dfef2 - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > dfef2		Structured version Visualization version GIF version

Theorem dfef2 26916

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 11197	. . . . . . . 8 ⊢ 1 ∈ ℂ
3		simpl 482	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℕ) → 𝐴 ∈ ℂ)
4		nncn 12251	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℕ → 𝑥 ∈ ℂ)
5	4	adantl 481	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℕ) → 𝑥 ∈ ℂ)
6		nnne0 12277	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℕ → 𝑥 ≠ 0)
7	6	adantl 481	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℕ) → 𝑥 ≠ 0)
8	3, 5, 7	divcld 12021	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℕ) → (𝐴 / 𝑥) ∈ ℂ)
9		addcl 11221	. . . . . . . 8 ⊢ ((1 ∈ ℂ ∧ (𝐴 / 𝑥) ∈ ℂ) → (1 + (𝐴 / 𝑥)) ∈ ℂ)
10	2, 8, 9	sylancr 586	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℕ) → (1 + (𝐴 / 𝑥)) ∈ ℂ)
11		nnnn0 12510	. . . . . . . 8 ⊢ (𝑥 ∈ ℕ → 𝑥 ∈ ℕ₀)
12	11	adantl 481	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℕ) → 𝑥 ∈ ℕ₀)
13		cxpexp 26615	. . . . . . 7 ⊢ (((1 + (𝐴 / 𝑥)) ∈ ℂ ∧ 𝑥 ∈ ℕ₀) → ((1 + (𝐴 / 𝑥))↑_𝑐𝑥) = ((1 + (𝐴 / 𝑥))↑𝑥))
14	10, 12, 13	syl2anc 583	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℕ) → ((1 + (𝐴 / 𝑥))↑_𝑐𝑥) = ((1 + (𝐴 / 𝑥))↑𝑥))
15	14	mpteq2dva 5248	. . . . 5 ⊢ (𝐴 ∈ ℂ → (𝑥 ∈ ℕ ↦ ((1 + (𝐴 / 𝑥))↑_𝑐𝑥)) = (𝑥 ∈ ℕ ↦ ((1 + (𝐴 / 𝑥))↑𝑥)))
16		nnrp 13018	. . . . . . . 8 ⊢ (𝑥 ∈ ℕ → 𝑥 ∈ ℝ⁺)
17	16	ssriv 3984	. . . . . . 7 ⊢ ℕ ⊆ ℝ⁺
18	17	a1i 11	. . . . . 6 ⊢ (𝐴 ∈ ℂ → ℕ ⊆ ℝ⁺)
19		eqid 2728	. . . . . . 7 ⊢ (0(ball‘(abs ∘ − ))(1 / ((abs‘𝐴) + 1))) = (0(ball‘(abs ∘ − ))(1 / ((abs‘𝐴) + 1)))
20	19	efrlim 26914	. . . . . 6 ⊢ (𝐴 ∈ ℂ → (𝑥 ∈ ℝ⁺ ↦ ((1 + (𝐴 / 𝑥))↑_𝑐𝑥)) ⇝_𝑟 (exp‘𝐴))
21	18, 20	rlimres2 15538	. . . . 5 ⊢ (𝐴 ∈ ℂ → (𝑥 ∈ ℕ ↦ ((1 + (𝐴 / 𝑥))↑_𝑐𝑥)) ⇝_𝑟 (exp‘𝐴))
22	15, 21	eqbrtrrd 5172	. . . 4 ⊢ (𝐴 ∈ ℂ → (𝑥 ∈ ℕ ↦ ((1 + (𝐴 / 𝑥))↑𝑥)) ⇝_𝑟 (exp‘𝐴))
23		nnuz 12896	. . . . 5 ⊢ ℕ = (ℤ_≥‘1)
24		1zzd 12624	. . . . 5 ⊢ (𝐴 ∈ ℂ → 1 ∈ ℤ)
25	10, 12	expcld 14143	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℕ) → ((1 + (𝐴 / 𝑥))↑𝑥) ∈ ℂ)
26	25	fmpttd 7125	. . . . 5 ⊢ (𝐴 ∈ ℂ → (𝑥 ∈ ℕ ↦ ((1 + (𝐴 / 𝑥))↑𝑥)):ℕ⟶ℂ)
27	23, 24, 26	rlimclim 15523	. . . 4 ⊢ (𝐴 ∈ ℂ → ((𝑥 ∈ ℕ ↦ ((1 + (𝐴 / 𝑥))↑𝑥)) ⇝_𝑟 (exp‘𝐴) ↔ (𝑥 ∈ ℕ ↦ ((1 + (𝐴 / 𝑥))↑𝑥)) ⇝ (exp‘𝐴)))
28	22, 27	mpbid 231	. . 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 12249	. . . . 5 ⊢ ℕ ∈ V
31	30	mptex 7235	. . . 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 12624	. . 3 ⊢ (𝜑 → 1 ∈ ℤ)
35		oveq2 7428	. . . . . . . 8 ⊢ (𝑥 = 𝑘 → (𝐴 / 𝑥) = (𝐴 / 𝑘))
36	35	oveq2d 7436	. . . . . . 7 ⊢ (𝑥 = 𝑘 → (1 + (𝐴 / 𝑥)) = (1 + (𝐴 / 𝑘)))
37		id 22	. . . . . . 7 ⊢ (𝑥 = 𝑘 → 𝑥 = 𝑘)
38	36, 37	oveq12d 7438	. . . . . 6 ⊢ (𝑥 = 𝑘 → ((1 + (𝐴 / 𝑥))↑𝑥) = ((1 + (𝐴 / 𝑘))↑𝑘))
39		eqid 2728	. . . . . 6 ⊢ (𝑥 ∈ ℕ ↦ ((1 + (𝐴 / 𝑥))↑𝑥)) = (𝑥 ∈ ℕ ↦ ((1 + (𝐴 / 𝑥))↑𝑥))
40		ovex 7453	. . . . . 6 ⊢ ((1 + (𝐴 / 𝑘))↑𝑘) ∈ V
41	38, 39, 40	fvmpt 7005	. . . . 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 15544	. 2 ⊢ (𝜑 → ((𝑥 ∈ ℕ ↦ ((1 + (𝐴 / 𝑥))↑𝑥)) ⇝ (exp‘𝐴) ↔ 𝐹 ⇝ (exp‘𝐴)))
46	29, 45	mpbid 231	1 ⊢ (𝜑 → 𝐹 ⇝ (exp‘𝐴))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1534 ∈ wcel 2099 ≠ wne 2937 Vcvv 3471 ⊆ wss 3947 class class class wbr 5148 ↦ cmpt 5231 ∘ ccom 5682 ‘cfv 6548 (class class class)co 7420 ℂcc 11137 0cc0 11139 1c1 11140 + caddc 11142 − cmin 11475 / cdiv 11902 ℕcn 12243 ℕ₀cn0 12503 ℝ⁺crp 13007 ↑cexp 14059 abscabs 15214 ⇝ cli 15461 ⇝_𝑟 crli 15462 expce 16038 ballcbl 21266 ↑_𝑐ccxp 26502

