Theorem absexp 11590

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 6009	. . . . . 6 |- ( j = 0 -> ( A ^ j ) = ( A ^ 0 ) )
2	1	fveq2d 5631	. . . . 5 |- ( j = 0 -> ( abs ` ( A ^ j ) ) = ( abs ` ( A ^ 0 ) ) )
3		oveq2 6009	. . . . 5 |- ( j = 0 -> ( ( abs ` A ) ^ j ) = ( ( abs ` A ) ^ 0 ) )
4	2, 3	eqeq12d 2244	. . . 4 |- ( j = 0 -> ( ( abs ` ( A ^ j ) ) = ( ( abs ` A ) ^ j ) <-> ( abs ` ( A ^ 0 ) ) = ( ( abs ` A ) ^ 0 ) ) )
5	4	imbi2d 230	. . 3 |- ( j = 0 -> ( ( A e. CC -> ( abs ` ( A ^ j ) ) = ( ( abs ` A ) ^ j ) ) <-> ( A e. CC -> ( abs ` ( A ^ 0 ) ) = ( ( abs ` A ) ^ 0 ) ) ) )
6		oveq2 6009	. . . . . 6 |- ( j = k -> ( A ^ j ) = ( A ^ k ) )
7	6	fveq2d 5631	. . . . 5 |- ( j = k -> ( abs ` ( A ^ j ) ) = ( abs ` ( A ^ k ) ) )
8		oveq2 6009	. . . . 5 |- ( j = k -> ( ( abs ` A ) ^ j ) = ( ( abs ` A ) ^ k ) )
9	7, 8	eqeq12d 2244	. . . 4 |- ( j = k -> ( ( abs ` ( A ^ j ) ) = ( ( abs ` A ) ^ j ) <-> ( abs ` ( A ^ k ) ) = ( ( abs ` A ) ^ k ) ) )
10	9	imbi2d 230	. . 3 |- ( j = k -> ( ( A e. CC -> ( abs ` ( A ^ j ) ) = ( ( abs ` A ) ^ j ) ) <-> ( A e. CC -> ( abs ` ( A ^ k ) ) = ( ( abs ` A ) ^ k ) ) ) )
11		oveq2 6009	. . . . . 6 |- ( j = ( k + 1 ) -> ( A ^ j ) = ( A ^ ( k + 1 ) ) )
12	11	fveq2d 5631	. . . . 5 |- ( j = ( k + 1 ) -> ( abs ` ( A ^ j ) ) = ( abs ` ( A ^ ( k + 1 ) ) ) )
13		oveq2 6009	. . . . 5 |- ( j = ( k + 1 ) -> ( ( abs ` A ) ^ j ) = ( ( abs ` A ) ^ ( k + 1 ) ) )
14	12, 13	eqeq12d 2244	. . . 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 e. CC -> ( abs ` ( A ^ j ) ) = ( ( abs ` A ) ^ j ) ) <-> ( A e. CC -> ( abs ` ( A ^ ( k + 1 ) ) ) = ( ( abs ` A ) ^ ( k + 1 ) ) ) ) )
16		oveq2 6009	. . . . . 6 |- ( j = N -> ( A ^ j ) = ( A ^ N ) )
17	16	fveq2d 5631	. . . . 5 |- ( j = N -> ( abs ` ( A ^ j ) ) = ( abs ` ( A ^ N ) ) )
18		oveq2 6009	. . . . 5 |- ( j = N -> ( ( abs ` A ) ^ j ) = ( ( abs ` A ) ^ N ) )
19	17, 18	eqeq12d 2244	. . . 4 |- ( j = N -> ( ( abs ` ( A ^ j ) ) = ( ( abs ` A ) ^ j ) <-> ( abs ` ( A ^ N ) ) = ( ( abs ` A ) ^ N ) ) )
20	19	imbi2d 230	. . 3 |- ( j = N -> ( ( A e. CC -> ( abs ` ( A ^ j ) ) = ( ( abs ` A ) ^ j ) ) <-> ( A e. CC -> ( abs ` ( A ^ N ) ) = ( ( abs ` A ) ^ N ) ) ) )
21		abs1 11583	. . . 4 |- ( abs ` 1 ) = 1
22		exp0 10765	. . . . 5 |- ( A e. CC -> ( A ^ 0 ) = 1 )
23	22	fveq2d 5631	. . . 4 |- ( A e. CC -> ( abs ` ( A ^ 0 ) ) = ( abs ` 1 ) )
24		abscl 11562	. . . . . 6 |- ( A e. CC -> ( abs ` A ) e. RR )
25	24	recnd 8175	. . . . 5 |- ( A e. CC -> ( abs ` A ) e. CC )
26	25	exp0d 10889	. . . 4 |- ( A e. CC -> ( ( abs ` A ) ^ 0 ) = 1 )
27	21, 23, 26	3eqtr4a 2288	. . 3 |- ( A e. CC -> ( abs ` ( A ^ 0 ) ) = ( ( abs ` A ) ^ 0 ) )
28		oveq1 6008	. . . . . . . 8 |- ( ( abs ` ( A ^ k ) ) = ( ( abs ` A ) ^ k ) -> ( ( abs ` ( A ^ k ) ) x. ( abs ` A ) ) = ( ( ( abs ` A ) ^ k ) x. ( abs ` A ) ) )
29	28	adantl 277	. . . . . . 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 10768	. . . . . . . . . 10 |- ( ( A e. CC /\ k e. NN0 ) -> ( A ^ ( k + 1 ) ) = ( ( A ^ k ) x. A ) )
31	30	fveq2d 5631	. . . . . . . . 9 |- ( ( A e. CC /\ k e. NN0 ) -> ( abs ` ( A ^ ( k + 1 ) ) ) = ( abs ` ( ( A ^ k ) x. A ) ) )
32		expcl 10779	. . . . . . . . . 10 |- ( ( A e. CC /\ k e. NN0 ) -> ( A ^ k ) e. CC )
33		simpl 109	. . . . . . . . . 10 |- ( ( A e. CC /\ k e. NN0 ) -> A e. CC )
34		absmul 11580	. . . . . . . . . 10 |- ( ( ( A ^ k ) e. CC /\ A e. CC ) -> ( abs ` ( ( A ^ k ) x. A ) ) = ( ( abs ` ( A ^ k ) ) x. ( abs ` A ) ) )
35	32, 33, 34	syl2anc 411	. . . . . . . . 9 |- ( ( A e. CC /\ k e. NN0 ) -> ( abs ` ( ( A ^ k ) x. A ) ) = ( ( abs ` ( A ^ k ) ) x. ( abs ` A ) ) )
36	31, 35	eqtrd 2262	. . . . . . . 8 |- ( ( A e. CC /\ k e. NN0 ) -> ( abs ` ( A ^ ( k + 1 ) ) ) = ( ( abs ` ( A ^ k ) ) x. ( abs ` A ) ) )
37	36	adantr 276	. . . . . . 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 10768	. . . . . . . . 9 |- ( ( ( abs ` A ) e. CC /\ k e. NN0 ) -> ( ( abs ` A ) ^ ( k + 1 ) ) = ( ( ( abs ` A ) ^ k ) x. ( abs ` A ) ) )
39	25, 38	sylan 283	. . . . . . . 8 |- ( ( A e. CC /\ k e. NN0 ) -> ( ( abs ` A ) ^ ( k + 1 ) ) = ( ( ( abs ` A ) ^ k ) x. ( abs ` A ) ) )
40	39	adantr 276	. . . . . . 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 2272	. . . . . 6 |- ( ( ( A e. CC /\ k e. NN0 ) /\ ( abs ` ( A ^ k ) ) = ( ( abs ` A ) ^ k ) ) -> ( abs ` ( A ^ ( k + 1 ) ) ) = ( ( abs ` A ) ^ ( k + 1 ) ) )
42	41	exp31 364	. . . . 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 9561	. 2 |- ( N e. NN0 -> ( A e. CC -> ( abs ` ( A ^ N ) ) = ( ( abs ` A ) ^ N ) ) )
46	45	impcom 125	1 |- ( ( A e. CC /\ N e. NN0 ) -> ( abs ` ( A ^ N ) ) = ( ( abs ` A ) ^ N ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1395   e. wcel 2200   ` cfv 5318  (class class class)co 6001   CCcc 7997   0cc0 7999   1c1 8000   + caddc 8002   x. cmul 8004   NN0cn0 9369   ^cexp 10760   abscabs 11508

