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Theorem fprodabs 11579
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:    k, M    k, N    k, Z    ph, k
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, 2eleqtrdi 2263 . 2  |-  ( ph  ->  N  e.  ( ZZ>= `  M ) )
4 oveq2 5861 . . . . . . 7  |-  ( a  =  M  ->  ( M ... a )  =  ( M ... M
) )
54prodeq1d 11527 . . . . . 6  |-  ( a  =  M  ->  prod_ k  e.  ( M ... a ) A  = 
prod_ k  e.  ( M ... M ) A )
65fveq2d 5500 . . . . 5  |-  ( a  =  M  ->  ( abs `  prod_ k  e.  ( M ... a ) A )  =  ( abs `  prod_ k  e.  ( M ... M
) A ) )
74prodeq1d 11527 . . . . 5  |-  ( a  =  M  ->  prod_ k  e.  ( M ... a ) ( abs `  A )  =  prod_ k  e.  ( M ... M ) ( abs `  A ) )
86, 7eqeq12d 2185 . . . 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 229 . . 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 5861 . . . . . . 7  |-  ( a  =  n  ->  ( M ... a )  =  ( M ... n
) )
1110prodeq1d 11527 . . . . . 6  |-  ( a  =  n  ->  prod_ k  e.  ( M ... a ) A  = 
prod_ k  e.  ( M ... n ) A )
1211fveq2d 5500 . . . . 5  |-  ( a  =  n  ->  ( abs `  prod_ k  e.  ( M ... a ) A )  =  ( abs `  prod_ k  e.  ( M ... n
) A ) )
1310prodeq1d 11527 . . . . 5  |-  ( a  =  n  ->  prod_ k  e.  ( M ... a ) ( abs `  A )  =  prod_ k  e.  ( M ... n ) ( abs `  A ) )
1412, 13eqeq12d 2185 . . . 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 229 . . 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 5861 . . . . . . 7  |-  ( a  =  ( n  + 
1 )  ->  ( M ... a )  =  ( M ... (
n  +  1 ) ) )
1716prodeq1d 11527 . . . . . 6  |-  ( a  =  ( n  + 
1 )  ->  prod_ k  e.  ( M ... a ) A  = 
prod_ k  e.  ( M ... ( n  + 
1 ) ) A )
1817fveq2d 5500 . . . . 5  |-  ( a  =  ( n  + 
1 )  ->  ( abs `  prod_ k  e.  ( M ... a ) A )  =  ( abs `  prod_ k  e.  ( M ... (
n  +  1 ) ) A ) )
1916prodeq1d 11527 . . . . 5  |-  ( a  =  ( n  + 
1 )  ->  prod_ k  e.  ( M ... a ) ( abs `  A )  =  prod_ k  e.  ( M ... ( n  +  1
) ) ( abs `  A ) )
2018, 19eqeq12d 2185 . . . 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 229 . . 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 5861 . . . . . . 7  |-  ( a  =  N  ->  ( M ... a )  =  ( M ... N
) )
2322prodeq1d 11527 . . . . . 6  |-  ( a  =  N  ->  prod_ k  e.  ( M ... a ) A  = 
prod_ k  e.  ( M ... N ) A )
2423fveq2d 5500 . . . . 5  |-  ( a  =  N  ->  ( abs `  prod_ k  e.  ( M ... a ) A )  =  ( abs `  prod_ k  e.  ( M ... N
) A ) )
2522prodeq1d 11527 . . . . 5  |-  ( a  =  N  ->  prod_ k  e.  ( M ... a ) ( abs `  A )  =  prod_ k  e.  ( M ... N ) ( abs `  A ) )