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-inf2 9665 ax-cnex 11195 ax-resscn 11196 ax-1cn 11197 ax-icn 11198 ax-addcl 11199 ax-addrcl 11200 ax-mulcl 11201 ax-mulrcl 11202 ax-mulcom 11203 ax-addass 11204 ax-mulass 11205 ax-distr 11206 ax-i2m1 11207 ax-1ne0 11208 ax-1rid 11209 ax-rnegex 11210 ax-rrecex 11211 ax-cnre 11212 ax-pre-lttri 11213 ax-pre-lttrn 11214 ax-pre-ltadd 11215 ax-pre-mulgt0 11216 ax-pre-sup 11217 ax-addf 11218

This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3373 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-tp 4634 df-op 4636 df-uni 4909 df-int 4950 df-iun 4998 df-iin 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-se 5634 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-isom 6557 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-of 7685 df-om 7871 df-1st 7993 df-2nd 7994 df-supp 8166 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-1o 8487 df-2o 8488 df-er 8725 df-map 8847 df-pm 8848 df-ixp 8917 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-fsupp 9387 df-fi 9435 df-sup 9466 df-inf 9467 df-oi 9534 df-card 9963 df-pnf 11281 df-mnf 11282 df-xr 11283 df-ltxr 11284 df-le 11285 df-sub 11477 df-neg 11478 df-div 11903 df-nn 12244 df-2 12306 df-3 12307 df-4 12308 df-5 12309 df-6 12310 df-7 12311 df-8 12312 df-9 12313 df-n0 12504 df-z 12590 df-dec 12709 df-uz 12854 df-q 12964 df-rp 13008 df-xneg 13125 df-xadd 13126 df-xmul 13127 df-ioo 13361 df-ioc 13362 df-ico 13363 df-icc 13364 df-fz 13518 df-fzo 13661 df-fl 13790 df-mod 13868 df-seq 14000 df-exp 14060 df-fac 14266 df-bc 14295 df-hash 14323 df-shft 15047 df-cj 15079 df-re 15080 df-im 15081 df-sqrt 15215 df-abs 15216 df-limsup 15448 df-clim 15465 df-rlim 15466 df-sum 15666 df-ef 16044 df-sin 16046 df-cos 16047 df-tan 16048 df-pi 16049 df-struct 17116 df-sets 17133 df-slot 17151 df-ndx 17163 df-base 17181 df-ress 17210 df-plusg 17246 df-mulr 17247 df-starv 17248 df-sca 17249 df-vsca 17250 df-ip 17251 df-tset 17252 df-ple 17253 df-ds 17255 df-unif 17256 df-hom 17257 df-cco 17258 df-rest 17404 df-topn 17405 df-0g 17423 df-gsum 17424 df-topgen 17425 df-pt 17426 df-prds 17429 df-xrs 17484 df-qtop 17489 df-imas 17490 df-xps 17492 df-mre 17566 df-mrc 17567 df-acs 17569 df-mgm 18600 df-sgrp 18679 df-mnd 18695 df-submnd 18741 df-mulg 19024 df-cntz 19268 df-cmn 19737 df-psmet 21271 df-xmet 21272 df-met 21273 df-bl 21274 df-mopn 21275 df-fbas 21276 df-fg 21277 df-cnfld 21280 df-top 22809 df-topon 22826 df-topsp 22848 df-bases 22862 df-cld 22936 df-ntr 22937 df-cls 22938 df-nei 23015 df-lp 23053 df-perf 23054 df-cn 23144 df-cnp 23145 df-haus 23232 df-cmp 23304 df-tx 23479 df-hmeo 23672 df-fil 23763 df-fm 23855 df-flim 23856 df-flf 23857 df-xms 24239 df-ms 24240 df-tms 24241 df-cncf 24811 df-limc 25808 df-dv 25809 df-log 26503 df-cxp 26504

This theorem is referenced by: (None)

W3C validator