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4199  ax-sep 4202  ax-nul 4210  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-iinf 4680  ax-cnex 8090  ax-resscn 8091  ax-1cn 8092  ax-1re 8093  ax-icn 8094  ax-addcl 8095  ax-addrcl 8096  ax-mulcl 8097  ax-mulrcl 8098  ax-addcom 8099  ax-mulcom 8100  ax-addass 8101  ax-mulass 8102  ax-distr 8103  ax-i2m1 8104  ax-0lt1 8105  ax-1rid 8106  ax-0id 8107  ax-rnegex 8108  ax-precex 8109  ax-cnre 8110  ax-pre-ltirr 8111  ax-pre-ltwlin 8112  ax-pre-lttrn 8113  ax-pre-apti 8114  ax-pre-ltadd 8115  ax-pre-mulgt0 8116  ax-pre-mulext 8117  ax-arch 8118  ax-caucvg 8119

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-if 3603  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-tr 4183  df-id 4384  df-po 4387  df-iso 4388  df-iord 4457  df-on 4459  df-ilim 4460  df-suc 4462  df-iom 4683  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-riota 5954  df-ov 6004  df-oprab 6005  df-mpo 6006  df-1st 6286  df-2nd 6287  df-recs 6451  df-frec 6537  df-pnf 8183  df-mnf 8184  df-xr 8185  df-ltxr 8186  df-le 8187  df-sub 8319  df-neg 8320  df-reap 8722  df-ap 8729  df-div 8820  df-inn 9111  df-2 9169  df-3 9170  df-4 9171  df-n0 9370  df-z 9447  df-uz 9723  df-rp 9850  df-seqfrec 10670  df-exp 10761  df-cj 11353  df-re 11354  df-im 11355  df-rsqrt 11509  df-abs 11510

This theorem is referenced by:  absexpzap  11591  abssq  11592  sqabs  11593  absexpd  11703  expcnvap0  12013  expcnv  12015  eftabs  12167  efaddlem  12185