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Theorem prodfdivap 11323
Description: The quotient of two products. (Contributed by Scott Fenton, 15-Jan-2018.) (Revised by Jim Kingdon, 24-Mar-2024.)
Hypotheses
Ref Expression
prodfdiv.1  |-  ( ph  ->  N  e.  ( ZZ>= `  M ) )
prodfdivap.2  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )
)  ->  ( F `  k )  e.  CC )
prodfdivap.3  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )
)  ->  ( G `  k )  e.  CC )
prodfdivap.4  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )
)  ->  ( G `  k ) #  0 )
prodfdivap.5  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )
)  ->  ( H `  k )  =  ( ( F `  k
)  /  ( G `
 k ) ) )
Assertion
Ref Expression
prodfdivap  |-  ( 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 prodfdivap
Dummy variable  n is distinct from all other variables.
StepHypRef Expression
1 prodfdiv.1 . . . 4  |-  ( ph  ->  N  e.  ( ZZ>= `  M ) )
2 prodfdivap.3 . . . 4  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )
)  ->  ( G `  k )  e.  CC )
3 elfzuz 9809 . . . . 5  |-  ( k  e.  ( M ... N )  ->  k  e.  ( ZZ>= `  M )
)
4 prodfdivap.4 . . . . 5  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )
)  ->  ( G `  k ) #  0 )
53, 4sylan2 284 . . . 4  |-  ( (
ph  /\  k  e.  ( M ... N ) )  ->  ( G `  k ) #  0 )
6 eqid 2139 . . . . . 6  |-  ( n  e.  ( ZZ>= `  M
)  |->  ( 1  / 
( G `  n
) ) )  =  ( n  e.  (
ZZ>= `  M )  |->  ( 1  /  ( G `
 n ) ) )
7 fveq2 5421 . . . . . . 7  |-  ( n  =  k  ->  ( G `  n )  =  ( G `  k ) )
87oveq2d 5790 . . . . . 6  |-  ( n  =  k  ->  (
1  /  ( G `
 n ) )  =  ( 1  / 
( G `  k
) ) )
9 simpr 109 . . . . . 6  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )
)  ->  k  e.  ( ZZ>= `  M )
)
102, 4recclapd 8548 . . . . . 6  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )
)  ->  ( 1  /  ( G `  k ) )  e.  CC )
116, 8, 9, 10fvmptd3 5514 . . . . 5  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )
)  ->  ( (
n  e.  ( ZZ>= `  M )  |->  ( 1  /  ( G `  n ) ) ) `
 k )  =  ( 1  /  ( G `  k )
) )
123, 11sylan2 284 . . . 4  |-  ( (
ph  /\  k  e.  ( M ... N ) )  ->  ( (
n  e.  ( ZZ>= `  M )  |->  ( 1  /  ( G `  n ) ) ) `
 k )  =  ( 1  /  ( G `  k )
) )
1311, 10eqeltrd 2216 . . . 4  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )
)  ->  ( (
n  e.  ( ZZ>= `  M )  |->  ( 1  /  ( G `  n ) ) ) `
 k )  e.  CC )
141, 2, 5, 12, 13prodfrecap 11322 . . 3  |-  ( ph  ->  (  seq M (  x.  ,  ( n  e.  ( ZZ>= `  M
)  |->  ( 1  / 
( G `  n
) ) ) ) `
 N )  =  ( 1  /  (  seq M (  x.  ,  G ) `  N
) ) )
1514oveq2d 5790 . 2  |-  ( ph  ->  ( (  seq M
(  x.  ,  F
) `  N )  x.  (  seq M (  x.  ,  ( n  e.  ( ZZ>= `  M
)  |->  ( 1  / 
( G `  n
) ) ) ) `
 N ) )  =  ( (  seq M (  x.  ,  F ) `  N
)  x.  ( 1  /  (  seq M
(  x.  ,  G
) `  N )
) ) )
16 prodfdivap.2 . . 3  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )
)  ->  ( F `  k )  e.  CC )
17 eleq1w 2200 . . . . . . . . 9  |-  ( k  =  n  ->  (
k  e.  ( ZZ>= `  M )  <->  n  e.  ( ZZ>= `  M )
) )
1817anbi2d 459 . . . . . . . 8  |-  ( k  =  n  ->  (
( ph  /\  k  e.  ( ZZ>= `  M )
)  <->  ( ph  /\  n  e.  ( ZZ>= `  M ) ) ) )
19 fveq2 5421 . . . . . . . . 9  |-  ( k  =  n  ->  ( G `  k )  =  ( G `  n ) )
2019eleq1d 2208 . . . . . . . 8  |-  ( k  =  n  ->  (
( G `  k
)  e.  CC  <->  ( G `  n )  e.  CC ) )
2118, 20imbi12d 233 . . . . . . 7  |-  ( k  =  n  ->  (
( ( ph  /\  k  e.  ( ZZ>= `  M ) )  -> 
( G `  k
)  e.  CC )  <-> 
( ( ph  /\  n  e.  ( ZZ>= `  M ) )  -> 
( G `  n
)  e.  CC ) ) )
2221, 2chvarvv 1880 . . . . . 6  |-  ( (
ph  /\  n  e.  ( ZZ>= `  M )
)  ->  ( G `  n )  e.  CC )
2319breq1d 3939 . . . . . . . 8  |-  ( k  =  n  ->  (
( G `  k
) #  0  <->  ( G `  n ) #  0 ) )
2418, 23imbi12d 233 . . . . . . 7  |-  ( k  =  n  ->  (
( ( ph  /\  k  e.  ( ZZ>= `  M ) )  -> 
( G `  k
) #  0 )  <->  ( ( ph  /\  n  e.  (
ZZ>= `  M ) )  ->  ( G `  n ) #  0 ) ) )
