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Mirrors > Home > MPE Home > Th. List > eff | Structured version Visualization version GIF version |
Description: Domain and codomain of the exponential function. (Contributed by Paul Chapman, 22-Oct-2007.) (Proof shortened by Mario Carneiro, 28-Apr-2014.) |
Ref | Expression |
---|---|
eff | ⊢ exp:ℂ⟶ℂ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-ef 15506 | . 2 ⊢ exp = (𝑥 ∈ ℂ ↦ Σ𝑘 ∈ ℕ0 ((𝑥↑𝑘) / (!‘𝑘))) | |
2 | nn0uz 12355 | . . 3 ⊢ ℕ0 = (ℤ≥‘0) | |
3 | 0zd 12067 | . . 3 ⊢ (𝑥 ∈ ℂ → 0 ∈ ℤ) | |
4 | eqid 2738 | . . . . 5 ⊢ (𝑛 ∈ ℕ0 ↦ ((𝑥↑𝑛) / (!‘𝑛))) = (𝑛 ∈ ℕ0 ↦ ((𝑥↑𝑛) / (!‘𝑛))) | |
5 | 4 | eftval 15515 | . . . 4 ⊢ (𝑘 ∈ ℕ0 → ((𝑛 ∈ ℕ0 ↦ ((𝑥↑𝑛) / (!‘𝑛)))‘𝑘) = ((𝑥↑𝑘) / (!‘𝑘))) |
6 | 5 | adantl 485 | . . 3 ⊢ ((𝑥 ∈ ℂ ∧ 𝑘 ∈ ℕ0) → ((𝑛 ∈ ℕ0 ↦ ((𝑥↑𝑛) / (!‘𝑛)))‘𝑘) = ((𝑥↑𝑘) / (!‘𝑘))) |
7 | eftcl 15512 | . . 3 ⊢ ((𝑥 ∈ ℂ ∧ 𝑘 ∈ ℕ0) → ((𝑥↑𝑘) / (!‘𝑘)) ∈ ℂ) | |
8 | 4 | efcllem 15516 | . . 3 ⊢ (𝑥 ∈ ℂ → seq0( + , (𝑛 ∈ ℕ0 ↦ ((𝑥↑𝑛) / (!‘𝑛)))) ∈ dom ⇝ ) |
9 | 2, 3, 6, 7, 8 | isumcl 15202 | . 2 ⊢ (𝑥 ∈ ℂ → Σ𝑘 ∈ ℕ0 ((𝑥↑𝑘) / (!‘𝑘)) ∈ ℂ) |
10 | 1, 9 | fmpti 6880 | 1 ⊢ exp:ℂ⟶ℂ |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1542 ∈ wcel 2113 ↦ cmpt 5107 ⟶wf 6329 ‘cfv 6333 (class class class)co 7164 ℂcc 10606 0cc0 10608 / cdiv 11368 ℕ0cn0 11969 ↑cexp 13514 !cfa 13718 Σcsu 15128 expce 15500 |
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 1916 ax-6 1974 ax-7 2019 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2161 ax-12 2178 ax-ext 2710 ax-rep 5151 ax-sep 5164 ax-nul 5171 ax-pow 5229 ax-pr 5293 ax-un 7473 ax-inf2 9170 ax-cnex 10664 ax-resscn 10665 ax-1cn 10666 ax-icn 10667 ax-addcl 10668 ax-addrcl 10669 ax-mulcl 10670 ax-mulrcl 10671 ax-mulcom 10672 ax-addass 10673 ax-mulass 10674 ax-distr 10675 ax-i2m1 10676 ax-1ne0 10677 ax-1rid 10678 ax-rnegex 10679 ax-rrecex 10680 ax-cnre 10681 ax-pre-lttri 10682 ax-pre-lttrn 10683 ax-pre-ltadd 10684 ax-pre-mulgt0 10685 ax-pre-sup 10686 |
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 2074 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 3399 df-sbc 3680 df-csb 3789 df-dif 3844 df-un 3846 df-in 3848 df-ss 3858 df-pss 3860 df-nul 4210 df-if 4412 df-pw 4487 df-sn 4514 df-pr 4516 df-tp 4518 df-op 4520 df-uni 4794 df-int 4834 df-iun 4880 df-br 5028 df-opab 5090 df-mpt 5108 df-tr 5134 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-se 5479 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6123 df-ord 6169 df-on 6170 df-lim 6171 df-suc 6172 df-iota 6291 df-fun 6335 df-fn 6336 df-f 6337 df-f1 6338 df-fo 6339 df-f1o 6340 df-fv 6341 df-isom 6342 df-riota 7121 df-ov 7167 df-oprab 7168 df-mpo 7169 df-om 7594 df-1st 7707 df-2nd 7708 df-wrecs 7969 df-recs 8030 df-rdg 8068 df-1o 8124 df-er 8313 df-pm 8433 df-en 8549 df-dom 8550 df-sdom 8551 df-fin 8552 df-sup 8972 df-inf 8973 df-oi 9040 df-card 9434 df-pnf 10748 df-mnf 10749 df-xr 10750 df-ltxr 10751 df-le 10752 df-sub 10943 df-neg 10944 df-div 11369 df-nn 11710 df-2 11772 df-3 11773 df-n0 11970 df-z 12056 df-uz 12318 df-rp 12466 df-ico 12820 df-fz 12975 df-fzo 13118 df-fl 13246 df-seq 13454 df-exp 13515 df-fac 13719 df-hash 13776 df-shft 14509 df-cj 14541 df-re 14542 df-im 14543 df-sqrt 14677 df-abs 14678 df-limsup 14911 df-clim 14928 df-rlim 14929 df-sum 15129 df-ef 15506 |
This theorem is referenced by: efcl 15521 eff2 15537 reeff1 15558 dveflem 24723 dvef 24724 dvsincos 24725 efcn 25182 efcvx 25188 pige3ALT 25256 efabl 25286 efsubm 25287 dvrelog 25372 dvlog 25386 efopn 25393 dvcxp1 25473 dvcxp2 25474 dvcncxp1 25476 gamf 25772 gamcvg2lem 25788 itgexpif 32148 iprodefisumlem 33269 seff 41449 dvsef 41472 expgrowthi 41473 expgrowth 41475 |
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