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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 10636 | . 2 ⊢ 0 ∈ ℂ | |
2 | exp0 13436 | . 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 1536 ∈ wcel 2113 (class class class)co 7159 ℂcc 10538 0cc0 10540 1c1 10541 ↑cexp 13432 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-sep 5206 ax-nul 5213 ax-pr 5333 ax-1cn 10598 ax-icn 10599 ax-addcl 10600 ax-addrcl 10601 ax-mulcl 10602 ax-i2m1 10608 ax-rnegex 10611 ax-cnre 10613 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ral 3146 df-rex 3147 df-rab 3150 df-v 3499 df-sbc 3776 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-nul 4295 df-if 4471 df-sn 4571 df-pr 4573 df-op 4577 df-uni 4842 df-br 5070 df-opab 5132 df-mpt 5150 df-id 5463 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-pred 6151 df-iota 6317 df-fun 6360 df-fv 6366 df-ov 7162 df-oprab 7163 df-mpo 7164 df-wrecs 7950 df-recs 8011 df-rdg 8049 df-neg 10876 df-z 11985 df-seq 13373 df-exp 13433 |
This theorem is referenced by: faclbnd 13653 faclbnd3 13655 faclbnd4lem3 13658 facubnd 13663 ef0lem 15435 coefv0 24841 tayl0 24953 cxpexp 25254 musum 25771 logexprlim 25804 nn0expgcd 39190 etransclem14 42540 exple2lt6 44419 |
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