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Mirrors > Home > MPE Home > Th. List > expp1d | Structured version Visualization version GIF version |
Description: Value of a complex number raised to a nonnegative integer power plus one. Part of Definition 10-4.1 of [Gleason] p. 134. (Contributed by Mario Carneiro, 28-May-2016.) |
Ref | Expression |
---|---|
expcld.1 | ⊢ (𝜑 → 𝐴 ∈ ℂ) |
expcld.2 | ⊢ (𝜑 → 𝑁 ∈ ℕ0) |
Ref | Expression |
---|---|
expp1d | ⊢ (𝜑 → (𝐴↑(𝑁 + 1)) = ((𝐴↑𝑁) · 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | expcld.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ ℂ) | |
2 | expcld.2 | . 2 ⊢ (𝜑 → 𝑁 ∈ ℕ0) | |
3 | expp1 13185 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0) → (𝐴↑(𝑁 + 1)) = ((𝐴↑𝑁) · 𝐴)) | |
4 | 1, 2, 3 | syl2anc 579 | 1 ⊢ (𝜑 → (𝐴↑(𝑁 + 1)) = ((𝐴↑𝑁) · 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1601 ∈ wcel 2107 (class class class)co 6922 ℂcc 10270 1c1 10273 + caddc 10275 · cmul 10277 ℕ0cn0 11642 ↑cexp 13178 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 ax-cnex 10328 ax-resscn 10329 ax-1cn 10330 ax-icn 10331 ax-addcl 10332 ax-addrcl 10333 ax-mulcl 10334 ax-mulrcl 10335 ax-mulcom 10336 ax-addass 10337 ax-mulass 10338 ax-distr 10339 ax-i2m1 10340 ax-1ne0 10341 ax-1rid 10342 ax-rnegex 10343 ax-rrecex 10344 ax-cnre 10345 ax-pre-lttri 10346 ax-pre-lttrn 10347 ax-pre-ltadd 10348 ax-pre-mulgt0 10349 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-pss 3808 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4672 df-iun 4755 df-br 4887 df-opab 4949 df-mpt 4966 df-tr 4988 df-id 5261 df-eprel 5266 df-po 5274 df-so 5275 df-fr 5314 df-we 5316 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-pred 5933 df-ord 5979 df-on 5980 df-lim 5981 df-suc 5982 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-riota 6883 df-ov 6925 df-oprab 6926 df-mpt2 6927 df-om 7344 df-2nd 7446 df-wrecs 7689 df-recs 7751 df-rdg 7789 df-er 8026 df-en 8242 df-dom 8243 df-sdom 8244 df-pnf 10413 df-mnf 10414 df-xr 10415 df-ltxr 10416 df-le 10417 df-sub 10608 df-neg 10609 df-nn 11375 df-n0 11643 df-z 11729 df-uz 11993 df-seq 13120 df-exp 13179 |
This theorem is referenced by: facubnd 13405 hashmap 13536 binomlem 14965 incexclem 14972 geoserg 15002 cvgrat 15018 efcllem 15210 oexpneg 15473 pwp1fsum 15521 bitsp1 15559 bitsmod 15564 bitsinv1lem 15569 sadcaddlem 15585 sadadd2lem 15587 rplpwr 15682 eulerthlem2 15891 prmdiv 15894 vfermltlALT 15911 pcprendvds2 15950 pcpremul 15952 prmpwdvds 16012 2expltfac 16198 plyco 24434 dgrcolem1 24466 ftalem5 25255 bposlem5 25465 pntlemq 25742 pntlemr 25743 pntlemj 25744 ostth2lem2 25775 ostth2lem3 25776 rusgrnumwwlks 27354 rusgrnumwwlksOLD 27355 ex-ind-dvds 27893 nexple 30669 faclimlem3 32225 faclim2 32228 nn0prpwlem 32905 mzpexpmpt 38272 pell14qrexpclnn0 38394 expmordi 38475 jm2.17a 38490 jm2.17b 38491 jm2.17c 38492 jm2.18 38518 cnsrexpcl 38698 inductionexd 39413 binomcxplemnotnn0 39515 stoweidlem3 41151 stoweidlem19 41167 stirlinglem4 41225 stirlinglem7 41228 etransclem23 41405 sqrtpwpw2p 42475 fmtnorec2lem 42479 fmtnorec4 42486 fmtnoprmfac1lem 42501 fmtnoprmfac2 42504 fmtnofac1 42507 lighneallem3 42549 oexpnegALTV 42617 tgoldbachlt 42733 dignn0flhalflem2 43429 dignn0ehalf 43430 nn0sumshdiglemA 43432 nn0sumshdiglemB 43433 |
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