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Mirrors > Home > ILE Home > Th. List > expap0 | GIF version |
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 8940 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.) |
Ref | Expression |
---|---|
expap0 | ⊢ ((A ∈ ℂ ∧ 𝑁 ∈ ℕ) → ((A↑𝑁) # 0 ↔ A # 0)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oveq2 5463 | . . . . . 6 ⊢ (𝑗 = 1 → (A↑𝑗) = (A↑1)) | |
2 | 1 | breq1d 3765 | . . . . 5 ⊢ (𝑗 = 1 → ((A↑𝑗) # 0 ↔ (A↑1) # 0)) |
3 | 2 | bibi1d 222 | . . . 4 ⊢ (𝑗 = 1 → (((A↑𝑗) # 0 ↔ A # 0) ↔ ((A↑1) # 0 ↔ A # 0))) |
4 | 3 | imbi2d 219 | . . 3 ⊢ (𝑗 = 1 → ((A ∈ ℂ → ((A↑𝑗) # 0 ↔ A # 0)) ↔ (A ∈ ℂ → ((A↑1) # 0 ↔ A # 0)))) |
5 | oveq2 5463 | . . . . . 6 ⊢ (𝑗 = 𝑘 → (A↑𝑗) = (A↑𝑘)) | |
6 | 5 | breq1d 3765 | . . . . 5 ⊢ (𝑗 = 𝑘 → ((A↑𝑗) # 0 ↔ (A↑𝑘) # 0)) |
7 | 6 | bibi1d 222 | . . . 4 ⊢ (𝑗 = 𝑘 → (((A↑𝑗) # 0 ↔ A # 0) ↔ ((A↑𝑘) # 0 ↔ A # 0))) |
8 | 7 | imbi2d 219 | . . 3 ⊢ (𝑗 = 𝑘 → ((A ∈ ℂ → ((A↑𝑗) # 0 ↔ A # 0)) ↔ (A ∈ ℂ → ((A↑𝑘) # 0 ↔ A # 0)))) |
9 | oveq2 5463 | . . . . . 6 ⊢ (𝑗 = (𝑘 + 1) → (A↑𝑗) = (A↑(𝑘 + 1))) | |
10 | 9 | breq1d 3765 | . . . . 5 ⊢ (𝑗 = (𝑘 + 1) → ((A↑𝑗) # 0 ↔ (A↑(𝑘 + 1)) # 0)) |
11 | 10 | bibi1d 222 | . . . 4 ⊢ (𝑗 = (𝑘 + 1) → (((A↑𝑗) # 0 ↔ A # 0) ↔ ((A↑(𝑘 + 1)) # 0 ↔ A # 0))) |
12 | 11 | imbi2d 219 | . . 3 ⊢ (𝑗 = (𝑘 + 1) → ((A ∈ ℂ → ((A↑𝑗) # 0 ↔ A # 0)) ↔ (A ∈ ℂ → ((A↑(𝑘 + 1)) # 0 ↔ A # 0)))) |
13 | oveq2 5463 | . . . . . 6 ⊢ (𝑗 = 𝑁 → (A↑𝑗) = (A↑𝑁)) | |
14 | 13 | breq1d 3765 | . . . . 5 ⊢ (𝑗 = 𝑁 → ((A↑𝑗) # 0 ↔ (A↑𝑁) # 0)) |
15 | 14 | bibi1d 222 | . . . 4 ⊢ (𝑗 = 𝑁 → (((A↑𝑗) # 0 ↔ A # 0) ↔ ((A↑𝑁) # 0 ↔ A # 0))) |
16 | 15 | imbi2d 219 | . . 3 ⊢ (𝑗 = 𝑁 → ((A ∈ ℂ → ((A↑𝑗) # 0 ↔ A # 0)) ↔ (A ∈ ℂ → ((A↑𝑁) # 0 ↔ A # 0)))) |
17 | exp1 8915 | . . . 4 ⊢ (A ∈ ℂ → (A↑1) = A) | |
18 | 17 | breq1d 3765 | . . 3 ⊢ (A ∈ ℂ → ((A↑1) # 0 ↔ A # 0)) |
19 | nnnn0 7964 | . . . . . . . . 9 ⊢ (𝑘 ∈ ℕ → 𝑘 ∈ ℕ0) | |
20 | expp1 8916 | . . . . . . . . . . 11 ⊢ ((A ∈ ℂ ∧ 𝑘 ∈ ℕ0) → (A↑(𝑘 + 1)) = ((A↑𝑘) · A)) | |
21 | 20 | breq1d 3765 | . . . . . . . . . 10 ⊢ ((A ∈ ℂ ∧ 𝑘 ∈ ℕ0) → ((A↑(𝑘 + 1)) # 0 ↔ ((A↑𝑘) · A) # 0)) |
