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Mathbox for Alexander van der Vekens |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > nnoALTV | Structured version Visualization version GIF version |
Description: An alternate characterization of an odd number greater than 1. (Contributed by AV, 2-Jun-2020.) (Revised by AV, 21-Jun-2020.) |
Ref | Expression |
---|---|
nnoALTV | ⊢ ((𝑁 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ Odd ) → ((𝑁 − 1) / 2) ∈ ℕ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oddm1div2z 46856 | . . 3 ⊢ (𝑁 ∈ Odd → ((𝑁 − 1) / 2) ∈ ℤ) | |
2 | 1 | adantl 481 | . 2 ⊢ ((𝑁 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ Odd ) → ((𝑁 − 1) / 2) ∈ ℤ) |
3 | eluz2b1 12904 | . . . 4 ⊢ (𝑁 ∈ (ℤ≥‘2) ↔ (𝑁 ∈ ℤ ∧ 1 < 𝑁)) | |
4 | 1red 11216 | . . . . . . 7 ⊢ (𝑁 ∈ ℤ → 1 ∈ ℝ) | |
5 | zre 12563 | . . . . . . 7 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℝ) | |
6 | 4, 5 | posdifd 11802 | . . . . . 6 ⊢ (𝑁 ∈ ℤ → (1 < 𝑁 ↔ 0 < (𝑁 − 1))) |
7 | 6 | biimpa 476 | . . . . 5 ⊢ ((𝑁 ∈ ℤ ∧ 1 < 𝑁) → 0 < (𝑁 − 1)) |
8 | peano2zm 12606 | . . . . . . . . 9 ⊢ (𝑁 ∈ ℤ → (𝑁 − 1) ∈ ℤ) | |
9 | 8 | zred 12667 | . . . . . . . 8 ⊢ (𝑁 ∈ ℤ → (𝑁 − 1) ∈ ℝ) |
10 | 2re 12287 | . . . . . . . . 9 ⊢ 2 ∈ ℝ | |
11 | 10 | a1i 11 | . . . . . . . 8 ⊢ (𝑁 ∈ ℤ → 2 ∈ ℝ) |
12 | 2pos 12316 | . . . . . . . . 9 ⊢ 0 < 2 | |
13 | 12 | a1i 11 | . . . . . . . 8 ⊢ (𝑁 ∈ ℤ → 0 < 2) |
14 | 9, 11, 13 | 3jca 1125 | . . . . . . 7 ⊢ (𝑁 ∈ ℤ → ((𝑁 − 1) ∈ ℝ ∧ 2 ∈ ℝ ∧ 0 < 2)) |
15 | 14 | adantr 480 | . . . . . 6 ⊢ ((𝑁 ∈ ℤ ∧ 1 < 𝑁) → ((𝑁 − 1) ∈ ℝ ∧ 2 ∈ ℝ ∧ 0 < 2)) |
16 | gt0div 12081 | . . . . . 6 ⊢ (((𝑁 − 1) ∈ ℝ ∧ 2 ∈ ℝ ∧ 0 < 2) → (0 < (𝑁 − 1) ↔ 0 < ((𝑁 − 1) / 2))) | |
17 | 15, 16 | syl 17 | . . . . 5 ⊢ ((𝑁 ∈ ℤ ∧ 1 < 𝑁) → (0 < (𝑁 − 1) ↔ 0 < ((𝑁 − 1) / 2))) |
18 | 7, 17 | mpbid 231 | . . . 4 ⊢ ((𝑁 ∈ ℤ ∧ 1 < 𝑁) → 0 < ((𝑁 − 1) / 2)) |
19 | 3, 18 | sylbi 216 | . . 3 ⊢ (𝑁 ∈ (ℤ≥‘2) → 0 < ((𝑁 − 1) / 2)) |
20 | 19 | adantr 480 | . 2 ⊢ ((𝑁 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ Odd ) → 0 < ((𝑁 − 1) / 2)) |
21 | elnnz 12569 | . 2 ⊢ (((𝑁 − 1) / 2) ∈ ℕ ↔ (((𝑁 − 1) / 2) ∈ ℤ ∧ 0 < ((𝑁 − 1) / 2))) | |
22 | 2, 20, 21 | sylanbrc 582 | 1 ⊢ ((𝑁 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ Odd ) → ((𝑁 − 1) / 2) ∈ ℕ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 ∧ w3a 1084 ∈ wcel 2098 class class class wbr 5141 ‘cfv 6536 (class class class)co 7404 ℝcr 11108 0cc0 11109 1c1 11110 < clt 11249 − cmin 11445 / cdiv 11872 ℕcn 12213 2c2 12268 ℤcz 12559 ℤ≥cuz 12823 Odd codd 46847 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7721 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-pre-mulgt0 11186 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6293 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-om 7852 df-2nd 7972 df-frecs 8264 df-wrecs 8295 df-recs 8369 df-rdg 8408 df-er 8702 df-en 8939 df-dom 8940 df-sdom 8941 df-pnf 11251 df-mnf 11252 df-xr 11253 df-ltxr 11254 df-le 11255 df-sub 11447 df-neg 11448 df-div 11873 df-nn 12214 df-2 12276 df-n0 12474 df-z 12560 df-uz 12824 df-odd 46849 |
This theorem is referenced by: (None) |
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