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Theorem rpdivcxp 15433

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

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

**Proof of Theorem rpdivcxp**
Step	Hyp	Ref	Expression
1		simp2 1001	. . . . 5 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐶 ∈ ℂ) → 𝐵 ∈ ℝ⁺)
2	1	rpreccld 9842	. . . 4 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐶 ∈ ℂ) → (1 / 𝐵) ∈ ℝ⁺)
3		rpmulcxp 15431	. . . 4 ⊢ ((𝐴 ∈ ℝ⁺ ∧ (1 / 𝐵) ∈ ℝ⁺ ∧ 𝐶 ∈ ℂ) → ((𝐴 · (1 / 𝐵))↑_𝑐𝐶) = ((𝐴↑_𝑐𝐶) · ((1 / 𝐵)↑_𝑐𝐶)))
4	2, 3	syld3an2 1297	. . 3 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐶 ∈ ℂ) → ((𝐴 · (1 / 𝐵))↑_𝑐𝐶) = ((𝐴↑_𝑐𝐶) · ((1 / 𝐵)↑_𝑐𝐶)))
5		cxprec 15432	. . . . 5 ⊢ ((𝐵 ∈ ℝ⁺ ∧ 𝐶 ∈ ℂ) → ((1 / 𝐵)↑_𝑐𝐶) = (1 / (𝐵↑_𝑐𝐶)))
6	5	3adant1 1018	. . . 4 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐶 ∈ ℂ) → ((1 / 𝐵)↑_𝑐𝐶) = (1 / (𝐵↑_𝑐𝐶)))
7	6	oveq2d 5970	. . 3 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐶 ∈ ℂ) → ((𝐴↑_𝑐𝐶) · ((1 / 𝐵)↑_𝑐𝐶)) = ((𝐴↑_𝑐𝐶) · (1 / (𝐵↑_𝑐𝐶))))
8	4, 7	eqtrd 2239	. 2 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐶 ∈ ℂ) → ((𝐴 · (1 / 𝐵))↑_𝑐𝐶) = ((𝐴↑_𝑐𝐶) · (1 / (𝐵↑_𝑐𝐶))))
9		simp1 1000	. . . . 5 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐶 ∈ ℂ) → 𝐴 ∈ ℝ⁺)
10	9	rpcnd 9833	. . . 4 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐶 ∈ ℂ) → 𝐴 ∈ ℂ)
11	1	rpcnd 9833	. . . 4 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐶 ∈ ℂ) → 𝐵 ∈ ℂ)
12	1	rpap0d 9837	. . . 4 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐶 ∈ ℂ) → 𝐵 # 0)
13	10, 11, 12	divrecapd 8879	. . 3 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐶 ∈ ℂ) → (𝐴 / 𝐵) = (𝐴 · (1 / 𝐵)))
14	13	oveq1d 5969	. 2 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐶 ∈ ℂ) → ((𝐴 / 𝐵)↑_𝑐𝐶) = ((𝐴 · (1 / 𝐵))↑_𝑐𝐶))
15		rpcncxpcl 15424	. . . 4 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐶 ∈ ℂ) → (𝐴↑_𝑐𝐶) ∈ ℂ)
16	15	3adant2 1019	. . 3 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐶 ∈ ℂ) → (𝐴↑_𝑐𝐶) ∈ ℂ)
17		rpcncxpcl 15424	. . . 4 ⊢ ((𝐵 ∈ ℝ⁺ ∧ 𝐶 ∈ ℂ) → (𝐵↑_𝑐𝐶) ∈ ℂ)
18	17	3adant1 1018	. . 3 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐶 ∈ ℂ) → (𝐵↑_𝑐𝐶) ∈ ℂ)
19		cxpap0 15426	. . . 4 ⊢ ((𝐵 ∈ ℝ⁺ ∧ 𝐶 ∈ ℂ) → (𝐵↑_𝑐𝐶) # 0)
20	19	3adant1 1018	. . 3 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐶 ∈ ℂ) → (𝐵↑_𝑐𝐶) # 0)
21	16, 18, 20	divrecapd 8879	. 2 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐶 ∈ ℂ) → ((𝐴↑_𝑐𝐶) / (𝐵↑_𝑐𝐶)) = ((𝐴↑_𝑐𝐶) · (1 / (𝐵↑_𝑐𝐶))))
22	8, 14, 21	3eqtr4d 2249	1 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐶 ∈ ℂ) → ((𝐴 / 𝐵)↑_𝑐𝐶) = ((𝐴↑_𝑐𝐶) / (𝐵↑_𝑐𝐶)))

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ w3a 981 = wceq 1373 ∈ wcel 2177 class class class wbr 4048 (class class class)co 5954 ℂcc 7936 0cc0 7938 1c1 7939 · cmul 7943 # cap 8667 / cdiv 8758 ℝ⁺crp 9788 ↑_𝑐ccxp 15379

