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Mirrors > Home > MPE Home > Th. List > dfef2 | Structured version Visualization version GIF version |
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.) |
Ref | Expression |
---|---|
dfef2.1 | ⊢ (𝜑 → 𝐹 ∈ 𝑉) |
dfef2.2 | ⊢ (𝜑 → 𝐴 ∈ ℂ) |
dfef2.3 | ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐹‘𝑘) = ((1 + (𝐴 / 𝑘))↑𝑘)) |
Ref | Expression |
---|---|
dfef2 | ⊢ (𝜑 → 𝐹 ⇝ (exp‘𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfef2.2 | . . 3 ⊢ (𝜑 → 𝐴 ∈ ℂ) | |
2 | ax-1cn 10595 | . . . . . . . 8 ⊢ 1 ∈ ℂ | |
3 | simpl 485 | . . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℕ) → 𝐴 ∈ ℂ) | |
4 | nncn 11646 | . . . . . . . . . 10 ⊢ (𝑥 ∈ ℕ → 𝑥 ∈ ℂ) | |
5 | 4 | adantl 484 | . . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℕ) → 𝑥 ∈ ℂ) |
6 | nnne0 11672 | . . . . . . . . . 10 ⊢ (𝑥 ∈ ℕ → 𝑥 ≠ 0) | |
7 | 6 | adantl 484 | . . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℕ) → 𝑥 ≠ 0) |
8 | 3, 5, 7 | divcld 11416 | . . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℕ) → (𝐴 / 𝑥) ∈ ℂ) |
9 | addcl 10619 | . . . . . . . 8 ⊢ ((1 ∈ ℂ ∧ (𝐴 / 𝑥) ∈ ℂ) → (1 + (𝐴 / 𝑥)) ∈ ℂ) | |
10 | 2, 8, 9 | sylancr 589 | . . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℕ) → (1 + (𝐴 / 𝑥)) ∈ ℂ) |
11 | nnnn0 11905 | . . . . . . . 8 ⊢ (𝑥 ∈ ℕ → 𝑥 ∈ ℕ0) | |
12 | 11 | adantl 484 | . . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℕ) → 𝑥 ∈ ℕ0) |
13 | cxpexp 25251 | . . . . . . 7 ⊢ (((1 + (𝐴 / 𝑥)) ∈ ℂ ∧ 𝑥 ∈ ℕ0) → ((1 + (𝐴 / 𝑥))↑𝑐𝑥) = ((1 + (𝐴 / 𝑥))↑𝑥)) | |
14 | 10, 12, 13 | syl2anc 586 | . . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℕ) → ((1 + (𝐴 / 𝑥))↑𝑐𝑥) = ((1 + (𝐴 / 𝑥))↑𝑥)) |
15 | 14 | mpteq2dva 5161 | . . . . 5 ⊢ (𝐴 ∈ ℂ → (𝑥 ∈ ℕ ↦ ((1 + (𝐴 / 𝑥))↑𝑐𝑥)) = (𝑥 ∈ ℕ ↦ ((1 + (𝐴 / 𝑥))↑𝑥))) |
16 | nnrp 12401 | . . . . . . . 8 ⊢ (𝑥 ∈ ℕ → 𝑥 ∈ ℝ+) | |
17 | 16 | ssriv 3971 | . . . . . . 7 ⊢ ℕ ⊆ ℝ+ |
18 | 17 | a1i 11 | . . . . . 6 ⊢ (𝐴 ∈ ℂ → ℕ ⊆ ℝ+) |
19 | eqid 2821 | . . . . . . 7 ⊢ (0(ball‘(abs ∘ − ))(1 / ((abs‘𝐴) + 1))) = (0(ball‘(abs ∘ − ))(1 / ((abs‘𝐴) + 1))) | |
20 | 19 | efrlim 25547 | . . . . . 6 ⊢ (𝐴 ∈ ℂ → (𝑥 ∈ ℝ+ ↦ ((1 + (𝐴 / 𝑥))↑𝑐𝑥)) ⇝𝑟 (exp‘𝐴)) |
21 | 18, 20 | rlimres2 14918 | . . . . 5 ⊢ (𝐴 ∈ ℂ → (𝑥 ∈ ℕ ↦ ((1 + (𝐴 / 𝑥))↑𝑐𝑥)) ⇝𝑟 (exp‘𝐴)) |
