dfef2 - Metamath Proof Explorer

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Theorem dfef2 26911

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 11073	. . . . . . . 8 ⊢ 1 ∈ ℂ
3		simpl 482	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℕ) → 𝐴 ∈ ℂ)
4		nncn 12142	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℕ → 𝑥 ∈ ℂ)
5	4	adantl 481	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℕ) → 𝑥 ∈ ℂ)
6		nnne0 12168	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℕ → 𝑥 ≠ 0)
7	6	adantl 481	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℕ) → 𝑥 ≠ 0)
8	3, 5, 7	divcld 11906	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℕ) → (𝐴 / 𝑥) ∈ ℂ)
9		addcl 11097	. . . . . . . 8 ⊢ ((1 ∈ ℂ ∧ (𝐴 / 𝑥) ∈ ℂ) → (1 + (𝐴 / 𝑥)) ∈ ℂ)
10	2, 8, 9	sylancr 587	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℕ) → (1 + (𝐴 / 𝑥)) ∈ ℂ)
11		nnnn0 12397	. . . . . . . 8 ⊢ (𝑥 ∈ ℕ → 𝑥 ∈ ℕ₀)
12	11	adantl 481	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℕ) → 𝑥 ∈ ℕ₀)
13		cxpexp 26607	. . . . . . 7 ⊢ (((1 + (𝐴 / 𝑥)) ∈ ℂ ∧ 𝑥 ∈ ℕ₀) → ((1 + (𝐴 / 𝑥))↑_𝑐𝑥) = ((1 + (𝐴 / 𝑥))↑𝑥))
14	10, 12, 13	syl2anc 584	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℕ) → ((1 + (𝐴 / 𝑥))↑_𝑐𝑥) = ((1 + (𝐴 / 𝑥))↑𝑥))
15	14	mpteq2dva 5188	. . . . 5 ⊢ (𝐴 ∈ ℂ → (𝑥 ∈ ℕ ↦ ((1 + (𝐴 / 𝑥))↑_𝑐𝑥)) = (𝑥 ∈ ℕ ↦ ((1 + (𝐴 / 𝑥))↑𝑥)))
16		nnrp 12906	. . . . . . . 8 ⊢ (𝑥 ∈ ℕ → 𝑥 ∈ ℝ⁺)
17	16	ssriv 3934	. . . . . . 7 ⊢ ℕ ⊆ ℝ⁺
18	17	a1i 11	. . . . . 6 ⊢ (𝐴 ∈ ℂ → ℕ ⊆ ℝ⁺)
19		eqid 2733	. . . . . . 7 ⊢ (0(ball‘(abs ∘ − ))(1 / ((abs‘𝐴) + 1))) = (0(ball‘(abs ∘ − ))(1 / ((abs‘𝐴) + 1)))
20	19	efrlim 26909	. . . . . 6 ⊢ (𝐴 ∈ ℂ → (𝑥 ∈ ℝ⁺ ↦ ((1 + (𝐴 / 𝑥))↑_𝑐𝑥)) ⇝_𝑟 (exp‘𝐴))
21	18, 20	rlimres2 15472	. . . . 5 ⊢ (𝐴 ∈ ℂ → (𝑥 ∈ ℕ ↦ ((1 + (𝐴 / 𝑥))↑_𝑐𝑥)) ⇝_𝑟 (exp‘𝐴))
22	15, 21	eqbrtrrd 5119	. . . 4 ⊢ (𝐴 ∈ ℂ → (𝑥 ∈ ℕ ↦ ((1 + (𝐴 / 𝑥))↑𝑥)) ⇝_𝑟 (exp‘𝐴))
