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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 14106. (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 12525 | . 2 ⊢ (𝑁 ∈ ℕ0 ↔ (𝑁 ∈ ℕ ∨ 𝑁 = 0)) | |
2 | seqp1 14053 | . . . . . . 7 ⊢ (𝑁 ∈ (ℤ≥‘1) → (seq1( · , (ℕ × {𝐴}))‘(𝑁 + 1)) = ((seq1( · , (ℕ × {𝐴}))‘𝑁) · ((ℕ × {𝐴})‘(𝑁 + 1)))) | |
3 | nnuz 12918 | . . . . . . 7 ⊢ ℕ = (ℤ≥‘1) | |
4 | 2, 3 | eleq2s 2856 | . . . . . 6 ⊢ (𝑁 ∈ ℕ → (seq1( · , (ℕ × {𝐴}))‘(𝑁 + 1)) = ((seq1( · , (ℕ × {𝐴}))‘𝑁) · ((ℕ × {𝐴})‘(𝑁 + 1)))) |
5 | 4 | adantl 481 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) → (seq1( · , (ℕ × {𝐴}))‘(𝑁 + 1)) = ((seq1( · , (ℕ × {𝐴}))‘𝑁) · ((ℕ × {𝐴})‘(𝑁 + 1)))) |
6 | peano2nn 12275 | . . . . . . 7 ⊢ (𝑁 ∈ ℕ → (𝑁 + 1) ∈ ℕ) | |
7 | fvconst2g 7221 | . . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ (𝑁 + 1) ∈ ℕ) → ((ℕ × {𝐴})‘(𝑁 + 1)) = 𝐴) | |
8 | 6, 7 | sylan2 593 | . . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) → ((ℕ × {𝐴})‘(𝑁 + 1)) = 𝐴) |
9 | 8 | oveq2d 7446 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) → ((seq1( · , (ℕ × {𝐴}))‘𝑁) · ((ℕ × {𝐴})‘(𝑁 + 1))) = ((seq1( · , (ℕ × {𝐴}))‘𝑁) · 𝐴)) |
10 | 5, 9 | eqtrd 2774 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) → (seq1( · , (ℕ × {𝐴}))‘(𝑁 + 1)) = ((seq1( · , (ℕ × {𝐴}))‘𝑁) · 𝐴)) |
11 | expnnval 14101 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ (𝑁 + 1) ∈ ℕ) → (𝐴↑(𝑁 + 1)) = (seq1( · , (ℕ × {𝐴}))‘(𝑁 + 1))) | |
12 | 6, 11 | sylan2 593 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) → (𝐴↑(𝑁 + 1)) = (seq1( · , (ℕ × {𝐴}))‘(𝑁 + 1))) |
13 | expnnval 14101 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) → (𝐴↑𝑁) = (seq1( · , (ℕ × {𝐴}))‘𝑁)) | |
14 | 13 | oveq1d 7445 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) → ((𝐴↑𝑁) · 𝐴) = ((seq1( · , (ℕ × {𝐴}))‘𝑁) · 𝐴)) |
15 | 10, 12, 14 | 3eqtr4d 2784 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) → (𝐴↑(𝑁 + 1)) = ((𝐴↑𝑁) · 𝐴)) |
16 | exp1 14104 | . . . . . 6 ⊢ (𝐴 ∈ ℂ → (𝐴↑1) = 𝐴) | |
17 | mullid 11257 | . . . . . 6 ⊢ (𝐴 ∈ ℂ → (1 · 𝐴) = 𝐴) | |
18 | 16, 17 | eqtr4d 2777 | . . . . 5 ⊢ (𝐴 ∈ ℂ → (𝐴↑1) = (1 · 𝐴)) |
19 | 18 | adantr 480 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 = 0) → (𝐴↑1) = (1 · 𝐴)) |
20 | simpr 484 | . . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 = 0) → 𝑁 = 0) | |
21 | 20 | oveq1d 7445 | . . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 = 0) → (𝑁 + 1) = (0 + 1)) |
22 | 0p1e1 12385 | . . . . . 6 ⊢ (0 + 1) = 1 | |
23 | 21, 22 | eqtrdi 2790 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 = 0) → (𝑁 + 1) = 1) |
24 | 23 | oveq2d 7446 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 = 0) → (𝐴↑(𝑁 + 1)) = (𝐴↑1)) |
25 | oveq2 7438 | . . . . . 6 ⊢ (𝑁 = 0 → (𝐴↑𝑁) = (𝐴↑0)) | |
26 | exp0 14102 | . . . . . 6 ⊢ (𝐴 ∈ ℂ → (𝐴↑0) = 1) | |
27 | 25, 26 | sylan9eqr 2796 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 = 0) → (𝐴↑𝑁) = 1) |
