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Mirrors > Home > ILE Home > Th. List > flodddiv4t2lthalf | GIF version |
Description: The floor of an odd number divided by 4, multiplied by 2 is less than the half of the odd number. (Contributed by AV, 4-Jul-2021.) |
Ref | Expression |
---|---|
flodddiv4t2lthalf | ⊢ ((𝑁 ∈ ℤ ∧ ¬ 2 ∥ 𝑁) → ((⌊‘(𝑁 / 4)) · 2) < (𝑁 / 2)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | flodddiv4lt 11633 | . . 3 ⊢ ((𝑁 ∈ ℤ ∧ ¬ 2 ∥ 𝑁) → (⌊‘(𝑁 / 4)) < (𝑁 / 4)) | |
2 | 4nn 8883 | . . . . . . . 8 ⊢ 4 ∈ ℕ | |
3 | znq 9416 | . . . . . . . 8 ⊢ ((𝑁 ∈ ℤ ∧ 4 ∈ ℕ) → (𝑁 / 4) ∈ ℚ) | |
4 | 2, 3 | mpan2 421 | . . . . . . 7 ⊢ (𝑁 ∈ ℤ → (𝑁 / 4) ∈ ℚ) |
5 | 4 | flqcld 10050 | . . . . . 6 ⊢ (𝑁 ∈ ℤ → (⌊‘(𝑁 / 4)) ∈ ℤ) |
6 | 5 | zred 9173 | . . . . 5 ⊢ (𝑁 ∈ ℤ → (⌊‘(𝑁 / 4)) ∈ ℝ) |
7 | 6 | adantr 274 | . . . 4 ⊢ ((𝑁 ∈ ℤ ∧ ¬ 2 ∥ 𝑁) → (⌊‘(𝑁 / 4)) ∈ ℝ) |
8 | qre 9417 | . . . . . 6 ⊢ ((𝑁 / 4) ∈ ℚ → (𝑁 / 4) ∈ ℝ) | |
9 | 4, 8 | syl 14 | . . . . 5 ⊢ (𝑁 ∈ ℤ → (𝑁 / 4) ∈ ℝ) |
10 | 9 | adantr 274 | . . . 4 ⊢ ((𝑁 ∈ ℤ ∧ ¬ 2 ∥ 𝑁) → (𝑁 / 4) ∈ ℝ) |
11 | 2re 8790 | . . . . . 6 ⊢ 2 ∈ ℝ | |
12 | 2pos 8811 | . . . . . 6 ⊢ 0 < 2 | |
13 | 11, 12 | pm3.2i 270 | . . . . 5 ⊢ (2 ∈ ℝ ∧ 0 < 2) |
14 | 13 | a1i 9 | . . . 4 ⊢ ((𝑁 ∈ ℤ ∧ ¬ 2 ∥ 𝑁) → (2 ∈ ℝ ∧ 0 < 2)) |
15 | ltmul1 8354 | . . . 4 ⊢ (((⌊‘(𝑁 / 4)) ∈ ℝ ∧ (𝑁 / 4) ∈ ℝ ∧ (2 ∈ ℝ ∧ 0 < 2)) → ((⌊‘(𝑁 / 4)) < (𝑁 / 4) ↔ ((⌊‘(𝑁 / 4)) · 2) < ((𝑁 / 4) · 2))) | |
16 | 7, 10, 14, 15 | syl3anc 1216 | . . 3 ⊢ ((𝑁 ∈ ℤ ∧ ¬ 2 ∥ 𝑁) → ((⌊‘(𝑁 / 4)) < (𝑁 / 4) ↔ ((⌊‘(𝑁 / 4)) · 2) < ((𝑁 / 4) · 2))) |
17 | 1, 16 | mpbid 146 | . 2 ⊢ ((𝑁 ∈ ℤ ∧ ¬ 2 ∥ 𝑁) → ((⌊‘(𝑁 / 4)) · 2) < ((𝑁 / 4) · 2)) |
18 | zcn 9059 | . . . . . 6 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℂ) | |
19 | 18 | halfcld 8964 | . . . . 5 ⊢ (𝑁 ∈ ℤ → (𝑁 / 2) ∈ ℂ) |
20 | 2cnd 8793 | . . . . 5 ⊢ (𝑁 ∈ ℤ → 2 ∈ ℂ) | |
21 | 2ap0 8813 | . . . . . 6 ⊢ 2 # 0 | |
22 | 21 | a1i 9 | . . . . 5 ⊢ (𝑁 ∈ ℤ → 2 # 0) |
23 | 19, 20, 22 | divcanap1d 8551 | . . . 4 ⊢ (𝑁 ∈ ℤ → (((𝑁 / 2) / 2) · 2) = (𝑁 / 2)) |
24 | 18, 20, 20, 22, 22 | divdivap1d 8582 | . . . . . 6 ⊢ (𝑁 ∈ ℤ → ((𝑁 / 2) / 2) = (𝑁 / (2 · 2))) |
25 | 2t2e4 8874 | . . . . . . . 8 ⊢ (2 · 2) = 4 | |
