Theorem absexp 8119

Description: Absolute value of natural number exponentiation.

Assertion

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

Proof of Theorem absexp

Step	Hyp	Ref	Expression
1		opreq2 4890	. . . . . 6 |- (j = 0 -> (A^j) = (A^0))
2	1	fveq2d 4685	. . . . 5 |- (j = 0 -> (abs` (A^j)) = (abs` (A^0)))
3		opreq2 4890	. . . . 5 |- (j = 0 -> ((abs` A)^j) = ((abs` A)^0))
4	2, 3	eqeq12d 1899	. . . 4 |- (j = 0 -> ((abs` (A^j)) = ((abs` A)^j) <-> (abs` (A^0)) = ((abs` A)^0)))
5	4	imbi2d 674	. . 3 |- (j = 0 -> ((A e. CC -> (abs` (A^j)) = ((abs` A)^j)) <-> (A e. CC -> (abs` (A^0)) = ((abs` A)^0))))
6		opreq2 4890	. . . . . 6 |- (j = k -> (A^j) = (A^k))
7	6	fveq2d 4685	. . . . 5 |- (j = k -> (abs` (A^j)) = (abs` (A^k)))
8		opreq2 4890	. . . . 5 |- (j = k -> ((abs` A)^j) = ((abs` A)^k))
9	7, 8	eqeq12d 1899	. . . 4 |- (j = k -> ((abs` (A^j)) = ((abs` A)^j) <-> (abs` (A^k)) = ((abs` A)^k)))
10	9	imbi2d 674	. . 3 |- (j = k -> ((A e. CC -> (abs` (A^j)) = ((abs` A)^j)) <-> (A e. CC -> (abs` (A^k)) = ((abs` A)^k))))
11		opreq2 4890	. . . . . 6 |- (j = (k + 1) -> (A^j) = (A^(k + 1)))
12	11	fveq2d 4685	. . . . 5 |- (j = (k + 1) -> (abs` (A^j)) = (abs` (A^(k + 1))))
13		opreq2 4890	. . . . 5 |- (j = (k + 1) -> ((abs` A)^j) = ((abs` A)^(k + 1)))
14	12, 13	eqeq12d 1899	. . . 4 |- (j = (k + 1) -> ((abs` (A^j)) = ((abs` A)^j) <-> (abs` (A^(k + 1))) = ((abs` A)^(k + 1))))
15	14	imbi2d 674	. . 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 4890	. . . . . 6 |- (j = N -> (A^j) = (A^N))
17	16	fveq2d 4685	. . . . 5 |- (j = N -> (abs` (A^j)) = (abs` (A^N)))
18		opreq2 4890	. . . . 5 |- (j = N -> ((abs` A)^j) = ((abs` A)^N))
19	17, 18	eqeq12d 1899	. . . 4 |- (j = N -> ((abs` (A^j)) = ((abs` A)^j) <-> (abs` (A^N)) = ((abs` A)^N)))
20	19	imbi2d 674	. . 3 |- (j = N -> ((A e. CC -> (abs` (A^j)) = ((abs` A)^j)) <-> (A e. CC -> (abs` (A^N)) = ((abs` A)^N))))
21		0re 6603	. . . . . 6 |- 0 e. RR
22		1re 6598	. . . . . 6 |- 1 e. RR
23		lt01 6871	. . . . . 6 |- 0 < 1
24	21, 22, 23	ltleii 6756	. . . . 5 |- 0 <_ 1
25	22	absidi 8112	. . . . 5 |- (0 <_ 1 -> (abs` 1) = 1)
26	24, 25	ax-mp 7	. . . 4 |- (abs` 1) = 1
27		exp0 7814	. . . . 5 |- (A e. CC -> (A^0) = 1)
28	27	fveq2d 4685	. . . 4 |- (A e. CC -> (abs` (A^0)) = (abs` 1))
29		abscl 8084	. . . . . 6 |- (A e. CC -> (abs` A) e. RR)
30	29	recnd 6468	. . . . 5 |- (A e. CC -> (abs` A) e. CC)
31		exp0 7814	. . . . 5 |- ((abs` A) e. CC -> ((abs` A)^0) = 1)
32	30, 31	syl 12	. . . 4 |- (A e. CC -> ((abs` A)^0) = 1)
33	26, 28, 32	3eqtr4a 1954	. . 3 |- (A e. CC -> (abs` (A^0)) = ((abs` A)^0))
34		opreq1 4889	. . . . . . . 8 |- ((abs` (A^k)) = ((abs` A)^k) -> ((abs` (A^k)) x. (abs` A)) = (((abs` A)^k) x. (abs` A)))
35	34	adantl 424	. . . . . . 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		expp1 7817	. . . . . . . . . 10 |- ((A e. CC /\ k e. NN0) -> (A^(k + 1)) = ((A^k) x. A))
37	36	fveq2d 4685	. . . . . . . . 9 |- ((A e. CC /\ k e. NN0) -> (abs` (A^(k + 1))) = (abs` ((A^k) x. A)))
38		expcl 7824	. . . . . . . . . 10 |- ((A e. CC /\ k e. NN0) -> (A^k) e. CC)
39		simpl 346	. . . . . . . . . 10 |- ((A e. CC /\ k e. NN0) -> A e. CC)
40		absmul 8109	. . . . . . . . . 10 |- (((A^k) e. CC /\ A e. CC) -> (abs` ((A^k) x. A)) = ((abs` (A^k)) x. (abs` A)))
41	38, 39, 40	syl11anc 524	. . . . . . . . 9 |- ((A e. CC /\ k e. NN0) -> (abs` ((A^k) x. A)) = ((abs` (A^k)) x. (abs` A)))
42	37, 41	eqtrd 1925	. . . . . . . 8 |- ((A e. CC /\ k e. NN0) -> (abs` (A^(k + 1))) = ((abs` (A^k)) x. (abs` A)))
43	42	adantr 425	. . . . . . 7 |- (((A e. CC /\ k e. NN0) /\ (abs` (A^k)) = ((abs` A)^k)) -> (abs` (A^(k + 1))) = ((abs` (A^k)) x. (abs` A)))
44		expp1 7817	. . . . . . . . 9 |- (((abs` A) e. CC /\ k e. NN0) -> ((abs` A)^(k + 1)) = (((abs` A)^k) x. (abs` A)))
45	44, 30	sylan 497	. . . . . . . 8 |- ((A e. CC /\ k e. NN0) -> ((abs` A)^(k + 1)) = (((abs` A)^k) x. (abs` A)))
46	45	adantr 425	. . . . . . 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 1937	. . . . . 6 |- (((A e. CC /\ k e. NN0) /\ (abs` (A^k)) = ((abs` A)^k)) -> (abs` (A^(k + 1))) = ((abs` A)^(k + 1)))
48	47	exp31 407	. . . . 5 |- (A e. CC -> (k e. NN0 -> ((abs` (A^k)) = ((abs` A)^k) -> (abs` (A^(k + 1))) = ((abs` A)^(k + 1)))))
49	48	com12 14	. . . 4 |- (k e. NN0 -> (A e. CC -> ((abs` (A^k)) = ((abs` A)^k) -> (abs` (A^(k + 1))) = ((abs` A)^(k + 1)))))
50	49	a2d 16	. . 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 7424	. 2 |- (N e. NN0 -> (A e. CC -> (abs` (A^N)) = ((abs` A)^N)))
52	51	impcom 378	1 |- ((A e. CC /\ N e. NN0) -> (abs` (A^N)) = ((abs` A)^N))

