dfef2 - Metamath Proof Explorer

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Theorem dfef2 25710

Description: The limit of the sequence (1 + 𝐴 / 𝑘)↑𝑘 as 𝑘 goes to +∞ is (exp‘𝐴). This is another common definition of e. (Contributed by Mario Carneiro, 1-Mar-2015.)

**Hypotheses**
Ref	Expression
dfef2.1	⊢ (𝜑 → 𝐹 ∈ 𝑉)
dfef2.2	⊢ (𝜑 → 𝐴 ∈ ℂ)
dfef2.3	⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐹‘𝑘) = ((1 + (𝐴 / 𝑘))↑𝑘))

**Assertion**
Ref	Expression
dfef2	⊢ (𝜑 → 𝐹 ⇝ (exp‘𝐴))

Distinct variable groups: 𝐴,𝑘 𝑘,𝐹 𝜑,𝑘 Allowed substitution hint: 𝑉(𝑘)

Proof of Theorem dfef2 Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfef2.2	. . 3 ⊢ (𝜑 → 𝐴 ∈ ℂ)
2		ax-1cn 10675	. . . . . . . 8 ⊢ 1 ∈ ℂ
3		simpl 486	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℕ) → 𝐴 ∈ ℂ)
4		nncn 11726	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℕ → 𝑥 ∈ ℂ)
5	4	adantl 485	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℕ) → 𝑥 ∈ ℂ)
6		nnne0 11752	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℕ → 𝑥 ≠ 0)
7	6	adantl 485	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℕ) → 𝑥 ≠ 0)
8	3, 5, 7	divcld 11496	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℕ) → (𝐴 / 𝑥) ∈ ℂ)
9		addcl 10699	. . . . . . . 8 ⊢ ((1 ∈ ℂ ∧ (𝐴 / 𝑥) ∈ ℂ) → (1 + (𝐴 / 𝑥)) ∈ ℂ)
10	2, 8, 9	sylancr 590	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℕ) → (1 + (𝐴 / 𝑥)) ∈ ℂ)
11		nnnn0 11985	. . . . . . . 8 ⊢ (𝑥 ∈ ℕ → 𝑥 ∈ ℕ₀)
12	11	adantl 485	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℕ) → 𝑥 ∈ ℕ₀)
13		cxpexp 25413	. . . . . . 7 ⊢ (((1 + (𝐴 / 𝑥)) ∈ ℂ ∧ 𝑥 ∈ ℕ₀) → ((1 + (𝐴 / 𝑥))↑_𝑐𝑥) = ((1 + (𝐴 / 𝑥))↑𝑥))
14	10, 12, 13	syl2anc 587	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℕ) → ((1 + (𝐴 / 𝑥))↑_𝑐𝑥) = ((1 + (𝐴 / 𝑥))↑𝑥))
15	14	mpteq2dva 5125	. . . . 5 ⊢ (𝐴 ∈ ℂ → (𝑥 ∈ ℕ ↦ ((1 + (𝐴 / 𝑥))↑_𝑐𝑥)) = (𝑥 ∈ ℕ ↦ ((1 + (𝐴 / 𝑥))↑𝑥)))
16		nnrp 12485	. . . . . . . 8 ⊢ (𝑥 ∈ ℕ → 𝑥 ∈ ℝ⁺)
17	16	ssriv 3881	. . . . . . 7 ⊢ ℕ ⊆ ℝ⁺
18	17	a1i 11	. . . . . 6 ⊢ (𝐴 ∈ ℂ → ℕ ⊆ ℝ⁺)
19		eqid 2738	. . . . . . 7 ⊢ (0(ball‘(abs ∘ − ))(1 / ((abs‘𝐴) + 1))) = (0(ball‘(abs ∘ − ))(1 / ((abs‘𝐴) + 1)))
20	19	efrlim 25709	. . . . . 6 ⊢ (𝐴 ∈ ℂ → (𝑥 ∈ ℝ⁺ ↦ ((1 + (𝐴 / 𝑥))↑_𝑐𝑥)) ⇝_𝑟 (exp‘𝐴))
21	18, 20	rlimres2 15010	. . . . 5 ⊢ (𝐴 ∈ ℂ → (𝑥 ∈ ℕ ↦ ((1 + (𝐴 / 𝑥))↑_𝑐𝑥)) ⇝_𝑟 (exp‘𝐴))
