rpmulcxp - Intuitionistic Logic Explorer

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Theorem rpmulcxp 16142

Description: Complex exponentiation of a product. Proposition 10-4.2(c) of [Gleason] p. 135. (Contributed by Mario Carneiro, 2-Aug-2014.)

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

**Proof of Theorem rpmulcxp**
Step	Hyp	Ref	Expression
1		simp1 1028	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐶 ∈ ℂ) → 𝐴 ∈ ℝ⁺)
2		simp2 1029	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐶 ∈ ℂ) → 𝐵 ∈ ℝ⁺)
3	1, 2	relogmuld 16114	. . . . . 6 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐶 ∈ ℂ) → (log‘(𝐴 · 𝐵)) = ((log‘𝐴) + (log‘𝐵)))
4	3	oveq2d 6104	. . . . 5 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐶 ∈ ℂ) → (𝐶 · (log‘(𝐴 · 𝐵))) = (𝐶 · ((log‘𝐴) + (log‘𝐵))))
5		simp3 1030	. . . . . 6 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐶 ∈ ℂ) → 𝐶 ∈ ℂ)
6	1	relogcld 16112	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐶 ∈ ℂ) → (log‘𝐴) ∈ ℝ)
7	6	recnd 8360	. . . . . 6 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐶 ∈ ℂ) → (log‘𝐴) ∈ ℂ)
8	2	relogcld 16112	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐶 ∈ ℂ) → (log‘𝐵) ∈ ℝ)
9	8	recnd 8360	. . . . . 6 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐶 ∈ ℂ) → (log‘𝐵) ∈ ℂ)
10	5, 7, 9	adddid 8356	. . . . 5 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐶 ∈ ℂ) → (𝐶 · ((log‘𝐴) + (log‘𝐵))) = ((𝐶 · (log‘𝐴)) + (𝐶 · (log‘𝐵))))
11	4, 10	eqtrd 2271	. . . 4 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐶 ∈ ℂ) → (𝐶 · (log‘(𝐴 · 𝐵))) = ((𝐶 · (log‘𝐴)) + (𝐶 · (log‘𝐵))))
12	11	fveq2d 5702	. . 3 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐶 ∈ ℂ) → (exp‘(𝐶 · (log‘(𝐴 · 𝐵)))) = (exp‘((𝐶 · (log‘𝐴)) + (𝐶 · (log‘𝐵)))))
13	5, 7	mulcld 8352	. . . 4 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐶 ∈ ℂ) → (𝐶 · (log‘𝐴)) ∈ ℂ)
14	5, 9	mulcld 8352	. . . 4 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐶 ∈ ℂ) → (𝐶 · (log‘𝐵)) ∈ ℂ)
15		efadd 12467	. . . 4 ⊢ (((𝐶 · (log‘𝐴)) ∈ ℂ ∧ (𝐶 · (log‘𝐵)) ∈ ℂ) → (exp‘((𝐶 · (log‘𝐴)) + (𝐶 · (log‘𝐵)))) = ((exp‘(𝐶 · (log‘𝐴))) · (exp‘(𝐶 · (log‘𝐵)))))
16	13, 14, 15	syl2anc 415	. . 3 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐶 ∈ ℂ) → (exp‘((𝐶 · (log‘𝐴)) + (𝐶 · (log‘𝐵)))) = ((exp‘(𝐶 · (log‘𝐴))) · (exp‘(𝐶 · (log‘𝐵)))))
17	12, 16	eqtrd 2271	. 2 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐶 ∈ ℂ) → (exp‘(𝐶 · (log‘(𝐴 · 𝐵)))) = ((exp‘(𝐶 · (log‘𝐴))) · (exp‘(𝐶 · (log‘𝐵)))))
18	1, 2	rpmulcld 10131	. . 3 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐶 ∈ ℂ) → (𝐴 · 𝐵) ∈ ℝ⁺)
19		rpcxpef 16127	. . 3 ⊢ (((𝐴 · 𝐵) ∈ ℝ⁺ ∧ 𝐶 ∈ ℂ) → ((𝐴 · 𝐵)↑_𝑐𝐶) = (exp‘(𝐶 · (log‘(𝐴 · 𝐵)))))
20	18, 5, 19	syl2anc 415	. 2 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐶 ∈ ℂ) → ((𝐴 · 𝐵)↑_𝑐𝐶) = (exp‘(𝐶 · (log‘(𝐴 · 𝐵)))))
21		rpcxpef 16127	. . . 4 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐶 ∈ ℂ) → (𝐴↑_𝑐𝐶) = (exp‘(𝐶 · (log‘𝐴))))
22	1, 5, 21	syl2anc 415	. . 3 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐶 ∈ ℂ) → (𝐴↑_𝑐𝐶) = (exp‘(𝐶 · (log‘𝐴))))
23		rpcxpef 16127	. . . 4 ⊢ ((𝐵 ∈ ℝ⁺ ∧ 𝐶 ∈ ℂ) → (𝐵↑_𝑐𝐶) = (exp‘(𝐶 · (log‘𝐵))))
24	2, 5, 23	syl2anc 415	. . 3 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐶 ∈ ℂ) → (𝐵↑_𝑐𝐶) = (exp‘(𝐶 · (log‘𝐵))))
25	22, 24	oveq12d 6106	. 2 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐶 ∈ ℂ) → ((𝐴↑_𝑐𝐶) · (𝐵↑_𝑐𝐶)) = ((exp‘(𝐶 · (log‘𝐴))) · (exp‘(𝐶 · (log‘𝐵)))))
26	17, 20, 25	3eqtr4d 2281	1 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝐶 ∈ ℂ) → ((𝐴 · 𝐵)↑_𝑐𝐶) = ((𝐴↑_𝑐𝐶) · (𝐵↑_𝑐𝐶)))

