clim2div - Metamath Proof Explorer

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Theorem clim2div 15841

Description: The limit of an infinite product with an initial segment removed. (Contributed by Scott Fenton, 20-Dec-2017.)

**Hypotheses**
Ref	Expression
clim2div.1	⊢ 𝑍 = (ℤ_≥‘𝑀)
clim2div.2	⊢ (𝜑 → 𝑁 ∈ 𝑍)
clim2div.3	⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℂ)
clim2div.4	⊢ (𝜑 → seq𝑀( · , 𝐹) ⇝ 𝐴)
clim2div.5	⊢ (𝜑 → (seq𝑀( · , 𝐹)‘𝑁) ≠ 0)

**Assertion**
Ref	Expression
clim2div	⊢ (𝜑 → seq(𝑁 + 1)( · , 𝐹) ⇝ (𝐴 / (seq𝑀( · , 𝐹)‘𝑁)))

Distinct variable groups: 𝑘,𝐹 𝜑,𝑘 𝑘,𝑀 𝑘,𝑁 𝑘,𝑍 Allowed substitution hint: 𝐴(𝑘)

Proof of Theorem clim2div Dummy variables 𝑗 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2726	. . 3 ⊢ (ℤ_≥‘(𝑁 + 1)) = (ℤ_≥‘(𝑁 + 1))
2		clim2div.2	. . . . 5 ⊢ (𝜑 → 𝑁 ∈ 𝑍)
3		eluzelz 12836	. . . . . 6 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → 𝑁 ∈ ℤ)
4		clim2div.1	. . . . . 6 ⊢ 𝑍 = (ℤ_≥‘𝑀)
5	3, 4	eleq2s 2845	. . . . 5 ⊢ (𝑁 ∈ 𝑍 → 𝑁 ∈ ℤ)
6	2, 5	syl 17	. . . 4 ⊢ (𝜑 → 𝑁 ∈ ℤ)
7	6	peano2zd 12673	. . 3 ⊢ (𝜑 → (𝑁 + 1) ∈ ℤ)
8		clim2div.4	. . 3 ⊢ (𝜑 → seq𝑀( · , 𝐹) ⇝ 𝐴)
9		eluzel2 12831	. . . . . . . 8 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → 𝑀 ∈ ℤ)
10	9, 4	eleq2s 2845	. . . . . . 7 ⊢ (𝑁 ∈ 𝑍 → 𝑀 ∈ ℤ)
11	2, 10	syl 17	. . . . . 6 ⊢ (𝜑 → 𝑀 ∈ ℤ)
12		clim2div.3	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℂ)
13	4, 11, 12	prodf 15839	. . . . 5 ⊢ (𝜑 → seq𝑀( · , 𝐹):𝑍⟶ℂ)
14	13, 2	ffvelcdmd 7081	. . . 4 ⊢ (𝜑 → (seq𝑀( · , 𝐹)‘𝑁) ∈ ℂ)
15		clim2div.5	. . . 4 ⊢ (𝜑 → (seq𝑀( · , 𝐹)‘𝑁) ≠ 0)
16	14, 15	reccld 11987	. . 3 ⊢ (𝜑 → (1 / (seq𝑀( · , 𝐹)‘𝑁)) ∈ ℂ)
17		seqex 13974	. . . 4 ⊢ seq(𝑁 + 1)( · , 𝐹) ∈ V
18	17	a1i 11	. . 3 ⊢ (𝜑 → seq(𝑁 + 1)( · , 𝐹) ∈ V)
19	2, 4	eleqtrdi 2837	. . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘𝑀))
20		peano2uz 12889	. . . . . . 7 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → (𝑁 + 1) ∈ (ℤ_≥‘𝑀))
21	19, 20	syl 17	. . . . . 6 ⊢ (𝜑 → (𝑁 + 1) ∈ (ℤ_≥‘𝑀))
22	21, 4	eleqtrrdi 2838	. . . . 5 ⊢ (𝜑 → (𝑁 + 1) ∈ 𝑍)
23	4	uztrn2 12845	. . . . 5 ⊢ (((𝑁 + 1) ∈ 𝑍 ∧ 𝑗 ∈ (ℤ_≥‘(𝑁 + 1))) → 𝑗 ∈ 𝑍)
24	22, 23	sylan 579	. . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘(𝑁 + 1))) → 𝑗 ∈ 𝑍)
