Theorem absexp 10737

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

Assertion

Ref	Expression
absexp	|- ( ( A e. CC /\ N e. NN0 ) -> ( 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 5734	. . . . . 6 |- ( j = 0 -> ( A ^ j ) = ( A ^ 0 ) )
2	1	fveq2d 5377	. . . . 5 |- ( j = 0 -> ( abs ` ( A ^ j ) ) = ( abs ` ( A ^ 0 ) ) )
3		oveq2 5734	. . . . 5 |- ( j = 0 -> ( ( abs ` A ) ^ j ) = ( ( abs ` A ) ^ 0 ) )
4	2, 3	eqeq12d 2127	. . . 4 |- ( j = 0 -> ( ( abs ` ( A ^ j ) ) = ( ( abs ` A ) ^ j ) <-> ( abs ` ( A ^ 0 ) ) = ( ( abs ` A ) ^ 0 ) ) )
5	4	imbi2d 229	. . 3 |- ( j = 0 -> ( ( A e. CC -> ( abs ` ( A ^ j ) ) = ( ( abs ` A ) ^ j ) ) <-> ( A e. CC -> ( abs ` ( A ^ 0 ) ) = ( ( abs ` A ) ^ 0 ) ) ) )
6		oveq2 5734	. . . . . 6 |- ( j = k -> ( A ^ j ) = ( A ^ k ) )
7	6	fveq2d 5377	. . . . 5 |- ( j = k -> ( abs ` ( A ^ j ) ) = ( abs ` ( A ^ k ) ) )
8		oveq2 5734	. . . . 5 |- ( j = k -> ( ( abs ` A ) ^ j ) = ( ( abs ` A ) ^ k ) )
9	7, 8	eqeq12d 2127	. . . 4 |- ( j = k -> ( ( abs ` ( A ^ j ) ) = ( ( abs ` A ) ^ j ) <-> ( abs ` ( A ^ k ) ) = ( ( abs ` A ) ^ k ) ) )
10	9	imbi2d 229	. . 3 |- ( j = k -> ( ( A e. CC -> ( abs ` ( A ^ j ) ) = ( ( abs ` A ) ^ j ) ) <-> ( A e. CC -> ( abs ` ( A ^ k ) ) = ( ( abs ` A ) ^ k ) ) ) )
11		oveq2 5734	. . . . . 6 |- ( j = ( k + 1 ) -> ( A ^ j ) = ( A ^ ( k + 1 ) ) )
12	11	fveq2d 5377	. . . . 5 |- ( j = ( k + 1 ) -> ( abs ` ( A ^ j ) ) = ( abs ` ( A ^ ( k + 1 ) ) ) )
13		oveq2 5734	. . . . 5 |- ( j = ( k + 1 ) -> ( ( abs ` A ) ^ j ) = ( ( abs ` A ) ^ ( k + 1 ) ) )
14	12, 13	eqeq12d 2127	. . . 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 e. CC -> ( abs ` ( A ^ j ) ) = ( ( abs ` A ) ^ j ) ) <-> ( A e. CC -> ( abs ` ( A ^ ( k + 1 ) ) ) = ( ( abs ` A ) ^ ( k + 1 ) ) ) ) )
16		oveq2 5734	. . . . . 6 |- ( j = N -> ( A ^ j ) = ( A ^ N ) )
17	16	fveq2d 5377	. . . . 5 |- ( j = N -> ( abs ` ( A ^ j ) ) = ( abs ` ( A ^ N ) ) )
18		oveq2 5734	. . . . 5 |- ( j = N -> ( ( abs ` A ) ^ j ) = ( ( abs ` A ) ^ N ) )
19	17, 18	eqeq12d 2127	. . . 4 |- ( j = N -> ( ( abs ` ( A ^ j ) ) = ( ( abs ` A ) ^ j ) <-> ( abs ` ( A ^ N ) ) = ( ( abs ` A ) ^ N ) ) )
20	19	imbi2d 229	. . 3 |- ( j = N -> ( ( A e. CC -> ( abs ` ( A ^ j ) ) = ( ( abs ` A ) ^ j ) ) <-> ( A e. CC -> ( abs ` ( A ^ N ) ) = ( ( abs ` A ) ^ N ) ) ) )
21		abs1 10730	. . . 4 |- ( abs ` 1 ) = 1
22		exp0 10184	. . . . 5 |- ( A e. CC -> ( A ^ 0 ) = 1 )
23	22	fveq2d 5377	. . . 4 |- ( A e. CC -> ( abs ` ( A ^ 0 ) ) = ( abs ` 1 ) )
24		abscl 10709	. . . . . 6 |- ( A e. CC -> ( abs ` A ) e. RR )
25	24	recnd 7712	. . . . 5 |- ( A e. CC -> ( abs ` A ) e. CC )
26	25	exp0d 10305	. . . 4 |- ( A e. CC -> ( ( abs ` A ) ^ 0 ) = 1 )
27	21, 23, 26	3eqtr4a 2171	. . 3 |- ( A e. CC -> ( abs ` ( A ^ 0 ) ) = ( ( abs ` A ) ^ 0 ) )
28		oveq1 5733	. . . . . . . 8 |- ( ( abs ` ( A ^ k ) ) = ( ( abs ` A ) ^ k ) -> ( ( abs ` ( A ^ k ) ) x. ( abs ` A ) ) = ( ( ( abs ` A ) ^ k ) x. ( abs ` A ) ) )
