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Theorem fprodabs 11524
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 2250 . 2  |-  ( ph  ->  N  e.  ( ZZ>= `  M ) )
4 oveq2 5834 . . . . . . 7  |-  ( a  =  M  ->  ( M ... a )  =  ( M ... M
) )
54prodeq1d 11472 . . . . . 6  |-  ( a  =  M  ->  prod_ k  e.  ( M ... a ) A  = 
prod_ k  e.  ( M ... M ) A )
65fveq2d 5474 . . . . 5  |-  ( a  =  M  ->  ( abs `  prod_ k  e.  ( M ... a ) A )  =  ( abs `  prod_ k  e.  ( M ... M
) A ) )
74prodeq1d 11472 . . . . 5  |-  ( a  =  M  ->  prod_ k  e.  ( M ... a ) ( abs `  A )  =  prod_ k  e.  ( M ... M ) ( abs `  A ) )
86, 7eqeq12d 2172 . . . 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 5834 . . . . . . 7  |-  ( a  =  n  ->  ( M ... a )  =  ( M ... n
) )
1110prodeq1d 11472 . . . . . 6  |-  ( a  =  n  ->  prod_ k  e.  ( M ... a ) A  = 
prod_ k  e.  ( M ... n ) A )
1211fveq2d 5474 . . . . 5  |-  ( a  =  n  ->  ( abs `  prod_ k  e.  ( M ... a ) A )  =  ( abs `  prod_ k  e.  ( M ... n
) A ) )
1310prodeq1d 11472 . . . . 5  |-  ( a  =  n  ->  prod_ k  e.  ( M ... a ) ( abs `  A )  =  prod_ k  e.  ( M ... n ) ( abs `  A ) )
1412, 13eqeq12d 2172 . . . 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 5834 . . . . . . 7  |-  ( a  =  ( n  + 
1 )  ->  ( M ... a )  =  ( M ... (
n  +  1 ) ) )
1716prodeq1d 11472 . . . . . 6  |-  ( a  =  ( n  + 
1 )  ->  prod_ k  e.  ( M ... a ) A  = 
prod_ k  e.  ( M ... ( n  + 
1 ) ) A )
1817fveq2d 5474 . . . . 5  |-  ( a  =  ( n  + 
1 )  ->  ( abs `  prod_ k  e.  ( M ... a ) A )  =  ( abs `  prod_ k  e.  ( M ... (
n  +  1 ) ) A ) )
1916prodeq1d 11472 . . . . 5  |-  ( a  =  ( n  + 
1 )  ->  prod_ k  e.  ( M ... a ) ( abs `  A )  =  prod_ k  e.  ( M ... ( n  +  1
) ) ( abs `  A ) )
2018, 19eqeq12d 2172 . . . 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 5834 . . . . . . 7  |-  ( a  =  N  ->  ( M ... a )  =  ( M ... N
) )
2322prodeq1d 11472 . . . . . 6  |-  ( a  =  N  ->  prod_ k  e.  ( M ... a ) A  = 
prod_ k  e.  ( M ... N ) A )
2423fveq2d 5474 . . . . 5  |-  ( a  =  N  ->  ( abs `  prod_ k  e.  ( M ... a ) A )  =  ( abs `  prod_ k  e.  ( M ... N
) A ) )
2522prodeq1d 11472 . . . . 5  |-  ( a  =  N  ->  prod_ k  e.  ( M ... a ) ( abs `  A )  =  prod_ k  e.  ( M ... N ) ( abs `  A ) )
2624, 25eqeq12d 2172 . . . 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 5507 . . . . . 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 9974 . . . . . . . 8  |-  ( M  e.  ZZ  ->  ( M ... M )  =  { M } )
3130adantl 275 . . . . . . 7  |-  ( (
ph  /\  M  e.  ZZ )  ->  ( M ... M )  =  { M } )
3231prodeq1d 11472 . . . . . 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 9458 . . . . . . . . . . . 12  |-  ( M  e.  ZZ  ->  M  e.  ( ZZ>= `  M )
)
3534, 2eleqtrrdi 2251 . . . . . . . . . . 11  |-  ( M  e.  ZZ  ->  M  e.  Z )
36 fprodabs.3 . . . . . . . . . . . . 13  |-  ( (
ph  /\  k  e.  Z )  ->  A  e.  CC )
3736ralrimiva 2530 . . . . . . . . . . . 12  |-  ( ph  ->  A. k  e.  Z  A  e.  CC )
38 nfcsb1v 3064 . . . . . . . . . . . . . 14  |-  F/_ k [_ M  /  k ]_ A
3938nfel1 2310 . . . . . . . . . . . . 13  |-  F/ k
[_ M  /  k ]_ A  e.  CC
40 csbeq1a 3040 . . . . . . . . . . . . . 14  |-  ( k  =  M  ->  A  =  [_ M  /  k ]_ A )
4140eleq1d 2226 . . . . . . . . . . . . 13  |-  ( k  =  M  ->  ( A  e.  CC  <->  [_ M  / 
k ]_ A  e.  CC ) )
4239, 41rspc 2810 . . . . . . . . . . . 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 11092 . . . . . . . . 9  |-  ( (
