Theorem absexp 9965

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 5540	. . . . . 6 ⊢ (𝑗 = 0 → (𝐴↑𝑗) = (𝐴↑0))
2	1	fveq2d 5202	. . . . 5 ⊢ (𝑗 = 0 → (abs‘(𝐴↑𝑗)) = (abs‘(𝐴↑0)))
3		oveq2 5540	. . . . 5 ⊢ (𝑗 = 0 → ((abs‘𝐴)↑𝑗) = ((abs‘𝐴)↑0))
4	2, 3	eqeq12d 2095	. . . 4 ⊢ (𝑗 = 0 → ((abs‘(𝐴↑𝑗)) = ((abs‘𝐴)↑𝑗) ↔ (abs‘(𝐴↑0)) = ((abs‘𝐴)↑0)))
5	4	imbi2d 228	. . 3 ⊢ (𝑗 = 0 → ((𝐴 ∈ ℂ → (abs‘(𝐴↑𝑗)) = ((abs‘𝐴)↑𝑗)) ↔ (𝐴 ∈ ℂ → (abs‘(𝐴↑0)) = ((abs‘𝐴)↑0))))
6		oveq2 5540	. . . . . 6 ⊢ (𝑗 = 𝑘 → (𝐴↑𝑗) = (𝐴↑𝑘))
7	6	fveq2d 5202	. . . . 5 ⊢ (𝑗 = 𝑘 → (abs‘(𝐴↑𝑗)) = (abs‘(𝐴↑𝑘)))
8		oveq2 5540	. . . . 5 ⊢ (𝑗 = 𝑘 → ((abs‘𝐴)↑𝑗) = ((abs‘𝐴)↑𝑘))
9	7, 8	eqeq12d 2095	. . . 4 ⊢ (𝑗 = 𝑘 → ((abs‘(𝐴↑𝑗)) = ((abs‘𝐴)↑𝑗) ↔ (abs‘(𝐴↑𝑘)) = ((abs‘𝐴)↑𝑘)))
10	9	imbi2d 228	. . 3 ⊢ (𝑗 = 𝑘 → ((𝐴 ∈ ℂ → (abs‘(𝐴↑𝑗)) = ((abs‘𝐴)↑𝑗)) ↔ (𝐴 ∈ ℂ → (abs‘(𝐴↑𝑘)) = ((abs‘𝐴)↑𝑘))))
11		oveq2 5540	. . . . . 6 ⊢ (𝑗 = (𝑘 + 1) → (𝐴↑𝑗) = (𝐴↑(𝑘 + 1)))
12	11	fveq2d 5202	. . . . 5 ⊢ (𝑗 = (𝑘 + 1) → (abs‘(𝐴↑𝑗)) = (abs‘(𝐴↑(𝑘 + 1))))
13		oveq2 5540	. . . . 5 ⊢ (𝑗 = (𝑘 + 1) → ((abs‘𝐴)↑𝑗) = ((abs‘𝐴)↑(𝑘 + 1)))
14	12, 13	eqeq12d 2095	. . . 4 ⊢ (𝑗 = (𝑘 + 1) → ((abs‘(𝐴↑𝑗)) = ((abs‘𝐴)↑𝑗) ↔ (abs‘(𝐴↑(𝑘 + 1))) = ((abs‘𝐴)↑(𝑘 + 1))))
15	14	imbi2d 228	. . 3 ⊢ (𝑗 = (𝑘 + 1) → ((𝐴 ∈ ℂ → (abs‘(𝐴↑𝑗)) = ((abs‘𝐴)↑𝑗)) ↔ (𝐴 ∈ ℂ → (abs‘(𝐴↑(𝑘 + 1))) = ((abs‘𝐴)↑(𝑘 + 1)))))
16		oveq2 5540	. . . . . 6 ⊢ (𝑗 = 𝑁 → (𝐴↑𝑗) = (𝐴↑𝑁))
17	16	fveq2d 5202	. . . . 5 ⊢ (𝑗 = 𝑁 → (abs‘(𝐴↑𝑗)) = (abs‘(𝐴↑𝑁)))
18		oveq2 5540	. . . . 5 ⊢ (𝑗 = 𝑁 → ((abs‘𝐴)↑𝑗) = ((abs‘𝐴)↑𝑁))
19	17, 18	eqeq12d 2095	. . . 4 ⊢ (𝑗 = 𝑁 → ((abs‘(𝐴↑𝑗)) = ((abs‘𝐴)↑𝑗) ↔ (abs‘(𝐴↑𝑁)) = ((abs‘𝐴)↑𝑁)))
20	19	imbi2d 228	. . 3 ⊢ (𝑗 = 𝑁 → ((𝐴 ∈ ℂ → (abs‘(𝐴↑𝑗)) = ((abs‘𝐴)↑𝑗)) ↔ (𝐴 ∈ ℂ → (abs‘(𝐴↑𝑁)) = ((abs‘𝐴)↑𝑁))))
21		abs1 9958	. . . 4 ⊢ (abs‘1) = 1
22		exp0 9480	. . . . 5 ⊢ (𝐴 ∈ ℂ → (𝐴↑0) = 1)
23	22	fveq2d 5202	. . . 4 ⊢ (𝐴 ∈ ℂ → (abs‘(𝐴↑0)) = (abs‘1))
24		abscl 9937	. . . . . 6 ⊢ (𝐴 ∈ ℂ → (abs‘𝐴) ∈ ℝ)
25	24	recnd 7147	. . . . 5 ⊢ (𝐴 ∈ ℂ → (abs‘𝐴) ∈ ℂ)
