Users' Mathboxes Mathbox for Scott Fenton < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  fprodabs Unicode version

Theorem fprodabs 25250
Description: The absolute value of a finite product. (Contributed by Scott Fenton, 25-Dec-2017.)
Hypotheses
Ref Expression
fprodabs.1  |-  Z  =  ( ZZ>= `  M )
fprodabs.2  |-  ( ph  ->  N  e.  Z )
fprodabs.3  |-  ( (
ph  /\  k  e.  Z )  ->  A  e.  CC )
Assertion
Ref Expression
fprodabs  |-  ( ph  ->  ( abs `  prod_ k  e.  ( M ... N ) A )  =  prod_ k  e.  ( M ... N ) ( abs `  A
) )
Distinct variable groups:    ph, k    k, M    k, N    k, Z
Allowed substitution hint:    A( k)

Proof of Theorem fprodabs
Dummy variables  a  n are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fprodabs.2 . . 3  |-  ( ph  ->  N  e.  Z )
2 fprodabs.1 . . 3  |-  Z  =  ( ZZ>= `  M )
31, 2syl6eleq 2494 . 2  |-  ( ph  ->  N  e.  ( ZZ>= `  M ) )
4 oveq2 6048 . . . . . . 7  |-  ( a  =  M  ->  ( M ... a )  =  ( M ... M
) )
54prodeq1d 25200 . . . . . 6  |-  ( a  =  M  ->  prod_ k  e.  ( M ... a ) A  = 
prod_ k  e.  ( M ... M ) A )
65fveq2d 5691 . . . . 5  |-  ( a  =  M  ->  ( abs `  prod_ k  e.  ( M ... a ) A )  =  ( abs `  prod_ k  e.  ( M ... M
) A ) )
74prodeq1d 25200 . . . . 5  |-  ( a  =  M  ->  prod_ k  e.  ( M ... a ) ( abs `  A )  =  prod_ k  e.  ( M ... M ) ( abs `  A ) )
86, 7eqeq12d 2418 . . . 4  |-  ( a  =  M  ->  (
( abs `  prod_ k  e.  ( M ... a ) A )  =  prod_ k  e.  ( M ... a ) ( abs `  A
)  <->  ( abs `  prod_ k  e.  ( M ... M ) A )  =  prod_ k  e.  ( M ... M ) ( abs `  A
) ) )
98imbi2d 308 . . 3  |-  ( a  =  M  ->  (
( ph  ->  ( abs `  prod_ k  e.  ( M ... a ) A )  =  prod_ k  e.  ( M ... a ) ( abs `  A ) )  <->  ( ph  ->  ( abs `  prod_ k  e.  ( M ... M ) A )  =  prod_ k  e.  ( M ... M ) ( abs `  A
) ) ) )
10 oveq2 6048 . . . . . . 7  |-  ( a  =  n  ->  ( M ... a )  =  ( M ... n
) )
1110prodeq1d 25200 . . . . . 6  |-  ( a  =  n  ->  prod_ k  e.  ( M ... a ) A  = 
prod_ k  e.  ( M ... n ) A )
1211fveq2d 5691 . . . . 5  |-  ( a  =  n  ->  ( abs `  prod_ k  e.  ( M ... a ) A )  =  ( abs `  prod_ k  e.  ( M ... n
) A ) )
1310prodeq1d 25200 . . . . 5  |-  ( a  =  n  ->  prod_ k  e.  ( M ... a ) ( abs `  A )  =  prod_ k  e.  ( M ... n ) ( abs `  A ) )
1412, 13eqeq12d 2418 . . . 4  |-  ( a  =  n  ->  (
( abs `  prod_ k  e.  ( M ... a ) A )  =  prod_ k  e.  ( M ... a ) ( abs `  A
)  <->  ( abs `  prod_ k  e.  ( M ... n ) A )  =  prod_ k  e.  ( M ... n ) ( abs `  A
) ) )
1514imbi2d 308 . . 3  |-  ( a  =  n  ->  (
( ph  ->  ( abs `  prod_ k  e.  ( M ... a ) A )  =  prod_ k  e.  ( M ... a ) ( abs `  A ) )  <->  ( ph  ->  ( abs `  prod_ k  e.  ( M ... n ) A )  =  prod_ k  e.  ( M ... n ) ( abs `  A
) ) ) )
16 oveq2 6048 . . . . . . 7  |-  ( a  =  ( n  + 
1 )  ->  ( M ... a )  =  ( M ... (
