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Mirrors > Home > MPE Home > Th. List > oddp1d2 | Structured version Visualization version 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 12693 and zeo2 12694. (Contributed by AV, 22-Jun-2021.) |
Ref | Expression |
---|---|
oddp1d2 | ⊢ (𝑁 ∈ ℤ → (¬ 2 ∥ 𝑁 ↔ ((𝑁 + 1) / 2) ∈ ℤ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oddp1even 16340 | . 2 ⊢ (𝑁 ∈ ℤ → (¬ 2 ∥ 𝑁 ↔ 2 ∥ (𝑁 + 1))) | |
2 | 2z 12639 | . . 3 ⊢ 2 ∈ ℤ | |
3 | 2ne0 12361 | . . 3 ⊢ 2 ≠ 0 | |
4 | peano2z 12648 | . . 3 ⊢ (𝑁 ∈ ℤ → (𝑁 + 1) ∈ ℤ) | |
5 | dvdsval2 16253 | . . 3 ⊢ ((2 ∈ ℤ ∧ 2 ≠ 0 ∧ (𝑁 + 1) ∈ ℤ) → (2 ∥ (𝑁 + 1) ↔ ((𝑁 + 1) / 2) ∈ ℤ)) | |
6 | 2, 3, 4, 5 | mp3an12i 1462 | . 2 ⊢ (𝑁 ∈ ℤ → (2 ∥ (𝑁 + 1) ↔ ((𝑁 + 1) / 2) ∈ ℤ)) |
7 | 1, 6 | bitrd 278 | 1 ⊢ (𝑁 ∈ ℤ → (¬ 2 ∥ 𝑁 ↔ ((𝑁 + 1) / 2) ∈ ℤ)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∈ wcel 2099 ≠ wne 2930 class class class wbr 5145 (class class class)co 7415 0cc0 11148 1c1 11149 + caddc 11151 / cdiv 11911 2c2 12312 ℤcz 12603 ∥ cdvds 16250 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-sep 5296 ax-nul 5303 ax-pow 5361 ax-pr 5425 ax-un 7737 ax-resscn 11205 ax-1cn 11206 ax-icn 11207 ax-addcl 11208 ax-addrcl 11209 ax-mulcl 11210 ax-mulrcl 11211 ax-mulcom 11212 ax-addass 11213 ax-mulass 11214 ax-distr 11215 ax-i2m1 11216 ax-1ne0 11217 ax-1rid 11218 ax-rnegex 11219 ax-rrecex 11220 ax-cnre 11221 ax-pre-lttri 11222 ax-pre-lttrn 11223 ax-pre-ltadd 11224 ax-pre-mulgt0 11225 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3365 df-reu 3366 df-rab 3421 df-v 3466 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3968 df-nul 4325 df-if 4526 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4908 df-iun 4997 df-br 5146 df-opab 5208 df-mpt 5229 df-tr 5263 df-id 5572 df-eprel 5578 df-po 5586 df-so 5587 df-fr 5629 df-we 5631 df-xp 5680 df-rel 5681 df-cnv 5682 df-co 5683 df-dm 5684 df-rn 5685 df-res 5686 df-ima 5687 df-pred 6304 df-ord 6370 df-on 6371 df-lim 6372 df-suc 6373 df-iota 6497 df-fun 6547 df-fn 6548 df-f 6549 df-f1 6550 df-fo 6551 df-f1o 6552 df-fv 6553 df-riota 7371 df-ov 7418 df-oprab 7419 df-mpo 7420 df-om 7868 df-2nd 7995 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-er 8725 df-en 8966 df-dom 8967 df-sdom 8968 df-pnf 11290 df-mnf 11291 df-xr 11292 df-ltxr 11293 df-le 11294 df-sub 11486 df-neg 11487 df-div 11912 df-nn 12258 df-2 12320 df-n0 12518 df-z 12604 df-dvds 16251 |
This theorem is referenced by: oddm1d2 16356 nn0oddm1d2 16381 nnoddm1d2 16382 |
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