26	25	exp0d 9599	. . . 4 ⊢ (𝐴 ∈ ℂ → ((abs‘𝐴)↑0) = 1)
27	21, 23, 26	3eqtr4a 2139	. . 3 ⊢ (𝐴 ∈ ℂ → (abs‘(𝐴↑0)) = ((abs‘𝐴)↑0))
28		oveq1 5539	. . . . . . . 8 ⊢ ((abs‘(𝐴↑𝑘)) = ((abs‘𝐴)↑𝑘) → ((abs‘(𝐴↑𝑘)) · (abs‘𝐴)) = (((abs‘𝐴)↑𝑘) · (abs‘𝐴)))
29	28	adantl 271	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℕ0) ∧ (abs‘(𝐴↑𝑘)) = ((abs‘𝐴)↑𝑘)) → ((abs‘(𝐴↑𝑘)) · (abs‘𝐴)) = (((abs‘𝐴)↑𝑘) · (abs‘𝐴)))
30		expp1 9483	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℕ0) → (𝐴↑(𝑘 + 1)) = ((𝐴↑𝑘) · 𝐴))
31	30	fveq2d 5202	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℕ0) → (abs‘(𝐴↑(𝑘 + 1))) = (abs‘((𝐴↑𝑘) · 𝐴)))
32		expcl 9494	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℕ0) → (𝐴↑𝑘) ∈ ℂ)
33		simpl 107	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℕ0) → 𝐴 ∈ ℂ)
34		absmul 9955	. . . . . . . . . 10 ⊢ (((𝐴↑𝑘) ∈ ℂ ∧ 𝐴 ∈ ℂ) → (abs‘((𝐴↑𝑘) · 𝐴)) = ((abs‘(𝐴↑𝑘)) · (abs‘𝐴)))
35	32, 33, 34	syl2anc 403	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℕ0) → (abs‘((𝐴↑𝑘) · 𝐴)) = ((abs‘(𝐴↑𝑘)) · (abs‘𝐴)))
36	31, 35	eqtrd 2113	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℕ0) → (abs‘(𝐴↑(𝑘 + 1))) = ((abs‘(𝐴↑𝑘)) · (abs‘𝐴)))
37	36	adantr 270	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℕ0) ∧ (abs‘(𝐴↑𝑘)) = ((abs‘𝐴)↑𝑘)) → (abs‘(𝐴↑(𝑘 + 1))) = ((abs‘(𝐴↑𝑘)) · (abs‘𝐴)))
38		expp1 9483	. . . . . . . . 9 ⊢ (((abs‘𝐴) ∈ ℂ ∧ 𝑘 ∈ ℕ0) → ((abs‘𝐴)↑(𝑘 + 1)) = (((abs‘𝐴)↑𝑘) · (abs‘𝐴)))
39	25, 38	sylan 277	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℕ0) → ((abs‘𝐴)↑(𝑘 + 1)) = (((abs‘𝐴)↑𝑘) · (abs‘𝐴)))
40	39	adantr 270	. . . . . . 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 2123	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℕ0) ∧ (abs‘(𝐴↑𝑘)) = ((abs‘𝐴)↑𝑘)) → (abs‘(𝐴↑(𝑘 + 1))) = ((abs‘𝐴)↑(𝑘 + 1)))
42	41	exp31 356	. . . . 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 8461	. 2 ⊢ (𝑁 ∈ ℕ0 → (𝐴 ∈ ℂ → (abs‘(𝐴↑𝑁)) = ((abs‘𝐴)↑𝑁)))
46	45	impcom 123	1 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0) → (abs‘(𝐴↑𝑁)) = ((abs‘𝐴)↑𝑁))

