . . 8  |-  ( ( ( ph  /\  (
n  e.  Z  /\  m  e.  Z )  /\  ( ( y  =/=  0  /\  seq n
(  x.  ,  F
)  ~~>  y )  /\  ( z  =/=  0  /\  seq m (  x.  ,  G )  ~~>  z ) ) )  /\  m  <_  n )  ->  ph )
4342, 30sylan 473 . . . . . . 7  |-  ( ( ( ( ph  /\  ( n  e.  Z  /\  m  e.  Z
)  /\  ( (
y  =/=  0  /\ 
seq n (  x.  ,  F )  ~~>  y )  /\  ( z  =/=  0  /\  seq m
(  x.  ,  G
)  ~~>  z ) ) )  /\  m  <_  n )  /\  k  e.  Z )  ->  ( G `  k )  e.  CC )
4442, 28sylan 473 . . . . . . 7  |-  ( ( ( ( ph  /\  ( n  e.  Z  /\  m  e.  Z
)  /\  ( (
y  =/=  0  /\ 
seq n (  x.  ,  F )  ~~>  y )  /\  ( z  =/=  0  /\  seq m
(  x.  ,  G
)  ~~>  z ) ) )  /\  m  <_  n )  /\  k  e.  Z )  ->  ( F `  k )  e.  CC )
45 simpr 462 . . . . . . 7  |-  ( ( ( ph  /\  (
n  e.  Z  /\  m  e.  Z )  /\  ( ( y  =/=  0  /\  seq n
(  x.  ,  F
)  ~~>  y )  /\  ( z  =/=  0  /\  seq m (  x.  ,  G )  ~~>  z ) ) )  /\  m  <_  n )  ->  m  <_  n )
4628, 30mulcomd 9663 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  Z )  ->  (
( F `  k
)  x.  ( G `
 k ) )  =  ( ( G `
 k )  x.  ( F `  k
) ) )
4733, 46eqtrd 2470 . . . . . . . 8  |-  ( (
ph  /\  k  e.  Z )  ->  ( H `  k )  =  ( ( G `
 k )  x.  ( F `  k
) ) )
4842, 47sylan 473 . . . . . . 7  |-  ( ( ( ( ph  /\  ( n  e.  Z  /\  m  e.  Z
)  /\  ( (
y  =/=  0  /\ 
seq n (  x.  ,  F )  ~~>  y )  /\  ( z  =/=  0  /\  seq m
(  x.  ,  G
)  ~~>  z ) ) )  /\  m  <_  n )  /\  k  e.  Z )  ->  ( H `  k )  =  ( ( G `
 k )  x.  ( F `  k
) ) )
498, 36, 37, 38, 39, 40, 41, 43, 44, 45, 48ntrivcvgmullem 13935 . . . . . 6  |-  ( ( ( ph  /\  (
n  e.  Z  /\  m  e.  Z )  /\  ( ( y  =/=  0  /\  seq n
(  x.  ,  F
)  ~~>  y )  /\  ( z  =/=  0  /\  seq m (  x.  ,  G )  ~~>  z ) ) )  /\  m  <_  n )  ->  E. p  e.  Z  E. w
( w  =/=  0  /\  seq p (  x.  ,  H )  ~~>  w ) )
5013, 16, 35, 49lecasei 9739 . . . . 5  |-  ( (
ph  /\  ( n  e.  Z  /\  m  e.  Z )  /\  (
( y  =/=  0  /\  seq n (  x.  ,  F )  ~~>  y )  /\  ( z  =/=  0  /\  seq m
(  x.  ,  G
)  ~~>  z ) ) )  ->  E. p  e.  Z  E. w
( w  =/=  0  /\  seq p (  x.  ,  H )  ~~>  w ) )
51503expia 1207 . . . 4  |-  ( (
ph  /\  ( n  e.  Z  /\  m  e.  Z ) )  -> 
( ( ( y  =/=  0  /\  seq n (  x.  ,  F )  ~~>  y )  /\  ( z  =/=  0  /\  seq m
(  x.  ,  G
)  ~~>  z ) )  ->  E. p  e.  Z  E. w ( w  =/=  0  /\  seq p
(  x.  ,  H
)  ~~>  w ) ) )
5251exlimdvv 1772 . . 3  |-  ( (
ph  /\  ( n  e.  Z  /\  m  e.  Z ) )  -> 
( E. y E. z ( ( y  =/=  0  /\  seq n (  x.  ,  F )  ~~>  y )  /\  ( z  =/=  0  /\  seq m
(  x.  ,  G
)  ~~>  z ) )  ->  E. p  e.  Z  E. w ( w  =/=  0  /\  seq p
(  x.  ,  H
)  ~~>  w ) ) )
5352rexlimdvva 2931 . 2  |-  ( ph  ->  ( E. n  e.  Z  E. m  e.  Z  E. y E. z ( ( y  =/=  0  /\  seq n (  x.  ,  F )  ~~>  y )  /\  ( z  =/=  0  /\  seq m
(  x.  ,  G
)  ~~>  z ) )  ->  E. p  e.  Z  E. w ( w  =/=  0  /\  seq p
(  x.  ,  H
)  ~~>  w ) ) )
547, 53mpd 15 1  |-  ( ph  ->  E. p  e.  Z  E. w ( w  =/=  0  /\  seq p
(  x.  ,  H
)  ~~>  w ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370    /\ w3a 982    = wceq 1437   E.wex 1659    e. wcel 1870    =/= wne 2625   E.wrex 2783   class class class wbr 4426   ` cfv 5601  (class class class)co 6305   CCcc 9536   0cc0 9538    x. cmul 9543    <_ cle 9675   ZZcz 10937   ZZ>=cuz 11159    seqcseq 12210    ~~> cli 13526
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597  ax-inf2 8146  ax-cnex 9594  ax-resscn 9595  ax-1cn 9596  ax-icn 9597  ax-addcl 9598  ax-addrcl 9599  ax-mulcl 9600  ax-mulrcl 9601  ax-mulcom 9602  ax-addass 9603  ax-mulass 9604  ax-distr 9605  ax-i2m1 9606  ax-1ne0 9607  ax-1rid 9608  ax-rnegex 9609  ax-rrecex 9610  ax-cnre 9611  ax-pre-lttri 9612  ax-pre-lttrn 9613  ax-pre-ltadd 9614  ax-pre-mulgt0 9615  ax-pre-sup 9616
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-nel 2628  df-ral 2787  df-rex 2788  df-reu 2789  df-rmo 2790  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-pss 3458  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-tp 4007  df-op 4009  df-uni 4223  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-tr 4521  df-eprel 4765  df-id 4769  df-po 4775  df-so 4776  df-fr 4813  df-we 4815  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-om 6707  df-1st 6807  df-2nd 6808  df-wrecs 7036  df-recs 7098  df-rdg 7136  df-er 7371  df-en 7578  df-dom 7579  df-sdom 7580  df-sup 7962  df-pnf 9676  df-mnf 9677  df-xr 9678  df-ltxr 9679  df-le 9680  df-sub 9861  df-neg 9862  df-div 10269  df-nn 10610  df-2 10668  df-3 10669  df-n0 10870  df-z 10938  df-uz 11160  df-rp 11303  df-fz 11783  df-fzo 11914  df-seq 12211  df-exp 12270  df-cj 13141  df-re 13142  df-im 13143  df-sqrt 13277  df-abs 13278  df-clim 13530
This theorem is referenced by:  iprodmul  14034
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