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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.) |
Ref | Expression |
---|---|
2tnp1ge0ge0 | ⊢ (𝑁 ∈ ℤ → (0 ≤ ((2 · 𝑁) + 1) ↔ 0 ≤ 𝑁)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 2z 11409 | . . . . . . 7 ⊢ 2 ∈ ℤ | |
2 | 1 | a1i 11 | . . . . . 6 ⊢ (𝑁 ∈ ℤ → 2 ∈ ℤ) |
3 | id 22 | . . . . . 6 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℤ) | |
4 | 2, 3 | zmulcld 11488 | . . . . 5 ⊢ (𝑁 ∈ ℤ → (2 · 𝑁) ∈ ℤ) |
5 | 4 | peano2zd 11485 | . . . 4 ⊢ (𝑁 ∈ ℤ → ((2 · 𝑁) + 1) ∈ ℤ) |
6 | 5 | zred 11482 | . . 3 ⊢ (𝑁 ∈ ℤ → ((2 · 𝑁) + 1) ∈ ℝ) |
7 | 2re 11090 | . . . 4 ⊢ 2 ∈ ℝ | |
8 | 7 | a1i 11 | . . 3 ⊢ (𝑁 ∈ ℤ → 2 ∈ ℝ) |
9 | 2pos 11112 | . . . 4 ⊢ 0 < 2 | |
10 | 9 | a1i 11 | . . 3 ⊢ (𝑁 ∈ ℤ → 0 < 2) |
11 | ge0div 10890 | . . 3 ⊢ ((((2 · 𝑁) + 1) ∈ ℝ ∧ 2 ∈ ℝ ∧ 0 < 2) → (0 ≤ ((2 · 𝑁) + 1) ↔ 0 ≤ (((2 · 𝑁) + 1) / 2))) | |
12 | 6, 8, 10, 11 | syl3anc 1326 | . 2 ⊢ (𝑁 ∈ ℤ → (0 ≤ ((2 · 𝑁) + 1) ↔ 0 ≤ (((2 · 𝑁) + 1) / 2))) |
13 | 4 | zcnd 11483 | . . . . 5 ⊢ (𝑁 ∈ ℤ → (2 · 𝑁) ∈ ℂ) |
14 | 1cnd 10056 | . . . . 5 ⊢ (𝑁 ∈ ℤ → 1 ∈ ℂ) | |
15 | 2cnne0 11242 | . . . . . 6 ⊢ (2 ∈ ℂ ∧ 2 ≠ 0) | |
16 | 15 | a1i 11 | . . . . 5 ⊢ (𝑁 ∈ ℤ → (2 ∈ ℂ ∧ 2 ≠ 0)) |
17 | divdir 10710 | . . . . 5 ⊢ (((2 · 𝑁) ∈ ℂ ∧ 1 ∈ ℂ ∧ (2 ∈ ℂ ∧ 2 ≠ 0)) → (((2 · 𝑁) + 1) / 2) = (((2 · 𝑁) / 2) + (1 / 2))) | |
18 | 13, 14, 16, 17 | syl3anc 1326 | . . . 4 ⊢ (𝑁 ∈ ℤ → (((2 · 𝑁) + 1) / 2) = (((2 · 𝑁) / 2) + (1 / 2))) |
19 | zcn 11382 | . . . . . 6 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℂ) | |
20 | 2cnd 11093 | . . . . . 6 ⊢ (𝑁 ∈ ℤ → 2 ∈ ℂ) | |
21 | 2ne0 11113 | . . . . . . 7 ⊢ 2 ≠ 0 | |
22 | 21 | a1i 11 | . . . . . 6 ⊢ (𝑁 ∈ ℤ → 2 ≠ 0) |
23 | 19, 20, 22 | divcan3d 10806 | . . . . 5 ⊢ (𝑁 ∈ ℤ → ((2 · 𝑁) / 2) = 𝑁) |
24 | 23 | oveq1d 6665 | . . . 4 ⊢ (𝑁 ∈ ℤ → (((2 · 𝑁) / 2) + (1 / 2)) = (𝑁 + (1 / 2))) |
25 | 18, 24 | eqtrd 2656 | . . 3 ⊢ (𝑁 ∈ ℤ → (((2 · 𝑁) + 1) / 2) = (𝑁 + (1 / 2))) |
26 | 25 | breq2d 4665 | . 2 ⊢ (𝑁 ∈ ℤ → (0 ≤ (((2 · 𝑁) + 1) / 2) ↔ 0 ≤ (𝑁 + (1 / 2)))) |
27 | zre 11381 | . . . 4 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℝ) | |
28 | halfre 11246 | . . . . 5 ⊢ (1 / 2) ∈ ℝ | |
29 | 28 | a1i 11 | . . . 4 ⊢ (𝑁 ∈ ℤ → (1 / 2) ∈ ℝ) |
30 | 27, 29 | readdcld 10069 | . . 3 ⊢ (𝑁 ∈ ℤ → (𝑁 + (1 / 2)) ∈ ℝ) |
31 | halfge0 11249 | . . . 4 ⊢ 0 ≤ (1 / 2) | |
32 | 27, 29 | addge01d 10615 | . . . 4 ⊢ (𝑁 ∈ ℤ → (0 ≤ (1 / 2) ↔ 𝑁 ≤ (𝑁 + (1 / 2)))) |
33 | 31, 32 | mpbii 223 | . . 3 ⊢ (𝑁 ∈ ℤ → 𝑁 ≤ (𝑁 + (1 / 2))) |
34 | 1red 10055 | . . . 4 ⊢ (𝑁 ∈ ℤ → 1 ∈ ℝ) | |
35 | halflt1 11250 | . . . . 5 ⊢ (1 / 2) < 1 | |
36 | 35 | a1i 11 | . . . 4 ⊢ (𝑁 ∈ ℤ → (1 / 2) < 1) |
37 | 29, 34, 27, 36 | ltadd2dd 10196 | . . 3 ⊢ (𝑁 ∈ ℤ → (𝑁 + (1 / 2)) < (𝑁 + 1)) |
38 | btwnzge0 12629 | . . 3 ⊢ ((((𝑁 + (1 / 2)) ∈ ℝ ∧ 𝑁 ∈ ℤ) ∧ (𝑁 ≤ (𝑁 + (1 / 2)) ∧ (𝑁 + (1 / 2)) < (𝑁 + 1))) → (0 ≤ (𝑁 + (1 / 2)) ↔ 0 ≤ 𝑁)) | |
39 | 30, 3, 33, 37, 38 | syl22anc 1327 | . 2 ⊢ (𝑁 ∈ ℤ → (0 ≤ (𝑁 + (1 / 2)) ↔ 0 ≤ 𝑁)) |
40 | 12, 26, 39 | 3bitrd 294 | 1 ⊢ (𝑁 ∈ ℤ → (0 ≤ ((2 · 𝑁) + 1) ↔ 0 ≤ 𝑁)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ≠ wne 2794 class class class wbr 4653 (class class class)co 6650 ℂcc 9934 ℝcr 9935 0cc0 9936 1c1 9937 + caddc 9939 · cmul 9941 < clt 10074 ≤ cle 10075 / cdiv 10684 2c2 11070 ℤcz 11377 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-sup 8348 df-inf 8349 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-n0 11293 df-z 11378 df-uz 11688 df-fl 12593 |
This theorem is referenced by: oddnn02np1 15072 |
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