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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 11649 | . . . 4 ⊢ 1 ∈ ℕ | |
2 | expnnval 13433 | . . . 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 12013 | . . . 4 ⊢ 1 ∈ ℤ | |
5 | seq1 13383 | . . . 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 | syl6eq 2872 | . 2 ⊢ (𝐴 ∈ ℂ → (𝐴↑1) = ((ℕ × {𝐴})‘1)) |
8 | fvconst2g 6964 | . . 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 2856 | 1 ⊢ (𝐴 ∈ ℂ → (𝐴↑1) = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2114 {csn 4567 × cxp 5553 ‘cfv 6355 (class class class)co 7156 ℂcc 10535 1c1 10538 · cmul 10542 ℕcn 11638 ℤcz 11982 seqcseq 13370 ↑cexp 13430 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-nn 11639 df-n0 11899 df-z 11983 df-uz 12245 df-seq 13371 df-exp 13431 |
This theorem is referenced by: expp1 13437 expn1 13440 expcllem 13441 expeq0 13460 expp1z 13479 expm1 13480 sqval 13482 exp1d 13506 expmordi 13532 expnbnd 13594 digit1 13599 faclbnd4lem1 13654 climcndslem1 15204 climcndslem2 15205 geoisum1 15235 bpoly1 15405 ef4p 15466 efgt1p2 15467 efgt1p 15468 rpnnen2lem3 15569 modxp1i 16406 numexp1 16413 psgnpmtr 18638 lt6abl 19015 cphipval 23846 iblcnlem1 24388 itgcnlem 24390 dvexp 24550 dveflem 24576 plyid 24799 coeidp 24853 dgrid 24854 cxp1 25254 1cubrlem 25419 1cubr 25420 log2ublem3 25526 basellem5 25662 perfectlem2 25806 logdivsum 26109 log2sumbnd 26120 ipval2 28484 0dp2dp 30585 subfacval2 32434 dvasin 34993 areacirclem1 34997 1t10e1p1e11 43559 fmtnoge3 43741 fmtno0 43751 fmtno1 43752 lighneallem2 43820 lighneallem3 43821 41prothprmlem2 43832 perfectALTVlem2 43936 8exp8mod9 43950 tgblthelfgott 44029 exple2lt6 44461 pw2m1lepw2m1 44624 logbpw2m1 44676 nnpw2pmod 44692 |
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