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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 (0exp0e1 10524) , 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 9264 | . . 3 ⊢ (𝐴 ∈ ℂ → 0 ∈ ℤ) | |
2 | 0le0 9007 | . . . . 5 ⊢ 0 ≤ 0 | |
3 | 2 | a1i 9 | . . . 4 ⊢ (𝐴 ∈ ℂ → 0 ≤ 0) |
4 | 3 | olcd 734 | . . 3 ⊢ (𝐴 ∈ ℂ → (𝐴 # 0 ∨ 0 ≤ 0)) |
5 | exp3val 10521 | . . 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 1339 | . 2 ⊢ (𝐴 ∈ ℂ → (𝐴↑0) = if(0 = 0, 1, if(0 < 0, (seq1( · , (ℕ × {𝐴}))‘0), (1 / (seq1( · , (ℕ × {𝐴}))‘-0))))) |
7 | eqid 2177 | . . 3 ⊢ 0 = 0 | |
8 | 7 | iftruei 3540 | . 2 ⊢ if(0 = 0, 1, if(0 < 0, (seq1( · , (ℕ × {𝐴}))‘0), (1 / (seq1( · , (ℕ × {𝐴}))‘-0)))) = 1 |
9 | 6, 8 | eqtrdi 2226 | 1 ⊢ (𝐴 ∈ ℂ → (𝐴↑0) = 1) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∨ wo 708 = wceq 1353 ∈ wcel 2148 ifcif 3534 {csn 3592 class class class wbr 4003 × cxp 4624 ‘cfv 5216 (class class class)co 5874 ℂcc 7808 0cc0 7810 1c1 7811 · cmul 7815 < clt 7991 ≤ cle 7992 -cneg 8128 # cap 8537 / cdiv 8628 ℕcn 8918 ℤcz 9252 seqcseq 10444 ↑cexp 10518 |
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 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-coll 4118 ax-sep 4121 ax-nul 4129 ax-pow 4174 ax-pr 4209 ax-un 4433 ax-setind 4536 ax-iinf 4587 ax-cnex 7901 ax-resscn 7902 ax-1cn 7903 ax-1re 7904 ax-icn 7905 ax-addcl 7906 ax-addrcl 7907 ax-mulcl 7908 ax-mulrcl 7909 ax-addcom 7910 ax-mulcom 7911 ax-addass 7912 ax-mulass 7913 ax-distr 7914 ax-i2m1 7915 ax-0lt1 7916 ax-1rid 7917 ax-0id 7918 ax-rnegex 7919 ax-precex 7920 ax-cnre 7921 ax-pre-ltirr 7922 ax-pre-ltwlin 7923 ax-pre-lttrn 7924 ax-pre-apti 7925 ax-pre-ltadd 7926 ax-pre-mulgt0 7927 ax-pre-mulext 7928 |
This theorem depends on definitions: df-bi 117 df-dc 835 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-reu 2462 df-rmo 2463 df-rab 2464 df-v 2739 df-sbc 2963 df-csb 3058 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-nul 3423 df-if 3535 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3810 df-int 3845 df-iun 3888 df-br 4004 df-opab 4065 df-mpt 4066 df-tr 4102 df-id 4293 df-po 4296 df-iso 4297 df-iord 4366 df-on 4368 df-ilim 4369 df-suc 4371 df-iom 4590 df-xp 4632 df-rel 4633 df-cnv 4634 df-co 4635 df-dm 4636 df-rn 4637 df-res 4638 df-ima 4639 df-iota 5178 df-fun 5218 df-fn 5219 df-f 5220 df-f1 5221 df-fo 5222 df-f1o 5223 df-fv 5224 df-riota 5830 df-ov 5877 df-oprab 5878 df-mpo 5879 df-1st 6140 df-2nd 6141 df-recs 6305 df-frec 6391 df-pnf 7993 df-mnf 7994 df-xr 7995 df-ltxr 7996 df-le 7997 df-sub 8129 df-neg 8130 df-reap 8531 df-ap 8538 df-div 8629 df-inn 8919 df-n0 9176 df-z 9253 df-uz 9528 df-seqfrec 10445 df-exp 10519 |
This theorem is referenced by: 0exp0e1 10524 expp1 10526 expnegap0 10527 expcllem 10530 mulexp 10558 expadd 10561 expmul 10564 leexp1a 10574 exple1 10575 bernneq 10640 modqexp 10646 exp0d 10647 cjexp 10901 resqrexlemcalc3 11024 absexp 11087 binom 11491 ege2le3 11678 eft0val 11700 demoivreALT 11780 cnfldexp 13407 expcncf 14028 dvexp 14111 dvexp2 14112 |
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