dfef2 - Metamath Proof Explorer

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Theorem dfef2 24917

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 10195	. . . . . . . 8 ⊢ 1 ∈ ℂ
3		simpl 468	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℕ) → 𝐴 ∈ ℂ)
4		nncn 11229	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℕ → 𝑥 ∈ ℂ)
5	4	adantl 467	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℕ) → 𝑥 ∈ ℂ)
6		nnne0 11254	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℕ → 𝑥 ≠ 0)
7	6	adantl 467	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℕ) → 𝑥 ≠ 0)
8	3, 5, 7	divcld 11002	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℕ) → (𝐴 / 𝑥) ∈ ℂ)
9		addcl 10219	. . . . . . . 8 ⊢ ((1 ∈ ℂ ∧ (𝐴 / 𝑥) ∈ ℂ) → (1 + (𝐴 / 𝑥)) ∈ ℂ)
10	2, 8, 9	sylancr 567	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℕ) → (1 + (𝐴 / 𝑥)) ∈ ℂ)
11		nnnn0 11500	. . . . . . . 8 ⊢ (𝑥 ∈ ℕ → 𝑥 ∈ ℕ₀)
12	11	adantl 467	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℕ) → 𝑥 ∈ ℕ₀)
13		cxpexp 24634	. . . . . . 7 ⊢ (((1 + (𝐴 / 𝑥)) ∈ ℂ ∧ 𝑥 ∈ ℕ₀) → ((1 + (𝐴 / 𝑥))↑_𝑐𝑥) = ((1 + (𝐴 / 𝑥))↑𝑥))
14	10, 12, 13	syl2anc 565	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℕ) → ((1 + (𝐴 / 𝑥))↑_𝑐𝑥) = ((1 + (𝐴 / 𝑥))↑𝑥))
15	14	mpteq2dva 4876	. . . . 5 ⊢ (𝐴 ∈ ℂ → (𝑥 ∈ ℕ ↦ ((1 + (𝐴 / 𝑥))↑_𝑐𝑥)) = (𝑥 ∈ ℕ ↦ ((1 + (𝐴 / 𝑥))↑𝑥)))
16		nnrp 12044	. . . . . . . 8 ⊢ (𝑥 ∈ ℕ → 𝑥 ∈ ℝ⁺)
17	16	ssriv 3754	. . . . . . 7 ⊢ ℕ ⊆ ℝ⁺
18	17	a1i 11	. . . . . 6 ⊢ (𝐴 ∈ ℂ → ℕ ⊆ ℝ⁺)
19		eqid 2770	. . . . . . 7 ⊢ (0(ball‘(abs ∘ − ))(1 / ((abs‘𝐴) + 1))) = (0(ball‘(abs ∘ − ))(1 / ((abs‘𝐴) + 1)))
20	19	efrlim 24916	. . . . . 6 ⊢ (𝐴 ∈ ℂ → (𝑥 ∈ ℝ⁺ ↦ ((1 + (𝐴 / 𝑥))↑_𝑐𝑥)) ⇝_𝑟 (exp‘𝐴))
21	18, 20	rlimres2 14499	. . . . 5 ⊢ (𝐴 ∈ ℂ → (𝑥 ∈ ℕ ↦ ((1 + (𝐴 / 𝑥))↑_𝑐𝑥)) ⇝_𝑟 (exp‘𝐴))
22	15, 21	eqbrtrrd 4808	. . . 4 ⊢ (𝐴 ∈ ℂ → (𝑥 ∈ ℕ ↦ ((1 + (𝐴 / 𝑥))↑𝑥)) ⇝_𝑟 (exp‘𝐴))
23		nnuz 11924	. . . . 5 ⊢ ℕ = (ℤ_≥‘1)
24		1zzd 11609	. . . . 5 ⊢ (𝐴 ∈ ℂ → 1 ∈ ℤ)
