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Mirrors > Home > MPE Home > Th. List > abscxp2 | Structured version Visualization version GIF version |
Description: Absolute value of a power, when the exponent is real. (Contributed by Mario Carneiro, 15-Sep-2014.) |
Ref | Expression |
---|---|
abscxp2 | ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℝ) → (abs‘(𝐴↑𝑐𝐵)) = ((abs‘𝐴)↑𝑐𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0red 10041 | . . . . 5 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℝ) ∧ 𝐴 = 0) → 0 ∈ ℝ) | |
2 | 0le0 11110 | . . . . . 6 ⊢ 0 ≤ 0 | |
3 | 2 | a1i 11 | . . . . 5 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℝ) ∧ 𝐴 = 0) → 0 ≤ 0) |
4 | simplr 792 | . . . . 5 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℝ) ∧ 𝐴 = 0) → 𝐵 ∈ ℝ) | |
5 | recxpcl 24421 | . . . . 5 ⊢ ((0 ∈ ℝ ∧ 0 ≤ 0 ∧ 𝐵 ∈ ℝ) → (0↑𝑐𝐵) ∈ ℝ) | |
6 | 1, 3, 4, 5 | syl3anc 1326 | . . . 4 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℝ) ∧ 𝐴 = 0) → (0↑𝑐𝐵) ∈ ℝ) |
7 | cxpge0 24429 | . . . . 5 ⊢ ((0 ∈ ℝ ∧ 0 ≤ 0 ∧ 𝐵 ∈ ℝ) → 0 ≤ (0↑𝑐𝐵)) | |
8 | 1, 3, 4, 7 | syl3anc 1326 | . . . 4 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℝ) ∧ 𝐴 = 0) → 0 ≤ (0↑𝑐𝐵)) |
9 | 6, 8 | absidd 14161 | . . 3 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℝ) ∧ 𝐴 = 0) → (abs‘(0↑𝑐𝐵)) = (0↑𝑐𝐵)) |
10 | simpr 477 | . . . . 5 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℝ) ∧ 𝐴 = 0) → 𝐴 = 0) | |
11 | 10 | oveq1d 6665 | . . . 4 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℝ) ∧ 𝐴 = 0) → (𝐴↑𝑐𝐵) = (0↑𝑐𝐵)) |
12 | 11 | fveq2d 6195 | . . 3 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℝ) ∧ 𝐴 = 0) → (abs‘(𝐴↑𝑐𝐵)) = (abs‘(0↑𝑐𝐵))) |
13 | 10 | abs00bd 14031 | . . . 4 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℝ) ∧ 𝐴 = 0) → (abs‘𝐴) = 0) |
14 | 13 | oveq1d 6665 | . . 3 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℝ) ∧ 𝐴 = 0) → ((abs‘𝐴)↑𝑐𝐵) = (0↑𝑐𝐵)) |
15 | 9, 12, 14 | 3eqtr4d 2666 | . 2 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℝ) ∧ 𝐴 = 0) → (abs‘(𝐴↑𝑐𝐵)) = ((abs‘𝐴)↑𝑐𝐵)) |
16 | simplr 792 | . . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℝ) ∧ 𝐴 ≠ 0) → 𝐵 ∈ ℝ) | |
17 | 16 | recnd 10068 | . . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℝ) ∧ 𝐴 ≠ 0) → 𝐵 ∈ ℂ) |
18 | logcl 24315 | . . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (log‘𝐴) ∈ ℂ) | |
19 | 18 | adantlr 751 | . . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℝ) ∧ 𝐴 ≠ 0) → (log‘𝐴) ∈ ℂ) |
20 | 17, 19 | mulcld 10060 | . . . . 5 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℝ) ∧ 𝐴 ≠ 0) → (𝐵 · (log‘𝐴)) ∈ ℂ) |
21 | absef 14927 | . . . . 5 ⊢ ((𝐵 · (log‘𝐴)) ∈ ℂ → (abs‘(exp‘(𝐵 · (log‘𝐴)))) = (exp‘(ℜ‘(𝐵 · (log‘𝐴))))) | |
