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Mirrors > Home > MPE Home > Th. List > fldiv4lem1div2 | Structured version Visualization version GIF version |
Description: The floor of a positive integer divided by 4 is less than or equal to the half of the integer minus 1. (Contributed by AV, 9-Jul-2021.) |
Ref | Expression |
---|---|
fldiv4lem1div2 | ⊢ (𝑁 ∈ ℕ → (⌊‘(𝑁 / 4)) ≤ ((𝑁 − 1) / 2)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elnn1uz2 12888 | . 2 ⊢ (𝑁 ∈ ℕ ↔ (𝑁 = 1 ∨ 𝑁 ∈ (ℤ≥‘2))) | |
2 | 1lt4 12367 | . . . . . 6 ⊢ 1 < 4 | |
3 | 1nn0 12467 | . . . . . . 7 ⊢ 1 ∈ ℕ0 | |
4 | 4nn 12274 | . . . . . . 7 ⊢ 4 ∈ ℕ | |
5 | divfl0 13768 | . . . . . . 7 ⊢ ((1 ∈ ℕ0 ∧ 4 ∈ ℕ) → (1 < 4 ↔ (⌊‘(1 / 4)) = 0)) | |
6 | 3, 4, 5 | mp2an 690 | . . . . . 6 ⊢ (1 < 4 ↔ (⌊‘(1 / 4)) = 0) |
7 | 2, 6 | mpbi 229 | . . . . 5 ⊢ (⌊‘(1 / 4)) = 0 |
8 | 1re 11193 | . . . . . . 7 ⊢ 1 ∈ ℝ | |
9 | 4re 12275 | . . . . . . 7 ⊢ 4 ∈ ℝ | |
10 | 4ne0 12299 | . . . . . . 7 ⊢ 4 ≠ 0 | |
11 | redivcl 11912 | . . . . . . . . 9 ⊢ ((1 ∈ ℝ ∧ 4 ∈ ℝ ∧ 4 ≠ 0) → (1 / 4) ∈ ℝ) | |
12 | 11 | flcld 13742 | . . . . . . . 8 ⊢ ((1 ∈ ℝ ∧ 4 ∈ ℝ ∧ 4 ≠ 0) → (⌊‘(1 / 4)) ∈ ℤ) |
13 | 12 | zred 12645 | . . . . . . 7 ⊢ ((1 ∈ ℝ ∧ 4 ∈ ℝ ∧ 4 ≠ 0) → (⌊‘(1 / 4)) ∈ ℝ) |
14 | 8, 9, 10, 13 | mp3an 1461 | . . . . . 6 ⊢ (⌊‘(1 / 4)) ∈ ℝ |
15 | 14 | eqlei 11303 | . . . . 5 ⊢ ((⌊‘(1 / 4)) = 0 → (⌊‘(1 / 4)) ≤ 0) |
16 | 7, 15 | mp1i 13 | . . . 4 ⊢ (𝑁 = 1 → (⌊‘(1 / 4)) ≤ 0) |
17 | fvoveq1 7413 | . . . 4 ⊢ (𝑁 = 1 → (⌊‘(𝑁 / 4)) = (⌊‘(1 / 4))) | |
18 | oveq1 7397 | . . . . . . 7 ⊢ (𝑁 = 1 → (𝑁 − 1) = (1 − 1)) | |
19 | 1m1e0 12263 | . . . . . . 7 ⊢ (1 − 1) = 0 | |
20 | 18, 19 | eqtrdi 2787 | . . . . . 6 ⊢ (𝑁 = 1 → (𝑁 − 1) = 0) |
21 | 20 | oveq1d 7405 | . . . . 5 ⊢ (𝑁 = 1 → ((𝑁 − 1) / 2) = (0 / 2)) |
22 | 2cnne0 12401 | . . . . . 6 ⊢ (2 ∈ ℂ ∧ 2 ≠ 0) | |
23 | div0 11881 | . . . . . 6 ⊢ ((2 ∈ ℂ ∧ 2 ≠ 0) → (0 / 2) = 0) | |
24 | 22, 23 | ax-mp 5 | . . . . 5 ⊢ (0 / 2) = 0 |
25 | 21, 24 | eqtrdi 2787 | . . . 4 ⊢ (𝑁 = 1 → ((𝑁 − 1) / 2) = 0) |
26 | 16, 17, 25 | 3brtr4d 5170 | . . 3 ⊢ (𝑁 = 1 → (⌊‘(𝑁 / 4)) ≤ ((𝑁 − 1) / 2)) |
