rpcxpadd - Intuitionistic Logic Explorer

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Theorem rpcxpadd 15040

Description: Sum of exponents law for complex exponentiation. (Contributed by Mario Carneiro, 2-Aug-2014.) (Revised by Jim Kingdon, 13-Jun-2024.)

**Assertion**
Ref	Expression
rpcxpadd	⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → (𝐴↑_𝑐(𝐵 + 𝐶)) = ((𝐴↑_𝑐𝐵) · (𝐴↑_𝑐𝐶)))

**Proof of Theorem rpcxpadd**
Step	Hyp	Ref	Expression
1		simp2 1000	. . . . 5 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → 𝐵 ∈ ℂ)
2		simp3 1001	. . . . 5 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → 𝐶 ∈ ℂ)
3		relogcl 14997	. . . . . . 7 ⊢ (𝐴 ∈ ℝ⁺ → (log‘𝐴) ∈ ℝ)
4	3	3ad2ant1 1020	. . . . . 6 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → (log‘𝐴) ∈ ℝ)
5	4	recnd 8048	. . . . 5 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → (log‘𝐴) ∈ ℂ)
6	1, 2, 5	adddird 8045	. . . 4 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐵 + 𝐶) · (log‘𝐴)) = ((𝐵 · (log‘𝐴)) + (𝐶 · (log‘𝐴))))
7	6	fveq2d 5558	. . 3 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → (exp‘((𝐵 + 𝐶) · (log‘𝐴))) = (exp‘((𝐵 · (log‘𝐴)) + (𝐶 · (log‘𝐴)))))
8	1, 5	mulcld 8040	. . . 4 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → (𝐵 · (log‘𝐴)) ∈ ℂ)
9	2, 5	mulcld 8040	. . . 4 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → (𝐶 · (log‘𝐴)) ∈ ℂ)
10		efadd 11818	. . . 4 ⊢ (((𝐵 · (log‘𝐴)) ∈ ℂ ∧ (𝐶 · (log‘𝐴)) ∈ ℂ) → (exp‘((𝐵 · (log‘𝐴)) + (𝐶 · (log‘𝐴)))) = ((exp‘(𝐵 · (log‘𝐴))) · (exp‘(𝐶 · (log‘𝐴)))))
11	8, 9, 10	syl2anc 411	. . 3 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → (exp‘((𝐵 · (log‘𝐴)) + (𝐶 · (log‘𝐴)))) = ((exp‘(𝐵 · (log‘𝐴))) · (exp‘(𝐶 · (log‘𝐴)))))
12	7, 11	eqtrd 2226	. 2 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → (exp‘((𝐵 + 𝐶) · (log‘𝐴))) = ((exp‘(𝐵 · (log‘𝐴))) · (exp‘(𝐶 · (log‘𝐴)))))
13		simp1 999	. . 3 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → 𝐴 ∈ ℝ⁺)
14	1, 2	addcld 8039	. . 3 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → (𝐵 + 𝐶) ∈ ℂ)
15		rpcxpef 15029	. . 3 ⊢ ((𝐴 ∈ ℝ⁺ ∧ (𝐵 + 𝐶) ∈ ℂ) → (𝐴↑_𝑐(𝐵 + 𝐶)) = (exp‘((𝐵 + 𝐶) · (log‘𝐴))))
16	13, 14, 15	syl2anc 411	. 2 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → (𝐴↑_𝑐(𝐵 + 𝐶)) = (exp‘((𝐵 + 𝐶) · (log‘𝐴))))
17		rpcxpef 15029	. . . 4 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℂ) → (𝐴↑_𝑐𝐵) = (exp‘(𝐵 · (log‘𝐴))))
18	13, 1, 17	syl2anc 411	. . 3 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → (𝐴↑_𝑐𝐵) = (exp‘(𝐵 · (log‘𝐴))))
19		rpcxpef 15029	. . . 4 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐶 ∈ ℂ) → (𝐴↑_𝑐𝐶) = (exp‘(𝐶 · (log‘𝐴))))
20	13, 2, 19	syl2anc 411	. . 3 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → (𝐴↑_𝑐𝐶) = (exp‘(𝐶 · (log‘𝐴))))
21	18, 20	oveq12d 5936	. 2 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴↑_𝑐𝐵) · (𝐴↑_𝑐𝐶)) = ((exp‘(𝐵 · (log‘𝐴))) · (exp‘(𝐶 · (log‘𝐴)))))
22	12, 16, 21	3eqtr4d 2236	1 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → (𝐴↑_𝑐(𝐵 + 𝐶)) = ((𝐴↑_𝑐𝐵) · (𝐴↑_𝑐𝐶)))