2524, 4chvarvv 1880 . . . . . 6  |-  ( (
ph  /\  n  e.  ( ZZ>= `  M )
)  ->  ( G `  n ) #  0 )
2622, 25recclapd 8548 . . . . 5  |-  ( (
ph  /\  n  e.  ( ZZ>= `  M )
)  ->  ( 1  /  ( G `  n ) )  e.  CC )
2726fmpttd 5575 . . . 4  |-  ( ph  ->  ( n  e.  (
ZZ>= `  M )  |->  ( 1  /  ( G `
 n ) ) ) : ( ZZ>= `  M ) --> CC )
2827ffvelrnda 5555 . . 3  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )
)  ->  ( (
n  e.  ( ZZ>= `  M )  |->  ( 1  /  ( G `  n ) ) ) `
 k )  e.  CC )
2916, 2, 4divrecapd 8560 . . . 4  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )
)  ->  ( ( F `  k )  /  ( G `  k ) )  =  ( ( F `  k )  x.  (
1  /  ( G `
 k ) ) ) )
30 prodfdivap.5 . . . 4  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )
)  ->  ( H `  k )  =  ( ( F `  k
)  /  ( G `
 k ) ) )
3111oveq2d 5790 . . . 4  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )
)  ->  ( ( F `  k )  x.  ( ( n  e.  ( ZZ>= `  M )  |->  ( 1  /  ( G `  n )
) ) `  k
) )  =  ( ( F `  k
)  x.  ( 1  /  ( G `  k ) ) ) )
3229, 30, 313eqtr4d 2182 . . 3  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )
)  ->  ( H `  k )  =  ( ( F `  k
)  x.  ( ( n  e.  ( ZZ>= `  M )  |->  ( 1  /  ( G `  n ) ) ) `
 k ) ) )
331, 16, 28, 32prod3fmul 11317 . 2  |-  ( ph  ->  (  seq M (  x.  ,  H ) `
 N )  =  ( (  seq M
(  x.  ,  F
) `  N )  x.  (  seq M (  x.  ,  ( n  e.  ( ZZ>= `  M
)  |->  ( 1  / 
( G `  n
) ) ) ) `
 N ) ) )
34 eqid 2139 . . . . 5  |-  ( ZZ>= `  M )  =  (
ZZ>= `  M )
35 eluzel2 9338 . . . . . 6  |-  ( N  e.  ( ZZ>= `  M
)  ->  M  e.  ZZ )
361, 35syl 14 . . . . 5  |-  ( ph  ->  M  e.  ZZ )
3734, 36, 16prodf 11314 . . . 4  |-  ( ph  ->  seq M (  x.  ,  F ) : ( ZZ>= `  M ) --> CC )
3837, 1ffvelrnd 5556 . . 3  |-  ( ph  ->  (  seq M (  x.  ,  F ) `
 N )  e.  CC )
3934, 36, 2prodf 11314 . . . 4  |-  ( ph  ->  seq M (  x.  ,  G ) : ( ZZ>= `  M ) --> CC )
4039, 1ffvelrnd 5556 . . 3  |-  ( ph  ->  (  seq M (  x.  ,  G ) `
 N )  e.  CC )
411, 2, 5prodfap0 11321 . . 3  |-  ( ph  ->  (  seq M (  x.  ,  G ) `
 N ) #  0 )
4238, 40, 41divrecapd 8560 . 2  |-  ( ph  ->  ( (  seq M
(  x.  ,  F
) `  N )  /  (  seq M (  x.  ,  G ) `
 N ) )  =  ( (  seq M (  x.  ,  F ) `  N
)  x.  ( 1  /  (  seq M
(  x.  ,  G
) `  N )
) ) )
4315, 33, 423eqtr4d 2182 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 103    = wceq 1331    e. wcel 1480   class class class wbr 3929    |-> cmpt 3989   ` cfv 5123  (class class class)co 5774   CCcc 7625   0cc0 7627   1c1 7628    x. cmul 7632   # cap 8350    / cdiv 8439   ZZcz 9061   ZZ>=cuz 9333   ...cfz 9797    seqcseq 10225
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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-iinf 4502  ax-cnex 7718  ax-resscn 7719  ax-1cn 7720  ax-1re 7721  ax-icn 7722  ax-addcl 7723  ax-addrcl 7724  ax-mulcl 7725  ax-mulrcl 7726  ax-addcom 7727  ax-mulcom 7728  ax-addass 7729  ax-mulass 7730  ax-distr 7731  ax-i2m1 7732  ax-0lt1 7733  ax-1rid 7734  ax-0id 7735  ax-rnegex 7736  ax-precex 7737  ax-cnre 7738  ax-pre-ltirr 7739  ax-pre-ltwlin 7740  ax-pre-lttrn 7741  ax-pre-apti 7742  ax-pre-ltadd 7743  ax-pre-mulgt0 7744  ax-pre-mulext 7745
This theorem depends on definitions:  df-bi 116  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-nel 2404  df-ral 2421  df-rex 2422  df-reu 2423  df-rmo 2424  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-id 4215  df-po 4218  df-iso 4219  df-iord 4288  df-on 4290  df-ilim 4291  df-suc 4293  df-iom 4505  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-riota 5730  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-recs 6202  df-frec 6288  df-pnf 7809  df-mnf 7810  df-xr 7811  df-ltxr 7812  df-le 7813  df-sub 7942  df-neg 7943  df-reap 8344  df-ap 8351  df-div 8440  df-inn 8728  df-n0 8985  df-z 9062  df-uz 9334  df-fz 9798  df-fzo 9927  df-seqfrec 10226
This theorem is referenced by: (None)
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