Theorem absexp 11043

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 5861	. . . . . 6 ⊢ (𝑗 = 0 → (𝐴↑𝑗) = (𝐴↑0))
2	1	fveq2d 5500	. . . . 5 ⊢ (𝑗 = 0 → (abs‘(𝐴↑𝑗)) = (abs‘(𝐴↑0)))
3		oveq2 5861	. . . . 5 ⊢ (𝑗 = 0 → ((abs‘𝐴)↑𝑗) = ((abs‘𝐴)↑0))
4	2, 3	eqeq12d 2185	. . . 4 ⊢ (𝑗 = 0 → ((abs‘(𝐴↑𝑗)) = ((abs‘𝐴)↑𝑗) ↔ (abs‘(𝐴↑0)) = ((abs‘𝐴)↑0)))
5	4	imbi2d 229	. . 3 ⊢ (𝑗 = 0 → ((𝐴 ∈ ℂ → (abs‘(𝐴↑𝑗)) = ((abs‘𝐴)↑𝑗)) ↔ (𝐴 ∈ ℂ → (abs‘(𝐴↑0)) = ((abs‘𝐴)↑0))))
6		oveq2 5861	. . . . . 6 ⊢ (𝑗 = 𝑘 → (𝐴↑𝑗) = (𝐴↑𝑘))
7	6	fveq2d 5500	. . . . 5 ⊢ (𝑗 = 𝑘 → (abs‘(𝐴↑𝑗)) = (abs‘(𝐴↑𝑘)))
8		oveq2 5861	. . . . 5 ⊢ (𝑗 = 𝑘 → ((abs‘𝐴)↑𝑗) = ((abs‘𝐴)↑𝑘))
9	7, 8	eqeq12d 2185	. . . 4 ⊢ (𝑗 = 𝑘 → ((abs‘(𝐴↑𝑗)) = ((abs‘𝐴)↑𝑗) ↔ (abs‘(𝐴↑𝑘)) = ((abs‘𝐴)↑𝑘)))
10	9	imbi2d 229	. . 3 ⊢ (𝑗 = 𝑘 → ((𝐴 ∈ ℂ → (abs‘(𝐴↑𝑗)) = ((abs‘𝐴)↑𝑗)) ↔ (𝐴 ∈ ℂ → (abs‘(𝐴↑𝑘)) = ((abs‘𝐴)↑𝑘))))
11		oveq2 5861	. . . . . 6 ⊢ (𝑗 = (𝑘 + 1) → (𝐴↑𝑗) = (𝐴↑(𝑘 + 1)))
12	11	fveq2d 5500	. . . . 5 ⊢ (𝑗 = (𝑘 + 1) → (abs‘(𝐴↑𝑗)) = (abs‘(𝐴↑(𝑘 + 1))))
13		oveq2 5861	. . . . 5 ⊢ (𝑗 = (𝑘 + 1) → ((abs‘𝐴)↑𝑗) = ((abs‘𝐴)↑(𝑘 + 1)))
14	12, 13	eqeq12d 2185	. . . 4 ⊢ (𝑗 = (𝑘 + 1) → ((abs‘(𝐴↑𝑗)) = ((abs‘𝐴)↑𝑗) ↔ (abs‘(𝐴↑(𝑘 + 1))) = ((abs‘𝐴)↑(𝑘 + 1))))
15	14	imbi2d 229	. . 3 ⊢ (𝑗 = (𝑘 + 1) → ((𝐴 ∈ ℂ → (abs‘(𝐴↑𝑗)) = ((abs‘𝐴)↑𝑗)) ↔ (𝐴 ∈ ℂ → (abs‘(𝐴↑(𝑘 + 1))) = ((abs‘𝐴)↑(𝑘 + 1)))))
16		oveq2 5861	. . . . . 6 ⊢ (𝑗 = 𝑁 → (𝐴↑𝑗) = (𝐴↑𝑁))
17	16	fveq2d 5500	. . . . 5 ⊢ (𝑗 = 𝑁 → (abs‘(𝐴↑𝑗)) = (abs‘(𝐴↑𝑁)))
18		oveq2 5861	. . . . 5 ⊢ (𝑗 = 𝑁 → ((abs‘𝐴)↑𝑗) = ((abs‘𝐴)↑𝑁))
19	17, 18	eqeq12d 2185	. . . 4 ⊢ (𝑗 = 𝑁 → ((abs‘(𝐴↑𝑗)) = ((abs‘𝐴)↑𝑗) ↔ (abs‘(𝐴↑𝑁)) = ((abs‘𝐴)↑𝑁)))
20	19	imbi2d 229	. . 3 ⊢ (𝑗 = 𝑁 → ((𝐴 ∈ ℂ → (abs‘(𝐴↑𝑗)) = ((abs‘𝐴)↑𝑗)) ↔ (𝐴 ∈ ℂ → (abs‘(𝐴↑𝑁)) = ((abs‘𝐴)↑𝑁))))
21		abs1 11036	. . . 4 ⊢ (abs‘1) = 1
22		exp0 10480	. . . . 5 ⊢ (𝐴 ∈ ℂ → (𝐴↑0) = 1)
23	22	fveq2d 5500	. . . 4 ⊢ (𝐴 ∈ ℂ → (abs‘(𝐴↑0)) = (abs‘1))
24		abscl 11015	. . . . . 6 ⊢ (𝐴 ∈ ℂ → (abs‘𝐴) ∈ ℝ)
25	24	recnd 7948	. . . . 5 ⊢ (𝐴 ∈ ℂ → (abs‘𝐴) ∈ ℂ)
26	25	exp0d 10603	. . . 4 ⊢ (𝐴 ∈ ℂ → ((abs‘𝐴)↑0) = 1)
27	21, 23, 26	3eqtr4a 2229	. . 3 ⊢ (𝐴 ∈ ℂ → (abs‘(𝐴↑0)) = ((abs‘𝐴)↑0))
28		oveq1 5860	. . . . . . . 8 ⊢ ((abs‘(𝐴↑𝑘)) = ((abs‘𝐴)↑𝑘) → ((abs‘(𝐴↑𝑘)) · (abs‘𝐴)) = (((abs‘𝐴)↑𝑘) · (abs‘𝐴)))