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ w3a 980 = wceq 1364 ∈ wcel 2164 ‘cfv 5254 (class class class)co 5918 ℂcc 7870 ℝcr 7871 + caddc 7875 · cmul 7877 ℝ⁺crp 9719 expce 11785 logclog 14991 ↑_𝑐ccxp 14992

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 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2166 ax-14 2167 ax-ext 2175 ax-coll 4144 ax-sep 4147 ax-nul 4155 ax-pow 4203 ax-pr 4238 ax-un 4464 ax-setind 4569 ax-iinf 4620 ax-cnex 7963 ax-resscn 7964 ax-1cn 7965 ax-1re 7966 ax-icn 7967 ax-addcl 7968 ax-addrcl 7969 ax-mulcl 7970 ax-mulrcl 7971 ax-addcom 7972 ax-mulcom 7973 ax-addass 7974 ax-mulass 7975 ax-distr 7976 ax-i2m1 7977 ax-0lt1 7978 ax-1rid 7979 ax-0id 7980 ax-rnegex 7981 ax-precex 7982 ax-cnre 7983 ax-pre-ltirr 7984 ax-pre-ltwlin 7985 ax-pre-lttrn 7986 ax-pre-apti 7987 ax-pre-ltadd 7988 ax-pre-mulgt0 7989 ax-pre-mulext 7990 ax-arch 7991 ax-caucvg 7992 ax-pre-suploc 7993 ax-addf 7994 ax-mulf 7995

This theorem depends on definitions: df-bi 117 df-stab 832 df-dc 836 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2045 df-mo 2046 df-clab 2180 df-cleq 2186 df-clel 2189 df-nfc 2325 df-ne 2365 df-nel 2460 df-ral 2477 df-rex 2478 df-reu 2479 df-rmo 2480 df-rab 2481 df-v 2762 df-sbc 2986 df-csb 3081 df-dif 3155 df-un 3157 df-in 3159 df-ss 3166 df-nul 3447 df-if 3558 df-pw 3603 df-sn 3624 df-pr 3625 df-op 3627 df-uni 3836 df-int 3871 df-iun 3914 df-disj 4007 df-br 4030 df-opab 4091 df-mpt 4092 df-tr 4128 df-id 4324 df-po 4327 df-iso 4328 df-iord 4397 df-on 4399 df-ilim 4400 df-suc 4402 df-iom 4623 df-xp 4665 df-rel 4666 df-cnv 4667 df-co 4668 df-dm 4669 df-rn 4670 df-res 4671 df-ima 4672 df-iota 5215 df-fun 5256 df-fn 5257 df-f 5258 df-f1 5259 df-fo 5260 df-f1o 5261 df-fv 5262 df-isom 5263 df-riota 5873 df-ov 5921 df-oprab 5922 df-mpo 5923 df-of 6130 df-1st 6193 df-2nd 6194 df-recs 6358 df-irdg 6423 df-frec 6444 df-1o 6469 df-oadd 6473 df-er 6587 df-map 6704 df-pm 6705 df-en 6795 df-dom 6796 df-fin 6797 df-sup 7043 df-inf 7044 df-pnf 8056 df-mnf 8057 df-xr 8058 df-ltxr 8059 df-le 8060 df-sub 8192 df-neg 8193 df-reap 8594 df-ap 8601 df-div 8692 df-inn 8983 df-2 9041 df-3 9042 df-4 9043 df-n0 9241 df-z 9318 df-uz 9593 df-q 9685 df-rp 9720 df-xneg 9838 df-xadd 9839 df-ioo 9958 df-ico 9960 df-icc 9961 df-fz 10075 df-fzo 10209 df-seqfrec 10519 df-exp 10610 df-fac 10797 df-bc 10819 df-ihash 10847 df-shft 10959 df-cj 10986 df-re 10987 df-im 10988 df-rsqrt 11142 df-abs 11143 df-clim 11422 df-sumdc 11497 df-ef 11791 df-e 11792 df-rest 12852 df-topgen 12871 df-psmet 14039 df-xmet 14040 df-met 14041 df-bl 14042 df-mopn 14043 df-top 14166 df-topon 14179 df-bases 14211 df-ntr 14264 df-cn 14356 df-cnp 14357 df-tx 14421 df-cncf 14726 df-limced 14810 df-dvap 14811 df-relog 14993 df-rpcxp 14994

This theorem is referenced by: rpcxpp1 15041 rpcxpneg 15042 rpcxpsub 15043 rpcxpsqrt 15056

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