Theorem expap0 10821

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 10822 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 6021	. . . . . 6 |- ( j = 1 -> ( A ^ j ) = ( A ^ 1 ) )
2	1	breq1d 4096	. . . . 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 6021	. . . . . 6 |- ( j = k -> ( A ^ j ) = ( A ^ k ) )
6	5	breq1d 4096	. . . . 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 6021	. . . . . 6 |- ( j = ( k + 1 ) -> ( A ^ j ) = ( A ^ ( k + 1 ) ) )
10	9	breq1d 4096	. . . . 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 6021	. . . . . 6 |- ( j = N -> ( A ^ j ) = ( A ^ N ) )
14	13	breq1d 4096	. . . . 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 10797	. . . 4 |- ( A e. CC -> ( A ^ 1 ) = A )
18	17	breq1d 4096	. . 3 |- ( A e. CC -> ( ( A ^ 1 ) # 0 <-> A # 0 ) )
19		nnnn0 9399	. . . . . . . . 9 |- ( k e. NN -> k e. NN0 )
20		expp1 10798	. . . . . . . . . . 11 |- ( ( A e. CC /\ k e. NN0 ) -> ( A ^ ( k + 1 ) ) = ( ( A ^ k ) x. A ) )
21	20	breq1d 4096	. . . . . . . . . 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 10809	. . . . . . . . 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 8827	. . . . . . 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 9149	. 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 4086  (class class class)co 6013   CCcc 8020   0cc0 8022   1c1 8023   + caddc 8025   x. cmul 8027   # cap 8751   NNcn 9133   NN0cn0 9392   ^cexp 10790

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 4202  ax-sep 4205  ax-nul 4213  ax-pow 4262  ax-pr 4297  ax-un 4528  ax-setind 4633  ax-iinf 4684  ax-cnex 8113  ax-resscn 8114  ax-1cn 8115  ax-1re 8116  ax-icn 8117  ax-addcl 8118  ax-addrcl 8119  ax-mulcl 8120  ax-mulrcl 8121  ax-addcom 8122  ax-mulcom 8123  ax-addass 8124  ax-mulass 8125  ax-distr 8126  ax-i2m1 8127  ax-0lt1 8128  ax-1rid 8129  ax-0id 8130  ax-rnegex 8131  ax-precex 8132  ax-cnre 8133  ax-pre-ltirr 8134  ax-pre-ltwlin 8135  ax-pre-lttrn 8136  ax-pre-apti 8137  ax-pre-ltadd 8138  ax-pre-mulgt0 8139  ax-pre-mulext 8140

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 2802  df-sbc 3030  df-csb 3126  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-nul 3493  df-if 3604  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-int 3927  df-iun 3970  df-br 4087  df-opab 4149  df-mpt 4150  df-tr 4186  df-id 4388  df-po 4391  df-iso 4392  df-iord 4461  df-on 4463  df-ilim 4464  df-suc 4466  df-iom 4687  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-ima 4736  df-iota 5284  df-fun 5326  df-fn 5327  df-f 5328  df-f1 5329  df-fo 5330  df-f1o 5331  df-fv 5332  df-riota 5966  df-ov 6016  df-oprab 6017  df-mpo 6018  df-1st 6298  df-2nd 6299  df-recs 6466  df-frec 6552  df-pnf 8206  df-mnf 8207  df-xr 8208  df-ltxr 8209  df-le 8210  df-sub 8342  df-neg 8343  df-reap 8745  df-ap 8752  df-div 8843  df-inn 9134  df-n0 9393  df-z 9470  df-uz 9746  df-seqfrec 10700  df-exp 10791

This theorem is referenced by:  expeq0  10822  abs00ap  11613