22 | 21 | ancoms 255 | . . . . . . . . 9 ⊢ ((𝑘 ∈ ℕ0 ∧ A ∈ ℂ) → ((A↑(𝑘 + 1)) # 0 ↔ ((A↑𝑘) · A) # 0)) |
23 | 19, 22 | sylan 267 | . . . . . . . 8 ⊢ ((𝑘 ∈ ℕ ∧ A ∈ ℂ) → ((A↑(𝑘 + 1)) # 0 ↔ ((A↑𝑘) · A) # 0)) |
24 | 23 | adantr 261 | . . . . . . 7 ⊢ (((𝑘 ∈ ℕ ∧ A ∈ ℂ) ∧ ((A↑𝑘) # 0 ↔ A # 0)) → ((A↑(𝑘 + 1)) # 0 ↔ ((A↑𝑘) · A) # 0)) |
25 | simplr 482 | . . . . . . . . 9 ⊢ (((𝑘 ∈ ℕ ∧ A ∈ ℂ) ∧ ((A↑𝑘) # 0 ↔ A # 0)) → A ∈ ℂ) | |
26 | 19 | ad2antrr 457 | . . . . . . . . 9 ⊢ (((𝑘 ∈ ℕ ∧ A ∈ ℂ) ∧ ((A↑𝑘) # 0 ↔ A # 0)) → 𝑘 ∈ ℕ0) |
27 | expcl 8927 | . . . . . . . . 9 ⊢ ((A ∈ ℂ ∧ 𝑘 ∈ ℕ0) → (A↑𝑘) ∈ ℂ) | |
28 | 25, 26, 27 | syl2anc 391 | . . . . . . . 8 ⊢ (((𝑘 ∈ ℕ ∧ A ∈ ℂ) ∧ ((A↑𝑘) # 0 ↔ A # 0)) → (A↑𝑘) ∈ ℂ) |
29 | 28, 25 | mulap0bd 7420 | . . . . . . 7 ⊢ (((𝑘 ∈ ℕ ∧ A ∈ ℂ) ∧ ((A↑𝑘) # 0 ↔ A # 0)) → (((A↑𝑘) # 0 ∧ A # 0) ↔ ((A↑𝑘) · A) # 0)) |
30 | anbi1 439 | . . . . . . . 8 ⊢ (((A↑𝑘) # 0 ↔ A # 0) → (((A↑𝑘) # 0 ∧ A # 0) ↔ (A # 0 ∧ A # 0))) | |
31 | 30 | adantl 262 | . . . . . . 7 ⊢ (((𝑘 ∈ ℕ ∧ A ∈ ℂ) ∧ ((A↑𝑘) # 0 ↔ A # 0)) → (((A↑𝑘) # 0 ∧ A # 0) ↔ (A # 0 ∧ A # 0))) |
32 | 24, 29, 31 | 3bitr2d 205 | . . . . . 6 ⊢ (((𝑘 ∈ ℕ ∧ A ∈ ℂ) ∧ ((A↑𝑘) # 0 ↔ A # 0)) → ((A↑(𝑘 + 1)) # 0 ↔ (A # 0 ∧ A # 0))) |
33 | anidm 376 | . . . . . 6 ⊢ ((A # 0 ∧ A # 0) ↔ A # 0) | |
34 | 32, 33 | syl6bb 185 | . . . . 5 ⊢ (((𝑘 ∈ ℕ ∧ A ∈ ℂ) ∧ ((A↑𝑘) # 0 ↔ A # 0)) → ((A↑(𝑘 + 1)) # 0 ↔ A # 0)) |
35 | 34 | exp31 346 | . . . 4 ⊢ (𝑘 ∈ ℕ → (A ∈ ℂ → (((A↑𝑘) # 0 ↔ A # 0) → ((A↑(𝑘 + 1)) # 0 ↔ A # 0)))) |
36 | 35 | a2d 23 | . . 3 ⊢ (𝑘 ∈ ℕ → ((A ∈ ℂ → ((A↑𝑘) # 0 ↔ A # 0)) → (A ∈ ℂ → ((A↑(𝑘 + 1)) # 0 ↔ A # 0)))) |
37 | 4, 8, 12, 16, 18, 36 | nnind 7711 | . 2 ⊢ (𝑁 ∈ ℕ → (A ∈ ℂ → ((A↑𝑁) # 0 ↔ A # 0))) |
38 | 37 | impcom 116 | 1 ⊢ ((A ∈ ℂ ∧ 𝑁 ∈ ℕ) → ((A↑𝑁) # 0 ↔ A # 0)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 ↔ wb 98 = wceq 1242 ∈ wcel 1390 class class class wbr 3755 (class class class)co 5455 ℂcc 6709 0cc0 6711 1c1 6712 + caddc 6714 · cmul 6716 # cap 7365 ℕcn 7695 ℕ0cn0 7957 ↑cexp 8908 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-13 1401 ax-14 1402 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 ax-coll 3863 ax-sep 3866 ax-nul 3874 ax-pow 3918 