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| Mirrors > Home > MPE Home > Th. List > numexp0 | Structured version Visualization version GIF version | ||
| Description: Calculate an integer power. (Contributed by Mario Carneiro, 17-Apr-2015.) |
| Ref | Expression |
|---|---|
| numexp.1 | ⊢ 𝐴 ∈ ℕ0 |
| Ref | Expression |
|---|---|
| numexp0 | ⊢ (𝐴↑0) = 1 |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | numexp.1 | . . 3 ⊢ 𝐴 ∈ ℕ0 | |
| 2 | 1 | nn0cni 12538 | . 2 ⊢ 𝐴 ∈ ℂ |
| 3 | exp0 14106 | . 2 ⊢ (𝐴 ∈ ℂ → (𝐴↑0) = 1) | |
| 4 | 2, 3 | ax-mp 5 | 1 ⊢ (𝐴↑0) = 1 |
| Colors of variables: wff setvar class |
| Syntax hints: = wceq 1540 ∈ wcel 2108 (class class class)co 7431 ℂcc 11153 0cc0 11155 1c1 11156 ℕ0cn0 12526 ↑cexp 14102 |
| 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 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 ax-un 7755 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-i2m1 11223 ax-rnegex 11226 ax-cnre 11228 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-neg 11495 df-nn 12267 df-n0 12527 df-z 12614 df-seq 14043 df-exp 14103 |
| This theorem is referenced by: decsplit0b 17117 dchrisum0flb 27554 ex-ind-dvds 30480 hgt750lemd 34663 hgt750lem 34666 itcovalt2lem1 48596 |
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