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Theorem fprodabs 29071
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 2539 . 2  |-  ( ph  ->  N  e.  ( ZZ>= `  M ) )
4 oveq2 6285 . . . . . . 7  |-  ( a  =  M  ->  ( M ... a )  =  ( M ... M
) )
54prodeq1d 29021 . . . . . 6  |-  ( a  =  M  ->  prod_ k  e.  ( M ... a ) A  = 
prod_ k  e.  ( M ... M ) A )
65fveq2d 5856 . . . . 5  |-  ( a  =  M  ->  ( abs `  prod_ k  e.  ( M ... a ) A )  =  ( abs `  prod_ k  e.  ( M ... M
) A ) )
74prodeq1d 29021 . . . . 5  |-  ( a  =  M  ->  prod_ k  e.  ( M ... a ) ( abs `  A )  =  prod_ k  e.  ( M ... M ) ( abs `  A ) )
86, 7eqeq12d 2463 . . . 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 316 . . 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 6285 . . . . . . 7  |-  ( a  =  n  ->  ( M ... a )  =  ( M ... n
) )
1110prodeq1d 29021 . . . . . 6  |-  ( a  =  n  ->  prod_ k  e.  ( M ... a ) A  = 
prod_ k  e.  ( M ... n ) A )
1211fveq2d 5856 . . . . 5  |-  ( a  =  n  ->  ( abs `  prod_ k  e.  ( M ... a ) A )  =  ( abs `  prod_ k  e.  ( M ... n
) A ) )
1310prodeq1d 29021 . . . . 5  |-  ( a  =  n  ->  prod_ k  e.  ( M ... a ) ( abs `  A )  =  prod_ k  e.  ( M ... n ) ( abs `  A ) )
1412, 13eqeq12d 2463 . . . 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 316 . . 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 6285 . . . . . . 7  |-  ( a  =  ( n  + 
1 )  ->  ( M ... a )  =  ( M ... (
n  +  1 ) ) )
1716prodeq1d 29021 . . . . . 6  |-  ( a  =  ( n  + 
1 )  ->  prod_ k  e.  ( M ... a ) A  = 
prod_ k  e.  ( M ... ( n  + 
1 ) ) A )
1817fveq2d 5856 . . . . 5  |-  ( a  =  ( n  + 
1 )  ->  ( abs `  prod_ k  e.  ( M ... a ) A )  =  ( abs `  prod_ k  e.  ( M ... (
n  +  1 ) ) A ) )
1916prodeq1d 29021 . . . . 5  |-  ( a  =  ( n  + 
1 )  ->  prod_ k  e.  ( M ... a ) ( abs `  A )  =  prod_ k  e.  ( M ... ( n  +  1
) ) ( abs `  A ) )
2018, 19eqeq12d 2463 . . . 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 316 . . 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 6285 . . . . . . 7  |-  ( a  =  N  ->  ( M ... a )  =  ( M ... N
) )
2322prodeq1d 29021 . . . . . 6  |-  ( a  =  N  ->  prod_ k  e.  ( M ... a ) A  = 
prod_ k  e.  ( M ... N ) A )
2423fveq2d 5856 . . . . 5  |-  ( a  =  N  ->  ( abs `  prod_ k  e.  ( M ... a ) A )  =  ( abs `  prod_ k  e.  ( M ... N
) A ) )
2522prodeq1d 29021 . . . . 5  |-  ( a  =  N  ->  prod_ k  e.  ( M ... a ) ( abs `  A )  =  prod_ k  e.  ( M ... N ) ( abs `  A ) )
2624, 25eqeq12d 2463 . . . 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 316 . . 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 5889 . . . . . 6  |-  ( M  e.  ZZ  ->  [_ M  /  k ]_ ( abs `  A )  =  ( abs `  [_ M  /  k ]_ A
) )
2928adantl 466 . . . . 5  |-  ( (
ph  /\  M  e.  ZZ )  ->  [_ M  /  k ]_ ( abs `  A )  =  ( abs `  [_ M  /  k ]_ A
) )
30 fzsn 11729 . . . . . . . 8  |-  ( M  e.  ZZ  ->  ( M ... M )  =  { M } )
3130adantl 466 . . . . . . 7  |-  ( (
ph  /\  M  e.  ZZ )  ->  ( M ... M )  =  { M } )
3231prodeq1d 29021 . . . . . 6  |-  ( (
ph  /\  M  e.  ZZ )  ->  prod_ k  e.  ( M ... M
