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Mirrors > Home > ILE Home > Th. List > abscxp | GIF version |
Description: Absolute value of a power, when the base is real. (Contributed by Mario Carneiro, 15-Sep-2014.) |
Ref | Expression |
---|---|
abscxp | ⊢ ((𝐴 ∈ ℝ+ ∧ 𝐵 ∈ ℂ) → (abs‘(𝐴↑𝑐𝐵)) = (𝐴↑𝑐(ℜ‘𝐵))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpr 110 | . . . . 5 ⊢ ((𝐴 ∈ ℝ+ ∧ 𝐵 ∈ ℂ) → 𝐵 ∈ ℂ) | |
2 | relogcl 13950 | . . . . . . 7 ⊢ (𝐴 ∈ ℝ+ → (log‘𝐴) ∈ ℝ) | |
3 | 2 | recnd 7976 | . . . . . 6 ⊢ (𝐴 ∈ ℝ+ → (log‘𝐴) ∈ ℂ) |
4 | 3 | adantr 276 | . . . . 5 ⊢ ((𝐴 ∈ ℝ+ ∧ 𝐵 ∈ ℂ) → (log‘𝐴) ∈ ℂ) |
5 | 1, 4 | mulcld 7968 | . . . 4 ⊢ ((𝐴 ∈ ℝ+ ∧ 𝐵 ∈ ℂ) → (𝐵 · (log‘𝐴)) ∈ ℂ) |
6 | absef 11761 | . . . 4 ⊢ ((𝐵 · (log‘𝐴)) ∈ ℂ → (abs‘(exp‘(𝐵 · (log‘𝐴)))) = (exp‘(ℜ‘(𝐵 · (log‘𝐴))))) | |
7 | 5, 6 | syl 14 | . . 3 ⊢ ((𝐴 ∈ ℝ+ ∧ 𝐵 ∈ ℂ) → (abs‘(exp‘(𝐵 · (log‘𝐴)))) = (exp‘(ℜ‘(𝐵 · (log‘𝐴))))) |
8 | remul2 10866 | . . . . . 6 ⊢ (((log‘𝐴) ∈ ℝ ∧ 𝐵 ∈ ℂ) → (ℜ‘((log‘𝐴) · 𝐵)) = ((log‘𝐴) · (ℜ‘𝐵))) | |
9 | 2, 8 | sylan 283 | . . . . 5 ⊢ ((𝐴 ∈ ℝ+ ∧ 𝐵 ∈ ℂ) → (ℜ‘((log‘𝐴) · 𝐵)) = ((log‘𝐴) · (ℜ‘𝐵))) |
10 | 1, 4 | mulcomd 7969 | . . . . . 6 ⊢ ((𝐴 ∈ ℝ+ ∧ 𝐵 ∈ ℂ) → (𝐵 · (log‘𝐴)) = ((log‘𝐴) · 𝐵)) |
11 | 10 | fveq2d 5515 | . . . . 5 ⊢ ((𝐴 ∈ ℝ+ ∧ 𝐵 ∈ ℂ) → (ℜ‘(𝐵 · (log‘𝐴))) = (ℜ‘((log‘𝐴) · 𝐵))) |
12 | recl 10846 | . . . . . . . 8 ⊢ (𝐵 ∈ ℂ → (ℜ‘𝐵) ∈ ℝ) | |
13 | 12 | adantl 277 | . . . . . . 7 ⊢ ((𝐴 ∈ ℝ+ ∧ 𝐵 ∈ ℂ) → (ℜ‘𝐵) ∈ ℝ) |
14 | 13 | recnd 7976 | . . . . . 6 ⊢ ((𝐴 ∈ ℝ+ ∧ 𝐵 ∈ ℂ) → (ℜ‘𝐵) ∈ ℂ) |
15 | 14, 4 | mulcomd 7969 | . . . . 5 ⊢ ((𝐴 ∈ ℝ+ ∧ 𝐵 ∈ ℂ) → ((ℜ‘𝐵) · (log‘𝐴)) = ((log‘𝐴) · (ℜ‘𝐵))) |
16 | 9, 11, 15 | 3eqtr4d 2220 | . . . 4 ⊢ ((𝐴 ∈ ℝ+ ∧ 𝐵 ∈ ℂ) → (ℜ‘(𝐵 · (log‘𝐴))) = ((ℜ‘𝐵) · (log‘𝐴))) |
17 | 16 | fveq2d 5515 | . . 3 ⊢ ((𝐴 ∈ ℝ+ ∧ 𝐵 ∈ ℂ) → (exp‘(ℜ‘(𝐵 · (log‘𝐴)))) = (exp‘((ℜ‘𝐵) · (log‘𝐴)))) |
18 | 7, 17 | eqtrd 2210 | . 2 ⊢ ((𝐴 ∈ ℝ+ ∧ 𝐵 ∈ ℂ) → (abs‘(exp‘(𝐵 · (log‘𝐴)))) = (exp‘((ℜ‘𝐵) · (log‘𝐴)))) |
19 | rpcxpef 13982 | . . 3 ⊢ ((𝐴 ∈ ℝ+ ∧ 𝐵 ∈ ℂ) → (𝐴↑𝑐𝐵) = (exp‘(𝐵 · (log‘𝐴)))) | |
20 | 19 | fveq2d 5515 | . 2 ⊢ ((𝐴 ∈ ℝ+ ∧ 𝐵 ∈ ℂ) → (abs‘(𝐴↑𝑐𝐵)) = (abs‘(exp‘(𝐵 · (log‘𝐴))))) |
21 | rpcxpef 13982 | . . 3 ⊢ ((𝐴 ∈ ℝ+ ∧ (ℜ‘𝐵) ∈ ℂ) → (𝐴↑𝑐(ℜ‘𝐵)) = (exp‘((ℜ‘𝐵) · (log‘𝐴)))) | |
22 | 14, 21 | syldan 282 | . 2 ⊢ ((𝐴 ∈ ℝ+ ∧ 𝐵 ∈ ℂ) → (𝐴↑𝑐(ℜ‘𝐵)) = (exp‘((ℜ‘𝐵) · (log‘𝐴)))) |
