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Mirrors > Home > MPE Home > Th. List > expp1 | 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. When 𝐴 is nonzero, this holds for all integers 𝑁, see expneg 14032. (Contributed by NM, 20-May-2005.) (Revised by Mario Carneiro, 2-Jul-2013.) |
Ref | Expression |
---|---|
expp1 | ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0) → (𝐴↑(𝑁 + 1)) = ((𝐴↑𝑁) · 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elnn0 12471 | . 2 ⊢ (𝑁 ∈ ℕ0 ↔ (𝑁 ∈ ℕ ∨ 𝑁 = 0)) | |
2 | seqp1 13978 | . . . . . . 7 ⊢ (𝑁 ∈ (ℤ≥‘1) → (seq1( · , (ℕ × {𝐴}))‘(𝑁 + 1)) = ((seq1( · , (ℕ × {𝐴}))‘𝑁) · ((ℕ × {𝐴})‘(𝑁 + 1)))) | |
3 | nnuz 12862 | . . . . . . 7 ⊢ ℕ = (ℤ≥‘1) | |
4 | 2, 3 | eleq2s 2852 | . . . . . 6 ⊢ (𝑁 ∈ ℕ → (seq1( · , (ℕ × {𝐴}))‘(𝑁 + 1)) = ((seq1( · , (ℕ × {𝐴}))‘𝑁) · ((ℕ × {𝐴})‘(𝑁 + 1)))) |
5 | 4 | adantl 483 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) → (seq1( · , (ℕ × {𝐴}))‘(𝑁 + 1)) = ((seq1( · , (ℕ × {𝐴}))‘𝑁) · ((ℕ × {𝐴})‘(𝑁 + 1)))) |
6 | peano2nn 12221 | . . . . . . 7 ⊢ (𝑁 ∈ ℕ → (𝑁 + 1) ∈ ℕ) | |
7 | fvconst2g 7200 | . . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ (𝑁 + 1) ∈ ℕ) → ((ℕ × {𝐴})‘(𝑁 + 1)) = 𝐴) | |
8 | 6, 7 | sylan2 594 | . . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) → ((ℕ × {𝐴})‘(𝑁 + 1)) = 𝐴) |
9 | 8 | oveq2d 7422 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) → ((seq1( · , (ℕ × {𝐴}))‘𝑁) · ((ℕ × {𝐴})‘(𝑁 + 1))) = ((seq1( · , (ℕ × {𝐴}))‘𝑁) · 𝐴)) |
10 | 5, 9 | eqtrd 2773 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) → (seq1( · , (ℕ × {𝐴}))‘(𝑁 + 1)) = ((seq1( · , (ℕ × {𝐴}))‘𝑁) · 𝐴)) |
11 | expnnval 14027 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ (𝑁 + 1) ∈ ℕ) → (𝐴↑(𝑁 + 1)) = (seq1( · , (ℕ × {𝐴}))‘(𝑁 + 1))) | |
12 | 6, 11 | sylan2 594 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) → (𝐴↑(𝑁 + 1)) = (seq1( · , (ℕ × {𝐴}))‘(𝑁 + 1))) |
13 | expnnval 14027 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) → (𝐴↑𝑁) = (seq1( · , (ℕ × {𝐴}))‘𝑁)) | |
14 | 13 | oveq1d 7421 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) → ((𝐴↑𝑁) · 𝐴) = ((seq1( · , (ℕ × {𝐴}))‘𝑁) · 𝐴)) |
15 | 10, 12, 14 | 3eqtr4d 2783 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) → (𝐴↑(𝑁 + 1)) = ((𝐴↑𝑁) · 𝐴)) |
16 | exp1 14030 | . . . . . 6 ⊢ (𝐴 ∈ ℂ → (𝐴↑1) = 𝐴) | |
17 | mullid 11210 | . . . . . 6 ⊢ (𝐴 ∈ ℂ → (1 · 𝐴) = 𝐴) | |
18 | 16, 17 | eqtr4d 2776 | . . . . 5 ⊢ (𝐴 ∈ ℂ → (𝐴↑1) = (1 · 𝐴)) |
19 | 18 | adantr 482 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 = 0) → (𝐴↑1) = (1 · 𝐴)) |
20 | simpr 486 | . . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 = 0) → 𝑁 = 0) | |
21 | 20 | oveq1d 7421 | . . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 = 0) → (𝑁 + 1) = (0 + 1)) |
22 | 0p1e1 12331 | . . . . . 6 ⊢ (0 + 1) = 1 | |
23 | 21, 22 | eqtrdi 2789 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 = 0) → (𝑁 + 1) = 1) |
24 | 23 | oveq2d 7422 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 = 0) → (𝐴↑(𝑁 + 1)) = (𝐴↑1)) |
25 | oveq2 7414 | . . . . . 6 ⊢ (𝑁 = 0 → (𝐴↑𝑁) = (𝐴↑0)) | |
