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Mirrors > Home > MPE Home > Th. List > eftabs | Structured version Visualization version GIF version |
Description: The absolute value of a term in the series expansion of the exponential function. (Contributed by Paul Chapman, 23-Nov-2007.) |
Ref | Expression |
---|---|
eftabs | ⊢ ((𝐴 ∈ ℂ ∧ 𝐾 ∈ ℕ0) → (abs‘((𝐴↑𝐾) / (!‘𝐾))) = (((abs‘𝐴)↑𝐾) / (!‘𝐾))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | expcl 14080 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐾 ∈ ℕ0) → (𝐴↑𝐾) ∈ ℂ) | |
2 | faccl 14278 | . . . . 5 ⊢ (𝐾 ∈ ℕ0 → (!‘𝐾) ∈ ℕ) | |
3 | 2 | adantl 480 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐾 ∈ ℕ0) → (!‘𝐾) ∈ ℕ) |
4 | 3 | nncnd 12261 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐾 ∈ ℕ0) → (!‘𝐾) ∈ ℂ) |
5 | facne0 14281 | . . . 4 ⊢ (𝐾 ∈ ℕ0 → (!‘𝐾) ≠ 0) | |
6 | 5 | adantl 480 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐾 ∈ ℕ0) → (!‘𝐾) ≠ 0) |
7 | 1, 4, 6 | absdivd 15438 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐾 ∈ ℕ0) → (abs‘((𝐴↑𝐾) / (!‘𝐾))) = ((abs‘(𝐴↑𝐾)) / (abs‘(!‘𝐾)))) |
8 | absexp 15287 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐾 ∈ ℕ0) → (abs‘(𝐴↑𝐾)) = ((abs‘𝐴)↑𝐾)) | |
9 | 3 | nnred 12260 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐾 ∈ ℕ0) → (!‘𝐾) ∈ ℝ) |
10 | 3 | nnnn0d 12565 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐾 ∈ ℕ0) → (!‘𝐾) ∈ ℕ0) |
11 | 10 | nn0ge0d 12568 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐾 ∈ ℕ0) → 0 ≤ (!‘𝐾)) |
12 | 9, 11 | absidd 15405 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐾 ∈ ℕ0) → (abs‘(!‘𝐾)) = (!‘𝐾)) |
13 | 8, 12 | oveq12d 7437 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐾 ∈ ℕ0) → ((abs‘(𝐴↑𝐾)) / (abs‘(!‘𝐾))) = (((abs‘𝐴)↑𝐾) / (!‘𝐾))) |
14 | 7, 13 | eqtrd 2765 | 1 ⊢ ((𝐴 ∈ ℂ ∧ 𝐾 ∈ ℕ0) → (abs‘((𝐴↑𝐾) / (!‘𝐾))) = (((abs‘𝐴)↑𝐾) / (!‘𝐾))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1533 ∈ wcel 2098 ≠ wne 2929 ‘cfv 6549 (class class class)co 7419 ℂcc 11138 0cc0 11140 / cdiv 11903 ℕcn 12245 ℕ0cn0 12505 ↑cexp 14062 !cfa 14268 abscabs 15217 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 ax-cnex 11196 ax-resscn 11197 ax-1cn 11198 ax-icn 11199 ax-addcl 11200 ax-addrcl 11201 ax-mulcl 11202 ax-mulrcl 11203 ax-mulcom 11204 ax-addass 11205 ax-mulass 11206 ax-distr 11207 ax-i2m1 11208 ax-1ne0 11209 ax-1rid 11210 ax-rnegex 11211 ax-rrecex 11212 ax-cnre 11213 ax-pre-lttri 11214 ax-pre-lttrn 11215 ax-pre-ltadd 11216 ax-pre-mulgt0 11217 ax-pre-sup 11218 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-rmo 3363 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3964 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-iun 4999 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6307 df-ord 6374 df-on 6375 df-lim 6376 df-suc 6377 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-riota 7375 df-ov 7422 df-oprab 7423 df-mpo 7424 df-om 7872 df-2nd 7995 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-er 8725 df-en 8965 df-dom 8966 df-sdom 8967 df-sup 9467 df-pnf 11282 df-mnf 11283 df-xr 11284 df-ltxr 11285 df-le 11286 df-sub 11478 df-neg 11479 df-div 11904 df-nn 12246 df-2 12308 df-3 12309 df-n0 12506 df-z 12592 df-uz 12856 df-rp 13010 df-seq 14003 df-exp 14063 df-fac 14269 df-cj 15082 df-re 15083 df-im 15084 df-sqrt 15218 df-abs 15219 |
This theorem is referenced by: efcllem 16057 eftlub 16089 |
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