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

Theorem prodfdiv 28925
Description: The quotient of two infinite products. (Contributed by Scott Fenton, 15-Jan-2018.)
Hypotheses
Ref Expression
prodfdiv.1  |-  ( ph  ->  N  e.  ( ZZ>= `  M ) )
prodfdiv.2  |-  ( (
ph  /\  k  e.  ( M ... N ) )  ->  ( F `  k )  e.  CC )
prodfdiv.3  |-  ( (
ph  /\  k  e.  ( M ... N ) )  ->  ( G `  k )  e.  CC )
prodfdiv.4  |-  ( (
ph  /\  k  e.  ( M ... N ) )  ->  ( G `  k )  =/=  0
)
prodfdiv.5  |-  ( (
ph  /\  k  e.  ( M ... N ) )  ->  ( H `  k )  =  ( ( F `  k
)  /  ( G `
 k ) ) )
Assertion
Ref Expression
prodfdiv  |-  ( ph  ->  (  seq M (  x.  ,  H ) `
 N )  =  ( (  seq M
(  x.  ,  F
) `  N )  /  (  seq M (  x.  ,  G ) `
 N ) ) )
Distinct variable groups:    k, F    k, G    k, H    ph, k    k, M    k, N

Proof of Theorem prodfdiv
Dummy variables  x  n are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 prodfdiv.1 . . . 4  |-  ( ph  ->  N  e.  ( ZZ>= `  M ) )
2 prodfdiv.3 . . . 4  |-  ( (
ph  /\  k  e.  ( M ... N ) )  ->  ( G `  k )  e.  CC )
3 prodfdiv.4 . . . 4  |-  ( (
ph  /\  k  e.  ( M ... N ) )  ->  ( G `  k )  =/=  0
)
4 fveq2 5871 . . . . . . 7  |-  ( n  =  k  ->  ( G `  n )  =  ( G `  k ) )
54oveq2d 6310 . . . . . 6  |-  ( n  =  k  ->  (
1  /  ( G `
 n ) )  =  ( 1  / 
( G `  k
) ) )
6 eqid 2467 . . . . . 6  |-  ( n  e.  ( M ... N )  |->  ( 1  /  ( G `  n ) ) )  =  ( n  e.  ( M ... N
)  |->  ( 1  / 
( G `  n
) ) )
7 ovex 6319 . . . . . 6  |-  ( 1  /  ( G `  k ) )  e. 
_V
85, 6, 7fvmpt 5956 . . . . 5  |-  ( k  e.  ( M ... N )  ->  (
( n  e.  ( M ... N ) 
|->  ( 1  /  ( G `  n )
) ) `  k
)  =  ( 1  /  ( G `  k ) ) )
98adantl 466 . . . 4  |-  ( (
ph  /\  k  e.  ( M ... N ) )  ->  ( (
n  e.  ( M ... N )  |->  ( 1  /  ( G `
 n ) ) ) `  k )  =  ( 1  / 
( G `  k
) ) )
101, 2, 3, 9prodfrec 28924 . . 3  |-  ( ph  ->  (  seq M (  x.  ,  ( n  e.  ( M ... N )  |->  ( 1  /  ( G `  n ) ) ) ) `  N )  =  ( 1  / 
(  seq M (  x.  ,  G ) `  N ) ) )
1110oveq2d 6310 . 2  |-  ( ph  ->  ( (  seq M
(  x.  ,  F
) `  N )  x.  (  seq M (  x.  ,  ( n  e.  ( M ... N )  |->  ( 1  /  ( G `  n ) ) ) ) `  N ) )  =  ( (  seq M (  x.  ,  F ) `  N )  x.  (
1  /  (  seq M (  x.  ,  G ) `  N
) ) ) )
12 prodfdiv.2 . . 3  |-  ( (
ph  /\  k  e.  ( M ... N ) )  ->  ( F `  k )  e.  CC )
13 eleq1 2539 . . . . . . . . 9  |-  ( k  =  n  ->  (
