Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 13636. (Contributed by David A. Wheeler, 8-Dec-2018.) |
Ref | Expression |
---|---|
0exp0e1 | ⊢ (0↑0) = 1 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0cn 10822 | . 2 ⊢ 0 ∈ ℂ | |
2 | exp0 13636 | . 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 1543 ∈ wcel 2110 (class class class)co 7210 ℂcc 10724 0cc0 10726 1c1 10727 ↑cexp 13632 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-sep 5189 ax-nul 5196 ax-pr 5319 ax-1cn 10784 ax-icn 10785 ax-addcl 10786 ax-addrcl 10787 ax-mulcl 10788 ax-i2m1 10794 ax-rnegex 10797 ax-cnre 10799 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ral 3063 df-rex 3064 df-rab 3067 df-v 3407 df-sbc 3692 df-dif 3866 df-un 3868 df-in 3870 df-ss 3880 df-nul 4235 df-if 4437 df-sn 4539 df-pr 4541 df-op 4545 df-uni 4817 df-br 5051 df-opab 5113 df-mpt 5133 df-id 5452 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6157 df-iota 6335 df-fun 6379 df-fv 6385 df-ov 7213 df-oprab 7214 df-mpo 7215 df-wrecs 8044 df-recs 8105 df-rdg 8143 df-neg 11062 df-z 12174 df-seq 13572 df-exp 13633 |
This theorem is referenced by: faclbnd 13853 faclbnd3 13855 faclbnd4lem3 13858 facubnd 13863 ef0lem 15637 coefv0 25139 tayl0 25251 cxpexp 25553 musum 26070 logexprlim 26103 nn0expgcd 40041 etransclem14 43462 exple2lt6 45371 |
Copyright terms: Public domain | W3C validator |