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Theorem prodfdivap 11510
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 (𝜑𝑁 ∈ (ℤ𝑀))
prodfdivap.2 ((𝜑𝑘 ∈ (ℤ𝑀)) → (𝐹𝑘) ∈ ℂ)
prodfdivap.3 ((𝜑𝑘 ∈ (ℤ𝑀)) → (𝐺𝑘) ∈ ℂ)
prodfdivap.4 ((𝜑𝑘 ∈ (ℤ𝑀)) → (𝐺𝑘) # 0)
prodfdivap.5 ((𝜑𝑘 ∈ (ℤ𝑀)) → (𝐻𝑘) = ((𝐹𝑘) / (𝐺𝑘)))
Assertion
Ref Expression
prodfdivap (𝜑 → (seq𝑀( · , 𝐻)‘𝑁) = ((seq𝑀( · , 𝐹)‘𝑁) / (seq𝑀( · , 𝐺)‘𝑁)))
Distinct variable groups:   𝑘,𝐹   𝑘,𝐺   𝑘,𝐻   𝜑,𝑘   𝑘,𝑀   𝑘,𝑁

Proof of Theorem prodfdivap
Dummy variable 𝑛 is distinct from all other variables.
StepHypRef Expression
1 prodfdiv.1 . . . 4 (𝜑𝑁 ∈ (ℤ𝑀))
2 prodfdivap.3 . . . 4 ((𝜑𝑘 ∈ (ℤ𝑀)) → (𝐺𝑘) ∈ ℂ)
3 elfzuz 9977 . . . . 5 (𝑘 ∈ (𝑀...𝑁) → 𝑘 ∈ (ℤ𝑀))
4 prodfdivap.4 . . . . 5 ((𝜑𝑘 ∈ (ℤ𝑀)) → (𝐺𝑘) # 0)
53, 4sylan2 284 . . . 4 ((𝜑𝑘 ∈ (𝑀...𝑁)) → (𝐺𝑘) # 0)
6 eqid 2170 . . . . . 6 (𝑛 ∈ (ℤ𝑀) ↦ (1 / (𝐺𝑛))) = (𝑛 ∈ (ℤ𝑀) ↦ (1 / (𝐺𝑛)))
7 fveq2 5496 . . . . . . 7 (𝑛 = 𝑘 → (𝐺𝑛) = (𝐺𝑘))
87oveq2d 5869 . . . . . 6 (𝑛 = 𝑘 → (1 / (𝐺𝑛)) = (1 / (𝐺𝑘)))
9 simpr 109 . . . . . 6 ((𝜑𝑘 ∈ (ℤ𝑀)) → 𝑘 ∈ (ℤ𝑀))
102, 4recclapd 8698 . . . . . 6 ((𝜑𝑘 ∈ (ℤ𝑀)) → (1 / (𝐺𝑘)) ∈ ℂ)
116, 8, 9, 10fvmptd3 5589 . . . . 5 ((𝜑𝑘 ∈ (ℤ𝑀)) → ((𝑛 ∈ (ℤ𝑀) ↦ (1 / (𝐺𝑛)))‘𝑘) = (1 / (𝐺𝑘)))
123, 11sylan2 284 . . . 4 ((𝜑𝑘 ∈ (𝑀...𝑁)) → ((𝑛 ∈ (ℤ𝑀) ↦ (1 / (𝐺𝑛)))‘𝑘) = (1 / (𝐺𝑘)))
1311, 10eqeltrd 2247 . . . 4 ((𝜑𝑘 ∈ (ℤ𝑀)) → ((𝑛 ∈ (ℤ𝑀) ↦ (1 / (𝐺𝑛)))‘𝑘) ∈ ℂ)
141, 2, 5, 12, 13prodfrecap 11509 . . 3 (𝜑 → (seq𝑀( · , (𝑛 ∈ (ℤ𝑀) ↦ (1 / (𝐺𝑛))))‘𝑁) = (1 / (seq𝑀( · , 𝐺)‘𝑁)))
1514oveq2d 5869 . 2 (𝜑 → ((seq𝑀( · , 𝐹)‘𝑁) · (seq𝑀( · , (𝑛 ∈ (ℤ𝑀) ↦ (1 / (𝐺𝑛))))‘𝑁)) = ((seq𝑀( · , 𝐹)‘𝑁) · (1 / (seq𝑀( · , 𝐺)‘𝑁))))
16 prodfdivap.2 . . 3 ((𝜑𝑘 ∈ (ℤ𝑀)) → (𝐹𝑘) ∈ ℂ)
17 eleq1w 2231 . . . . . . . . 9 (𝑘 = 𝑛 → (𝑘 ∈ (ℤ𝑀) ↔ 𝑛 ∈ (ℤ𝑀)))
