Theorem exprecOLD 7838

Description: Nonnegative integer exponentiation of a reciprocal.

Assertion

Ref	Expression
exprecOLD	|- ((A e. CC /\ N e. NN0 /\ A =/= 0) -> ((1 / A)^N) = (1 / (A^N)))

Proof of Theorem exprecOLD

Step	Hyp	Ref	Expression
1		opreq2 4890	. . . . . . 7 |- (j = 0 -> ((1 / A)^j) = ((1 / A)^0))
2		opreq2 4890	. . . . . . . 8 |- (j = 0 -> (A^j) = (A^0))
3	2	opreq2d 4898	. . . . . . 7 |- (j = 0 -> (1 / (A^j)) = (1 / (A^0)))
4	1, 3	eqeq12d 1899	. . . . . 6 |- (j = 0 -> (((1 / A)^j) = (1 / (A^j)) <-> ((1 / A)^0) = (1 / (A^0))))
5	4	imbi2d 674	. . . . 5 |- (j = 0 -> (((A e. CC /\ A =/= 0) -> ((1 / A)^j) = (1 / (A^j))) <-> ((A e. CC /\ A =/= 0) -> ((1 / A)^0) = (1 / (A^0)))))
6		opreq2 4890	. . . . . . 7 |- (j = k -> ((1 / A)^j) = ((1 / A)^k))
7		opreq2 4890	. . . . . . . 8 |- (j = k -> (A^j) = (A^k))
8	7	opreq2d 4898	. . . . . . 7 |- (j = k -> (1 / (A^j)) = (1 / (A^k)))
9	6, 8	eqeq12d 1899	. . . . . 6 |- (j = k -> (((1 / A)^j) = (1 / (A^j)) <-> ((1 / A)^k) = (1 / (A^k))))
10	9	imbi2d 674	. . . . 5 |- (j = k -> (((A e. CC /\ A =/= 0) -> ((1 / A)^j) = (1 / (A^j))) <-> ((A e. CC /\ A =/= 0) -> ((1 / A)^k) = (1 / (A^k)))))
11		opreq2 4890	. . . . . . 7 |- (j = (k + 1) -> ((1 / A)^j) = ((1 / A)^(k + 1)))
12		opreq2 4890	. . . . . . . 8 |- (j = (k + 1) -> (A^j) = (A^(k + 1)))
13	12	opreq2d 4898	. . . . . . 7 |- (j = (k + 1) -> (1 / (A^j)) = (1 / (A^(k + 1))))
14	11, 13	eqeq12d 1899	. . . . . 6 |- (j = (k + 1) -> (((1 / A)^j) = (1 / (A^j)) <-> ((1 / A)^(k + 1)) = (1 / (A^(k + 1)))))
15	14	imbi2d 674	. . . . 5 |- (j = (k + 1) -> (((A e. CC /\ A =/= 0) -> ((1 / A)^j) = (1 / (A^j))) <-> ((A e. CC /\ A =/= 0) -> ((1 / A)^(k + 1)) = (1 / (A^(k + 1))))))
16		opreq2 4890	. . . . . . 7 |- (j = N -> ((1 / A)^j) = ((1 / A)^N))
17		opreq2 4890	. . . . . . . 8 |- (j = N -> (A^j) = (A^N))
18	17	opreq2d 4898	. . . . . . 7 |- (j = N -> (1 / (A^j)) = (1 / (A^N)))
19	16, 18	eqeq12d 1899	. . . . . 6 |- (j = N -> (((1 / A)^j) = (1 / (A^j)) <-> ((1 / A)^N) = (1 / (A^N))))
20	19	imbi2d 674	. . . . 5 |- (j = N -> (((A e. CC /\ A =/= 0) -> ((1 / A)^j) = (1 / (A^j))) <-> ((A e. CC /\ A =/= 0) -> ((1 / A)^N) = (1 / (A^N)))))
21		reccl 6904	. . . . . . 7 |- ((A e. CC /\ A =/= 0) -> (1 / A) e. CC)
22		exp0 7814	. . . . . . 7 |- ((1 / A) e. CC -> ((1 / A)^0) = 1)
