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Mirrors > Home > MPE Home > Th. List > cxp1d | Structured version Visualization version GIF version |
Description: Value of the complex power function at one. (Contributed by Mario Carneiro, 30-May-2016.) |
Ref | Expression |
---|---|
cxp0d.1 | ⊢ (𝜑 → 𝐴 ∈ ℂ) |
Ref | Expression |
---|---|
cxp1d | ⊢ (𝜑 → (𝐴↑𝑐1) = 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cxp0d.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ ℂ) | |
2 | cxp1 24815 | . 2 ⊢ (𝐴 ∈ ℂ → (𝐴↑𝑐1) = 𝐴) | |
3 | 1, 2 | syl 17 | 1 ⊢ (𝜑 → (𝐴↑𝑐1) = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1658 ∈ wcel 2166 (class class class)co 6904 ℂcc 10249 1c1 10252 ↑𝑐ccxp 24700 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2390 ax-ext 2802 ax-rep 4993 ax-sep 5004 ax-nul 5012 ax-pow 5064 ax-pr 5126 ax-un 7208 ax-inf2 8814 ax-cnex 10307 ax-resscn 10308 ax-1cn 10309 ax-icn 10310 ax-addcl 10311 ax-addrcl 10312 ax-mulcl 10313 ax-mulrcl 10314 ax-mulcom 10315 ax-addass 10316 ax-mulass 10317 ax-distr 10318 ax-i2m1 10319 ax-1ne0 10320 ax-1rid 10321 ax-rnegex 10322 ax-rrecex 10323 ax-cnre 10324 ax-pre-lttri 10325 ax-pre-lttrn 10326 ax-pre-ltadd 10327 ax-pre-mulgt0 10328 ax-pre-sup 10329 ax-addf 10330 ax-mulf 10331 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3or 1114 df-3an 1115 df-tru 1662 df-fal 1672 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2604 df-eu 2639 df-clab 2811 df-cleq 2817 df-clel 2820 df-nfc 2957 df-ne 2999 df-nel 3102 df-ral 3121 df-rex 3122 df-reu 3123 df-rmo 3124 df-rab 3125 df-v 3415 df-sbc 3662 df-csb 3757 df-dif 3800 df-un 3802 df-in 3804 df-ss 3811 df-pss 3813 df-nul 4144 df-if 4306 df-pw 4379 df-sn 4397 df-pr 4399 df-tp 4401 df-op 4403 df-uni 4658 df-int 4697 df-iun 4741 df-iin 4742 df-br 4873 df-opab 4935 df-mpt 4952 df-tr 4975 df-id 5249 df-eprel 5254 df-po 5262 df-so 5263 df-fr 5300 df-se 5301 df-we 5302 df-xp 5347 df-rel 5348 df-cnv 5349 df-co 5350 df-dm 5351 df-rn 5352 df-res 5353 df-ima 5354 df-pred 5919 df-ord 5965 df-on 5966 df-lim 5967 df-suc 5968 df-iota 6085 df-fun 6124 df-fn 6125 df-f 6126 df-f1 6127 df-fo 6128 df-f1o 6129 df-fv 6130 df-isom 6131 df-riota 6865 df-ov 6907 df-oprab 6908 df-mpt2 6909 df-of 7156 df-om 7326 df-1st 7427 df-2nd 7428 df-supp 7559 df-wrecs 7671 df-recs 7733 df-rdg 7771 df-1o 7825 df-2o 7826 df-oadd 7829 df-er 8008 df-map 8123 df-pm 8124 df-ixp 8175 df-en 8222 df-dom 8223 df-sdom 8224 df-fin 8225 df-fsupp 8544 df-fi 8585 df-sup 8616 df-inf 8617 df-oi 8683 df-card 9077 df-cda 9304 df-pnf 10392 df-mnf 10393 df-xr 10394 df-ltxr 10395 df-le 10396 df-sub 10586 df-neg 10587 df-div 11009 df-nn 11350 df-2 11413 df-3 11414 df-4 11415 df-5 11416 df-6 11417 df-7 11418 df-8 11419 df-9 11420 df-n0 11618 df-z 11704 df-dec 11821 df-uz 11968 df-q 12071 df-rp 12112 df-xneg 12231 df-xadd 12232 df-xmul 12233 df-ioo 12466 df-ioc 12467 df-ico 12468 df-icc 12469 df-fz 12619 df-fzo 12760 df-fl 12887 df-mod 12963 df-seq 13095 df-exp 13154 df-fac 13353 df-bc 13382 df-hash 13410 df-shft 14183 df-cj 14215 df-re 14216 df-im 14217 df-sqrt 14351 df-abs 14352 df-limsup 14578 df-clim 14595 df-rlim 14596 df-sum 14793 df-ef 15169 df-sin 15171 df-cos 15172 df-pi 15174 df-struct 16223 df-ndx 16224 df-slot 16225 df-base 16227 df-sets 16228 df-ress 16229 df-plusg 16317 df-mulr 16318 df-starv 16319 df-sca 16320 df-vsca 16321 df-ip 16322 df-tset 16323 df-ple 16324 df-ds 16326 df-unif 16327 df-hom 16328 df-cco 16329 df-rest 16435 df-topn 16436 df-0g 16454 df-gsum 16455 df-topgen 16456 df-pt 16457 df-prds 16460 df-xrs 16514 df-qtop 16519 df-imas 16520 df-xps 16522 df-mre 16598 df-mrc 16599 df-acs 16601 df-mgm 17594 df-sgrp 17636 df-mnd 17647 df-submnd 17688 df-mulg 17894 df-cntz 18099 df-cmn 18547 df-psmet 20097 df-xmet 20098 df-met 20099 df-bl 20100 df-mopn 20101 df-fbas 20102 df-fg 20103 df-cnfld 20106 df-top 21068 df-topon 21085 df-topsp 21107 df-bases 21120 df-cld 21193 df-ntr 21194 df-cls 21195 df-nei 21272 df-lp 21310 df-perf 21311 df-cn 21401 df-cnp 21402 df-haus 21489 df-tx 21735 df-hmeo 21928 df-fil 22019 df-fm 22111 df-flim 22112 df-flf 22113 df-xms 22494 df-ms 22495 df-tms 22496 df-cncf 23050 df-limc 24028 df-dv 24029 df-log 24701 df-cxp 24702 |
This theorem is referenced by: dvcxp1 24882 dvcncxp1 24885 cxpcn3lem 24889 cxpaddlelem 24893 relogbdiv 24918 cxplim 25110 rlimcxp 25112 cxp2lim 25115 zetacvg 25153 ftalem5 25215 1sgmprm 25336 chpchtsum 25356 logfacrlim 25361 logexprlim 25362 perfectlem2 25367 dchrabs 25397 chtppilimlem1 25574 chtppilimlem2 25575 dchrvmasumlema 25601 logdivsum 25634 logdivsqrle 31276 binomcxplemnotnn0 39394 perfectALTVlem2 42460 young2d 43446 |
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