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Mirrors > Home > MPE Home > Th. List > 1cxp | Structured version Visualization version GIF version |
Description: Value of the complex power function at one. (Contributed by Mario Carneiro, 2-Aug-2014.) |
Ref | Expression |
---|---|
1cxp | ⊢ (𝐴 ∈ ℂ → (1↑𝑐𝐴) = 1) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax-1cn 10075 | . . 3 ⊢ 1 ∈ ℂ | |
2 | ax-1ne0 10086 | . . 3 ⊢ 1 ≠ 0 | |
3 | cxpef 24499 | . . 3 ⊢ ((1 ∈ ℂ ∧ 1 ≠ 0 ∧ 𝐴 ∈ ℂ) → (1↑𝑐𝐴) = (exp‘(𝐴 · (log‘1)))) | |
4 | 1, 2, 3 | mp3an12 1495 | . 2 ⊢ (𝐴 ∈ ℂ → (1↑𝑐𝐴) = (exp‘(𝐴 · (log‘1)))) |
5 | log1 24420 | . . . . . 6 ⊢ (log‘1) = 0 | |
6 | 5 | oveq2i 6744 | . . . . 5 ⊢ (𝐴 · (log‘1)) = (𝐴 · 0) |
7 | mul01 10296 | . . . . 5 ⊢ (𝐴 ∈ ℂ → (𝐴 · 0) = 0) | |
8 | 6, 7 | syl5eq 2738 | . . . 4 ⊢ (𝐴 ∈ ℂ → (𝐴 · (log‘1)) = 0) |
9 | 8 | fveq2d 6276 | . . 3 ⊢ (𝐴 ∈ ℂ → (exp‘(𝐴 · (log‘1))) = (exp‘0)) |
10 | ef0 14909 | . . 3 ⊢ (exp‘0) = 1 | |
11 | 9, 10 | syl6eq 2742 | . 2 ⊢ (𝐴 ∈ ℂ → (exp‘(𝐴 · (log‘1))) = 1) |
12 | 4, 11 | eqtrd 2726 | 1 ⊢ (𝐴 ∈ ℂ → (1↑𝑐𝐴) = 1) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1564 ∈ wcel 2071 ≠ wne 2864 ‘cfv 5969 (class class class)co 6733 ℂcc 10015 0cc0 10017 1c1 10018 · cmul 10022 expce 14880 logclog 24389 ↑𝑐ccxp 24390 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1818 ax-5 1920 ax-6 1986 ax-7 2022 ax-8 2073 ax-9 2080 ax-10 2100 ax-11 2115 ax-12 2128 ax-13 2323 ax-ext 2672 ax-rep 4847 ax-sep 4857 ax-nul 4865 ax-pow 4916 ax-pr 4979 ax-un 7034 ax-inf2 8619 ax-cnex 10073 ax-resscn 10074 ax-1cn 10075 ax-icn 10076 ax-addcl 10077 ax-addrcl 10078 ax-mulcl 10079 ax-mulrcl 10080 ax-mulcom 10081 ax-addass 10082 ax-mulass 10083 ax-distr 10084 ax-i2m1 10085 ax-1ne0 10086 ax-1rid 10087 ax-rnegex 10088 ax-rrecex 10089 ax-cnre 10090 ax-pre-lttri 10091 ax-pre-lttrn 10092 ax-pre-ltadd 10093 ax-pre-mulgt0 10094 ax-pre-sup 10095 ax-addf 10096 ax-mulf 10097 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1073 df-3an 1074 df-tru 1567 df-fal 1570 df-ex 1786 df-nf 1791 df-sb 1979 df-eu 2543 df-mo 2544 df-clab 2679 df-cleq 2685 df-clel 2688 df-nfc 2823 df-ne 2865 df-nel 2968 df-ral 2987 df-rex 2988 df-reu 2989 df-rmo 2990 df-rab 2991 df-v 3274 df-sbc 3510 df-csb 3608 df-dif 3651 df-un 3653 df-in 3655 df-ss 3662 df-pss 3664 df-nul 3992 df-if 4163 df-pw 4236 df-sn 4254 df-pr 4256 df-tp 4258 df-op 4260 df-uni 4513 df-int 4552 df-iun 4598 df-iin 4599 df-br 4729 df-opab 4789 df-mpt 4806 df-tr 4829 df-id 5096 df-eprel 5101 