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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 13171 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐾 ∈ ℕ0) → (𝐴↑𝐾) ∈ ℂ) | |
2 | faccl 13362 | . . . . 5 ⊢ (𝐾 ∈ ℕ0 → (!‘𝐾) ∈ ℕ) | |
3 | 2 | adantl 475 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐾 ∈ ℕ0) → (!‘𝐾) ∈ ℕ) |
4 | 3 | nncnd 11367 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐾 ∈ ℕ0) → (!‘𝐾) ∈ ℂ) |
5 | facne0 13365 | . . . 4 ⊢ (𝐾 ∈ ℕ0 → (!‘𝐾) ≠ 0) | |
6 | 5 | adantl 475 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐾 ∈ ℕ0) → (!‘𝐾) ≠ 0) |
7 | 1, 4, 6 | absdivd 14570 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐾 ∈ ℕ0) → (abs‘((𝐴↑𝐾) / (!‘𝐾))) = ((abs‘(𝐴↑𝐾)) / (abs‘(!‘𝐾)))) |
8 | absexp 14420 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐾 ∈ ℕ0) → (abs‘(𝐴↑𝐾)) = ((abs‘𝐴)↑𝐾)) | |
9 | 3 | nnred 11366 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐾 ∈ ℕ0) → (!‘𝐾) ∈ ℝ) |
10 | 3 | nnnn0d 11677 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐾 ∈ ℕ0) → (!‘𝐾) ∈ ℕ0) |
11 | 10 | nn0ge0d 11680 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐾 ∈ ℕ0) → 0 ≤ (!‘𝐾)) |
12 | 9, 11 | absidd 14537 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐾 ∈ ℕ0) → (abs‘(!‘𝐾)) = (!‘𝐾)) |
13 | 8, 12 | oveq12d 6922 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐾 ∈ ℕ0) → ((abs‘(𝐴↑𝐾)) / (abs‘(!‘𝐾))) = (((abs‘𝐴)↑𝐾) / (!‘𝐾))) |
14 | 7, 13 | eqtrd 2860 | 1 ⊢ ((𝐴 ∈ ℂ ∧ 𝐾 ∈ ℕ0) → (abs‘((𝐴↑𝐾) / (!‘𝐾))) = (((abs‘𝐴)↑𝐾) / (!‘𝐾))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 = wceq 1658 ∈ wcel 2166 ≠ wne 2998 ‘cfv 6122 (class class class)co 6904 ℂcc 10249 0cc0 10251 / cdiv 11008 ℕcn 11349 ℕ0cn0 11617 ↑cexp 13153 !cfa 13352 abscabs 14350 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2390 ax-ext 2802 ax-sep 5004 ax-nul 5012 ax-pow 5064 ax-pr 5126 ax-un 7208 ax-cnex 10307 ax-resscn 10308 ax-1cn 10309 ax-icn 10310 ax-addcl 10311 ax-addrcl 10312 ax-mulcl 10313 ax-mulrcl 10314 ax-mulcom 10315 ax-addass 10316 ax-mulass 10317 ax-distr 10318 ax-i2m1 10319 ax-1ne0 10320 ax-1rid 10321 ax-rnegex 10322 ax-rrecex 10323 ax-cnre 10324 ax-pre-lttri 10325 ax-pre-lttrn 10326 ax-pre-ltadd 10327 ax-pre-mulgt0 10328 ax-pre-sup 10329 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3or 1114 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2604 df-eu 2639 df-clab 2811 df-cleq 2817 df-clel 2820 df-nfc 2957 df-ne 2999 df-nel 3102 df-ral 3121 df-rex 3122 df-reu 3123 df-rmo 3124 df-rab 3125 df-v 3415 df-sbc 3662 df-csb 3757 df-dif 3800 df-un 3802 df-in 3804 df-ss 3811 df-pss 3813 df-nul 4144 df-if 4306 df-pw 4379 df-sn 4397 df-pr 4399 df-tp 4401 df-op 4403 df-uni 4658 df-iun 4741 df-br 4873 df-opab 4935 df-mpt 4952 df-tr 4975 df-id 5249 df-eprel 5254 df-po 5262 df-so 5263 df-fr 5300 df-we 5302 df-xp 5347 df-rel 5348 df-cnv 5349 df-co 5350 df-dm 5351 df-rn 5352 df-res 5353 df-ima 5354 df-pred 5919 df-ord 5965 df-on 5966 df-lim 5967 df-suc 5968 df-iota 6085 df-fun 6124 df-fn 6125 df-f 6126 df-f1 6127 df-fo 6128 df-f1o 6129 df-fv 6130 df-riota 6865 df-ov 6907 df-oprab 6908 df-mpt2 6909 df-om 7326 df-2nd 7428 df-wrecs 7671 df-recs 7733 df-rdg 7771 df-er 8008 df-en 8222 df-dom 8223 df-sdom 8224 df-sup 8616 df-pnf 10392 df-mnf 10393 df-xr 10394 df-ltxr 10395 df-le 10396 df-sub 10586 df-neg 10587 df-div 11009 df-nn 11350 df-2 11413 df-3 11414 df-n0 11618 df-z 11704 df-uz 11968 df-rp 12112 df-seq 13095 df-exp 13154 df-fac 13353 df-cj 14215 df-re 14216 df-im 14217 df-sqrt 14351 df-abs 14352 |
This theorem is referenced by: efcllem 15179 eftlub 15210 |
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