22 | 20, 21 | syl 17 | . . . 4 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℝ) ∧ 𝐴 ≠ 0) → (abs‘(exp‘(𝐵 · (log‘𝐴)))) = (exp‘(ℜ‘(𝐵 · (log‘𝐴))))) |
23 | 16, 19 | remul2d 13967 | . . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℝ) ∧ 𝐴 ≠ 0) → (ℜ‘(𝐵 · (log‘𝐴))) = (𝐵 · (ℜ‘(log‘𝐴)))) |
24 | relog 24343 | . . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (ℜ‘(log‘𝐴)) = (log‘(abs‘𝐴))) | |
25 | 24 | adantlr 751 | . . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℝ) ∧ 𝐴 ≠ 0) → (ℜ‘(log‘𝐴)) = (log‘(abs‘𝐴))) |
26 | 25 | oveq2d 6666 | . . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℝ) ∧ 𝐴 ≠ 0) → (𝐵 · (ℜ‘(log‘𝐴))) = (𝐵 · (log‘(abs‘𝐴)))) |
27 | 23, 26 | eqtrd 2656 | . . . . 5 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℝ) ∧ 𝐴 ≠ 0) → (ℜ‘(𝐵 · (log‘𝐴))) = (𝐵 · (log‘(abs‘𝐴)))) |
28 | 27 | fveq2d 6195 | . . . 4 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℝ) ∧ 𝐴 ≠ 0) → (exp‘(ℜ‘(𝐵 · (log‘𝐴)))) = (exp‘(𝐵 · (log‘(abs‘𝐴))))) |
29 | 22, 28 | eqtrd 2656 | . . 3 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℝ) ∧ 𝐴 ≠ 0) → (abs‘(exp‘(𝐵 · (log‘𝐴)))) = (exp‘(𝐵 · (log‘(abs‘𝐴))))) |
30 | simpll 790 | . . . . 5 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℝ) ∧ 𝐴 ≠ 0) → 𝐴 ∈ ℂ) | |
31 | simpr 477 | . . . . 5 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℝ) ∧ 𝐴 ≠ 0) → 𝐴 ≠ 0) | |
32 | cxpef 24411 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐵 ∈ ℂ) → (𝐴↑𝑐𝐵) = (exp‘(𝐵 · (log‘𝐴)))) | |
33 | 30, 31, 17, 32 | syl3anc 1326 | . . . 4 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℝ) ∧ 𝐴 ≠ 0) → (𝐴↑𝑐𝐵) = (exp‘(𝐵 · (log‘𝐴)))) |
34 | 33 | fveq2d 6195 | . . 3 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℝ) ∧ 𝐴 ≠ 0) → (abs‘(𝐴↑𝑐𝐵)) = (abs‘(exp‘(𝐵 · (log‘𝐴))))) |
35 | 30 | abscld 14175 | . . . . 5 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℝ) ∧ 𝐴 ≠ 0) → (abs‘𝐴) ∈ ℝ) |
36 | 35 | recnd 10068 | . . . 4 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℝ) ∧ 𝐴 ≠ 0) → (abs‘𝐴) ∈ ℂ) |
37 | abs00 14029 | . . . . . . 7 ⊢ (𝐴 ∈ ℂ → ((abs‘𝐴) = 0 ↔ 𝐴 = 0)) | |
38 | 37 | adantr 481 | . . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℝ) → ((abs‘𝐴) = 0 ↔ 𝐴 = 0)) |
39 | 38 | necon3bid 2838 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℝ) → ((abs‘𝐴) ≠ 0 ↔ 𝐴 ≠ 0)) |
40 | 39 | biimpar 502 | . . . 4 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℝ) ∧ 𝐴 ≠ 0) → (abs‘𝐴) ≠ 0) |
41 | cxpef 24411 | . . . 4 ⊢ (((abs‘𝐴) ∈ ℂ ∧ (abs‘𝐴) ≠ 0 ∧ 𝐵 ∈ ℂ) → ((abs‘𝐴)↑𝑐𝐵) = (exp‘(𝐵 · (log‘(abs‘𝐴))))) | |
42 | 36, 40, 17, 41 | syl3anc 1326 | . . 3 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℝ) ∧ 𝐴 ≠ 0) → ((abs‘𝐴)↑𝑐𝐵) = (exp‘(𝐵 · (log‘(abs‘𝐴))))) |
43 | 29, 34, 42 | 3eqtr4d 2666 | . 2 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℝ) ∧ 𝐴 ≠ 0) → (abs‘(𝐴↑𝑐𝐵)) = ((abs‘𝐴)↑𝑐𝐵)) |
