Theorem expap0 10931

Description: Positive integer exponentiation is apart from zero iff its base is apart from zero. That it is easier to prove this first, and then prove expeq0 10932 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." (Remark of [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 6058	. . . . . 6 |- ( j = 1 -> ( A ^ j ) = ( A ^ 1 ) )
2	1	breq1d 4119	. . . . 5 |- ( j = 1 -> ( ( A ^ j ) # 0 <-> ( A ^ 1 ) # 0 ) )
3	2	bibi1d 233	. . . 4 |- ( j = 1 -> ( ( ( A ^ j ) # 0 <-> A # 0 ) <-> ( ( A ^ 1 ) # 0 <-> A # 0 ) ) )
4	3	imbi2d 230	. . 3 |- ( j = 1 -> ( ( A e. CC -> ( ( A ^ j ) # 0 <-> A # 0 ) ) <-> ( A e. CC -> ( ( A ^ 1 ) # 0 <-> A # 0 ) ) ) )
5		oveq2 6058	. . . . . 6 |- ( j = k -> ( A ^ j ) = ( A ^ k ) )
6	5	breq1d 4119	. . . . 5 |- ( j = k -> ( ( A ^ j ) # 0 <-> ( A ^ k ) # 0 ) )
7	6	bibi1d 233	. . . 4 |- ( j = k -> ( ( ( A ^ j ) # 0 <-> A # 0 ) <-> ( ( A ^ k ) # 0 <-> A # 0 ) ) )
8	7	imbi2d 230	. . 3 |- ( j = k -> ( ( A e. CC -> ( ( A ^ j ) # 0 <-> A # 0 ) ) <-> ( A e. CC -> ( ( A ^ k ) # 0 <-> A # 0 ) ) ) )
9		oveq2 6058	. . . . . 6 |- ( j = ( k + 1 ) -> ( A ^ j ) = ( A ^ ( k + 1 ) ) )
10	9	breq1d 4119	. . . . 5 |- ( j = ( k + 1 ) -> ( ( A ^ j ) # 0 <-> ( A ^ ( k + 1 ) ) # 0 ) )
11	10	bibi1d 233	. . . 4 |- ( j = ( k + 1 ) -> ( ( ( A ^ j ) # 0 <-> A # 0 ) <-> ( ( A ^ ( k + 1 ) ) # 0 <-> A # 0 ) ) )
12	11	imbi2d 230	. . 3 |- ( j = ( k + 1 ) -> ( ( A e. CC -> ( ( A ^ j ) # 0 <-> A # 0 ) ) <-> ( A e. CC -> ( ( A ^ ( k + 1 ) ) # 0 <-> A # 0 ) ) ) )
13		oveq2 6058	. . . . . 6 |- ( j = N -> ( A ^ j ) = ( A ^ N ) )
14	13	breq1d 4119	. . . . 5 |- ( j = N -> ( ( A ^ j ) # 0 <-> ( A ^ N ) # 0 ) )
15	14	bibi1d 233	. . . 4 |- ( j = N -> ( ( ( A ^ j ) # 0 <-> A # 0 ) <-> ( ( A ^ N ) # 0 <-> A # 0 ) ) )
16	15	imbi2d 230	. . 3 |- ( j = N -> ( ( A e. CC -> ( ( A ^ j ) # 0 <-> A # 0 ) ) <-> ( A e. CC -> ( ( A ^ N ) # 0 <-> A # 0 ) ) ) )
17		exp1 10907	. . . 4 |- ( A e. CC -> ( A ^ 1 ) = A )
18	17	breq1d 4119	. . 3 |- ( A e. CC -> ( ( A ^ 1 ) # 0 <-> A # 0 ) )
19		nnnn0 9503	. . . . . . . . 9 |- ( k e. NN -> k e. NN0 )
20		expp1 10908	. . . . . . . . . . 11 |- ( ( A e. CC /\ k e. NN0 ) -> ( A ^ ( k + 1 ) ) = ( ( A ^ k ) x. A ) )
21	20	breq1d 4119	. . . . . . . . . 10 |- ( ( A e. CC /\ k e. NN0 ) -> ( ( A ^ ( k + 1 ) ) # 0 <-> ( ( A ^ k ) x. A ) # 0 ) )
22	21	ancoms 268	. . . . . . . . 9 |- ( ( k e. NN0 /\ A e. CC ) -> ( ( A ^ ( k + 1 ) ) # 0 <-> ( ( A ^ k ) x. A ) # 0 ) )
23	19, 22	sylan 283	. . . . . . . 8 |- ( ( k e. NN /\ A e. CC ) -> ( ( A ^ ( k + 1 ) ) # 0 <-> ( ( A ^ k ) x. A ) # 0 ) )
24	23	adantr 276	. . . . . . 7 |- ( ( ( k e. NN /\ A e. CC ) /\ ( ( A ^ k ) # 0 <-> A # 0 ) ) -> ( ( A ^ ( k + 1 ) ) # 0 <-> ( ( A ^ k ) x. A ) # 0 ) )
25		simplr 529	. . . . . . . . 9 |- ( ( ( k e. NN /\ A e. CC ) /\ ( ( A ^ k ) # 0 <-> A # 0 ) ) -> A e. CC )
26	19	ad2antrr 488	. . . . . . . . 9 |- ( ( ( k e. NN /\ A e. CC ) /\ ( ( A ^ k ) # 0 <-> A # 0 ) ) -> k e. NN0 )
27		expcl 10919	. . . . . . . . 9 |- ( ( A e. CC /\ k e. NN0 ) -> ( A ^ k ) e. CC )
28	25, 26, 27	syl2anc 411	. . . . . . . 8 |- ( ( ( k e. NN /\ A e. CC ) /\ ( ( A ^ k ) # 0 <-> A # 0 ) ) -> ( A ^ k ) e. CC )
29	28, 25	mulap0bd 8931	. . . . . . 7 |- ( ( ( k e. NN /\ A e. CC ) /\ ( ( A ^ k ) # 0 <-> A # 0 ) ) -> ( ( ( A ^ k ) # 0 /\ A # 0 ) <-> ( ( A ^ k ) x. A ) # 0 ) )
30		anbi1 466	. . . . . . . 8 |- ( ( ( A ^ k ) # 0 <-> A # 0 ) -> ( ( ( A ^ k ) # 0 /\ A # 0 ) <-> ( A # 0 /\ A # 0 ) ) )
31	30	adantl 277	. . . . . . 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 216	. . . . . 6 |- ( ( ( k e. NN /\ A e. CC ) /\ ( ( A ^ k ) # 0 <-> A # 0 ) ) -> ( ( A ^ ( k + 1 ) ) # 0 <-> ( A # 0 /\ A # 0 ) ) )
33		anidm 396	. . . . . 6 |- ( ( A # 0 /\ A # 0 ) <-> A # 0 )
34	32, 33	bitrdi 196	. . . . 5 |- ( ( ( k e. NN /\ A e. CC ) /\ ( ( A ^ k ) # 0 <-> A # 0 ) ) -> ( ( A ^ ( k + 1 ) ) # 0 <-> A # 0 ) )
35	34	exp31 364	. . . 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 9253	. 2 |- ( N e. NN -> ( A e. CC -> ( ( A ^ N ) # 0 <-> A # 0 ) ) )
38	37	impcom 125	1 |- ( ( A e. CC /\ N e. NN ) -> ( ( A ^ N ) # 0 <-> A # 0 ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1398   e. wcel 2203   class class class wbr 4109  (class class class)co 6050   CCcc 8125   0cc0 8127   1c1 8128   + caddc 8130   x. cmul 8132   # cap 8855   NNcn 9237   NN0cn0 9496   ^cexp 10900

