Theorem absexpt 6811

Description: Absolute value of natural number exponentiation.

Assertion

Ref	Expression
absexpt	|- ((A e. CC /\ N e. NN0) -> (abs` (A^N)) = ((abs` A)^N))

Proof of Theorem absexpt

Step	Hyp	Ref	Expression
1		opreq2 3960	. . . . . 6 |- (j = 0 -> (A^j) = (A^0))
2	1	fveq2d 3719	. . . . 5 |- (j = 0 -> (abs` (A^j)) = (abs` (A^0)))
3		opreq2 3960	. . . . 5 |- (j = 0 -> ((abs` A)^j) = ((abs` A)^0))
4	2, 3	eqeq12d 1486	. . . 4 |- (j = 0 -> ((abs` (A^j)) = ((abs` A)^j) <-> (abs` (A^0)) = ((abs` A)^0)))
5	4	imbi2d 611	. . 3 |- (j = 0 -> ((A e. CC -> (abs` (A^j)) = ((abs` A)^j)) <-> (A e. CC -> (abs` (A^0)) = ((abs` A)^0))))
6		opreq2 3960	. . . . . 6 |- (j = k -> (A^j) = (A^k))
7	6	fveq2d 3719	. . . . 5 |- (j = k -> (abs` (A^j)) = (abs` (A^k)))
8		opreq2 3960	. . . . 5 |- (j = k -> ((abs` A)^j) = ((abs` A)^k))
9	7, 8	eqeq12d 1486	. . . 4 |- (j = k -> ((abs` (A^j)) = ((abs` A)^j) <-> (abs` (A^k)) = ((abs` A)^k)))
10	9	imbi2d 611	. . 3 |- (j = k -> ((A e. CC -> (abs` (A^j)) = ((abs` A)^j)) <-> (A e. CC -> (abs` (A^k)) = ((abs` A)^k))))
11		opreq2 3960	. . . . . 6 |- (j = (k + 1) -> (A^j) = (A^(k + 1)))
12	11	fveq2d 3719	. . . . 5 |- (j = (k + 1) -> (abs` (A^j)) = (abs` (A^(k + 1))))
13		opreq2 3960	. . . . 5 |- (j = (k + 1) -> ((abs` A)^j) = ((abs` A)^(k + 1)))
14	12, 13	eqeq12d 1486	. . . 4 |- (j = (k + 1) -> ((abs` (A^j)) = ((abs` A)^j) <-> (abs` (A^(k + 1))) = ((abs` A)^(k + 1))))
15	14	imbi2d 611	. . 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		opreq2 3960	. . . . . 6 |- (j = N -> (A^j) = (A^N))
17	16	fveq2d 3719	. . . . 5 |- (j = N -> (abs` (A^j)) = (abs` (A^N)))
18		opreq2 3960	. . . . 5 |- (j = N -> ((abs` A)^j) = ((abs` A)^N))
19	17, 18	eqeq12d 1486	. . . 4 |- (j = N -> ((abs` (A^j)) = ((abs` A)^j) <-> (abs` (A^N)) = ((abs` A)^N)))
20	19	imbi2d 611	. . 3 |- (j = N -> ((A e. CC -> (abs` (A^j)) = ((abs` A)^j)) <-> (A e. CC -> (abs` (A^N)) = ((abs` A)^N))))
21		0re 5420	. . . . . 6 |- 0 e. RR
22		1re 5415	. . . . . 6 |- 1 e. RR
23		lt01 5661	. . . . . 6 |- 0 < 1
24	21, 22, 23	ltlei 5562	. . . . 5 |- 0 <_ 1
25	22	absid 6804	. . . . 5 |- (0 <_ 1 -> (abs` 1) = 1)
26	24, 25	ax-mp 7	. . . 4 |- (abs` 1) = 1
27		exp0t 6511	. . . . 5 |- (A e. CC -> (A^0) = 1)
28	27	fveq2d 3719	. . . 4 |- (A e. CC -> (abs` (A^0)) = (abs` 1))
29		absclt 6776	. . . . . 6 |- (A e. CC -> (abs` A) e. RR)
30	29	recnd 5295	. . . . 5 |- (A e. CC -> (abs` A) e. CC)
31		exp0t 6511	. . . . 5 |- ((abs` A) e. CC -> ((abs` A)^0) = 1)
32	30, 31	syl 10	. . . 4 |- (A e. CC -> ((abs` A)^0) = 1)
33	26, 28, 32	3eqtr4a 1529	. . 3 |- (A e. CC -> (abs` (A^0)) = ((abs` A)^0))
34		opreq1 3959	. . . . . . . 8 |- ((abs` (A^k)) = ((abs` A)^k) -> ((abs` (A^k)) x. (abs` A)) = (((abs` A)^k) x. (abs` A)))
35	34	adantl 388	. . . . . . 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)))
36		expp1t 6514	. . . . . . . . . 10 |- ((A e. CC /\ k e. NN0) -> (A^(k + 1)) = ((A^k) x. A))
37	36	fveq2d 3719	. . . . . . . . 9 |- ((A e. CC /\ k e. NN0) -> (abs` (A^(k + 1))) = (abs` ((A^k) x. A)))
38		absmult 6801	. . . . . . . . . 10 |- (((A^k) e. CC /\ A e. CC) -> (abs` ((A^k) x. A)) = ((abs` (A^k)) x. (abs` A)))
39		expclt 6521	. . . . . . . . . 10 |- ((A e. CC /\ k e. NN0) -> (A^k) e. CC)
40		pm3.26 319	. . . . . . . . . 10 |- ((A e. CC /\ k e. NN0) -> A e. CC)
41	38, 39, 40	sylanc 471	. . . . . . . . 9 |- ((A e. CC /\ k e. NN0) -> (abs` ((A^k) x. A)) = ((abs` (A^k)) x. (abs` A)))
42	37, 41	eqtrd 1504	. . . . . . . 8 |- ((A e. CC /\ k e. NN0) -> (abs` (A^(k + 1))) = ((abs` (A^k)) x. (abs` A)))
43	42	adantr 389	. . . . . . 7 |- (((A e. CC /\ k e. NN0) /\ (abs` (A^k)) = ((abs` A)^k)) -> (abs` (A^(k + 1))) = ((abs` (A^k)) x. (abs` A)))
44		expp1t 6514	. . . . . . . . 9 |- (((abs` A) e. CC /\ k e. NN0) -> ((abs` A)^(k + 1)) = (((abs` A)^k) x. (abs` A)))
45	44, 30	sylan 448	. . . . . . . 8 |- ((A e. CC /\ k e. NN0) -> ((abs` A)^(k + 1)) = (((abs` A)^k) x. (abs` A)))
46	45	adantr 389	. . . . . . 7 |- (((A e. CC /\ k e. NN0) /\ (abs` (A^k)) = ((abs` A)^k)) -> ((abs` A)^(k + 1)) = (((abs` A)^k) x. (abs` A)))
47	35, 43, 46	3eqtr4d 1514	. . . . . 6 |- (((A e. CC /\ k e. NN0) /\ (abs` (A^k)) = ((abs` A)^k)) -> (abs` (A^(k + 1))) = ((abs` A)^(k + 1)))
48	47	exp31 376	. . . . 5 |- (A e. CC -> (k e. NN0 -> ((abs` (A^k)) = ((abs` A)^k) -> (abs` (A^(k + 1))) = ((abs` A)^(k + 1)))))
49	48	com12 11	. . . 4 |- (k e. NN0 -> (A e. CC -> ((abs` (A^k)) = ((abs` A)^k) -> (abs` (A^(k + 1))) = ((abs` A)^(k + 1)))))
50	49	a2d 13	. . 3 |- (k e. NN0 -> ((A e. CC -> (abs` (A^k)) = ((abs` A)^k)) -> (A e. CC -> (abs` (A^(k + 1))) = ((abs` A)^(k + 1)))))
51	5, 10, 15, 20, 33, 50	nn0ind 6168	. 2 |- (N e. NN0 -> (A e. CC -> (abs` (A^N)) = ((abs` A)^N)))
52	51	impcom 351	1 |- ((A e. CC /\ N e. NN0) -> (abs` (A^N)) = ((abs` A)^N))

