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Mirrors > Home > MPE Home > Th. List > 0exp0e1 | Structured version Visualization version GIF version |
Description: The zeroth power of zero equals one. See comment of exp0 14103. (Contributed by David A. Wheeler, 8-Dec-2018.) |
Ref | Expression |
---|---|
0exp0e1 | ⊢ (0↑0) = 1 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0cn 11251 | . 2 ⊢ 0 ∈ ℂ | |
2 | exp0 14103 | . 2 ⊢ (0 ∈ ℂ → (0↑0) = 1) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ (0↑0) = 1 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ∈ wcel 2106 (class class class)co 7431 ℂcc 11151 0cc0 11153 1c1 11154 ↑cexp 14099 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-i2m1 11221 ax-rnegex 11224 ax-cnre 11226 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-sbc 3792 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-iota 6516 df-fun 6565 df-fv 6571 df-ov 7434 df-oprab 7435 df-mpo 7436 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-neg 11493 df-z 12612 df-seq 14040 df-exp 14100 |
This theorem is referenced by: faclbnd 14326 faclbnd3 14328 faclbnd4lem3 14331 facubnd 14336 ef0lem 16111 nn0expgcd 16598 coefv0 26302 tayl0 26418 cxpexp 26725 musum 27249 logexprlim 27284 etransclem14 46204 exple2lt6 48209 |
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