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Theorem divcxp 26714

Description: Complex exponentiation of a quotient. (Contributed by Mario Carneiro, 8-Sep-2014.)

**Assertion**
Ref	Expression
divcxp	⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐶 ∈ ℂ) → ((𝐴 / 𝐵)↑_𝑐𝐶) = ((𝐴↑_𝑐𝐶) / (𝐵↑_𝑐𝐶)))

**Proof of Theorem divcxp**
Step	Hyp	Ref	Expression
1		simp1l 1194	. . . 4 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐶 ∈ ℂ) → 𝐴 ∈ ℝ)
2		simp1r 1195	. . . 4 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐶 ∈ ℂ) → 0 ≤ 𝐴)
3		simp2 1134	. . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐶 ∈ ℂ) → 𝐵 ∈ ℝ⁺)
4	3	rpreccld 13080	. . . . 5 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐶 ∈ ℂ) → (1 / 𝐵) ∈ ℝ⁺)
5	4	rpred 13070	. . . 4 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐶 ∈ ℂ) → (1 / 𝐵) ∈ ℝ)
6	4	rpge0d 13074	. . . 4 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐶 ∈ ℂ) → 0 ≤ (1 / 𝐵))
7		simp3 1135	. . . 4 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐶 ∈ ℂ) → 𝐶 ∈ ℂ)
8		mulcxp 26712	. . . 4 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ ((1 / 𝐵) ∈ ℝ ∧ 0 ≤ (1 / 𝐵)) ∧ 𝐶 ∈ ℂ) → ((𝐴 · (1 / 𝐵))↑_𝑐𝐶) = ((𝐴↑_𝑐𝐶) · ((1 / 𝐵)↑_𝑐𝐶)))
9	1, 2, 5, 6, 7, 8	syl221anc 1378	. . 3 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐶 ∈ ℂ) → ((𝐴 · (1 / 𝐵))↑_𝑐𝐶) = ((𝐴↑_𝑐𝐶) · ((1 / 𝐵)↑_𝑐𝐶)))
10		cxprec 26713	. . . . 5 ⊢ ((𝐵 ∈ ℝ⁺ ∧ 𝐶 ∈ ℂ) → ((1 / 𝐵)↑_𝑐𝐶) = (1 / (𝐵↑_𝑐𝐶)))
11	3, 7, 10	syl2anc 582	. . . 4 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐶 ∈ ℂ) → ((1 / 𝐵)↑_𝑐𝐶) = (1 / (𝐵↑_𝑐𝐶)))
12	11	oveq2d 7440	. . 3 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐶 ∈ ℂ) → ((𝐴↑_𝑐𝐶) · ((1 / 𝐵)↑_𝑐𝐶)) = ((𝐴↑_𝑐𝐶) · (1 / (𝐵↑_𝑐𝐶))))
13	9, 12	eqtrd 2766	. 2 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐶 ∈ ℂ) → ((𝐴 · (1 / 𝐵))↑_𝑐𝐶) = ((𝐴↑_𝑐𝐶) · (1 / (𝐵↑_𝑐𝐶))))
14	1	recnd 11292	. . . 4 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐶 ∈ ℂ) → 𝐴 ∈ ℂ)
15	3	rpcnd 13072	. . . 4 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐶 ∈ ℂ) → 𝐵 ∈ ℂ)
16	3	rpne0d 13075	. . . 4 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐶 ∈ ℂ) → 𝐵 ≠ 0)
17	14, 15, 16	divrecd 12044	. . 3 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐶 ∈ ℂ) → (𝐴 / 𝐵) = (𝐴 · (1 / 𝐵)))
18	17	oveq1d 7439	. 2 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐶 ∈ ℂ) → ((𝐴 / 𝐵)↑_𝑐𝐶) = ((𝐴 · (1 / 𝐵))↑_𝑐𝐶))
19		cxpcl 26701	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐶 ∈ ℂ) → (𝐴↑_𝑐𝐶) ∈ ℂ)
20	14, 7, 19	syl2anc 582	. . 3 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐶 ∈ ℂ) → (𝐴↑_𝑐𝐶) ∈ ℂ)
21		cxpcl 26701	. . . 4 ⊢ ((𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → (𝐵↑_𝑐𝐶) ∈ ℂ)
22	15, 7, 21	syl2anc 582	. . 3 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐶 ∈ ℂ) → (𝐵↑_𝑐𝐶) ∈ ℂ)
23		cxpne0 26704	. . . 4 ⊢ ((𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ∧ 𝐶 ∈ ℂ) → (𝐵↑_𝑐𝐶) ≠ 0)
24	15, 16, 7, 23	syl3anc 1368	. . 3 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐶 ∈ ℂ) → (𝐵↑_𝑐𝐶) ≠ 0)
25	20, 22, 24	divrecd 12044	. 2 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐶 ∈ ℂ) → ((𝐴↑_𝑐𝐶) / (𝐵↑_𝑐𝐶)) = ((𝐴↑_𝑐𝐶) · (1 / (𝐵↑_𝑐𝐶))))
26	13, 18, 25	3eqtr4d 2776	1 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐶 ∈ ℂ) → ((𝐴 / 𝐵)↑_𝑐𝐶) = ((𝐴↑_𝑐𝐶) / (𝐵↑_𝑐𝐶)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 394 ∧ w3a 1084 = wceq 1534 ∈ wcel 2099 ≠ wne 2930 class class class wbr 5153 (class class class)co 7424 ℂcc 11156 ℝcr 11157 0cc0 11158 1c1 11159 · cmul 11163 ≤ cle 11299 / cdiv 11921 ℝ⁺crp 13028 ↑_𝑐ccxp 26582