Colors of variables: wff set class

This proof depends on syntax axioms: → wi 4 ∧ w3a 1009 = wceq 1402 ∈ wcel 2209 ‘cfv 5380 (class class class)co 6088 ℂcc 8183 + caddc 8188 · cmul 8190 ℝ⁺crp 10071 expce 12434 logclog 16085 ↑_𝑐ccxp 16086

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 623 ax-in2 624 ax-io 721 ax-5 1500 ax-7 1501 ax-gen 1502 ax-ie1 1546 ax-ie2 1547 ax-8 1557 ax-10 1558 ax-11 1559 ax-i12 1560 ax-bndl 1562 ax-4 1563 ax-17 1579 ax-i9 1583 ax-ial 1587 ax-i5r 1588 ax-14 2212 ax-ext 2220 ax-coll 4248 ax-sep 4251 ax-nul 4261 ax-pow 4313 ax-pr 4348 ax-un 4580 ax-setind 4686 ax-iinf 4737 ax-cnex 8276 ax-resscn 8277 ax-1cn 8278 ax-1re 8279 ax-icn 8280 ax-addcl 8281 ax-addrcl 8282 ax-mulcl 8283 ax-mulrcl 8284 ax-addcom 8285 ax-mulcom 8286 ax-addass 8287 ax-mulass 8288 ax-distr 8289 ax-i2m1 8290 ax-0lt1 8291 ax-1rid 8292 ax-0id 8293 ax-rnegex 8294 ax-precex 8295 ax-cnre 8296 ax-pre-ltirr 8297 ax-pre-ltwlin 8298 ax-pre-lttrn 8299 ax-pre-apti 8300 ax-pre-ltadd 8301 ax-pre-mulgt0 8302 ax-pre-mulext 8303 ax-arch 8304 ax-caucvg 8305 ax-pre-suploc 8306 ax-addf 8307 ax-mulf 8308

This proof depends on definitions: df-bi 117 df-stab 843 df-dc 847 df-3or 1010 df-3an 1011 df-tru 1405 df-fal 1408 df-nf 1514 df-sb 1816 df-eu 2089 df-mo 2090 df-clab 2225 df-cleq 2231 df-clel 2234 df-nfc 2381 df-ne 2421 df-nel 2516 df-ral 2533 df-rex 2534 df-reu 2535 df-rmo 2536 df-rab 2537 df-v 2823 df-sbc 3052 df-csb 3148 df-dif 3222 df-un 3224 df-in 3226 df-ss 3233 df-nul 3521 df-if 3639 df-pw 3690 df-sn 3715 df-pr 3716 df-op 3718 df-uni 3936 df-int 3971 df-iun 4014 df-disj 4107 df-br 4132 df-opab 4194 df-mpt 4195 df-tr 4232 df-id 4440 df-po 4443 df-iso 4444 df-iord 4513 df-on 4515 df-ilim 4516 df-suc 4518 df-iom 4740 df-xp 4782 df-rel 4783 df-cnv 4784 df-co 4785 df-dm 4786 df-rn 4787 df-res 4788 df-ima 4789 df-iota 5340 df-fun 5382 df-fn 5383 df-f 5384 df-f1 5385 df-fo 5386 df-f1o 5387 df-fv 5388 df-isom 5389 df-riota 6041 df-ov 6091 df-oprab 6092 df-mpo 6093 df-of 6305 df-1st 6378 df-2nd 6379 df-recs 6580 df-irdg 6645 df-frec 6666 df-1o 6691 df-oadd 6695 df-er 6811 df-map 6928 df-pm 6929 df-en 7027 df-dom 7028 df-fin 7029 df-sup 7330 df-inf 7331 df-pnf 8368 df-mnf 8369 df-xr 8370 df-ltxr 8371 df-le 8372 df-sub 8506 df-neg 8507 df-reap 8911 df-ap 8918 df-div 9011 df-inn 9314 df-2 9372 df-3 9373 df-4 9374 df-n0 9575 df-z 9656 df-uz 9938 df-q 10036 df-rp 10072 df-xneg 10191 df-xadd 10192 df-ioo 10311 df-ico 10313 df-icc 10314 df-fz 10429 df-fzo 10567 df-seqfrec 10906 df-exp 10997 df-fac 11186 df-bc 11208 df-ihash 11237 df-shft 11602 df-cj 11629 df-re 11630 df-im 11631 df-rsqrt 11786 df-abs 11787 df-clim 12070 df-sumdc 12145 df-ef 12440 df-e 12441 df-rest 13654 df-topgen 13673 df-psmet 14989 df-xmet 14990 df-met 14991 df-bl 14992 df-mopn 14993 df-top 15224 df-topon 15237 df-bases 15269 df-ntr 15322 df-cn 15414 df-cnp 15415 df-tx 15479 df-cncf 15797 df-limced 15882 df-dvap 15883 df-relog 16087 df-rpcxp 16088

This theorem is used by: cxprec 16143 rpdivcxp 16144 sgmmul 16290

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