2624, 25eqeq12d 2185 . . . 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 229 . . 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 5533 . . . . . 6  |-  ( M  e.  ZZ  ->  [_ M  /  k ]_ ( abs `  A )  =  ( abs `  [_ M  /  k ]_ A
) )
2928adantl 275 . . . . 5  |-  ( (
ph  /\  M  e.  ZZ )  ->  [_ M  /  k ]_ ( abs `  A )  =  ( abs `  [_ M  /  k ]_ A
) )
30 fzsn 10022 . . . . . . . 8  |-  ( M  e.  ZZ  ->  ( M ... M )  =  { M } )
3130adantl 275 . . . . . . 7  |-  ( (
ph  /\  M  e.  ZZ )  ->  ( M ... M )  =  { M } )
3231prodeq1d 11527 . . . . . 6  |-  ( (
ph  /\  M  e.  ZZ )  ->  prod_ k  e.  ( M ... M
) ( abs `  A
)  =  prod_ k  e.  { M }  ( abs `  A ) )
33 simpr 109 . . . . . . 7  |-  ( (
ph  /\  M  e.  ZZ )  ->  M  e.  ZZ )
34 uzid 9501 . . . . . . . . . . . 12  |-  ( M  e.  ZZ  ->  M  e.  ( ZZ>= `  M )
)
3534, 2eleqtrrdi 2264 . . . . . . . . . . 11  |-  ( M  e.  ZZ  ->  M  e.  Z )
36 fprodabs.3 . . . . . . . . . . . . 13  |-  ( (
ph  /\  k  e.  Z )  ->  A  e.  CC )
3736ralrimiva 2543 . . . . . . . . . . . 12  |-  ( ph  ->  A. k  e.  Z  A  e.  CC )
38 nfcsb1v 3082 . . . . . . . . . . . . . 14  |-  F/_ k [_ M  /  k ]_ A
3938nfel1 2323 . . . . . . . . . . . . 13  |-  F/ k
[_ M  /  k ]_ A  e.  CC
40 csbeq1a 3058 . . . . . . . . . . . . . 14  |-  ( k  =  M  ->  A  =  [_ M  /  k ]_ A )
4140eleq1d 2239 . . . . . . . . . . . . 13  |-  ( k  =  M  ->  ( A  e.  CC  <->  [_ M  / 
k ]_ A  e.  CC ) )
4239, 41rspc 2828 . . . . . . . . . . . 12  |-  ( M  e.  Z  ->  ( A. k  e.  Z  A  e.  CC  ->  [_ M  /  k ]_ A  e.  CC )
)
4337, 42mpan9 279 . . . . . . . . . . 11  |-  ( (
ph  /\  M  e.  Z )  ->  [_ M  /  k ]_ A  e.  CC )
4435, 43sylan2 284 . . . . . . . . . 10  |-  ( (
ph  /\  M  e.  ZZ )  ->  [_ M  /  k ]_ A  e.  CC )
4544abscld 11145 . . . . . . . . 9  |-  ( (
ph  /\  M  e.  ZZ )  ->  ( abs `  [_ M  /  k ]_ A )  e.  RR )
4645recnd 7948 . . . . . . . 8  |-  ( (
ph  /\  M  e.  ZZ )  ->  ( abs `  [_ M  /  k ]_ A )  e.  CC )
4729, 46eqeltrd 2247 . . . . . . 7  |-  ( (
ph  /\  M  e.  ZZ )  ->  [_ M  /  k ]_ ( abs `  A )  e.  CC )
48 prodsns 11566 . . . . . . 7  |-  ( ( M  e.  ZZ  /\  [_ M  /  k ]_ ( abs `  A )  e.  CC )  ->  prod_ k  e.  { M }  ( abs `  A
)  =  [_ M  /  k ]_ ( abs `  A ) )
4933, 47, 48syl2anc 409 . . . . . 6  |-  ( (
ph  /\  M  e.  ZZ )  ->  prod_ k  e.  { M }  ( abs `  A )  = 
[_ M  /  k ]_ ( abs `  A
) )
5032, 49eqtrd 2203 . . . . 5  |-  ( (
ph  /\  M  e.  ZZ )  ->  prod_ k  e.  ( M ... M
) ( abs `  A
)  =  [_ M  /  k ]_ ( abs `  A ) )
5130prodeq1d 11527 . . . . . . . 8  |-  ( M  e.  ZZ  ->  prod_ k  e.  ( M ... M ) A  = 