Syntax hints:   → wi 4   ∧ wa 102   = wceq 1284   ∈ wcel 1433  ‘cfv 4922  (class class class)co 5532  ℂcc 6979  0cc0 6981  1c1 6982   + caddc 6984   · cmul 6986  ℕ₀cn0 8288  ↑cexp 9475  abscabs 9883

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-13 1444  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-coll 3893  ax-sep 3896  ax-nul 3904  ax-pow 3948  ax-pr 3964  ax-un 4188  ax-setind 4280  ax-iinf 4329  ax-cnex 7067  ax-resscn 7068  ax-1cn 7069  ax-1re 7070  ax-icn 7071  ax-addcl 7072  ax-addrcl 7073  ax-mulcl 7074  ax-mulrcl 7075  ax-addcom 7076  ax-mulcom 7077  ax-addass 7078  ax-mulass 7079  ax-distr 7080  ax-i2m1 7081  ax-0lt1 7082  ax-1rid 7083  ax-0id 7084  ax-rnegex 7085  ax-precex 7086  ax-cnre 7087  ax-pre-ltirr 7088  ax-pre-ltwlin 7089  ax-pre-lttrn 7090  ax-pre-apti 7091  ax-pre-ltadd 7092  ax-pre-mulgt0 7093  ax-pre-mulext 7094  ax-arch 7095  ax-caucvg 7096

This theorem depends on definitions:  df-bi 115  df-dc 776  df-3or 920  df-3an 921  df-tru 1287  df-fal 1290  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ne 2246  df-nel 2340  df-ral 2353  df-rex 2354  df-reu 2355  df-rmo 2356  df-rab 2357  df-v 2603  df-sbc 2816  df-csb 2909  df-dif 2975  df-un 2977  df-in 2979  df-ss 2986  df-nul 3252  df-if 3352  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-int 3637  df-iun 3680  df-br 3786  df-opab 3840  df-mpt 3841  df-tr 3876  df-id 4048  df-po 4051  df-iso 4052  df-iord 4121  df-on 4123  df-suc 4126  df-iom 4332  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-rn 4374  df-res 4375  df-ima 4376  df-iota 4887  df-fun 4924  df-fn 4925  df-f 4926  df-f1 4927  df-fo 4928  df-f1o 4929  df-fv 4930  df-riota 5488  df-ov 5535  df-oprab 5536  df-mpt2 5537  df-1st 5787  df-2nd 5788  df-recs 5943  df-frec 6001  df-pnf 7155  df-mnf 7156  df-xr 7157  df-ltxr 7158  df-le 7159  df-sub 7281  df-neg 7282  df-reap 7675  df-ap 7682  df-div 7761  df-inn 8040  df-2 8098  df-3 8099  df-4 8100  df-n0 8289  df-z 8352  df-uz 8620  df-rp 8735  df-iseq 9432  df-iexp 9476  df-cj 9729  df-re 9730  df-im 9731  df-rsqrt 9884  df-abs 9885

This theorem is referenced by:  absexpzap  9966  abssq  9967  sqabs  9968  absexpd  10078