abscxp - Intuitionistic Logic Explorer

	Intuitionistic Logic Explorer		< Previous Next > Nearby theorems
Mirrors > Home > ILE Home > Th. List > abscxp		GIF version

Theorem abscxp 13485

Description: Absolute value of a power, when the base is real. (Contributed by Mario Carneiro, 15-Sep-2014.)

**Assertion**
Ref	Expression
abscxp	⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℂ) → (abs‘(𝐴↑_𝑐𝐵)) = (𝐴↑_𝑐(ℜ‘𝐵)))

**Proof of Theorem abscxp**
Step	Hyp	Ref	Expression
1		simpr 109	. . . . 5 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℂ) → 𝐵 ∈ ℂ)
2		relogcl 13433	. . . . . . 7 ⊢ (𝐴 ∈ ℝ⁺ → (log‘𝐴) ∈ ℝ)
3	2	recnd 7927	. . . . . 6 ⊢ (𝐴 ∈ ℝ⁺ → (log‘𝐴) ∈ ℂ)
4	3	adantr 274	. . . . 5 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℂ) → (log‘𝐴) ∈ ℂ)
5	1, 4	mulcld 7919	. . . 4 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℂ) → (𝐵 · (log‘𝐴)) ∈ ℂ)
6		absef 11710	. . . 4 ⊢ ((𝐵 · (log‘𝐴)) ∈ ℂ → (abs‘(exp‘(𝐵 · (log‘𝐴)))) = (exp‘(ℜ‘(𝐵 · (log‘𝐴)))))
7	5, 6	syl 14	. . 3 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℂ) → (abs‘(exp‘(𝐵 · (log‘𝐴)))) = (exp‘(ℜ‘(𝐵 · (log‘𝐴)))))
8		remul2 10815	. . . . . 6 ⊢ (((log‘𝐴) ∈ ℝ ∧ 𝐵 ∈ ℂ) → (ℜ‘((log‘𝐴) · 𝐵)) = ((log‘𝐴) · (ℜ‘𝐵)))
9	2, 8	sylan 281	. . . . 5 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℂ) → (ℜ‘((log‘𝐴) · 𝐵)) = ((log‘𝐴) · (ℜ‘𝐵)))
10	1, 4	mulcomd 7920	. . . . . 6 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℂ) → (𝐵 · (log‘𝐴)) = ((log‘𝐴) · 𝐵))
11	10	fveq2d 5490	. . . . 5 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℂ) → (ℜ‘(𝐵 · (log‘𝐴))) = (ℜ‘((log‘𝐴) · 𝐵)))
12		recl 10795	. . . . . . . 8 ⊢ (𝐵 ∈ ℂ → (ℜ‘𝐵) ∈ ℝ)
13	12	adantl 275	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℂ) → (ℜ‘𝐵) ∈ ℝ)
14	13	recnd 7927	. . . . . 6 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℂ) → (ℜ‘𝐵) ∈ ℂ)
15	14, 4	mulcomd 7920	. . . . 5 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℂ) → ((ℜ‘𝐵) · (log‘𝐴)) = ((log‘𝐴) · (ℜ‘𝐵)))
16	9, 11, 15	3eqtr4d 2208	. . . 4 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℂ) → (ℜ‘(𝐵 · (log‘𝐴))) = ((ℜ‘𝐵) · (log‘𝐴)))
17	16	fveq2d 5490	. . 3 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℂ) → (exp‘(ℜ‘(𝐵 · (log‘𝐴)))) = (exp‘((ℜ‘𝐵) · (log‘𝐴))))
18	7, 17	eqtrd 2198	. 2 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℂ) → (abs‘(exp‘(𝐵 · (log‘𝐴)))) = (exp‘((ℜ‘𝐵) · (log‘𝐴))))
19		rpcxpef 13465	. . 3 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℂ) → (𝐴↑_𝑐𝐵) = (exp‘(𝐵 · (log‘𝐴))))
20	19	fveq2d 5490	. 2 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℂ) → (abs‘(𝐴↑_𝑐𝐵)) = (abs‘(exp‘(𝐵 · (log‘𝐴)))))
21		rpcxpef 13465	. . 3 ⊢ ((𝐴 ∈ ℝ⁺ ∧ (ℜ‘𝐵) ∈ ℂ) → (𝐴↑_𝑐(ℜ‘𝐵)) = (exp‘((ℜ‘𝐵) · (log‘𝐴))))
22	14, 21	syldan 280	. 2 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℂ) → (𝐴↑_𝑐(ℜ‘𝐵)) = (exp‘((ℜ‘𝐵) · (log‘𝐴))))
23	18, 20, 22	3eqtr4d 2208	1 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℂ) → (abs‘(𝐴↑_𝑐𝐵)) = (𝐴↑_𝑐(ℜ‘𝐵)))

