ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  prodfdivap Unicode version

Theorem prodfdivap 11574
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 10040 . . . . 5  |-  ( k  e.  ( M ... N )  ->  k  e.  ( ZZ>= `  M )
)
4 prodfdivap.4 . . . . 5  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )
)  ->  ( G `  k ) #  0 )
53, 4sylan2 286 . . . 4  |-  ( (
ph  /\  k  e.  ( M ... N ) )  ->  ( G `  k ) #  0 )
6 eqid 2189 . . . . . 6  |-  ( n  e.  ( ZZ>= `  M
)  |->  ( 1  / 
( G `  n
) ) )  =  ( n  e.  (
ZZ>= `  M )  |->  ( 1  /  ( G `
 n ) ) )
7 fveq2 5530 . . . . . . 7  |-  ( n  =  k  ->  ( G `  n )  =  ( G `  k ) )
87oveq2d 5907 . . . . . 6  |-  ( n  =  k  ->  (
1  /  ( G `
 n ) )  =  ( 1  / 
( G `  k
) ) )
9 simpr 110 . . . . . 6  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )
)  ->  k  e.  ( ZZ>= `  M )
)
102, 4recclapd 8757 . . . . . 6  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )
)  ->  ( 1  /  ( G `  k ) )  e.  CC )
116, 8, 9, 10fvmptd3 5625 . . . . 5  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )
)  ->  ( (
n  e.  ( ZZ>= `  M )  |->  ( 1  /  ( G `  n ) ) ) `
 k )  =  ( 1  /  ( G `  k )
) )
123, 11sylan2 286 . . . 4  |-  ( (
ph  /\  k  e.  ( M ... N ) )  ->  ( (
n  e.  ( ZZ>= `  M )  |->  ( 1  /  ( G `  n ) ) ) `
 k )  =  ( 1  /  ( G `  k )
) )
1311, 10eqeltrd 2266 . . . 4  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )
)  ->  ( (
n  e.  ( ZZ>= `  M )  |->  ( 1  /  ( G `  n ) ) ) `
 k )  e.  CC )
141, 2, 5, 12, 13prodfrecap 11573 . . 3  |-  ( ph  ->  (  seq M (  x.  ,  ( n  e.  ( ZZ>= `  M
)  |->  ( 1  / 
( G `  n
) ) ) ) `
 N )  =  ( 1  /  (  seq M (  x.  ,  G ) `  N
) ) )
1514oveq2d 5907 . 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 2250 . . . . . . . . 9  |-  ( k  =  n  ->  (
k  e.  ( ZZ>= `  M )  <->  n  e.  ( ZZ>= `  M )
) )
1817anbi2d 464 . . . . . . . 8  |-  ( k  =  n  ->  (
( ph  /\  k  e.  ( ZZ>= `  M )
)  <->  ( ph  /\  n  e.  ( ZZ>= `  M ) ) ) )
19 fveq2 5530 . . . . . . . . 9  |-  ( k  =  n  ->  ( G `  k )  =  ( G `  n ) )
2019eleq1d 2258 . . . . . . . 8  |-  ( k  =  n  ->  (
( G `  k
)  e.  CC  <->  ( G `  n )  e.  CC ) )
2118, 20imbi12d 234 . . . . . . 7  |-  ( k  =  n  ->  (
( ( ph  /\  k  e.  ( ZZ>= `  M ) )  -> 
( G `  k
)  e.  CC )  <-> 
( ( ph  /\  n  e.  ( ZZ>= `  M ) )  -> 
( G `  n
)  e.  CC ) ) )
2221, 2chvarvv 1920 . . . . . 6  |-  ( (
ph  /\  n  e.  ( ZZ>= `  M )
)  ->  ( G `  n )  e.  CC )
2319breq1d 4028 . . . . . . . 8  |-  ( k  =  n  ->  (
( G `  k
) #  0  <->  ( G `  n ) #  0 ) )
2418, 23imbi12d 234 . . . . . . 7  |-  ( k  =  n  ->  (
( ( ph  /\  k  e.  ( ZZ>= `  M ) )  -> 
( G `  k
) #  0 )  <->  ( ( ph  /\  n  e.  (
ZZ>= `  M ) )  ->  ( G `  n ) #  0 ) ) )
2524, 4chvarvv 1920 . . . . . 6  |-  ( (
ph  /\  n  e.  ( ZZ>= `  M )
)  ->  ( G `  n ) #  0 )
2622, 25recclapd 8757 . . . . 5  |-  ( (
ph  /\  n  e.  ( ZZ>= `  M )
)  ->  ( 1  /  ( G `  n ) )  e.  CC )
2726fmpttd 5687 . . . 4  |-  ( ph  ->  ( n  e.  (
ZZ>= `  M )  |->  ( 1  /  ( G `
 n ) ) ) : ( ZZ>= `  M ) --> CC )
