Theorem expap0 9506

Description: Positive integer exponentiation is apart from zero iff its mantissa is apart from zero. That it is easier to prove this first, and then prove expeq0 9507 in terms of it, rather than the other way around, is perhaps an illustration of the maxim "In constructive analysis, the apartness is more basic [ than ] equality." ([Geuvers], p. 1). (Contributed by Jim Kingdon, 10-Jun-2020.)

Assertion

Ref	Expression
expap0	|- ( ( A e. CC /\ N e. NN ) -> ( ( A ^ N ) # 0 <-> A # 0 ) )

Proof of Theorem expap0

Dummy variables j k are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		oveq2 5540	. . . . . 6 |- ( j = 1 -> ( A ^ j ) = ( A ^ 1 ) )
2	1	breq1d 3795	. . . . 5 |- ( j = 1 -> ( ( A ^ j ) # 0 <-> ( A ^ 1 ) # 0 ) )
3	2	bibi1d 231	. . . 4 |- ( j = 1 -> ( ( ( A ^ j ) # 0 <-> A # 0 ) <-> ( ( A ^ 1 ) # 0 <-> A # 0 ) ) )
4	3	imbi2d 228	. . 3 |- ( j = 1 -> ( ( A e. CC -> ( ( A ^ j ) # 0 <-> A # 0 ) ) <-> ( A e. CC -> ( ( A ^ 1 ) # 0 <-> A # 0 ) ) ) )
5		oveq2 5540	. . . . . 6 |- ( j = k -> ( A ^ j ) = ( A ^ k ) )
6	5	breq1d 3795	. . . . 5 |- ( j = k -> ( ( A ^ j ) # 0 <-> ( A ^ k ) # 0 ) )
7	6	bibi1d 231	. . . 4 |- ( j = k -> ( ( ( A ^ j ) # 0 <-> A # 0 ) <-> ( ( A ^ k ) # 0 <-> A # 0 ) ) )
8	7	imbi2d 228	. . 3 |- ( j = k -> ( ( A e. CC -> ( ( A ^ j ) # 0 <-> A # 0 ) ) <-> ( A e. CC -> ( ( A ^ k ) # 0 <-> A # 0 ) ) ) )
9		oveq2 5540	. . . . . 6 |- ( j = ( k + 1 ) -> ( A ^ j ) = ( A ^ ( k + 1 ) ) )
10	9	breq1d 3795	. . . . 5 |- ( j = ( k + 1 ) -> ( ( A ^ j ) # 0 <-> ( A ^ ( k + 1 ) ) # 0 ) )
11	10	bibi1d 231	. . . 4 |- ( j = ( k + 1 ) -> ( ( ( A ^ j ) # 0 <-> A # 0 ) <-> ( ( A ^ ( k + 1 ) ) # 0 <-> A # 0 ) ) )
12	11	imbi2d 228	. . 3 |- ( j = ( k + 1 ) -> ( ( A e. CC -> ( ( A ^ j ) # 0 <-> A # 0 ) ) <-> ( A e. CC -> ( ( A ^ ( k + 1 ) ) # 0 <-> A # 0 ) ) ) )
13		oveq2 5540	. . . . . 6 |- ( j = N -> ( A ^ j ) = ( A ^ N ) )
14	13	breq1d 3795	. . . . 5 |- ( j = N -> ( ( A ^ j ) # 0 <-> ( A ^ N ) # 0 ) )
15	14	bibi1d 231	. . . 4 |- ( j = N -> ( ( ( A ^ j ) # 0 <-> A # 0 ) <-> ( ( A ^ N ) # 0 <-> A # 0 ) ) )
16	15	imbi2d 228	. . 3 |- ( j = N -> ( ( A e. CC -> ( ( A ^ j ) # 0 <-> A # 0 ) ) <-> ( A e. CC -> ( ( A ^ N ) # 0 <-> A # 0 ) ) ) )
17		exp1 9482	. . . 4 |- ( A e. CC -> ( A ^ 1 ) = A )
18	17	breq1d 3795	. . 3 |- ( A e. CC -> ( ( A ^ 1 ) # 0 <-> A # 0 ) )
19		nnnn0 8295	. . . . . . . . 9 |- ( k e. NN -> k e. NN0 )
20		expp1 9483	. . . . . . . . . . 11 |- ( ( A e. CC /\ k e. NN0 ) -> ( A ^ ( k + 1 ) ) = ( ( A ^ k ) x. A ) )
21	20	breq1d 3795	. . . . . . . . . 10 |- ( ( A e. CC /\ k e. NN0 ) -> ( ( A ^ ( k + 1 ) ) # 0 <-> ( ( A ^ k ) x. A ) # 0 ) )
22	21	ancoms 264	. . . . . . . . 9 |- ( ( k e. NN0 /\ A e. CC ) -> ( ( A ^ ( k + 1 ) ) # 0 <-> ( ( A ^ k ) x. A ) # 0 ) )
23	19, 22	sylan 277	. . . . . . . 8 |- ( ( k e. NN /\ A e. CC ) -> ( ( A ^ ( k + 1 ) ) # 0 <-> ( ( A ^ k ) x. A ) # 0 ) )
24	23	adantr 270	. . . . . . 7 |- ( ( ( k e. NN /\ A e. CC ) /\ ( ( A ^ k ) # 0 <-> A # 0 ) ) -> ( ( A ^ ( k + 1 ) ) # 0 <-> ( ( A ^ k ) x. A ) # 0 ) )
25		simplr 496	. . . . . . . . 9 |- ( ( ( k e. NN /\ A e. CC ) /\ ( ( A ^ k ) # 0 <-> A # 0 ) ) -> A e. CC )
26	19	ad2antrr 471	. . . . . . . . 9 |- ( ( ( k e. NN /\ A e. CC ) /\ ( ( A ^ k ) # 0 <-> A # 0 ) ) -> k e. NN0 )
27		expcl 9494	. . . . . . . . 9 |- ( ( A e. CC /\ k e. NN0 ) -> ( A ^ k ) e. CC )
28	25, 26, 27	syl2anc 403	. . . . . . . 8 |- ( ( ( k e. NN /\ A e. CC ) /\ ( ( A ^ k ) # 0 <-> A # 0 ) ) -> ( A ^ k ) e. CC )
29	28, 25	mulap0bd 7747	. . . . . . 7 |- ( ( ( k e. NN /\ A e. CC ) /\ ( ( A ^ k ) # 0 <-> A # 0 ) ) -> ( ( ( A ^ k ) # 0 /\ A # 0 ) <-> ( ( A ^ k ) x. A ) # 0 ) )
30		anbi1 453	. . . . . . . 8 |- ( ( ( A ^ k ) # 0 <-> A # 0 ) -> ( ( ( A ^ k ) # 0 /\ A # 0 ) <-> ( A # 0 /\ A # 0 ) ) )
31	30	adantl 271	. . . . . . 7 |- ( ( ( k e. NN /\ A e. CC ) /\ ( ( A ^ k ) # 0 <-> A # 0 ) ) -> ( ( ( A ^ k ) # 0 /\ A # 0 ) <-> ( A # 0 /\ A # 0 ) ) )
32	24, 29, 31	3bitr2d 214	. . . . . 6 |- ( ( ( k e. NN /\ A e. CC ) /\ ( ( A ^ k ) # 0 <-> A # 0 ) ) -> ( ( A ^ ( k + 1 ) ) # 0 <-> ( A # 0 /\ A # 0 ) ) )
33		anidm 388	. . . . . 6 |- ( ( A # 0 /\ A # 0 ) <-> A # 0 )
34	32, 33	syl6bb 194	. . . . 5 |- ( ( ( k e. NN /\ A e. CC ) /\ ( ( A ^ k ) # 0 <-> A # 0 ) ) -> ( ( A ^ ( k + 1 ) ) # 0 <-> A # 0 ) )
35	34	exp31 356	. . . 4 |- ( k e. NN -> ( A e. CC -> ( ( ( A ^ k ) # 0 <-> A # 0 ) -> ( ( A ^ ( k + 1 ) ) # 0 <-> A # 0 ) ) ) )
36	35	a2d 26	. . 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 8055	. 2 |- ( N e. NN -> ( A e. CC -> ( ( A ^ N ) # 0 <-> A # 0 ) ) )
38	37	impcom 123	1 |- ( ( A e. CC /\ N e. NN ) -> ( ( A ^ N ) # 0 <-> A # 0 ) )