22 | 15, 21 | eqbrtrrd 5090 | . . . 4 ⊢ (𝐴 ∈ ℂ → (𝑥 ∈ ℕ ↦ ((1 + (𝐴 / 𝑥))↑𝑥)) ⇝𝑟 (exp‘𝐴)) |
23 | nnuz 12282 | . . . . 5 ⊢ ℕ = (ℤ≥‘1) | |
24 | 1zzd 12014 | . . . . 5 ⊢ (𝐴 ∈ ℂ → 1 ∈ ℤ) | |
25 | 10, 12 | expcld 13511 | . . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℕ) → ((1 + (𝐴 / 𝑥))↑𝑥) ∈ ℂ) |
26 | 25 | fmpttd 6879 | . . . . 5 ⊢ (𝐴 ∈ ℂ → (𝑥 ∈ ℕ ↦ ((1 + (𝐴 / 𝑥))↑𝑥)):ℕ⟶ℂ) |
27 | 23, 24, 26 | rlimclim 14903 | . . . 4 ⊢ (𝐴 ∈ ℂ → ((𝑥 ∈ ℕ ↦ ((1 + (𝐴 / 𝑥))↑𝑥)) ⇝𝑟 (exp‘𝐴) ↔ (𝑥 ∈ ℕ ↦ ((1 + (𝐴 / 𝑥))↑𝑥)) ⇝ (exp‘𝐴))) |
28 | 22, 27 | mpbid 234 | . . 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 11644 | . . . . 5 ⊢ ℕ ∈ V | |
31 | 30 | mptex 6986 | . . . 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 12014 | . . 3 ⊢ (𝜑 → 1 ∈ ℤ) | |
35 | oveq2 7164 | . . . . . . . 8 ⊢ (𝑥 = 𝑘 → (𝐴 / 𝑥) = (𝐴 / 𝑘)) | |
36 | 35 | oveq2d 7172 | . . . . . . 7 ⊢ (𝑥 = 𝑘 → (1 + (𝐴 / 𝑥)) = (1 + (𝐴 / 𝑘))) |
37 | id 22 | . . . . . . 7 ⊢ (𝑥 = 𝑘 → 𝑥 = 𝑘) | |
38 | 36, 37 | oveq12d 7174 | . . . . . 6 ⊢ (𝑥 = 𝑘 → ((1 + (𝐴 / 𝑥))↑𝑥) = ((1 + (𝐴 / 𝑘))↑𝑘)) |
39 | eqid 2821 | . . . . . 6 ⊢ (𝑥 ∈ ℕ ↦ ((1 + (𝐴 / 𝑥))↑𝑥)) = (𝑥 ∈ ℕ ↦ ((1 + (𝐴 / 𝑥))↑𝑥)) | |
40 | ovex 7189 | . . . . . 6 ⊢ ((1 + (𝐴 / 𝑘))↑𝑘) ∈ V | |
41 | 38, 39, 40 | fvmpt 6768 | . . . . 5 ⊢ (𝑘 ∈ ℕ → ((𝑥 ∈ ℕ ↦ ((1 + (𝐴 / 𝑥))↑𝑥))‘𝑘) = ((1 + (𝐴 / 𝑘))↑𝑘)) |
42 | 41 | adantl 484 | . . . 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 2859 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((𝑥 ∈ ℕ ↦ ((1 + (𝐴 / 𝑥))↑𝑥))‘𝑘) = (𝐹‘𝑘)) |
45 | 23, 32, 33, 34, 44 | climeq 14924 | . 2 ⊢ (𝜑 → ((𝑥 ∈ ℕ ↦ ((1 + (𝐴 / 𝑥))↑𝑥)) ⇝ (exp‘𝐴) ↔ 𝐹 ⇝ (exp‘𝐴))) |
46 | 29, 45 | mpbid 234 | 1 ⊢ (𝜑 → 𝐹 ⇝ (exp‘𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ≠ wne 3016 Vcvv 3494 ⊆ wss 3936 class class class wbr 5066 ↦ cmpt 5146 ∘ ccom 5559 ‘cfv 6355 (class class class)co 7156 ℂcc 10535 0cc0 10537 1c1 10538 + caddc 10540 − cmin 10870 / cdiv 11297 ℕcn 11638 ℕ0cn0 11898 ℝ+crp 12390 ↑cexp 13430 abscabs 14593 ⇝ cli 14841 ⇝𝑟 crli 14842 expce 15415 ballcbl 20532 ↑𝑐ccxp 25139 |
