Theorem absexp 11845

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 6093	. . . . . 6 ⊢ (𝑗 = 0 → (𝐴↑𝑗) = (𝐴↑0))
2	1	fveq2d 5699	. . . . 5 ⊢ (𝑗 = 0 → (abs‘(𝐴↑𝑗)) = (abs‘(𝐴↑0)))
3		oveq2 6093	. . . . 5 ⊢ (𝑗 = 0 → ((abs‘𝐴)↑𝑗) = ((abs‘𝐴)↑0))
4	2, 3	eqeq12d 2253	. . . 4 ⊢ (𝑗 = 0 → ((abs‘(𝐴↑𝑗)) = ((abs‘𝐴)↑𝑗) ↔ (abs‘(𝐴↑0)) = ((abs‘𝐴)↑0)))
5	4	imbi2d 230	. . 3 ⊢ (𝑗 = 0 → ((𝐴 ∈ ℂ → (abs‘(𝐴↑𝑗)) = ((abs‘𝐴)↑𝑗)) ↔ (𝐴 ∈ ℂ → (abs‘(𝐴↑0)) = ((abs‘𝐴)↑0))))
6		oveq2 6093	. . . . . 6 ⊢ (𝑗 = 𝑘 → (𝐴↑𝑗) = (𝐴↑𝑘))
7	6	fveq2d 5699	. . . . 5 ⊢ (𝑗 = 𝑘 → (abs‘(𝐴↑𝑗)) = (abs‘(𝐴↑𝑘)))
8		oveq2 6093	. . . . 5 ⊢ (𝑗 = 𝑘 → ((abs‘𝐴)↑𝑗) = ((abs‘𝐴)↑𝑘))
9	7, 8	eqeq12d 2253	. . . 4 ⊢ (𝑗 = 𝑘 → ((abs‘(𝐴↑𝑗)) = ((abs‘𝐴)↑𝑗) ↔ (abs‘(𝐴↑𝑘)) = ((abs‘𝐴)↑𝑘)))
10	9	imbi2d 230	. . 3 ⊢ (𝑗 = 𝑘 → ((𝐴 ∈ ℂ → (abs‘(𝐴↑𝑗)) = ((abs‘𝐴)↑𝑗)) ↔ (𝐴 ∈ ℂ → (abs‘(𝐴↑𝑘)) = ((abs‘𝐴)↑𝑘))))
11		oveq2 6093	. . . . . 6 ⊢ (𝑗 = (𝑘 + 1) → (𝐴↑𝑗) = (𝐴↑(𝑘 + 1)))
12	11	fveq2d 5699	. . . . 5 ⊢ (𝑗 = (𝑘 + 1) → (abs‘(𝐴↑𝑗)) = (abs‘(𝐴↑(𝑘 + 1))))
13		oveq2 6093	. . . . 5 ⊢ (𝑗 = (𝑘 + 1) → ((abs‘𝐴)↑𝑗) = ((abs‘𝐴)↑(𝑘 + 1)))
14	12, 13	eqeq12d 2253	. . . 4 ⊢ (𝑗 = (𝑘 + 1) → ((abs‘(𝐴↑𝑗)) = ((abs‘𝐴)↑𝑗) ↔ (abs‘(𝐴↑(𝑘 + 1))) = ((abs‘𝐴)↑(𝑘 + 1))))
15	14	imbi2d 230	. . 3 ⊢ (𝑗 = (𝑘 + 1) → ((𝐴 ∈ ℂ → (abs‘(𝐴↑𝑗)) = ((abs‘𝐴)↑𝑗)) ↔ (𝐴 ∈ ℂ → (abs‘(𝐴↑(𝑘 + 1))) = ((abs‘𝐴)↑(𝑘 + 1)))))
16		oveq2 6093	. . . . . 6 ⊢ (𝑗 = 𝑁 → (𝐴↑𝑗) = (𝐴↑𝑁))
17	16	fveq2d 5699	. . . . 5 ⊢ (𝑗 = 𝑁 → (abs‘(𝐴↑𝑗)) = (abs‘(𝐴↑𝑁)))
18		oveq2 6093	. . . . 5 ⊢ (𝑗 = 𝑁 → ((abs‘𝐴)↑𝑗) = ((abs‘𝐴)↑𝑁))
19	17, 18	eqeq12d 2253	. . . 4 ⊢ (𝑗 = 𝑁 → ((abs‘(𝐴↑𝑗)) = ((abs‘𝐴)↑𝑗) ↔ (abs‘(𝐴↑𝑁)) = ((abs‘𝐴)↑𝑁)))
20	19	imbi2d 230	. . 3 ⊢ (𝑗 = 𝑁 → ((𝐴 ∈ ℂ → (abs‘(𝐴↑𝑗)) = ((abs‘𝐴)↑𝑗)) ↔ (𝐴 ∈ ℂ → (abs‘(𝐴↑𝑁)) = ((abs‘𝐴)↑𝑁))))
21		abs1 11838	. . . 4 ⊢ (abs‘1) = 1
22		exp0 10980	. . . . 5 ⊢ (𝐴 ∈ ℂ → (𝐴↑0) = 1)
23	22	fveq2d 5699	. . . 4 ⊢ (𝐴 ∈ ℂ → (abs‘(𝐴↑0)) = (abs‘1))
24		abscl 11817	. . . . . 6 ⊢ (𝐴 ∈ ℂ → (abs‘𝐴) ∈ ℝ)
25	24	recnd 8354	. . . . 5 ⊢ (𝐴 ∈ ℂ → (abs‘𝐴) ∈ ℂ)
26	25	exp0d 11105	. . . 4 ⊢ (𝐴 ∈ ℂ → ((abs‘𝐴)↑0) = 1)
27	21, 23, 26	3eqtr4a 2297	. . 3 ⊢ (𝐴 ∈ ℂ → (abs‘(𝐴↑0)) = ((abs‘𝐴)↑0))
28		oveq1 6092	. . . . . . . 8 ⊢ ((abs‘(𝐴↑𝑘)) = ((abs‘𝐴)↑𝑘) → ((abs‘(𝐴↑𝑘)) · (abs‘𝐴)) = (((abs‘𝐴)↑𝑘) · (abs‘𝐴)))
29	28	adantl 277	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℕ0) ∧ (abs‘(𝐴↑𝑘)) = ((abs‘𝐴)↑𝑘)) → ((abs‘(𝐴↑𝑘)) · (abs‘𝐴)) = (((abs‘𝐴)↑𝑘) · (abs‘𝐴)))
30		expp1 10983	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℕ0) → (𝐴↑(𝑘 + 1)) = ((𝐴↑𝑘) · 𝐴))
31	30	fveq2d 5699	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℕ0) → (abs‘(𝐴↑(𝑘 + 1))) = (abs‘((𝐴↑𝑘) · 𝐴)))
32		expcl 10994	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℕ0) → (𝐴↑𝑘) ∈ ℂ)
33		simpl 109	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℕ0) → 𝐴 ∈ ℂ)
34		absmul 11835	. . . . . . . . . 10 ⊢ (((𝐴↑𝑘) ∈ ℂ ∧ 𝐴 ∈ ℂ) → (abs‘((𝐴↑𝑘) · 𝐴)) = ((abs‘(𝐴↑𝑘)) · (abs‘𝐴)))
35	32, 33, 34	syl2anc 415	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℕ0) → (abs‘((𝐴↑𝑘) · 𝐴)) = ((abs‘(𝐴↑𝑘)) · (abs‘𝐴)))
36	31, 35	eqtrd 2271	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℕ0) → (abs‘(𝐴↑(𝑘 + 1))) = ((abs‘(𝐴↑𝑘)) · (abs‘𝐴)))
37	36	adantr 276	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℕ0) ∧ (abs‘(𝐴↑𝑘)) = ((abs‘𝐴)↑𝑘)) → (abs‘(𝐴↑(𝑘 + 1))) = ((abs‘(𝐴↑𝑘)) · (abs‘𝐴)))
38		expp1 10983	. . . . . . . . 9 ⊢ (((abs‘𝐴) ∈ ℂ ∧ 𝑘 ∈ ℕ0) → ((abs‘𝐴)↑(𝑘 + 1)) = (((abs‘𝐴)↑𝑘) · (abs‘𝐴)))
39	25, 38	sylan 283	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℕ0) → ((abs‘𝐴)↑(𝑘 + 1)) = (((abs‘𝐴)↑𝑘) · (abs‘𝐴)))
40	39	adantr 276	. . . . . . 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 2281	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℕ0) ∧ (abs‘(𝐴↑𝑘)) = ((abs‘𝐴)↑𝑘)) → (abs‘(𝐴↑(𝑘 + 1))) = ((abs‘𝐴)↑(𝑘 + 1)))
42	41	exp31 364	. . . . 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 9760	. 2 ⊢ (𝑁 ∈ ℕ0 → (𝐴 ∈ ℂ → (abs‘(𝐴↑𝑁)) = ((abs‘𝐴)↑𝑁)))
46	45	impcom 125	1 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0) → (abs‘(𝐴↑𝑁)) = ((abs‘𝐴)↑𝑁))