23	21, 22	syl 12	. . . . . 6 |- ((A e. CC /\ A =/= 0) -> ((1 / A)^0) = 1)
24		exp0 7814	. . . . . . . . 9 |- (A e. CC -> (A^0) = 1)
25	24	opreq2d 4898	. . . . . . . 8 |- (A e. CC -> (1 / (A^0)) = (1 / 1))
26		ax1cn 6422	. . . . . . . . 9 |- 1 e. CC
27	26	div1i 6948	. . . . . . . 8 |- (1 / 1) = 1
28	25, 27	syl6eq 1944	. . . . . . 7 |- (A e. CC -> (1 / (A^0)) = 1)
29	28	adantr 425	. . . . . 6 |- ((A e. CC /\ A =/= 0) -> (1 / (A^0)) = 1)
30	23, 29	eqtr4d 1928	. . . . 5 |- ((A e. CC /\ A =/= 0) -> ((1 / A)^0) = (1 / (A^0)))
31		opreq1 4889	. . . . . . . . . . 11 |- (((1 / A)^k) = (1 / (A^k)) -> (((1 / A)^k) x. (1 / A)) = ((1 / (A^k)) x. (1 / A)))
32	31	ad2antll 443	. . . . . . . . . 10 |- (((A e. CC /\ k e. NN0) /\ (A =/= 0 /\ ((1 / A)^k) = (1 / (A^k)))) -> (((1 / A)^k) x. (1 / A)) = ((1 / (A^k)) x. (1 / A)))
33		expp1 7817	. . . . . . . . . . . . 13 |- (((1 / A) e. CC /\ k e. NN0) -> ((1 / A)^(k + 1)) = (((1 / A)^k) x. (1 / A)))
34	33, 21	sylan 497	. . . . . . . . . . . 12 |- (((A e. CC /\ A =/= 0) /\ k e. NN0) -> ((1 / A)^(k + 1)) = (((1 / A)^k) x. (1 / A)))
35	34	an1rs 547	. . . . . . . . . . 11 |- (((A e. CC /\ k e. NN0) /\ A =/= 0) -> ((1 / A)^(k + 1)) = (((1 / A)^k) x. (1 / A)))
36	35	adantrr 431	. . . . . . . . . 10 |- (((A e. CC /\ k e. NN0) /\ (A =/= 0 /\ ((1 / A)^k) = (1 / (A^k)))) -> ((1 / A)^(k + 1)) = (((1 / A)^k) x. (1 / A)))
37		expp1 7817	. . . . . . . . . . . . . 14 |- ((A e. CC /\ k e. NN0) -> (A^(k + 1)) = ((A^k) x. A))
38	37	opreq2d 4898	. . . . . . . . . . . . 13 |- ((A e. CC /\ k e. NN0) -> (1 / (A^(k + 1))) = (1 / ((A^k) x. A)))
39	38	adantr 425	. . . . . . . . . . . 12 |- (((A e. CC /\ k e. NN0) /\ A =/= 0) -> (1 / (A^(k + 1))) = (1 / ((A^k) x. A)))
40		expcl 7824	. . . . . . . . . . . . . . . . 17 |- ((A e. CC /\ k e. NN0) -> (A^k) e. CC)
41	40, 26	jctil 316	. . . . . . . . . . . . . . . 16 |- ((A e. CC /\ k e. NN0) -> (1 e. CC /\ (A^k) e. CC))
42		simpl 346	. . . . . . . . . . . . . . . . 17 |- ((A e. CC /\ k e. NN0) -> A e. CC)
43	42, 26	jctil 316	. . . . . . . . . . . . . . . 16 |- ((A e. CC /\ k e. NN0) -> (1 e. CC /\ A e. CC))
44	41, 43	jca 310	. . . . . . . . . . . . . . 15 |- ((A e. CC /\ k e. NN0) -> ((1 e. CC /\ (A^k) e. CC) /\ (1 e. CC /\ A e. CC)))
