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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 | โข (๐ด โ โ โ (๐ดโ1) = ๐ด) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1nn 12219 | . . . 4 โข 1 โ โ | |
2 | expnnval 14026 | . . . 4 โข ((๐ด โ โ โง 1 โ โ) โ (๐ดโ1) = (seq1( ยท , (โ ร {๐ด}))โ1)) | |
3 | 1, 2 | mpan2 689 | . . 3 โข (๐ด โ โ โ (๐ดโ1) = (seq1( ยท , (โ ร {๐ด}))โ1)) |
4 | 1z 12588 | . . . 4 โข 1 โ โค | |
5 | seq1 13975 | . . . 4 โข (1 โ โค โ (seq1( ยท , (โ ร {๐ด}))โ1) = ((โ ร {๐ด})โ1)) | |
6 | 4, 5 | ax-mp 5 | . . 3 โข (seq1( ยท , (โ ร {๐ด}))โ1) = ((โ ร {๐ด})โ1) |
7 | 3, 6 | eqtrdi 2788 | . 2 โข (๐ด โ โ โ (๐ดโ1) = ((โ ร {๐ด})โ1)) |
8 | fvconst2g 7199 | . . 3 โข ((๐ด โ โ โง 1 โ โ) โ ((โ ร {๐ด})โ1) = ๐ด) | |
9 | 1, 8 | mpan2 689 | . 2 โข (๐ด โ โ โ ((โ ร {๐ด})โ1) = ๐ด) |
10 | 7, 9 | eqtrd 2772 | 1 โข (๐ด โ โ โ (๐ดโ1) = ๐ด) |
Colors of variables: wff setvar class |
Syntax hints: โ wi 4 = wceq 1541 โ wcel 2106 {csn 4627 ร cxp 5673 โcfv 6540 (class class class)co 7405 โcc 11104 1c1 11107 ยท cmul 11111 โcn 12208 โคcz 12554 seqcseq 13962 โcexp 14023 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7852 df-2nd 7972 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-er 8699 df-en 8936 df-dom 8937 df-sdom 8938 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-nn 12209 df-n0 12469 df-z 12555 df-uz 12819 df-seq 13963 df-exp 14024 |
This theorem is referenced by: expp1 14030 expn1 14033 expcllem 14034 expeq0 14054 expp1z 14073 expm1 14074 sqval 14076 exp1d 14102 expmordi 14128 expnbnd 14191 digit1 14196 faclbnd4lem1 14249 climcndslem1 15791 climcndslem2 15792 geoisum1 15821 bpoly1 15991 ef4p 16052 efgt1p2 16053 efgt1p 16054 rpnnen2lem3 16155 modxp1i 16999 numexp1 17006 psgnpmtr 19372 lt6abl 19757 cphipval 24751 iblcnlem1 25296 itgcnlem 25298 dvexp 25461 dveflem 25487 plyid 25714 coeidp 25768 dgrid 25769 cxp1 26170 1cubrlem 26335 1cubr 26336 log2ublem3 26442 basellem5 26578 perfectlem2 26722 logdivsum 27025 log2sumbnd 27036 ipval2 29947 0dp2dp 32062 subfacval2 34166 dvasin 36560 areacirclem1 36564 1t10e1p1e11 46004 fmtnoge3 46184 fmtno0 46194 fmtno1 46195 lighneallem2 46260 lighneallem3 46261 41prothprmlem2 46272 perfectALTVlem2 46376 8exp8mod9 46390 tgblthelfgott 46469 exple2lt6 46993 pw2m1lepw2m1 47154 logbpw2m1 47206 nnpw2pmod 47222 |
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