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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 14109 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0) → (𝐴↑(𝑁 + 1)) = ((𝐴↑𝑁) · 𝐴)) | |
| 4 | 1, 2, 3 | syl2anc 584 | 1 ⊢ (𝜑 → (𝐴↑(𝑁 + 1)) = ((𝐴↑𝑁) · 𝐴)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 (class class class)co 7431 ℂcc 11153 1c1 11156 + caddc 11158 · cmul 11160 ℕ0cn0 12526 ↑cexp 14102 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-nn 12267 df-n0 12527 df-z 12614 df-uz 12879 df-seq 14043 df-exp 14103 |
| This theorem is referenced by: expmordi 14207 facubnd 14339 hashmap 14474 binomlem 15865 incexclem 15872 geoserg 15902 cvgrat 15919 efcllem 16113 oexpneg 16382 pwp1fsum 16428 bitsp1 16468 bitsmod 16473 bitsinv1lem 16478 sadcaddlem 16494 sadadd2lem 16496 rplpwr 16595 eulerthlem2 16819 prmdiv 16822 vfermltlALT 16840 pcprendvds2 16879 pcpremul 16881 prmpwdvds 16942 2expltfac 17130 plyco 26280 dgrcolem1 26313 ftalem5 27120 bposlem5 27332 pntlemq 27645 pntlemr 27646 pntlemj 27647 ostth2lem2 27678 ostth2lem3 27679 rusgrnumwwlks 29994 ex-ind-dvds 30480 nexple 32833 2exple2exp 32834 fldext2rspun 33732 fldext2chn 33769 faclimlem3 35745 faclim2 35748 nn0prpwlem 36323 3lexlogpow5ineq5 42061 nicomachus 42346 abvexp 42542 3cubeslem2 42696 3cubeslem3l 42697 3cubeslem3r 42698 mzpexpmpt 42756 pell14qrexpclnn0 42877 jm2.17a 42972 jm2.17b 42973 jm2.17c 42974 jm2.18 43000 cnsrexpcl 43177 inductionexd 44168 binomcxplemnotnn0 44375 stoweidlem3 46018 stoweidlem19 46034 stirlinglem4 46092 stirlinglem7 46095 etransclem23 46272 sqrtpwpw2p 47525 fmtnorec2lem 47529 fmtnorec4 47536 fmtnoprmfac1lem 47551 fmtnoprmfac2 47554 fmtnofac1 47557 lighneallem3 47594 oexpnegALTV 47664 fppr2odd 47718 tgoldbachlt 47803 dignn0flhalflem2 48537 dignn0ehalf 48538 nn0sumshdiglemA 48540 nn0sumshdiglemB 48541 itcovalt2lem2lem2 48595 |
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