45	44	adantr 425	. . . . . . . . . . . . . 14 |- (((A e. CC /\ k e. NN0) /\ A =/= 0) -> ((1 e. CC /\ (A^k) e. CC) /\ (1 e. CC /\ A e. CC)))
46		simp1 876	. . . . . . . . . . . . . . . 16 |- ((A e. CC /\ k e. NN0 /\ A =/= 0) -> A e. CC)
47		simp3 878	. . . . . . . . . . . . . . . 16 |- ((A e. CC /\ k e. NN0 /\ A =/= 0) -> A =/= 0)
48		simp2 877	. . . . . . . . . . . . . . . 16 |- ((A e. CC /\ k e. NN0 /\ A =/= 0) -> k e. NN0)
49		expne0i 7830	. . . . . . . . . . . . . . . 16 |- ((A e. CC /\ A =/= 0 /\ k e. NN0) -> (A^k) =/= 0)
50	46, 47, 48, 49	syl111anc 1100	. . . . . . . . . . . . . . 15 |- ((A e. CC /\ k e. NN0 /\ A =/= 0) -> (A^k) =/= 0)
51	50	3expa 1067	. . . . . . . . . . . . . 14 |- (((A e. CC /\ k e. NN0) /\ A =/= 0) -> (A^k) =/= 0)
52		simpr 350	. . . . . . . . . . . . . 14 |- (((A e. CC /\ k e. NN0) /\ A =/= 0) -> A =/= 0)
53		simplll 452	. . . . . . . . . . . . . . 15 |- ((((1 e. CC /\ (A^k) e. CC) /\ (1 e. CC /\ A e. CC)) /\ ((A^k) =/= 0 /\ A =/= 0)) -> 1 e. CC)
54		simplr 449	. . . . . . . . . . . . . . . 16 |- (((1 e. CC /\ (A^k) e. CC) /\ (1 e. CC /\ A e. CC)) -> (A^k) e. CC)
55		simpl 346	. . . . . . . . . . . . . . . 16 |- (((A^k) =/= 0 /\ A =/= 0) -> (A^k) =/= 0)
56	54, 55	anim12i 360	. . . . . . . . . . . . . . 15 |- ((((1 e. CC /\ (A^k) e. CC) /\ (1 e. CC /\ A e. CC)) /\ ((A^k) =/= 0 /\ A =/= 0)) -> ((A^k) e. CC /\ (A^k) =/= 0))
57		simprr 451	. . . . . . . . . . . . . . . 16 |- (((1 e. CC /\ (A^k) e. CC) /\ (1 e. CC /\ A e. CC)) -> A e. CC)
58		simpr 350	. . . . . . . . . . . . . . . 16 |- (((A^k) =/= 0 /\ A =/= 0) -> A =/= 0)
59	57, 58	anim12i 360	. . . . . . . . . . . . . . 15 |- ((((1 e. CC /\ (A^k) e. CC) /\ (1 e. CC /\ A e. CC)) /\ ((A^k) =/= 0 /\ A =/= 0)) -> (A e. CC /\ A =/= 0))
60		divmuldiv 6956	. . . . . . . . . . . . . . 15 |- (((1 e. CC /\ 1 e. CC) /\ (((A^k) e. CC /\ (A^k) =/= 0) /\ (A e. CC /\ A =/= 0))) -> ((1 / (A^k)) x. (1 / A)) = ((1 x. 1) / ((A^k) x. A)))
61	53, 53, 56, 59, 60	syl22anc 1101	. . . . . . . . . . . . . 14 |- ((((1 e. CC /\ (A^k) e. CC) /\ (1 e. CC /\ A e. CC)) /\ ((A^k) =/= 0 /\ A =/= 0)) -> ((1 / (A^k)) x. (1 / A)) = ((1 x. 1) / ((A^k) x. A)))
62	45, 51, 52, 61	syl12anc 1098	. . . . . . . . . . . . 13 |- (((A e. CC /\ k e. NN0) /\ A =/= 0) -> ((1 / (A^k)) x. (1 / A)) = ((1 x. 1) / ((A^k) x. A)))