29	28	adantl 275	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℕ0) ∧ (abs‘(𝐴↑𝑘)) = ((abs‘𝐴)↑𝑘)) → ((abs‘(𝐴↑𝑘)) · (abs‘𝐴)) = (((abs‘𝐴)↑𝑘) · (abs‘𝐴)))
30		expp1 10483	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℕ0) → (𝐴↑(𝑘 + 1)) = ((𝐴↑𝑘) · 𝐴))
31	30	fveq2d 5500	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℕ0) → (abs‘(𝐴↑(𝑘 + 1))) = (abs‘((𝐴↑𝑘) · 𝐴)))
32		expcl 10494	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℕ0) → (𝐴↑𝑘) ∈ ℂ)
33		simpl 108	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℕ0) → 𝐴 ∈ ℂ)
34		absmul 11033	. . . . . . . . . 10 ⊢ (((𝐴↑𝑘) ∈ ℂ ∧ 𝐴 ∈ ℂ) → (abs‘((𝐴↑𝑘) · 𝐴)) = ((abs‘(𝐴↑𝑘)) · (abs‘𝐴)))
35	32, 33, 34	syl2anc 409	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℕ0) → (abs‘((𝐴↑𝑘) · 𝐴)) = ((abs‘(𝐴↑𝑘)) · (abs‘𝐴)))
36	31, 35	eqtrd 2203	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℕ0) → (abs‘(𝐴↑(𝑘 + 1))) = ((abs‘(𝐴↑𝑘)) · (abs‘𝐴)))
37	36	adantr 274	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℕ0) ∧ (abs‘(𝐴↑𝑘)) = ((abs‘𝐴)↑𝑘)) → (abs‘(𝐴↑(𝑘 + 1))) = ((abs‘(𝐴↑𝑘)) · (abs‘𝐴)))
38		expp1 10483	. . . . . . . . 9 ⊢ (((abs‘𝐴) ∈ ℂ ∧ 𝑘 ∈ ℕ0) → ((abs‘𝐴)↑(𝑘 + 1)) = (((abs‘𝐴)↑𝑘) · (abs‘𝐴)))
39	25, 38	sylan 281	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℕ0) → ((abs‘𝐴)↑(𝑘 + 1)) = (((abs‘𝐴)↑𝑘) · (abs‘𝐴)))
40	39	adantr 274	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℕ0) ∧ (abs‘(𝐴↑𝑘)) = ((abs‘𝐴)↑𝑘)) → ((abs‘𝐴)↑(𝑘 + 1)) = (((abs‘𝐴)↑𝑘) · (abs‘𝐴)))
41	29, 37, 40	3eqtr4d 2213	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℕ0) ∧ (abs‘(𝐴↑𝑘)) = ((abs‘𝐴)↑𝑘)) → (abs‘(𝐴↑(𝑘 + 1))) = ((abs‘𝐴)↑(𝑘 + 1)))
42	41	exp31 362	. . . . 5 ⊢ (𝐴 ∈ ℂ → (𝑘 ∈ ℕ0 → ((abs‘(𝐴↑𝑘)) = ((abs‘𝐴)↑𝑘) → (abs‘(𝐴↑(𝑘 + 1))) = ((abs‘𝐴)↑(𝑘 + 1)))))
43	42	com12 30	. . . 4 ⊢ (𝑘 ∈ ℕ0 → (𝐴 ∈ ℂ → ((abs‘(𝐴↑𝑘)) = ((abs‘𝐴)↑𝑘) → (abs‘(𝐴↑(𝑘 + 1))) = ((abs‘𝐴)↑(𝑘 + 1)))))
44	43	a2d 26	. . 3 ⊢ (𝑘 ∈ ℕ0 → ((𝐴 ∈ ℂ → (abs‘(𝐴↑𝑘)) = ((abs‘𝐴)↑𝑘)) → (𝐴 ∈ ℂ → (abs‘(𝐴↑(𝑘 + 1))) = ((abs‘𝐴)↑(𝑘 + 1)))))
45	5, 10, 15, 20, 27, 44	nn0ind 9326	. 2 ⊢ (𝑁 ∈ ℕ0 → (𝐴 ∈ ℂ → (abs‘(𝐴↑𝑁)) = ((abs‘𝐴)↑𝑁)))
46	45	impcom 124	1 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0) → (abs‘(𝐴↑𝑁)) = ((abs‘𝐴)↑𝑁))

