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Mirrors > Home > MPE Home > Th. List > flodddiv4t2lthalf | Structured version Visualization version 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.) (Proof shortened by AV, 10-Jul-2022.) |
Ref | Expression |
---|---|
flodddiv4t2lthalf | ⊢ ((𝑁 ∈ ℤ ∧ ¬ 2 ∥ 𝑁) → ((⌊‘(𝑁 / 4)) · 2) < (𝑁 / 2)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | flodddiv4lt 15756 | . . 3 ⊢ ((𝑁 ∈ ℤ ∧ ¬ 2 ∥ 𝑁) → (⌊‘(𝑁 / 4)) < (𝑁 / 4)) | |
2 | zre 11973 | . . . . . . . 8 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℝ) | |
3 | 4re 11709 | . . . . . . . . 9 ⊢ 4 ∈ ℝ | |
4 | 3 | a1i 11 | . . . . . . . 8 ⊢ (𝑁 ∈ ℤ → 4 ∈ ℝ) |
5 | 4ne0 11733 | . . . . . . . . 9 ⊢ 4 ≠ 0 | |
6 | 5 | a1i 11 | . . . . . . . 8 ⊢ (𝑁 ∈ ℤ → 4 ≠ 0) |
7 | 2, 4, 6 | redivcld 11457 | . . . . . . 7 ⊢ (𝑁 ∈ ℤ → (𝑁 / 4) ∈ ℝ) |
8 | 7 | flcld 13163 | . . . . . 6 ⊢ (𝑁 ∈ ℤ → (⌊‘(𝑁 / 4)) ∈ ℤ) |
9 | 8 | zred 12075 | . . . . 5 ⊢ (𝑁 ∈ ℤ → (⌊‘(𝑁 / 4)) ∈ ℝ) |
10 | 2rp 12382 | . . . . . 6 ⊢ 2 ∈ ℝ+ | |
11 | 10 | a1i 11 | . . . . 5 ⊢ (𝑁 ∈ ℤ → 2 ∈ ℝ+) |
12 | 9, 7, 11 | ltmul1d 12460 | . . . 4 ⊢ (𝑁 ∈ ℤ → ((⌊‘(𝑁 / 4)) < (𝑁 / 4) ↔ ((⌊‘(𝑁 / 4)) · 2) < ((𝑁 / 4) · 2))) |
13 | 12 | adantr 484 | . . 3 ⊢ ((𝑁 ∈ ℤ ∧ ¬ 2 ∥ 𝑁) → ((⌊‘(𝑁 / 4)) < (𝑁 / 4) ↔ ((⌊‘(𝑁 / 4)) · 2) < ((𝑁 / 4) · 2))) |
14 | 1, 13 | mpbid 235 | . 2 ⊢ ((𝑁 ∈ ℤ ∧ ¬ 2 ∥ 𝑁) → ((⌊‘(𝑁 / 4)) · 2) < ((𝑁 / 4) · 2)) |
15 | zcn 11974 | . . . . . 6 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℂ) | |
16 | 15 | halfcld 11870 | . . . . 5 ⊢ (𝑁 ∈ ℤ → (𝑁 / 2) ∈ ℂ) |
17 | 2cnd 11703 | . . . . 5 ⊢ (𝑁 ∈ ℤ → 2 ∈ ℂ) | |
18 | 2ne0 11729 | . . . . . 6 ⊢ 2 ≠ 0 | |
19 | 18 | a1i 11 | . . . . 5 ⊢ (𝑁 ∈ ℤ → 2 ≠ 0) |
20 | 16, 17, 19 | divcan1d 11406 | . . . 4 ⊢ (𝑁 ∈ ℤ → (((𝑁 / 2) / 2) · 2) = (𝑁 / 2)) |
21 | 2cnne0 11835 | . . . . . . . 8 ⊢ (2 ∈ ℂ ∧ 2 ≠ 0) | |
22 | 21 | a1i 11 | . . . . . . 7 ⊢ (𝑁 ∈ ℤ → (2 ∈ ℂ ∧ 2 ≠ 0)) |
23 | divdiv1 11340 | . . . . . . 7 ⊢ ((𝑁 ∈ ℂ ∧ (2 ∈ ℂ ∧ 2 ≠ 0) ∧ (2 ∈ ℂ ∧ 2 ≠ 0)) → ((𝑁 / 2) / 2) = (𝑁 / (2 · 2))) | |
24 | 15, 22, 22, 23 | syl3anc 1368 | . . . . . 6 ⊢ (𝑁 ∈ ℤ → ((𝑁 / 2) / 2) = (𝑁 / (2 · 2))) |
25 | 2t2e4 11789 | . . . . . . . 8 ⊢ (2 · 2) = 4 | |
26 | 25 | a1i 11 | . . . . . . 7 ⊢ (𝑁 ∈ ℤ → (2 · 2) = 4) |
27 | 26 | oveq2d 7151 | . . . . . 6 ⊢ (𝑁 ∈ ℤ → (𝑁 / (2 · 2)) = (𝑁 / 4)) |
28 | 24, 27 | eqtrd 2833 | . . . . 5 ⊢ (𝑁 ∈ ℤ → ((𝑁 / 2) / 2) = (𝑁 / 4)) |
29 | 28 | oveq1d 7150 | . . . 4 ⊢ (𝑁 ∈ ℤ → (((𝑁 / 2) / 2) · 2) = ((𝑁 / 4) · 2)) |
30 | 20, 29 | eqtr3d 2835 | . . 3 ⊢ (𝑁 ∈ ℤ → (𝑁 / 2) = ((𝑁 / 4) · 2)) |
31 | 30 | adantr 484 | . 2 ⊢ ((𝑁 ∈ ℤ ∧ ¬ 2 ∥ 𝑁) → (𝑁 / 2) = ((𝑁 / 4) · 2)) |
32 | 14, 31 | breqtrrd 5058 | 1 ⊢ ((𝑁 ∈ ℤ ∧ ¬ 2 ∥ 𝑁) → ((⌊‘(𝑁 / 4)) · 2) < (𝑁 / 2)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ≠ wne 2987 class class class wbr 5030 ‘cfv 6324 (class class class)co 7135 ℂcc 10524 ℝcr 10525 0cc0 10526 · cmul 10531 < clt 10664 / cdiv 11286 2c2 11680 4c4 11682 ℤcz 11969 ℝ+crp 12377 ⌊cfl 13155 ∥ cdvds 15599 |
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-3 11689 df-4 11690 df-n0 11886 df-z 11970 df-uz 12232 df-rp 12378 df-fl 13157 df-dvds 15600 |
This theorem is referenced by: gausslemma2dlem0e 25944 |
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