26 | exp0 14028 | . . . . . 6 ⊢ (𝐴 ∈ ℂ → (𝐴↑0) = 1) | |
27 | 25, 26 | sylan9eqr 2795 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 = 0) → (𝐴↑𝑁) = 1) |
28 | 27 | oveq1d 7421 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 = 0) → ((𝐴↑𝑁) · 𝐴) = (1 · 𝐴)) |
29 | 19, 24, 28 | 3eqtr4d 2783 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 = 0) → (𝐴↑(𝑁 + 1)) = ((𝐴↑𝑁) · 𝐴)) |
30 | 15, 29 | jaodan 957 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ (𝑁 ∈ ℕ ∨ 𝑁 = 0)) → (𝐴↑(𝑁 + 1)) = ((𝐴↑𝑁) · 𝐴)) |
31 | 1, 30 | sylan2b 595 | 1 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0) → (𝐴↑(𝑁 + 1)) = ((𝐴↑𝑁) · 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 ∨ wo 846 = wceq 1542 ∈ wcel 2107 {csn 4628 × cxp 5674 ‘cfv 6541 (class class class)co 7406 ℂcc 11105 0cc0 11107 1c1 11108 + caddc 11110 · cmul 11112 ℕcn 12209 ℕ0cn0 12469 ℤ≥cuz 12819 seqcseq 13963 ↑cexp 14024 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7722 ax-cnex 11163 ax-resscn 11164 ax-1cn 11165 ax-icn 11166 ax-addcl 11167 ax-addrcl 11168 ax-mulcl 11169 ax-mulrcl 11170 ax-mulcom 11171 ax-addass 11172 ax-mulass 11173 ax-distr 11174 ax-i2m1 11175 ax-1ne0 11176 ax-1rid 11177 ax-rnegex 11178 ax-rrecex 11179 ax-cnre 11180 ax-pre-lttri 11181 ax-pre-lttrn 11182 ax-pre-ltadd 11183 ax-pre-mulgt0 11184 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6298 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6493 df-fun 6543 df-fn 6544 df-f 6545 df-f1 6546 df-fo 6547 df-f1o 6548 df-fv 6549 df-riota 7362 df-ov 7409 df-oprab 7410 df-mpo 7411 df-om 7853 df-2nd 7973 df-frecs 8263 df-wrecs 8294 df-recs 8368 df-rdg 8407 df-er 8700 df-en 8937 df-dom 8938 df-sdom 8939 df-pnf 11247 df-mnf 11248 df-xr 11249 df-ltxr 11250 df-le 11251 df-sub 11443 df-neg 11444 df-nn 12210 df-n0 12470 df-z 12556 df-uz 12820 df-seq 13964 df-exp 14025 |
This theorem is referenced by: expcllem 14035 expm1t 14053 expeq0 14055 mulexp 14064 expadd 14067 expmul 14070 sqval 14077 expp1d 14109 leexp2r 14136 leexp1a 14137 cu2 14161 i3 14164 binom3 14184 bernneq 14189 modexp 14198 faclbnd 14247 faclbnd2 14248 faclbnd4lem1 14250 faclbnd6 14256 cjexp 15094 absexp 15248 binomlem 15772 climcndslem1 15792 climcndslem2 15793 pwdif 15811 geolim 15813 geo2sum 15816 efexp 16041 demoivreALT 16141 rpnnen2lem11 16164 pwp1fsum 16331 prmdvdsexp 16649 pcexp 16789 prmreclem6 16851 decexp2 17005 numexpp1 17008 2exp7 17018 cnfldexp 20971 expcn 24380 mbfi1fseqlem5 25229 dvexp 25462 aaliou3lem2 25848 tangtx 26007 cxpmul2 26189 mcubic 26342 cubic2 26343 binom4 26345 dquartlem2 26347 quart1lem 26350 quart1 26351 quartlem1 26352 log2cnv 26439 log2ublem2 26442 log2ub 26444 basellem3 26577 chtublem 26704 perfectlem1 26722 perfectlem2 26723 bclbnd 26773 bposlem8 26784 dchrisum0flblem1 27001 pntlemo 27100 qabvexp 27119 psgnfzto1st 32252 oddpwdc 33342 hgt750lem 33652 subfacval2 34167 sinccvglem 34646 gg-expcn 35153 heiborlem6 36673 bfplem1 36679 3lexlogpow5ineq1 40908 perfectALTVlem1 46376 perfectALTVlem2 46377 altgsumbcALT 46983 |
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