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

Theorem prodfdivap 11568
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 10034 . . . . 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 2187 . . . . . 6  |-  ( n  e.  ( ZZ>= `  M
)  |->  ( 1  / 
( G `  n
) ) )  =  ( n  e.  (
ZZ>= `  M )  |->  ( 1  /  ( G `
 n ) ) )
7 fveq2 5527 . . . . . . 7  |-  ( n  =  k  ->  ( G `  n )  =  ( G `  k ) )
87oveq2d 5904 . . . . . 6  |-  ( n  =  k  ->  (
1  /  ( G `
 n ) )  =  ( 1  / 
( G `  k
) ) )
9 simpr 110 . . . . . 6  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )
)  ->  k  e.  ( ZZ>= `  M )
)
102, 4recclapd 8751 . . . . . 6  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )
)  ->  ( 1  /  ( G `  k ) )  e.  CC )
116, 8, 9, 10fvmptd3 5622 . . . . 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 2264 . . . 4  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )
)  ->  ( (
n  e.  ( ZZ>= `  M )  |->  ( 1  /  ( G `  n ) ) ) `
 k )  e.  CC )
141, 2, 5, 12, 13prodfrecap 11567 . . 3  |-  ( ph  ->  (  seq M (  x.  ,  ( n  e.  ( ZZ>= `  M
)  |->  ( 1  / 
( G `  n
) ) ) ) `
 N )  =  ( 1  /  (  seq M (  x.  ,  G ) `  N
) ) )
1514oveq2d 5904 . 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 2248 . . . . . . . . 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 5527 . . . . . . . . 9  |-  ( k  =  n  ->  ( G `  k )  =  ( G `  n ) )
2019eleq1d 2256 . . . . . . . 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 1918 . . . . . 6  |-  ( (
ph  /\  n  e.  ( ZZ>= `  M )
)  ->  ( G `  n )  e.  CC )
2319breq1d 4025 . . . . . . . 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 1918 . . . . . 6  |-  ( (
ph  /\  n  e.  ( ZZ>= `  M )
)  ->  ( G `  n ) #  0 )
2622, 25recclapd 8751 . . . . 5  |-  ( (
ph  /\  n  e.  ( ZZ>= `  M )
)  ->  ( 1  /  ( G `  n ) )  e.  CC )
2726fmpttd 5684 . . . 4  |-  ( ph  ->  ( n  e.  (
ZZ>= `  M )  |->  ( 1  /  ( G `
 n ) ) ) : ( ZZ>= `  M ) --> CC )
2827ffvelcdmda 5664 . . 3  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )
)  ->  ( (
n  e.  ( ZZ>= `  M )  |->  ( 1  /  ( G `  n ) ) ) `
 k )  e.  CC )
2916, 2, 4divrecapd 8763 . . . 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 5904 . . . 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 2230 . . 3  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )
)  ->  ( H `  k )  =  ( ( F `  k
)  x.  ( ( n  e.  ( ZZ>= `  M )  |->  ( 1  /  ( G `  n ) ) ) `
 k ) ) )
331, 16, 28, 32prod3fmul 11562 . 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 2187 . . . . 5  |-  ( ZZ>= `  M )  =  (
ZZ>= `  M )
35 eluzel2 9546 . . . . . 6  |-  ( N  e.  ( ZZ>= `  M
)  ->  M  e.  ZZ )
361, 35syl 14 . . . . 5  |-  ( ph  ->  M  e.  ZZ )
3734, 36, 16prodf 11559 . . . 4  |-  ( ph  ->  seq M (  x.  ,  F ) : ( ZZ>= `  M ) --> CC )
3837, 1ffvelcdmd 5665 . . 3  |-  ( ph  ->  (  seq M (  x.  ,  F ) `
 N )  e.  CC )
3934, 36, 2prodf 11559 . . . 4  |-  ( ph  ->  seq M (  x.  ,  G ) : ( ZZ>= `  M ) --> CC )
4039, 1ffvelcdmd 5665 . . 3  |-  ( ph  ->  (  seq M (  x.  ,  G ) `
 N )  e.  CC )
411, 2, 5prodfap0 11566 . . 3  |-  ( ph  ->  (  seq M (  x.  ,  G ) `
 N ) #  0 )
4238, 40, 41divrecapd 8763 . 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 2230 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 1363    e. wcel 2158   class class class wbr 4015    |-> cmpt 4076   ` cfv 5228  (class class class)co 5888   CCcc 7822   0cc0 7824   1c1 7825    x. cmul 7829   # cap 8551    / cdiv 8642   ZZcz 9266   ZZ>=cuz 9541   ...cfz 10021    seqcseq 10458
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 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-coll 4130  ax-sep 4133  ax-nul 4141  ax-pow 4186  ax-pr 4221  ax-un 4445  ax-setind 4548  ax-iinf 4599  ax-cnex 7915  ax-resscn 7916  ax-1cn 7917  ax-1re 7918  ax-icn 7919  ax-addcl 7920  ax-addrcl 7921  ax-mulcl 7922  ax-mulrcl 7923  ax-addcom 7924  ax-mulcom 7925  ax-addass 7926  ax-mulass 7927  ax-distr 7928  ax-i2m1 7929  ax-0lt1 7930  ax-1rid 7931  ax-0id 7932  ax-rnegex 7933  ax-precex 7934  ax-cnre 7935  ax-pre-ltirr 7936  ax-pre-ltwlin 7937  ax-pre-lttrn 7938  ax-pre-apti 7939  ax-pre-ltadd 7940  ax-pre-mulgt0 7941  ax-pre-mulext 7942
This theorem depends on definitions:  df-bi 117  df-3or 980  df-3an 981  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ne 2358  df-nel 2453  df-ral 2470  df-rex 2471  df-reu 2472  df-rmo 2473  df-rab 2474  df-v 2751  df-sbc 2975  df-csb 3070  df-dif 3143  df-un 3145  df-in 3147  df-ss 3154  df-nul 3435  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-int 3857  df-iun 3900  df-br 4016  df-opab 4077  df-mpt 4078  df-tr 4114  df-id 4305  df-po 4308  df-iso 4309  df-iord 4378  df-on 4380  df-ilim 4381  df-suc 4383  df-iom 4602  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-rn 4649  df-res 4650  df-ima 4651  df-iota 5190  df-fun 5230  df-fn 5231  df-f 5232  df-f1 5233  df-fo 5234  df-f1o 5235  df-fv 5236  df-riota 5844  df-ov 5891  df-oprab 5892  df-mpo 5893  df-1st 6154  df-2nd 6155  df-recs 6319  df-frec 6405  df-pnf 8007  df-mnf 8008  df-xr 8009  df-ltxr 8010  df-le 8011  df-sub 8143  df-neg 8144  df-reap 8545  df-ap 8552  df-div 8643  df-inn 8933  df-n0 9190  df-z 9267  df-uz 9542  df-fz 10022  df-fzo 10156  df-seqfrec 10459
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator