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Mirrors > Home > MPE Home > Th. List > Mathboxes > nnpw2evenALTV | Structured version Visualization version GIF version |
Description: 2 to the power of a positive integer is even. (Contributed by AV, 2-Jun-2020.) (Revised by AV, 20-Jun-2020.) |
Ref | Expression |
---|---|
nnpw2evenALTV | ⊢ (𝑁 ∈ ℕ → (2↑𝑁) ∈ Even ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 2z 12097 | . . 3 ⊢ 2 ∈ ℤ | |
2 | nnnn0 11985 | . . 3 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℕ0) | |
3 | zexpcl 13538 | . . 3 ⊢ ((2 ∈ ℤ ∧ 𝑁 ∈ ℕ0) → (2↑𝑁) ∈ ℤ) | |
4 | 1, 2, 3 | sylancr 590 | . 2 ⊢ (𝑁 ∈ ℕ → (2↑𝑁) ∈ ℤ) |
5 | iddvdsexp 15727 | . . 3 ⊢ ((2 ∈ ℤ ∧ 𝑁 ∈ ℕ) → 2 ∥ (2↑𝑁)) | |
6 | 1, 5 | mpan 690 | . 2 ⊢ (𝑁 ∈ ℕ → 2 ∥ (2↑𝑁)) |
7 | iseven2 44666 | . 2 ⊢ ((2↑𝑁) ∈ Even ↔ ((2↑𝑁) ∈ ℤ ∧ 2 ∥ (2↑𝑁))) | |
8 | 4, 6, 7 | sylanbrc 586 | 1 ⊢ (𝑁 ∈ ℕ → (2↑𝑁) ∈ Even ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2114 class class class wbr 5030 (class class class)co 7172 ℕcn 11718 2c2 11773 ℕ0cn0 11978 ℤcz 12064 ↑cexp 13523 ∥ cdvds 15701 Even ceven 44639 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2710 ax-sep 5167 ax-nul 5174 ax-pow 5232 ax-pr 5296 ax-un 7481 ax-cnex 10673 ax-resscn 10674 ax-1cn 10675 ax-icn 10676 ax-addcl 10677 ax-addrcl 10678 ax-mulcl 10679 ax-mulrcl 10680 ax-mulcom 10681 ax-addass 10682 ax-mulass 10683 ax-distr 10684 ax-i2m1 10685 ax-1ne0 10686 ax-1rid 10687 ax-rnegex 10688 ax-rrecex 10689 ax-cnre 10690 ax-pre-lttri 10691 ax-pre-lttrn 10692 ax-pre-ltadd 10693 ax-pre-mulgt0 10694 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-nel 3039 df-ral 3058 df-rex 3059 df-reu 3060 df-rmo 3061 df-rab 3062 df-v 3400 df-sbc 3681 df-csb 3791 df-dif 3846 df-un 3848 df-in 3850 df-ss 3860 df-pss 3862 df-nul 4212 df-if 4415 df-pw 4490 df-sn 4517 df-pr 4519 df-tp 4521 df-op 4523 df-uni 4797 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5429 df-eprel 5434 df-po 5442 df-so 5443 df-fr 5483 df-we 5485 df-xp 5531 df-rel 5532 df-cnv 5533 df-co 5534 df-dm 5535 df-rn 5536 df-res 5537 df-ima 5538 df-pred 6129 df-ord 6175 df-on 6176 df-lim 6177 df-suc 6178 df-iota 6297 df-fun 6341 df-fn 6342 df-f 6343 df-f1 6344 df-fo 6345 df-f1o 6346 df-fv 6347 df-riota 7129 df-ov 7175 df-oprab 7176 df-mpo 7177 df-om 7602 df-2nd 7717 df-wrecs 7978 df-recs 8039 df-rdg 8077 df-er 8322 df-en 8558 df-dom 8559 df-sdom 8560 df-pnf 10757 df-mnf 10758 df-xr 10759 df-ltxr 10760 df-le 10761 df-sub 10952 df-neg 10953 df-div 11378 df-nn 11719 df-2 11781 df-n0 11979 df-z 12065 df-uz 12327 df-seq 13463 df-exp 13524 df-dvds 15702 df-even 44641 |
This theorem is referenced by: perfectALTVlem1 44736 |
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