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Mirrors > Home > ILE Home > Th. List > evennn2n | GIF version |
Description: A positive integer is even iff it is twice another positive integer. (Contributed by AV, 12-Aug-2021.) |
Ref | Expression |
---|---|
evennn2n | ⊢ (𝑁 ∈ ℕ → (2 ∥ 𝑁 ↔ ∃𝑛 ∈ ℕ (2 · 𝑛) = 𝑁)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eleq1 2252 | . . . . . . . 8 ⊢ ((2 · 𝑛) = 𝑁 → ((2 · 𝑛) ∈ ℕ ↔ 𝑁 ∈ ℕ)) | |
2 | simpr 110 | . . . . . . . . . 10 ⊢ (((2 · 𝑛) ∈ ℕ ∧ 𝑛 ∈ ℤ) → 𝑛 ∈ ℤ) | |
3 | 2re 9018 | . . . . . . . . . . . 12 ⊢ 2 ∈ ℝ | |
4 | 3 | a1i 9 | . . . . . . . . . . 11 ⊢ (((2 · 𝑛) ∈ ℕ ∧ 𝑛 ∈ ℤ) → 2 ∈ ℝ) |
5 | zre 9286 | . . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℤ → 𝑛 ∈ ℝ) | |
6 | 5 | adantl 277 | . . . . . . . . . . 11 ⊢ (((2 · 𝑛) ∈ ℕ ∧ 𝑛 ∈ ℤ) → 𝑛 ∈ ℝ) |
7 | 0le2 9038 | . . . . . . . . . . . 12 ⊢ 0 ≤ 2 | |
8 | 7 | a1i 9 | . . . . . . . . . . 11 ⊢ (((2 · 𝑛) ∈ ℕ ∧ 𝑛 ∈ ℤ) → 0 ≤ 2) |
9 | nngt0 8973 | . . . . . . . . . . . 12 ⊢ ((2 · 𝑛) ∈ ℕ → 0 < (2 · 𝑛)) | |
10 | 9 | adantr 276 | . . . . . . . . . . 11 ⊢ (((2 · 𝑛) ∈ ℕ ∧ 𝑛 ∈ ℤ) → 0 < (2 · 𝑛)) |
11 | prodgt0 8838 | . . . . . . . . . . 11 ⊢ (((2 ∈ ℝ ∧ 𝑛 ∈ ℝ) ∧ (0 ≤ 2 ∧ 0 < (2 · 𝑛))) → 0 < 𝑛) | |
12 | 4, 6, 8, 10, 11 | syl22anc 1250 | . . . . . . . . . 10 ⊢ (((2 · 𝑛) ∈ ℕ ∧ 𝑛 ∈ ℤ) → 0 < 𝑛) |
13 | elnnz 9292 | . . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ ↔ (𝑛 ∈ ℤ ∧ 0 < 𝑛)) | |
14 | 2, 12, 13 | sylanbrc 417 | . . . . . . . . 9 ⊢ (((2 · 𝑛) ∈ ℕ ∧ 𝑛 ∈ ℤ) → 𝑛 ∈ ℕ) |
15 | 14 | ex 115 | . . . . . . . 8 ⊢ ((2 · 𝑛) ∈ ℕ → (𝑛 ∈ ℤ → 𝑛 ∈ ℕ)) |
16 | 1, 15 | biimtrrdi 164 | . . . . . . 7 ⊢ ((2 · 𝑛) = 𝑁 → (𝑁 ∈ ℕ → (𝑛 ∈ ℤ → 𝑛 ∈ ℕ))) |
17 | 16 | com13 80 | . . . . . 6 ⊢ (𝑛 ∈ ℤ → (𝑁 ∈ ℕ → ((2 · 𝑛) = 𝑁 → 𝑛 ∈ ℕ))) |
18 | 17 | impcom 125 | . . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ 𝑛 ∈ ℤ) → ((2 · 𝑛) = 𝑁 → 𝑛 ∈ ℕ)) |
19 | 18 | pm4.71rd 394 | . . . 4 ⊢ ((𝑁 ∈ ℕ ∧ 𝑛 ∈ ℤ) → ((2 · 𝑛) = 𝑁 ↔ (𝑛 ∈ ℕ ∧ (2 · 𝑛) = 𝑁))) |
20 | 19 | bicomd 141 | . . 3 ⊢ ((𝑁 ∈ ℕ ∧ 𝑛 ∈ ℤ) → ((𝑛 ∈ ℕ ∧ (2 · 𝑛) = 𝑁) ↔ (2 · 𝑛) = 𝑁)) |