prod_ k  e.  { M } A )
5251adantl 275 . . . . . . 7  |-  ( (
ph  /\  M  e.  ZZ )  ->  prod_ k  e.  ( M ... M
) A  =  prod_ k  e.  { M } A )
53 prodsns 11566 . . . . . . . 8  |-  ( ( M  e.  ZZ  /\  [_ M  /  k ]_ A  e.  CC )  ->  prod_ k  e.  { M } A  =  [_ M  /  k ]_ A
)
5433, 44, 53syl2anc 409 . . . . . . 7  |-  ( (
ph  /\  M  e.  ZZ )  ->  prod_ k  e.  { M } A  =  [_ M  /  k ]_ A )
5552, 54eqtrd 2203 . . . . . 6  |-  ( (
ph  /\  M  e.  ZZ )  ->  prod_ k  e.  ( M ... M
) A  =  [_ M  /  k ]_ A
)
5655fveq2d 5500 . . . . 5  |-  ( (
ph  /\  M  e.  ZZ )  ->  ( abs `  prod_ k  e.  ( M ... M ) A )  =  ( abs `  [_ M  /  k ]_ A
) )
5729, 50, 563eqtr4rd 2214 . . . 4  |-  ( (
ph  /\  M  e.  ZZ )  ->  ( abs `  prod_ k  e.  ( M ... M ) A )  =  prod_ k  e.  ( M ... M ) ( abs `  A ) )
5857expcom 115 . . 3  |-  ( M  e.  ZZ  ->  ( ph  ->  ( abs `  prod_ k  e.  ( M ... M ) A )  =  prod_ k  e.  ( M ... M ) ( abs `  A
) ) )
59 simp3 994 . . . . . . . 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 peano2uz 9542 . . . . . . . . . . 11  |-  ( n  e.  ( ZZ>= `  M
)  ->  ( n  +  1 )  e.  ( ZZ>= `  M )
)
61 csbfv2g 5533 . . . . . . . . . . 11  |-  ( ( n  +  1 )  e.  ( ZZ>= `  M
)  ->  [_ ( n  +  1 )  / 
k ]_ ( abs `  A
)  =  ( abs `  [_ ( n  + 
1 )  /  k ]_ A ) )
6260, 61syl 14 . . . . . . . . . 10  |-  ( n  e.  ( ZZ>= `  M
)  ->  [_ ( n  +  1 )  / 
k ]_ ( abs `  A
)  =  ( abs `  [_ ( n  + 
1 )  /  k ]_ A ) )
6362eqcomd 2176 . . . . . . . . 9  |-  ( n  e.  ( ZZ>= `  M
)  ->  ( abs ` 
[_ ( n  + 
1 )  /  k ]_ A )  =  [_ ( n  +  1
)  /  k ]_ ( abs `  A ) )
64633ad2ant2 1014 . . . . . . . 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 5871 . . . . . . 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 109 . . . . . . . . . . 11  |-  ( (
ph  /\  n  e.  ( ZZ>= `  M )
)  ->  n  e.  ( ZZ>= `  M )
)
67 elfzuz 9977 . . . . . . . . . . . . . 14  |-  ( k  e.  ( M ... ( n  +  1
) )  ->  k  e.  ( ZZ>= `  M )
)
6867, 2eleqtrrdi 2264 . . . . . . . . . . . . 13  |-  ( k  e.  ( M ... ( n  +  1
) )  ->  k  e.  Z )
6968, 36sylan2 284 . . . . . . . . . . . 12  |-  ( (
ph  /\  k  e.  ( M ... ( n  +  1 ) ) )  ->  A  e.  CC )
7069adantlr 474 . . . . . . . . . . 11  |-  ( ( ( ph  /\  n  e.  ( ZZ>= `  M )
)  /\  k  e.  ( M ... ( n  +  1 ) ) )  ->  A  e.  CC )
7166, 70fprodp1s 11565 . . . . . . . . . 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 5500 . . . . . . . . 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 eluzel2 9492 . . . . . . . . . . . . 13  |-  ( n  e.  ( ZZ>= `  M