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103   = wceq 1284   e. wcel 1433   class class class wbr 3785  (class class class)co 5532   CCcc 6979   0cc0 6981   1c1 6982   + caddc 6984   x. cmul 6986   # cap 7681   NNcn 8039   NN0cn0 8288   ^cexp 9475

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-13 1444  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-coll 3893  ax-sep 3896  ax-nul 3904  ax-pow 3948  ax-pr 3964  ax-un 4188  ax-setind 4280  ax-iinf 4329  ax-cnex 7067  ax-resscn 7068  ax-1cn 7069  ax-1re 7070  ax-icn 7071  ax-addcl 7072  ax-addrcl 7073  ax-mulcl 7074  ax-mulrcl 7075  ax-addcom 7076  ax-mulcom 7077  ax-addass 7078  ax-mulass 7079  ax-distr 7080  ax-i2m1 7081  ax-0lt1 7082  ax-1rid 7083  ax-0id 7084  ax-rnegex 7085  ax-precex 7086  ax-cnre 7087  ax-pre-ltirr 7088  ax-pre-ltwlin 7089  ax-pre-lttrn 7090  ax-pre-apti 7091  ax-pre-ltadd 7092  ax-pre-mulgt0 7093  ax-pre-mulext 7094

This theorem depends on definitions:  df-bi 115  df-dc 776  df-3or 920  df-3an 921  df-tru 1287  df-fal 1290  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ne 2246  df-nel 2340  df-ral 2353  df-rex 2354  df-reu 2355  df-rmo 2356  df-rab 2357  df-v 2603  df-sbc 2816  df-csb 2909  df-dif 2975  df-un 2977  df-in 2979  df-ss 2986  df-nul 3252  df-if 3352  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-int 3637  df-iun 3680  df-br 3786  df-opab 3840  df-mpt 3841  df-tr 3876  df-id 4048  df-po 4051  df-iso 4052  df-iord 4121  df-on 4123  df-suc 4126  df-iom 4332  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-rn 4374  df-res 4375  df-ima 4376  df-iota 4887  df-fun 4924  df-fn 4925  df-f 4926  df-f1 4927  df-fo 4928  df-f1o 4929  df-fv 4930  df-riota 5488  df-ov 5535  df-oprab 5536  df-mpt2 5537  df-1st 5787  df-2nd 5788  df-recs 5943  df-frec 6001  df-pnf 7155  df-mnf 7156  df-xr 7157  df-ltxr 7158  df-le 7159  df-sub 7281  df-neg 7282  df-reap 7675  df-ap 7682  df-div 7761  df-inn 8040  df-n0 8289  df-z 8352  df-uz 8620  df-iseq 9432  df-iexp 9476

This theorem is referenced by:  expeq0  9507  abs00ap  9948