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Mirrors > Home > ILE Home > Th. List > oddp1even | GIF version |
Description: An integer is odd iff its successor is even. (Contributed by Mario Carneiro, 5-Sep-2016.) |
Ref | Expression |
---|---|
oddp1even | ⊢ (𝑁 ∈ ℤ → (¬ 2 ∥ 𝑁 ↔ 2 ∥ (𝑁 + 1))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oddm1even 10274 | . 2 ⊢ (𝑁 ∈ ℤ → (¬ 2 ∥ 𝑁 ↔ 2 ∥ (𝑁 − 1))) | |
2 | 2z 8379 | . . 3 ⊢ 2 ∈ ℤ | |
3 | peano2zm 8389 | . . 3 ⊢ (𝑁 ∈ ℤ → (𝑁 − 1) ∈ ℤ) | |
4 | dvdsadd 10238 | . . 3 ⊢ ((2 ∈ ℤ ∧ (𝑁 − 1) ∈ ℤ) → (2 ∥ (𝑁 − 1) ↔ 2 ∥ (2 + (𝑁 − 1)))) | |
5 | 2, 3, 4 | sylancr 405 | . 2 ⊢ (𝑁 ∈ ℤ → (2 ∥ (𝑁 − 1) ↔ 2 ∥ (2 + (𝑁 − 1)))) |
6 | 2cnd 8112 | . . . . 5 ⊢ (𝑁 ∈ ℤ → 2 ∈ ℂ) | |
7 | zcn 8356 | . . . . 5 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℂ) | |
8 | 1cnd 7135 | . . . . 5 ⊢ (𝑁 ∈ ℤ → 1 ∈ ℂ) | |
9 | 6, 7, 8 | addsub12d 7442 | . . . 4 ⊢ (𝑁 ∈ ℤ → (2 + (𝑁 − 1)) = (𝑁 + (2 − 1))) |
10 | 2m1e1 8156 | . . . . 5 ⊢ (2 − 1) = 1 | |
11 | 10 | oveq2i 5543 | . . . 4 ⊢ (𝑁 + (2 − 1)) = (𝑁 + 1) |
12 | 9, 11 | syl6eq 2129 | . . 3 ⊢ (𝑁 ∈ ℤ → (2 + (𝑁 − 1)) = (𝑁 + 1)) |
13 | 12 | breq2d 3797 | . 2 ⊢ (𝑁 ∈ ℤ → (2 ∥ (2 + (𝑁 − 1)) ↔ 2 ∥ (𝑁 + 1))) |
14 | 1, 5, 13 | 3bitrd 212 | 1 ⊢ (𝑁 ∈ ℤ → (¬ 2 ∥ 𝑁 ↔ 2 ∥ (𝑁 + 1))) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 103 ∈ wcel 1433 class class class wbr 3785 (class class class)co 5532 1c1 6982 + caddc 6984 − cmin 7279 2c2 8089 ℤcz 8351 ∥ cdvds 10195 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-13 1444 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-pow 3948 ax-pr 3964 ax-un 4188 ax-setind 4280 ax-cnex 7067 ax-resscn 7068 ax-1cn 7069 ax-1re 7070 ax-icn 7071 ax-addcl 7072 ax-addrcl 7073 ax-mulcl 7074 ax-mulrcl 7075 ax-addcom 7076 ax-mulcom 7077 ax-addass 7078 ax-mulass 7079 ax-distr 7080 ax-i2m1 7081 ax-0lt1 7082 ax-1rid 7083 ax-0id 7084 ax-rnegex 7085 ax-precex 7086 ax-cnre 7087 ax-pre-ltirr 7088 ax-pre-ltwlin 7089 ax-pre-lttrn 7090 ax-pre-apti 7091 ax-pre-ltadd 7092 ax-pre-mulgt0 7093 ax-pre-mulext 7094 |
This theorem depends on definitions: df-bi 115 df-3or 920 df-3an 921 df-tru 1287 df-fal 1290 df-xor 1307 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ne 2246 df-nel 2340 df-ral 2353 df-rex 2354 df-reu 2355 df-rmo 2356 df-rab 2357 df-v 2603 df-sbc 2816 df-dif 2975 df-un 2977 df-in 2979 df-ss 2986 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-int 3637 df-br 3786 df-opab 3840 df-id 4048 df-po 4051 df-iso 4052 df-xp 4369 df-rel 4370 df-cnv 4371 df-co 4372 df-dm 4373 df-iota 4887 df-fun 4924 df-fv 4930 df-riota 5488 df-ov 5535 df-oprab 5536 df-mpt2 5537 df-pnf 7155 df-mnf 7156 df-xr 7157 df-ltxr 7158 df-le 7159 df-sub 7281 df-neg 7282 df-reap 7675 df-ap 7682 df-div 7761 df-inn 8040 df-2 8098 df-n0 8289 df-z 8352 df-dvds 10196 |
This theorem is referenced by: zeo5 10288 oddp1d2 10290 n2dvdsm1 10313 2sqpwodd 10554 |
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