n  +  1 ) ) )
1716prodeq1d 25200 . . . . . 6  |-  ( a  =  ( n  + 
1 )  ->  prod_ k  e.  ( M ... a ) A  = 
prod_ k  e.  ( M ... ( n  + 
1 ) ) A )
1817fveq2d 5691 . . . . 5  |-  ( a  =  ( n  + 
1 )  ->  ( abs `  prod_ k  e.  ( M ... a ) A )  =  ( abs `  prod_ k  e.  ( M ... (
n  +  1 ) ) A ) )
1916prodeq1d 25200 . . . . 5  |-  ( a  =  ( n  + 
1 )  ->  prod_ k  e.  ( M ... a ) ( abs `  A )  =  prod_ k  e.  ( M ... ( n  +  1
) ) ( abs `  A ) )
2018, 19eqeq12d 2418 . . . 4  |-  ( a  =  ( n  + 
1 )  ->  (
( abs `  prod_ k  e.  ( M ... a ) A )  =  prod_ k  e.  ( M ... a ) ( abs `  A
)  <->  ( abs `  prod_ k  e.  ( M ... ( n  +  1
) ) A )  =  prod_ k  e.  ( M ... ( n  +  1 ) ) ( abs `  A
) ) )
2120imbi2d 308 . . 3  |-  ( a  =  ( n  + 
1 )  ->  (
( ph  ->  ( abs `  prod_ k  e.  ( M ... a ) A )  =  prod_ k  e.  ( M ... a ) ( abs `  A ) )  <->  ( ph  ->  ( abs `  prod_ k  e.  ( M ... ( n  +  1
) ) A )  =  prod_ k  e.  ( M ... ( n  +  1 ) ) ( abs `  A
) ) ) )
22 oveq2 6048 . . . . . . 7  |-  ( a  =  N  ->  ( M ... a )  =  ( M ... N
) )
2322prodeq1d 25200 . . . . . 6  |-  ( a  =  N  ->  prod_ k  e.  ( M ... a ) A  = 
prod_ k  e.  ( M ... N ) A )
2423fveq2d 5691 . . . . 5  |-  ( a  =  N  ->  ( abs `  prod_ k  e.  ( M ... a ) A )  =  ( abs `  prod_ k  e.  ( M ... N
) A ) )
2522prodeq1d 25200 . . . . 5  |-  ( a  =  N  ->  prod_ k  e.  ( M ... a ) ( abs `  A )  =  prod_ k  e.  ( M ... N ) ( abs `  A ) )
2624, 25eqeq12d 2418 . . . 4  |-  ( a  =  N  ->  (
( abs `  prod_ k  e.  ( M ... a ) A )  =  prod_ k  e.  ( M ... a ) ( abs `  A
)  <->  ( abs `  prod_ k  e.  ( M ... N ) A )  =  prod_ k  e.  ( M ... N ) ( abs `  A
) ) )
2726imbi2d 308 . . 3  |-  ( a  =  N  ->  (
( ph  ->  ( abs `  prod_ k  e.  ( M ... a ) A )  =  prod_ k  e.  ( M ... a ) ( abs `  A ) )  <->  ( ph  ->  ( abs `  prod_ k  e.  ( M ... N ) A )  =  prod_ k  e.  ( M ... N ) ( abs `  A
) ) ) )
28 csbfv2g 5699 . . . . . 6  |-  ( M  e.  ZZ  ->  [_ M  /  k ]_ ( abs `  A )  =  ( abs `  [_ M  /  k ]_ A
) )
2928adantl 453 . . . . 5  |-  ( (
ph  /\  M  e.  ZZ )  ->  [_ M  /  k ]_ ( abs `  A )  =  ( abs `  [_ M  /  k ]_ A
) )
30 fzsn 11050 . . . . . . . 8  |-  ( M  e.  ZZ  ->  ( M ... M )  =  { M } )
3130adantl 453 . . . . . . 7  |-  ( (
ph  /\  M  e.  ZZ )  ->  ( M ... M )  =  { M } )
3231prodeq1d 25200 . . . . . 6  |-  ( (
ph  /\  M  e.  ZZ )  ->  prod_ k  e.  ( M ... M
) ( abs `  A
)  =  prod_ k  e.  { M }  ( abs `  A ) )
33 simpr 448 . . . . . . 7  |-  ( (
ph  /\  M  e.  ZZ )  ->  M  e.  ZZ )
34 uzid 10456 . . . . . . . . . . . 12  |-  ( M  e.  ZZ  ->  M  e.  ( ZZ>= `  M )
)
3534, 2syl6eleqr 2495 . . . . . . . . . . 11  |-  ( M  e.  ZZ  ->  M  e.  Z )
36 fprodabs.3 . . . . . . . . . . . . 13  |-  ( (
ph  /\  k  e.  Z )  ->  A  e.  CC )