) ( abs `  A
)  =  prod_ k  e.  { M }  ( abs `  A ) )
33 simpr 461 . . . . . . 7  |-  ( (
ph  /\  M  e.  ZZ )  ->  M  e.  ZZ )
34 uzid 11099 . . . . . . . . . . . 12  |-  ( M  e.  ZZ  ->  M  e.  ( ZZ>= `  M )
)
3534, 2syl6eleqr 2540 . . . . . . . . . . 11  |-  ( M  e.  ZZ  ->  M  e.  Z )
36 fprodabs.3 . . . . . . . . . . . . 13  |-  ( (
ph  /\  k  e.  Z )  ->  A  e.  CC )
3736ralrimiva 2855 . . . . . . . . . . . 12  |-  ( ph  ->  A. k  e.  Z  A  e.  CC )
38 nfcsb1v 3433 . . . . . . . . . . . . . 14  |-  F/_ k [_ M  /  k ]_ A
3938nfel1 2619 . . . . . . . . . . . . 13  |-  F/ k
[_ M  /  k ]_ A  e.  CC
40 csbeq1a 3426 . . . . . . . . . . . . . 14  |-  ( k  =  M  ->  A  =  [_ M  /  k ]_ A )
4140eleq1d 2510 . . . . . . . . . . . . 13  |-  ( k  =  M  ->  ( A  e.  CC  <->  [_ M  / 
k ]_ A  e.  CC ) )
4239, 41rspc 3188 . . . . . . . . . . . 12  |-  ( M  e.  Z  ->  ( A. k  e.  Z  A  e.  CC  ->  [_ M  /  k ]_ A  e.  CC )
)
4337, 42mpan9 469 . . . . . . . . . . 11  |-  ( (
ph  /\  M  e.  Z )  ->  [_ M  /  k ]_ A  e.  CC )
4435, 43sylan2 474 . . . . . . . . . 10  |-  ( (
ph  /\  M  e.  ZZ )  ->  [_ M  /  k ]_ A  e.  CC )
4544abscld 13241 . . . . . . . . 9  |-  ( (
ph  /\  M  e.  ZZ )  ->  ( abs `  [_ M  /  k ]_ A )  e.  RR )
4645recnd 9620 . . . . . . . 8  |-  ( (
ph  /\  M  e.  ZZ )  ->  ( abs `  [_ M  /  k ]_ A )  e.  CC )
4729, 46eqeltrd 2529 . . . . . . 7  |-  ( (
ph  /\  M  e.  ZZ )  ->  [_ M  /  k ]_ ( abs `  A )  e.  CC )
48 prodsns 29069 . . . . . . 7  |-  ( ( M  e.  ZZ  /\  [_ M  /  k ]_ ( abs `  A )  e.  CC )  ->  prod_ k  e.  { M }  ( abs `  A
)  =  [_ M  /  k ]_ ( abs `  A ) )
4933, 47, 48syl2anc 661 . . . . . 6  |-  ( (
ph  /\  M  e.  ZZ )  ->  prod_ k  e.  { M }  ( abs `  A )  = 
[_ M  /  k ]_ ( abs `  A
) )
5032, 49eqtrd 2482 . . . . 5  |-  ( (
ph  /\  M  e.  ZZ )  ->  prod_ k  e.  ( M ... M
) ( abs `  A
)  =  [_ M  /  k ]_ ( abs `  A ) )
5130prodeq1d 29021 . . . . . . . 8  |-  ( M  e.  ZZ  ->  prod_ k  e.  ( M ... M ) A  = 
prod_ k  e.  { M } A )
5251adantl 466 . . . . . . 7  |-  ( (
ph  /\  M  e.  ZZ )  ->  prod_ k  e.  ( M ... M
) A  =  prod_ k  e.  { M } A )
53 prodsns 29069 . . . . . . . 8  |-  ( ( M  e.  ZZ  /\  [_ M  /  k ]_ A  e.  CC )  ->  prod_ k  e.  { M } A  =  [_ M  /  k ]_ A
)
5433, 44, 53syl2anc 661 . . . . . . 7  |-  ( (
ph  /\  M  e.  ZZ )  ->  prod_ k  e.  { M } A  =  [_ M  /  k ]_ A )
5552, 54eqtrd 2482 . . . . . 6  |-  ( (
ph  /\  M  e.  ZZ )  ->  prod_ k  e.  ( M ... M
) A  =  [_ M  /  k ]_ A
)
5655fveq2d 5856 . . . . 5  |-  ( (
ph  /\  M  e.  ZZ )  ->  ( abs `  prod_ k  e.  ( M ... M ) A )  =  ( abs `  [_ M  /  k ]_ A
) )
5729, 50, 563eqtr4rd 2493 . . . 4  |-  ( (
ph  /\  M  e.  ZZ )  ->  ( abs `  prod_ k  e.  ( M ... M ) A )  =  prod_ k  e.  ( M ... M ) ( abs `  A ) )
5857expcom 435 . . 3  |-  ( M  e.  ZZ  ->  ( ph  ->  ( abs `  prod_ k  e.  ( M ... M ) A )  =  prod_ k  e.  ( M ... M ) ( abs `  A
) ) )
59 simp3 997 . . . . . . . 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 6305 . . . . . . . . . . 11  |-  ( n  +  1 )  e. 