)  ->  M  e.  ZZ )
7473adantl 275 . . . . . . . . . . . 12  |-  ( (
ph  /\  n  e.  ( ZZ>= `  M )
)  ->  M  e.  ZZ )
75 eluzelz 9496 . . . . . . . . . . . . 13  |-  ( n  e.  ( ZZ>= `  M
)  ->  n  e.  ZZ )
7675adantl 275 . . . . . . . . . . . 12  |-  ( (
ph  /\  n  e.  ( ZZ>= `  M )
)  ->  n  e.  ZZ )
7774, 76fzfigd 10387 . . . . . . . . . . 11  |-  ( (
ph  /\  n  e.  ( ZZ>= `  M )
)  ->  ( M ... n )  e.  Fin )
78 elfzuz 9977 . . . . . . . . . . . . . 14  |-  ( k  e.  ( M ... n )  ->  k  e.  ( ZZ>= `  M )
)
7978, 2eleqtrrdi 2264 . . . . . . . . . . . . 13  |-  ( k  e.  ( M ... n )  ->  k  e.  Z )
8079, 36sylan2 284 . . . . . . . . . . . 12  |-  ( (
ph  /\  k  e.  ( M ... n ) )  ->  A  e.  CC )
8180adantlr 474 . . . . . . . . . . 11  |-  ( ( ( ph  /\  n  e.  ( ZZ>= `  M )
)  /\  k  e.  ( M ... n ) )  ->  A  e.  CC )
8277, 81fprodcl 11570 . . . . . . . . . 10  |-  ( (
ph  /\  n  e.  ( ZZ>= `  M )
)  ->  prod_ k  e.  ( M ... n
) A  e.  CC )
8360, 2eleqtrrdi 2264 . . . . . . . . . . 11  |-  ( n  e.  ( ZZ>= `  M
)  ->  ( n  +  1 )  e.  Z )
84 nfcsb1v 3082 . . . . . . . . . . . . . 14  |-  F/_ k [_ ( n  +  1 )  /  k ]_ A
8584nfel1 2323 . . . . . . . . . . . . 13  |-  F/ k
[_ ( n  + 
1 )  /  k ]_ A  e.  CC
86 csbeq1a 3058 . . . . . . . . . . . . . 14  |-  ( k  =  ( n  + 
1 )  ->  A  =  [_ ( n  + 
1 )  /  k ]_ A )
8786eleq1d 2239 . . . . . . . . . . . . 13  |-  ( k  =  ( n  + 
1 )  ->  ( A  e.  CC  <->  [_ ( n  +  1 )  / 
k ]_ A  e.  CC ) )
8885, 87rspc 2828 . . . . . . . . . . . 12  |-  ( ( n  +  1 )  e.  Z  ->  ( A. k  e.  Z  A  e.  CC  ->  [_ ( n  +  1 )  /  k ]_ A  e.  CC )
)
8937, 88mpan9 279 . . . . . . . . . . 11  |-  ( (
ph  /\  ( n  +  1 )  e.  Z )  ->  [_ (
n  +  1 )  /  k ]_ A  e.  CC )
9083, 89sylan2 284 . . . . . . . . . 10  |-  ( (
ph  /\  n  e.  ( ZZ>= `  M )
)  ->  [_ ( n  +  1 )  / 
k ]_ A  e.  CC )
9182, 90absmuld 11158 . . . . . . . . 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
) ) )
9272, 91eqtrd 2203 . . . . . . . 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
) ) )
93923adant3 1012 . . . . . . 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
) ) )
9470abscld 11145 . . . . . . . . . 10  |-  ( ( ( ph  /\  n  e.  ( ZZ>= `  M )
)  /\  k  e.  ( M ... ( n  +  1 ) ) )  ->  ( abs `  A )  e.  RR )
9594recnd 7948 . . . . . . . . 9  |-  ( ( ( ph  /\  n  e.  ( ZZ>= `  M )
)  /\  k  e.  ( M ... ( n  +  1 ) ) )  ->  ( abs `  A )  e.  CC )
9666, 95fprodp1s 11565 . . . . . . . 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
) ) )
97963adant3 1012 . . . . . . 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