25	13	ffvelcdmda 7080	. . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (seq𝑀( · , 𝐹)‘𝑗) ∈ ℂ)
26	24, 25	syldan 590	. . 3 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘(𝑁 + 1))) → (seq𝑀( · , 𝐹)‘𝑗) ∈ ℂ)
27		mulcl 11196	. . . . . . . 8 ⊢ ((𝑘 ∈ ℂ ∧ 𝑥 ∈ ℂ) → (𝑘 · 𝑥) ∈ ℂ)
28	27	adantl 481	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘(𝑁 + 1))) ∧ (𝑘 ∈ ℂ ∧ 𝑥 ∈ ℂ)) → (𝑘 · 𝑥) ∈ ℂ)
29		mulass 11200	. . . . . . . 8 ⊢ ((𝑘 ∈ ℂ ∧ 𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → ((𝑘 · 𝑥) · 𝑦) = (𝑘 · (𝑥 · 𝑦)))
30	29	adantl 481	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘(𝑁 + 1))) ∧ (𝑘 ∈ ℂ ∧ 𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ)) → ((𝑘 · 𝑥) · 𝑦) = (𝑘 · (𝑥 · 𝑦)))
31		simpr 484	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘(𝑁 + 1))) → 𝑗 ∈ (ℤ_≥‘(𝑁 + 1)))
32	19	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘(𝑁 + 1))) → 𝑁 ∈ (ℤ_≥‘𝑀))
33		elfzuz 13503	. . . . . . . . . 10 ⊢ (𝑘 ∈ (𝑀...𝑗) → 𝑘 ∈ (ℤ_≥‘𝑀))
34	33, 4	eleqtrrdi 2838	. . . . . . . . 9 ⊢ (𝑘 ∈ (𝑀...𝑗) → 𝑘 ∈ 𝑍)
35	34, 12	sylan2 592	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑗)) → (𝐹‘𝑘) ∈ ℂ)
36	35	adantlr 712	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘(𝑁 + 1))) ∧ 𝑘 ∈ (𝑀...𝑗)) → (𝐹‘𝑘) ∈ ℂ)
37	28, 30, 31, 32, 36	seqsplit 14006	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘(𝑁 + 1))) → (seq𝑀( · , 𝐹)‘𝑗) = ((seq𝑀( · , 𝐹)‘𝑁) · (seq(𝑁 + 1)( · , 𝐹)‘𝑗)))
38	37	eqcomd 2732	. . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘(𝑁 + 1))) → ((seq𝑀( · , 𝐹)‘𝑁) · (seq(𝑁 + 1)( · , 𝐹)‘𝑗)) = (seq𝑀( · , 𝐹)‘𝑗))
39	14	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘(𝑁 + 1))) → (seq𝑀( · , 𝐹)‘𝑁) ∈ ℂ)
40	4	uztrn2 12845	. . . . . . . . . 10 ⊢ (((𝑁 + 1) ∈ 𝑍 ∧ 𝑘 ∈ (ℤ_≥‘(𝑁 + 1))) → 𝑘 ∈ 𝑍)
41	22, 40	sylan 579	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘(𝑁 + 1))) → 𝑘 ∈ 𝑍)
42	41, 12	syldan 590	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘(𝑁 + 1))) → (𝐹‘𝑘) ∈ ℂ)
43	1, 7, 42	prodf 15839	. . . . . . 7 ⊢ (𝜑 → seq(𝑁 + 1)( · , 𝐹):(ℤ_≥‘(𝑁 + 1))⟶ℂ)
44	43	ffvelcdmda 7080	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘(𝑁 + 1))) → (seq(𝑁 + 1)( · , 𝐹)‘𝑗) ∈ ℂ)
45	15	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘(𝑁 + 1))) → (seq𝑀( · , 𝐹)‘𝑁) ≠ 0)