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 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-inf2 9104 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 ax-pre-sup 10615 ax-addf 10616 ax-mulf 10617 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-fal 1550 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-iin 4922 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-se 5515 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-isom 6364 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-of 7409 df-om 7581 df-1st 7689 df-2nd 7690 df-supp 7831 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-2o 8103 df-oadd 8106 df-er 8289 df-map 8408 df-pm 8409 df-ixp 8462 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-fsupp 8834 df-fi 8875 df-sup 8906 df-inf 8907 df-oi 8974 df-card 9368 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-div 11298 df-nn 11639 df-2 11701 df-3 11702 df-4 11703 df-5 11704 df-6 11705 df-7 11706 df-8 11707 df-9 11708 df-n0 11899 df-z 11983 df-dec 12100 df-uz 12245 df-q 12350 df-rp 12391 df-xneg 12508 df-xadd 12509 df-xmul 12510 df-ioo 12743 df-ioc 12744 df-ico 12745 df-icc 12746 df-fz 12894 df-fzo 13035 df-fl 13163 df-mod 13239 df-seq 13371 df-exp 13431 df-fac 13635 df-bc 13664 df-hash 13692 df-shft 14426 df-cj 14458 df-re 14459 df-im 14460 df-sqrt 14594 df-abs 14595 df-limsup 14828 df-clim 14845 df-rlim 14846 df-sum 15043 df-ef 15421 df-sin 15423 df-cos 15424 df-tan 15425 df-pi 15426 df-struct 16485 df-ndx 16486 df-slot 16487 df-base 16489 df-sets 16490 df-ress 16491 df-plusg 16578 df-mulr 16579 df-starv 16580 df-sca 16581 df-vsca 16582 df-ip 16583 df-tset 16584 df-ple 16585 df-ds 16587 df-unif 16588 df-hom 16589 df-cco 16590 df-rest 16696 df-topn 16697 df-0g 16715 df-gsum 16716 df-topgen 16717 df-pt 16718 df-prds 16721 df-xrs 16775 df-qtop 16780 df-imas 16781 df-xps 16783 df-mre 16857 df-mrc 16858 df-acs 16860 df-mgm 17852 df-sgrp 17901 df-mnd 17912 df-submnd 17957 df-mulg 18225 df-cntz 18447 df-cmn 18908 df-psmet 20537 df-xmet 20538 df-met 20539 df-bl 20540 df-mopn 20541 df-fbas 20542 df-fg 20543 df-cnfld 20546 df-top 21502 df-topon 21519 df-topsp 21541 df-bases 21554 df-cld 21627 df-ntr 21628 df-cls 21629 df-nei 21706 df-lp 21744 df-perf 21745 df-cn 21835 df-cnp 21836 df-haus 21923 df-cmp 21995 df-tx 22170 df-hmeo 22363 df-fil 22454 df-fm 22546 df-flim 22547 df-flf 22548 df-xms 22930 df-ms 22931 df-tms 22932 df-cncf 23486 df-limc 24464 df-dv 24465 df-log 25140 df-cxp 25141 |
This theorem is referenced by: (None) |
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