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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. (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 11171 | . 2 ⊢ (𝑁 ∈ ℕ0 ↔ (𝑁 ∈ ℕ ∨ 𝑁 = 0)) | |
2 | seqp1 12678 | . . . . . . 7 ⊢ (𝑁 ∈ (ℤ≥‘1) → (seq1( · , (ℕ × {𝐴}))‘(𝑁 + 1)) = ((seq1( · , (ℕ × {𝐴}))‘𝑁) · ((ℕ × {𝐴})‘(𝑁 + 1)))) | |
3 | nnuz 11599 | . . . . . . 7 ⊢ ℕ = (ℤ≥‘1) | |
4 | 2, 3 | eleq2s 2706 | . . . . . 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 10909 | . . . . . . 7 ⊢ (𝑁 ∈ ℕ → (𝑁 + 1) ∈ ℕ) | |
7 | fvconst2g 6372 | . . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ (𝑁 + 1) ∈ ℕ) → ((ℕ × {𝐴})‘(𝑁 + 1)) = 𝐴) | |
8 | 6, 7 | sylan2 490 | . . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) → ((ℕ × {𝐴})‘(𝑁 + 1)) = 𝐴) |
9 | 8 | oveq2d 6565 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) → ((seq1( · , (ℕ × {𝐴}))‘𝑁) · ((ℕ × {𝐴})‘(𝑁 + 1))) = ((seq1( · , (ℕ × {𝐴}))‘𝑁) · 𝐴)) |
10 | 5, 9 | eqtrd 2644 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) → (seq1( · , (ℕ × {𝐴}))‘(𝑁 + 1)) = ((seq1( · , (ℕ × {𝐴}))‘𝑁) · 𝐴)) |
11 | expnnval 12725 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ (𝑁 + 1) ∈ ℕ) → (𝐴↑(𝑁 + 1)) = (seq1( · , (ℕ × {𝐴}))‘(𝑁 + 1))) | |
12 | 6, 11 | sylan2 490 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) → (𝐴↑(𝑁 + 1)) = (seq1( · , (ℕ × {𝐴}))‘(𝑁 + 1))) |
13 | expnnval 12725 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) → (𝐴↑𝑁) = (seq1( · , (ℕ × {𝐴}))‘𝑁)) | |
14 | 13 | oveq1d 6564 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) → ((𝐴↑𝑁) · 𝐴) = ((seq1( · , (ℕ × {𝐴}))‘𝑁) · 𝐴)) |
15 | 10, 12, 14 | 3eqtr4d 2654 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) → (𝐴↑(𝑁 + 1)) = ((𝐴↑𝑁) · 𝐴)) |
16 | exp1 12728 | . . . . . 6 ⊢ (𝐴 ∈ ℂ → (𝐴↑1) = 𝐴) | |
17 | mulid2 9917 | . . . . . 6 ⊢ (𝐴 ∈ ℂ → (1 · 𝐴) = 𝐴) | |
18 | 16, 17 | eqtr4d 2647 | . . . . 5 ⊢ (𝐴 ∈ ℂ → (𝐴↑1) = (1 · 𝐴)) |
19 | 18 | adantr 480 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 = 0) → (𝐴↑1) = (1 · 𝐴)) |
20 | simpr 476 | . . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 = 0) → 𝑁 = 0) | |
21 | 20 | oveq1d 6564 | . . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 = 0) → (𝑁 + 1) = (0 + 1)) |
22 | 0p1e1 11009 | . . . . . 6 ⊢ (0 + 1) = 1 | |
23 | 21, 22 | syl6eq 2660 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 = 0) → (𝑁 + 1) = 1) |
24 | 23 | oveq2d 6565 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 = 0) → (𝐴↑(𝑁 + 1)) = (𝐴↑1)) |
25 | oveq2 6557 | . . . . . 6 ⊢ (𝑁 = 0 → (𝐴↑𝑁) = (𝐴↑0)) | |
26 | exp0 12726 | . . . . . 6 ⊢ (𝐴 ∈ ℂ → (𝐴↑0) = 1) | |
27 | 25, 26 | sylan9eqr 2666 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 = 0) → (𝐴↑𝑁) = 1) |
28 | 27 | oveq1d 6564 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 = 0) → ((𝐴↑𝑁) · 𝐴) = (1 · 𝐴)) |
29 | 19, 24, 28 | 3eqtr4d 2654 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 = 0) → (𝐴↑(𝑁 + 1)) = ((𝐴↑𝑁) · 𝐴)) |
30 | 15, 29 | jaodan 822 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ (𝑁 ∈ ℕ ∨ 𝑁 = 0)) → (𝐴↑(𝑁 + 1)) = ((𝐴↑𝑁) · 𝐴)) |
31 | 1, 30 | sylan2b 491 | 1 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0) → (𝐴↑(𝑁 + 1)) = ((𝐴↑𝑁) · 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∨ wo 382 ∧ wa 383 = wceq 1475 ∈ wcel 1977 {csn 4125 × cxp 5036 ‘cfv 5804 (class class class)co 6549 ℂcc 9813 0cc0 9815 1c1 9816 + caddc 9818 · cmul 9820 ℕcn 10897 ℕ0cn0 11169 ℤ≥cuz 11563 seqcseq 12663 ↑cexp 12722 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-n0 11170 df-z 11255 df-uz 11564 df-seq 12664 df-exp 12723 |
This theorem is referenced by: expcllem 12733 expm1t 12750 expeq0 12752 mulexp 12761 expadd 12764 expmul 12767 leexp2r 12780 leexp1a 12781 sqval 12784 cu2 12825 i3 12828 binom3 12847 bernneq 12852 modexp 12861 expp1d 12871 faclbnd 12939 faclbnd2 12940 faclbnd4lem1 12942 faclbnd6 12948 cjexp 13738 absexp 13892 binomlem 14400 climcndslem1 14420 climcndslem2 14421 geolim 14440 geo2sum 14443 efexp 14670 demoivreALT 14770 rpnnen2lem11 14792 pwp1fsum 14952 prmdvdsexp 15265 pcexp 15402 prmreclem6 15463 decexp2 15617 numexpp1 15620 cnfldexp 19598 expcn 22483 mbfi1fseqlem5 23292 dvexp 23522 aaliou3lem2 23902 tangtx 24061 cxpmul2 24235 mcubic 24374 cubic2 24375 binom4 24377 dquartlem2 24379 quart1lem 24382 quart1 24383 quartlem1 24384 log2cnv 24471 log2ublem2 24474 log2ub 24476 basellem3 24609 chtublem 24736 perfectlem1 24754 perfectlem2 24755 bclbnd 24805 bposlem8 24816 dchrisum0flblem1 24997 pntlemo 25096 qabvexp 25115 rusgranumwlks 26483 psgnfzto1st 29186 oddpwdc 29743 subfacval2 30423 sinccvglem 30820 heiborlem6 32785 bfplem1 32791 pwdif 40039 2exp7 40052 perfectALTVlem1 40164 perfectALTVlem2 40165 altgsumbcALT 41924 |
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