3736ralrimiva 2749 . . . . . . . . . . . 12  |-  ( ph  ->  A. k  e.  Z  A  e.  CC )
38 nfcsb1v 3243 . . . . . . . . . . . . . 14  |-  F/_ k [_ M  /  k ]_ A
3938nfel1 2550 . . . . . . . . . . . . 13  |-  F/ k
[_ M  /  k ]_ A  e.  CC
40 csbeq1a 3219 . . . . . . . . . . . . . 14  |-  ( k  =  M  ->  A  =  [_ M  /  k ]_ A )
4140eleq1d 2470 . . . . . . . . . . . . 13  |-  ( k  =  M  ->  ( A  e.  CC  <->  [_ M  / 
k ]_ A  e.  CC ) )
4239, 41rspc 3006 . . . . . . . . . . . 12  |-  ( M  e.  Z  ->  ( A. k  e.  Z  A  e.  CC  ->  [_ M  /  k ]_ A  e.  CC )
)
4337, 42mpan9 456 . . . . . . . . . . 11  |-  ( (
ph  /\  M  e.  Z )  ->  [_ M  /  k ]_ A  e.  CC )
4435, 43sylan2 461 . . . . . . . . . 10  |-  ( (
ph  /\  M  e.  ZZ )  ->  [_ M  /  k ]_ A  e.  CC )
4544abscld 12193 . . . . . . . . 9  |-  ( (
ph  /\  M  e.  ZZ )  ->  ( abs `  [_ M  /  k ]_ A )  e.  RR )
4645recnd 9070 . . . . . . . 8  |-  ( (
ph  /\  M  e.  ZZ )  ->  ( abs `  [_ M  /  k ]_ A )  e.  CC )
4729, 46eqeltrd 2478 . . . . . . 7  |-  ( (
ph  /\  M  e.  ZZ )  ->  [_ M  /  k ]_ ( abs `  A )  e.  CC )
48 prodsns 25248 . . . . . . 7  |-  ( ( M  e.  ZZ  /\  [_ M  /  k ]_ ( abs `  A )  e.  CC )  ->  prod_ k  e.  { M }  ( abs `  A
)  =  [_ M  /  k ]_ ( abs `  A ) )
4933, 47, 48syl2anc 643 . . . . . 6  |-  ( (
ph  /\  M  e.  ZZ )  ->  prod_ k  e.  { M }  ( abs `  A )  = 
[_ M  /  k ]_ ( abs `  A
) )
5032, 49eqtrd 2436 . . . . 5  |-  ( (
ph  /\  M  e.  ZZ )  ->  prod_ k  e.  ( M ... M
) ( abs `  A
)  =  [_ M  /  k ]_ ( abs `  A ) )
5130prodeq1d 25200 . . . . . . . 8  |-  ( M  e.  ZZ  ->  prod_ k  e.  ( M ... M ) A  = 
prod_ k  e.  { M } A )
5251adantl 453 . . . . . . 7  |-  ( (
ph  /\  M  e.  ZZ )  ->  prod_ k  e.  ( M ... M
) A  =  prod_ k  e.  { M } A )
53 prodsns 25248 . . . . . . . 8  |-  ( ( M  e.  ZZ  /\  [_ M  /  k ]_ A  e.  CC )  ->  prod_ k  e.  { M } A  =  [_ M  /  k ]_ A
)
5433, 44, 53syl2anc 643 . . . . . . 7  |-  ( (
ph  /\  M  e.  ZZ )  ->  prod_ k  e.  { M } A  =  [_ M  /  k ]_ A )
5552, 54eqtrd 2436 . . . . . 6  |-  ( (
ph  /\  M  e.  ZZ )  ->  prod_ k  e.  ( M ... M
) A  =  [_ M  /  k ]_ A
)
5655fveq2d 5691 . . . . 5  |-  ( (
ph  /\  M  e.  ZZ )  ->  ( abs `  prod_ k  e.  ( M ... M ) A )  =  ( abs `  [_ M  /  k ]_ A
) )
5729, 50, 563eqtr4rd 2447 . . . 4  |-  ( (
ph  /\  M  e.  ZZ )  ->  ( abs `  prod_ k  e.  ( M ... M ) A )  =  prod_ k  e.  ( M ... M ) ( abs `  A ) )
5857expcom 425 . . 3  |-  ( M  e.  ZZ  ->  ( ph  ->  ( abs `  prod_ k  e.  ( M ... M ) A )  =  prod_ k  e.  ( M ... M ) ( abs `  A
) ) )
59 simp3 959 . . . . . . . 8  |-  ( (
ph  /\  n  e.  ( ZZ>= `  M )  /\  ( abs `  prod_ k  e.  ( M ... n ) A )  =  prod_ k  e.  ( M ... n ) ( abs `  A
) )  ->  ( abs `  prod_ k  e.  ( M ... n ) A )  =  prod_ k  e.  ( M ... n ) ( abs `  A ) )
60 ovex 6065 . . . . . . . . . . 11  |-  ( n  +  1 )  e. 
