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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 15766 | . . 3 ⊢ ((𝑁 ∈ ℤ ∧ ¬ 2 ∥ 𝑁) → (⌊‘(𝑁 / 4)) < (𝑁 / 4)) | |
2 | zre 11986 | . . . . . . . 8 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℝ) | |
3 | 4re 11722 | . . . . . . . . 9 ⊢ 4 ∈ ℝ | |
4 | 3 | a1i 11 | . . . . . . . 8 ⊢ (𝑁 ∈ ℤ → 4 ∈ ℝ) |
5 | 4ne0 11746 | . . . . . . . . 9 ⊢ 4 ≠ 0 | |
6 | 5 | a1i 11 | . . . . . . . 8 ⊢ (𝑁 ∈ ℤ → 4 ≠ 0) |
7 | 2, 4, 6 | redivcld 11468 | . . . . . . 7 ⊢ (𝑁 ∈ ℤ → (𝑁 / 4) ∈ ℝ) |
8 | 7 | flcld 13169 | . . . . . 6 ⊢ (𝑁 ∈ ℤ → (⌊‘(𝑁 / 4)) ∈ ℤ) |
9 | 8 | zred 12088 | . . . . 5 ⊢ (𝑁 ∈ ℤ → (⌊‘(𝑁 / 4)) ∈ ℝ) |
10 | 2rp 12395 | . . . . . 6 ⊢ 2 ∈ ℝ+ | |
11 | 10 | a1i 11 | . . . . 5 ⊢ (𝑁 ∈ ℤ → 2 ∈ ℝ+) |
12 | 9, 7, 11 | ltmul1d 12473 | . . . 4 ⊢ (𝑁 ∈ ℤ → ((⌊‘(𝑁 / 4)) < (𝑁 / 4) ↔ ((⌊‘(𝑁 / 4)) · 2) < ((𝑁 / 4) · 2))) |
13 | 12 | adantr 483 | . . 3 ⊢ ((𝑁 ∈ ℤ ∧ ¬ 2 ∥ 𝑁) → ((⌊‘(𝑁 / 4)) < (𝑁 / 4) ↔ ((⌊‘(𝑁 / 4)) · 2) < ((𝑁 / 4) · 2))) |
14 | 1, 13 | mpbid 234 | . 2 ⊢ ((𝑁 ∈ ℤ ∧ ¬ 2 ∥ 𝑁) → ((⌊‘(𝑁 / 4)) · 2) < ((𝑁 / 4) · 2)) |
15 | zcn 11987 | . . . . . 6 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℂ) | |
16 | 15 | halfcld 11883 | . . . . 5 ⊢ (𝑁 ∈ ℤ → (𝑁 / 2) ∈ ℂ) |
17 | 2cnd 11716 | . . . . 5 ⊢ (𝑁 ∈ ℤ → 2 ∈ ℂ) | |
18 | 2ne0 11742 | . . . . . 6 ⊢ 2 ≠ 0 | |
19 | 18 | a1i 11 | . . . . 5 ⊢ (𝑁 ∈ ℤ → 2 ≠ 0) |
20 | 16, 17, 19 | divcan1d 11417 | . . . 4 ⊢ (𝑁 ∈ ℤ → (((𝑁 / 2) / 2) · 2) = (𝑁 / 2)) |
21 | 2cnne0 11848 | . . . . . . . 8 ⊢ (2 ∈ ℂ ∧ 2 ≠ 0) | |
22 | 21 | a1i 11 | . . . . . . 7 ⊢ (𝑁 ∈ ℤ → (2 ∈ ℂ ∧ 2 ≠ 0)) |
23 | divdiv1 11351 | . . . . . . 7 ⊢ ((𝑁 ∈ ℂ ∧ (2 ∈ ℂ ∧ 2 ≠ 0) ∧ (2 ∈ ℂ ∧ 2 ≠ 0)) → ((𝑁 / 2) / 2) = (𝑁 / (2 · 2))) | |
24 | 15, 22, 22, 23 | syl3anc 1367 | . . . . . 6 ⊢ (𝑁 ∈ ℤ → ((𝑁 / 2) / 2) = (𝑁 / (2 · 2))) |
25 | 2t2e4 11802 | . . . . . . . 8 ⊢ (2 · 2) = 4 | |
26 | 25 | a1i 11 | . . . . . . 7 ⊢ (𝑁 ∈ ℤ → (2 · 2) = 4) |
27 | 26 | oveq2d 7172 | . . . . . 6 ⊢ (𝑁 ∈ ℤ → (𝑁 / (2 · 2)) = (𝑁 / 4)) |
28 | 24, 27 | eqtrd 2856 | . . . . 5 ⊢ (𝑁 ∈ ℤ → ((𝑁 / 2) / 2) = (𝑁 / 4)) |
29 | 28 | oveq1d 7171 | . . . 4 ⊢ (𝑁 ∈ ℤ → (((𝑁 / 2) / 2) · 2) = ((𝑁 / 4) · 2)) |
30 | 20, 29 | eqtr3d 2858 | . . 3 ⊢ (𝑁 ∈ ℤ → (𝑁 / 2) = ((𝑁 / 4) · 2)) |
31 | 30 | adantr 483 | . 2 ⊢ ((𝑁 ∈ ℤ ∧ ¬ 2 ∥ 𝑁) → (𝑁 / 2) = ((𝑁 / 4) · 2)) |
32 | 14, 31 | breqtrrd 5094 | 1 ⊢ ((𝑁 ∈ ℤ ∧ ¬ 2 ∥ 𝑁) → ((⌊‘(𝑁 / 4)) · 2) < (𝑁 / 2)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ≠ wne 3016 class class class wbr 5066 ‘cfv 6355 (class class class)co 7156 ℂcc 10535 ℝcr 10536 0cc0 10537 · cmul 10542 < clt 10675 / cdiv 11297 2c2 11693 4c4 11695 ℤcz 11982 ℝ+crp 12390 ⌊cfl 13161 ∥ cdvds 15607 |
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-3 11702 df-4 11703 df-n0 11899 df-z 11983 df-uz 12245 df-rp 12391 df-fl 13163 df-dvds 15608 |
This theorem is referenced by: gausslemma2dlem0e 25936 |
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