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Mirrors > Home > ILE Home > Th. List > oddm1even | GIF version |
Description: An integer is odd iff its predecessor is even. (Contributed by Mario Carneiro, 5-Sep-2016.) |
Ref | Expression |
---|---|
oddm1even | ⊢ (𝑁 ∈ ℤ → (¬ 2 ∥ 𝑁 ↔ 2 ∥ (𝑁 − 1))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl 108 | . . . . . 6 ⊢ ((𝑁 ∈ ℤ ∧ 𝑛 ∈ ℤ) → 𝑁 ∈ ℤ) | |
2 | 1 | zcnd 9335 | . . . . 5 ⊢ ((𝑁 ∈ ℤ ∧ 𝑛 ∈ ℤ) → 𝑁 ∈ ℂ) |
3 | 1cnd 7936 | . . . . 5 ⊢ ((𝑁 ∈ ℤ ∧ 𝑛 ∈ ℤ) → 1 ∈ ℂ) | |
4 | 2cnd 8951 | . . . . . 6 ⊢ ((𝑁 ∈ ℤ ∧ 𝑛 ∈ ℤ) → 2 ∈ ℂ) | |
5 | simpr 109 | . . . . . . 7 ⊢ ((𝑁 ∈ ℤ ∧ 𝑛 ∈ ℤ) → 𝑛 ∈ ℤ) | |
6 | 5 | zcnd 9335 | . . . . . 6 ⊢ ((𝑁 ∈ ℤ ∧ 𝑛 ∈ ℤ) → 𝑛 ∈ ℂ) |
7 | 4, 6 | mulcld 7940 | . . . . 5 ⊢ ((𝑁 ∈ ℤ ∧ 𝑛 ∈ ℤ) → (2 · 𝑛) ∈ ℂ) |
8 | 2, 3, 7 | subadd2d 8249 | . . . 4 ⊢ ((𝑁 ∈ ℤ ∧ 𝑛 ∈ ℤ) → ((𝑁 − 1) = (2 · 𝑛) ↔ ((2 · 𝑛) + 1) = 𝑁)) |
9 | eqcom 2172 | . . . . 5 ⊢ ((𝑁 − 1) = (2 · 𝑛) ↔ (2 · 𝑛) = (𝑁 − 1)) | |
10 | 4, 6 | mulcomd 7941 | . . . . . 6 ⊢ ((𝑁 ∈ ℤ ∧ 𝑛 ∈ ℤ) → (2 · 𝑛) = (𝑛 · 2)) |
11 | 10 | eqeq1d 2179 | . . . . 5 ⊢ ((𝑁 ∈ ℤ ∧ 𝑛 ∈ ℤ) → ((2 · 𝑛) = (𝑁 − 1) ↔ (𝑛 · 2) = (𝑁 − 1))) |
12 | 9, 11 | syl5bb 191 | . . . 4 ⊢ ((𝑁 ∈ ℤ ∧ 𝑛 ∈ ℤ) → ((𝑁 − 1) = (2 · 𝑛) ↔ (𝑛 · 2) = (𝑁 − 1))) |
13 | 8, 12 | bitr3d 189 | . . 3 ⊢ ((𝑁 ∈ ℤ ∧ 𝑛 ∈ ℤ) → (((2 · 𝑛) + 1) = 𝑁 ↔ (𝑛 · 2) = (𝑁 − 1))) |
14 | 13 | rexbidva 2467 | . 2 ⊢ (𝑁 ∈ ℤ → (∃𝑛 ∈ ℤ ((2 · 𝑛) + 1) = 𝑁 ↔ ∃𝑛 ∈ ℤ (𝑛 · 2) = (𝑁 − 1))) |
15 | odd2np1 11832 | . 2 ⊢ (𝑁 ∈ ℤ → (¬ 2 ∥ 𝑁 ↔ ∃𝑛 ∈ ℤ ((2 · 𝑛) + 1) = 𝑁)) | |
16 | 2z 9240 | . . 3 ⊢ 2 ∈ ℤ | |
17 | peano2zm 9250 | . . 3 ⊢ (𝑁 ∈ ℤ → (𝑁 − 1) ∈ ℤ) | |
18 | divides 11751 | . . 3 ⊢ ((2 ∈ ℤ ∧ (𝑁 − 1) ∈ ℤ) → (2 ∥ (𝑁 − 1) ↔ ∃𝑛 ∈ ℤ (𝑛 · 2) = (𝑁 − 1))) | |
19 | 16, 17, 18 | sylancr 412 | . 2 ⊢ (𝑁 ∈ ℤ → (2 ∥ (𝑁 − 1) ↔ ∃𝑛 ∈ ℤ (𝑛 · 2) = (𝑁 − 1))) |
20 | 14, 15, 19 | 3bitr4d 219 | 1 ⊢ (𝑁 ∈ ℤ → (¬ 2 ∥ 𝑁 ↔ 2 ∥ (𝑁 − 1))) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 103 ↔ wb 104 = wceq 1348 ∈ wcel 2141 ∃wrex 2449 class class class wbr 3989 (class class class)co 5853 1c1 7775 + caddc 7777 · cmul 7779 − cmin 8090 2c2 8929 ℤcz 9212 ∥ cdvds 11749 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-sep 4107 ax-pow 4160 ax-pr 4194 ax-un 4418 ax-setind 4521 ax-cnex 7865 ax-resscn 7866 ax-1cn 7867 ax-1re 7868 ax-icn 7869 ax-addcl 7870 ax-addrcl 7871 ax-mulcl 7872 ax-mulrcl 7873 ax-addcom 7874 ax-mulcom 7875 ax-addass 7876 ax-mulass 7877 ax-distr 7878 ax-i2m1 7879 ax-0lt1 7880 ax-1rid 7881 ax-0id 7882 ax-rnegex 7883 ax-precex 7884 ax-cnre 7885 ax-pre-ltirr 7886 ax-pre-ltwlin 7887 ax-pre-lttrn 7888 ax-pre-apti 7889 ax-pre-ltadd 7890 ax-pre-mulgt0 7891 ax-pre-mulext 7892 |
This theorem depends on definitions: df-bi 116 df-3or 974 df-3an 975 df-tru 1351 df-fal 1354 df-xor 1371 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-nel 2436 df-ral 2453 df-rex 2454 df-reu 2455 df-rmo 2456 df-rab 2457 df-v 2732 df-sbc 2956 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-int 3832 df-br 3990 df-opab 4051 df-id 4278 df-po 4281 df-iso 4282 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-iota 5160 df-fun 5200 df-fv 5206 df-riota 5809 df-ov 5856 df-oprab 5857 df-mpo 5858 df-pnf 7956 df-mnf 7957 df-xr 7958 df-ltxr 7959 df-le 7960 df-sub 8092 df-neg 8093 df-reap 8494 df-ap 8501 df-div 8590 df-inn 8879 df-2 8937 df-n0 9136 df-z 9213 df-dvds 11750 |
This theorem is referenced by: oddp1even 11835 n2dvds3 11874 oddennn 12347 |
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