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Theorem absexp 15255

Description: Absolute value of positive integer exponentiation. (Contributed by NM, 5-Jan-2006.)

**Assertion**
Ref	Expression
absexp	⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ₀) → (abs‘(𝐴↑𝑁)) = ((abs‘𝐴)↑𝑁))

Proof of Theorem absexp Dummy variables 𝑗 𝑘 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		oveq2 7412	. . . 4 ⊢ (𝑗 = 0 → (𝐴↑𝑗) = (𝐴↑0))
2	1	fveq2d 6888	. . 3 ⊢ (𝑗 = 0 → (abs‘(𝐴↑𝑗)) = (abs‘(𝐴↑0)))
3		oveq2 7412	. . 3 ⊢ (𝑗 = 0 → ((abs‘𝐴)↑𝑗) = ((abs‘𝐴)↑0))
4	2, 3	eqeq12d 2742	. 2 ⊢ (𝑗 = 0 → ((abs‘(𝐴↑𝑗)) = ((abs‘𝐴)↑𝑗) ↔ (abs‘(𝐴↑0)) = ((abs‘𝐴)↑0)))
5		oveq2 7412	. . . 4 ⊢ (𝑗 = 𝑘 → (𝐴↑𝑗) = (𝐴↑𝑘))
6	5	fveq2d 6888	. . 3 ⊢ (𝑗 = 𝑘 → (abs‘(𝐴↑𝑗)) = (abs‘(𝐴↑𝑘)))
7		oveq2 7412	. . 3 ⊢ (𝑗 = 𝑘 → ((abs‘𝐴)↑𝑗) = ((abs‘𝐴)↑𝑘))
8	6, 7	eqeq12d 2742	. 2 ⊢ (𝑗 = 𝑘 → ((abs‘(𝐴↑𝑗)) = ((abs‘𝐴)↑𝑗) ↔ (abs‘(𝐴↑𝑘)) = ((abs‘𝐴)↑𝑘)))
9		oveq2 7412	. . . 4 ⊢ (𝑗 = (𝑘 + 1) → (𝐴↑𝑗) = (𝐴↑(𝑘 + 1)))
10	9	fveq2d 6888	. . 3 ⊢ (𝑗 = (𝑘 + 1) → (abs‘(𝐴↑𝑗)) = (abs‘(𝐴↑(𝑘 + 1))))
11		oveq2 7412	. . 3 ⊢ (𝑗 = (𝑘 + 1) → ((abs‘𝐴)↑𝑗) = ((abs‘𝐴)↑(𝑘 + 1)))
12	10, 11	eqeq12d 2742	. 2 ⊢ (𝑗 = (𝑘 + 1) → ((abs‘(𝐴↑𝑗)) = ((abs‘𝐴)↑𝑗) ↔ (abs‘(𝐴↑(𝑘 + 1))) = ((abs‘𝐴)↑(𝑘 + 1))))
13		oveq2 7412	. . . 4 ⊢ (𝑗 = 𝑁 → (𝐴↑𝑗) = (𝐴↑𝑁))
14	13	fveq2d 6888	. . 3 ⊢ (𝑗 = 𝑁 → (abs‘(𝐴↑𝑗)) = (abs‘(𝐴↑𝑁)))
15		oveq2 7412	. . 3 ⊢ (𝑗 = 𝑁 → ((abs‘𝐴)↑𝑗) = ((abs‘𝐴)↑𝑁))
16	14, 15	eqeq12d 2742	. 2 ⊢ (𝑗 = 𝑁 → ((abs‘(𝐴↑𝑗)) = ((abs‘𝐴)↑𝑗) ↔ (abs‘(𝐴↑𝑁)) = ((abs‘𝐴)↑𝑁)))
17		abs1 15248	. . 3 ⊢ (abs‘1) = 1
18		exp0 14034	. . . 4 ⊢ (𝐴 ∈ ℂ → (𝐴↑0) = 1)
19	18	fveq2d 6888	. . 3 ⊢ (𝐴 ∈ ℂ → (abs‘(𝐴↑0)) = (abs‘1))
20		abscl 15229	. . . . 5 ⊢ (𝐴 ∈ ℂ → (abs‘𝐴) ∈ ℝ)
21	20	recnd 11243	. . . 4 ⊢ (𝐴 ∈ ℂ → (abs‘𝐴) ∈ ℂ)
22	21	exp0d 14108	. . 3 ⊢ (𝐴 ∈ ℂ → ((abs‘𝐴)↑0) = 1)
23	17, 19, 22	3eqtr4a 2792	. 2 ⊢ (𝐴 ∈ ℂ → (abs‘(𝐴↑0)) = ((abs‘𝐴)↑0))
24		oveq1 7411	. . . 4 ⊢ ((abs‘(𝐴↑𝑘)) = ((abs‘𝐴)↑𝑘) → ((abs‘(𝐴↑𝑘)) · (abs‘𝐴)) = (((abs‘𝐴)↑𝑘) · (abs‘𝐴)))
25	24	adantl 481	. . 3 ⊢ (((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℕ₀) ∧ (abs‘(𝐴↑𝑘)) = ((abs‘𝐴)↑𝑘)) → ((abs‘(𝐴↑𝑘)) · (abs‘𝐴)) = (((abs‘𝐴)↑𝑘) · (abs‘𝐴)))
26		expp1 14037	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℕ₀) → (𝐴↑(𝑘 + 1)) = ((𝐴↑𝑘) · 𝐴))
27	26	fveq2d 6888	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℕ₀) → (abs‘(𝐴↑(𝑘 + 1))) = (abs‘((𝐴↑𝑘) · 𝐴)))
28		expcl 14048	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℕ₀) → (𝐴↑𝑘) ∈ ℂ)
