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Mirrors > Home > MPE Home > Th. List > exp0 | Structured version Visualization version GIF version |
Description: Value of a complex number raised to the zeroth power. Under our definition, 0↑0 = 1 (0exp0e1 14029), following standard convention, for instance 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 | 0z 12566 | . . 3 ⊢ 0 ∈ ℤ | |
2 | expval 14026 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 0 ∈ ℤ) → (𝐴↑0) = if(0 = 0, 1, if(0 < 0, (seq1( · , (ℕ × {𝐴}))‘0), (1 / (seq1( · , (ℕ × {𝐴}))‘-0))))) | |
3 | 1, 2 | mpan2 688 | . 2 ⊢ (𝐴 ∈ ℂ → (𝐴↑0) = if(0 = 0, 1, if(0 < 0, (seq1( · , (ℕ × {𝐴}))‘0), (1 / (seq1( · , (ℕ × {𝐴}))‘-0))))) |
4 | eqid 2724 | . . 3 ⊢ 0 = 0 | |
5 | 4 | iftruei 4527 | . 2 ⊢ if(0 = 0, 1, if(0 < 0, (seq1( · , (ℕ × {𝐴}))‘0), (1 / (seq1( · , (ℕ × {𝐴}))‘-0)))) = 1 |
6 | 3, 5 | eqtrdi 2780 | 1 ⊢ (𝐴 ∈ ℂ → (𝐴↑0) = 1) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 ifcif 4520 {csn 4620 class class class wbr 5138 × cxp 5664 ‘cfv 6533 (class class class)co 7401 ℂcc 11104 0cc0 11106 1c1 11107 · cmul 11111 < clt 11245 -cneg 11442 / cdiv 11868 ℕcn 12209 ℤcz 12555 seqcseq 13963 ↑cexp 14024 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-sep 5289 ax-nul 5296 ax-pr 5417 ax-1cn 11164 ax-addrcl 11167 ax-rnegex 11177 ax-cnre 11179 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-ral 3054 df-rex 3063 df-rab 3425 df-v 3468 df-sbc 3770 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-nul 4315 df-if 4521 df-sn 4621 df-pr 4623 df-op 4627 df-uni 4900 df-br 5139 df-opab 5201 df-mpt 5222 df-id 5564 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-pred 6290 df-iota 6485 df-fun 6535 df-fv 6541 df-ov 7404 df-oprab 7405 df-mpo 7406 df-frecs 8261 df-wrecs 8292 df-recs 8366 df-rdg 8405 df-neg 11444 df-z 12556 df-seq 13964 df-exp 14025 |
This theorem is referenced by: 0exp0e1 14029 expp1 14031 expneg 14032 expcllem 14035 mulexp 14064 expadd 14067 expmul 14070 exp0d 14102 leexp1a 14137 exple1 14138 bernneq 14189 modexp 14198 faclbnd4lem1 14250 faclbnd4lem3 14252 faclbnd4lem4 14253 cjexp 15094 absexp 15248 binom 15773 incexclem 15779 incexc 15780 climcndslem1 15792 pwdif 15811 fprodconst 15919 fallfac0 15969 bpoly0 15991 ege2le3 16030 eft0val 16052 demoivreALT 16141 pwp1fsum 16331 bits0 16366 0bits 16377 bitsinv1 16380 sadcadd 16396 smumullem 16430 numexp0 17008 psgnunilem4 19407 psgn0fv0 19421 psgnsn 19430 psgnprfval1 19432 cnfldexp 21262 expmhm 21298 expcn 24712 expcnOLD 24714 iblcnlem1 25639 itgcnlem 25641 dvexp 25807 dvexp2 25808 plyconst 26060 0dgr 26099 0dgrb 26100 aaliou3lem2 26197 cxp0 26520 1cubr 26690 log2ublem3 26796 basellem2 26930 basellem5 26933 lgsquad2lem2 27234 0dp2dp 32542 oddpwdc 33842 breprexp 34134 subfacval2 34667 fwddifn0 35631 stoweidlem19 45220 fmtno0 46693 bits0ALTV 46832 0dig2nn0e 47486 0dig2nn0o 47487 nn0sumshdiglemA 47493 nn0sumshdiglemB 47494 nn0sumshdiglem1 47495 nn0sumshdiglem2 47496 |
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