25	10, 12	expcld 13214	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℕ) → ((1 + (𝐴 / 𝑥))↑𝑥) ∈ ℂ)
26		eqid 2770	. . . . . 6 ⊢ (𝑥 ∈ ℕ ↦ ((1 + (𝐴 / 𝑥))↑𝑥)) = (𝑥 ∈ ℕ ↦ ((1 + (𝐴 / 𝑥))↑𝑥))
27	25, 26	fmptd 6527	. . . . 5 ⊢ (𝐴 ∈ ℂ → (𝑥 ∈ ℕ ↦ ((1 + (𝐴 / 𝑥))↑𝑥)):ℕ⟶ℂ)
28	23, 24, 27	rlimclim 14484	. . . 4 ⊢ (𝐴 ∈ ℂ → ((𝑥 ∈ ℕ ↦ ((1 + (𝐴 / 𝑥))↑𝑥)) ⇝_𝑟 (exp‘𝐴) ↔ (𝑥 ∈ ℕ ↦ ((1 + (𝐴 / 𝑥))↑𝑥)) ⇝ (exp‘𝐴)))
29	22, 28	mpbid 222	. . 3 ⊢ (𝐴 ∈ ℂ → (𝑥 ∈ ℕ ↦ ((1 + (𝐴 / 𝑥))↑𝑥)) ⇝ (exp‘𝐴))
30	1, 29	syl 17	. 2 ⊢ (𝜑 → (𝑥 ∈ ℕ ↦ ((1 + (𝐴 / 𝑥))↑𝑥)) ⇝ (exp‘𝐴))
31		nnex 11227	. . . . 5 ⊢ ℕ ∈ V
32	31	mptex 6629	. . . 4 ⊢ (𝑥 ∈ ℕ ↦ ((1 + (𝐴 / 𝑥))↑𝑥)) ∈ V
33	32	a1i 11	. . 3 ⊢ (𝜑 → (𝑥 ∈ ℕ ↦ ((1 + (𝐴 / 𝑥))↑𝑥)) ∈ V)
34		dfef2.1	. . 3 ⊢ (𝜑 → 𝐹 ∈ 𝑉)
35		1zzd 11609	. . 3 ⊢ (𝜑 → 1 ∈ ℤ)
36		oveq2 6800	. . . . . . . 8 ⊢ (𝑥 = 𝑘 → (𝐴 / 𝑥) = (𝐴 / 𝑘))
37	36	oveq2d 6808	. . . . . . 7 ⊢ (𝑥 = 𝑘 → (1 + (𝐴 / 𝑥)) = (1 + (𝐴 / 𝑘)))
38		id 22	. . . . . . 7 ⊢ (𝑥 = 𝑘 → 𝑥 = 𝑘)
39	37, 38	oveq12d 6810	. . . . . 6 ⊢ (𝑥 = 𝑘 → ((1 + (𝐴 / 𝑥))↑𝑥) = ((1 + (𝐴 / 𝑘))↑𝑘))
40		ovex 6822	. . . . . 6 ⊢ ((1 + (𝐴 / 𝑘))↑𝑘) ∈ V
41	39, 26, 40	fvmpt 6424	. . . . 5 ⊢ (𝑘 ∈ ℕ → ((𝑥 ∈ ℕ ↦ ((1 + (𝐴 / 𝑥))↑𝑥))‘𝑘) = ((1 + (𝐴 / 𝑘))↑𝑘))
42	41	adantl 467	. . . 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 2807	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((𝑥 ∈ ℕ ↦ ((1 + (𝐴 / 𝑥))↑𝑥))‘𝑘) = (𝐹‘𝑘))
45	23, 33, 34, 35, 44	climeq 14505	. 2 ⊢ (𝜑 → ((𝑥 ∈ ℕ ↦ ((1 + (𝐴 / 𝑥))↑𝑥)) ⇝ (exp‘𝐴) ↔ 𝐹 ⇝ (exp‘𝐴)))
46	30, 45	mpbid 222	1 ⊢ (𝜑 → 𝐹 ⇝ (exp‘𝐴))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 382 = wceq 1630 ∈ wcel 2144 ≠ wne 2942 Vcvv 3349 ⊆ wss 3721 class class class wbr 4784 ↦ cmpt 4861 ∘ ccom 5253 ‘cfv 6031 (class class class)co 6792 ℂcc 10135 0cc0 10137 1c1 10138 + caddc 10140 − cmin 10467 / cdiv 10885 ℕcn 11221 ℕ₀cn0 11493 ℝ⁺crp 12034 ↑cexp 13066 abscabs 14181 ⇝ cli 14422 ⇝_𝑟 crli 14423 expce 14997 ballcbl 19947 ↑_𝑐ccxp 24522

