Theorem absexp 10851

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 5782	. . . . . 6 |- ( j = 0 -> ( A ^ j ) = ( A ^ 0 ) )
2	1	fveq2d 5425	. . . . 5 |- ( j = 0 -> ( abs ` ( A ^ j ) ) = ( abs ` ( A ^ 0 ) ) )
3		oveq2 5782	. . . . 5 |- ( j = 0 -> ( ( abs ` A ) ^ j ) = ( ( abs ` A ) ^ 0 ) )
4	2, 3	eqeq12d 2154	. . . 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 5782	. . . . . 6 |- ( j = k -> ( A ^ j ) = ( A ^ k ) )
7	6	fveq2d 5425	. . . . 5 |- ( j = k -> ( abs ` ( A ^ j ) ) = ( abs ` ( A ^ k ) ) )
8		oveq2 5782	. . . . 5 |- ( j = k -> ( ( abs ` A ) ^ j ) = ( ( abs ` A ) ^ k ) )
9	7, 8	eqeq12d 2154	. . . 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 5782	. . . . . 6 |- ( j = ( k + 1 ) -> ( A ^ j ) = ( A ^ ( k + 1 ) ) )
12	11	fveq2d 5425	. . . . 5 |- ( j = ( k + 1 ) -> ( abs ` ( A ^ j ) ) = ( abs ` ( A ^ ( k + 1 ) ) ) )
13		oveq2 5782	. . . . 5 |- ( j = ( k + 1 ) -> ( ( abs ` A ) ^ j ) = ( ( abs ` A ) ^ ( k + 1 ) ) )
14	12, 13	eqeq12d 2154	. . . 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 5782	. . . . . 6 |- ( j = N -> ( A ^ j ) = ( A ^ N ) )
17	16	fveq2d 5425	. . . . 5 |- ( j = N -> ( abs ` ( A ^ j ) ) = ( abs ` ( A ^ N ) ) )
18		oveq2 5782	. . . . 5 |- ( j = N -> ( ( abs ` A ) ^ j ) = ( ( abs ` A ) ^ N ) )
19	17, 18	eqeq12d 2154	. . . 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 10844	. . . 4 |- ( abs ` 1 ) = 1
22		exp0 10297	. . . . 5 |- ( A e. CC -> ( A ^ 0 ) = 1 )
23	22	fveq2d 5425	. . . 4 |- ( A e. CC -> ( abs ` ( A ^ 0 ) ) = ( abs ` 1 ) )
24		abscl 10823	. . . . . 6 |- ( A e. CC -> ( abs ` A ) e. RR )
25	24	recnd 7794	. . . . 5 |- ( A e. CC -> ( abs ` A ) e. CC )
26	25	exp0d 10418	. . . 4 |- ( A e. CC -> ( ( abs ` A ) ^ 0 ) = 1 )
27	21, 23, 26	3eqtr4a 2198	. . 3 |- ( A e. CC -> ( abs ` ( A ^ 0 ) ) = ( ( abs ` A ) ^ 0 ) )
28		oveq1 5781	. . . . . . . 8 |- ( ( abs ` ( A ^ k ) ) = ( ( abs ` A ) ^ k ) -> ( ( abs ` ( A ^ k ) ) x. ( abs ` A ) ) = ( ( ( abs ` A ) ^ k ) x. ( abs ` A ) ) )
29	28	adantl 275	. . . . . . 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 10300	. . . . . . . . . 10 |- ( ( A e. CC /\ k e. NN0 ) -> ( A ^ ( k + 1 ) ) = ( ( A ^ k ) x. A ) )
31	30	fveq2d 5425	. . . . . . . . 9 |- ( ( A e. CC /\ k e. NN0 ) -> ( abs ` ( A ^ ( k + 1 ) ) ) = ( abs ` ( ( A ^ k ) x. A ) ) )
32		expcl 10311	. . . . . . . . . 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 10841	. . . . . . . . . 10 |- ( ( ( A ^ k ) e. CC /\ A e. CC ) -> ( abs ` ( ( A ^ k ) x. A ) ) = ( ( abs ` ( A ^ k ) ) x. ( abs ` A ) ) )
35	32, 33, 34	syl2anc 408	. . . . . . . . 9 |- ( ( A e. CC /\ k e. NN0 ) -> ( abs ` ( ( A ^ k ) x. A ) ) = ( ( abs ` ( A ^ k ) ) x. ( abs ` A ) ) )
36	31, 35	eqtrd 2172	. . . . . . . 8 |- ( ( A e. CC /\ k e. NN0 ) -> ( abs ` ( A ^ ( k + 1 ) ) ) = ( ( abs ` ( A ^ k ) ) x. ( abs ` A ) ) )
37	36	adantr 274	. . . . . . 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 10300	. . . . . . . . 9 |- ( ( ( abs ` A ) e. CC /\ k e. NN0 ) -> ( ( abs ` A ) ^ ( k + 1 ) ) = ( ( ( abs ` A ) ^ k ) x. ( abs ` A ) ) )
39	25, 38	sylan 281	. . . . . . . 8 |- ( ( A e. CC /\ k e. NN0 ) -> ( ( abs ` A ) ^ ( k + 1 ) ) = ( ( ( abs ` A ) ^ k ) x. ( abs ` A ) ) )
40	39	adantr 274	. . . . . . 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 2182	. . . . . 6 |- ( ( ( A e. CC /\ k e. NN0 ) /\ ( abs ` ( A ^ k ) ) = ( ( abs ` A ) ^ k ) ) -> ( abs ` ( A ^ ( k + 1 ) ) ) = ( ( abs ` A ) ^ ( k + 1 ) ) )
42	41	exp31 361	. . . . 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 9165	. 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 1331   e. wcel 1480   ` cfv 5123  (class class class)co 5774   CCcc 7618   0cc0 7620   1c1 7621   + caddc 7623   x. cmul 7625   NN0cn0 8977   ^cexp 10292   abscabs 10769

