Theorem expeq0t 6586

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

Assertion

Ref	Expression
expeq0t	|- ((A e. CC /\ N e. NN) -> ((A^N) = 0 <-> A = 0))

Proof of Theorem expeq0t

Step	Hyp	Ref	Expression
1		opreq2 3975	. . . . . 6 |- (j = 1 -> (A^j) = (A^1))
2	1	eqeq1d 1486	. . . . 5 |- (j = 1 -> ((A^j) = 0 <-> (A^1) = 0))
3	2	bibi1d 621	. . . 4 |- (j = 1 -> (((A^j) = 0 <-> A = 0) <-> ((A^1) = 0 <-> A = 0)))
4	3	imbi2d 614	. . 3 |- (j = 1 -> ((A e. CC -> ((A^j) = 0 <-> A = 0)) <-> (A e. CC -> ((A^1) = 0 <-> A = 0))))
5		opreq2 3975	. . . . . 6 |- (j = k -> (A^j) = (A^k))
6	5	eqeq1d 1486	. . . . 5 |- (j = k -> ((A^j) = 0 <-> (A^k) = 0))
7	6	bibi1d 621	. . . 4 |- (j = k -> (((A^j) = 0 <-> A = 0) <-> ((A^k) = 0 <-> A = 0)))
8	7	imbi2d 614	. . 3 |- (j = k -> ((A e. CC -> ((A^j) = 0 <-> A = 0)) <-> (A e. CC -> ((A^k) = 0 <-> A = 0))))
9		opreq2 3975	. . . . . 6 |- (j = (k + 1) -> (A^j) = (A^(k + 1)))
10	9	eqeq1d 1486	. . . . 5 |- (j = (k + 1) -> ((A^j) = 0 <-> (A^(k + 1)) = 0))
11	10	bibi1d 621	. . . 4 |- (j = (k + 1) -> (((A^j) = 0 <-> A = 0) <-> ((A^(k + 1)) = 0 <-> A = 0)))
12	11	imbi2d 614	. . 3 |- (j = (k + 1) -> ((A e. CC -> ((A^j) = 0 <-> A = 0)) <-> (A e. CC -> ((A^(k + 1)) = 0 <-> A = 0))))
13		opreq2 3975	. . . . . 6 |- (j = N -> (A^j) = (A^N))
14	13	eqeq1d 1486	. . . . 5 |- (j = N -> ((A^j) = 0 <-> (A^N) = 0))
15	14	bibi1d 621	. . . 4 |- (j = N -> (((A^j) = 0 <-> A = 0) <-> ((A^N) = 0 <-> A = 0)))
16	15	imbi2d 614	. . 3 |- (j = N -> ((A e. CC -> ((A^j) = 0 <-> A = 0)) <-> (A e. CC -> ((A^N) = 0 <-> A = 0))))
17		exp1t 6574	. . . 4 |- (A e. CC -> (A^1) = A)
18	17	eqeq1d 1486	. . 3 |- (A e. CC -> ((A^1) = 0 <-> A = 0))
19		expp1t 6575	. . . . . . . . . 10 |- ((A e. CC /\ k e. NN0) -> (A^(k + 1)) = ((A^k) x. A))
20	19	eqeq1d 1486	. . . . . . . . 9 |- ((A e. CC /\ k e. NN0) -> ((A^(k + 1)) = 0 <-> ((A^k) x. A) = 0))
21		mul0ort 5708	. . . . . . . . . 10 |- (((A^k) e. CC /\ A e. CC) -> (((A^k) x. A) = 0 <-> ((A^k) = 0 \/ A = 0)))
22		expclt 6582	. . . . . . . . . 10 |- ((A e. CC /\ k e. NN0) -> (A^k) e. CC)
23		pm3.26 319	. . . . . . . . . 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 530	. . . . . . . 8 |- ((A e. CC /\ k e. NN0) -> ((A^(k + 1)) = 0 <-> ((A^k) = 0 \/ A = 0)))
26		nnnn0t 6108	. . . . . . . 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 148	. . . . . . . . 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 426	. . . . . . . 8 |- (((A^k) = 0 <-> A = 0) -> (((A^k) = 0 \/ A = 0) -> A = 0))
31		olc 268	. . . . . . . 8 |- (A = 0 -> ((A^k) = 0 \/ A = 0))
32	30, 31	impbid1 519	. . . . . . 7 |- (((A^k) = 0 <-> A = 0) -> (((A^k) = 0 \/ A = 0) <-> A = 0))
33	27, 32	sylan9bb 542	. . . . . 6 |- (((A e. CC /\ k e. NN) /\ ((A^k) = 0 <-> A = 0)) -> ((A^(k + 1)) = 0 <-> A = 0))
34	33	exp31 378	. . . . 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 5939	. 2 |- (N e. NN -> (A e. CC -> ((A^N) = 0 <-> A = 0)))
38	37	impcom 351	1 |- ((A e. CC /\ N e. NN) -> ((A^N) = 0 <-> A = 0))

