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Mirrors > Home > MPE Home > Th. List > expm1t | Structured version Visualization version GIF version |
Description: Exponentiation in terms of predecessor exponent. (Contributed by NM, 19-Dec-2005.) |
Ref | Expression |
---|---|
expm1t | ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) → (𝐴↑𝑁) = ((𝐴↑(𝑁 − 1)) · 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nncn 11066 | . . . . 5 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℂ) | |
2 | ax-1cn 10032 | . . . . 5 ⊢ 1 ∈ ℂ | |
3 | npcan 10328 | . . . . 5 ⊢ ((𝑁 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑁 − 1) + 1) = 𝑁) | |
4 | 1, 2, 3 | sylancl 695 | . . . 4 ⊢ (𝑁 ∈ ℕ → ((𝑁 − 1) + 1) = 𝑁) |
5 | 4 | oveq2d 6706 | . . 3 ⊢ (𝑁 ∈ ℕ → (𝐴↑((𝑁 − 1) + 1)) = (𝐴↑𝑁)) |
6 | 5 | adantl 481 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) → (𝐴↑((𝑁 − 1) + 1)) = (𝐴↑𝑁)) |
7 | nnm1nn0 11372 | . . 3 ⊢ (𝑁 ∈ ℕ → (𝑁 − 1) ∈ ℕ0) | |
8 | expp1 12907 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ (𝑁 − 1) ∈ ℕ0) → (𝐴↑((𝑁 − 1) + 1)) = ((𝐴↑(𝑁 − 1)) · 𝐴)) | |
9 | 7, 8 | sylan2 490 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) → (𝐴↑((𝑁 − 1) + 1)) = ((𝐴↑(𝑁 − 1)) · 𝐴)) |
10 | 6, 9 | eqtr3d 2687 | 1 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) → (𝐴↑𝑁) = ((𝐴↑(𝑁 − 1)) · 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1523 ∈ wcel 2030 (class class class)co 6690 ℂcc 9972 1c1 9975 + caddc 9977 · cmul 9979 − cmin 10304 ℕcn 11058 ℕ0cn0 11330 ↑cexp 12900 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 ax-un 6991 ax-cnex 10030 ax-resscn 10031 ax-1cn 10032 ax-icn 10033 ax-addcl 10034 ax-addrcl 10035 ax-mulcl 10036 ax-mulrcl 10037 ax-mulcom 10038 ax-addass 10039 ax-mulass 10040 ax-distr 10041 ax-i2m1 10042 ax-1ne0 10043 ax-1rid 10044 ax-rnegex 10045 ax-rrecex 10046 ax-cnre 10047 ax-pre-lttri 10048 ax-pre-lttrn 10049 ax-pre-ltadd 10050 ax-pre-mulgt0 10051 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1055 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-nel 2927 df-ral 2946 df-rex 2947 df-reu 2948 df-rab 2950 df-v 3233 df-sbc 3469 df-csb 3567 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-pss 3623 df-nul 3949 df-if 4120 df-pw 4193 df-sn 4211 df-pr 4213 df-tp 4215 df-op 4217 df-uni 4469 df-iun 4554 df-br 4686 df-opab 4746 df-mpt 4763 df-tr 4786 df-id 5053 df-eprel 5058 df-po 5064 df-so 5065 df-fr 5102 df-we 5104 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 df-pred 5718 df-ord 5764 df-on 5765 df-lim 5766 df-suc 5767 df-iota 5889 df-fun 5928 df-fn 5929 df-f 5930 df-f1 5931 df-fo 5932 df-f1o 5933 df-fv 5934 df-riota 6651 df-ov 6693 df-oprab 6694 df-mpt2 6695 df-om 7108 df-2nd 7211 df-wrecs 7452 df-recs 7513 df-rdg 7551 df-er 7787 df-en 7998 df-dom 7999 df-sdom 8000 df-pnf 10114 df-mnf 10115 df-xr 10116 df-ltxr 10117 df-le 10118 df-sub 10306 df-neg 10307 df-nn 11059 df-n0 11331 df-z 11416 df-uz 11726 df-seq 12842 df-exp 12901 |
This theorem is referenced by: expgt1 12938 digit1 13038 faclbnd4lem1 13120 binomlem 14605 incexc 14613 iddvdsexp 15052 bitsfzolem 15203 phiprmpw 15528 fermltl 15536 prmpwdvds 15655 dvexp 23761 abelthlem6 24235 ftalem1 24844 perfect1 24998 perfect 25001 knoppndvlem2 32629 jm2.18 37872 jm2.22 37879 itgsinexp 40488 fmtnoodd 41770 perfectALTV 41957 |
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