k  e.  ( M ... N )  <->  n  e.  ( M ... N ) ) )
1413anbi2d 703 . . . . . . . 8  |-  ( k  =  n  ->  (
( ph  /\  k  e.  ( M ... N
) )  <->  ( ph  /\  n  e.  ( M ... N ) ) ) )
15 fveq2 5871 . . . . . . . . 9  |-  ( k  =  n  ->  ( G `  k )  =  ( G `  n ) )
1615eleq1d 2536 . . . . . . . 8  |-  ( k  =  n  ->  (
( G `  k
)  e.  CC  <->  ( G `  n )  e.  CC ) )
1714, 16imbi12d 320 . . . . . . 7  |-  ( k  =  n  ->  (
( ( ph  /\  k  e.  ( M ... N ) )  -> 
( G `  k
)  e.  CC )  <-> 
( ( ph  /\  n  e.  ( M ... N ) )  -> 
( G `  n
)  e.  CC ) ) )
1817, 2chvarv 1983 . . . . . 6  |-  ( (
ph  /\  n  e.  ( M ... N ) )  ->  ( G `  n )  e.  CC )
1915neeq1d 2744 . . . . . . . 8  |-  ( k  =  n  ->  (
( G `  k
)  =/=  0  <->  ( G `  n )  =/=  0 ) )
2014, 19imbi12d 320 . . . . . . 7  |-  ( k  =  n  ->  (
( ( ph  /\  k  e.  ( M ... N ) )  -> 
( G `  k
)  =/=  0 )  <-> 
( ( ph  /\  n  e.  ( M ... N ) )  -> 
( G `  n
)  =/=  0 ) ) )
2120, 3chvarv 1983 . . . . . 6  |-  ( (
ph  /\  n  e.  ( M ... N ) )  ->  ( G `  n )  =/=  0
)
2218, 21reccld 10323 . . . . 5  |-  ( (
ph  /\  n  e.  ( M ... N ) )  ->  ( 1  /  ( G `  n ) )  e.  CC )
2322, 6fmptd 6055 . . . 4  |-  ( ph  ->  ( n  e.  ( M ... N ) 
|->  ( 1  /  ( G `  n )
) ) : ( M ... N ) --> CC )
2423ffvelrnda 6031 . . 3  |-  ( (
ph  /\  k  e.  ( M ... N ) )  ->  ( (
n  e.  ( M ... N )  |->  ( 1  /  ( G `
 n ) ) ) `  k )  e.  CC )
2512, 2, 3divrecd 10333 . . . 4  |-  ( (
ph  /\  k  e.  ( M ... N ) )  ->  ( ( F `  k )  /  ( G `  k ) )  =  ( ( F `  k )  x.  (
1  /  ( G `
 k ) ) ) )
26 prodfdiv.5 . . . 4  |-  ( (
ph  /\  k  e.  ( M ... N ) )  ->  ( H `  k )  =  ( ( F `  k
)  /  ( G `
 k ) ) )
279oveq2d 6310 . . . 4  |-  ( (
ph  /\  k  e.  ( M ... N ) )  ->  ( ( F `  k )  x.  ( ( n  e.  ( M ... N
)  |->  ( 1  / 
( G `  n
) ) ) `  k ) )  =  ( ( F `  k )  x.  (
1  /  ( G `
 k ) ) ) )
2825, 26, 273eqtr4d 2518 . . 3  |-  ( (
ph  /\  k  e.  ( M ... N ) )  ->  ( H `  k )  =  ( ( F `  k
)  x.  ( ( n  e.  ( M ... N )  |->  ( 1  /  ( G `
 n ) ) ) `  k ) ) )
291, 12, 24, 28prodfmul 28919 . 2  |-  ( ph  ->  (  seq M (  x.  ,  H ) `
 N )  =  ( (  seq M
(  x.  ,  F
) `  N )  x.  (  seq M (  x.  ,  ( n  e.  ( M ... N )  |->  ( 1  /  ( G `  n ) ) ) ) `  N ) ) )
30 mulcl 9586 . . . . 5  |-  ( ( k  e.  CC  /\  x  e.  CC )  ->  ( k  x.  x
)  e.  CC )
3130adantl 466 . . . 4  |-  ( (