23		nnuz 12779	. . . . 5 ⊢ ℕ = (ℤ_≥‘1)
24		1zzd 12511	. . . . 5 ⊢ (𝐴 ∈ ℂ → 1 ∈ ℤ)
25	10, 12	expcld 14057	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℕ) → ((1 + (𝐴 / 𝑥))↑𝑥) ∈ ℂ)
26	25	fmpttd 7056	. . . . 5 ⊢ (𝐴 ∈ ℂ → (𝑥 ∈ ℕ ↦ ((1 + (𝐴 / 𝑥))↑𝑥)):ℕ⟶ℂ)
27	23, 24, 26	rlimclim 15457	. . . 4 ⊢ (𝐴 ∈ ℂ → ((𝑥 ∈ ℕ ↦ ((1 + (𝐴 / 𝑥))↑𝑥)) ⇝_𝑟 (exp‘𝐴) ↔ (𝑥 ∈ ℕ ↦ ((1 + (𝐴 / 𝑥))↑𝑥)) ⇝ (exp‘𝐴)))
28	22, 27	mpbid 232	. . 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 12140	. . . . 5 ⊢ ℕ ∈ V
31	30	mptex 7165	. . . 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 12511	. . 3 ⊢ (𝜑 → 1 ∈ ℤ)
35		oveq2 7362	. . . . . . . 8 ⊢ (𝑥 = 𝑘 → (𝐴 / 𝑥) = (𝐴 / 𝑘))
36	35	oveq2d 7370	. . . . . . 7 ⊢ (𝑥 = 𝑘 → (1 + (𝐴 / 𝑥)) = (1 + (𝐴 / 𝑘)))
37		id 22	. . . . . . 7 ⊢ (𝑥 = 𝑘 → 𝑥 = 𝑘)
38	36, 37	oveq12d 7372	. . . . . 6 ⊢ (𝑥 = 𝑘 → ((1 + (𝐴 / 𝑥))↑𝑥) = ((1 + (𝐴 / 𝑘))↑𝑘))
39		eqid 2733	. . . . . 6 ⊢ (𝑥 ∈ ℕ ↦ ((1 + (𝐴 / 𝑥))↑𝑥)) = (𝑥 ∈ ℕ ↦ ((1 + (𝐴 / 𝑥))↑𝑥))
40		ovex 7387	. . . . . 6 ⊢ ((1 + (𝐴 / 𝑘))↑𝑘) ∈ V
41	38, 39, 40	fvmpt 6937	. . . . 5 ⊢ (𝑘 ∈ ℕ → ((𝑥 ∈ ℕ ↦ ((1 + (𝐴 / 𝑥))↑𝑥))‘𝑘) = ((1 + (𝐴 / 𝑘))↑𝑘))
42	41	adantl 481	. . . 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 2771	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((𝑥 ∈ ℕ ↦ ((1 + (𝐴 / 𝑥))↑𝑥))‘𝑘) = (𝐹‘𝑘))
45	23, 32, 33, 34, 44	climeq 15478	. 2 ⊢ (𝜑 → ((𝑥 ∈ ℕ ↦ ((1 + (𝐴 / 𝑥))↑𝑥)) ⇝ (exp‘𝐴) ↔ 𝐹 ⇝ (exp‘𝐴)))
46	29, 45	mpbid 232	1 ⊢ (𝜑 → 𝐹 ⇝ (exp‘𝐴))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1541 ∈ wcel 2113 ≠ wne 2929 Vcvv 3437 ⊆ wss 3898 class class class wbr 5095 ↦ cmpt 5176 ∘ ccom 5625 ‘cfv 6488 (class class class)co 7354 ℂcc 11013 0cc0 11015 1c1 11016 + caddc 11018 − cmin 11353 / cdiv 11783 ℕcn 12134 ℕ₀cn0 12390 ℝ⁺crp 12894 ↑cexp 13972 abscabs 15145 ⇝ cli 15395 ⇝_𝑟 crli 15396 expce 15972 ballcbl 21282 ↑_𝑐ccxp 26494