28 | 27 | oveq1d 7445 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 = 0) → ((𝐴↑𝑁) · 𝐴) = (1 · 𝐴)) |
29 | 19, 24, 28 | 3eqtr4d 2784 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 = 0) → (𝐴↑(𝑁 + 1)) = ((𝐴↑𝑁) · 𝐴)) |
30 | 15, 29 | jaodan 959 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ (𝑁 ∈ ℕ ∨ 𝑁 = 0)) → (𝐴↑(𝑁 + 1)) = ((𝐴↑𝑁) · 𝐴)) |
31 | 1, 30 | sylan2b 594 | 1 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0) → (𝐴↑(𝑁 + 1)) = ((𝐴↑𝑁) · 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∨ wo 847 = wceq 1536 ∈ wcel 2105 {csn 4630 × cxp 5686 ‘cfv 6562 (class class class)co 7430 ℂcc 11150 0cc0 11152 1c1 11153 + caddc 11155 · cmul 11157 ℕcn 12263 ℕ0cn0 12523 ℤ≥cuz 12875 seqcseq 14038 ↑cexp 14098 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 ax-cnex 11208 ax-resscn 11209 ax-1cn 11210 ax-icn 11211 ax-addcl 11212 ax-addrcl 11213 ax-mulcl 11214 ax-mulrcl 11215 ax-mulcom 11216 ax-addass 11217 ax-mulass 11218 ax-distr 11219 ax-i2m1 11220 ax-1ne0 11221 ax-1rid 11222 ax-rnegex 11223 ax-rrecex 11224 ax-cnre 11225 ax-pre-lttri 11226 ax-pre-lttrn 11227 ax-pre-ltadd 11228 ax-pre-mulgt0 11229 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-iun 4997 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-we 5642 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-pred 6322 df-ord 6388 df-on 6389 df-lim 6390 df-suc 6391 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-riota 7387 df-ov 7433 df-oprab 7434 df-mpo 7435 df-om 7887 df-2nd 8013 df-frecs 8304 df-wrecs 8335 df-recs 8409 df-rdg 8448 df-er 8743 df-en 8984 df-dom 8985 df-sdom 8986 df-pnf 11294 df-mnf 11295 df-xr 11296 df-ltxr 11297 df-le 11298 df-sub 11491 df-neg 11492 df-nn 12264 df-n0 12524 df-z 12611 df-uz 12876 df-seq 14039 df-exp 14099 |
This theorem is referenced by: expcllem 14109 expm1t 14127 expeq0 14129 mulexp 14138 expadd 14141 expmul 14144 sqval 14151 expp1d 14183 leexp2r 14210 leexp1a 14211 cu2 14235 i3 14238 binom3 14259 bernneq 14264 modexp 14273 faclbnd 14325 faclbnd2 14326 faclbnd4lem1 14328 faclbnd6 14334 cjexp 15185 absexp 15339 binomlem 15861 climcndslem1 15881 climcndslem2 15882 pwdif 15900 geolim 15902 geo2sum 15905 efexp 16133 demoivreALT 16233 rpnnen2lem11 16256 pwp1fsum 16424 prmdvdsexp 16748 pcexp 16892 prmreclem6 16954 decexp2 17108 numexpp1 17111 2exp7 17121 cnfldexp 21434 expcn 24909 expcnOLD 24911 mbfi1fseqlem5 25768 dvexp 26005 aaliou3lem2 26399 tangtx 26561 cxpmul2 26745 mcubic 26904 cubic2 26905 binom4 26907 dquartlem2 26909 quart1lem 26912 quart1 26913 quartlem1 26914 log2cnv 27001 log2ublem2 27004 log2ub 27006 basellem3 27140 chtublem 27269 perfectlem1 27287 perfectlem2 27288 bclbnd 27338 bposlem8 27349 dchrisum0flblem1 27566 pntlemo 27665 qabvexp 27684 psgnfzto1st 33107 oddpwdc 34335 hgt750lem 34644 subfacval2 35171 sinccvglem 35656 heiborlem6 37802 bfplem1 37808 3lexlogpow5ineq1 42035 perfectALTVlem1 47645 perfectALTVlem2 47646 altgsumbcALT 48197 |
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