_V
61 csbfv2g 5699 . . . . . . . . . . 11  |-  ( ( n  +  1 )  e.  _V  ->  [_ (
n  +  1 )  /  k ]_ ( abs `  A )  =  ( abs `  [_ (
n  +  1 )  /  k ]_ A
) )
6260, 61ax-mp 8 . . . . . . . . . 10  |-  [_ (
n  +  1 )  /  k ]_ ( abs `  A )  =  ( abs `  [_ (
n  +  1 )  /  k ]_ A
)
6362eqcomi 2408 . . . . . . . . 9  |-  ( abs `  [_ ( n  + 
1 )  /  k ]_ A )  =  [_ ( n  +  1
)  /  k ]_ ( abs `  A )
6463a1i 11 . . . . . . . 8  |-  ( (
ph  /\  n  e.  ( ZZ>= `  M )  /\  ( abs `  prod_ k  e.  ( M ... n ) A )  =  prod_ k  e.  ( M ... n ) ( abs `  A
) )  ->  ( abs `  [_ ( n  +  1 )  / 
k ]_ A )  = 
[_ ( n  + 
1 )  /  k ]_ ( abs `  A
) )
6559, 64oveq12d 6058 . . . . . . 7  |-  ( (
ph  /\  n  e.  ( ZZ>= `  M )  /\  ( abs `  prod_ k  e.  ( M ... n ) A )  =  prod_ k  e.  ( M ... n ) ( abs `  A
) )  ->  (
( abs `  prod_ k  e.  ( M ... n ) A )  x.  ( abs `  [_ (
n  +  1 )  /  k ]_ A
) )  =  (
prod_ k  e.  ( M ... n ) ( abs `  A )  x.  [_ ( n  +  1 )  / 
k ]_ ( abs `  A
) ) )
66 simpr 448 . . . . . . . . . . 11  |-  ( (
ph  /\  n  e.  ( ZZ>= `  M )
)  ->  n  e.  ( ZZ>= `  M )
)
67 elfzuz 11011 . . . . . . . . . . . . . 14  |-  ( k  e.  ( M ... ( n  +  1
) )  ->  k  e.  ( ZZ>= `  M )
)
6867, 2syl6eleqr 2495 . . . . . . . . . . . . 13  |-  ( k  e.  ( M ... ( n  +  1
) )  ->  k  e.  Z )
6968, 36sylan2 461 . . . . . . . . . . . 12  |-  ( (
ph  /\  k  e.  ( M ... ( n  +  1 ) ) )  ->  A  e.  CC )
7069adantlr 696 . . . . . . . . . . 11  |-  ( ( ( ph  /\  n  e.  ( ZZ>= `  M )
)  /\  k  e.  ( M ... ( n  +  1 ) ) )  ->  A  e.  CC )
7166, 70fprodp1s 25247 . . . . . . . . . 10  |-  ( (
ph  /\  n  e.  ( ZZ>= `  M )
)  ->  prod_ k  e.  ( M ... (
n  +  1 ) ) A  =  (
prod_ k  e.  ( M ... n ) A  x.  [_ ( n  +  1 )  / 
k ]_ A ) )
7271fveq2d 5691 . . . . . . . . 9  |-  ( (
ph  /\  n  e.  ( ZZ>= `  M )
)  ->  ( abs ` 
prod_ k  e.  ( M ... ( n  + 
1 ) ) A )  =  ( abs `  ( prod_ k  e.  ( M ... n ) A  x.  [_ (
n  +  1 )  /  k ]_ A
) ) )
73 fzfid 11267 . . . . . . . . . . 11  |-  ( (
ph  /\  n  e.  ( ZZ>= `  M )
)  ->  ( M ... n )  e.  Fin )
74 elfzuz 11011 . . . . . . . . . . . . . 14  |-  ( k  e.  ( M ... n )  ->  k  e.  ( ZZ>= `  M )
)
7574, 2syl6eleqr 2495 . . . . . . . . . . . . 13  |-  ( k  e.  ( M ... n )  ->  k  e.  Z )
7675, 36sylan2 461 . . . . . . . . . . . 12  |-  ( (