_V
61 csbfv2g 5889 . . . . . . . . . . 11  |-  ( ( n  +  1 )  e.  _V  ->  [_ (
n  +  1 )  /  k ]_ ( abs `  A )  =  ( abs `  [_ (
n  +  1 )  /  k ]_ A
) )
6260, 61ax-mp 5 . . . . . . . . . 10  |-  [_ (
n  +  1 )  /  k ]_ ( abs `  A )  =  ( abs `  [_ (
n  +  1 )  /  k ]_ A
)
6362eqcomi 2454 . . . . . . . . 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 6295 . . . . . . 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 461 . . . . . . . . . . 11  |-  ( (
ph  /\  n  e.  ( ZZ>= `  M )
)  ->  n  e.  ( ZZ>= `  M )
)
67 elfzuz 11688 . . . . . . . . . . . . . 14  |-  ( k  e.  ( M ... ( n  +  1
) )  ->  k  e.  ( ZZ>= `  M )
)
6867, 2syl6eleqr 2540 . . . . . . . . . . . . 13  |-  ( k  e.  ( M ... ( n  +  1
) )  ->  k  e.  Z )
6968, 36sylan2 474 . . . . . . . . . . . 12  |-  ( (
ph  /\  k  e.  ( M ... ( n  +  1 ) ) )  ->  A  e.  CC )
7069adantlr 714 . . . . . . . . . . 11  |-  ( ( ( ph  /\  n  e.  ( ZZ>= `  M )
)  /\  k  e.  ( M ... ( n  +  1 ) ) )  ->  A  e.  CC )
7166, 70fprodp1s 29068 . . . . . . . . . 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 5856 . . . . . . . . 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 12057 . . . . . . . . . . 11  |-  ( (
ph  /\  n  e.  ( ZZ>= `  M )
)  ->  ( M ... n )  e.  Fin )
74 elfzuz 11688 . . . . . . . . . . . . . 14  |-  ( k  e.  ( M ... n )  ->  k  e.  ( ZZ>= `  M )
)
7574, 2syl6eleqr 2540 . . . . . . . . . . . . 13  |-  ( k  e.  ( M ... n )  ->  k  e.  Z )
7675, 36sylan2 474 . . . . . . . . . . . 12  |-  ( (
ph  /\  k  e.  ( M ... n ) )  ->  A  e.  CC )
7776adantlr 714 . . . . . . . . . . 11  |-  ( ( ( ph  /\  n  e.  ( ZZ>= `  M )
)  /\  k  e.  ( M ... n ) )  ->  A  e.  CC )
7873, 77fprodcl 29052 . . . . . . . . . 10  |-  ( (
ph  /\  n  e.  ( ZZ>= `  M )
)  ->  prod_ k  e.  ( M ... n
) A  e.  CC )
79 peano2uz 11138 . . . . . . . . . . . 12  |-  ( n  e.  ( ZZ>= `  M
)  ->  ( n  +  1 )  e.  ( ZZ>= `  M )
)
8079, 2syl6eleqr 2540 . . . . . . . . . . 11  |-  ( n  e.  ( ZZ>= `  M
)  ->  ( n  +  1 )  e.  Z )
81 nfcsb1v 3433 . . . . . . . . . . . . . 14  |-  F/_ k [_ ( n  +  1 )  /  k ]_ A
8281nfel1 2619 . . . . . . . . . . . . 13  |-  F/ k
[_ ( n  + 
1 )  /  k ]_ A  e.  CC
83 csbeq1a 3426 . . . . . . . . . . . . . 14  |-  ( k  =  ( n  + 
1 )  ->  A  =  [_ ( n  + 
1 )  /  k ]_ A )
8483eleq1d 2510 . . . . . . . . . . . . 13  |-  ( k  =  ( n  + 
1 )  ->  ( A  e.  CC  <->  [_ ( n  +  1 )  / 
k ]_ A  e.  CC ) )
8582, 84rspc 3188 . . . . . . . . . . . 12  |-  ( ( n  +  1 )  e.  Z  ->  ( A. k  e.  Z  A  e.  CC  ->  [_ ( n  +  1 )  /  k ]_ A  e.  CC )
)
8637, 85mpan9 469 . . . . . . . . . . 11  |-  ( (
ph  /\  ( n  +  1 )  e.  Z )  ->  [_ (
n  +  1 )  /  k ]_ A  e.  CC )
8780, 86sylan2 474 . . . . . . . . . 10  |-  ( (
ph  /\  n  e.  ( ZZ>= `  M )
)  ->  [_ ( n  +  1 )  / 
k ]_ A  e.  CC )
8878, 87absmuld 13259 . . . . . . . . 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 2482 . . . . . . . 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 1015 . . . . . . 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 13241 . . . . . . . . . 10  |-  ( ( ( ph  /\  n  e.  ( ZZ>= `  M )
)  /\  k  e.  ( M ... ( n  +  1 ) ) )  ->  ( abs `  A )  e.  RR )
9291recnd 9620 . . . . . . . . 9  |-  ( ( ( ph  /\  n  e.  ( ZZ>= `  M )
