Theorem prodfdivap 12333

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	⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀))
prodfdivap.2	⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑘) ∈ ℂ)
prodfdivap.3	⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → (𝐺‘𝑘) ∈ ℂ)
prodfdivap.4	⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → (𝐺‘𝑘) # 0)
prodfdivap.5	⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → (𝐻‘𝑘) = ((𝐹‘𝑘) / (𝐺‘𝑘)))

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.

Step	Hyp	Ref	Expression
1		prodfdiv.1	. . . 4 ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀))
2		prodfdivap.3	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → (𝐺‘𝑘) ∈ ℂ)
3		elfzuz 10435	. . . . 5 ⊢ (𝑘 ∈ (𝑀...𝑁) → 𝑘 ∈ (ℤ≥‘𝑀))
4		prodfdivap.4	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → (𝐺‘𝑘) # 0)
5	3, 4	sylan2 286	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝐺‘𝑘) # 0)
6		eqid 2238	. . . . . 6 ⊢ (𝑛 ∈ (ℤ≥‘𝑀) ↦ (1 / (𝐺‘𝑛))) = (𝑛 ∈ (ℤ≥‘𝑀) ↦ (1 / (𝐺‘𝑛)))
7		fveq2 5695	. . . . . . 7 ⊢ (𝑛 = 𝑘 → (𝐺‘𝑛) = (𝐺‘𝑘))
8	7	oveq2d 6101	. . . . . 6 ⊢ (𝑛 = 𝑘 → (1 / (𝐺‘𝑛)) = (1 / (𝐺‘𝑘)))
9		simpr 110	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → 𝑘 ∈ (ℤ≥‘𝑀))
10	2, 4	recclapd 9114	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → (1 / (𝐺‘𝑘)) ∈ ℂ)
11	6, 8, 9, 10	fvmptd3 5799	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → ((𝑛 ∈ (ℤ≥‘𝑀) ↦ (1 / (𝐺‘𝑛)))‘𝑘) = (1 / (𝐺‘𝑘)))
12	3, 11	sylan2 286	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → ((𝑛 ∈ (ℤ≥‘𝑀) ↦ (1 / (𝐺‘𝑛)))‘𝑘) = (1 / (𝐺‘𝑘)))
13	11, 10	eqeltrd 2315	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → ((𝑛 ∈ (ℤ≥‘𝑀) ↦ (1 / (𝐺‘𝑛)))‘𝑘) ∈ ℂ)
14	1, 2, 5, 12, 13	prodfrecap 12332	. . 3 ⊢ (𝜑 → (seq𝑀( · , (𝑛 ∈ (ℤ≥‘𝑀) ↦ (1 / (𝐺‘𝑛))))‘𝑁) = (1 / (seq𝑀( · , 𝐺)‘𝑁)))
15	14	oveq2d 6101	. 2 ⊢ (𝜑 → ((seq𝑀( · , 𝐹)‘𝑁) · (seq𝑀( · , (𝑛 ∈ (ℤ≥‘𝑀) ↦ (1 / (𝐺‘𝑛))))‘𝑁)) = ((seq𝑀( · , 𝐹)‘𝑁) · (1 / (seq𝑀( · , 𝐺)‘𝑁))))
16		prodfdivap.2	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑘) ∈ ℂ)
17		eleq1w 2299	. . . . . . . . 9 ⊢ (𝑘 = 𝑛 → (𝑘 ∈ (ℤ≥‘𝑀) ↔ 𝑛 ∈ (ℤ≥‘𝑀)))
18	17	anbi2d 468	. . . . . . . 8 ⊢ (𝑘 = 𝑛 → ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) ↔ (𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀))))
19		fveq2 5695	. . . . . . . . 9 ⊢ (𝑘 = 𝑛 → (𝐺‘𝑘) = (𝐺‘𝑛))
20	19	eleq1d 2307	. . . . . . . 8 ⊢ (𝑘 = 𝑛 → ((𝐺‘𝑘) ∈ ℂ ↔ (𝐺‘𝑛) ∈ ℂ))
21	18, 20	imbi12d 234	. . . . . . 7 ⊢ (𝑘 = 𝑛 → (((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → (𝐺‘𝑘) ∈ ℂ) ↔ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀)) → (𝐺‘𝑛) ∈ ℂ)))
22	21, 2	chvarvv 1964	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀)) → (𝐺‘𝑛) ∈ ℂ)
23	19	breq1d 4140	. . . . . . . 8 ⊢ (𝑘 = 𝑛 → ((𝐺‘𝑘) # 0 ↔ (𝐺‘𝑛) # 0))
24	18, 23	imbi12d 234	. . . . . . 7 ⊢ (𝑘 = 𝑛 → (((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → (𝐺‘𝑘) # 0) ↔ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀)) → (𝐺‘𝑛) # 0)))
25	24, 4	chvarvv 1964	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀)) → (𝐺‘𝑛) # 0)
26	22, 25	recclapd 9114	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀)) → (1 / (𝐺‘𝑛)) ∈ ℂ)
27	26	fmpttd 5863	. . . 4 ⊢ (𝜑 → (𝑛 ∈ (ℤ≥‘𝑀) ↦ (1 / (𝐺‘𝑛))):(ℤ≥‘𝑀)⟶ℂ)
28	27	ffvelcdmda 5843	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → ((𝑛 ∈ (ℤ≥‘𝑀) ↦ (1 / (𝐺‘𝑛)))‘𝑘) ∈ ℂ)
29	16, 2, 4	divrecapd 9126	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → ((𝐹‘𝑘) / (𝐺‘𝑘)) = ((𝐹‘𝑘) · (1 / (𝐺‘𝑘))))
30		prodfdivap.5	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → (𝐻‘𝑘) = ((𝐹‘𝑘) / (𝐺‘𝑘)))
31	11	oveq2d 6101	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → ((𝐹‘𝑘) · ((𝑛 ∈ (ℤ≥‘𝑀) ↦ (1 / (𝐺‘𝑛)))‘𝑘)) = ((𝐹‘𝑘) · (1 / (𝐺‘𝑘))))
32	29, 30, 31	3eqtr4d 2281	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → (𝐻‘𝑘) = ((𝐹‘𝑘) · ((𝑛 ∈ (ℤ≥‘𝑀) ↦ (1 / (𝐺‘𝑛)))‘𝑘)))
33	1, 16, 28, 32	prod3fmul 12327	. 2 ⊢ (𝜑 → (seq𝑀( · , 𝐻)‘𝑁) = ((seq𝑀( · , 𝐹)‘𝑁) · (seq𝑀( · , (𝑛 ∈ (ℤ≥‘𝑀) ↦ (1 / (𝐺‘𝑛))))‘𝑁)))
34		eqid 2238	. . . . 5 ⊢ (ℤ≥‘𝑀) = (ℤ≥‘𝑀)
35		eluzel2 9936	. . . . . 6 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑀 ∈ ℤ)
36	1, 35	syl 14	. . . . 5 ⊢ (𝜑 → 𝑀 ∈ ℤ)
37	34, 36, 16	prodf 12324	. . . 4 ⊢ (𝜑 → seq𝑀( · , 𝐹):(ℤ≥‘𝑀)⟶ℂ)
38	37, 1	ffvelcdmd 5844	. . 3 ⊢ (𝜑 → (seq𝑀( · , 𝐹)‘𝑁) ∈ ℂ)
39	34, 36, 2	prodf 12324	. . . 4 ⊢ (𝜑 → seq𝑀( · , 𝐺):(ℤ≥‘𝑀)⟶ℂ)
40	39, 1	ffvelcdmd 5844	. . 3 ⊢ (𝜑 → (seq𝑀( · , 𝐺)‘𝑁) ∈ ℂ)
41	1, 2, 5	prodfap0 12331	. . 3 ⊢ (𝜑 → (seq𝑀( · , 𝐺)‘𝑁) # 0)
42	38, 40, 41	divrecapd 9126	. 2 ⊢ (𝜑 → ((seq𝑀( · , 𝐹)‘𝑁) / (seq𝑀( · , 𝐺)‘𝑁)) = ((seq𝑀( · , 𝐹)‘𝑁) · (1 / (seq𝑀( · , 𝐺)‘𝑁))))
43	15, 33, 42	3eqtr4d 2281	1 ⊢ (𝜑 → (seq𝑀( · , 𝐻)‘𝑁) = ((seq𝑀( · , 𝐹)‘𝑁) / (seq𝑀( · , 𝐺)‘𝑁)))