22	15, 21	eqbrtrrd 5054	. . . 4 ⊢ (𝐴 ∈ ℂ → (𝑥 ∈ ℕ ↦ ((1 + (𝐴 / 𝑥))↑𝑥)) ⇝_𝑟 (exp‘𝐴))
23		nnuz 12365	. . . . 5 ⊢ ℕ = (ℤ_≥‘1)
24		1zzd 12096	. . . . 5 ⊢ (𝐴 ∈ ℂ → 1 ∈ ℤ)
25	10, 12	expcld 13604	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℕ) → ((1 + (𝐴 / 𝑥))↑𝑥) ∈ ℂ)
26	25	fmpttd 6891	. . . . 5 ⊢ (𝐴 ∈ ℂ → (𝑥 ∈ ℕ ↦ ((1 + (𝐴 / 𝑥))↑𝑥)):ℕ⟶ℂ)
27	23, 24, 26	rlimclim 14995	. . . 4 ⊢ (𝐴 ∈ ℂ → ((𝑥 ∈ ℕ ↦ ((1 + (𝐴 / 𝑥))↑𝑥)) ⇝_𝑟 (exp‘𝐴) ↔ (𝑥 ∈ ℕ ↦ ((1 + (𝐴 / 𝑥))↑𝑥)) ⇝ (exp‘𝐴)))
28	22, 27	mpbid 235	. . 3 ⊢ (𝐴 ∈ ℂ → (𝑥 ∈ ℕ ↦ ((1 + (𝐴 / 𝑥))↑𝑥)) ⇝ (exp‘𝐴))
29	1, 28	syl 17	. 2 ⊢ (𝜑 → (𝑥 ∈ ℕ ↦ ((1 + (𝐴 / 𝑥))↑𝑥)) ⇝ (exp‘𝐴))
30		nnex 11724	. . . . 5 ⊢ ℕ ∈ V
31	30	mptex 6998	. . . 4 ⊢ (𝑥 ∈ ℕ ↦ ((1 + (𝐴 / 𝑥))↑𝑥)) ∈ V
32	31	a1i 11	. . 3 ⊢ (𝜑 → (𝑥 ∈ ℕ ↦ ((1 + (𝐴 / 𝑥))↑𝑥)) ∈ V)
33		dfef2.1	. . 3 ⊢ (𝜑 → 𝐹 ∈ 𝑉)
34		1zzd 12096	. . 3 ⊢ (𝜑 → 1 ∈ ℤ)
35		oveq2 7180	. . . . . . . 8 ⊢ (𝑥 = 𝑘 → (𝐴 / 𝑥) = (𝐴 / 𝑘))
36	35	oveq2d 7188	. . . . . . 7 ⊢ (𝑥 = 𝑘 → (1 + (𝐴 / 𝑥)) = (1 + (𝐴 / 𝑘)))
37		id 22	. . . . . . 7 ⊢ (𝑥 = 𝑘 → 𝑥 = 𝑘)
38	36, 37	oveq12d 7190	. . . . . 6 ⊢ (𝑥 = 𝑘 → ((1 + (𝐴 / 𝑥))↑𝑥) = ((1 + (𝐴 / 𝑘))↑𝑘))
39		eqid 2738	. . . . . 6 ⊢ (𝑥 ∈ ℕ ↦ ((1 + (𝐴 / 𝑥))↑𝑥)) = (𝑥 ∈ ℕ ↦ ((1 + (𝐴 / 𝑥))↑𝑥))
40		ovex 7205	. . . . . 6 ⊢ ((1 + (𝐴 / 𝑘))↑𝑘) ∈ V
41	38, 39, 40	fvmpt 6777	. . . . 5 ⊢ (𝑘 ∈ ℕ → ((𝑥 ∈ ℕ ↦ ((1 + (𝐴 / 𝑥))↑𝑥))‘𝑘) = ((1 + (𝐴 / 𝑘))↑𝑘))
42	41	adantl 485	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((𝑥 ∈ ℕ ↦ ((1 + (𝐴 / 𝑥))↑𝑥))‘𝑘) = ((1 + (𝐴 / 𝑘))↑𝑘))
43		dfef2.3	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐹‘𝑘) = ((1 + (𝐴 / 𝑘))↑𝑘))
44	42, 43	eqtr4d 2776	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((𝑥 ∈ ℕ ↦ ((1 + (𝐴 / 𝑥))↑𝑥))‘𝑘) = (𝐹‘𝑘))
45	23, 32, 33, 34, 44	climeq 15016	. 2 ⊢ (𝜑 → ((𝑥 ∈ ℕ ↦ ((1 + (𝐴 / 𝑥))↑𝑥)) ⇝ (exp‘𝐴) ↔ 𝐹 ⇝ (exp‘𝐴)))
46	29, 45	mpbid 235	1 ⊢ (𝜑 → 𝐹 ⇝ (exp‘𝐴))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 399 = wceq 1542 ∈ wcel 2114 ≠ wne 2934 Vcvv 3398 ⊆ wss 3843 class class class wbr 5030 ↦ cmpt 5110 ∘ ccom 5529 ‘cfv 6339 (class class class)co 7172 ℂcc 10615 0cc0 10617 1c1 10618 + caddc 10620 − cmin 10950 / cdiv 11377 ℕcn 11718 ℕ₀cn0 11978 ℝ⁺crp 12474 ↑cexp 13523 abscabs 14685 ⇝ cli 14933 ⇝_𝑟 crli 14934 expce 15509 ballcbl 20206 ↑_𝑐ccxp 25301

