expp1d - Metamath Proof Explorer

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Theorem expp1d 14072

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 13993	. 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 2109 (class class class)co 7353 ℂcc 11026 1c1 11029 + caddc 11031 · cmul 11033 ℕ₀cn0 12402 ↑cexp 13986

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 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-sep 5238 ax-nul 5248 ax-pow 5307 ax-pr 5374 ax-un 7675 ax-cnex 11084 ax-resscn 11085 ax-1cn 11086 ax-icn 11087 ax-addcl 11088 ax-addrcl 11089 ax-mulcl 11090 ax-mulrcl 11091 ax-mulcom 11092 ax-addass 11093 ax-mulass 11094 ax-distr 11095 ax-i2m1 11096 ax-1ne0 11097 ax-1rid 11098 ax-rnegex 11099 ax-rrecex 11100 ax-cnre 11101 ax-pre-lttri 11102 ax-pre-lttrn 11103 ax-pre-ltadd 11104 ax-pre-mulgt0 11105

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-reu 3346 df-rab 3397 df-v 3440 df-sbc 3745 df-csb 3854 df-dif 3908 df-un 3910 df-in 3912 df-ss 3922 df-pss 3925 df-nul 4287 df-if 4479 df-pw 4555 df-sn 4580 df-pr 4582 df-op 4586 df-uni 4862 df-iun 4946 df-br 5096 df-opab 5158 df-mpt 5177 df-tr 5203 df-id 5518 df-eprel 5523 df-po 5531 df-so 5532 df-fr 5576 df-we 5578 df-xp 5629 df-rel 5630 df-cnv 5631 df-co 5632 df-dm 5633 df-rn 5634 df-res 5635 df-ima 5636 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 7310 df-ov 7356 df-oprab 7357 df-mpo 7358 df-om 7807 df-2nd 7932 df-frecs 8221 df-wrecs 8252 df-recs 8301 df-rdg 8339 df-er 8632 df-en 8880 df-dom 8881 df-sdom 8882 df-pnf 11170 df-mnf 11171 df-xr 11172 df-ltxr 11173 df-le 11174 df-sub 11367 df-neg 11368 df-nn 12147 df-n0 12403 df-z 12490 df-uz 12754 df-seq 13927 df-exp 13987

This theorem is referenced by: expmordi 14092 facubnd 14225 hashmap 14360 binomlem 15754 incexclem 15761 geoserg 15791 cvgrat 15808 efcllem 16002 oexpneg 16274 pwp1fsum 16320 bitsp1 16360 bitsmod 16365 bitsinv1lem 16370 sadcaddlem 16386 sadadd2lem 16388 rplpwr 16487 eulerthlem2 16711 prmdiv 16714 vfermltlALT 16732 pcprendvds2 16771 pcpremul 16773 prmpwdvds 16834 2expltfac 17022 plyco 26162 dgrcolem1 26195 ftalem5 27003 bposlem5 27215 pntlemq 27528 pntlemr 27529 pntlemj 27530 ostth2lem2 27561 ostth2lem3 27562 rusgrnumwwlks 29937 ex-ind-dvds 30423 nexple 32802 2exple2exp 32803 oexpled 32805 fldext2rspun 33653 fldext2chn 33694 faclimlem3 35717 faclim2 35720 nn0prpwlem 36295 3lexlogpow5ineq5 42033 nicomachus 42285 abvexp 42505 3cubeslem2 42658 3cubeslem3l 42659 3cubeslem3r 42660 mzpexpmpt 42718 pell14qrexpclnn0 42839 jm2.17a 42933 jm2.17b 42934 jm2.17c 42935 jm2.18 42961 cnsrexpcl 43138 inductionexd 44128 binomcxplemnotnn0 44329 stoweidlem3 45985 stoweidlem19 46001 stirlinglem4 46059 stirlinglem7 46062 etransclem23 46239 sqrtpwpw2p 47523 fmtnorec2lem 47527 fmtnorec4 47534 fmtnoprmfac1lem 47549 fmtnoprmfac2 47552 fmtnofac1 47555 lighneallem3 47592 oexpnegALTV 47662 fppr2odd 47716 tgoldbachlt 47801 dignn0flhalflem2 48602 dignn0ehalf 48603 nn0sumshdiglemA 48605 nn0sumshdiglemB 48606 itcovalt2lem2lem2 48660

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