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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 2903 | . . . . . . . 8 ⊢ ((2 · 𝑛) = 𝑁 → ((2 · 𝑛) ∈ ℕ0 ↔ 𝑁 ∈ ℕ0)) | |
2 | simpr 487 | . . . . . . . . . 10 ⊢ (((2 · 𝑛) ∈ ℕ0 ∧ 𝑛 ∈ ℤ) → 𝑛 ∈ ℤ) | |
3 | 2rp 12397 | . . . . . . . . . . . 12 ⊢ 2 ∈ ℝ+ | |
4 | 3 | a1i 11 | . . . . . . . . . . 11 ⊢ (((2 · 𝑛) ∈ ℕ0 ∧ 𝑛 ∈ ℤ) → 2 ∈ ℝ+) |
5 | zre 11988 | . . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℤ → 𝑛 ∈ ℝ) | |
6 | 5 | adantl 484 | . . . . . . . . . . 11 ⊢ (((2 · 𝑛) ∈ ℕ0 ∧ 𝑛 ∈ ℤ) → 𝑛 ∈ ℝ) |
7 | nn0ge0 11925 | . . . . . . . . . . . 12 ⊢ ((2 · 𝑛) ∈ ℕ0 → 0 ≤ (2 · 𝑛)) | |
8 | 7 | adantr 483 | . . . . . . . . . . 11 ⊢ (((2 · 𝑛) ∈ ℕ0 ∧ 𝑛 ∈ ℤ) → 0 ≤ (2 · 𝑛)) |
9 | 4, 6, 8 | prodge0rd 12499 | . . . . . . . . . 10 ⊢ (((2 · 𝑛) ∈ ℕ0 ∧ 𝑛 ∈ ℤ) → 0 ≤ 𝑛) |
10 | elnn0z 11997 | . . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ0 ↔ (𝑛 ∈ ℤ ∧ 0 ≤ 𝑛)) | |
11 | 2, 9, 10 | sylanbrc 585 | . . . . . . . . 9 ⊢ (((2 · 𝑛) ∈ ℕ0 ∧ 𝑛 ∈ ℤ) → 𝑛 ∈ ℕ0) |
12 | 11 | ex 415 | . . . . . . . 8 ⊢ ((2 · 𝑛) ∈ ℕ0 → (𝑛 ∈ ℤ → 𝑛 ∈ ℕ0)) |
13 | 1, 12 | syl6bir 256 | . . . . . . 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 410 | . . . . 5 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑛 ∈ ℤ) → ((2 · 𝑛) = 𝑁 → 𝑛 ∈ ℕ0)) |
16 | 15 | pm4.71rd 565 | . . . 4 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑛 ∈ ℤ) → ((2 · 𝑛) = 𝑁 ↔ (𝑛 ∈ ℕ0 ∧ (2 · 𝑛) = 𝑁))) |
17 | 16 | bicomd 225 | . . 3 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑛 ∈ ℤ) → ((𝑛 ∈ ℕ0 ∧ (2 · 𝑛) = 𝑁) ↔ (2 · 𝑛) = 𝑁)) |
18 | 17 | rexbidva 3299 | . 2 ⊢ (𝑁 ∈ ℕ0 → (∃𝑛 ∈ ℤ (𝑛 ∈ ℕ0 ∧ (2 · 𝑛) = 𝑁) ↔ ∃𝑛 ∈ ℤ (2 · 𝑛) = 𝑁)) |
19 | nn0ssz 12006 | . . 3 ⊢ ℕ0 ⊆ ℤ | |
20 | rexss 4041 | . . 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 15694 | . . 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 314 | 1 ⊢ (𝑁 ∈ ℕ0 → (2 ∥ 𝑁 ↔ ∃𝑛 ∈ ℕ0 (2 · 𝑛) = 𝑁)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1536 ∈ wcel 2113 ∃wrex 3142 ⊆ wss 3939 class class class wbr 5069 (class class class)co 7159 ℝcr 10539 0cc0 10540 · cmul 10545 ≤ cle 10679 2c2 11695 ℕ0cn0 11900 ℤcz 11984 ℝ+crp 12392 ∥ cdvds 15610 |
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-pow 5269 ax-pr 5333 ax-un 7464 ax-resscn 10597 ax-1cn 10598 ax-icn 10599 ax-addcl 10600 ax-addrcl 10601 ax-mulcl 10602 ax-mulrcl 10603 ax-mulcom 10604 ax-addass 10605 ax-mulass 10606 ax-distr 10607 ax-i2m1 10608 ax-1ne0 10609 ax-1rid 10610 ax-rnegex 10611 ax-rrecex 10612 ax-cnre 10613 ax-pre-lttri 10614 ax-pre-lttrn 10615 ax-pre-ltadd 10616 ax-pre-mulgt0 10617 |
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-ne 3020 df-nel 3127 df-ral 3146 df-rex 3147 df-reu 3148 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-pss 3957 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-tp 4575 df-op 4577 df-uni 4842 df-iun 4924 df-br 5070 df-opab 5132 df-mpt 5150 df-tr 5176 df-id 5463 df-eprel 5468 df-po 5477 df-so 5478 df-fr 5517 df-we 5519 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-ord 6197 df-on 6198 df-lim 6199 df-suc 6200 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7117 df-ov 7162 df-oprab 7163 df-mpo 7164 df-om 7584 df-wrecs 7950 df-recs 8011 df-rdg 8049 df-er 8292 df-en 8513 df-dom 8514 df-sdom 8515 df-pnf 10680 df-mnf 10681 df-xr 10682 df-ltxr 10683 df-le 10684 df-sub 10875 df-neg 10876 df-nn 11642 df-2 11703 df-n0 11901 df-z 11985 df-rp 12393 df-dvds 15611 |
This theorem is referenced by: wrdt2ind 30631 |
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