29	28	adantl 273	. . . . . . 7 |- ( ( ( A e. CC /\ k e. NN0 ) /\ ( abs ` ( A ^ k ) ) = ( ( abs ` A ) ^ k ) ) -> ( ( abs ` ( A ^ k ) ) x. ( abs ` A ) ) = ( ( ( abs ` A ) ^ k ) x. ( abs ` A ) ) )
30		expp1 10187	. . . . . . . . . 10 |- ( ( A e. CC /\ k e. NN0 ) -> ( A ^ ( k + 1 ) ) = ( ( A ^ k ) x. A ) )
31	30	fveq2d 5377	. . . . . . . . 9 |- ( ( A e. CC /\ k e. NN0 ) -> ( abs ` ( A ^ ( k + 1 ) ) ) = ( abs ` ( ( A ^ k ) x. A ) ) )
32		expcl 10198	. . . . . . . . . 10 |- ( ( A e. CC /\ k e. NN0 ) -> ( A ^ k ) e. CC )
33		simpl 108	. . . . . . . . . 10 |- ( ( A e. CC /\ k e. NN0 ) -> A e. CC )
34		absmul 10727	. . . . . . . . . 10 |- ( ( ( A ^ k ) e. CC /\ A e. CC ) -> ( abs ` ( ( A ^ k ) x. A ) ) = ( ( abs ` ( A ^ k ) ) x. ( abs ` A ) ) )
35	32, 33, 34	syl2anc 406	. . . . . . . . 9 |- ( ( A e. CC /\ k e. NN0 ) -> ( abs ` ( ( A ^ k ) x. A ) ) = ( ( abs ` ( A ^ k ) ) x. ( abs ` A ) ) )
36	31, 35	eqtrd 2145	. . . . . . . 8 |- ( ( A e. CC /\ k e. NN0 ) -> ( abs ` ( A ^ ( k + 1 ) ) ) = ( ( abs ` ( A ^ k ) ) x. ( abs ` A ) ) )
37	36	adantr 272	. . . . . . 7 |- ( ( ( A e. CC /\ k e. NN0 ) /\ ( abs ` ( A ^ k ) ) = ( ( abs ` A ) ^ k ) ) -> ( abs ` ( A ^ ( k + 1 ) ) ) = ( ( abs ` ( A ^ k ) ) x. ( abs ` A ) ) )
38		expp1 10187	. . . . . . . . 9 |- ( ( ( abs ` A ) e. CC /\ k e. NN0 ) -> ( ( abs ` A ) ^ ( k + 1 ) ) = ( ( ( abs ` A ) ^ k ) x. ( abs ` A ) ) )
39	25, 38	sylan 279	. . . . . . . 8 |- ( ( A e. CC /\ k e. NN0 ) -> ( ( abs ` A ) ^ ( k + 1 ) ) = ( ( ( abs ` A ) ^ k ) x. ( abs ` A ) ) )
40	39	adantr 272	. . . . . . 7 |- ( ( ( A e. CC /\ k e. NN0 ) /\ ( abs ` ( A ^ k ) ) = ( ( abs ` A ) ^ k ) ) -> ( ( abs ` A ) ^ ( k + 1 ) ) = ( ( ( abs ` A ) ^ k ) x. ( abs ` A ) ) )
41	29, 37, 40	3eqtr4d 2155	. . . . . 6 |- ( ( ( A e. CC /\ k e. NN0 ) /\ ( abs ` ( A ^ k ) ) = ( ( abs ` A ) ^ k ) ) -> ( abs ` ( A ^ ( k + 1 ) ) ) = ( ( abs ` A ) ^ ( k + 1 ) ) )
42	41	exp31 359	. . . . 5 |- ( A e. CC -> ( k e. NN0 -> ( ( abs ` ( A ^ k ) ) = ( ( abs ` A ) ^ k ) -> ( abs ` ( A ^ ( k + 1 ) ) ) = ( ( abs ` A ) ^ ( k + 1 ) ) ) ) )
43	42	com12 30	. . . 4 |- ( k e. NN0 -> ( A e. CC -> ( ( abs ` ( A ^ k ) ) = ( ( abs ` A ) ^ k ) -> ( abs ` ( A ^ ( k + 1 ) ) ) = ( ( abs ` A ) ^ ( k + 1 ) ) ) ) )
44	43	a2d 26	. . 3 |- ( k e. NN0 -> ( ( A e. CC -> ( abs ` ( A ^ k ) ) = ( ( abs ` A ) ^ k ) ) -> ( A e. CC -> ( abs ` ( A ^ ( k + 1 ) ) ) = ( ( abs ` A ) ^ ( k + 1 ) ) ) ) )
45	5, 10, 15, 20, 27, 44	nn0ind 9063	. 2 |- ( N e. NN0 -> ( A e. CC -> ( abs ` ( A ^ N ) ) = ( ( abs ` A ) ^ N ) ) )
46	45	impcom 124	1 |- ( ( A e. CC /\ N e. NN0 ) -> ( abs ` ( A ^ N ) ) = ( ( abs ` A ) ^ N ) )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1312   e. wcel 1461   ` cfv 5079  (class class class)co 5726   CCcc 7539   0cc0 7541   1c1 7542   + caddc 7544   x. cmul 7546   NN0cn0 8875   ^cexp 10179   abscabs 10655