27 | fldiv4lem1div2uz2 13780 | . . 3 ⊢ (𝑁 ∈ (ℤ≥‘2) → (⌊‘(𝑁 / 4)) ≤ ((𝑁 − 1) / 2)) | |
28 | 26, 27 | jaoi 855 | . 2 ⊢ ((𝑁 = 1 ∨ 𝑁 ∈ (ℤ≥‘2)) → (⌊‘(𝑁 / 4)) ≤ ((𝑁 − 1) / 2)) |
29 | 1, 28 | sylbi 216 | 1 ⊢ (𝑁 ∈ ℕ → (⌊‘(𝑁 / 4)) ≤ ((𝑁 − 1) / 2)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 ∨ wo 845 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ≠ wne 2939 class class class wbr 5138 ‘cfv 6529 (class class class)co 7390 ℂcc 11087 ℝcr 11088 0cc0 11089 1c1 11090 < clt 11227 ≤ cle 11228 − cmin 11423 / cdiv 11850 ℕcn 12191 2c2 12246 4c4 12248 ℕ0cn0 12451 ℤ≥cuz 12801 ⌊cfl 13734 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2702 ax-sep 5289 ax-nul 5296 ax-pow 5353 ax-pr 5417 ax-un 7705 ax-cnex 11145 ax-resscn 11146 ax-1cn 11147 ax-icn 11148 ax-addcl 11149 ax-addrcl 11150 ax-mulcl 11151 ax-mulrcl 11152 ax-mulcom 11153 ax-addass 11154 ax-mulass 11155 ax-distr 11156 ax-i2m1 11157 ax-1ne0 11158 ax-1rid 11159 ax-rnegex 11160 ax-rrecex 11161 ax-cnre 11162 ax-pre-lttri 11163 ax-pre-lttrn 11164 ax-pre-ltadd 11165 ax-pre-mulgt0 11166 ax-pre-sup 11167 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3430 df-v 3472 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3960 df-nul 4316 df-if 4520 df-pw 4595 df-sn 4620 df-pr 4622 df-op 4626 df-uni 4899 df-iun 4989 df-br 5139 df-opab 5201 df-mpt 5222 df-tr 5256 df-id 5564 df-eprel 5570 df-po 5578 df-so 5579 df-fr 5621 df-we 5623 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-pred 6286 df-ord 6353 df-on 6354 df-lim 6355 df-suc 6356 df-iota 6481 df-fun 6531 df-fn 6532 df-f 6533 df-f1 6534 df-fo 6535 df-f1o 6536 df-fv 6537 df-riota 7346 df-ov 7393 df-oprab 7394 df-mpo 7395 df-om 7836 df-2nd 7955 df-frecs 8245 df-wrecs 8276 df-recs 8350 df-rdg 8389 df-er 8683 df-en 8920 df-dom 8921 df-sdom 8922 df-sup 9416 df-inf 9417 df-pnf 11229 df-mnf 11230 df-xr 11231 df-ltxr 11232 df-le 11233 df-sub 11425 df-neg 11426 df-div 11851 df-nn 12192 df-2 12254 df-3 12255 df-4 12256 df-n0 12452 df-z 12538 df-uz 12802 df-rp 12954 df-fl 13736 |
This theorem is referenced by: gausslemma2dlem0g 26787 |
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