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2182 ax-ext 2705 ax-rep 5221 ax-sep 5238 ax-nul 5248 ax-pow 5307 ax-pr 5374 ax-un 7676 ax-inf2 9540 ax-cnex 11071 ax-resscn 11072 ax-1cn 11073 ax-icn 11074 ax-addcl 11075 ax-addrcl 11076 ax-mulcl 11077 ax-mulrcl 11078 ax-mulcom 11079 ax-addass 11080 ax-mulass 11081 ax-distr 11082 ax-i2m1 11083 ax-1ne0 11084 ax-1rid 11085 ax-rnegex 11086 ax-rrecex 11087 ax-cnre 11088 ax-pre-lttri 11089 ax-pre-lttrn 11090 ax-pre-ltadd 11091 ax-pre-mulgt0 11092 ax-pre-sup 11093 ax-addf 11094

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2725 df-clel 2808 df-nfc 2882 df-ne 2930 df-nel 3034 df-ral 3049 df-rex 3058 df-rmo 3347 df-reu 3348 df-rab 3397 df-v 3439 df-sbc 3738 df-csb 3847 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-pss 3918 df-nul 4283 df-if 4477 df-pw 4553 df-sn 4578 df-pr 4580 df-tp 4582 df-op 4584 df-uni 4861 df-int 4900 df-iun 4945 df-iin 4946 df-br 5096 df-opab 5158 df-mpt 5177 df-tr 5203 df-id 5516 df-eprel 5521 df-po 5529 df-so 5530 df-fr 5574 df-se 5575 df-we 5576 df-xp 5627 df-rel 5628 df-cnv 5629 df-co 5630 df-dm 5631 df-rn 5632 df-res 5633 df-ima 5634 df-pred 6255 df-ord 6316 df-on 6317 df-lim 6318 df-suc 6319 df-iota 6444 df-fun 6490 df-fn 6491 df-f 6492 df-f1 6493 df-fo 6494 df-f1o 6495 df-fv 6496 df-isom 6497 df-riota 7311 df-ov 7357 df-oprab 7358 df-mpo 7359 df-of 7618 df-om 7805 df-1st 7929 df-2nd 7930 df-supp 8099 df-frecs 8219 df-wrecs 8250 df-recs 8299 df-rdg 8337 df-1o 8393 df-2o 8394 df-er 8630 df-map 8760 df-pm 8761 df-ixp 8830 df-en 8878 df-dom 8879 df-sdom 8880 df-fin 8881 df-fsupp 9255 df-fi 9304 df-sup 9335 df-inf 9336 df-oi 9405 df-card 9841 df-pnf 11157 df-mnf 11158 df-xr 11159 df-ltxr 11160 df-le 11161 df-sub 11355 df-neg 11356 df-div 11784 df-nn 12135 df-2 12197 df-3 12198 df-4 12199 df-5 12200 df-6 12201 df-7 12202 df-8 12203 df-9 12204 df-n0 12391 df-z 12478 df-dec 12597 df-uz 12741 df-q 12851 df-rp 12895 df-xneg 13015 df-xadd 13016 df-xmul 13017 df-ioo 13253 df-ioc 13254 df-ico 13255 df-icc 13256 df-fz 13412 df-fzo 13559 df-fl 13700 df-mod 13778 df-seq 13913 df-exp 13973 df-fac 14185 df-bc 14214 df-hash 14242 df-shft 14978 df-cj 15010 df-re 15011 df-im 15012 df-sqrt 15146 df-abs 15147 df-limsup 15382 df-clim 15399 df-rlim 15400 df-sum 15598 df-ef 15978 df-sin 15980 df-cos 15981 df-tan 15982 df-pi 15983 df-struct 17062 df-sets 17079 df-slot 17097 df-ndx 17109 df-base 17125 df-ress 17146 df-plusg 17178 df-mulr 17179 df-starv 17180 df-sca 17181 df-vsca 17182 df-ip 17183 df-tset 17184 df-ple 17185 df-ds 17187 df-unif 17188 df-hom 17189 df-cco 17190 df-rest 17330 df-topn 17331 df-0g 17349 df-gsum 17350 df-topgen 17351 df-pt 17352 df-prds 17355 df-xrs 17410 df-qtop 17415 df-imas 17416 df-xps 17418 df-mre 17492 df-mrc 17493 df-acs 17495 df-mgm 18552 df-sgrp 18631 df-mnd 18647 df-submnd 18696 df-mulg 18985 df-cntz 19233 df-cmn 19698 df-psmet 21287 df-xmet 21288 df-met 21289 df-bl 21290 df-mopn 21291 df-fbas 21292 df-fg 21293 df-cnfld 21296 df-top 22812 df-topon 22829 df-topsp 22851 df-bases 22864 df-cld 22937 df-ntr 22938 df-cls 22939 df-nei 23016 df-lp 23054 df-perf 23055 df-cn 23145 df-cnp 23146 df-haus 23233 df-cmp 23305 df-tx 23480 df-hmeo 23673 df-fil 23764 df-fm 23856 df-flim 23857 df-flf 23858 df-xms 24238 df-ms 24239 df-tms 24240 df-cncf 24801 df-limc 25797 df-dv 25798 df-log 26495 df-cxp 26496

This theorem is referenced by: (None)

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