Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > evennn02n | Structured version Visualization version GIF version |
Description: A nonnegative integer is even iff it is twice another nonnegative integer. (Contributed by AV, 12-Aug-2021.) (Proof shortened by AV, 10-Jul-2022.) |
Ref | Expression |
---|---|
evennn02n | ⊢ (𝑁 ∈ ℕ0 → (2 ∥ 𝑁 ↔ ∃𝑛 ∈ ℕ0 (2 · 𝑛) = 𝑁)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eleq1 2821 | . . . . . . . 8 ⊢ ((2 · 𝑛) = 𝑁 → ((2 · 𝑛) ∈ ℕ0 ↔ 𝑁 ∈ ℕ0)) | |
2 | simpr 488 | . . . . . . . . . 10 ⊢ (((2 · 𝑛) ∈ ℕ0 ∧ 𝑛 ∈ ℤ) → 𝑛 ∈ ℤ) | |
3 | 2rp 12489 | . . . . . . . . . . . 12 ⊢ 2 ∈ ℝ+ | |
4 | 3 | a1i 11 | . . . . . . . . . . 11 ⊢ (((2 · 𝑛) ∈ ℕ0 ∧ 𝑛 ∈ ℤ) → 2 ∈ ℝ+) |
5 | zre 12078 | . . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℤ → 𝑛 ∈ ℝ) | |
6 | 5 | adantl 485 | . . . . . . . . . . 11 ⊢ (((2 · 𝑛) ∈ ℕ0 ∧ 𝑛 ∈ ℤ) → 𝑛 ∈ ℝ) |
7 | nn0ge0 12013 | . . . . . . . . . . . 12 ⊢ ((2 · 𝑛) ∈ ℕ0 → 0 ≤ (2 · 𝑛)) | |
8 | 7 | adantr 484 | . . . . . . . . . . 11 ⊢ (((2 · 𝑛) ∈ ℕ0 ∧ 𝑛 ∈ ℤ) → 0 ≤ (2 · 𝑛)) |
9 | 4, 6, 8 | prodge0rd 12591 | . . . . . . . . . 10 ⊢ (((2 · 𝑛) ∈ ℕ0 ∧ 𝑛 ∈ ℤ) → 0 ≤ 𝑛) |
10 | elnn0z 12087 | . . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ0 ↔ (𝑛 ∈ ℤ ∧ 0 ≤ 𝑛)) | |
11 | 2, 9, 10 | sylanbrc 586 | . . . . . . . . 9 ⊢ (((2 · 𝑛) ∈ ℕ0 ∧ 𝑛 ∈ ℤ) → 𝑛 ∈ ℕ0) |
12 | 11 | ex 416 | . . . . . . . 8 ⊢ ((2 · 𝑛) ∈ ℕ0 → (𝑛 ∈ ℤ → 𝑛 ∈ ℕ0)) |
13 | 1, 12 | syl6bir 257 | . . . . . . 7 ⊢ ((2 · 𝑛) = 𝑁 → (𝑁 ∈ ℕ0 → (𝑛 ∈ ℤ → 𝑛 ∈ ℕ0))) |
14 | 13 | com13 88 | . . . . . 6 ⊢ (𝑛 ∈ ℤ → (𝑁 ∈ ℕ0 → ((2 · 𝑛) = 𝑁 → 𝑛 ∈ ℕ0))) |
15 | 14 | impcom 411 | . . . . 5 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑛 ∈ ℤ) → ((2 · 𝑛) = 𝑁 → 𝑛 ∈ ℕ0)) |
16 | 15 | pm4.71rd 566 | . . . 4 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑛 ∈ ℤ) → ((2 · 𝑛) = 𝑁 ↔ (𝑛 ∈ ℕ0 ∧ (2 · 𝑛) = 𝑁))) |
17 | 16 | bicomd 226 | . . 3 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑛 ∈ ℤ) → ((𝑛 ∈ ℕ0 ∧ (2 · 𝑛) = 𝑁) ↔ (2 · 𝑛) = 𝑁)) |
18 | 17 | rexbidva 3207 | . 2 ⊢ (𝑁 ∈ ℕ0 → (∃𝑛 ∈ ℤ (𝑛 ∈ ℕ0 ∧ (2 · 𝑛) = 𝑁) ↔ ∃𝑛 ∈ ℤ (2 · 𝑛) = 𝑁)) |
19 | nn0ssz 12096 | . . 3 ⊢ ℕ0 ⊆ ℤ | |
20 | rexss 3958 | . . 3 ⊢ (ℕ0 ⊆ ℤ → (∃𝑛 ∈ ℕ0 (2 · 𝑛) = 𝑁 ↔ ∃𝑛 ∈ ℤ (𝑛 ∈ ℕ0 ∧ (2 · 𝑛) = 𝑁))) | |
21 | 19, 20 | mp1i 13 | . 2 ⊢ (𝑁 ∈ ℕ0 → (∃𝑛 ∈ ℕ0 (2 · 𝑛) = 𝑁 ↔ ∃𝑛 ∈ ℤ (𝑛 ∈ ℕ0 ∧ (2 · 𝑛) = 𝑁))) |
22 | even2n 15799 | . . 3 ⊢ (2 ∥ 𝑁 ↔ ∃𝑛 ∈ ℤ (2 · 𝑛) = 𝑁) | |
23 | 22 | a1i 11 | . 2 ⊢ (𝑁 ∈ ℕ0 → (2 ∥ 𝑁 ↔ ∃𝑛 ∈ ℤ (2 · 𝑛) = 𝑁)) |
24 | 18, 21, 23 | 3bitr4rd 315 | 1 ⊢ (𝑁 ∈ ℕ0 → (2 ∥ 𝑁 ↔ ∃𝑛 ∈ ℕ0 (2 · 𝑛) = 𝑁)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1542 ∈ wcel 2114 ∃wrex 3055 ⊆ wss 3853 class class class wbr 5040 (class class class)co 7182 ℝcr 10626 0cc0 10627 · cmul 10632 ≤ cle 10766 2c2 11783 ℕ0cn0 11988 ℤcz 12074 ℝ+crp 12484 ∥ cdvds 15711 |
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 2711 ax-sep 5177 ax-nul 5184 ax-pow 5242 ax-pr 5306 ax-un 7491 ax-resscn 10684 ax-1cn 10685 ax-icn 10686 ax-addcl 10687 ax-addrcl 10688 ax-mulcl 10689 ax-mulrcl 10690 ax-mulcom 10691 ax-addass 10692 ax-mulass 10693 ax-distr 10694 ax-i2m1 10695 ax-1ne0 10696 ax-1rid 10697 ax-rnegex 10698 ax-rrecex 10699 ax-cnre 10700 ax-pre-lttri 10701 ax-pre-lttrn 10702 ax-pre-ltadd 10703 ax-pre-mulgt0 10704 |
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 2541 df-eu 2571 df-clab 2718 df-cleq 2731 df-clel 2812 df-nfc 2882 df-ne 2936 df-nel 3040 df-ral 3059 df-rex 3060 df-reu 3061 df-rab 3063 df-v 3402 df-sbc 3686 df-csb 3801 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4222 df-if 4425 df-pw 4500 df-sn 4527 df-pr 4529 df-tp 4531 df-op 4533 df-uni 4807 df-iun 4893 df-br 5041 df-opab 5103 df-mpt 5121 df-tr 5147 df-id 5439 df-eprel 5444 df-po 5452 df-so 5453 df-fr 5493 df-we 5495 df-xp 5541 df-rel 5542 df-cnv 5543 df-co 5544 df-dm 5545 df-rn 5546 df-res 5547 df-ima 5548 df-pred 6139 df-ord 6185 df-on 6186 df-lim 6187 df-suc 6188 df-iota 6307 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7139 df-ov 7185 df-oprab 7186 df-mpo 7187 df-om 7612 df-wrecs 7988 df-recs 8049 df-rdg 8087 df-er 8332 df-en 8568 df-dom 8569 df-sdom 8570 df-pnf 10767 df-mnf 10768 df-xr 10769 df-ltxr 10770 df-le 10771 df-sub 10962 df-neg 10963 df-nn 11729 df-2 11791 df-n0 11989 df-z 12075 df-rp 12485 df-dvds 15712 |
This theorem is referenced by: wrdt2ind 30812 |
Copyright terms: Public domain | W3C validator |