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 103 = wceq 1343 ∈ wcel 2136 ‘cfv 5188 (class class class)co 5842 ℂcc 7751 ℝcr 7752 · cmul 7758 ℝ⁺crp 9589 ℜcre 10782 abscabs 10939 expce 11583 logclog 13427 ↑_𝑐ccxp 13428

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 604 ax-in2 605 ax-io 699 ax-5 1435 ax-7 1436 ax-gen 1437 ax-ie1 1481 ax-ie2 1482 ax-8 1492 ax-10 1493 ax-11 1494 ax-i12 1495 ax-bndl 1497 ax-4 1498 ax-17 1514 ax-i9 1518 ax-ial 1522 ax-i5r 1523 ax-13 2138 ax-14 2139 ax-ext 2147 ax-coll 4097 ax-sep 4100 ax-nul 4108 ax-pow 4153 ax-pr 4187 ax-un 4411 ax-setind 4514 ax-iinf 4565 ax-cnex 7844 ax-resscn 7845 ax-1cn 7846 ax-1re 7847 ax-icn 7848 ax-addcl 7849 ax-addrcl 7850 ax-mulcl 7851 ax-mulrcl 7852 ax-addcom 7853 ax-mulcom 7854 ax-addass 7855 ax-mulass 7856 ax-distr 7857 ax-i2m1 7858 ax-0lt1 7859 ax-1rid 7860 ax-0id 7861 ax-rnegex 7862 ax-precex 7863 ax-cnre 7864 ax-pre-ltirr 7865 ax-pre-ltwlin 7866 ax-pre-lttrn 7867 ax-pre-apti 7868 ax-pre-ltadd 7869 ax-pre-mulgt0 7870 ax-pre-mulext 7871 ax-arch 7872 ax-caucvg 7873 ax-pre-suploc 7874 ax-addf 7875 ax-mulf 7876

This theorem depends on definitions: df-bi 116 df-stab 821 df-dc 825 df-3or 969 df-3an 970 df-tru 1346 df-fal 1349 df-nf 1449 df-sb 1751 df-eu 2017 df-mo 2018 df-clab 2152 df-cleq 2158 df-clel 2161 df-nfc 2297 df-ne 2337 df-nel 2432 df-ral 2449 df-rex 2450 df-reu 2451 df-rmo 2452 df-rab 2453 df-v 2728 df-sbc 2952 df-csb 3046 df-dif 3118 df-un 3120 df-in 3122 df-ss 3129 df-nul 3410 df-if 3521 df-pw 3561 df-sn 3582 df-pr 3583 df-op 3585 df-uni 3790 df-int 3825 df-iun 3868 df-disj 3960 df-br 3983 df-opab 4044 df-mpt 4045 df-tr 4081 df-id 4271 df-po 4274 df-iso 4275 df-iord 4344 df-on 4346 df-ilim 4347 df-suc 4349 df-iom 4568 df-xp 4610 df-rel 4611 df-cnv 4612 df-co 4613 df-dm 4614 df-rn 4615 df-res 4616 df-ima 4617 df-iota 5153 df-fun 5190 df-fn 5191 df-f 5192 df-f1 5193 df-fo 5194 df-f1o 5195 df-fv 5196 df-isom 5197 df-riota 5798 df-ov 5845 df-oprab 5846 df-mpo 5847 df-of 6050 df-1st 6108 df-2nd 6109 df-recs 6273 df-irdg 6338 df-frec 6359 df-1o 6384 df-oadd 6388 df-er 6501 df-map 6616 df-pm 6617 df-en 6707 df-dom 6708 df-fin 6709 df-sup 6949 df-inf 6950 df-pnf 7935 df-mnf 7936 df-xr 7937 df-ltxr 7938 df-le 7939 df-sub 8071 df-neg 8072 df-reap 8473 df-ap 8480 df-div 8569 df-inn 8858 df-2 8916 df-3 8917 df-4 8918 df-n0 9115 df-z 9192 df-uz 9467 df-q 9558 df-rp 9590 df-xneg 9708 df-xadd 9709 df-ioo 9828 df-ico 9830 df-icc 9831 df-fz 9945 df-fzo 10078 df-seqfrec 10381 df-exp 10455 df-fac 10639 df-bc 10661 df-ihash 10689 df-shft 10757 df-cj 10784 df-re 10785 df-im 10786 df-rsqrt 10940 df-abs 10941 df-clim 11220 df-sumdc 11295 df-ef 11589 df-e 11590 df-sin 11591 df-cos 11592 df-rest 12558 df-topgen 12577 df-psmet 12637 df-xmet 12638 df-met 12639 df-bl 12640 df-mopn 12641 df-top 12646 df-topon 12659 df-bases 12691 df-ntr 12746 df-cn 12838 df-cnp 12839 df-tx 12903 df-cncf 13208 df-limced 13275 df-dvap 13276 df-relog 13429 df-rpcxp 13430

This theorem is referenced by: (None)

W3C validator