Theorem expap0 10786

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 10787 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 6008	. . . . . 6 |- ( j = 1 -> ( A ^ j ) = ( A ^ 1 ) )
2	1	breq1d 4092	. . . . 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 6008	. . . . . 6 |- ( j = k -> ( A ^ j ) = ( A ^ k ) )
6	5	breq1d 4092	. . . . 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 6008	. . . . . 6 |- ( j = ( k + 1 ) -> ( A ^ j ) = ( A ^ ( k + 1 ) ) )
10	9	breq1d 4092	. . . . 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 6008	. . . . . 6 |- ( j = N -> ( A ^ j ) = ( A ^ N ) )
14	13	breq1d 4092	. . . . 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 10762	. . . 4 |- ( A e. CC -> ( A ^ 1 ) = A )
18	17	breq1d 4092	. . 3 |- ( A e. CC -> ( ( A ^ 1 ) # 0 <-> A # 0 ) )
19		nnnn0 9372	. . . . . . . . 9 |- ( k e. NN -> k e. NN0 )
20		expp1 10763	. . . . . . . . . . 11 |- ( ( A e. CC /\ k e. NN0 ) -> ( A ^ ( k + 1 ) ) = ( ( A ^ k ) x. A ) )
21	20	breq1d 4092	. . . . . . . . . 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 528	. . . . . . . . 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 10774	. . . . . . . . 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 8800	. . . . . . 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 9122	. 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 1395   e. wcel 2200   class class class wbr 4082  (class class class)co 6000   CCcc 7993   0cc0 7995   1c1 7996   + caddc 7998   x. cmul 8000   # cap 8724   NNcn 9106   NN0cn0 9365   ^cexp 10755

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4198  ax-sep 4201  ax-nul 4209  ax-pow 4257  ax-pr 4292  ax-un 4523  ax-setind 4628  ax-iinf 4679  ax-cnex 8086  ax-resscn 8087  ax-1cn 8088  ax-1re 8089  ax-icn 8090  ax-addcl 8091  ax-addrcl 8092  ax-mulcl 8093  ax-mulrcl 8094  ax-addcom 8095  ax-mulcom 8096  ax-addass 8097  ax-mulass 8098  ax-distr 8099  ax-i2m1 8100  ax-0lt1 8101  ax-1rid 8102  ax-0id 8103  ax-rnegex 8104  ax-precex 8105  ax-cnre 8106  ax-pre-ltirr 8107  ax-pre-ltwlin 8108  ax-pre-lttrn 8109  ax-pre-apti 8110  ax-pre-ltadd 8111  ax-pre-mulgt0 8112  ax-pre-mulext 8113

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-if 3603  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-int 3923  df-iun 3966  df-br 4083  df-opab 4145  df-mpt 4146  df-tr 4182  df-id 4383  df-po 4386  df-iso 4387  df-iord 4456  df-on 4458  df-ilim 4459  df-suc 4461  df-iom 4682  df-xp 4724  df-rel 4725  df-cnv 4726  df-co 4727  df-dm 4728  df-rn 4729  df-res 4730  df-ima 4731  df-iota 5277  df-fun 5319  df-fn 5320  df-f 5321  df-f1 5322  df-fo 5323  df-f1o 5324  df-fv 5325  df-riota 5953  df-ov 6003  df-oprab 6004  df-mpo 6005  df-1st 6284  df-2nd 6285  df-recs 6449  df-frec 6535  df-pnf 8179  df-mnf 8180  df-xr 8181  df-ltxr 8182  df-le 8183  df-sub 8315  df-neg 8316  df-reap 8718  df-ap 8725  df-div 8816  df-inn 9107  df-n0 9366  df-z 9443  df-uz 9719  df-seqfrec 10665  df-exp 10756

This theorem is referenced by:  expeq0  10787  abs00ap  11568