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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 15949 | . 2 ⊢ exp = (𝑥 ∈ ℂ ↦ Σ𝑘 ∈ ℕ0 ((𝑥↑𝑘) / (!‘𝑘))) | |
2 | nn0uz 12804 | . . 3 ⊢ ℕ0 = (ℤ≥‘0) | |
3 | 0zd 12510 | . . 3 ⊢ (𝑥 ∈ ℂ → 0 ∈ ℤ) | |
4 | eqid 2736 | . . . . 5 ⊢ (𝑛 ∈ ℕ0 ↦ ((𝑥↑𝑛) / (!‘𝑛))) = (𝑛 ∈ ℕ0 ↦ ((𝑥↑𝑛) / (!‘𝑛))) | |
5 | 4 | eftval 15958 | . . . 4 ⊢ (𝑘 ∈ ℕ0 → ((𝑛 ∈ ℕ0 ↦ ((𝑥↑𝑛) / (!‘𝑛)))‘𝑘) = ((𝑥↑𝑘) / (!‘𝑘))) |
6 | 5 | adantl 482 | . . 3 ⊢ ((𝑥 ∈ ℂ ∧ 𝑘 ∈ ℕ0) → ((𝑛 ∈ ℕ0 ↦ ((𝑥↑𝑛) / (!‘𝑛)))‘𝑘) = ((𝑥↑𝑘) / (!‘𝑘))) |
7 | eftcl 15955 | . . 3 ⊢ ((𝑥 ∈ ℂ ∧ 𝑘 ∈ ℕ0) → ((𝑥↑𝑘) / (!‘𝑘)) ∈ ℂ) | |
8 | 4 | efcllem 15959 | . . 3 ⊢ (𝑥 ∈ ℂ → seq0( + , (𝑛 ∈ ℕ0 ↦ ((𝑥↑𝑛) / (!‘𝑛)))) ∈ dom ⇝ ) |
9 | 2, 3, 6, 7, 8 | isumcl 15645 | . 2 ⊢ (𝑥 ∈ ℂ → Σ𝑘 ∈ ℕ0 ((𝑥↑𝑘) / (!‘𝑘)) ∈ ℂ) |
10 | 1, 9 | fmpti 7059 | 1 ⊢ exp:ℂ⟶ℂ |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1541 ∈ wcel 2106 ↦ cmpt 5188 ⟶wf 6492 ‘cfv 6496 (class class class)co 7356 ℂcc 11048 0cc0 11050 / cdiv 11811 ℕ0cn0 12412 ↑cexp 13966 !cfa 14172 Σcsu 15569 expce 15943 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-rep 5242 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7671 ax-inf2 9576 ax-cnex 11106 ax-resscn 11107 ax-1cn 11108 ax-icn 11109 ax-addcl 11110 ax-addrcl 11111 ax-mulcl 11112 ax-mulrcl 11113 ax-mulcom 11114 ax-addass 11115 ax-mulass 11116 ax-distr 11117 ax-i2m1 11118 ax-1ne0 11119 ax-1rid 11120 ax-rnegex 11121 ax-rrecex 11122 ax-cnre 11123 ax-pre-lttri 11124 ax-pre-lttrn 11125 ax-pre-ltadd 11126 ax-pre-mulgt0 11127 ax-pre-sup 11128 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3065 df-rex 3074 df-rmo 3353 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-pss 3929 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-op 4593 df-uni 4866 df-int 4908 df-iun 4956 df-br 5106 df-opab 5168 df-mpt 5189 df-tr 5223 df-id 5531 df-eprel 5537 df-po 5545 df-so 5546 df-fr 5588 df-se 5589 df-we 5590 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-pred 6253 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-isom 6505 df-riota 7312 df-ov 7359 df-oprab 7360 df-mpo 7361 df-om 7802 df-1st 7920 df-2nd 7921 df-frecs 8211 df-wrecs 8242 df-recs 8316 df-rdg 8355 df-1o 8411 df-er 8647 df-pm 8767 df-en 8883 df-dom 8884 df-sdom 8885 df-fin 8886 df-sup 9377 df-inf 9378 df-oi 9445 df-card 9874 df-pnf 11190 df-mnf 11191 df-xr 11192 df-ltxr 11193 df-le 11194 df-sub 11386 df-neg 11387 df-div 11812 df-nn 12153 df-2 12215 df-3 12216 df-n0 12413 df-z 12499 df-uz 12763 df-rp 12915 df-ico 13269 df-fz 13424 df-fzo 13567 df-fl 13696 df-seq 13906 df-exp 13967 df-fac 14173 df-hash 14230 df-shft 14951 df-cj 14983 df-re 14984 df-im 14985 df-sqrt 15119 df-abs 15120 df-limsup 15352 df-clim 15369 df-rlim 15370 df-sum 15570 df-ef 15949 |
This theorem is referenced by: efcl 15964 eff2 15980 reeff1 16001 dveflem 25341 dvef 25342 dvsincos 25343 efcn 25800 efcvx 25806 pige3ALT 25874 efabl 25904 efsubm 25905 dvrelog 25990 dvlog 26004 efopn 26011 dvcxp1 26091 dvcxp2 26092 dvcncxp1 26094 gamf 26390 gamcvg2lem 26406 itgexpif 33159 iprodefisumlem 34253 seff 42570 dvsef 42593 expgrowthi 42594 expgrowth 42596 |
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