Syntax hints:   -> wi 3   /\ wa 240   = wceq 1298   e. wcel 1300   class class class wbr 3338  ` cfv 3998  (class class class)co 4884  CCcc 6384  0cc0 6386  1c1 6387   + caddc 6389   x. cmul 6391   <_ cle 6448  NN0cn0 6450  ^cexp 7811  abscabs 8000

This theorem is referenced by:  sqabs 8120  expcnv 8494  efaddlem10 8609  eftabsi 8637  absefm1lei 8677

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790  ax-inf2 5731

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3or 859  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-nel 2020  df-ral 2109  df-rex 2110  df-reu 2111  df-rab 2112  df-v 2294  df-sbc 2454  df-csb 2541  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-pss 2607  df-nul 2876  df-if 2983  df-pw 3035  df-sn 3049  df-pr 3050  df-tp 3052  df-op 3053  df-uni 3178  df-int 3215  df-iun 3257  df-br 3339  df-opab 3396  df-tr 3412  df-eprel 3583  df-id 3586  df-po 3591  df-so 3604  df-fr 3625  df-we 3644  df-ord 3660  df-on 3661  df-lim 3662  df-suc 3663  df-om 3950  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-f1 4011  df-fo 4012  df-f1o 4013  df-fv 4014  df-opr 4886  df-oprab 4887  df-mpt 5006  df-1st 5020  df-2nd 5021  df-iota 5089  df-rdg 5140  df-1o 5177  df-oadd 5179  df-omul 5180  df-er 5318  df-ec 5320  df-qs 5323  df-en 5427  df-dom 5428  df-sdom 5429  df-undef 5556  df-riota 5560  df-sup 5664  df-ni 6152  df-pli 6153  df-mi 6154  df-lti 6155  df-plpq 6187  df-mpq 6188  df-enq 6189  df-nq 6190  df-plq 6191  df-mq 6192  df-rq 6193  df-ltq 6194  df-1q 6195  df-np 6238  df-1p 6239  df-plp 6240  df-mp 6241  df-ltp 6242  df-plpr 6316  df-mpr 6317  df-enr 6318  df-nr 6319  df-plr 6320  df-mr 6321  df-ltr 6322  df-0r 6323  df-1r 6324  df-m1r 6325  df-c 6392  df-0 6393  df-1 6394  df-i 6395  df-r 6396  df-plus 6397  df-mul 6398  df-lt 6399  df-sub 6511  df-neg 6513  df-pnf 6654  df-mnf 6655  df-xr 6656  df-ltxr 6657  df-le 6658  df-div 6892  df-n 7108  df-2 7154  df-n0 7309  df-z 7345  df-seq1 7721  df-exp 7812  df-sqr 7920  df-re 8001  df-im 8002  df-cj 8003  df-abs 8004

Copyright terms: Public domain