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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 16455 | . . 3 ⊢ ((𝑁 ∈ ℤ ∧ ¬ 2 ∥ 𝑁) → (⌊‘(𝑁 / 4)) < (𝑁 / 4)) | |
| 2 | zre 12619 | . . . . . . . 8 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℝ) | |
| 3 | 4re 12351 | . . . . . . . . 9 ⊢ 4 ∈ ℝ | |
| 4 | 3 | a1i 11 | . . . . . . . 8 ⊢ (𝑁 ∈ ℤ → 4 ∈ ℝ) | 
| 5 | 4ne0 12375 | . . . . . . . . 9 ⊢ 4 ≠ 0 | |
| 6 | 5 | a1i 11 | . . . . . . . 8 ⊢ (𝑁 ∈ ℤ → 4 ≠ 0) | 
| 7 | 2, 4, 6 | redivcld 12096 | . . . . . . 7 ⊢ (𝑁 ∈ ℤ → (𝑁 / 4) ∈ ℝ) | 
| 8 | 7 | flcld 13839 | . . . . . 6 ⊢ (𝑁 ∈ ℤ → (⌊‘(𝑁 / 4)) ∈ ℤ) | 
| 9 | 8 | zred 12724 | . . . . 5 ⊢ (𝑁 ∈ ℤ → (⌊‘(𝑁 / 4)) ∈ ℝ) | 
| 10 | 2rp 13040 | . . . . . 6 ⊢ 2 ∈ ℝ+ | |
| 11 | 10 | a1i 11 | . . . . 5 ⊢ (𝑁 ∈ ℤ → 2 ∈ ℝ+) | 
| 12 | 9, 7, 11 | ltmul1d 13119 | . . . 4 ⊢ (𝑁 ∈ ℤ → ((⌊‘(𝑁 / 4)) < (𝑁 / 4) ↔ ((⌊‘(𝑁 / 4)) · 2) < ((𝑁 / 4) · 2))) | 
| 13 | 12 | adantr 480 | . . 3 ⊢ ((𝑁 ∈ ℤ ∧ ¬ 2 ∥ 𝑁) → ((⌊‘(𝑁 / 4)) < (𝑁 / 4) ↔ ((⌊‘(𝑁 / 4)) · 2) < ((𝑁 / 4) · 2))) | 
| 14 | 1, 13 | mpbid 232 | . 2 ⊢ ((𝑁 ∈ ℤ ∧ ¬ 2 ∥ 𝑁) → ((⌊‘(𝑁 / 4)) · 2) < ((𝑁 / 4) · 2)) | 
| 15 | zcn 12620 | . . . . . 6 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℂ) | |
| 16 | 15 | halfcld 12513 | . . . . 5 ⊢ (𝑁 ∈ ℤ → (𝑁 / 2) ∈ ℂ) | 
| 17 | 2cnd 12345 | . . . . 5 ⊢ (𝑁 ∈ ℤ → 2 ∈ ℂ) | |
| 18 | 2ne0 12371 | . . . . . 6 ⊢ 2 ≠ 0 | |
| 19 | 18 | a1i 11 | . . . . 5 ⊢ (𝑁 ∈ ℤ → 2 ≠ 0) | 
| 20 | 16, 17, 19 | divcan1d 12045 | . . . 4 ⊢ (𝑁 ∈ ℤ → (((𝑁 / 2) / 2) · 2) = (𝑁 / 2)) | 
| 21 | 2cnne0 12477 | . . . . . . . 8 ⊢ (2 ∈ ℂ ∧ 2 ≠ 0) | |
| 22 | 21 | a1i 11 | . . . . . . 7 ⊢ (𝑁 ∈ ℤ → (2 ∈ ℂ ∧ 2 ≠ 0)) | 
| 23 | divdiv1 11979 | . . . . . . 7 ⊢ ((𝑁 ∈ ℂ ∧ (2 ∈ ℂ ∧ 2 ≠ 0) ∧ (2 ∈ ℂ ∧ 2 ≠ 0)) → ((𝑁 / 2) / 2) = (𝑁 / (2 · 2))) | |
| 24 | 15, 22, 22, 23 | syl3anc 1372 | . . . . . 6 ⊢ (𝑁 ∈ ℤ → ((𝑁 / 2) / 2) = (𝑁 / (2 · 2))) | 
| 25 | 2t2e4 12431 | . . . . . . . 8 ⊢ (2 · 2) = 4 | |
| 26 | 25 | a1i 11 | . . . . . . 7 ⊢ (𝑁 ∈ ℤ → (2 · 2) = 4) | 
| 27 | 26 | oveq2d 7448 | . . . . . 6 ⊢ (𝑁 ∈ ℤ → (𝑁 / (2 · 2)) = (𝑁 / 4)) | 
