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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.) |
Ref | Expression |
---|---|
flodddiv4t2lthalf | ⊢ ((𝑁 ∈ ℤ ∧ ¬ 2 ∥ 𝑁) → ((⌊‘(𝑁 / 4)) · 2) < (𝑁 / 2)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | flodddiv4lt 14977 | . . 3 ⊢ ((𝑁 ∈ ℤ ∧ ¬ 2 ∥ 𝑁) → (⌊‘(𝑁 / 4)) < (𝑁 / 4)) | |
2 | zre 11258 | . . . . . . . 8 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℝ) | |
3 | 4re 10974 | . . . . . . . . 9 ⊢ 4 ∈ ℝ | |
4 | 3 | a1i 11 | . . . . . . . 8 ⊢ (𝑁 ∈ ℤ → 4 ∈ ℝ) |
5 | 4ne0 10994 | . . . . . . . . 9 ⊢ 4 ≠ 0 | |
6 | 5 | a1i 11 | . . . . . . . 8 ⊢ (𝑁 ∈ ℤ → 4 ≠ 0) |
7 | 2, 4, 6 | redivcld 10732 | . . . . . . 7 ⊢ (𝑁 ∈ ℤ → (𝑁 / 4) ∈ ℝ) |
8 | 7 | flcld 12461 | . . . . . 6 ⊢ (𝑁 ∈ ℤ → (⌊‘(𝑁 / 4)) ∈ ℤ) |
9 | 8 | zred 11358 | . . . . 5 ⊢ (𝑁 ∈ ℤ → (⌊‘(𝑁 / 4)) ∈ ℝ) |
10 | 9 | adantr 480 | . . . 4 ⊢ ((𝑁 ∈ ℤ ∧ ¬ 2 ∥ 𝑁) → (⌊‘(𝑁 / 4)) ∈ ℝ) |
11 | 7 | adantr 480 | . . . 4 ⊢ ((𝑁 ∈ ℤ ∧ ¬ 2 ∥ 𝑁) → (𝑁 / 4) ∈ ℝ) |
12 | 2re 10967 | . . . . . 6 ⊢ 2 ∈ ℝ | |
13 | 2pos 10989 | . . . . . 6 ⊢ 0 < 2 | |
14 | 12, 13 | pm3.2i 470 | . . . . 5 ⊢ (2 ∈ ℝ ∧ 0 < 2) |
15 | 14 | a1i 11 | . . . 4 ⊢ ((𝑁 ∈ ℤ ∧ ¬ 2 ∥ 𝑁) → (2 ∈ ℝ ∧ 0 < 2)) |
16 | ltmul1 10752 | . . . 4 ⊢ (((⌊‘(𝑁 / 4)) ∈ ℝ ∧ (𝑁 / 4) ∈ ℝ ∧ (2 ∈ ℝ ∧ 0 < 2)) → ((⌊‘(𝑁 / 4)) < (𝑁 / 4) ↔ ((⌊‘(𝑁 / 4)) · 2) < ((𝑁 / 4) · 2))) | |
17 | 10, 11, 15, 16 | syl3anc 1318 | . . 3 ⊢ ((𝑁 ∈ ℤ ∧ ¬ 2 ∥ 𝑁) → ((⌊‘(𝑁 / 4)) < (𝑁 / 4) ↔ ((⌊‘(𝑁 / 4)) · 2) < ((𝑁 / 4) · 2))) |
18 | 1, 17 | mpbid 221 | . 2 ⊢ ((𝑁 ∈ ℤ ∧ ¬ 2 ∥ 𝑁) → ((⌊‘(𝑁 / 4)) · 2) < ((𝑁 / 4) · 2)) |
19 | zcn 11259 | . . . . . 6 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℂ) | |
20 | 19 | halfcld 11154 | . . . . 5 ⊢ (𝑁 ∈ ℤ → (𝑁 / 2) ∈ ℂ) |
21 | 2cnd 10970 | . . . . 5 ⊢ (𝑁 ∈ ℤ → 2 ∈ ℂ) | |
22 | 2ne0 10990 | . . . . . 6 ⊢ 2 ≠ 0 | |
23 | 22 | a1i 11 | . . . . 5 ⊢ (𝑁 ∈ ℤ → 2 ≠ 0) |
24 | 20, 21, 23 | divcan1d 10681 | . . . 4 ⊢ (𝑁 ∈ ℤ → (((𝑁 / 2) / 2) · 2) = (𝑁 / 2)) |
25 | 2cnne0 11119 | . . . . . . . 8 ⊢ (2 ∈ ℂ ∧ 2 ≠ 0) | |
26 | 25 | a1i 11 | . . . . . . 7 ⊢ (𝑁 ∈ ℤ → (2 ∈ ℂ ∧ 2 ≠ 0)) |
27 | divdiv1 10615 | . . . . . . 7 ⊢ ((𝑁 ∈ ℂ ∧ (2 ∈ ℂ ∧ 2 ≠ 0) ∧ (2 ∈ ℂ ∧ 2 ≠ 0)) → ((𝑁 / 2) / 2) = (𝑁 / (2 · 2))) | |
28 | 19, 26, 26, 27 | syl3anc 1318 | . . . . . 6 ⊢ (𝑁 ∈ ℤ → ((𝑁 / 2) / 2) = (𝑁 / (2 · 2))) |
29 | 2t2e4 11054 | . . . . . . . 8 ⊢ (2 · 2) = 4 | |
30 | 29 | a1i 11 | . . . . . . 7 ⊢ (𝑁 ∈ ℤ → (2 · 2) = 4) |
31 | 30 | oveq2d 6565 | . . . . . 6 ⊢ (𝑁 ∈ ℤ → (𝑁 / (2 · 2)) = (𝑁 / 4)) |
32 | 28, 31 | eqtrd 2644 | . . . . 5 ⊢ (𝑁 ∈ ℤ → ((𝑁 / 2) / 2) = (𝑁 / 4)) |
33 | 32 | oveq1d 6564 | . . . 4 ⊢ (𝑁 ∈ ℤ → (((𝑁 / 2) / 2) · 2) = ((𝑁 / 4) · 2)) |
34 | 24, 33 | eqtr3d 2646 | . . 3 ⊢ (𝑁 ∈ ℤ → (𝑁 / 2) = ((𝑁 / 4) · 2)) |
35 | 34 | adantr 480 | . 2 ⊢ ((𝑁 ∈ ℤ ∧ ¬ 2 ∥ 𝑁) → (𝑁 / 2) = ((𝑁 / 4) · 2)) |
36 | 18, 35 | breqtrrd 4611 | 1 ⊢ ((𝑁 ∈ ℤ ∧ ¬ 2 ∥ 𝑁) → ((⌊‘(𝑁 / 4)) · 2) < (𝑁 / 2)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 class class class wbr 4583 ‘cfv 5804 (class class class)co 6549 ℂcc 9813 ℝcr 9814 0cc0 9815 · cmul 9820 < clt 9953 / cdiv 10563 2c2 10947 4c4 10949 ℤcz 11254 ⌊cfl 12453 ∥ cdvds 14821 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-sup 8231 df-inf 8232 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-3 10957 df-4 10958 df-n0 11170 df-z 11255 df-uz 11564 df-fl 12455 df-dvds 14822 |
This theorem is referenced by: gausslemma2dlem0e 24885 |
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