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Mirrors > Home > ILE Home > Th. List > oddm1d2 | GIF version |
Description: An integer is odd iff its predecessor divided by 2 is an integer. This is another representation of odd numbers without using the divides relation. (Contributed by AV, 18-Jun-2021.) (Proof shortened by AV, 22-Jun-2021.) |
Ref | Expression |
---|---|
oddm1d2 | ⊢ (𝑁 ∈ ℤ → (¬ 2 ∥ 𝑁 ↔ ((𝑁 − 1) / 2) ∈ ℤ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oddp1d2 12005 | . 2 ⊢ (𝑁 ∈ ℤ → (¬ 2 ∥ 𝑁 ↔ ((𝑁 + 1) / 2) ∈ ℤ)) | |
2 | zob 12006 | . 2 ⊢ (𝑁 ∈ ℤ → (((𝑁 + 1) / 2) ∈ ℤ ↔ ((𝑁 − 1) / 2) ∈ ℤ)) | |
3 | 1, 2 | bitrd 188 | 1 ⊢ (𝑁 ∈ ℤ → (¬ 2 ∥ 𝑁 ↔ ((𝑁 − 1) / 2) ∈ ℤ)) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 105 ∈ wcel 2160 class class class wbr 4025 (class class class)co 5906 1c1 7859 + caddc 7861 − cmin 8176 / cdiv 8677 2c2 9019 ℤcz 9303 ∥ cdvds 11904 |
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 4143 ax-pow 4199 ax-pr 4234 ax-un 4458 ax-setind 4561 ax-cnex 7949 ax-resscn 7950 ax-1cn 7951 ax-1re 7952 ax-icn 7953 ax-addcl 7954 ax-addrcl 7955 ax-mulcl 7956 ax-mulrcl 7957 ax-addcom 7958 ax-mulcom 7959 ax-addass 7960 ax-mulass 7961 ax-distr 7962 ax-i2m1 7963 ax-0lt1 7964 ax-1rid 7965 ax-0id 7966 ax-rnegex 7967 ax-precex 7968 ax-cnre 7969 ax-pre-ltirr 7970 ax-pre-ltwlin 7971 ax-pre-lttrn 7972 ax-pre-apti 7973 ax-pre-ltadd 7974 ax-pre-mulgt0 7975 ax-pre-mulext 7976 |
This theorem depends on definitions: df-bi 117 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-xor 1387 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 2758 df-sbc 2982 df-dif 3151 df-un 3153 df-in 3155 df-ss 3162 df-pw 3599 df-sn 3620 df-pr 3621 df-op 3623 df-uni 3832 df-int 3867 df-br 4026 df-opab 4087 df-id 4318 df-po 4321 df-iso 4322 df-xp 4657 df-rel 4658 df-cnv 4659 df-co 4660 df-dm 4661 df-iota 5203 df-fun 5244 df-fv 5250 df-riota 5861 df-ov 5909 df-oprab 5910 df-mpo 5911 df-pnf 8042 df-mnf 8043 df-xr 8044 df-ltxr 8045 df-le 8046 df-sub 8178 df-neg 8179 df-reap 8580 df-ap 8587 df-div 8678 df-inn 8969 df-2 9027 df-n0 9227 df-z 9304 df-dvds 11905 |
This theorem is referenced by: nnoddn2prm 12372 gausslemma2dlem5 15110 |
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