46	26, 39, 44, 45	divmuld 12016	. . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘(𝑁 + 1))) → (((seq𝑀( · , 𝐹)‘𝑗) / (seq𝑀( · , 𝐹)‘𝑁)) = (seq(𝑁 + 1)( · , 𝐹)‘𝑗) ↔ ((seq𝑀( · , 𝐹)‘𝑁) · (seq(𝑁 + 1)( · , 𝐹)‘𝑗)) = (seq𝑀( · , 𝐹)‘𝑗)))
47	38, 46	mpbird 257	. . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘(𝑁 + 1))) → ((seq𝑀( · , 𝐹)‘𝑗) / (seq𝑀( · , 𝐹)‘𝑁)) = (seq(𝑁 + 1)( · , 𝐹)‘𝑗))
48	26, 39, 45	divrec2d 11998	. . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘(𝑁 + 1))) → ((seq𝑀( · , 𝐹)‘𝑗) / (seq𝑀( · , 𝐹)‘𝑁)) = ((1 / (seq𝑀( · , 𝐹)‘𝑁)) · (seq𝑀( · , 𝐹)‘𝑗)))
49	47, 48	eqtr3d 2768	. . 3 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘(𝑁 + 1))) → (seq(𝑁 + 1)( · , 𝐹)‘𝑗) = ((1 / (seq𝑀( · , 𝐹)‘𝑁)) · (seq𝑀( · , 𝐹)‘𝑗)))
50	1, 7, 8, 16, 18, 26, 49	climmulc2 15587	. 2 ⊢ (𝜑 → seq(𝑁 + 1)( · , 𝐹) ⇝ ((1 / (seq𝑀( · , 𝐹)‘𝑁)) · 𝐴))
51		climcl 15449	. . . 4 ⊢ (seq𝑀( · , 𝐹) ⇝ 𝐴 → 𝐴 ∈ ℂ)
52	8, 51	syl 17	. . 3 ⊢ (𝜑 → 𝐴 ∈ ℂ)
53	52, 14, 15	divrec2d 11998	. 2 ⊢ (𝜑 → (𝐴 / (seq𝑀( · , 𝐹)‘𝑁)) = ((1 / (seq𝑀( · , 𝐹)‘𝑁)) · 𝐴))
54	50, 53	breqtrrd 5169	1 ⊢ (𝜑 → seq(𝑁 + 1)( · , 𝐹) ⇝ (𝐴 / (seq𝑀( · , 𝐹)‘𝑁)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ≠ wne 2934 Vcvv 3468 class class class wbr 5141 ‘cfv 6537 (class class class)co 7405 ℂcc 11110 0cc0 11112 1c1 11113 + caddc 11115 · cmul 11117 / cdiv 11875 ℤcz 12562 ℤ_≥cuz 12826 ...cfz 13490 seqcseq 13972 ⇝ cli 15434

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 ax-inf2 9638 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 ax-pre-sup 11190

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6294 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7853 df-1st 7974 df-2nd 7975 df-frecs 8267 df-wrecs 8298 df-recs 8372 df-rdg 8411 df-er 8705 df-en 8942 df-dom 8943 df-sdom 8944 df-sup 9439 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-div 11876 df-nn 12217 df-2 12279 df-3 12280 df-n0 12477 df-z 12563 df-uz 12827 df-rp 12981 df-fz 13491 df-seq 13973 df-exp 14033 df-cj 15052 df-re 15053 df-im 15054 df-sqrt 15188 df-abs 15189 df-clim 15438

This theorem is referenced by: ntrivcvgtail 15852

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