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 615 ax-in2 616 ax-io 711 ax-5 1471 ax-7 1472 ax-gen 1473 ax-ie1 1517 ax-ie2 1518 ax-8 1528 ax-10 1529 ax-11 1530 ax-i12 1531 ax-bndl 1533 ax-4 1534 ax-17 1550 ax-i9 1554 ax-ial 1558 ax-i5r 1559 ax-13 2179 ax-14 2180 ax-ext 2188 ax-coll 4164 ax-sep 4167 ax-nul 4175 ax-pow 4223 ax-pr 4258 ax-un 4485 ax-setind 4590 ax-iinf 4641 ax-cnex 8029 ax-resscn 8030 ax-1cn 8031 ax-1re 8032 ax-icn 8033 ax-addcl 8034 ax-addrcl 8035 ax-mulcl 8036 ax-mulrcl 8037 ax-addcom 8038 ax-mulcom 8039 ax-addass 8040 ax-mulass 8041 ax-distr 8042 ax-i2m1 8043 ax-0lt1 8044 ax-1rid 8045 ax-0id 8046 ax-rnegex 8047 ax-precex 8048 ax-cnre 8049 ax-pre-ltirr 8050 ax-pre-ltwlin 8051 ax-pre-lttrn 8052 ax-pre-apti 8053 ax-pre-ltadd 8054 ax-pre-mulgt0 8055 ax-pre-mulext 8056 ax-arch 8057 ax-caucvg 8058 ax-pre-suploc 8059 ax-addf 8060 ax-mulf 8061

This theorem depends on definitions: df-bi 117 df-stab 833 df-dc 837 df-3or 982 df-3an 983 df-tru 1376 df-fal 1379 df-nf 1485 df-sb 1787 df-eu 2058 df-mo 2059 df-clab 2193 df-cleq 2199 df-clel 2202 df-nfc 2338 df-ne 2378 df-nel 2473 df-ral 2490 df-rex 2491 df-reu 2492 df-rmo 2493 df-rab 2494 df-v 2775 df-sbc 3001 df-csb 3096 df-dif 3170 df-un 3172 df-in 3174 df-ss 3181 df-nul 3463 df-if 3574 df-pw 3620 df-sn 3641 df-pr 3642 df-op 3644 df-uni 3854 df-int 3889 df-iun 3932 df-disj 4025 df-br 4049 df-opab 4111 df-mpt 4112 df-tr 4148 df-id 4345 df-po 4348 df-iso 4349 df-iord 4418 df-on 4420 df-ilim 4421 df-suc 4423 df-iom 4644 df-xp 4686 df-rel 4687 df-cnv 4688 df-co 4689 df-dm 4690 df-rn 4691 df-res 4692 df-ima 4693 df-iota 5238 df-fun 5279 df-fn 5280 df-f 5281 df-f1 5282 df-fo 5283 df-f1o 5284 df-fv 5285 df-isom 5286 df-riota 5909 df-ov 5957 df-oprab 5958 df-mpo 5959 df-of 6168 df-1st 6236 df-2nd 6237 df-recs 6401 df-irdg 6466 df-frec 6487 df-1o 6512 df-oadd 6516 df-er 6630 df-map 6747 df-pm 6748 df-en 6838 df-dom 6839 df-fin 6840 df-sup 7098 df-inf 7099 df-pnf 8122 df-mnf 8123 df-xr 8124 df-ltxr 8125 df-le 8126 df-sub 8258 df-neg 8259 df-reap 8661 df-ap 8668 df-div 8759 df-inn 9050 df-2 9108 df-3 9109 df-4 9110 df-n0 9309 df-z 9386 df-uz 9662 df-q 9754 df-rp 9789 df-xneg 9907 df-xadd 9908 df-ioo 10027 df-ico 10029 df-icc 10030 df-fz 10144 df-fzo 10278 df-seqfrec 10606 df-exp 10697 df-fac 10884 df-bc 10906 df-ihash 10934 df-shft 11176 df-cj 11203 df-re 11204 df-im 11205 df-rsqrt 11359 df-abs 11360 df-clim 11640 df-sumdc 11715 df-ef 12009 df-e 12010 df-rest 13123 df-topgen 13142 df-psmet 14355 df-xmet 14356 df-met 14357 df-bl 14358 df-mopn 14359 df-top 14520 df-topon 14533 df-bases 14565 df-ntr 14618 df-cn 14710 df-cnp 14711 df-tx 14775 df-cncf 15093 df-limced 15178 df-dvap 15179 df-relog 15380 df-rpcxp 15381

This theorem is referenced by: (None)

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