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Mirrors > Home > MPE Home > Th. List > 0exp0e1 | Structured version Visualization version GIF version |
Description: 0↑0 = 1 (common case). 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 9911 | . 2 ⊢ 0 ∈ ℂ | |
2 | exp0 12726 | . 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 1475 ∈ wcel 1977 (class class class)co 6549 ℂcc 9813 0cc0 9815 1c1 9816 ↑cexp 12722 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-i2m1 9883 ax-1ne0 9884 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-iota 5768 df-fun 5806 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-neg 10148 df-z 11255 df-seq 12664 df-exp 12723 |
This theorem is referenced by: faclbnd 12939 faclbnd3 12941 faclbnd4lem3 12944 facubnd 12949 ef0lem 14648 coefv0 23808 tayl0 23920 cxpexp 24214 musum 24717 logexprlim 24750 etransclem14 39141 exple2lt6 41939 |
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