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1869 ax-4 1884 ax-5 1990 ax-6 2056 ax-7 2092 ax-8 2146 ax-9 2153 ax-10 2173 ax-11 2189 ax-12 2202 ax-13 2407 ax-ext 2750 ax-rep 4902 ax-sep 4912 ax-nul 4920 ax-pow 4971 ax-pr 5034 ax-un 7095 ax-inf2 8701 ax-cnex 10193 ax-resscn 10194 ax-1cn 10195 ax-icn 10196 ax-addcl 10197 ax-addrcl 10198 ax-mulcl 10199 ax-mulrcl 10200 ax-mulcom 10201 ax-addass 10202 ax-mulass 10203 ax-distr 10204 ax-i2m1 10205 ax-1ne0 10206 ax-1rid 10207 ax-rnegex 10208 ax-rrecex 10209 ax-cnre 10210 ax-pre-lttri 10211 ax-pre-lttrn 10212 ax-pre-ltadd 10213 ax-pre-mulgt0 10214 ax-pre-sup 10215 ax-addf 10216 ax-mulf 10217

This theorem depends on definitions: df-bi 197 df-an 383 df-or 827 df-3or 1071 df-3an 1072 df-tru 1633 df-fal 1636 df-ex 1852 df-nf 1857 df-sb 2049 df-eu 2621 df-mo 2622 df-clab 2757 df-cleq 2763 df-clel 2766 df-nfc 2901 df-ne 2943 df-nel 3046 df-ral 3065 df-rex 3066 df-reu 3067 df-rmo 3068 df-rab 3069 df-v 3351 df-sbc 3586 df-csb 3681 df-dif 3724 df-un 3726 df-in 3728 df-ss 3735 df-pss 3737 df-nul 4062 df-if 4224 df-pw 4297 df-sn 4315 df-pr 4317 df-tp 4319 df-op 4321 df-uni 4573 df-int 4610 df-iun 4654 df-iin 4655 df-br 4785 df-opab 4845 df-mpt 4862 df-tr 4885 df-id 5157 df-eprel 5162 df-po 5170 df-so 5171 df-fr 5208 df-se 5209 df-we 5210 df-xp 5255 df-rel 5256 df-cnv 5257 df-co 5258 df-dm 5259 df-rn 5260 df-res 5261 df-ima 5262 df-pred 5823 df-ord 5869 df-on 5870 df-lim 5871 df-suc 5872 df-iota 5994 df-fun 6033 df-fn 6034 df-f 6035 df-f1 6036 df-fo 6037 df-f1o 6038 df-fv 6039 df-isom 6040 df-riota 6753 df-ov 6795 df-oprab 6796 df-mpt2 6797 df-of 7043 df-om 7212 df-1st 7314 df-2nd 7315 df-supp 7446 df-wrecs 7558 df-recs 7620 df-rdg 7658 df-1o 7712 df-2o 7713 df-oadd 7716 df-er 7895 df-map 8010 df-pm 8011 df-ixp 8062 df-en 8109 df-dom 8110 df-sdom 8111 df-fin 8112 df-fsupp 8431 df-fi 8472 df-sup 8503 df-inf 8504 df-oi 8570 df-card 8964 df-cda 9191 df-pnf 10277 df-mnf 10278 df-xr 10279 df-ltxr 10280 df-le 10281 df-sub 10469 df-neg 10470 df-div 10886 df-nn 11222 df-2 11280 df-3 11281 df-4 11282 df-5 11283 df-6 11284 df-7 11285 df-8 11286 df-9 11287 df-n0 11494 df-z 11579 df-dec 11695 df-uz 11888 df-q 11991 df-rp 12035 df-xneg 12150 df-xadd 12151 df-xmul 12152 df-ioo 12383 df-ioc 12384 df-ico 12385 df-icc 12386 df-fz 12533 df-fzo 12673 df-fl 12800 df-mod 12876 df-seq 13008 df-exp 13067 df-fac 13264 df-bc 13293 df-hash 13321 df-shft 14014 df-cj 14046 df-re 14047 df-im 14048 df-sqrt 14182 df-abs 14183 df-limsup 14409 df-clim 14426 df-rlim 14427 df-sum 14624 df-ef 15003 df-sin 15005 df-cos 15006 df-tan 15007 df-pi 15008 df-struct 16065 df-ndx 16066 df-slot 16067 df-base 16069 df-sets 16070 df-ress 16071 df-plusg 16161 df-mulr 16162 df-starv 16163 df-sca 16164 df-vsca 16165 df-ip 16166 df-tset 16167 df-ple 16168 df-ds 16171 df-unif 16172 df-hom 16173 df-cco 16174 df-rest 16290 df-topn 16291 df-0g 16309 df-gsum 16310 df-topgen 16311 df-pt 16312 df-prds 16315 df-xrs 16369 df-qtop 16374 df-imas 16375 df-xps 16377 df-mre 16453 df-mrc 16454 df-acs 16456 df-mgm 17449 df-sgrp 17491 df-mnd 17502 df-submnd 17543 df-mulg 17748 df-cntz 17956 df-cmn 18401 df-psmet 19952 df-xmet 19953 df-met 19954 df-bl 19955 df-mopn 19956 df-fbas 19957 df-fg 19958 df-cnfld 19961 df-top 20918 df-topon 20935 df-topsp 20957 df-bases 20970 df-cld 21043 df-ntr 21044 df-cls 21045 df-nei 21122 df-lp 21160 df-perf 21161 df-cn 21251 df-cnp 21252 df-haus 21339 df-cmp 21410 df-tx 21585 df-hmeo 21778 df-fil 21869 df-fm 21961 df-flim 21962 df-flf 21963 df-xms 22344 df-ms 22345 df-tms 22346 df-cncf 22900 df-limc 23849 df-dv 23850 df-log 24523 df-cxp 24524

This theorem is referenced by: (None)

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