Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 12015 | . . . . . . 7 ⊢ 2 ∈ ℤ | |
2 | 1 | a1i 11 | . . . . . 6 ⊢ (𝑁 ∈ ℤ → 2 ∈ ℤ) |
3 | id 22 | . . . . . 6 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℤ) | |
4 | 2, 3 | zmulcld 12094 | . . . . 5 ⊢ (𝑁 ∈ ℤ → (2 · 𝑁) ∈ ℤ) |
5 | 4 | peano2zd 12091 | . . . 4 ⊢ (𝑁 ∈ ℤ → ((2 · 𝑁) + 1) ∈ ℤ) |
6 | 5 | zred 12088 | . . 3 ⊢ (𝑁 ∈ ℤ → ((2 · 𝑁) + 1) ∈ ℝ) |
7 | 2rp 12395 | . . . 4 ⊢ 2 ∈ ℝ+ | |
8 | 7 | a1i 11 | . . 3 ⊢ (𝑁 ∈ ℤ → 2 ∈ ℝ+) |
9 | 6, 8 | ge0divd 12470 | . 2 ⊢ (𝑁 ∈ ℤ → (0 ≤ ((2 · 𝑁) + 1) ↔ 0 ≤ (((2 · 𝑁) + 1) / 2))) |
10 | 4 | zcnd 12089 | . . . . 5 ⊢ (𝑁 ∈ ℤ → (2 · 𝑁) ∈ ℂ) |
11 | 1cnd 10636 | . . . . 5 ⊢ (𝑁 ∈ ℤ → 1 ∈ ℂ) | |
12 | 2cnne0 11848 | . . . . . 6 ⊢ (2 ∈ ℂ ∧ 2 ≠ 0) | |
13 | 12 | a1i 11 | . . . . 5 ⊢ (𝑁 ∈ ℤ → (2 ∈ ℂ ∧ 2 ≠ 0)) |
14 | divdir 11323 | . . . . 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 1367 | . . . 4 ⊢ (𝑁 ∈ ℤ → (((2 · 𝑁) + 1) / 2) = (((2 · 𝑁) / 2) + (1 / 2))) |
16 | zcn 11987 | . . . . . 6 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℂ) | |
17 | 2cnd 11716 | . . . . . 6 ⊢ (𝑁 ∈ ℤ → 2 ∈ ℂ) | |
18 | 2ne0 11742 | . . . . . . 7 ⊢ 2 ≠ 0 | |
19 | 18 | a1i 11 | . . . . . 6 ⊢ (𝑁 ∈ ℤ → 2 ≠ 0) |
20 | 16, 17, 19 | divcan3d 11421 | . . . . 5 ⊢ (𝑁 ∈ ℤ → ((2 · 𝑁) / 2) = 𝑁) |
21 | 20 | oveq1d 7171 | . . . 4 ⊢ (𝑁 ∈ ℤ → (((2 · 𝑁) / 2) + (1 / 2)) = (𝑁 + (1 / 2))) |
22 | 15, 21 | eqtrd 2856 | . . 3 ⊢ (𝑁 ∈ ℤ → (((2 · 𝑁) + 1) / 2) = (𝑁 + (1 / 2))) |
23 | 22 | breq2d 5078 | . 2 ⊢ (𝑁 ∈ ℤ → (0 ≤ (((2 · 𝑁) + 1) / 2) ↔ 0 ≤ (𝑁 + (1 / 2)))) |
24 | zre 11986 | . . . 4 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℝ) | |
25 | halfre 11852 | . . . . 5 ⊢ (1 / 2) ∈ ℝ | |
26 | 25 | a1i 11 | . . . 4 ⊢ (𝑁 ∈ ℤ → (1 / 2) ∈ ℝ) |
27 | 24, 26 | readdcld 10670 | . . 3 ⊢ (𝑁 ∈ ℤ → (𝑁 + (1 / 2)) ∈ ℝ) |
28 | halfge0 11855 | . . . 4 ⊢ 0 ≤ (1 / 2) | |
29 | 24, 26 | addge01d 11228 | . . . 4 ⊢ (𝑁 ∈ ℤ → (0 ≤ (1 / 2) ↔ 𝑁 ≤ (𝑁 + (1 / 2)))) |
30 | 28, 29 | mpbii 235 | . . 3 ⊢ (𝑁 ∈ ℤ → 𝑁 ≤ (𝑁 + (1 / 2))) |
31 | 1red 10642 | . . . 4 ⊢ (𝑁 ∈ ℤ → 1 ∈ ℝ) | |
32 | halflt1 11856 | . . . . 5 ⊢ (1 / 2) < 1 | |
33 | 32 | a1i 11 | . . . 4 ⊢ (𝑁 ∈ ℤ → (1 / 2) < 1) |
34 | 26, 31, 24, 33 | ltadd2dd 10799 | . . 3 ⊢ (𝑁 ∈ ℤ → (𝑁 + (1 / 2)) < (𝑁 + 1)) |
35 | btwnzge0 13199 | . . 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 836 | . 2 ⊢ (𝑁 ∈ ℤ → (0 ≤ (𝑁 + (1 / 2)) ↔ 0 ≤ 𝑁)) |
37 | 9, 23, 36 | 3bitrd 307 | 1 ⊢ (𝑁 ∈ ℤ → (0 ≤ ((2 · 𝑁) + 1) ↔ 0 ≤ 𝑁)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ≠ wne 3016 class class class wbr 5066 (class class class)co 7156 ℂcc 10535 ℝcr 10536 0cc0 10537 1c1 10538 + caddc 10540 · cmul 10542 < clt 10675 ≤ cle 10676 / cdiv 11297 2c2 11693 ℤcz 11982 ℝ+crp 12390 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 ax-pre-sup 10615 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-sup 8906 df-inf 8907 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-div 11298 df-nn 11639 df-2 11701 df-n0 11899 df-z 11983 df-uz 12245 df-rp 12391 df-fl 13163 |
This theorem is referenced by: oddnn02np1 15697 |
Copyright terms: Public domain | W3C validator |