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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-iinf 4502  ax-cnex 7711  ax-resscn 7712  ax-1cn 7713  ax-1re 7714  ax-icn 7715  ax-addcl 7716  ax-addrcl 7717  ax-mulcl 7718  ax-mulrcl 7719  ax-addcom 7720  ax-mulcom 7721  ax-addass 7722  ax-mulass 7723  ax-distr 7724  ax-i2m1 7725  ax-0lt1 7726  ax-1rid 7727  ax-0id 7728  ax-rnegex 7729  ax-precex 7730  ax-cnre 7731  ax-pre-ltirr 7732  ax-pre-ltwlin 7733  ax-pre-lttrn 7734  ax-pre-apti 7735  ax-pre-ltadd 7736  ax-pre-mulgt0 7737  ax-pre-mulext 7738  ax-arch 7739  ax-caucvg 7740

This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-nel 2404  df-ral 2421  df-rex 2422  df-reu 2423  df-rmo 2424  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-if 3475  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-id 4215  df-po 4218  df-iso 4219  df-iord 4288  df-on 4290  df-ilim 4291  df-suc 4293  df-iom 4505  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-riota 5730  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-recs 6202  df-frec 6288  df-pnf 7802  df-mnf 7803  df-xr 7804  df-ltxr 7805  df-le 7806  df-sub 7935  df-neg 7936  df-reap 8337  df-ap 8344  df-div 8433  df-inn 8721  df-2 8779  df-3 8780  df-4 8781  df-n0 8978  df-z 9055  df-uz 9327  df-rp 9442  df-seqfrec 10219  df-exp 10293  df-cj 10614  df-re 10615  df-im 10616  df-rsqrt 10770  df-abs 10771

This theorem is referenced by:  absexpzap  10852  abssq  10853  sqabs  10854  absexpd  10964  expcnvap0  11271  expcnv  11273  eftabs  11362  efaddlem  11380