1817anbi2d 461 . . . . . . . 8 (𝑘 = 𝑛 → ((𝜑𝑘 ∈ (ℤ𝑀)) ↔ (𝜑𝑛 ∈ (ℤ𝑀))))
19 fveq2 5496 . . . . . . . . 9 (𝑘 = 𝑛 → (𝐺𝑘) = (𝐺𝑛))
2019eleq1d 2239 . . . . . . . 8 (𝑘 = 𝑛 → ((𝐺𝑘) ∈ ℂ ↔ (𝐺𝑛) ∈ ℂ))
2118, 20imbi12d 233 . . . . . . 7 (𝑘 = 𝑛 → (((𝜑𝑘 ∈ (ℤ𝑀)) → (𝐺𝑘) ∈ ℂ) ↔ ((𝜑𝑛 ∈ (ℤ𝑀)) → (𝐺𝑛) ∈ ℂ)))
2221, 2chvarvv 1901 . . . . . 6 ((𝜑𝑛 ∈ (ℤ𝑀)) → (𝐺𝑛) ∈ ℂ)
2319breq1d 3999 . . . . . . . 8 (𝑘 = 𝑛 → ((𝐺𝑘) # 0 ↔ (𝐺𝑛) # 0))
2418, 23imbi12d 233 . . . . . . 7 (𝑘 = 𝑛 → (((𝜑𝑘 ∈ (ℤ𝑀)) → (𝐺𝑘) # 0) ↔ ((𝜑𝑛 ∈ (ℤ𝑀)) → (𝐺𝑛) # 0)))
2524, 4chvarvv 1901 . . . . . 6 ((𝜑𝑛 ∈ (ℤ𝑀)) → (𝐺𝑛) # 0)
2622, 25recclapd 8698 . . . . 5 ((𝜑𝑛 ∈ (ℤ𝑀)) → (1 / (𝐺𝑛)) ∈ ℂ)
2726fmpttd 5651 . . . 4 (𝜑 → (𝑛 ∈ (ℤ𝑀) ↦ (1 / (𝐺𝑛))):(ℤ𝑀)⟶ℂ)
2827ffvelrnda 5631 . . 3 ((𝜑𝑘 ∈ (ℤ𝑀)) → ((𝑛 ∈ (ℤ𝑀) ↦ (1 / (𝐺𝑛)))‘𝑘) ∈ ℂ)
2916, 2, 4divrecapd 8710 . . . 4 ((𝜑𝑘 ∈ (ℤ𝑀)) → ((𝐹𝑘) / (𝐺𝑘)) = ((𝐹𝑘) · (1 / (𝐺𝑘))))
30 prodfdivap.5 . . . 4 ((𝜑𝑘 ∈ (ℤ𝑀)) → (𝐻𝑘) = ((𝐹𝑘) / (𝐺𝑘)))
3111oveq2d 5869 . . . 4 ((𝜑𝑘 ∈ (ℤ𝑀)) → ((𝐹𝑘) · ((𝑛 ∈ (ℤ𝑀) ↦ (1 / (𝐺𝑛)))‘𝑘)) = ((𝐹𝑘) · (1 / (𝐺𝑘))))
3229, 30, 313eqtr4d 2213 . . 3 ((𝜑𝑘 ∈ (ℤ𝑀)) → (𝐻𝑘) = ((𝐹𝑘) · ((𝑛 ∈ (ℤ𝑀) ↦ (1 / (𝐺𝑛)))‘𝑘)))
331, 16, 28, 32prod3fmul 11504 . 2 (𝜑 → (seq𝑀( · , 𝐻)‘𝑁) = ((seq𝑀( · , 𝐹)‘𝑁) · (seq𝑀( · , (𝑛 ∈ (ℤ𝑀) ↦ (1 / (𝐺𝑛))))‘𝑁)))
34 eqid 2170 . . . . 5 (ℤ𝑀) = (ℤ𝑀)
35 eluzel2 9492 . . . . . 6 (𝑁 ∈ (ℤ𝑀) → 𝑀 ∈ ℤ)
361, 35syl 14 . . . . 5 (𝜑𝑀 ∈ ℤ)
3734, 36, 16prodf 11501 . . . 4 (𝜑 → seq𝑀( · , 𝐹):(ℤ𝑀)⟶ℂ)
3837, 1ffvelrnd 5632 . . 3 (𝜑 → (seq𝑀( · , 𝐹)‘𝑁) ∈ ℂ)
3934, 36, 2prodf 11501 . . . 4 (𝜑 → seq𝑀( · , 𝐺):(ℤ𝑀)⟶ℂ)
4039, 1ffvelrnd 5632 . . 3 (𝜑 → (seq𝑀( · , 𝐺)‘𝑁) ∈ ℂ)
411, 2, 5prodfap0 11508 . . 3 (𝜑 → (seq𝑀( · , 𝐺)‘𝑁) # 0)
4238, 40, 41divrecapd 8710 . 2 (𝜑 → ((seq𝑀( · , 𝐹)‘𝑁) / (seq𝑀( · , 𝐺)‘𝑁)) = ((seq𝑀( · , 𝐹)‘𝑁) · (1 / (seq𝑀( · , 𝐺)‘𝑁))))
