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Mirrors > Home > MPE Home > Th. List > 0exp0e1 | Structured version Visualization version GIF version |
Description: 0↑0 = 1. This is our convention. It follows the convention used by Gleason; see Part of Definition 10-4.1 of [Gleason] p. 134. (Contributed by David A. Wheeler, 8-Dec-2018.) |
Ref | Expression |
---|---|
0exp0e1 | ⊢ (0↑0) = 1 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0cn 10479 | . 2 ⊢ 0 ∈ ℂ | |
2 | exp0 13283 | . 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 1522 ∈ wcel 2081 (class class class)co 7016 ℂcc 10381 0cc0 10383 1c1 10384 ↑cexp 13279 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1777 ax-4 1791 ax-5 1888 ax-6 1947 ax-7 1992 ax-8 2083 ax-9 2091 ax-10 2112 ax-11 2126 ax-12 2141 ax-13 2344 ax-ext 2769 ax-sep 5094 ax-nul 5101 ax-pr 5221 ax-1cn 10441 ax-icn 10442 ax-addcl 10443 ax-addrcl 10444 ax-mulcl 10445 ax-i2m1 10451 ax-rnegex 10454 ax-cnre 10456 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1081 df-3an 1082 df-tru 1525 df-ex 1762 df-nf 1766 df-sb 2043 df-mo 2576 df-eu 2612 df-clab 2776 df-cleq 2788 df-clel 2863 df-nfc 2935 df-ral 3110 df-rex 3111 df-rab 3114 df-v 3439 df-sbc 3707 df-dif 3862 df-un 3864 df-in 3866 df-ss 3874 df-nul 4212 df-if 4382 df-sn 4473 df-pr 4475 df-op 4479 df-uni 4746 df-br 4963 df-opab 5025 df-mpt 5042 df-id 5348 df-xp 5449 df-rel 5450 df-cnv 5451 df-co 5452 df-dm 5453 df-rn 5454 df-res 5455 df-ima 5456 df-pred 6023 df-iota 6189 df-fun 6227 df-fv 6233 df-ov 7019 df-oprab 7020 df-mpo 7021 df-wrecs 7798 df-recs 7860 df-rdg 7898 df-neg 10720 df-z 11830 df-seq 13220 df-exp 13280 |
This theorem is referenced by: faclbnd 13500 faclbnd3 13502 faclbnd4lem3 13505 facubnd 13510 ef0lem 15265 coefv0 24521 tayl0 24633 cxpexp 24932 musum 25450 logexprlim 25483 nn0expgcd 38706 etransclem14 42075 exple2lt6 43892 |
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