ph  /\  M  e.  ZZ )  ->  ( abs `  [_ M  /  k ]_ A )  e.  RR )
4645recnd 7908 . . . . . . . 8  |-  ( (
ph  /\  M  e.  ZZ )  ->  ( abs `  [_ M  /  k ]_ A )  e.  CC )
4729, 46eqeltrd 2234 . . . . . . 7  |-  ( (
ph  /\  M  e.  ZZ )  ->  [_ M  /  k ]_ ( abs `  A )  e.  CC )
48 prodsns 11511 . . . . . . 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 2190 . . . . 5  |-  ( (
ph  /\  M  e.  ZZ )  ->  prod_ k  e.  ( M ... M
) ( abs `  A
)  =  [_ M  /  k ]_ ( abs `  A ) )
5130prodeq1d 11472 . . . . . . . 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 11511 . . . . . . . 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 2190 . . . . . 6  |-  ( (
ph  /\  M  e.  ZZ )  ->  prod_ k  e.  ( M ... M
) A  =  [_ M  /  k ]_ A
)
5655fveq2d 5474 . . . . 5  |-  ( (
ph  /\  M  e.  ZZ )  ->  ( abs `  prod_ k  e.  ( M ... M ) A )  =  ( abs `  [_ M  /  k ]_ A
) )
5729, 50, 563eqtr4rd 2201 . . . 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 984 . . . . . . . 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 9499 . . . . . . . . . . 11  |-  ( n  e.  ( ZZ>= `  M
)  ->  ( n  +  1 )  e.  ( ZZ>= `  M )
)
61 csbfv2g 5507 . . . . . . . . . . 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 2163 . . . . . . . . 9  |-  ( n  e.  ( ZZ>= `  M
)  ->  ( abs ` 
[_ ( n  + 
1 )  /  k ]_ A )  =  [_ ( n  +  1
)  /  k ]_ ( abs `  A ) )
64633ad2ant2 1004 . . . . . . . 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 5844 . . . . . . 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 9930 . . . . . . . . . . . . . 14  |-  ( k  e.  ( M ... ( n  +  1
) )  ->  k  e.  ( ZZ>= `  M )
)
6867, 2eleqtrrdi 2251 . . . . . . . . . . . . 13  |-  ( k  e.  ( M ... ( n  +  1
) )  ->  k  e.  Z )
6968, 36sylan2 284 . . . . . . . . . . . 12  |-  ( (
ph  /\  k  e.  ( M ... ( n  +  1 ) ) )  ->  A  e.  CC )
7069adantlr 469 . . . . . . . . . . 11  |-  ( ( ( ph  /\  n  e.  ( ZZ>= `  M )
)  /\  k  e.  ( M ... ( n  +  1 ) ) )  ->  A  e.  CC )
7166, 70fprodp1s 11510 . . . . . . . . . 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 5474 . . . . . . . . 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 9449 . . . . . . . . . . . . 13  |-  ( n  e.  ( ZZ>= `  M
)  ->  M  e.  ZZ )
7473adantl 275 . . . . . . . . . . . 12  |-  ( (
ph  /\  n  e.  ( ZZ>= `  M )
)  ->  M  e.  ZZ )
75 eluzelz 9453 . . . . . . . . . . . . 13  |-  ( n  e.  ( ZZ>= `  M
)  ->  n  e.  ZZ )
7675adantl 275 . . . . . . . . . . . 12  |-  ( (
ph  /\  n  e.  ( ZZ>= `  M )
)  ->  n  e.  ZZ )
7774, 76fzfigd 10339 . . . . . . . . . . 11  |-  ( (
ph  /\  n  e.  ( ZZ>= `  M )
)  ->  ( M ... n )  e.  Fin )
78 elfzuz 9930 . . . . . . . . . . . . . 14  |-  ( k  e.  ( M ... n )  ->  k  e.  ( ZZ>= `  M )
)
7978, 2eleqtrrdi 2251 . . . . . . . . . . . . 13  |-  ( k  e.  ( M ... n )  ->  k  e.  Z )
8079, 36sylan2 284 . . . . . . . . . . . 12  |-  ( (
ph  /\  k  e.  ( M ... n ) )  ->  A  e.  CC )
8180adantlr 469 . . . . . . . . . . 11  |-  ( ( ( ph  /\  n  e.  ( ZZ>= `  M )
)  /\  k  e.  ( M ... n ) )  ->  A  e.  CC )
8277, 81fprodcl 11515 . . . . . . . . . 10  |-  ( (
ph  /\  n  e.  ( ZZ>= `  M )
)  ->  prod_ k  e.  ( M ... n
) A  e.  CC )
8360, 2eleqtrrdi 2251 . . . . . . . . . . 11  |-  ( n  e.  ( ZZ>= `  M
)  ->  ( n  +  1 )  e.  Z )
84 nfcsb1v 3064 . . . . . . . . . . . . . 14  |-  F/_ k [_ ( n  +  1 )  /  k ]_ A
8584nfel1 2310 . . . . . . . . . . . . 13  |-  F/ k
[_ ( n  + 
1 )  /  k ]_ A  e.  CC
86 csbeq1a 3040 . . . . . . . . . . . . . 14  |-  ( k  =  ( n  + 
