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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 15017 | . 2 ⊢ exp = (𝑥 ∈ ℂ ↦ Σ𝑘 ∈ ℕ0 ((𝑥↑𝑘) / (!‘𝑘))) | |
2 | nn0uz 11935 | . . 3 ⊢ ℕ0 = (ℤ≥‘0) | |
3 | 0zd 11601 | . . 3 ⊢ (𝑥 ∈ ℂ → 0 ∈ ℤ) | |
4 | eqid 2760 | . . . . 5 ⊢ (𝑛 ∈ ℕ0 ↦ ((𝑥↑𝑛) / (!‘𝑛))) = (𝑛 ∈ ℕ0 ↦ ((𝑥↑𝑛) / (!‘𝑛))) | |
5 | 4 | eftval 15026 | . . . 4 ⊢ (𝑘 ∈ ℕ0 → ((𝑛 ∈ ℕ0 ↦ ((𝑥↑𝑛) / (!‘𝑛)))‘𝑘) = ((𝑥↑𝑘) / (!‘𝑘))) |
6 | 5 | adantl 473 | . . 3 ⊢ ((𝑥 ∈ ℂ ∧ 𝑘 ∈ ℕ0) → ((𝑛 ∈ ℕ0 ↦ ((𝑥↑𝑛) / (!‘𝑛)))‘𝑘) = ((𝑥↑𝑘) / (!‘𝑘))) |
7 | eftcl 15023 | . . 3 ⊢ ((𝑥 ∈ ℂ ∧ 𝑘 ∈ ℕ0) → ((𝑥↑𝑘) / (!‘𝑘)) ∈ ℂ) | |
8 | 4 | efcllem 15027 | . . 3 ⊢ (𝑥 ∈ ℂ → seq0( + , (𝑛 ∈ ℕ0 ↦ ((𝑥↑𝑛) / (!‘𝑛)))) ∈ dom ⇝ ) |
9 | 2, 3, 6, 7, 8 | isumcl 14711 | . 2 ⊢ (𝑥 ∈ ℂ → Σ𝑘 ∈ ℕ0 ((𝑥↑𝑘) / (!‘𝑘)) ∈ ℂ) |
10 | 1, 9 | fmpti 6547 | 1 ⊢ exp:ℂ⟶ℂ |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1632 ∈ wcel 2139 ↦ cmpt 4881 ⟶wf 6045 ‘cfv 6049 (class class class)co 6814 ℂcc 10146 0cc0 10148 / cdiv 10896 ℕ0cn0 11504 ↑cexp 13074 !cfa 13274 Σcsu 14635 expce 15011 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-8 2141 ax-9 2148 ax-10 2168 ax-11 2183 ax-12 2196 ax-13 2391 ax-ext 2740 ax-rep 4923 ax-sep 4933 ax-nul 4941 ax-pow 4992 ax-pr 5055 ax-un 7115 ax-inf2 8713 ax-cnex 10204 ax-resscn 10205 ax-1cn 10206 ax-icn 10207 ax-addcl 10208 ax-addrcl 10209 ax-mulcl 10210 ax-mulrcl 10211 ax-mulcom 10212 ax-addass 10213 ax-mulass 10214 ax-distr 10215 ax-i2m1 10216 ax-1ne0 10217 ax-1rid 10218 ax-rnegex 10219 ax-rrecex 10220 ax-cnre 10221 ax-pre-lttri 10222 ax-pre-lttrn 10223 ax-pre-ltadd 10224 ax-pre-mulgt0 10225 ax-pre-sup 10226 ax-addf 10227 ax-mulf 10228 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1073 df-3an 1074 df-tru 1635 df-fal 1638 df-ex 1854 df-nf 1859 df-sb 2047 df-eu 2611 df-mo 2612 df-clab 2747 df-cleq 2753 df-clel 2756 df-nfc 2891 df-ne 2933 df-nel 3036 df-ral 3055 df-rex 3056 df-reu 3057 df-rmo 3058 df-rab 3059 df-v 3342 df-sbc 3577 df-csb 3675 df-dif 3718 df-un 3720 df-in 3722 df-ss 3729 df-pss 3731 df-nul 4059 df-if 4231 df-pw 4304 df-sn 4322 df-pr 4324 df-tp 4326 df-op 4328 df-uni 4589 df-int 4628 df-iun 4674 df-br 4805 df-opab 4865 df-mpt 4882 df-tr 4905 df-id 5174 df-eprel 5179 df-po 5187 df-so 5188 df-fr 5225 df-se 5226 df-we 5227 df-xp 5272 df-rel 5273 df-cnv 5274 df-co 5275 df-dm 5276 df-rn 5277 df-res 5278 df-ima 5279 df-pred 5841 df-ord 5887 df-on 5888 df-lim 5889 df-suc 5890 df-iota 6012 df-fun 6051 df-fn 6052 df-f 6053 df-f1 6054 df-fo 6055 df-f1o 6056 df-fv 6057 df-isom 6058 df-riota 6775 df-ov 6817 df-oprab 6818 df-mpt2 6819 df-om 7232 df-1st 7334 df-2nd 7335 df-wrecs 7577 df-recs 7638 df-rdg 7676 df-1o 7730 df-oadd 7734 df-er 7913 df-pm 8028 df-en 8124 df-dom 8125 df-sdom 8126 df-fin 8127 df-sup 8515 df-inf 8516 df-oi 8582 df-card 8975 df-pnf 10288 df-mnf 10289 df-xr 10290 df-ltxr 10291 df-le 10292 df-sub 10480 df-neg 10481 df-div 10897 df-nn 11233 df-2 11291 df-3 11292 df-n0 11505 df-z 11590 df-uz 11900 df-rp 12046 df-ico 12394 df-fz 12540 df-fzo 12680 df-fl 12807 df-seq 13016 df-exp 13075 df-fac 13275 df-hash 13332 df-shft 14026 df-cj 14058 df-re 14059 df-im 14060 df-sqrt 14194 df-abs 14195 df-limsup 14421 df-clim 14438 df-rlim 14439 df-sum 14636 df-ef 15017 |
This theorem is referenced by: efcl 15032 eff2 15048 reeff1 15069 dveflem 23961 dvef 23962 dvsincos 23963 efcn 24416 efcvx 24422 pige3 24489 efabl 24516 efsubm 24517 dvrelog 24603 dvlog 24617 efopn 24624 dvcxp1 24701 dvcxp2 24702 dvcncxp1 24704 gamf 24989 gamcvg2lem 25005 itgexpif 31014 iprodefisumlem 31954 seff 39028 dvsef 39051 expgrowthi 39052 expgrowth 39054 |
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