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Mirrors > Home > MPE Home > Th. List > exp1 | Structured version Visualization version GIF version |
Description: Value of a complex number raised to the first power. (Contributed by NM, 20-Oct-2004.) (Revised by Mario Carneiro, 2-Jul-2013.) |
Ref | Expression |
---|---|
exp1 | ⊢ (𝐴 ∈ ℂ → (𝐴↑1) = 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1nn 12277 | . . . 4 ⊢ 1 ∈ ℕ | |
2 | expnnval 14086 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 1 ∈ ℕ) → (𝐴↑1) = (seq1( · , (ℕ × {𝐴}))‘1)) | |
3 | 1, 2 | mpan2 689 | . . 3 ⊢ (𝐴 ∈ ℂ → (𝐴↑1) = (seq1( · , (ℕ × {𝐴}))‘1)) |
4 | 1z 12646 | . . . 4 ⊢ 1 ∈ ℤ | |
5 | seq1 14036 | . . . 4 ⊢ (1 ∈ ℤ → (seq1( · , (ℕ × {𝐴}))‘1) = ((ℕ × {𝐴})‘1)) | |
6 | 4, 5 | ax-mp 5 | . . 3 ⊢ (seq1( · , (ℕ × {𝐴}))‘1) = ((ℕ × {𝐴})‘1) |
7 | 3, 6 | eqtrdi 2782 | . 2 ⊢ (𝐴 ∈ ℂ → (𝐴↑1) = ((ℕ × {𝐴})‘1)) |
8 | fvconst2g 7221 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 1 ∈ ℕ) → ((ℕ × {𝐴})‘1) = 𝐴) | |
9 | 1, 8 | mpan2 689 | . 2 ⊢ (𝐴 ∈ ℂ → ((ℕ × {𝐴})‘1) = 𝐴) |
10 | 7, 9 | eqtrd 2766 | 1 ⊢ (𝐴 ∈ ℂ → (𝐴↑1) = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1534 ∈ wcel 2099 {csn 4633 × cxp 5682 ‘cfv 6556 (class class class)co 7426 ℂcc 11158 1c1 11161 · cmul 11165 ℕcn 12266 ℤcz 12612 seqcseq 14023 ↑cexp 14083 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-sep 5306 ax-nul 5313 ax-pow 5371 ax-pr 5435 ax-un 7748 ax-cnex 11216 ax-resscn 11217 ax-1cn 11218 ax-icn 11219 ax-addcl 11220 ax-addrcl 11221 ax-mulcl 11222 ax-mulrcl 11223 ax-mulcom 11224 ax-addass 11225 ax-mulass 11226 ax-distr 11227 ax-i2m1 11228 ax-1ne0 11229 ax-1rid 11230 ax-rnegex 11231 ax-rrecex 11232 ax-cnre 11233 ax-pre-lttri 11234 ax-pre-lttrn 11235 ax-pre-ltadd 11236 ax-pre-mulgt0 11237 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3967 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4916 df-iun 5005 df-br 5156 df-opab 5218 df-mpt 5239 df-tr 5273 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5639 df-we 5641 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-pred 6314 df-ord 6381 df-on 6382 df-lim 6383 df-suc 6384 df-iota 6508 df-fun 6558 df-fn 6559 df-f 6560 df-f1 6561 df-fo 6562 df-f1o 6563 df-fv 6564 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-om 7879 df-2nd 8006 df-frecs 8298 df-wrecs 8329 df-recs 8403 df-rdg 8442 df-er 8736 df-en 8977 df-dom 8978 df-sdom 8979 df-pnf 11302 df-mnf 11303 df-xr 11304 df-ltxr 11305 df-le 11306 df-sub 11498 df-neg 11499 df-nn 12267 df-n0 12527 df-z 12613 df-uz 12877 df-seq 14024 df-exp 14084 |
This theorem is referenced by: expp1 14090 expn1 14093 expcllem 14094 expeq0 14114 expp1z 14133 expm1 14134 sqval 14136 exp1d 14162 expmordi 14188 expnbnd 14251 digit1 14256 faclbnd4lem1 14312 climcndslem1 15855 climcndslem2 15856 geoisum1 15885 bpoly1 16055 ef4p 16117 efgt1p2 16118 efgt1p 16119 rpnnen2lem3 16220 modxp1i 17074 numexp1 17081 psgnpmtr 19510 lt6abl 19895 cphipval 25265 iblcnlem1 25811 itgcnlem 25813 dvexp 25979 dveflem 26005 plyid 26239 coeidp 26294 dgrid 26295 cxp1 26701 1cubrlem 26872 1cubr 26873 log2ublem3 26979 basellem5 27116 perfectlem2 27262 logdivsum 27565 log2sumbnd 27576 ipval2 30643 0dp2dp 32772 subfacval2 35017 dvasin 37407 areacirclem1 37411 1t10e1p1e11 46941 fmtnoge3 47120 fmtno0 47130 fmtno1 47131 lighneallem2 47196 lighneallem3 47197 41prothprmlem2 47208 perfectALTVlem2 47312 8exp8mod9 47326 tgblthelfgott 47405 exple2lt6 47761 pw2m1lepw2m1 47921 logbpw2m1 47973 nnpw2pmod 47989 |
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