Theorem absexp 11660

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 6029	. . . . . 6 |- ( j = 0 -> ( A ^ j ) = ( A ^ 0 ) )
2	1	fveq2d 5644	. . . . 5 |- ( j = 0 -> ( abs ` ( A ^ j ) ) = ( abs ` ( A ^ 0 ) ) )
3		oveq2 6029	. . . . 5 |- ( j = 0 -> ( ( abs ` A ) ^ j ) = ( ( abs ` A ) ^ 0 ) )
4	2, 3	eqeq12d 2246	. . . 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 6029	. . . . . 6 |- ( j = k -> ( A ^ j ) = ( A ^ k ) )
7	6	fveq2d 5644	. . . . 5 |- ( j = k -> ( abs ` ( A ^ j ) ) = ( abs ` ( A ^ k ) ) )
8		oveq2 6029	. . . . 5 |- ( j = k -> ( ( abs ` A ) ^ j ) = ( ( abs ` A ) ^ k ) )
9	7, 8	eqeq12d 2246	. . . 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 6029	. . . . . 6 |- ( j = ( k + 1 ) -> ( A ^ j ) = ( A ^ ( k + 1 ) ) )
12	11	fveq2d 5644	. . . . 5 |- ( j = ( k + 1 ) -> ( abs ` ( A ^ j ) ) = ( abs ` ( A ^ ( k + 1 ) ) ) )
13		oveq2 6029	. . . . 5 |- ( j = ( k + 1 ) -> ( ( abs ` A ) ^ j ) = ( ( abs ` A ) ^ ( k + 1 ) ) )
14	12, 13	eqeq12d 2246	. . . 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 6029	. . . . . 6 |- ( j = N -> ( A ^ j ) = ( A ^ N ) )
17	16	fveq2d 5644	. . . . 5 |- ( j = N -> ( abs ` ( A ^ j ) ) = ( abs ` ( A ^ N ) ) )
18		oveq2 6029	. . . . 5 |- ( j = N -> ( ( abs ` A ) ^ j ) = ( ( abs ` A ) ^ N ) )
19	17, 18	eqeq12d 2246	. . . 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 11653	. . . 4 |- ( abs ` 1 ) = 1
22		exp0 10809	. . . . 5 |- ( A e. CC -> ( A ^ 0 ) = 1 )
23	22	fveq2d 5644	. . . 4 |- ( A e. CC -> ( abs ` ( A ^ 0 ) ) = ( abs ` 1 ) )
24		abscl 11632	. . . . . 6 |- ( A e. CC -> ( abs ` A ) e. RR )
25	24	recnd 8211	. . . . 5 |- ( A e. CC -> ( abs ` A ) e. CC )
26	25	exp0d 10933	. . . 4 |- ( A e. CC -> ( ( abs ` A ) ^ 0 ) = 1 )
27	21, 23, 26	3eqtr4a 2290	. . 3 |- ( A e. CC -> ( abs ` ( A ^ 0 ) ) = ( ( abs ` A ) ^ 0 ) )
28		oveq1 6028	. . . . . . . 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 10812	. . . . . . . . . 10 |- ( ( A e. CC /\ k e. NN0 ) -> ( A ^ ( k + 1 ) ) = ( ( A ^ k ) x. A ) )
31	30	fveq2d 5644	. . . . . . . . 9 |- ( ( A e. CC /\ k e. NN0 ) -> ( abs ` ( A ^ ( k + 1 ) ) ) = ( abs ` ( ( A ^ k ) x. A ) ) )
32		expcl 10823	. . . . . . . . . 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 11650	. . . . . . . . . 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 2264	. . . . . . . 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 10812	. . . . . . . . 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 2274	. . . . . 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 9597	. 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 1397   e. wcel 2202   ` cfv 5326  (class class class)co 6021   CCcc 8033   0cc0 8035   1c1 8036   + caddc 8038   x. cmul 8040   NN0cn0 9405   ^cexp 10804   abscabs 11578

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-iinf 4686  ax-cnex 8126  ax-resscn 8127  ax-1cn 8128  ax-1re 8129  ax-icn 8130  ax-addcl 8131  ax-addrcl 8132  ax-mulcl 8133  ax-mulrcl 8134  ax-addcom 8135  ax-mulcom 8136  ax-addass 8137  ax-mulass 8138  ax-distr 8139  ax-i2m1 8140  ax-0lt1 8141  ax-1rid 8142  ax-0id 8143  ax-rnegex 8144  ax-precex 8145  ax-cnre 8146  ax-pre-ltirr 8147  ax-pre-ltwlin 8148  ax-pre-lttrn 8149  ax-pre-apti 8150  ax-pre-ltadd 8151  ax-pre-mulgt0 8152  ax-pre-mulext 8153  ax-arch 8154  ax-caucvg 8155

This theorem depends on definitions:  df-bi 117  df-dc 842  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-reu 2517  df-rmo 2518  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-if 3606  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-tr 4188  df-id 4390  df-po 4393  df-iso 4394  df-iord 4463  df-on 4465  df-ilim 4466  df-suc 4468  df-iom 4689  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-riota 5974  df-ov 6024  df-oprab 6025  df-mpo 6026  df-1st 6306  df-2nd 6307  df-recs 6474  df-frec 6560  df-pnf 8219  df-mnf 8220  df-xr 8221  df-ltxr 8222  df-le 8223  df-sub 8355  df-neg 8356  df-reap 8758  df-ap 8765  df-div 8856  df-inn 9147  df-2 9205  df-3 9206  df-4 9207  df-n0 9406  df-z 9483  df-uz 9759  df-rp 9892  df-seqfrec 10714  df-exp 10805  df-cj 11423  df-re 11424  df-im 11425  df-rsqrt 11579  df-abs 11580

This theorem is referenced by:  absexpzap  11661  abssq  11662  sqabs  11663  absexpd  11773  expcnvap0  12084  expcnv  12086  eftabs  12238  efaddlem  12256