63	26	mulid1i 6485	. . . . . . . . . . . . . 14 |- (1 x. 1) = 1
64	63	opreq1i 4892	. . . . . . . . . . . . 13 |- ((1 x. 1) / ((A^k) x. A)) = (1 / ((A^k) x. A))
65	62, 64	syl6req 1945	. . . . . . . . . . . 12 |- (((A e. CC /\ k e. NN0) /\ A =/= 0) -> (1 / ((A^k) x. A)) = ((1 / (A^k)) x. (1 / A)))
66	39, 65	eqtrd 1925	. . . . . . . . . . 11 |- (((A e. CC /\ k e. NN0) /\ A =/= 0) -> (1 / (A^(k + 1))) = ((1 / (A^k)) x. (1 / A)))
67	66	adantrr 431	. . . . . . . . . 10 |- (((A e. CC /\ k e. NN0) /\ (A =/= 0 /\ ((1 / A)^k) = (1 / (A^k)))) -> (1 / (A^(k + 1))) = ((1 / (A^k)) x. (1 / A)))
68	32, 36, 67	3eqtr4d 1937	. . . . . . . . 9 |- (((A e. CC /\ k e. NN0) /\ (A =/= 0 /\ ((1 / A)^k) = (1 / (A^k)))) -> ((1 / A)^(k + 1)) = (1 / (A^(k + 1))))
69	68	exp43 415	. . . . . . . 8 |- (A e. CC -> (k e. NN0 -> (A =/= 0 -> (((1 / A)^k) = (1 / (A^k)) -> ((1 / A)^(k + 1)) = (1 / (A^(k + 1)))))))
70	69	com12 14	. . . . . . 7 |- (k e. NN0 -> (A e. CC -> (A =/= 0 -> (((1 / A)^k) = (1 / (A^k)) -> ((1 / A)^(k + 1)) = (1 / (A^(k + 1)))))))
71	70	imp3a 388	. . . . . 6 |- (k e. NN0 -> ((A e. CC /\ A =/= 0) -> (((1 / A)^k) = (1 / (A^k)) -> ((1 / A)^(k + 1)) = (1 / (A^(k + 1))))))
72	71	a2d 16	. . . . 5 |- (k e. NN0 -> (((A e. CC /\ A =/= 0) -> ((1 / A)^k) = (1 / (A^k))) -> ((A e. CC /\ A =/= 0) -> ((1 / A)^(k + 1)) = (1 / (A^(k + 1))))))
73	5, 10, 15, 20, 30, 72	nn0ind 7424	. . . 4 |- (N e. NN0 -> ((A e. CC /\ A =/= 0) -> ((1 / A)^N) = (1 / (A^N))))
74	73	exp3a 405	. . 3 |- (N e. NN0 -> (A e. CC -> (A =/= 0 -> ((1 / A)^N) = (1 / (A^N)))))
75	74	com12 14	. 2 |- (A e. CC -> (N e. NN0 -> (A =/= 0 -> ((1 / A)^N) = (1 / (A^N)))))
76	75	3imp 1061	1 |- ((A e. CC /\ N e. NN0 /\ A =/= 0) -> ((1 / A)^N) = (1 / (A^N)))

Syntax hints:   -> wi 3   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300   =/= wne 2017  (class class class)co 4884  CCcc 6384  0cc0 6386  1c1 6387   + caddc 6389   x. cmul 6391   / cdiv 6447  NN0cn0 6450  ^cexp 7811

This theorem is referenced by:  expord2 7849  exple1 7852  expcnvlem2 8489  0.999... 8508  erelem3 8583  ef1tllem 8643

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-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-n0 7309  df-z 7345  df-seq1 7721  df-exp 7812

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