ph  /\  k  e.  ( M ... n ) )  ->  A  e.  CC )
7776adantlr 696 . . . . . . . . . . 11  |-  ( ( ( ph  /\  n  e.  ( ZZ>= `  M )
)  /\  k  e.  ( M ... n ) )  ->  A  e.  CC )
7873, 77fprodcl 25231 . . . . . . . . . 10  |-  ( (
ph  /\  n  e.  ( ZZ>= `  M )
)  ->  prod_ k  e.  ( M ... n
) A  e.  CC )
79 peano2uz 10486 . . . . . . . . . . . 12  |-  ( n  e.  ( ZZ>= `  M
)  ->  ( n  +  1 )  e.  ( ZZ>= `  M )
)
8079, 2syl6eleqr 2495 . . . . . . . . . . 11  |-  ( n  e.  ( ZZ>= `  M
)  ->  ( n  +  1 )  e.  Z )
81 nfcsb1v 3243 . . . . . . . . . . . . . 14  |-  F/_ k [_ ( n  +  1 )  /  k ]_ A
8281nfel1 2550 . . . . . . . . . . . . 13  |-  F/ k
[_ ( n  + 
1 )  /  k ]_ A  e.  CC
83 csbeq1a 3219 . . . . . . . . . . . . . 14  |-  ( k  =  ( n  + 
1 )  ->  A  =  [_ ( n  + 
1 )  /  k ]_ A )
8483eleq1d 2470 . . . . . . . . . . . . 13  |-  ( k  =  ( n  + 
1 )  ->  ( A  e.  CC  <->  [_ ( n  +  1 )  / 
k ]_ A  e.  CC ) )
8582, 84rspc 3006 . . . . . . . . . . . 12  |-  ( ( n  +  1 )  e.  Z  ->  ( A. k  e.  Z  A  e.  CC  ->  [_ ( n  +  1 )  /  k ]_ A  e.  CC )
)
8637, 85mpan9 456 . . . . . . . . . . 11  |-  ( (
ph  /\  ( n  +  1 )  e.  Z )  ->  [_ (
n  +  1 )  /  k ]_ A  e.  CC )
8780, 86sylan2 461 . . . . . . . . . 10  |-  ( (
ph  /\  n  e.  ( ZZ>= `  M )
)  ->  [_ ( n  +  1 )  / 
k ]_ A  e.  CC )
8878, 87absmuld 12211 . . . . . . . . 9  |-  ( (
ph  /\  n  e.  ( ZZ>= `  M )
)  ->  ( abs `  ( prod_ k  e.  ( M ... n ) A  x.  [_ (
n  +  1 )  /  k ]_ A
) )  =  ( ( abs `  prod_ k  e.  ( M ... n ) A )  x.  ( abs `  [_ (
n  +  1 )  /  k ]_ A
) ) )
8972, 88eqtrd 2436 . . . . . . . 8  |-  ( (
ph  /\  n  e.  ( ZZ>= `  M )
)  ->  ( abs ` 
prod_ k  e.  ( M ... ( n  + 
1 ) ) A )  =  ( ( abs `  prod_ k  e.  ( M ... n
) A )  x.  ( abs `  [_ (
n  +  1 )  /  k ]_ A
) ) )
90893adant3 977 . . . . . . 7  |-  ( (
ph  /\  n  e.  ( ZZ>= `  M )  /\  ( abs `  prod_ k  e.  ( M ... n ) A )  =  prod_ k  e.  ( M ... n ) ( abs `  A
) )  ->  ( abs `  prod_ k  e.  ( M ... ( n  +  1 ) ) A )  =  ( ( abs `  prod_ k  e.  ( M ... n ) A )  x.  ( abs `  [_ (
n  +  1 )  /  k ]_ A
) ) )
9170abscld 12193 . . . . . . . . . 10  |-  ( ( ( ph  /\  n  e.  ( ZZ>= `  M )
)  /\  k  e.  ( M ... ( n  +  1 ) ) )  ->  ( abs `  A )  e.  RR )
9291recnd 9070 . . . . . . . . 9  |-  ( ( ( ph  /\  n  e.  ( ZZ>= `  M )
