![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 15133 | . 2 ⊢ exp = (𝑥 ∈ ℂ ↦ Σ𝑘 ∈ ℕ0 ((𝑥↑𝑘) / (!‘𝑘))) | |
2 | nn0uz 11965 | . . 3 ⊢ ℕ0 = (ℤ≥‘0) | |
3 | 0zd 11677 | . . 3 ⊢ (𝑥 ∈ ℂ → 0 ∈ ℤ) | |
4 | eqid 2800 | . . . . 5 ⊢ (𝑛 ∈ ℕ0 ↦ ((𝑥↑𝑛) / (!‘𝑛))) = (𝑛 ∈ ℕ0 ↦ ((𝑥↑𝑛) / (!‘𝑛))) | |
5 | 4 | eftval 15142 | . . . 4 ⊢ (𝑘 ∈ ℕ0 → ((𝑛 ∈ ℕ0 ↦ ((𝑥↑𝑛) / (!‘𝑛)))‘𝑘) = ((𝑥↑𝑘) / (!‘𝑘))) |
6 | 5 | adantl 474 | . . 3 ⊢ ((𝑥 ∈ ℂ ∧ 𝑘 ∈ ℕ0) → ((𝑛 ∈ ℕ0 ↦ ((𝑥↑𝑛) / (!‘𝑛)))‘𝑘) = ((𝑥↑𝑘) / (!‘𝑘))) |
7 | eftcl 15139 | . . 3 ⊢ ((𝑥 ∈ ℂ ∧ 𝑘 ∈ ℕ0) → ((𝑥↑𝑘) / (!‘𝑘)) ∈ ℂ) | |
8 | 4 | efcllem 15143 | . . 3 ⊢ (𝑥 ∈ ℂ → seq0( + , (𝑛 ∈ ℕ0 ↦ ((𝑥↑𝑛) / (!‘𝑛)))) ∈ dom ⇝ ) |
9 | 2, 3, 6, 7, 8 | isumcl 14830 | . 2 ⊢ (𝑥 ∈ ℂ → Σ𝑘 ∈ ℕ0 ((𝑥↑𝑘) / (!‘𝑘)) ∈ ℂ) |
10 | 1, 9 | fmpti 6609 | 1 ⊢ exp:ℂ⟶ℂ |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1653 ∈ wcel 2157 ↦ cmpt 4923 ⟶wf 6098 ‘cfv 6102 (class class class)co 6879 ℂcc 10223 0cc0 10225 / cdiv 10977 ℕ0cn0 11579 ↑cexp 13113 !cfa 13312 Σcsu 14756 expce 15127 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2378 ax-ext 2778 ax-rep 4965 ax-sep 4976 ax-nul 4984 ax-pow 5036 ax-pr 5098 ax-un 7184 ax-inf2 8789 ax-cnex 10281 ax-resscn 10282 ax-1cn 10283 ax-icn 10284 ax-addcl 10285 ax-addrcl 10286 ax-mulcl 10287 ax-mulrcl 10288 ax-mulcom 10289 ax-addass 10290 ax-mulass 10291 ax-distr 10292 ax-i2m1 10293 ax-1ne0 10294 ax-1rid 10295 ax-rnegex 10296 ax-rrecex 10297 ax-cnre 10298 ax-pre-lttri 10299 ax-pre-lttrn 10300 ax-pre-ltadd 10301 ax-pre-mulgt0 10302 ax-pre-sup 10303 ax-addf 10304 ax-mulf 10305 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3or 1109 df-3an 1110 df-tru 1657 df-fal 1667 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2592 df-eu 2610 df-clab 2787 df-cleq 2793 df-clel 2796 df-nfc 2931 df-ne 2973 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rmo 3098 df-rab 3099 df-v 3388 df-sbc 3635 df-csb 3730 df-dif 3773 df-un 3775 df-in 3777 df-ss 3784 df-pss 3786 df-nul 4117 df-if 4279 df-pw 4352 df-sn 4370 df-pr 4372 df-tp 4374 df-op 4376 df-uni 4630 df-int 4669 df-iun 4713 df-br 4845 df-opab 4907 df-mpt 4924 df-tr 4947 df-id 5221 df-eprel 5226 df-po 5234 df-so 5235 df-fr 5272 df-se 5273 df-we 5274 df-xp 5319 df-rel 5320 df-cnv 5321 df-co 5322 df-dm 5323 df-rn 5324 df-res 5325 df-ima 5326 df-pred 5899 df-ord 5945 df-on 5946 df-lim 5947 df-suc 5948 df-iota 6065 df-fun 6104 df-fn 6105 df-f 6106 df-f1 6107 df-fo 6108 df-f1o 6109 df-fv 6110 df-isom 6111 df-riota 6840 df-ov 6882 df-oprab 6883 df-mpt2 6884 df-om 7301 df-1st 7402 df-2nd 7403 df-wrecs 7646 df-recs 7708 df-rdg 7746 df-1o 7800 df-oadd 7804 df-er 7983 df-pm 8099 df-en 8197 df-dom 8198 df-sdom 8199 df-fin 8200 df-sup 8591 df-inf 8592 df-oi 8658 df-card 9052 df-pnf 10366 df-mnf 10367 df-xr 10368 df-ltxr 10369 df-le 10370 df-sub 10559 df-neg 10560 df-div 10978 df-nn 11314 df-2 11375 df-3 11376 df-n0 11580 df-z 11666 df-uz 11930 df-rp 12074 df-ico 12429 df-fz 12580 df-fzo 12720 df-fl 12847 df-seq 13055 df-exp 13114 df-fac 13313 df-hash 13370 df-shft 14147 df-cj 14179 df-re 14180 df-im 14181 df-sqrt 14315 df-abs 14316 df-limsup 14542 df-clim 14559 df-rlim 14560 df-sum 14757 df-ef 15133 |
This theorem is referenced by: efcl 15148 eff2 15164 reeff1 15185 dveflem 24082 dvef 24083 dvsincos 24084 efcn 24537 efcvx 24543 pige3 24610 efabl 24637 efsubm 24638 dvrelog 24723 dvlog 24737 efopn 24744 dvcxp1 24824 dvcxp2 24825 dvcncxp1 24827 gamf 25120 gamcvg2lem 25136 itgexpif 31203 iprodefisumlem 32139 seff 39285 dvsef 39308 expgrowthi 39309 expgrowth 39311 |
Copyright terms: Public domain | W3C validator |