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-rep 5290 ax-sep 5304 ax-nul 5311 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-inf2 9684 ax-cnex 11214 ax-resscn 11215 ax-1cn 11216 ax-icn 11217 ax-addcl 11218 ax-addrcl 11219 ax-mulcl 11220 ax-mulrcl 11221 ax-mulcom 11222 ax-addass 11223 ax-mulass 11224 ax-distr 11225 ax-i2m1 11226 ax-1ne0 11227 ax-1rid 11228 ax-rnegex 11229 ax-rrecex 11230 ax-cnre 11231 ax-pre-lttri 11232 ax-pre-lttrn 11233 ax-pre-ltadd 11234 ax-pre-mulgt0 11235 ax-pre-sup 11236 ax-addf 11237

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3967 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-tp 4638 df-op 4640 df-uni 4914 df-int 4955 df-iun 5003 df-iin 5004 df-br 5154 df-opab 5216 df-mpt 5237 df-tr 5271 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-se 5638 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6312 df-ord 6379 df-on 6380 df-lim 6381 df-suc 6382 df-iota 6506 df-fun 6556 df-fn 6557 df-f 6558 df-f1 6559 df-fo 6560 df-f1o 6561 df-fv 6562 df-isom 6563 df-riota 7380 df-ov 7427 df-oprab 7428 df-mpo 7429 df-of 7690 df-om 7877 df-1st 8003 df-2nd 8004 df-supp 8175 df-frecs 8296 df-wrecs 8327 df-recs 8401 df-rdg 8440 df-1o 8496 df-2o 8497 df-er 8734 df-map 8857 df-pm 8858 df-ixp 8927 df-en 8975 df-dom 8976 df-sdom 8977 df-fin 8978 df-fsupp 9406 df-fi 9454 df-sup 9485 df-inf 9486 df-oi 9553 df-card 9982 df-pnf 11300 df-mnf 11301 df-xr 11302 df-ltxr 11303 df-le 11304 df-sub 11496 df-neg 11497 df-div 11922 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-7 12332 df-8 12333 df-9 12334 df-n0 12525 df-z 12611 df-dec 12730 df-uz 12875 df-q 12985 df-rp 13029 df-xneg 13146 df-xadd 13147 df-xmul 13148 df-ioo 13382 df-ioc 13383 df-ico 13384 df-icc 13385 df-fz 13539 df-fzo 13682 df-fl 13812 df-mod 13890 df-seq 14022 df-exp 14082 df-fac 14291 df-bc 14320 df-hash 14348 df-shft 15072 df-cj 15104 df-re 15105 df-im 15106 df-sqrt 15240 df-abs 15241 df-limsup 15473 df-clim 15490 df-rlim 15491 df-sum 15691 df-ef 16069 df-sin 16071 df-cos 16072 df-pi 16074 df-struct 17149 df-sets 17166 df-slot 17184 df-ndx 17196 df-base 17214 df-ress 17243 df-plusg 17279 df-mulr 17280 df-starv 17281 df-sca 17282 df-vsca 17283 df-ip 17284 df-tset 17285 df-ple 17286 df-ds 17288 df-unif 17289 df-hom 17290 df-cco 17291 df-rest 17437 df-topn 17438 df-0g 17456 df-gsum 17457 df-topgen 17458 df-pt 17459 df-prds 17462 df-xrs 17517 df-qtop 17522 df-imas 17523 df-xps 17525 df-mre 17599 df-mrc 17600 df-acs 17602 df-mgm 18633 df-sgrp 18712 df-mnd 18728 df-submnd 18774 df-mulg 19062 df-cntz 19311 df-cmn 19780 df-psmet 21335 df-xmet 21336 df-met 21337 df-bl 21338 df-mopn 21339 df-fbas 21340 df-fg 21341 df-cnfld 21344 df-top 22887 df-topon 22904 df-topsp 22926 df-bases 22940 df-cld 23014 df-ntr 23015 df-cls 23016 df-nei 23093 df-lp 23131 df-perf 23132 df-cn 23222 df-cnp 23223 df-haus 23310 df-tx 23557 df-hmeo 23750 df-fil 23841 df-fm 23933 df-flim 23934 df-flf 23935 df-xms 24317 df-ms 24318 df-tms 24319 df-cncf 24889 df-limc 25886 df-dv 25887 df-log 26583 df-cxp 26584

This theorem is referenced by: divcxpd 26749 cxploglim2 27007

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