Syntax hints:   → wi 4   ∧ wa 103   = wceq 1348   ∈ wcel 2141  ‘cfv 5198  (class class class)co 5853  ℂcc 7772  0cc0 7774  1c1 7775   + caddc 7777   · cmul 7779  ℕ₀cn0 9135  ↑cexp 10475  abscabs 10961

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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-iinf 4572  ax-cnex 7865  ax-resscn 7866  ax-1cn 7867  ax-1re 7868  ax-icn 7869  ax-addcl 7870  ax-addrcl 7871  ax-mulcl 7872  ax-mulrcl 7873  ax-addcom 7874  ax-mulcom 7875  ax-addass 7876  ax-mulass 7877  ax-distr 7878  ax-i2m1 7879  ax-0lt1 7880  ax-1rid 7881  ax-0id 7882  ax-rnegex 7883  ax-precex 7884  ax-cnre 7885  ax-pre-ltirr 7886  ax-pre-ltwlin 7887  ax-pre-lttrn 7888  ax-pre-apti 7889  ax-pre-ltadd 7890  ax-pre-mulgt0 7891  ax-pre-mulext 7892  ax-arch 7893  ax-caucvg 7894

This theorem depends on definitions:  df-bi 116  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-reu 2455  df-rmo 2456  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-if 3527  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-tr 4088  df-id 4278  df-po 4281  df-iso 4282  df-iord 4351  df-on 4353  df-ilim 4354  df-suc 4356  df-iom 4575  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-riota 5809  df-ov 5856  df-oprab 5857  df-mpo 5858  df-1st 6119  df-2nd 6120  df-recs 6284  df-frec 6370  df-pnf 7956  df-mnf 7957  df-xr 7958  df-ltxr 7959  df-le 7960  df-sub 8092  df-neg 8093  df-reap 8494  df-ap 8501  df-div 8590  df-inn 8879  df-2 8937  df-3 8938  df-4 8939  df-n0 9136  df-z 9213  df-uz 9488  df-rp 9611  df-seqfrec 10402  df-exp 10476  df-cj 10806  df-re 10807  df-im 10808  df-rsqrt 10962  df-abs 10963

This theorem is referenced by:  absexpzap  11044  abssq  11045  sqabs  11046  absexpd  11156  expcnvap0  11465  expcnv  11467  eftabs  11619  efaddlem  11637