29		simpl 482	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℕ₀) → 𝐴 ∈ ℂ)
30		absmul 15245	. . . . . 6 ⊢ (((𝐴↑𝑘) ∈ ℂ ∧ 𝐴 ∈ ℂ) → (abs‘((𝐴↑𝑘) · 𝐴)) = ((abs‘(𝐴↑𝑘)) · (abs‘𝐴)))
31	28, 29, 30	syl2anc 583	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℕ₀) → (abs‘((𝐴↑𝑘) · 𝐴)) = ((abs‘(𝐴↑𝑘)) · (abs‘𝐴)))
32	27, 31	eqtrd 2766	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℕ₀) → (abs‘(𝐴↑(𝑘 + 1))) = ((abs‘(𝐴↑𝑘)) · (abs‘𝐴)))
33	32	adantr 480	. . 3 ⊢ (((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℕ₀) ∧ (abs‘(𝐴↑𝑘)) = ((abs‘𝐴)↑𝑘)) → (abs‘(𝐴↑(𝑘 + 1))) = ((abs‘(𝐴↑𝑘)) · (abs‘𝐴)))
34		expp1 14037	. . . . 5 ⊢ (((abs‘𝐴) ∈ ℂ ∧ 𝑘 ∈ ℕ₀) → ((abs‘𝐴)↑(𝑘 + 1)) = (((abs‘𝐴)↑𝑘) · (abs‘𝐴)))
35	21, 34	sylan 579	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℕ₀) → ((abs‘𝐴)↑(𝑘 + 1)) = (((abs‘𝐴)↑𝑘) · (abs‘𝐴)))
36	35	adantr 480	. . 3 ⊢ (((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℕ₀) ∧ (abs‘(𝐴↑𝑘)) = ((abs‘𝐴)↑𝑘)) → ((abs‘𝐴)↑(𝑘 + 1)) = (((abs‘𝐴)↑𝑘) · (abs‘𝐴)))
37	25, 33, 36	3eqtr4d 2776	. 2 ⊢ (((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℕ₀) ∧ (abs‘(𝐴↑𝑘)) = ((abs‘𝐴)↑𝑘)) → (abs‘(𝐴↑(𝑘 + 1))) = ((abs‘𝐴)↑(𝑘 + 1)))
38	4, 8, 12, 16, 23, 37	nn0indd 12660	1 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ₀) → (abs‘(𝐴↑𝑁)) = ((abs‘𝐴)↑𝑁))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1533 ∈ wcel 2098 ‘cfv 6536 (class class class)co 7404 ℂcc 11107 0cc0 11109 1c1 11110 + caddc 11112 · cmul 11114 ℕ₀cn0 12473 ↑cexp 14030 abscabs 15185

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7721 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-pre-mulgt0 11186 ax-pre-sup 11187

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6293 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-om 7852 df-2nd 7972 df-frecs 8264 df-wrecs 8295 df-recs 8369 df-rdg 8408 df-er 8702 df-en 8939 df-dom 8940 df-sdom 8941 df-sup 9436 df-pnf 11251 df-mnf 11252 df-xr 11253 df-ltxr 11254 df-le 11255 df-sub 11447 df-neg 11448 df-div 11873 df-nn 12214 df-2 12276 df-3 12277 df-n0 12474 df-z 12560 df-uz 12824 df-rp 12978 df-seq 13970 df-exp 14031 df-cj 15050 df-re 15051 df-im 15052 df-sqrt 15186 df-abs 15187

This theorem is referenced by: absexpz 15256 abssq 15257 sqabs 15258 absexpd 15403 expcnv 15814 eftabs 16023 efcllem 16025 efaddlem 16041 iblabsr 25710 iblmulc2 25711 abelthlem7 26326 efif1olem3 26429 efif1olem4 26430 logtayllem 26544 bndatandm 26812 ftalem1 26956 mule1 27031 iblmulc2nc 37064

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