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Mirrors > Home > ILE Home > Th. List > oddp1d2 | GIF version |
Description: An integer is odd iff its successor divided by 2 is an integer. This is a representation of odd numbers without using the divides relation, see zeo 9060 and zeo2 9061. (Contributed by AV, 22-Jun-2021.) |
Ref | Expression |
---|---|
oddp1d2 | ⊢ (𝑁 ∈ ℤ → (¬ 2 ∥ 𝑁 ↔ ((𝑁 + 1) / 2) ∈ ℤ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oddp1even 11421 | . 2 ⊢ (𝑁 ∈ ℤ → (¬ 2 ∥ 𝑁 ↔ 2 ∥ (𝑁 + 1))) | |
2 | 2z 8986 | . . 3 ⊢ 2 ∈ ℤ | |
3 | 2ne0 8722 | . . 3 ⊢ 2 ≠ 0 | |
4 | peano2z 8994 | . . 3 ⊢ (𝑁 ∈ ℤ → (𝑁 + 1) ∈ ℤ) | |
5 | dvdsval2 11344 | . . 3 ⊢ ((2 ∈ ℤ ∧ 2 ≠ 0 ∧ (𝑁 + 1) ∈ ℤ) → (2 ∥ (𝑁 + 1) ↔ ((𝑁 + 1) / 2) ∈ ℤ)) | |
6 | 2, 3, 4, 5 | mp3an12i 1302 | . 2 ⊢ (𝑁 ∈ ℤ → (2 ∥ (𝑁 + 1) ↔ ((𝑁 + 1) / 2) ∈ ℤ)) |
7 | 1, 6 | bitrd 187 | 1 ⊢ (𝑁 ∈ ℤ → (¬ 2 ∥ 𝑁 ↔ ((𝑁 + 1) / 2) ∈ ℤ)) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 104 ∈ wcel 1463 ≠ wne 2282 class class class wbr 3895 (class class class)co 5728 0cc0 7547 1c1 7548 + caddc 7550 / cdiv 8345 2c2 8681 ℤcz 8958 ∥ cdvds 11341 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 586 ax-in2 587 ax-io 681 ax-5 1406 ax-7 1407 ax-gen 1408 ax-ie1 1452 ax-ie2 1453 ax-8 1465 ax-10 1466 ax-11 1467 ax-i12 1468 ax-bndl 1469 ax-4 1470 ax-13 1474 ax-14 1475 ax-17 1489 ax-i9 1493 ax-ial 1497 ax-i5r 1498 ax-ext 2097 ax-sep 4006 ax-pow 4058 ax-pr 4091 ax-un 4315 ax-setind 4412 ax-cnex 7636 ax-resscn 7637 ax-1cn 7638 ax-1re 7639 ax-icn 7640 ax-addcl 7641 ax-addrcl 7642 ax-mulcl 7643 ax-mulrcl 7644 ax-addcom 7645 ax-mulcom 7646 ax-addass 7647 ax-mulass 7648 ax-distr 7649 ax-i2m1 7650 ax-0lt1 7651 ax-1rid 7652 ax-0id 7653 ax-rnegex 7654 ax-precex 7655 ax-cnre 7656 ax-pre-ltirr 7657 ax-pre-ltwlin 7658 ax-pre-lttrn 7659 ax-pre-apti 7660 ax-pre-ltadd 7661 ax-pre-mulgt0 7662 ax-pre-mulext 7663 |
This theorem depends on definitions: df-bi 116 df-3or 946 df-3an 947 df-tru 1317 df-fal 1320 df-xor 1337 df-nf 1420 df-sb 1719 df-eu 1978 df-mo 1979 df-clab 2102 df-cleq 2108 df-clel 2111 df-nfc 2244 df-ne 2283 df-nel 2378 df-ral 2395 df-rex 2396 df-reu 2397 df-rmo 2398 df-rab 2399 df-v 2659 df-sbc 2879 df-dif 3039 df-un 3041 df-in 3043 df-ss 3050 df-pw 3478 df-sn 3499 df-pr 3500 df-op 3502 df-uni 3703 df-int 3738 df-br 3896 df-opab 3950 df-id 4175 df-po 4178 df-iso 4179 df-xp 4505 df-rel 4506 df-cnv 4507 df-co 4508 df-dm 4509 df-iota 5046 df-fun 5083 df-fv 5089 df-riota 5684 df-ov 5731 df-oprab 5732 df-mpo 5733 df-pnf 7726 df-mnf 7727 df-xr 7728 df-ltxr 7729 df-le 7730 df-sub 7858 df-neg 7859 df-reap 8255 df-ap 8262 df-div 8346 df-inn 8631 df-2 8689 df-n0 8882 df-z 8959 df-dvds 11342 |
This theorem is referenced by: oddm1d2 11437 nn0oddm1d2 11454 nnoddm1d2 11455 |
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