df-po 5107 df-so 5108 df-fr 5145 df-se 5146 df-we 5147 df-xp 5192 df-rel 5193 df-cnv 5194 df-co 5195 df-dm 5196 df-rn 5197 df-res 5198 df-ima 5199 df-pred 5761 df-ord 5807 df-on 5808 df-lim 5809 df-suc 5810 df-iota 5932 df-fun 5971 df-fn 5972 df-f 5973 df-f1 5974 df-fo 5975 df-f1o 5976 df-fv 5977 df-isom 5978 df-riota 6694 df-ov 6736 df-oprab 6737 df-mpt2 6738 df-of 6982 df-om 7151 df-1st 7253 df-2nd 7254 df-supp 7384 df-wrecs 7495 df-recs 7556 df-rdg 7594 df-1o 7648 df-2o 7649 df-oadd 7652 df-er 7830 df-map 7944 df-pm 7945 df-ixp 7994 df-en 8041 df-dom 8042 df-sdom 8043 df-fin 8044 df-fsupp 8360 df-fi 8401 df-sup 8432 df-inf 8433 df-oi 8499 df-card 8846 df-cda 9071 df-pnf 10157 df-mnf 10158 df-xr 10159 df-ltxr 10160 df-le 10161 df-sub 10349 df-neg 10350 df-div 10766 df-nn 11102 df-2 11160 df-3 11161 df-4 11162 df-5 11163 df-6 11164 df-7 11165 df-8 11166 df-9 11167 df-n0 11374 df-z 11459 df-dec 11575 df-uz 11769 df-q 11871 df-rp 11915 df-xneg 12028 df-xadd 12029 df-xmul 12030 df-ioo 12261 df-ioc 12262 df-ico 12263 df-icc 12264 df-fz 12409 df-fzo 12549 df-fl 12676 df-mod 12752 df-seq 12885 df-exp 12944 df-fac 13144 df-bc 13173 df-hash 13201 df-shft 13895 df-cj 13927 df-re 13928 df-im 13929 df-sqrt 14063 df-abs 14064 df-limsup 14290 df-clim 14307 df-rlim 14308 df-sum 14505 df-ef 14886 df-sin 14888 df-cos 14889 df-pi 14891 df-struct 15950 df-ndx 15951 df-slot 15952 df-base 15954 df-sets 15955 df-ress 15956 df-plusg 16045 df-mulr 16046 df-starv 16047 df-sca 16048 df-vsca 16049 df-ip 16050 df-tset 16051 df-ple 16052 df-ds 16055 df-unif 16056 df-hom 16057 df-cco 16058 df-rest 16174 df-topn 16175 df-0g 16193 df-gsum 16194 df-topgen 16195 df-pt 16196 df-prds 16199 df-xrs 16253 df-qtop 16258 df-imas 16259 df-xps 16261 df-mre 16337 df-mrc 16338 df-acs 16340 df-mgm 17332 df-sgrp 17374 df-mnd 17385 df-submnd 17426 df-mulg 17631 df-cntz 17839 df-cmn 18284 df-psmet 19829 df-xmet 19830 df-met 19831 df-bl 19832 df-mopn 19833 df-fbas 19834 df-fg 19835 df-cnfld 19838 df-top 20790 df-topon 20807 df-topsp 20828 df-bases 20841 df-cld 20914 df-ntr 20915 df-cls 20916 df-nei 20993 df-lp 21031 df-perf 21032 df-cn 21122 df-cnp 21123 df-haus 21210 df-tx 21456 df-hmeo 21649 df-fil 21740 df-fm 21832 df-flim 21833 df-flf 21834 df-xms 22215 df-ms 22216 df-tms 22217 df-cncf 22771 df-limc 23718 df-dv 23719 df-log 24391 df-cxp 24392 |
This theorem is referenced by: cxprec 24520 cxplea 24530 1cxpd 24541 1cubr 24657 |
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