4315, 33, 423eqtr4d 2213 1 (𝜑 → (seq𝑀( · , 𝐻)‘𝑁) = ((seq𝑀( · , 𝐹)‘𝑁) / (seq𝑀( · , 𝐺)‘𝑁)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103   = wceq 1348  wcel 2141   class class class wbr 3989  cmpt 4050  cfv 5198  (class class class)co 5853  cc 7772  0cc0 7774  1c1 7775   · cmul 7779   # cap 8500   / cdiv 8589  cz 9212  cuz 9487  ...cfz 9965  seqcseq 10401
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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-iinf 4572  ax-cnex 7865  ax-resscn 7866  ax-1cn 7867  ax-1re 7868  ax-icn 7869  ax-addcl 7870  ax-addrcl 7871  ax-mulcl 7872  ax-mulrcl 7873  ax-addcom 7874  ax-mulcom 7875  ax-addass 7876  ax-mulass 7877  ax-distr 7878  ax-i2m1 7879  ax-0lt1 7880  ax-1rid 7881  ax-0id 7882  ax-rnegex 7883  ax-precex 7884  ax-cnre 7885  ax-pre-ltirr 7886  ax-pre-ltwlin 7887  ax-pre-lttrn 7888  ax-pre-apti 7889  ax-pre-ltadd 7890  ax-pre-mulgt0 7891  ax-pre-mulext 7892
This theorem depends on definitions:  df-bi 116  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-reu 2455  df-rmo 2456  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-tr 4088  df-id 4278  df-po 4281  df-iso 4282  df-iord 4351  df-on 4353  df-ilim 4354  df-suc 4356  df-iom 4575  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-riota 5809  df-ov 5856  df-oprab 5857  df-mpo 5858  df-1st 6119  df-2nd 6120  df-recs 6284  df-frec 6370  df-pnf 7956  df-mnf 7957  df-xr 7958  df-ltxr 7959  df-le 7960  df-sub 8092  df-neg 8093  df-reap 8494  df-ap 8501  df-div 8590  df-inn 8879  df-n0 9136  df-z 9213  df-uz 9488  df-fz 9966  df-fzo 10099  df-seqfrec 10402
This theorem is referenced by: (None)
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