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Mirrors > Home > ILE Home > Th. List > cxpexprp | GIF version |
Description: Relate the complex power function to the integer power function. (Contributed by Mario Carneiro, 2-Aug-2014.) (Revised by Jim Kingdon, 12-Jun-2024.) |
Ref | Expression |
---|---|
cxpexprp | ⊢ ((𝐴 ∈ ℝ+ ∧ 𝐵 ∈ ℤ) → (𝐴↑𝑐𝐵) = (𝐴↑𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | zcn 9204 | . . 3 ⊢ (𝐵 ∈ ℤ → 𝐵 ∈ ℂ) | |
2 | rpcxpef 13568 | . . 3 ⊢ ((𝐴 ∈ ℝ+ ∧ 𝐵 ∈ ℂ) → (𝐴↑𝑐𝐵) = (exp‘(𝐵 · (log‘𝐴)))) | |
3 | 1, 2 | sylan2 284 | . 2 ⊢ ((𝐴 ∈ ℝ+ ∧ 𝐵 ∈ ℤ) → (𝐴↑𝑐𝐵) = (exp‘(𝐵 · (log‘𝐴)))) |
4 | reexplog 13545 | . 2 ⊢ ((𝐴 ∈ ℝ+ ∧ 𝐵 ∈ ℤ) → (𝐴↑𝐵) = (exp‘(𝐵 · (log‘𝐴)))) | |
5 | 3, 4 | eqtr4d 2206 | 1 ⊢ ((𝐴 ∈ ℝ+ ∧ 𝐵 ∈ ℤ) → (𝐴↑𝑐𝐵) = (𝐴↑𝐵)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1348 ∈ wcel 2141 ‘cfv 5196 (class class class)co 5850 ℂcc 7759 · cmul 7766 ℤcz 9199 ℝ+crp 9597 ↑cexp 10462 expce 11592 logclog 13530 ↑𝑐ccxp 13531 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-coll 4102 ax-sep 4105 ax-nul 4113 ax-pow 4158 ax-pr 4192 ax-un 4416 ax-setind 4519 ax-iinf 4570 ax-cnex 7852 ax-resscn 7853 ax-1cn 7854 ax-1re 7855 ax-icn 7856 ax-addcl 7857 ax-addrcl 7858 ax-mulcl 7859 ax-mulrcl 7860 ax-addcom 7861 ax-mulcom 7862 ax-addass 7863 ax-mulass 7864 ax-distr 7865 ax-i2m1 7866 ax-0lt1 7867 ax-1rid 7868 ax-0id 7869 ax-rnegex 7870 ax-precex 7871 ax-cnre 7872 ax-pre-ltirr 7873 ax-pre-ltwlin 7874 ax-pre-lttrn 7875 ax-pre-apti 7876 ax-pre-ltadd 7877 ax-pre-mulgt0 7878 ax-pre-mulext 7879 ax-arch 7880 ax-caucvg 7881 ax-pre-suploc 7882 ax-addf 7883 ax-mulf 7884 |
This theorem depends on definitions: df-bi 116 df-stab 826 df-dc 830 df-3or 974 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-nel 2436 df-ral 2453 df-rex 2454 df-reu 2455 df-rmo 2456 df-rab 2457 df-v 2732 df-sbc 2956 df-csb 3050 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-nul 3415 df-if 3526 df-pw 3566 df-sn 3587 df-pr 3588 df-op 3590 df-uni 3795 df-int 3830 df-iun 3873 df-disj 3965 df-br 3988 df-opab 4049 df-mpt 4050 df-tr 4086 df-id 4276 df-po 4279 df-iso 4280 df-iord 4349 df-on 4351 df-ilim 4352 df-suc 4354 df-iom 4573 df-xp 4615 df-rel 4616 df-cnv 4617 df-co 4618 df-dm 4619 df-rn 4620 df-res 4621 df-ima 4622 df-iota 5158 df-fun 5198 df-fn 5199 df-f 5200 df-f1 5201 df-fo 5202 df-f1o 5203 df-fv 5204 df-isom 5205 df-riota 5806 df-ov 5853 df-oprab 5854 df-mpo 5855 df-of 6058 df-1st 6116 df-2nd 6117 df-recs 6281 df-irdg 6346 df-frec 6367 df-1o 6392 df-oadd 6396 df-er 6509 df-map 6624 df-pm 6625 df-en 6715 df-dom 6716 df-fin 6717 df-sup 6957 df-inf 6958 df-pnf 7943 df-mnf 7944 df-xr 7945 df-ltxr 7946 df-le 7947 df-sub 8079 df-neg 8080 df-reap 8481 df-ap 8488 df-div 8577 df-inn 8866 df-2 8924 df-3 8925 df-4 8926 df-n0 9123 df-z 9200 df-uz 9475 df-q 9566 df-rp 9598 df-xneg 9716 df-xadd 9717 df-ioo 9836 df-ico 9838 df-icc 9839 df-fz 9953 df-fzo 10086 df-seqfrec 10389 df-exp 10463 df-fac 10647 df-bc 10669 df-ihash 10697 df-shft 10766 df-cj 10793 df-re 10794 df-im 10795 df-rsqrt 10949 df-abs 10950 df-clim 11229 df-sumdc 11304 df-ef 11598 df-e 11599 df-rest 12568 df-topgen 12587 df-psmet 12740 df-xmet 12741 df-met 12742 df-bl 12743 df-mopn 12744 df-top 12749 df-topon 12762 df-bases 12794 df-ntr 12849 df-cn 12941 df-cnp 12942 df-tx 13006 df-cncf 13311 df-limced 13378 df-dvap 13379 df-relog 13532 df-rpcxp 13533 |
This theorem is referenced by: cxpexpnn 13570 rpcxp0 13572 rpcxp1 13573 rpcxproot 13587 rpcxpsqrtth 13603 ltexp2 13613 rplogbzexp 13625 logbgcd1irr 13638 logbgcd1irraplemap 13640 |
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