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Theorem prodfdiv 27340
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 5688 . . . . . . 7  |-  ( n  =  k  ->  ( G `  n )  =  ( G `  k ) )
54oveq2d 6106 . . . . . 6  |-  ( n  =  k  ->  (
1  /  ( G `
 n ) )  =  ( 1  / 
( G `  k
) ) )
6 eqid 2441 . . . . . 6  |-  ( n  e.  ( M ... N )  |->  ( 1  /  ( G `  n ) ) )  =  ( n  e.  ( M ... N
)  |->  ( 1  / 
( G `  n
) ) )
7 ovex 6115 . . . . . 6  |-  ( 1  /  ( G `  k ) )  e. 
_V
85, 6, 7fvmpt 5771 . . . . 5  |-  ( k  e.  ( M ... N )  ->  (
( n  e.  ( M ... N ) 
|->  ( 1  /  ( G `  n )
) ) `  k
)  =  ( 1  /  ( G `  k ) ) )
98adantl 463 . . . 4  |-  ( (
ph  /\  k  e.  ( M ... N ) )  ->  ( (
n  e.  ( M ... N )  |->  ( 1  /  ( G `
 n ) ) ) `  k )  =  ( 1  / 
( G `  k
) ) )
101, 2, 3, 9prodfrec 27339 . . 3  |-  ( ph  ->  (  seq M (  x.  ,  ( n  e.  ( M ... N )  |->  ( 1  /  ( G `  n ) ) ) ) `  N )  =  ( 1  / 
(  seq M (  x.  ,  G ) `  N ) ) )
1110oveq2d 6106 . 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 2501 . . . . . . . . 9  |-  ( k  =  n  ->  (
k  e.  ( M ... N )  <->  n  e.  ( M ... N ) ) )
1413anbi2d 698 . . . . . . . 8  |-  ( k  =  n  ->  (
( ph  /\  k  e.  ( M ... N
) )  <->  ( ph  /\  n  e.  ( M ... N ) ) ) )
15 fveq2 5688 . . . . . . . . 9  |-  ( k  =  n  ->  ( G `  k )  =  ( G `  n ) )
1615eleq1d 2507 . . . . . . . 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 1963 . . . . . 6  |-  ( (
ph  /\  n  e.  ( M ... N ) )  ->  ( G `  n )  e.  CC )
1915neeq1d 2619 . . . . . . . 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 1963 . . . . . 6  |-  ( (
ph  /\  n  e.  ( M ... N ) )  ->  ( G `  n )  =/=  0
)
2218, 21reccld 10096 . . . . 5  |-  ( (
ph  /\  n  e.  ( M ... N ) )  ->  ( 1  /  ( G `  n ) )  e.  CC )
2322, 6fmptd 5864 . . . 4  |-  ( ph  ->  ( n  e.  ( M ... N ) 
|->  ( 1  /  ( G `  n )
) ) : ( M ... N ) --> CC )
2423ffvelrnda 5840 . . 3  |-  ( (
ph  /\  k  e.  ( M ... N ) )  ->  ( (
n  e.  ( M ... N )  |->  ( 1  /  ( G `
 n ) ) ) `  k )  e.  CC )
2512, 2, 3divrecd 10106 . . . 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 6106 . . . 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 2483 . . 3  |-  ( (
ph  /\  k  e.  ( M ... N ) )  ->  ( H `  k )  =  ( ( F `  k
)  x.  ( ( n  e.  ( M ... N )  |->  ( 1  /  ( G `
 n ) ) ) `  k ) ) )
291, 12, 24, 28prodfmul 27334 . 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 9362 . . . . 5  |-  ( ( k  e.  CC  /\  x  e.  CC )  ->  ( k  x.  x
)  e.  CC )
3130adantl 463 . . . 4  |-  ( (
ph  /\  ( k  e.  CC  /\  x  e.  CC ) )  -> 
( k  x.  x
)  e.  CC )
321, 12, 31seqcl 11822 . . 3  |-  ( ph  ->  (  seq M (  x.  ,  F ) `
 N )  e.  CC )
331, 2, 31seqcl 11822 . . 3  |-  ( ph  ->  (  seq M (  x.  ,  G ) `
 N )  e.  CC )
341, 2, 3prodfn0 27338 . . 3  |-  ( ph  ->  (  seq M (  x.  ,  G ) `
 N )  =/=  0 )
3532, 33, 34divrecd 10106 . 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 2483 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 1364    e. wcel 1761    =/= wne 2604    e. cmpt 4347   ` cfv 5415  (class class class)co 6090   CCcc 9276   0cc0 9278   1c1 9279    x. cmul 9283    / cdiv 9989   ZZ>=cuz 10857   ...cfz 11433    seqcseq 11802
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-1st 6576  df-2nd 6577  df-recs 6828  df-rdg 6862  df-er 7097  df-en 7307  df-dom 7308  df-sdom 7309  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-div 9990  df-nn 10319  df-n0 10576  df-z 10643  df-uz 10858  df-fz 11434  df-fzo 11545  df-seq 11803
This theorem is referenced by:  fproddiv  27401
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