23 | 18, 20, 22 | 3eqtr4d 2220 | 1 ⊢ ((𝐴 ∈ ℝ+ ∧ 𝐵 ∈ ℂ) → (abs‘(𝐴↑𝑐𝐵)) = (𝐴↑𝑐(ℜ‘𝐵))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 = wceq 1353 ∈ wcel 2148 ‘cfv 5212 (class class class)co 5869 ℂcc 7800 ℝcr 7801 · cmul 7807 ℝ+crp 9640 ℜcre 10833 abscabs 10990 expce 11634 logclog 13944 ↑𝑐ccxp 13945 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-coll 4115 ax-sep 4118 ax-nul 4126 ax-pow 4171 ax-pr 4206 ax-un 4430 ax-setind 4533 ax-iinf 4584 ax-cnex 7893 ax-resscn 7894 ax-1cn 7895 ax-1re 7896 ax-icn 7897 ax-addcl 7898 ax-addrcl 7899 ax-mulcl 7900 ax-mulrcl 7901 ax-addcom 7902 ax-mulcom 7903 ax-addass 7904 ax-mulass 7905 ax-distr 7906 ax-i2m1 7907 ax-0lt1 7908 ax-1rid 7909 ax-0id 7910 ax-rnegex 7911 ax-precex 7912 ax-cnre 7913 ax-pre-ltirr 7914 ax-pre-ltwlin 7915 ax-pre-lttrn 7916 ax-pre-apti 7917 ax-pre-ltadd 7918 ax-pre-mulgt0 7919 ax-pre-mulext 7920 ax-arch 7921 ax-caucvg 7922 ax-pre-suploc 7923 ax-addf 7924 ax-mulf 7925 |
This theorem depends on definitions: df-bi 117 df-stab 831 df-dc 835 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-reu 2462 df-rmo 2463 df-rab 2464 df-v 2739 df-sbc 2963 df-csb 3058 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-nul 3423 df-if 3535 df-pw 3576 df-sn 3597 df-pr 3598 df-op 3600 df-uni 3808 df-int 3843 df-iun 3886 df-disj 3978 df-br 4001 df-opab 4062 df-mpt 4063 df-tr 4099 df-id 4290 df-po 4293 df-iso 4294 df-iord 4363 df-on 4365 df-ilim 4366 df-suc 4368 df-iom 4587 df-xp 4629 df-rel 4630 df-cnv 4631 df-co 4632 df-dm 4633 df-rn 4634 df-res 4635 df-ima 4636 df-iota 5174 df-fun 5214 df-fn 5215 df-f 5216 df-f1 5217 df-fo 5218 df-f1o 5219 df-fv 5220 df-isom 5221 df-riota 5825 df-ov 5872 df-oprab 5873 df-mpo 5874 df-of 6077 df-1st 6135 df-2nd 6136 df-recs 6300 df-irdg 6365 df-frec 6386 df-1o 6411 df-oadd 6415 df-er 6529 df-map 6644 df-pm 6645 df-en 6735 df-dom 6736 df-fin 6737 df-sup 6977 df-inf 6978 df-pnf 7984 df-mnf 7985 df-xr 7986 df-ltxr 7987 df-le 7988 df-sub 8120 df-neg 8121 df-reap 8522 df-ap 8529 df-div 8619 df-inn 8909 df-2 8967 df-3 8968 df-4 8969 df-n0 9166 df-z 9243 df-uz 9518 df-q 9609 df-rp 9641 df-xneg 9759 df-xadd 9760 df-ioo 9879 df-ico 9881 df-icc 9882 df-fz 9996 df-fzo 10129 df-seqfrec 10432 df-exp 10506 df-fac 10690 df-bc 10712 df-ihash 10740 df-shft 10808 df-cj 10835 df-re 10836 df-im 10837 df-rsqrt 10991 df-abs 10992 df-clim 11271 df-sumdc 11346 df-ef 11640 df-e 11641 df-sin 11642 df-cos 11643 df-rest 12638 df-topgen 12657 df-psmet 13154 df-xmet 13155 df-met 13156 df-bl 13157 df-mopn 13158 df-top 13163 df-topon 13176 df-bases 13208 df-ntr 13263 df-cn 13355 df-cnp 13356 df-tx 13420 df-cncf 13725 df-limced 13792 df-dvap 13793 df-relog 13946 df-rpcxp 13947 |
This theorem is referenced by: (None) |
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