This proof depends on syntax axioms:   → wi 4   ∧ wa 104   = wceq 1402   ∈ wcel 2209  ‘cfv 5377  (class class class)co 6085  ℂcc 8177  0cc0 8179  1c1 8180   + caddc 8182   · cmul 8184  ℕ₀cn0 9563  ↑cexp 10975  abscabs 11763

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 4246  ax-sep 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-iinf 4735  ax-cnex 8270  ax-resscn 8271  ax-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-mulrcl 8278  ax-addcom 8279  ax-mulcom 8280  ax-addass 8281  ax-mulass 8282  ax-distr 8283  ax-i2m1 8284  ax-0lt1 8285  ax-1rid 8286  ax-0id 8287  ax-rnegex 8288  ax-precex 8289  ax-cnre 8290  ax-pre-ltirr 8291  ax-pre-ltwlin 8292  ax-pre-lttrn 8293  ax-pre-apti 8294  ax-pre-ltadd 8295  ax-pre-mulgt0 8296  ax-pre-mulext 8297  ax-arch 8298  ax-caucvg 8299

This proof depends on definitions:  df-bi 117  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-br 4131  df-opab 4193  df-mpt 4194  df-tr 4230  df-id 4438  df-po 4441  df-iso 4442  df-iord 4511  df-on 4513  df-ilim 4514  df-suc 4516  df-iom 4738  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-recs 6576  df-frec 6662  df-pnf 8362  df-mnf 8363  df-xr 8364  df-ltxr 8365  df-le 8366  df-sub 8499  df-neg 8500  df-reap 8903  df-ap 8910  df-div 9003  df-inn 9305  df-2 9363  df-3 9364  df-4 9365  df-n0 9564  df-z 9645  df-uz 9922  df-rp 10055  df-seqfrec 10885  df-exp 10976  df-cj 11607  df-re 11608  df-im 11609  df-rsqrt 11764  df-abs 11765

This theorem is used by:  absexpzap  11846  abssq  11847  sqabs  11848  absexpd  11958  expcnvap0  12269  expcnv  12271  eftabs  12423  efaddlem  12441