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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 12218 | . . . 4 ⊢ 1 ∈ ℕ | |
2 | expnnval 14025 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 1 ∈ ℕ) → (𝐴↑1) = (seq1( · , (ℕ × {𝐴}))‘1)) | |
3 | 1, 2 | mpan2 690 | . . 3 ⊢ (𝐴 ∈ ℂ → (𝐴↑1) = (seq1( · , (ℕ × {𝐴}))‘1)) |
4 | 1z 12587 | . . . 4 ⊢ 1 ∈ ℤ | |
5 | seq1 13974 | . . . 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 2789 | . 2 ⊢ (𝐴 ∈ ℂ → (𝐴↑1) = ((ℕ × {𝐴})‘1)) |
8 | fvconst2g 7197 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 1 ∈ ℕ) → ((ℕ × {𝐴})‘1) = 𝐴) | |
9 | 1, 8 | mpan2 690 | . 2 ⊢ (𝐴 ∈ ℂ → ((ℕ × {𝐴})‘1) = 𝐴) |
10 | 7, 9 | eqtrd 2773 | 1 ⊢ (𝐴 ∈ ℂ → (𝐴↑1) = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2107 {csn 4626 × cxp 5672 ‘cfv 6539 (class class class)co 7403 ℂcc 11103 1c1 11106 · cmul 11110 ℕcn 12207 ℤcz 12553 seqcseq 13961 ↑cexp 14022 |
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 5297 ax-nul 5304 ax-pow 5361 ax-pr 5425 ax-un 7719 ax-cnex 11161 ax-resscn 11162 ax-1cn 11163 ax-icn 11164 ax-addcl 11165 ax-addrcl 11166 ax-mulcl 11167 ax-mulrcl 11168 ax-mulcom 11169 ax-addass 11170 ax-mulass 11171 ax-distr 11172 ax-i2m1 11173 ax-1ne0 11174 ax-1rid 11175 ax-rnegex 11176 ax-rrecex 11177 ax-cnre 11178 ax-pre-lttri 11179 ax-pre-lttrn 11180 ax-pre-ltadd 11181 ax-pre-mulgt0 11182 |
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 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-pss 3965 df-nul 4321 df-if 4527 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4907 df-iun 4997 df-br 5147 df-opab 5209 df-mpt 5230 df-tr 5264 df-id 5572 df-eprel 5578 df-po 5586 df-so 5587 df-fr 5629 df-we 5631 df-xp 5680 df-rel 5681 df-cnv 5682 df-co 5683 df-dm 5684 df-rn 5685 df-res 5686 df-ima 5687 df-pred 6296 df-ord 6363 df-on 6364 df-lim 6365 df-suc 6366 df-iota 6491 df-fun 6541 df-fn 6542 df-f 6543 df-f1 6544 df-fo 6545 df-f1o 6546 df-fv 6547 df-riota 7359 df-ov 7406 df-oprab 7407 df-mpo 7408 df-om 7850 df-2nd 7970 df-frecs 8260 df-wrecs 8291 df-recs 8365 df-rdg 8404 df-er 8698 df-en 8935 df-dom 8936 df-sdom 8937 df-pnf 11245 df-mnf 11246 df-xr 11247 df-ltxr 11248 df-le 11249 df-sub 11441 df-neg 11442 df-nn 12208 df-n0 12468 df-z 12554 df-uz 12818 df-seq 13962 df-exp 14023 |
This theorem is referenced by: expp1 14029 expn1 14032 expcllem 14033 expeq0 14053 expp1z 14072 expm1 14073 sqval 14075 exp1d 14101 expmordi 14127 expnbnd 14190 digit1 14195 faclbnd4lem1 14248 climcndslem1 15790 climcndslem2 15791 geoisum1 15820 bpoly1 15990 ef4p 16051 efgt1p2 16052 efgt1p 16053 rpnnen2lem3 16154 modxp1i 16998 numexp1 17005 psgnpmtr 19370 lt6abl 19754 cphipval 24741 iblcnlem1 25286 itgcnlem 25288 dvexp 25451 dveflem 25477 plyid 25704 coeidp 25758 dgrid 25759 cxp1 26160 1cubrlem 26325 1cubr 26326 log2ublem3 26432 basellem5 26568 perfectlem2 26712 logdivsum 27015 log2sumbnd 27026 ipval2 29937 0dp2dp 32052 subfacval2 34115 dvasin 36509 areacirclem1 36513 1t10e1p1e11 45952 fmtnoge3 46132 fmtno0 46142 fmtno1 46143 lighneallem2 46208 lighneallem3 46209 41prothprmlem2 46220 perfectALTVlem2 46324 8exp8mod9 46338 tgblthelfgott 46417 exple2lt6 46941 pw2m1lepw2m1 47102 logbpw2m1 47154 nnpw2pmod 47170 |
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