ax-pr 3935 ax-un 4136 ax-setind 4220 ax-iinf 4254 ax-cnex 6774 ax-resscn 6775 ax-1cn 6776 ax-1re 6777 ax-icn 6778 ax-addcl 6779 ax-addrcl 6780 ax-mulcl 6781 ax-mulrcl 6782 ax-addcom 6783 ax-mulcom 6784 ax-addass 6785 ax-mulass 6786 ax-distr 6787 ax-i2m1 6788 ax-1rid 6790 ax-0id 6791 ax-rnegex 6792 ax-precex 6793 ax-cnre 6794 ax-pre-ltirr 6795 ax-pre-ltwlin 6796 ax-pre-lttrn 6797 ax-pre-apti 6798 ax-pre-ltadd 6799 ax-pre-mulgt0 6800 ax-pre-mulext 6801 |
This theorem depends on definitions: df-bi 110 df-dc 742 df-3or 885 df-3an 886 df-tru 1245 df-fal 1248 df-nf 1347 df-sb 1643 df-eu 1900 df-mo 1901 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-ne 2203 df-nel 2204 df-ral 2305 df-rex 2306 df-reu 2307 df-rmo 2308 df-rab 2309 df-v 2553 df-sbc 2759 df-csb 2847 df-dif 2914 df-un 2916 df-in 2918 df-ss 2925 df-nul 3219 df-if 3326 df-pw 3353 df-sn 3373 df-pr 3374 df-op 3376 df-uni 3572 df-int 3607 df-iun 3650 df-br 3756 df-opab 3810 df-mpt 3811 df-tr 3846 df-eprel 4017 df-id 4021 df-po 4024 df-iso 4025 df-iord 4069 df-on 4071 df-suc 4074 df-iom 4257 df-xp 4294 df-rel 4295 df-cnv 4296 df-co 4297 df-dm 4298 df-rn 4299 df-res 4300 df-ima 4301 df-iota 4810 df-fun 4847 df-fn 4848 df-f 4849 df-f1 4850 df-fo 4851 df-f1o 4852 df-fv 4853 df-riota 5411 df-ov 5458 df-oprab 5459 df-mpt2 5460 df-1st 5709 df-2nd 5710 df-recs 5861 df-irdg 5897 df-frec 5918 df-1o 5940 df-2o 5941 df-oadd 5944 df-omul 5945 df-er 6042 df-ec 6044 df-qs 6048 df-ni 6288 df-pli 6289 df-mi 6290 df-lti 6291 df-plpq 6328 df-mpq 6329 df-enq 6331 df-nqqs 6332 df-plqqs 6333 df-mqqs 6334 df-1nqqs 6335 df-rq 6336 df-ltnqqs 6337 df-enq0 6407 df-nq0 6408 df-0nq0 6409 df-plq0 6410 df-mq0 6411 df-inp 6449 df-i1p 6450 df-iplp 6451 df-iltp 6453 df-enr 6654 df-nr 6655 df-ltr 6658 df-0r 6659 df-1r 6660 df-0 6718 df-1 6719 df-r 6721 df-lt 6724 df-pnf 6859 df-mnf 6860 df-xr 6861 df-ltxr 6862 df-le 6863 df-sub 6981 df-neg 6982 df-reap 7359 df-ap 7366 df-div 7434 df-inn 7696 df-n0 7958 df-z 8022 df-uz 8250 df-iseq 8893 df-iexp 8909 |
This theorem is referenced by: expeq0 8940 |
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