)  /\  k  e.  ( M ... ( n  +  1 ) ) )  ->  ( abs `  A )  e.  CC )
9366, 92fprodp1s 29068 . . . . . . . 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 1015 . . . . . . 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 2492 . . . . . 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 1194 . . . . 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 31 . . . 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 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
) ) ) )
999, 15, 21, 27, 58, 98uzind4 11143 . 2  |-  ( N  e.  ( ZZ>= `  M
)  ->  ( ph  ->  ( abs `  prod_ k  e.  ( M ... N ) A )  =  prod_ k  e.  ( M ... N ) ( abs `  A
) ) )
1003, 99mpcom 36 1  |-  ( ph  ->  ( abs `  prod_ k  e.  ( M ... N ) A )  =  prod_ k  e.  ( M ... N ) ( abs `  A
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 972    = wceq 1381    e. wcel 1802   A.wral 2791   _Vcvv 3093   [_csb 3417   {csn 4010   ` cfv 5574  (class class class)co 6277   CCcc 9488   1c1 9491    + caddc 9493    x. cmul 9495   ZZcz 10865   ZZ>=cuz 11085   ...cfz 11676   abscabs 13041   prod_cprod 29005
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-8 1804  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-rep 4544  ax-sep 4554  ax-nul 4562  ax-pow 4611  ax-pr 4672  ax-un 6573  ax-inf2 8056  ax-cnex 9546  ax-resscn 9547  ax-1cn 9548  ax-icn 9549  ax-addcl 9550  ax-addrcl 9551  ax-mulcl 9552  ax-mulrcl 9553  ax-mulcom 9554  ax-addass 9555  ax-mulass 9556  ax-distr 9557  ax-i2m1 9558  ax-1ne0 9559  ax-1rid 9560  ax-rnegex 9561  ax-rrecex 9562  ax-cnre 9563  ax-pre-lttri 9564  ax-pre-lttrn 9565  ax-pre-ltadd 9566  ax-pre-mulgt0 9567  ax-pre-sup 9568
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 973  df-3an 974  df-tru 1384  df-fal 1387  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-nel 2639  df-ral 2796  df-rex 2797  df-reu 2798  df-rmo 2799  df-rab 2800  df-v 3095  df-sbc 3312  df-csb 3418  df-dif 3461  df-un 3463  df-in 3465  df-ss 3472  df-pss 3474  df-nul 3768  df-if 3923  df-pw 3995  df-sn 4011  df-pr 4013  df-tp 4015  df-op 4017  df-uni 4231  df-int 4268  df-iun 4313  df-br 4434  df-opab 4492  df-mpt 4493  df-tr 4527  df-eprel 4777  df-id 4781  df-po 4786  df-so 4787  df-fr 4824  df-se 4825  df-we 4826  df-ord 4867  df-on 4868  df-lim 4869  df-suc 4870  df-xp 4991  df-rel 4992  df-cnv 4993  df-co 4994  df-dm 4995  df-rn 4996  df-res 4997  df-ima 4998  df-iota 5537  df-fun 5576  df-fn 5577  df-f 5578  df-f1 5579  df-fo 5580  df-f1o 5581  df-fv 5582  df-isom 5583  df-riota 6238  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-om 6682  df-1st 6781  df-2nd 6782  df-recs 7040  df-rdg 7074  df-1o 7128  df-oadd 7132  df-er 7309  df-en 7515  df-dom 7516  df-sdom 7517  df-fin 7518  df-sup 7899  df-oi 7933  df-card 8318  df-pnf 9628  df-mnf 9629  df-xr 9630  df-ltxr 9631  df-le 9632  df-sub 9807  df-neg 9808  df-div 10208  df-nn 10538  df-2 10595  df-3 10596  df-n0 10797  df-z 10866  df-uz 11086  df-rp 11225  df-fz 11677  df-fzo 11799  df-seq 12082  df-exp 12141  df-hash 12380  df-cj 12906  df-re 12907  df-im 12908  df-sqrt 13042  df-abs 13043  df-clim 13285  df-prod 29006
This theorem is referenced by: (None)
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