Theorem expeq0 6780

Description: Natural number exponentiation is 0 iff its mantissa is 0.

Assertion

Ref	Expression
expeq0	|- ((A e. CC /\ N e. NN) -> ((A^N) = 0 <-> A = 0))

Proof of Theorem expeq0

Step	Hyp	Ref	Expression
1		opreq2 4027	. . . . . 6 |- (j = 1 -> (A^j) = (A^1))
2	1	eqeq1d 1526	. . . . 5 |- (j = 1 -> ((A^j) = 0 <-> (A^1) = 0))
3	2	bibi1d 622	. . . 4 |- (j = 1 -> (((A^j) = 0 <-> A = 0) <-> ((A^1) = 0 <-> A = 0)))
4	3	imbi2d 615	. . 3 |- (j = 1 -> ((A e. CC -> ((A^j) = 0 <-> A = 0)) <-> (A e. CC -> ((A^1) = 0 <-> A = 0))))
5		opreq2 4027	. . . . . 6 |- (j = k -> (A^j) = (A^k))
6	5	eqeq1d 1526	. . . . 5 |- (j = k -> ((A^j) = 0 <-> (A^k) = 0))
7	6	bibi1d 622	. . . 4 |- (j = k -> (((A^j) = 0 <-> A = 0) <-> ((A^k) = 0 <-> A = 0)))
8	7	imbi2d 615	. . 3 |- (j = k -> ((A e. CC -> ((A^j) = 0 <-> A = 0)) <-> (A e. CC -> ((A^k) = 0 <-> A = 0))))
9		opreq2 4027	. . . . . 6 |- (j = (k + 1) -> (A^j) = (A^(k + 1)))
10	9	eqeq1d 1526	. . . . 5 |- (j = (k + 1) -> ((A^j) = 0 <-> (A^(k + 1)) = 0))
11	10	bibi1d 622	. . . 4 |- (j = (k + 1) -> (((A^j) = 0 <-> A = 0) <-> ((A^(k + 1)) = 0 <-> A = 0)))
12	11	imbi2d 615	. . 3 |- (j = (k + 1) -> ((A e. CC -> ((A^j) = 0 <-> A = 0)) <-> (A e. CC -> ((A^(k + 1)) = 0 <-> A = 0))))
13		opreq2 4027	. . . . . 6 |- (j = N -> (A^j) = (A^N))
14	13	eqeq1d 1526	. . . . 5 |- (j = N -> ((A^j) = 0 <-> (A^N) = 0))
15	14	bibi1d 622	. . . 4 |- (j = N -> (((A^j) = 0 <-> A = 0) <-> ((A^N) = 0 <-> A = 0)))
16	15	imbi2d 615	. . 3 |- (j = N -> ((A e. CC -> ((A^j) = 0 <-> A = 0)) <-> (A e. CC -> ((A^N) = 0 <-> A = 0))))
17		exp1 6768	. . . 4 |- (A e. CC -> (A^1) = A)
18	17	eqeq1d 1526	. . 3 |- (A e. CC -> ((A^1) = 0 <-> A = 0))
19		expp1 6769	. . . . . . . . . 10 |- ((A e. CC /\ k e. NN0) -> (A^(k + 1)) = ((A^k) x. A))
20	19	eqeq1d 1526	. . . . . . . . 9 |- ((A e. CC /\ k e. NN0) -> ((A^(k + 1)) = 0 <-> ((A^k) x. A) = 0))
21		mul0or 5848	. . . . . . . . . 10 |- (((A^k) e. CC /\ A e. CC) -> (((A^k) x. A) = 0 <-> ((A^k) = 0 \/ A = 0)))
22		expcl 6776	. . . . . . . . . 10 |- ((A e. CC /\ k e. NN0) -> (A^k) e. CC)
23		pm3.26 317	. . . . . . . . . 10 |- ((A e. CC /\ k e. NN0) -> A e. CC)
24	21, 22, 23	sylanc 473	. . . . . . . . 9 |- ((A e. CC /\ k e. NN0) -> (((A^k) x. A) = 0 <-> ((A^k) = 0 \/ A = 0)))
25	20, 24	bitrd 531	. . . . . . . 8 |- ((A e. CC /\ k e. NN0) -> ((A^(k + 1)) = 0 <-> ((A^k) = 0 \/ A = 0)))
26		nnnn0 6274	. . . . . . . 8 |- (k e. NN -> k e. NN0)
27	25, 26	sylan2 453	. . . . . . 7 |- ((A e. CC /\ k e. NN) -> ((A^(k + 1)) = 0 <-> ((A^k) = 0 \/ A = 0)))
28		bi1 146	. . . . . . . . 9 |- (((A^k) = 0 <-> A = 0) -> ((A^k) = 0 -> A = 0))
29		idd 61	. . . . . . . . 9 |- (((A^k) = 0 <-> A = 0) -> (A = 0 -> A = 0))
30	28, 29	jaod 424	. . . . . . . 8 |- (((A^k) = 0 <-> A = 0) -> (((A^k) = 0 \/ A = 0) -> A = 0))
31		olc 266	. . . . . . . 8 |- (A = 0 -> ((A^k) = 0 \/ A = 0))
32	30, 31	impbid1 520	. . . . . . 7 |- (((A^k) = 0 <-> A = 0) -> (((A^k) = 0 \/ A = 0) <-> A = 0))
33	27, 32	sylan9bb 543	. . . . . 6 |- (((A e. CC /\ k e. NN) /\ ((A^k) = 0 <-> A = 0)) -> ((A^(k + 1)) = 0 <-> A = 0))
34	33	exp31 376	. . . . 5 |- (A e. CC -> (k e. NN -> (((A^k) = 0 <-> A = 0) -> ((A^(k + 1)) = 0 <-> A = 0))))
35	34	com12 11	. . . 4 |- (k e. NN -> (A e. CC -> (((A^k) = 0 <-> A = 0) -> ((A^(k + 1)) = 0 <-> A = 0))))
36	35	a2d 13	. . 3 |- (k e. NN -> ((A e. CC -> ((A^k) = 0 <-> A = 0)) -> (A e. CC -> ((A^(k + 1)) = 0 <-> A = 0))))
37	4, 8, 12, 16, 18, 36	nnind 6082	. 2 |- (N e. NN -> (A e. CC -> ((A^N) = 0 <-> A = 0)))
38	37	impcom 349	1 |- ((A e. CC /\ N e. NN) -> ((A^N) = 0 <-> A = 0))