ph  /\  ( k  e.  CC  /\  x  e.  CC ) )  -> 
( k  x.  x
)  e.  CC )
321, 12, 31seqcl 12105 . . 3  |-  ( ph  ->  (  seq M (  x.  ,  F ) `
 N )  e.  CC )
331, 2, 31seqcl 12105 . . 3  |-  ( ph  ->  (  seq M (  x.  ,  G ) `
 N )  e.  CC )
341, 2, 3prodfn0 28923 . . 3  |-  ( ph  ->  (  seq M (  x.  ,  G ) `
 N )  =/=  0 )
3532, 33, 34divrecd 10333 . 2  |-  ( ph  ->  ( (  seq M
(  x.  ,  F
) `  N )  /  (  seq M (  x.  ,  G ) `
 N ) )  =  ( (  seq M (  x.  ,  F ) `  N
)  x.  ( 1  /  (  seq M
(  x.  ,  G
) `  N )
) ) )
3611, 29, 353eqtr4d 2518 1  |-  ( ph  ->  (  seq M (  x.  ,  H ) `
 N )  =  ( (  seq M
(  x.  ,  F
) `  N )  /  (  seq M (  x.  ,  G ) `
 N ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767    =/= wne 2662    |-> cmpt 4510   ` cfv 5593  (class class class)co 6294   CCcc 9500   0cc0 9502   1c1 9503    x. cmul 9507    / cdiv 10216   ZZ>=cuz 11092   ...cfz 11682    seqcseq 12085
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-sep 4573  ax-nul 4581  ax-pow 4630  ax-pr 4691  ax-un 6586  ax-cnex 9558  ax-resscn 9559  ax-1cn 9560  ax-icn 9561  ax-addcl 9562  ax-addrcl 9563  ax-mulcl 9564  ax-mulrcl 9565  ax-mulcom 9566  ax-addass 9567  ax-mulass 9568  ax-distr 9569  ax-i2m1 9570  ax-1ne0 9571  ax-1rid 9572  ax-rnegex 9573  ax-rrecex 9574  ax-cnre 9575  ax-pre-lttri 9576  ax-pre-lttrn 9577  ax-pre-ltadd 9578  ax-pre-mulgt0 9579
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 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4251  df-iun 4332  df-br 4453  df-opab 4511  df-mpt 4512  df-tr 4546  df-eprel 4796  df-id 4800  df-po 4805  df-so 4806  df-fr 4843  df-we 4845  df-ord 4886  df-on 4887  df-lim 4888  df-suc 4889  df-xp 5010  df-rel 5011  df-cnv 5012  df-co 5013  df-dm 5014  df-rn 5015  df-res 5016  df-ima 5017  df-iota 5556  df-fun 5595  df-fn 5596  df-f 5597  df-f1 5598  df-fo 5599  df-f1o 5600  df-fv 5601  df-riota 6255  df-ov 6297  df-oprab 6298  df-mpt2 6299  df-om 6695  df-1st 6794  df-2nd 6795  df-recs 7052  df-rdg 7086  df-er 7321  df-en 7527  df-dom 7528  df-sdom 7529  df-pnf 9640  df-mnf 9641  df-xr 9642  df-ltxr 9643  df-le 9644  df-sub 9817  df-neg 9818  df-div 10217  df-nn 10547  df-n0 10806  df-z 10875  df-uz 11093  df-fz 11683  df-fzo 11803  df-seq 12086
This theorem is referenced by:  fproddiv  28986
  Copyright terms: Public domain W3C validator