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Theorem ntrivcvgmul 28963
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 1957 . . . . 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 2970 . . . 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 3034 . . . 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 249 . . 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 664 . 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 11113 . . . . . . . . 9  |-  ( ZZ>= `  M )  C_  ZZ
108, 9eqsstri 3539 . . . . . . . 8  |-  Z  C_  ZZ
11 simp2l 1022 . . . . . . . 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 3507 . . . . . . 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 10978 . . . . . 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 1023 . . . . . . . 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 3507 . . . . . . 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 10978 . . . . . 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 1049 . . . . . . 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 1050 . . . . . . 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 1067 . . . . . . . 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 465 . . . . . . 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 1069 . . . . . . . 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 465 . . . . . . 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 1068 . . . . . . . 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 465 . . . . . . 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 1070 . . . . . . . 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 465 . . . . . . 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 999 . . . . . . . 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 471 . . . . . . 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 471 . . . . . . 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 461 . . . . . . 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 471 . . . . . . 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 28962 . . . . . 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 1050 . . . . . . 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 1049 . . . . . . 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 465 . . . . . . 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 465 . . . . . . 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 465 . . . . . . 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 465 . . . . . . 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 999 . . . . . . . 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 471 . . . . . . 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 471 . . . . . . 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 461 . . . . . . 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 9629 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  Z )  ->  (
( F `  k
)  x.  ( G `
 k ) )  =  ( ( G `
 k )  x.  ( F `  k
) ) )
4733, 46eqtrd 2508 . . . . . . . 8  |-  ( (
ph  /\  k  e.  Z )  ->  ( H `  k )  =  ( ( G `
 k )  x.  ( F `  k
) ) )
4842, 47sylan 471 . . . . . . 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 28962 . . . . . 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 9702 . . . . 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 1198 . . . 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 1701 . . 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 2966 . 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 369    /\ w3a 973    = wceq 1379   E.wex 1596    e. wcel 1767    =/= wne 2662   E.wrex 2818   class class class wbr 4453   ` cfv 5594  (class class class)co 6295   CCcc 9502   0cc0 9504    x. cmul 9509    <_ cle 9641   ZZcz 10876   ZZ>=cuz 11094    seqcseq 12087    ~~> cli 13287
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-inf2 8070  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581  ax-pre-sup 9582
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6696  df-1st 6795  df-2nd 6796  df-recs 7054  df-rdg 7088  df-er 7323  df-en 7529  df-dom 7530  df-sdom 7531  df-sup 7913  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-div 10219  df-nn 10549  df-2 10606  df-3 10607  df-n0 10808  df-z 10877  df-uz 11095  df-rp 11233  df-fz 11685  df-fzo 11805  df-seq 12088  df-exp 12147  df-cj 12912  df-re 12913  df-im 12914  df-sqrt 13048  df-abs 13049  df-clim 13291
This theorem is referenced by:  iprodmul  29049
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