Syntax hints:   -> wi 3   /\ wa 223   = wceq 954   e. wcel 956   class class class wbr 2614  ` cfv 3177  (class class class)co 3954  CCcc 5212  0cc0 5214  1c1 5215   + caddc 5217   x. cmul 5219   <_ cle 5275  NN0cn0 5277  ^cexp 6508  abscabs 6689

This theorem is referenced by:  expcnv 7176  efaddlem10 7297  eftabs 7325  absefm1le 7360

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-9 963  ax-10 964  ax-11 965  ax-12 966  ax-13 967  ax-14 968  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457  ax-rep 2688  ax-sep 2698  ax-nul 2705  ax-pow 2737  ax-pr 2774  ax-un 2861  ax-inf2 4605

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 775  df-3an 776  df-ex 979  df-sb 1170  df-eu 1380  df-mo 1381  df-clab 1462  df-cleq 1467  df-clel 1470  df-ne 1584  df-nel 1585  df-ral 1646  df-rex 1647  df-reu 1648  df-rab 1649  df-v 1808  df-sbc 1938  df-csb 1998  df-dif 2045  df-un 2046  df-in 2047  df-ss 2049  df-pss 2051  df-nul 2277  df-if 2358  df-pw 2398  df-sn 2408  df-pr 2409  df-tp 2411  df-op 2412  df-uni 2499  df-int 2529  df-iun 2563  df-br 2615  df-opab 2662  df-tr 2676  df-eprel 2827  df-id 2830  df-po 2835  df-so 2845  df-fr 2912  df-we 2929  df-ord 2946  df-on 2947  df-lim 2948  df-suc 2949  df-om 3127  df-xp