21 | 20 | rexbidva 2487 | . 2 ⊢ (𝑁 ∈ ℕ → (∃𝑛 ∈ ℤ (𝑛 ∈ ℕ ∧ (2 · 𝑛) = 𝑁) ↔ ∃𝑛 ∈ ℤ (2 · 𝑛) = 𝑁)) |
22 | nnssz 9299 | . . 3 ⊢ ℕ ⊆ ℤ | |
23 | rexss 3237 | . . 3 ⊢ (ℕ ⊆ ℤ → (∃𝑛 ∈ ℕ (2 · 𝑛) = 𝑁 ↔ ∃𝑛 ∈ ℤ (𝑛 ∈ ℕ ∧ (2 · 𝑛) = 𝑁))) | |
24 | 22, 23 | mp1i 10 | . 2 ⊢ (𝑁 ∈ ℕ → (∃𝑛 ∈ ℕ (2 · 𝑛) = 𝑁 ↔ ∃𝑛 ∈ ℤ (𝑛 ∈ ℕ ∧ (2 · 𝑛) = 𝑁))) |
25 | even2n 11910 | . . 3 ⊢ (2 ∥ 𝑁 ↔ ∃𝑛 ∈ ℤ (2 · 𝑛) = 𝑁) | |
26 | 25 | a1i 9 | . 2 ⊢ (𝑁 ∈ ℕ → (2 ∥ 𝑁 ↔ ∃𝑛 ∈ ℤ (2 · 𝑛) = 𝑁)) |
27 | 21, 24, 26 | 3bitr4rd 221 | 1 ⊢ (𝑁 ∈ ℕ → (2 ∥ 𝑁 ↔ ∃𝑛 ∈ ℕ (2 · 𝑛) = 𝑁)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 = wceq 1364 ∈ wcel 2160 ∃wrex 2469 ⊆ wss 3144 class class class wbr 4018 (class class class)co 5895 ℝcr 7839 0cc0 7840 · cmul 7845 < clt 8021 ≤ cle 8022 ℕcn 8948 2c2 8999 ℤcz 9282 ∥ cdvds 11825 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-sep 4136 ax-pow 4192 ax-pr 4227 ax-un 4451 ax-setind 4554 ax-cnex 7931 ax-resscn 7932 ax-1cn 7933 ax-1re 7934 ax-icn 7935 ax-addcl 7936 ax-addrcl 7937 ax-mulcl 7938 ax-mulrcl 7939 ax-addcom 7940 ax-mulcom 7941 ax-addass 7942 ax-mulass 7943 ax-distr 7944 ax-i2m1 7945 ax-0lt1 7946 ax-1rid 7947 ax-0id 7948 ax-rnegex 7949 ax-precex 7950 ax-cnre 7951 ax-pre-ltirr 7952 ax-pre-ltwlin 7953 ax-pre-lttrn 7954 ax-pre-apti 7955 ax-pre-ltadd 7956 ax-pre-mulgt0 7957 ax-pre-mulext 7958 |
This theorem depends on definitions: df-bi 117 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-nel 2456 df-ral 2473 df-rex 2474 df-reu 2475 df-rmo 2476 df-rab 2477 df-v 2754 df-sbc 2978 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-int 3860 df-br 4019 df-opab 4080 df-id 4311 df-po 4314 df-iso 4315 df-xp 4650 df-rel 4651 df-cnv 4652 df-co 4653 df-dm 4654 df-iota 5196 df-fun 5237 df-fv 5243 df-riota 5851 df-ov 5898 df-oprab 5899 df-mpo 5900 df-pnf 8023 df-mnf 8024 df-xr 8025 df-ltxr 8026 df-le 8027 df-sub 8159 df-neg 8160 df-reap 8561 df-ap 8568 df-div 8659 df-inn 8949 df-2 9007 df-n0 9206 df-z 9283 df-dvds 11826 |
This theorem is referenced by: (None) |
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