| 28 | 24, 27 | eqtrd 2776 | . . . . 5 ⊢ (𝑁 ∈ ℤ → ((𝑁 / 2) / 2) = (𝑁 / 4)) | 
| 29 | 28 | oveq1d 7447 | . . . 4 ⊢ (𝑁 ∈ ℤ → (((𝑁 / 2) / 2) · 2) = ((𝑁 / 4) · 2)) | 
| 30 | 20, 29 | eqtr3d 2778 | . . 3 ⊢ (𝑁 ∈ ℤ → (𝑁 / 2) = ((𝑁 / 4) · 2)) | 
| 31 | 30 | adantr 480 | . 2 ⊢ ((𝑁 ∈ ℤ ∧ ¬ 2 ∥ 𝑁) → (𝑁 / 2) = ((𝑁 / 4) · 2)) | 
| 32 | 14, 31 | breqtrrd 5170 | 1 ⊢ ((𝑁 ∈ ℤ ∧ ¬ 2 ∥ 𝑁) → ((⌊‘(𝑁 / 4)) · 2) < (𝑁 / 2)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1539 ∈ wcel 2107 ≠ wne 2939 class class class wbr 5142 ‘cfv 6560 (class class class)co 7432 ℂcc 11154 ℝcr 11155 0cc0 11156 · cmul 11161 < clt 11296 / cdiv 11921 2c2 12322 4c4 12324 ℤcz 12615 ℝ+crp 13035 ⌊cfl 13831 ∥ cdvds 16291 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 ax-cnex 11212 ax-resscn 11213 ax-1cn 11214 ax-icn 11215 ax-addcl 11216 ax-addrcl 11217 ax-mulcl 11218 ax-mulrcl 11219 ax-mulcom 11220 ax-addass 11221 ax-mulass 11222 ax-distr 11223 ax-i2m1 11224 ax-1ne0 11225 ax-1rid 11226 ax-rnegex 11227 ax-rrecex 11228 ax-cnre 11229 ax-pre-lttri 11230 ax-pre-lttrn 11231 ax-pre-ltadd 11232 ax-pre-mulgt0 11233 ax-pre-sup 11234 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3379 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-iun 4992 df-br 5143 df-opab 5205 df-mpt 5225 df-tr 5259 df-id 5577 df-eprel 5583 df-po 5591 df-so 5592 df-fr 5636 df-we 5638 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-pred 6320 df-ord 6386 df-on 6387 df-lim 6388 df-suc 6389 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-riota 7389 df-ov 7435 df-oprab 7436 df-mpo 7437 df-om 7889 df-2nd 8016 df-frecs 8307 df-wrecs 8338 df-recs 8412 df-rdg 8451 df-er 8746 df-en 8987 df-dom 8988 df-sdom 8989 df-sup 9483 df-inf 9484 df-pnf 11298 df-mnf 11299 df-xr 11300 df-ltxr 11301 df-le 11302 df-sub 11495 df-neg 11496 df-div 11922 df-nn 12268 df-2 12330 df-3 12331 df-4 12332 df-n0 12529 df-z 12616 df-uz 12880 df-rp 13036 df-fl 13833 df-dvds 16292 | 
| This theorem is referenced by: gausslemma2dlem0e 27405 | 
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