26 | 25 | a1i 9 | . . . . . . 7 ⊢ (𝑁 ∈ ℤ → (2 · 2) = 4) |
27 | 26 | oveq2d 5790 | . . . . . 6 ⊢ (𝑁 ∈ ℤ → (𝑁 / (2 · 2)) = (𝑁 / 4)) |
28 | 24, 27 | eqtrd 2172 | . . . . 5 ⊢ (𝑁 ∈ ℤ → ((𝑁 / 2) / 2) = (𝑁 / 4)) |
29 | 28 | oveq1d 5789 | . . . 4 ⊢ (𝑁 ∈ ℤ → (((𝑁 / 2) / 2) · 2) = ((𝑁 / 4) · 2)) |
30 | 23, 29 | eqtr3d 2174 | . . 3 ⊢ (𝑁 ∈ ℤ → (𝑁 / 2) = ((𝑁 / 4) · 2)) |
31 | 30 | adantr 274 | . 2 ⊢ ((𝑁 ∈ ℤ ∧ ¬ 2 ∥ 𝑁) → (𝑁 / 2) = ((𝑁 / 4) · 2)) |
32 | 17, 31 | breqtrrd 3956 | 1 ⊢ ((𝑁 ∈ ℤ ∧ ¬ 2 ∥ 𝑁) → ((⌊‘(𝑁 / 4)) · 2) < (𝑁 / 2)) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 103 ↔ wb 104 = wceq 1331 ∈ wcel 1480 class class class wbr 3929 ‘cfv 5123 (class class class)co 5774 ℝcr 7619 0cc0 7620 · cmul 7625 < clt 7800 # cap 8343 / cdiv 8432 ℕcn 8720 2c2 8771 4c4 8773 ℤcz 9054 ℚcq 9411 ⌊cfl 10041 ∥ cdvds 11493 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-cnex 7711 ax-resscn 7712 ax-1cn 7713 ax-1re 7714 ax-icn 7715 ax-addcl 7716 ax-addrcl 7717 ax-mulcl 7718 ax-mulrcl 7719 ax-addcom 7720 ax-mulcom 7721 ax-addass 7722 ax-mulass 7723 ax-distr 7724 ax-i2m1 7725 ax-0lt1 7726 ax-1rid 7727 ax-0id 7728 ax-rnegex 7729 ax-precex 7730 ax-cnre 7731 ax-pre-ltirr 7732 ax-pre-ltwlin 7733 ax-pre-lttrn 7734 ax-pre-apti 7735 ax-pre-ltadd 7736 ax-pre-mulgt0 7737 ax-pre-mulext 7738 ax-arch 7739 |
This theorem depends on definitions: df-bi 116 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-nel 2404 df-ral 2421 df-rex 2422 df-reu 2423 df-rmo 2424 df-rab 2425 df-v 2688 df-sbc 2910 df-csb 3004 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-int 3772 df-iun 3815 df-br 3930 df-opab 3990 df-mpt 3991 df-id 4215 df-po 4218 df-iso 4219 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-fv 5131 df-riota 5730 df-ov 5777 df-oprab 5778 df-mpo 5779 df-1st 6038 df-2nd 6039 df-pnf 7802 df-mnf 7803 df-xr 7804 df-ltxr 7805 df-le 7806 df-sub 7935 df-neg 7936 df-reap 8337 df-ap 8344 df-div 8433 df-inn 8721 df-2 8779 df-3 8780 df-4 8781 df-n0 8978 df-z 9055 df-q 9412 df-rp 9442 df-fl 10043 df-dvds 11494 |
This theorem is referenced by: (None) |
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