Syntax hints:   -> wi 3   <-> wb 144   \/ wo 220   /\ wa 221   = wceq 992   e. wcel 994  (class class class)co 4021  CCcc 5386  0cc0 5388  1c1 5389   + caddc 5391   x. cmul 5393  NNcn 5450  NN0cn0 5451  ^cexp 6763

This theorem is referenced by:  expne0 6781  0exp 6784  sqeq0 6810  divexp 11859

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 998  ax-gen 999  ax-8 1000  ax-9 1001  ax-10 1002  ax-11 1003  ax-12 1004  ax-13 1005  ax-14 1006  ax-17 1007  ax-4 1009  ax-5o 1011  ax-6o 1014  ax-9o 1159  ax-10o 1177  ax-16 1247  ax-11o 1255  ax-ext 1500  ax-rep 2767  ax-sep 2777  ax-nul 2784  ax-pow 2818  ax-pr 2855  ax-un 3089  ax-inf2 4770

This theorem depends on definitions:  df-bi 145  df-or 222  df-an 223  df-3or 782  df-3an 783  df-ex 1017  df-sb 1209  df-eu 1421  df-mo 1422  df-clab 1506  df-cleq 1511  df-clel 1514  df-ne 1630  df-nel 1631  df-ral 1695  df-rex 1696  df-reu 1697  df-rab 1698  df-v 1858  df-sbc 1987  df-csb 2052  df-dif 2101  df-un 2102  df-in 2103  df-ss 2105  df-pss 2107  df-nul 2333  df-if 2416  df-pw 2459  df-sn 2470  df-pr 2471  df-tp 2473  df-op 2474  df-uni 2570  df-int 2601  df-iun 2635  df-br 2693  df-opab 2741  df-tr 2755  df-eprel 2910  df-id 2913  df-po 2918  df-so 2929  df-fr 2947  df-we 2962  df-ord 2978  df-on 2979  df-lim 2980  df-suc 2981  df-om 3219  df-xp 3265  df-rel 3266  df-cnv 3267  df-co 3268  df-dm 3269  df-rn 3270  df-res 3271  df-ima 3272  df-fun 3273  df-fn 3274  df-f 3275  df-f1 3276  df-fo 3277  df-f1o 3278  df-fv 3279  df-opr 4023  df-oprab 4024  df-1st 4140  df-2nd 4141  df-rdg 4233  df-1o 4269  df-oadd 4271  df-omul 4272  df-er 4401  df-ec 4403  df-qs 4406  df-en 4509  df-dom 4510  df-sdom 4511  df-ni 5154  df-pli 5155  df-mi 5156  df-lti 5157  df-plpq 5189  df-mpq 5190  df-enq 5191  df-nq 5192  df-plq 5193  df-mq 5194  df-rq 5195  df-ltq 5196  df-1q 5197  df-np 5240  df-1p 5241  df-plp 5242  df-mp 5243  df-ltp 5244  df-plpr 5318  df-mpr 5319  df-enr 5320  df-nr 5321  df-plr 5322  df-mr 5323  df-ltr 5324  df-0r 5325  df-1r 5326  df-m1r 5327  df-c 5394  df-0 5395  df-1 5396  df-i 5397  df-r 5398  df-plus 5399  df-mul 5400  df-lt 5401  df-sub 5510  df-neg 5512  df-pnf 5641  df-mnf 5642  df-xr 5643  df-ltxr 5644  df-le 5645  df-n 6070  df-n0 6268  df-z 6304  df-seq1 6673  df-exp 6764

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