Syntax hints:   -> wi 3   <-> wb 146   \/ wo 222   /\ wa 223   = wceq 958   e. wcel 960  (class class class)co 3969  CCcc 5244  0cc0 5246  1c1 5247   + caddc 5249   x. cmul 5251  NNcn 5308  NN0cn0 5309  ^cexp 6569

This theorem is referenced by:  expne0t 6587  expne0tOLD 6588  0expt 6591  sqeq0t 6614

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-9 967  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-rep 2698  ax-sep 2708  ax-nul 2715  ax-pow 2748  ax-pr 2785  ax-un 2872  ax-inf2 4634

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 778  df-3an 779  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-nel 1591  df-ral 1652  df-rex 1653  df-reu 1654  df-rab 1655  df-v 1815  df-sbc 1945  df-csb 2005  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-pss 2058  df-nul 2284  df-if 2366  df-pw 2406  df-sn 2416  df-pr 2417  df-tp 2419  df-op 2420  df-uni 2508  df-int 2538  df-iun 2572  df-br 2625  df-opab 2672  df-tr 2686  df-eprel 2838  df-id 2841  df-po 2846  df-so 2856  df-fr 2923  df-we 2940  df-ord 2957  df-on 2958  df-lim 2959  df-suc 2960  df-om 3138  df-xp 3190  df-rel 3191  df-cnv 3192  df-co 3193  df-dm 3194  df-rn 3195  df-res 3196  df-ima 3197  df-fun 3198  df-fn 3199  df-f 3200  df-f1 3201  df-fo 3202  df-f1o 3203  df-fv 3204  df-rdg 3938  df-opr 3971  df-oprab 3972  df-1st 4085  df-2nd 4086  df-1o 4139  df-oadd 4141  df-omul 4142  df-er 4267  df-ec 4269  df-qs 4272  df-en 4374  df-dom 4375  df-sdom 4376  df-ni 5012  df-pli 5013  df-mi 5014  df-lti 5015  df-plpq 5047  df-mpq 5048  df-enq 5049  df-nq 5050  df-plq 5051  df-mq 5052  df-rq 5053  df-ltq 5054  df-1q 5055  df-np 5098  df-1p 5099  df-plp 5100  df-mp 5101  df-ltp 5102  df-plpr 5176  df-mpr 5177  df-enr 5178  df-nr 5179  df-plr 5180  df-mr 5181  df-ltr 5182  df-0r 5183  df-1r 5184  df-m1r 5185  df-c 5252  df-0 5253  df-1 5254  df-i 5255  df-r 5256  df-plus 5257  df-mul 5258  df-lt 5259  df-sub 5368  df-neg 5370  df-pnf 5499  df-mnf 5500  df-xr 5501  df-ltxr 5502  df-le 5503  df-n 5927  df-n0 6102  df-z 6138  df-seq1 6309  df-exp 6570

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