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1404  ax-7 1405  ax-gen 1406  ax-ie1 1450  ax-ie2 1451  ax-8 1463  ax-10 1464  ax-11 1465  ax-i12 1466  ax-bndl 1467  ax-4 1468  ax-13 1472  ax-14 1473  ax-17 1487  ax-i9 1491  ax-ial 1495  ax-i5r 1496  ax-ext 2095  ax-coll 4001  ax-sep 4004  ax-nul 4012  ax-pow 4056  ax-pr 4089  ax-un 4313  ax-setind 4410  ax-iinf 4460  ax-cnex 7630  ax-resscn 7631  ax-1cn 7632  ax-1re 7633  ax-icn 7634  ax-addcl 7635  ax-addrcl 7636  ax-mulcl 7637  ax-mulrcl 7638  ax-addcom 7639  ax-mulcom 7640  ax-addass 7641  ax-mulass 7642  ax-distr 7643  ax-i2m1 7644  ax-0lt1 7645  ax-1rid 7646  ax-0id 7647  ax-rnegex 7648  ax-precex 7649  ax-cnre 7650  ax-pre-ltirr 7651  ax-pre-ltwlin 7652  ax-pre-lttrn 7653  ax-pre-apti 7654  ax-pre-ltadd 7655  ax-pre-mulgt0 7656  ax-pre-mulext 7657  ax-arch 7658  ax-caucvg 7659

This theorem depends on definitions:  df-bi 116  df-dc 803  df-3or 944  df-3an 945  df-tru 1315  df-fal 1318  df-nf 1418  df-sb 1717  df-eu 1976  df-mo 1977  df-clab 2100  df-cleq 2106  df-clel 2109  df-nfc 2242  df-ne 2281  df-nel 2376  df-ral 2393  df-rex 2394  df-reu 2395  df-rmo 2396  df-rab 2397  df-v 2657  df-sbc 2877  df-csb 2970  df-dif 3037  df-un 3039  df-in 3041  df-ss 3048  df-nul 3328  df-if 3439  df-pw 3476  df-sn 3497  df-pr 3498  df-op 3500  df-uni 3701  df-int 3736  df-iun 3779  df-br 3894  df-opab 3948  df-mpt 3949  df-tr 3985  df-id 4173  df-po 4176  df-iso 4177  df-iord 4246  df-on 4248  df-ilim 4249  df-suc 4251  df-iom 4463  df-xp 4503  df-rel 4504  df-cnv 4505  df-co 4506  df-dm 4507  df-rn 4508  df-res 4509  df-ima 4510  df-iota 5044  df-fun 5081  df-fn 5082  df-f 5083  df-f1 5084  df-fo 5085  df-f1o 5086  df-fv 5087  df-riota 5682  df-ov 5729  df-oprab 5730  df-mpo 5731  df-1st 5990  df-2nd 5991  df-recs 6154  df-frec 6240  df-pnf 7720  df-mnf 7721  df-xr 7722  df-ltxr 7723  df-le 7724  df-sub 7852  df-neg 7853  df-reap 8249  df-ap 8256  df-div 8340  df-inn 8625  df-2 8683  df-3 8684  df-4 8685  df-n0 8876  df-z 8953  df-uz 9223  df-rp 9338  df-seqfrec 10106  df-exp 10180  df-cj 10501  df-re 10502  df-im 10503  df-rsqrt 10656  df-abs 10657

This theorem is referenced by:  absexpzap  10738  abssq  10739  sqabs  10740  absexpd  10850  expcnvap0  11157  expcnv  11159  eftabs  11207  efaddlem  11225