expp1d - Metamath Proof Explorer

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Theorem expp1d 14056

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.)

**Hypotheses**
Ref	Expression
expcld.1	⊢ (𝜑 → 𝐴 ∈ ℂ)
expcld.2	⊢ (𝜑 → 𝑁 ∈ ℕ₀)

**Assertion**
Ref	Expression
expp1d	⊢ (𝜑 → (𝐴↑(𝑁 + 1)) = ((𝐴↑𝑁) · 𝐴))

**Proof of Theorem expp1d**
Step	Hyp	Ref	Expression
1		expcld.1	. 2 ⊢ (𝜑 → 𝐴 ∈ ℂ)
2		expcld.2	. 2 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
3		expp1 13977	. 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 1541 ∈ wcel 2113 (class class class)co 7352 ℂcc 11011 1c1 11014 + caddc 11016 · cmul 11018 ℕ₀cn0 12388 ↑cexp 13970

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2182 ax-ext 2705 ax-sep 5236 ax-nul 5246 ax-pow 5305 ax-pr 5372 ax-un 7674 ax-cnex 11069 ax-resscn 11070 ax-1cn 11071 ax-icn 11072 ax-addcl 11073 ax-addrcl 11074 ax-mulcl 11075 ax-mulrcl 11076 ax-mulcom 11077 ax-addass 11078 ax-mulass 11079 ax-distr 11080 ax-i2m1 11081 ax-1ne0 11082 ax-1rid 11083 ax-rnegex 11084 ax-rrecex 11085 ax-cnre 11086 ax-pre-lttri 11087 ax-pre-lttrn 11088 ax-pre-ltadd 11089 ax-pre-mulgt0 11090

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2725 df-clel 2808 df-nfc 2882 df-ne 2930 df-nel 3034 df-ral 3049 df-rex 3058 df-reu 3348 df-rab 3397 df-v 3439 df-sbc 3738 df-csb 3847 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-pss 3918 df-nul 4283 df-if 4475 df-pw 4551 df-sn 4576 df-pr 4578 df-op 4582 df-uni 4859 df-iun 4943 df-br 5094 df-opab 5156 df-mpt 5175 df-tr 5201 df-id 5514 df-eprel 5519 df-po 5527 df-so 5528 df-fr 5572 df-we 5574 df-xp 5625 df-rel 5626 df-cnv 5627 df-co 5628 df-dm 5629 df-rn 5630 df-res 5631 df-ima 5632 df-pred 6253 df-ord 6314 df-on 6315 df-lim 6316 df-suc 6317 df-iota 6442 df-fun 6488 df-fn 6489 df-f 6490 df-f1 6491 df-fo 6492 df-f1o 6493 df-fv 6494 df-riota 7309 df-ov 7355 df-oprab 7356 df-mpo 7357 df-om 7803 df-2nd 7928 df-frecs 8217 df-wrecs 8248 df-recs 8297 df-rdg 8335 df-er 8628 df-en 8876 df-dom 8877 df-sdom 8878 df-pnf 11155 df-mnf 11156 df-xr 11157 df-ltxr 11158 df-le 11159 df-sub 11353 df-neg 11354 df-nn 12133 df-n0 12389 df-z 12476 df-uz 12739 df-seq 13911 df-exp 13971

This theorem is referenced by: expmordi 14076 facubnd 14209 hashmap 14344 binomlem 15738 incexclem 15745 geoserg 15775 cvgrat 15792 efcllem 15986 oexpneg 16258 pwp1fsum 16304 bitsp1 16344 bitsmod 16349 bitsinv1lem 16354 sadcaddlem 16370 sadadd2lem 16372 rplpwr 16471 eulerthlem2 16695 prmdiv 16698 vfermltlALT 16716 pcprendvds2 16755 pcpremul 16757 prmpwdvds 16818 2expltfac 17006 plyco 26174 dgrcolem1 26207 ftalem5 27015 bposlem5 27227 pntlemq 27540 pntlemr 27541 pntlemj 27542 ostth2lem2 27573 ostth2lem3 27574 rusgrnumwwlks 29957 ex-ind-dvds 30443 nexple 32832 2exple2exp 32833 oexpled 32835 fldext2rspun 33716 fldext2chn 33762 faclimlem3 35810 faclim2 35813 nn0prpwlem 36387 3lexlogpow5ineq5 42173 nicomachus 42430 abvexp 42650 3cubeslem2 42802 3cubeslem3l 42803 3cubeslem3r 42804 mzpexpmpt 42862 pell14qrexpclnn0 42983 jm2.17a 43077 jm2.17b 43078 jm2.17c 43079 jm2.18 43105 cnsrexpcl 43282 inductionexd 44272 binomcxplemnotnn0 44473 stoweidlem3 46125 stoweidlem19 46141 stirlinglem4 46199 stirlinglem7 46202 etransclem23 46379 sqrtpwpw2p 47662 fmtnorec2lem 47666 fmtnorec4 47673 fmtnoprmfac1lem 47688 fmtnoprmfac2 47691 fmtnofac1 47694 lighneallem3 47731 oexpnegALTV 47801 fppr2odd 47855 tgoldbachlt 47940 dignn0flhalflem2 48741 dignn0ehalf 48742 nn0sumshdiglemA 48744 nn0sumshdiglemB 48745 itcovalt2lem2lem2 48799

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