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Mirrors > Home > MPE Home > Th. List > 2tnp1ge0ge0 | Structured version Visualization version GIF version |
Description: Two times an integer plus one is not negative iff the integer is not negative. (Contributed by AV, 19-Jun-2021.) (Proof shortened by AV, 10-Jul-2022.) |
Ref | Expression |
---|---|
2tnp1ge0ge0 | ⊢ (𝑁 ∈ ℤ → (0 ≤ ((2 · 𝑁) + 1) ↔ 0 ≤ 𝑁)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 2z 12002 | . . . . . . 7 ⊢ 2 ∈ ℤ | |
2 | 1 | a1i 11 | . . . . . 6 ⊢ (𝑁 ∈ ℤ → 2 ∈ ℤ) |
3 | id 22 | . . . . . 6 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℤ) | |
4 | 2, 3 | zmulcld 12081 | . . . . 5 ⊢ (𝑁 ∈ ℤ → (2 · 𝑁) ∈ ℤ) |
5 | 4 | peano2zd 12078 | . . . 4 ⊢ (𝑁 ∈ ℤ → ((2 · 𝑁) + 1) ∈ ℤ) |
6 | 5 | zred 12075 | . . 3 ⊢ (𝑁 ∈ ℤ → ((2 · 𝑁) + 1) ∈ ℝ) |
7 | 2rp 12382 | . . . 4 ⊢ 2 ∈ ℝ+ | |
8 | 7 | a1i 11 | . . 3 ⊢ (𝑁 ∈ ℤ → 2 ∈ ℝ+) |
9 | 6, 8 | ge0divd 12457 | . 2 ⊢ (𝑁 ∈ ℤ → (0 ≤ ((2 · 𝑁) + 1) ↔ 0 ≤ (((2 · 𝑁) + 1) / 2))) |
10 | 4 | zcnd 12076 | . . . . 5 ⊢ (𝑁 ∈ ℤ → (2 · 𝑁) ∈ ℂ) |
11 | 1cnd 10625 | . . . . 5 ⊢ (𝑁 ∈ ℤ → 1 ∈ ℂ) | |
12 | 2cnne0 11835 | . . . . . 6 ⊢ (2 ∈ ℂ ∧ 2 ≠ 0) | |
13 | 12 | a1i 11 | . . . . 5 ⊢ (𝑁 ∈ ℤ → (2 ∈ ℂ ∧ 2 ≠ 0)) |
14 | divdir 11312 | . . . . 5 ⊢ (((2 · 𝑁) ∈ ℂ ∧ 1 ∈ ℂ ∧ (2 ∈ ℂ ∧ 2 ≠ 0)) → (((2 · 𝑁) + 1) / 2) = (((2 · 𝑁) / 2) + (1 / 2))) | |
15 | 10, 11, 13, 14 | syl3anc 1368 | . . . 4 ⊢ (𝑁 ∈ ℤ → (((2 · 𝑁) + 1) / 2) = (((2 · 𝑁) / 2) + (1 / 2))) |
16 | zcn 11974 | . . . . . 6 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℂ) | |
17 | 2cnd 11703 | . . . . . 6 ⊢ (𝑁 ∈ ℤ → 2 ∈ ℂ) | |
18 | 2ne0 11729 | . . . . . . 7 ⊢ 2 ≠ 0 | |
19 | 18 | a1i 11 | . . . . . 6 ⊢ (𝑁 ∈ ℤ → 2 ≠ 0) |
20 | 16, 17, 19 | divcan3d 11410 | . . . . 5 ⊢ (𝑁 ∈ ℤ → ((2 · 𝑁) / 2) = 𝑁) |
21 | 20 | oveq1d 7150 | . . . 4 ⊢ (𝑁 ∈ ℤ → (((2 · 𝑁) / 2) + (1 / 2)) = (𝑁 + (1 / 2))) |
22 | 15, 21 | eqtrd 2833 | . . 3 ⊢ (𝑁 ∈ ℤ → (((2 · 𝑁) + 1) / 2) = (𝑁 + (1 / 2))) |
23 | 22 | breq2d 5042 | . 2 ⊢ (𝑁 ∈ ℤ → (0 ≤ (((2 · 𝑁) + 1) / 2) ↔ 0 ≤ (𝑁 + (1 / 2)))) |
24 | zre 11973 | . . . 4 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℝ) | |
25 | halfre 11839 | . . . . 5 ⊢ (1 / 2) ∈ ℝ | |
26 | 25 | a1i 11 | . . . 4 ⊢ (𝑁 ∈ ℤ → (1 / 2) ∈ ℝ) |
27 | 24, 26 | readdcld 10659 | . . 3 ⊢ (𝑁 ∈ ℤ → (𝑁 + (1 / 2)) ∈ ℝ) |
28 | halfge0 11842 | . . . 4 ⊢ 0 ≤ (1 / 2) | |
29 | 24, 26 | addge01d 11217 | . . . 4 ⊢ (𝑁 ∈ ℤ → (0 ≤ (1 / 2) ↔ 𝑁 ≤ (𝑁 + (1 / 2)))) |
30 | 28, 29 | mpbii 236 | . . 3 ⊢ (𝑁 ∈ ℤ → 𝑁 ≤ (𝑁 + (1 / 2))) |
31 | 1red 10631 | . . . 4 ⊢ (𝑁 ∈ ℤ → 1 ∈ ℝ) | |
32 | halflt1 11843 | . . . . 5 ⊢ (1 / 2) < 1 | |
33 | 32 | a1i 11 | . . . 4 ⊢ (𝑁 ∈ ℤ → (1 / 2) < 1) |
34 | 26, 31, 24, 33 | ltadd2dd 10788 | . . 3 ⊢ (𝑁 ∈ ℤ → (𝑁 + (1 / 2)) < (𝑁 + 1)) |
35 | btwnzge0 13193 | . . 3 ⊢ ((((𝑁 + (1 / 2)) ∈ ℝ ∧ 𝑁 ∈ ℤ) ∧ (𝑁 ≤ (𝑁 + (1 / 2)) ∧ (𝑁 + (1 / 2)) < (𝑁 + 1))) → (0 ≤ (𝑁 + (1 / 2)) ↔ 0 ≤ 𝑁)) | |
36 | 27, 3, 30, 34, 35 | syl22anc 837 | . 2 ⊢ (𝑁 ∈ ℤ → (0 ≤ (𝑁 + (1 / 2)) ↔ 0 ≤ 𝑁)) |
37 | 9, 23, 36 | 3bitrd 308 | 1 ⊢ (𝑁 ∈ ℤ → (0 ≤ ((2 · 𝑁) + 1) ↔ 0 ≤ 𝑁)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ≠ wne 2987 class class class wbr 5030 (class class class)co 7135 ℂcc 10524 ℝcr 10525 0cc0 10526 1c1 10527 + caddc 10529 · cmul 10531 < clt 10664 ≤ cle 10665 / cdiv 11286 2c2 11680 ℤcz 11969 ℝ+crp 12377 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-pre-sup 10604 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-sup 8890 df-inf 8891 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-nn 11626 df-2 11688 df-n0 11886 df-z 11970 df-uz 12232 df-rp 12378 df-fl 13157 |
This theorem is referenced by: oddnn02np1 15689 |
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