44 | 15, 43 | pm2.61dane 2881 | 1 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℝ) → (abs‘(𝐴↑𝑐𝐵)) = ((abs‘𝐴)↑𝑐𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ≠ wne 2794 class class class wbr 4653 ‘cfv 5888 (class class class)co 6650 ℂcc 9934 ℝcr 9935 0cc0 9936 · cmul 9941 ≤ cle 10075 ℜcre 13837 abscabs 13974 expce 14792 logclog 24301 ↑𝑐ccxp 24302 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-inf2 8538 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014 ax-addf 10015 ax-mulf 10016 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-fal 1489 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-iin 4523 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-se 5074 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-isom 5897 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-of 6897 df-om 7066 df-1st 7168 df-2nd 7169 df-supp 7296 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-2o 7561 df-oadd 7564 df-er 7742 df-map 7859 df-pm 7860 df-ixp 7909 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-fsupp 8276 df-fi 8317 df-sup 8348 df-inf 8349 df-oi 8415 df-card 8765 df-cda 8990 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-7 11084 df-8 11085 df-9 11086 df-n0 11293 df-z 11378 df-dec 11494 df-uz 11688 df-q 11789 df-rp 11833 df-xneg 11946 df-xadd 11947 df-xmul 11948 df-ioo 12179 df-ioc 12180 df-ico 12181 df-icc 12182 df-fz 12327 df-fzo 12466 df-fl 12593 df-mod 12669 df-seq 12802 df-exp 12861 df-fac 13061 df-bc 13090 df-hash 13118 df-shft 13807 df-cj 13839 df-re 13840 df-im 13841 df-sqrt 13975 df-abs 13976 df-limsup 14202 df-clim 14219 df-rlim 14220 df-sum 14417 df-ef 14798 df-sin 14800 df-cos 14801 df-pi 14803 df-struct 15859 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-ress 15865 df-plusg 15954 df-mulr 15955 df-starv 15956 df-sca 15957 df-vsca 15958 df-ip 15959 df-tset 15960 df-ple 15961 df-ds 15964 df-unif 15965 df-hom 15966 df-cco 15967 df-rest 16083 df-topn 16084 df-0g 16102 df-gsum 16103 df-topgen 16104 df-pt 16105 df-prds 16108 df-xrs 16162 df-qtop 16167 df-imas 16168 df-xps 16170 df-mre 16246 df-mrc 16247 df-acs 16249 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-submnd 17336 df-mulg 17541 df-cntz 17750 df-cmn 18195 df-psmet 19738 df-xmet 19739 df-met 19740 df-bl 19741 df-mopn 19742 df-fbas 19743 df-fg 19744 df-cnfld 19747 df-top 20699 df-topon 20716 df-topsp 20737 df-bases 20750 df-cld 20823 df-ntr 20824 df-cls 20825 df-nei 20902 df-lp 20940 df-perf 20941 df-cn 21031 df-cnp 21032 df-haus 21119 df-tx 21365 df-hmeo 21558 df-fil 21650 df-fm 21742 df-flim 21743 df-flf 21744 df-xms 22125 df-ms 22126 df-tms 22127 df-cncf 22681 df-limc 23630 df-dv 23631 df-log 24303 df-cxp 24304 |
This theorem is referenced by: root1cj 24497 rlimcxp 24700 |
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