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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 14110. (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 12528 | . 2 ⊢ (𝑁 ∈ ℕ0 ↔ (𝑁 ∈ ℕ ∨ 𝑁 = 0)) | |
| 2 | seqp1 14057 | . . . . . . 7 ⊢ (𝑁 ∈ (ℤ≥‘1) → (seq1( · , (ℕ × {𝐴}))‘(𝑁 + 1)) = ((seq1( · , (ℕ × {𝐴}))‘𝑁) · ((ℕ × {𝐴})‘(𝑁 + 1)))) | |
| 3 | nnuz 12921 | . . . . . . 7 ⊢ ℕ = (ℤ≥‘1) | |
| 4 | 2, 3 | eleq2s 2859 | . . . . . 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 12278 | . . . . . . 7 ⊢ (𝑁 ∈ ℕ → (𝑁 + 1) ∈ ℕ) | |
| 7 | fvconst2g 7222 | . . . . . . 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 7447 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) → ((seq1( · , (ℕ × {𝐴}))‘𝑁) · ((ℕ × {𝐴})‘(𝑁 + 1))) = ((seq1( · , (ℕ × {𝐴}))‘𝑁) · 𝐴)) |
| 10 | 5, 9 | eqtrd 2777 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) → (seq1( · , (ℕ × {𝐴}))‘(𝑁 + 1)) = ((seq1( · , (ℕ × {𝐴}))‘𝑁) · 𝐴)) |
| 11 | expnnval 14105 | . . . . 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 14105 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) → (𝐴↑𝑁) = (seq1( · , (ℕ × {𝐴}))‘𝑁)) | |
| 14 | 13 | oveq1d 7446 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) → ((𝐴↑𝑁) · 𝐴) = ((seq1( · , (ℕ × {𝐴}))‘𝑁) · 𝐴)) |
| 15 | 10, 12, 14 | 3eqtr4d 2787 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) → (𝐴↑(𝑁 + 1)) = ((𝐴↑𝑁) · 𝐴)) |
| 16 | exp1 14108 | . . . . . 6 ⊢ (𝐴 ∈ ℂ → (𝐴↑1) = 𝐴) | |
| 17 | mullid 11260 | . . . . . 6 ⊢ (𝐴 ∈ ℂ → (1 · 𝐴) = 𝐴) | |
| 18 | 16, 17 | eqtr4d 2780 | . . . . 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 7446 | . . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 = 0) → (𝑁 + 1) = (0 + 1)) |
| 22 | 0p1e1 12388 | . . . . . 6 ⊢ (0 + 1) = 1 | |
| 23 | 21, 22 | eqtrdi 2793 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 = 0) → (𝑁 + 1) = 1) |
| 24 | 23 | oveq2d 7447 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 = 0) → (𝐴↑(𝑁 + 1)) = (𝐴↑1)) |
| 25 | oveq2 7439 | . . . . . 6 ⊢ (𝑁 = 0 → (𝐴↑𝑁) = (𝐴↑0)) | |
| 26 | exp0 14106 | . . . . . 6 ⊢ (𝐴 ∈ ℂ → (𝐴↑0) = 1) | |
| 27 | 25, 26 | sylan9eqr 2799 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 = 0) → (𝐴↑𝑁) = 1) |
| 28 | 27 | oveq1d 7446 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 = 0) → ((𝐴↑𝑁) · 𝐴) = (1 · 𝐴)) |
| 29 | 19, 24, 28 | 3eqtr4d 2787 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 = 0) → (𝐴↑(𝑁 + 1)) = ((𝐴↑𝑁) · 𝐴)) |
| 30 | 15, 29 | jaodan 960 | . 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 848 = wceq 1540 ∈ wcel 2108 {csn 4626 × cxp 5683 ‘cfv 6561 (class class class)co 7431 ℂcc 11153 0cc0 11155 1c1 11156 + caddc 11158 · cmul 11160 ℕcn 12266 ℕ0cn0 12526 ℤ≥cuz 12878 seqcseq 14042 ↑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: expcllem 14113 expm1t 14131 expeq0 14133 mulexp 14142 expadd 14145 expmul 14148 sqval 14155 expp1d 14187 leexp2r 14214 leexp1a 14215 cu2 14239 i3 14242 binom3 14263 bernneq 14268 modexp 14277 faclbnd 14329 faclbnd2 14330 faclbnd4lem1 14332 faclbnd6 14338 cjexp 15189 absexp 15343 binomlem 15865 climcndslem1 15885 climcndslem2 15886 pwdif 15904 geolim 15906 geo2sum 15909 efexp 16137 demoivreALT 16237 rpnnen2lem11 16260 pwp1fsum 16428 prmdvdsexp 16752 pcexp 16897 prmreclem6 16959 numexpp1 17115 2exp7 17125 cnfldexp 21417 expcn 24896 expcnOLD 24898 mbfi1fseqlem5 25754 dvexp 25991 aaliou3lem2 26385 tangtx 26547 cxpmul2 26731 mcubic 26890 cubic2 26891 binom4 26893 dquartlem2 26895 quart1lem 26898 quart1 26899 quartlem1 26900 log2cnv 26987 log2ublem2 26990 log2ub 26992 basellem3 27126 chtublem 27255 perfectlem1 27273 perfectlem2 27274 bclbnd 27324 bposlem8 27335 dchrisum0flblem1 27552 pntlemo 27651 qabvexp 27670 psgnfzto1st 33125 oddpwdc 34356 hgt750lem 34666 subfacval2 35192 sinccvglem 35677 heiborlem6 37823 bfplem1 37829 3lexlogpow5ineq1 42055 perfectALTVlem1 47708 perfectALTVlem2 47709 altgsumbcALT 48269 |
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