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2710 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5232 ax-pr 5296 ax-un 7481 ax-inf2 9179 ax-cnex 10673 ax-resscn 10674 ax-1cn 10675 ax-icn 10676 ax-addcl 10677 ax-addrcl 10678 ax-mulcl 10679 ax-mulrcl 10680 ax-mulcom 10681 ax-addass 10682 ax-mulass 10683 ax-distr 10684 ax-i2m1 10685 ax-1ne0 10686 ax-1rid 10687 ax-rnegex 10688 ax-rrecex 10689 ax-cnre 10690 ax-pre-lttri 10691 ax-pre-lttrn 10692 ax-pre-ltadd 10693 ax-pre-mulgt0 10694 ax-pre-sup 10695 ax-addf 10696 ax-mulf 10697

This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-nel 3039 df-ral 3058 df-rex 3059 df-reu 3060 df-rmo 3061 df-rab 3062 df-v 3400 df-sbc 3681 df-csb 3791 df-dif 3846 df-un 3848 df-in 3850 df-ss 3860 df-pss 3862 df-nul 4212 df-if 4415 df-pw 4490 df-sn 4517 df-pr 4519 df-tp 4521 df-op 4523 df-uni 4797 df-int 4837 df-iun 4883 df-iin 4884 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5429 df-eprel 5434 df-po 5442 df-so 5443 df-fr 5483 df-se 5484 df-we 5485 df-xp 5531 df-rel 5532 df-cnv 5533 df-co 5534 df-dm 5535 df-rn 5536 df-res 5537 df-ima 5538 df-pred 6129 df-ord 6175 df-on 6176 df-lim 6177 df-suc 6178 df-iota 6297 df-fun 6341 df-fn 6342 df-f 6343 df-f1 6344 df-fo 6345 df-f1o 6346 df-fv 6347 df-isom 6348 df-riota 7129 df-ov 7175 df-oprab 7176 df-mpo 7177 df-of 7427 df-om 7602 df-1st 7716 df-2nd 7717 df-supp 7859 df-wrecs 7978 df-recs 8039 df-rdg 8077 df-1o 8133 df-2o 8134 df-er 8322 df-map 8441 df-pm 8442 df-ixp 8510 df-en 8558 df-dom 8559 df-sdom 8560 df-fin 8561 df-fsupp 8909 df-fi 8950 df-sup 8981 df-inf 8982 df-oi 9049 df-card 9443 df-pnf 10757 df-mnf 10758 df-xr 10759 df-ltxr 10760 df-le 10761 df-sub 10952 df-neg 10953 df-div 11378 df-nn 11719 df-2 11781 df-3 11782 df-4 11783 df-5 11784 df-6 11785 df-7 11786 df-8 11787 df-9 11788 df-n0 11979 df-z 12065 df-dec 12182 df-uz 12327 df-q 12433 df-rp 12475 df-xneg 12592 df-xadd 12593 df-xmul 12594 df-ioo 12827 df-ioc 12828 df-ico 12829 df-icc 12830 df-fz 12984 df-fzo 13127 df-fl 13255 df-mod 13331 df-seq 13463 df-exp 13524 df-fac 13728 df-bc 13757 df-hash 13785 df-shft 14518 df-cj 14550 df-re 14551 df-im 14552 df-sqrt 14686 df-abs 14687 df-limsup 14920 df-clim 14937 df-rlim 14938 df-sum 15138 df-ef 15515 df-sin 15517 df-cos 15518 df-tan 15519 df-pi 15520 df-struct 16590 df-ndx 16591 df-slot 16592 df-base 16594 df-sets 16595 df-ress 16596 df-plusg 16683 df-mulr 16684 df-starv 16685 df-sca 16686 df-vsca 16687 df-ip 16688 df-tset 16689 df-ple 16690 df-ds 16692 df-unif 16693 df-hom 16694 df-cco 16695 df-rest 16801 df-topn 16802 df-0g 16820 df-gsum 16821 df-topgen 16822 df-pt 16823 df-prds 16826 df-xrs 16880 df-qtop 16885 df-imas 16886 df-xps 16888 df-mre 16962 df-mrc 16963 df-acs 16965 df-mgm 17970 df-sgrp 18019 df-mnd 18030 df-submnd 18075 df-mulg 18345 df-cntz 18567 df-cmn 19028 df-psmet 20211 df-xmet 20212 df-met 20213 df-bl 20214 df-mopn 20215 df-fbas 20216 df-fg 20217 df-cnfld 20220 df-top 21647 df-topon 21664 df-topsp 21686 df-bases 21699 df-cld 21772 df-ntr 21773 df-cls 21774 df-nei 21851 df-lp 21889 df-perf 21890 df-cn 21980 df-cnp 21981 df-haus 22068 df-cmp 22140 df-tx 22315 df-hmeo 22508 df-fil 22599 df-fm 22691 df-flim 22692 df-flf 22693 df-xms 23075 df-ms 23076 df-tms 23077 df-cncf 23632 df-limc 24620 df-dv 24621 df-log 25302 df-cxp 25303

This theorem is referenced by: (None)

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