)  /\  k  e.  ( M ... ( n  +  1 ) ) )  ->  ( abs `  A )  e.  CC )
9366, 92fprodp1s 25247 . . . . . . . 8  |-  ( (
ph  /\  n  e.  ( ZZ>= `  M )
)  ->  prod_ k  e.  ( M ... (
n  +  1 ) ) ( abs `  A
)  =  ( prod_
k  e.  ( M ... n ) ( abs `  A )  x.  [_ ( n  +  1 )  / 
k ]_ ( abs `  A
) ) )
94933adant3 977 . . . . . . 7  |-  ( (
ph  /\  n  e.  ( ZZ>= `  M )  /\  ( abs `  prod_ k  e.  ( M ... n ) A )  =  prod_ k  e.  ( M ... n ) ( abs `  A
) )  ->  prod_ k  e.  ( M ... ( n  +  1
) ) ( abs `  A )  =  (
prod_ k  e.  ( M ... n ) ( abs `  A )  x.  [_ ( n  +  1 )  / 
k ]_ ( abs `  A
) ) )
9565, 90, 943eqtr4d 2446 . . . . . 6  |-  ( (
ph  /\  n  e.  ( ZZ>= `  M )  /\  ( abs `  prod_ k  e.  ( M ... n ) A )  =  prod_ k  e.  ( M ... n ) ( abs `  A
) )  ->  ( abs `  prod_ k  e.  ( M ... ( n  +  1 ) ) A )  =  prod_ k  e.  ( M ... ( n  +  1
) ) ( abs `  A ) )
96953exp 1152 . . . . 5  |-  ( ph  ->  ( n  e.  (
ZZ>= `  M )  -> 
( ( abs `  prod_ k  e.  ( M ... n ) A )  =  prod_ k  e.  ( M ... n ) ( abs `  A
)  ->  ( abs ` 
prod_ k  e.  ( M ... ( n  + 
1 ) ) A )  =  prod_ k  e.  ( M ... (
n  +  1 ) ) ( abs `  A
) ) ) )
9796com12 29 . . . 4  |-  ( n  e.  ( ZZ>= `  M
)  ->  ( ph  ->  ( ( abs `  prod_ k  e.  ( M ... n ) A )  =  prod_ k  e.  ( M ... n ) ( abs `  A
)  ->  ( abs ` 
prod_ k  e.  ( M ... ( n  + 
1 ) ) A )  =  prod_ k  e.  ( M ... (
n  +  1 ) ) ( abs `  A
) ) ) )
9897a2d 24 . . 3  |-  ( n  e.  ( ZZ>= `  M
)  ->  ( ( ph  ->  ( abs `  prod_ k  e.  ( M ... n ) A )  =  prod_ k  e.  ( M ... n ) ( abs `  A
) )  ->  ( ph  ->  ( abs `  prod_ k  e.  ( M ... ( n  +  1
) ) A )  =  prod_ k  e.  ( M ... ( n  +  1 ) ) ( abs `  A
) ) ) )
999, 15, 21, 27, 58, 98uzind4 10490 . 2  |-  ( N  e.  ( ZZ>= `  M
)  ->  ( ph  ->  ( abs `  prod_ k  e.  ( M ... N ) A )  =  prod_ k  e.  ( M ... N ) ( abs `  A
) ) )
1003, 99mpcom 34 1  |-  ( ph  ->  ( abs `  prod_ k  e.  ( M ... N ) A )  =  prod_ k  e.  ( M ... N ) ( abs `  A
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721   A.wral 2666   _Vcvv 2916   [_csb 3211   {csn 3774   ` cfv 5413  (class class class)co 6040   CCcc 8944   1c1 8947    + caddc 8949    x. cmul 8951   ZZcz 10238   ZZ>=cuz 10444   ...cfz 10999   abscabs 11994   prod_cprod 25184
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  df-prod 25185
  Copyright terms: Public domain W3C validator