1 )  ->  A  =  [_ ( n  + 
1 )  /  k ]_ A )
8786eleq1d 2226 . . . . . . . . . . . . 13  |-  ( k  =  ( n  + 
1 )  ->  ( A  e.  CC  <->  [_ ( n  +  1 )  / 
k ]_ A  e.  CC ) )
8885, 87rspc 2810 . . . . . . . . . . . 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 11105 . . . . . . . . 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 2190 . . . . . . . 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 1002 . . . . . . 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 11092 . . . . . . . . . 10  |-  ( ( ( ph  /\  n  e.  ( ZZ>= `  M )
)  /\  k  e.  ( M ... ( n  +  1 ) ) )  ->  ( abs `  A )  e.  RR )
9594recnd 7908 . . . . . . . . 9  |-  ( ( ( ph  /\  n  e.  ( ZZ>= `  M )
)  /\  k  e.  ( M ... ( n  +  1 ) ) )  ->  ( abs `  A )  e.  CC )
9666, 95fprodp1s 11510 . . . . . . . 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 1002 . . . . . . 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 2200 . . . . . 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 1184 . . . . 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 9504 . 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 963    = wceq 1335    e. wcel 2128   A.wral 2435   [_csb 3031   {csn 3561   ` cfv 5172  (class class class)co 5826   CCcc 7732   1c1 7735    + caddc 7737    x. cmul 7739   ZZcz 9172   ZZ>=cuz 9444   ...cfz 9918   abscabs 10908   prod_cprod 11458
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 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-coll 4081  ax-sep 4084  ax-nul 4092  ax-pow 4137  ax-pr 4171  ax-un 4395  ax-setind 4498  ax-iinf 4549  ax-cnex 7825  ax-resscn 7826  ax-1cn 7827  ax-1re 7828  ax-icn 7829  ax-addcl 7830  ax-addrcl 7831  ax-mulcl 7832  ax-mulrcl 7833  ax-addcom 7834  ax-mulcom 7835  ax-addass 7836  ax-mulass 7837  ax-distr 7838  ax-i2m1 7839  ax-0lt1 7840  ax-1rid 7841  ax-0id 7842  ax-rnegex 7843  ax-precex 7844  ax-cnre 7845  ax-pre-ltirr 7846  ax-pre-ltwlin 7847  ax-pre-lttrn 7848  ax-pre-apti 7849  ax-pre-ltadd 7850  ax-pre-mulgt0 7851  ax-pre-mulext 7852  ax-arch 7853  ax-caucvg 7854
This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-nel 2423  df-ral 2440  df-rex 2441  df-reu 2442  df-rmo 2443  df-rab 2444  df-v 2714  df-sbc 2938  df-csb 3032  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-nul 3396  df-if 3507  df-pw 3546  df-sn 3567  df-pr 3568  df-op 3570  df-uni 3775  df-int 3810  df-iun 3853  df-br 3968  df-opab 4028  df-mpt 4029  df-tr 4065  df-id 4255  df-po 4258  df-iso 4259  df-iord 4328  df-on 4330  df-ilim 4331  df-suc 4333  df-iom 4552  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-iota 5137  df-fun 5174  df-fn 5175  df-f 5176  df-f1 5177  df-fo 5178  df-f1o 5179  df-fv 5180  df-isom 5181  df-riota 5782  df-ov 5829  df-oprab 5830  df-mpo 5831  df-1st 6090  df-2nd 6091  df-recs 6254  df-irdg 6319  df-frec 6340  df-1o 6365  df-oadd 6369  df-er 6482  df-en 6688  df-dom 6689  df-fin 6690  df-pnf 7916  df-mnf 7917  df-xr 7918  df-ltxr 7919  df-le 7920  df-sub 8052  df-neg 8053  df-reap 8454  df-ap 8461  df-div 8550  df-inn 8839  df-2 8897  df-3 8898  df-4 8899  df-n0 9096  df-z 9173  df-uz 9445  df-q 9535  df-rp 9567  df-fz 9919  df-fzo 10051  df-seqfrec 10354  df-exp 10428  df-ihash 10661  df-cj 10753  df-re 10754  df-im 10755  df-rsqrt 10909  df-abs 10910  df-clim 11187  df-proddc 11459
This theorem is referenced by: (None)
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