This proof depends on syntax axioms:   → wi 4   ∧ wa 104   = wceq 1402   ∈ wcel 2209   class class class wbr 4130   ↦ cmpt 4192  ‘cfv 5377  (class class class)co 6085  ℂcc 8178  0cc0 8180  1c1 8181   · cmul 8185   # cap 8912   / cdiv 9005  ℤcz 9649  ℤ_≥cuz 9931  ...cfz 10422  seqcseq 10899

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4246  ax-sep 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-iinf 4735  ax-cnex 8271  ax-resscn 8272  ax-1cn 8273  ax-1re 8274  ax-icn 8275  ax-addcl 8276  ax-addrcl 8277  ax-mulcl 8278  ax-mulrcl 8279  ax-addcom 8280  ax-mulcom 8281  ax-addass 8282  ax-mulass 8283  ax-distr 8284  ax-i2m1 8285  ax-0lt1 8286  ax-1rid 8287  ax-0id 8288  ax-rnegex 8289  ax-precex 8290  ax-cnre 8291  ax-pre-ltirr 8292  ax-pre-ltwlin 8293  ax-pre-lttrn 8294  ax-pre-apti 8295  ax-pre-ltadd 8296  ax-pre-mulgt0 8297  ax-pre-mulext 8298

This proof depends on definitions:  df-bi 117  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-tr 4230  df-id 4438  df-po 4441  df-iso 4442  df-iord 4511  df-on 4513  df-ilim 4514  df-suc 4516  df-iom 4738  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-recs 6576  df-frec 6662  df-pnf 8363  df-mnf 8364  df-xr 8365  df-ltxr 8366  df-le 8367  df-sub 8501  df-neg 8502  df-reap 8906  df-ap 8913  df-div 9006  df-inn 9308  df-n0 9569  df-z 9650  df-uz 9932  df-fz 10423  df-fzo 10561  df-seqfrec 10900

This theorem is used by: (None)