2827ffvelcdmda 5667 . . 3  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )
)  ->  ( (
n  e.  ( ZZ>= `  M )  |->  ( 1  /  ( G `  n ) ) ) `
 k )  e.  CC )
2916, 2, 4divrecapd 8769 . . . 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 5907 . . . 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 2232 . . 3  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )
)  ->  ( H `  k )  =  ( ( F `  k
)  x.  ( ( n  e.  ( ZZ>= `  M )  |->  ( 1  /  ( G `  n ) ) ) `
 k ) ) )
331, 16, 28, 32prod3fmul 11568 . 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 2189 . . . . 5  |-  ( ZZ>= `  M )  =  (
ZZ>= `  M )
35 eluzel2 9552 . . . . . 6  |-  ( N  e.  ( ZZ>= `  M
)  ->  M  e.  ZZ )
361, 35syl 14 . . . . 5  |-  ( ph  ->  M  e.  ZZ )
3734, 36, 16prodf 11565 . . . 4  |-  ( ph  ->  seq M (  x.  ,  F ) : ( ZZ>= `  M ) --> CC )
3837, 1ffvelcdmd 5668 . . 3  |-  ( ph  ->  (  seq M (  x.  ,  F ) `
 N )  e.  CC )
3934, 36, 2prodf 11565 . . . 4  |-  ( ph  ->  seq M (  x.  ,  G ) : ( ZZ>= `  M ) --> CC )
4039, 1ffvelcdmd 5668 . . 3  |-  ( ph  ->  (  seq M (  x.  ,  G ) `
 N )  e.  CC )
411, 2, 5prodfap0 11572 . . 3  |-  ( ph  ->  (  seq M (  x.  ,  G ) `
 N ) #  0 )
4238, 40, 41divrecapd 8769 . 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 2232 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 104    = wceq 1364    e. wcel 2160   class class class wbr 4018    |-> cmpt 4079   ` cfv 5231  (class class class)co 5891   CCcc 7828   0cc0 7830   1c1 7831    x. cmul 7835   # cap 8557    / cdiv 8648   ZZcz 9272   ZZ>=cuz 9547   ...cfz 10027    seqcseq 10464
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-coll 4133  ax-sep 4136  ax-nul 4144  ax-pow 4189  ax-pr 4224  ax-un 4448  ax-setind 4551  ax-iinf 4602  ax-cnex 7921  ax-resscn 7922  ax-1cn 7923  ax-1re 7924  ax-icn 7925  ax-addcl 7926  ax-addrcl 7927  ax-mulcl 7928  ax-mulrcl 7929  ax-addcom 7930  ax-mulcom 7931  ax-addass 7932  ax-mulass 7933  ax-distr 7934  ax-i2m1 7935  ax-0lt1 7936  ax-1rid 7937  ax-0id 7938  ax-rnegex 7939  ax-precex 7940  ax-cnre 7941  ax-pre-ltirr 7942  ax-pre-ltwlin 7943  ax-pre-lttrn 7944  ax-pre-apti 7945  ax-pre-ltadd 7946  ax-pre-mulgt0 7947  ax-pre-mulext 7948
This theorem depends on definitions:  df-bi 117  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-nel 2456  df-ral 2473  df-rex 2474  df-reu 2475  df-rmo 2476  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-tr 4117  df-id 4308  df-po 4311  df-iso 4312  df-iord 4381  df-on 4383  df-ilim 4384  df-suc 4386  df-iom 4605  df-xp 4647  df-rel 4648  df-cnv 4649  df-co 4650  df-dm 4651  df-rn 4652  df-res 4653  df-ima 4654  df-iota 5193  df-fun 5233  df-fn 5234  df-f 5235  df-f1 5236  df-fo 5237  df-f1o 5238  df-fv 5239  df-riota 5847  df-ov 5894  df-oprab 5895  df-mpo 5896  df-1st 6159  df-2nd 6160  df-recs 6324  df-frec 6410  df-pnf 8013  df-mnf 8014  df-xr 8015  df-ltxr 8016  df-le 8017  df-sub 8149  df-neg 8150  df-reap 8551  df-ap 8558  df-div 8649  df-inn 8939  df-n0 9196  df-z 9273  df-uz 9548  df-fz 10028  df-fzo 10162  df-seqfrec 10465
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator