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Theorem prodfdiv 25177
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 5687 . . . . . . 7  |-  ( n  =  k  ->  ( G `  n )  =  ( G `  k ) )
54oveq2d 6056 . . . . . 6  |-  ( n  =  k  ->  (
1  /  ( G `
 n ) )  =  ( 1  / 
( G `  k
) ) )
6 eqid 2404 . . . . . 6  |-  ( n  e.  ( M ... N )  |->  ( 1  /  ( G `  n ) ) )  =  ( n  e.  ( M ... N
)  |->  ( 1  / 
( G `  n
) ) )
7 ovex 6065 . . . . . 6  |-  ( 1  /  ( G `  k ) )  e. 
_V
85, 6, 7fvmpt 5765 . . . . 5  |-  ( k  e.  ( M ... N )  ->  (
( n  e.  ( M ... N ) 
|->  ( 1  /  ( G `  n )
) ) `  k
)  =  ( 1  /  ( G `  k ) ) )
98adantl 453 . . . 4  |-  ( (
ph  /\  k  e.  ( M ... N ) )  ->  ( (
n  e.  ( M ... N )  |->  ( 1  /  ( G `
 n ) ) ) `  k )  =  ( 1  / 
( G `  k
) ) )
101, 2, 3, 9prodfrec 25176 . . 3  |-  ( ph  ->  (  seq  M (  x.  ,  ( n  e.  ( M ... N )  |->  ( 1  /  ( G `  n ) ) ) ) `  N )  =  ( 1  / 
(  seq  M (  x.  ,  G ) `  N ) ) )
1110oveq2d 6056 . 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 2464 . . . . . . . . 9  |-  ( k  =  n  ->  (
k  e.  ( M ... N )  <->  n  e.  ( M ... N ) ) )
1413anbi2d 685 . . . . . . . 8  |-  ( k  =  n  ->  (
( ph  /\  k  e.  ( M ... N
) )  <->  ( ph  /\  n  e.  ( M ... N ) ) ) )
15 fveq2 5687 . . . . . . . . 9  |-  ( k  =  n  ->  ( G `  k )  =  ( G `  n ) )
1615eleq1d 2470 . . . . . . . 8  |-  ( k  =  n  ->  (
( G `  k
)  e.  CC  <->  ( G `  n )  e.  CC ) )
1714, 16imbi12d 312 . . . . . . 7  |-  ( k  =  n  ->  (
( ( ph  /\  k  e.  ( M ... N ) )  -> 
( G `  k
)  e.  CC )  <-> 
( ( ph  /\  n  e.  ( M ... N ) )  -> 
( G `  n
)  e.  CC ) ) )
1817, 2chvarv 2063 . . . . . 6  |-  ( (
ph  /\  n  e.  ( M ... N ) )  ->  ( G `  n )  e.  CC )
1915neeq1d 2580 . . . . . . . 8  |-  ( k  =  n  ->  (
( G `  k
)  =/=  0  <->  ( G `  n )  =/=  0 ) )
2014, 19imbi12d 312 . . . . . . 7  |-  ( k  =  n  ->  (
( ( ph  /\  k  e.  ( M ... N ) )  -> 
( G `  k
)  =/=  0 )  <-> 
( ( ph  /\  n  e.  ( M ... N ) )  -> 
( G `  n
)  =/=  0 ) ) )
2120, 3chvarv 2063 . . . . . 6  |-  ( (
ph  /\  n  e.  ( M ... N ) )  ->  ( G `  n )  =/=  0
)
2218, 21reccld 9739 . . . . 5  |-  ( (
ph  /\  n  e.  ( M ... N ) )  ->  ( 1  /  ( G `  n ) )  e.  CC )
2322, 6fmptd 5852 . . . 4  |-  ( ph  ->  ( n  e.  ( M ... N ) 
|->  ( 1  /  ( G `  n )
) ) : ( M ... N ) --> CC )
2423ffvelrnda 5829 . . 3  |-  ( (
ph  /\  k  e.  ( M ... N ) )  ->  ( (
n  e.  ( M ... N )  |->  ( 1  /  ( G `
 n ) ) ) `  k )  e.  CC )
2512, 2, 3divrecd 9749 . . . 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 6056 . . . 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 2446 . . 3  |-  ( (
ph  /\  k  e.  ( M ... N ) )  ->  ( H `  k )  =  ( ( F `  k
)  x.  ( ( n  e.  ( M ... N )  |->  ( 1  /  ( G `
 n ) ) ) `  k ) ) )
291, 12, 24, 28prodfmul 25171 . 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 9030 . . . . 5  |-  ( ( k  e.  CC  /\  x  e.  CC )  ->  ( k  x.  x
)  e.  CC )
3130adantl 453 . . . 4  |-  ( (
ph  /\  ( k  e.  CC  /\  x  e.  CC ) )  -> 
( k  x.  x
)  e.  CC )
321, 12, 31seqcl 11298 . . 3  |-  ( ph  ->  (  seq  M (  x.  ,  F ) `
 N )  e.  CC )
331, 2, 31seqcl 11298 . . 3  |-  ( ph  ->  (  seq  M (  x.  ,  G ) `
 N )  e.  CC )
341, 2, 3prodfn0 25175 . . 3  |-  ( ph  ->  (  seq  M (  x.  ,  G ) `
 N )  =/=  0 )
3532, 33, 34divrecd 9749 . 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 2446 1  |-  ( ph  ->  (  seq  M (  x.  ,  H ) `
 N )  =  ( (  seq  M
(  x.  ,  F
) `  N )  /  (  seq  M (  x.  ,  G ) `
 N ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1721    =/= wne 2567    e. cmpt 4226   ` cfv 5413  (class class class)co 6040   CCcc 8944   0cc0 8946   1c1 8947    x. cmul 8951    / cdiv 9633   ZZ>=cuz 10444   ...cfz 10999    seq cseq 11278
This theorem is referenced by:  fproddiv  25238
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-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  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
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-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-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-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-er 6864  df-en 7069  df-dom 7070  df-sdom 7071  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-n0 10178  df-z 10239  df-uz 10445  df-fz 11000  df-fzo 11091  df-seq 11279
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