) ) )
9865, 93, 973eqtr4d 2213 . . . . . 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 ) )
99983exp 1197 . . . . 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
) ) ) )
10099com12 30 . . . 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
) ) ) )
101100a2d 26 . . 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
) ) ) )
1029, 15, 21, 27, 58, 101uzind4 9547 . 2  |-  ( N  e.  ( ZZ>= `  M
)  ->  ( ph  ->  ( abs `  prod_ k  e.  ( M ... N ) A )  =  prod_ k  e.  ( M ... N ) ( abs `  A
) ) )
1033, 102mpcom 36 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 103    /\ w3a 973    = wceq 1348    e. wcel 2141   A.wral 2448   [_csb 3049   {csn 3583   ` cfv 5198  (class class class)co 5853   CCcc 7772   1c1 7775    + caddc 7777    x. cmul 7779   ZZcz 9212   ZZ>=cuz 9487   ...cfz 9965   abscabs 10961   prod_cprod 11513
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-iinf 4572  ax-cnex 7865  ax-resscn 7866  ax-1cn 7867  ax-1re 7868  ax-icn 7869  ax-addcl 7870  ax-addrcl 7871  ax-mulcl 7872  ax-mulrcl 7873  ax-addcom 7874  ax-mulcom 7875  ax-addass 7876  ax-mulass 7877  ax-distr 7878  ax-i2m1 7879  ax-0lt1 7880  ax-1rid 7881  ax-0id 7882  ax-rnegex 7883  ax-precex 7884  ax-cnre 7885  ax-pre-ltirr 7886  ax-pre-ltwlin 7887  ax-pre-lttrn 7888  ax-pre-apti 7889  ax-pre-ltadd 7890  ax-pre-mulgt0 7891  ax-pre-mulext 7892  ax-arch 7893  ax-caucvg 7894
This theorem depends on definitions:  df-bi 116  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-reu 2455  df-rmo 2456  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-if 3527  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-tr 4088  df-id 4278  df-po 4281  df-iso 4282  df-iord 4351  df-on 4353  df-ilim 4354  df-suc 4356  df-iom 4575  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-isom 5207  df-riota 5809  df-ov 5856  df-oprab 5857  df-mpo 5858  df-1st 6119  df-2nd 6120  df-recs 6284  df-irdg 6349  df-frec 6370  df-1o 6395  df-oadd 6399  df-er 6513  df-en 6719  df-dom 6720  df-fin 6721  df-pnf 7956  df-mnf 7957  df-xr 7958  df-ltxr 7959  df-le 7960  df-sub 8092  df-neg 8093  df-reap 8494  df-ap 8501  df-div 8590  df-inn 8879  df-2 8937  df-3 8938  df-4 8939  df-n0 9136  df-z 9213  df-uz 9488  df-q 9579  df-rp 9611  df-fz 9966  df-fzo 10099  df-seqfrec 10402  df-exp 10476  df-ihash 10710  df-cj 10806  df-re 10807  df-im 10808  df-rsqrt 10962  df-abs 10963  df-clim 11242  df-proddc 11514
This theorem is referenced by: (None)
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