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Mirrors > Home > ILE Home > Th. List > exp0 | GIF version |
Description: Value of a complex number raised to the 0th power. Note that under our definition, 0↑0 = 1, following the convention used by Gleason. Part of Definition 10-4.1 of [Gleason] p. 134. (Contributed by NM, 20-May-2004.) (Revised by Mario Carneiro, 4-Jun-2014.) |
Ref | Expression |
---|---|
exp0 | ⊢ (𝐴 ∈ ℂ → (𝐴↑0) = 1) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0zd 9024 | . . 3 ⊢ (𝐴 ∈ ℂ → 0 ∈ ℤ) | |
2 | 0le0 8773 | . . . . 5 ⊢ 0 ≤ 0 | |
3 | 2 | a1i 9 | . . . 4 ⊢ (𝐴 ∈ ℂ → 0 ≤ 0) |
4 | 3 | olcd 708 | . . 3 ⊢ (𝐴 ∈ ℂ → (𝐴 # 0 ∨ 0 ≤ 0)) |
5 | exp3val 10250 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 0 ∈ ℤ ∧ (𝐴 # 0 ∨ 0 ≤ 0)) → (𝐴↑0) = if(0 = 0, 1, if(0 < 0, (seq1( · , (ℕ × {𝐴}))‘0), (1 / (seq1( · , (ℕ × {𝐴}))‘-0))))) | |
6 | 1, 4, 5 | mpd3an23 1302 | . 2 ⊢ (𝐴 ∈ ℂ → (𝐴↑0) = if(0 = 0, 1, if(0 < 0, (seq1( · , (ℕ × {𝐴}))‘0), (1 / (seq1( · , (ℕ × {𝐴}))‘-0))))) |
7 | eqid 2117 | . . 3 ⊢ 0 = 0 | |
8 | 7 | iftruei 3450 | . 2 ⊢ if(0 = 0, 1, if(0 < 0, (seq1( · , (ℕ × {𝐴}))‘0), (1 / (seq1( · , (ℕ × {𝐴}))‘-0)))) = 1 |
9 | 6, 8 | syl6eq 2166 | 1 ⊢ (𝐴 ∈ ℂ → (𝐴↑0) = 1) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∨ wo 682 = wceq 1316 ∈ wcel 1465 ifcif 3444 {csn 3497 class class class wbr 3899 × cxp 4507 ‘cfv 5093 (class class class)co 5742 ℂcc 7586 0cc0 7588 1c1 7589 · cmul 7593 < clt 7768 ≤ cle 7769 -cneg 7902 # cap 8310 / cdiv 8399 ℕcn 8684 ℤcz 9012 seqcseq 10173 ↑cexp 10247 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 588 ax-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-13 1476 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-coll 4013 ax-sep 4016 ax-nul 4024 ax-pow 4068 ax-pr 4101 ax-un 4325 ax-setind 4422 ax-iinf 4472 ax-cnex 7679 ax-resscn 7680 ax-1cn 7681 ax-1re 7682 ax-icn 7683 ax-addcl 7684 ax-addrcl 7685 ax-mulcl 7686 ax-mulrcl 7687 ax-addcom 7688 ax-mulcom 7689 ax-addass 7690 ax-mulass 7691 ax-distr 7692 ax-i2m1 7693 ax-0lt1 7694 ax-1rid 7695 ax-0id 7696 ax-rnegex 7697 ax-precex 7698 ax-cnre 7699 ax-pre-ltirr 7700 ax-pre-ltwlin 7701 ax-pre-lttrn 7702 ax-pre-apti 7703 ax-pre-ltadd 7704 ax-pre-mulgt0 7705 ax-pre-mulext 7706 |
This theorem depends on definitions: df-bi 116 df-dc 805 df-3or 948 df-3an 949 df-tru 1319 df-fal 1322 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ne 2286 df-nel 2381 df-ral 2398 df-rex 2399 df-reu 2400 df-rmo 2401 df-rab 2402 df-v 2662 df-sbc 2883 df-csb 2976 df-dif 3043 df-un 3045 df-in 3047 df-ss 3054 df-nul 3334 df-if 3445 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-int 3742 df-iun 3785 df-br 3900 df-opab 3960 df-mpt 3961 df-tr 3997 df-id 4185 df-po 4188 df-iso 4189 df-iord 4258 df-on 4260 df-ilim 4261 df-suc 4263 df-iom 4475 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-rn 4520 df-res 4521 df-ima 4522 df-iota 5058 df-fun 5095 df-fn 5096 df-f 5097 df-f1 5098 df-fo 5099 df-f1o 5100 df-fv 5101 df-riota 5698 df-ov 5745 df-oprab 5746 df-mpo 5747 df-1st 6006 df-2nd 6007 df-recs 6170 df-frec 6256 df-pnf 7770 df-mnf 7771 df-xr 7772 df-ltxr 7773 df-le 7774 df-sub 7903 df-neg 7904 df-reap 8304 df-ap 8311 df-div 8400 df-inn 8685 df-n0 8936 df-z 9013 df-uz 9283 df-seqfrec 10174 df-exp 10248 |
This theorem is referenced by: 0exp0e1 10253 expp1 10255 expnegap0 10256 expcllem 10259 mulexp 10287 expadd 10290 expmul 10293 leexp1a 10303 exple1 10304 bernneq 10367 exp0d 10373 cjexp 10620 resqrexlemcalc3 10743 absexp 10806 binom 11208 ege2le3 11291 eft0val 11313 demoivreALT 11394 expcncf 12672 dvexp 12755 dvexp2 12756 |
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