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-coll 4225  ax-sep 4228  ax-nul 4236  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659  ax-iinf 4710  ax-cnex 8218  ax-resscn 8219  ax-1cn 8220  ax-1re 8221  ax-icn 8222  ax-addcl 8223  ax-addrcl 8224  ax-mulcl 8225  ax-mulrcl 8226  ax-addcom 8227  ax-mulcom 8228  ax-addass 8229  ax-mulass 8230  ax-distr 8231  ax-i2m1 8232  ax-0lt1 8233  ax-1rid 8234  ax-0id 8235  ax-rnegex 8236  ax-precex 8237  ax-cnre 8238  ax-pre-ltirr 8239  ax-pre-ltwlin 8240  ax-pre-lttrn 8241  ax-pre-apti 8242  ax-pre-ltadd 8243  ax-pre-mulgt0 8244  ax-pre-mulext 8245

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-nel 2508  df-ral 2525  df-rex 2526  df-reu 2527  df-rmo 2528  df-rab 2529  df-v 2815  df-sbc 3043  df-csb 3139  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-nul 3509  df-if 3621  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-int 3950  df-iun 3993  df-br 4110  df-opab 4172  df-mpt 4173  df-tr 4209  df-id 4414  df-po 4417  df-iso 4418  df-iord 4487  df-on 4489  df-ilim 4490  df-suc 4492  df-iom 4713  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-iota 5312  df-fun 5354  df-fn 5355  df-f 5356  df-f1 5357  df-fo 5358  df-f1o 5359  df-fv 5360  df-riota 6003  df-ov 6053  df-oprab 6054  df-mpo 6055  df-1st 6334  df-2nd 6335  df-recs 6536  df-frec 6622  df-pnf 8310  df-mnf 8311  df-xr 8312  df-ltxr 8313  df-le 8314  df-sub 8446  df-neg 8447  df-reap 8849  df-ap 8856  df-div 8947  df-inn 9238  df-n0 9497  df-z 9578  df-uz 9854  df-seqfrec 10810  df-exp 10901

This theorem is referenced by:  expeq0  10932  abs00ap  11747