Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 12324 | . 2 ⊢ (𝑁 ∈ ℕ ↔ (𝑁 = 1 ∨ 𝑁 ∈ (ℤ≥‘2))) | |
2 | 1lt4 11812 | . . . . . 6 ⊢ 1 < 4 | |
3 | 1nn0 11912 | . . . . . . 7 ⊢ 1 ∈ ℕ0 | |
4 | 4nn 11719 | . . . . . . 7 ⊢ 4 ∈ ℕ | |
5 | divfl0 13193 | . . . . . . 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 232 | . . . . 5 ⊢ (⌊‘(1 / 4)) = 0 |
8 | 1re 10640 | . . . . . . 7 ⊢ 1 ∈ ℝ | |
9 | 4re 11720 | . . . . . . 7 ⊢ 4 ∈ ℝ | |
10 | 4ne0 11744 | . . . . . . 7 ⊢ 4 ≠ 0 | |
11 | redivcl 11358 | . . . . . . . . 9 ⊢ ((1 ∈ ℝ ∧ 4 ∈ ℝ ∧ 4 ≠ 0) → (1 / 4) ∈ ℝ) | |
12 | 11 | flcld 13167 | . . . . . . . 8 ⊢ ((1 ∈ ℝ ∧ 4 ∈ ℝ ∧ 4 ≠ 0) → (⌊‘(1 / 4)) ∈ ℤ) |
13 | 12 | zred 12086 | . . . . . . 7 ⊢ ((1 ∈ ℝ ∧ 4 ∈ ℝ ∧ 4 ≠ 0) → (⌊‘(1 / 4)) ∈ ℝ) |
14 | 8, 9, 10, 13 | mp3an 1457 | . . . . . 6 ⊢ (⌊‘(1 / 4)) ∈ ℝ |
15 | 14 | eqlei 10749 | . . . . 5 ⊢ ((⌊‘(1 / 4)) = 0 → (⌊‘(1 / 4)) ≤ 0) |
16 | 7, 15 | mp1i 13 | . . . 4 ⊢ (𝑁 = 1 → (⌊‘(1 / 4)) ≤ 0) |
17 | fvoveq1 7178 | . . . 4 ⊢ (𝑁 = 1 → (⌊‘(𝑁 / 4)) = (⌊‘(1 / 4))) | |
18 | oveq1 7162 | . . . . . . 7 ⊢ (𝑁 = 1 → (𝑁 − 1) = (1 − 1)) | |
19 | 1m1e0 11708 | . . . . . . 7 ⊢ (1 − 1) = 0 | |
20 | 18, 19 | syl6eq 2872 | . . . . . 6 ⊢ (𝑁 = 1 → (𝑁 − 1) = 0) |
21 | 20 | oveq1d 7170 | . . . . 5 ⊢ (𝑁 = 1 → ((𝑁 − 1) / 2) = (0 / 2)) |
22 | 2cnne0 11846 | . . . . . 6 ⊢ (2 ∈ ℂ ∧ 2 ≠ 0) | |
23 | div0 11327 | . . . . . 6 ⊢ ((2 ∈ ℂ ∧ 2 ≠ 0) → (0 / 2) = 0) | |
24 | 22, 23 | ax-mp 5 | . . . . 5 ⊢ (0 / 2) = 0 |
25 | 21, 24 | syl6eq 2872 | . . . 4 ⊢ (𝑁 = 1 → ((𝑁 − 1) / 2) = 0) |
26 | 16, 17, 25 | 3brtr4d 5097 | . . 3 ⊢ (𝑁 = 1 → (⌊‘(𝑁 / 4)) ≤ ((𝑁 − 1) / 2)) |
27 | fldiv4lem1div2uz2 13205 | . . 3 ⊢ (𝑁 ∈ (ℤ≥‘2) → (⌊‘(𝑁 / 4)) ≤ ((𝑁 − 1) / 2)) | |
28 | 26, 27 | jaoi 853 | . 2 ⊢ ((𝑁 = 1 ∨ 𝑁 ∈ (ℤ≥‘2)) → (⌊‘(𝑁 / 4)) ≤ ((𝑁 − 1) / 2)) |
29 | 1, 28 | sylbi 219 | 1 ⊢ (𝑁 ∈ ℕ → (⌊‘(𝑁 / 4)) ≤ ((𝑁 − 1) / 2)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∨ wo 843 ∧ w3a 1083 = wceq 1533 ∈ wcel 2110 ≠ wne 3016 class class class wbr 5065 ‘cfv 6354 (class class class)co 7155 ℂcc 10534 ℝcr 10535 0cc0 10536 1c1 10537 < clt 10674 ≤ cle 10675 − cmin 10869 / cdiv 11296 ℕcn 11637 2c2 11691 4c4 11693 ℕ0cn0 11896 ℤ≥cuz 12242 ⌊cfl 13159 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 ax-cnex 10592 ax-resscn 10593 ax-1cn 10594 ax-icn 10595 ax-addcl 10596 ax-addrcl 10597 ax-mulcl 10598 ax-mulrcl 10599 ax-mulcom 10600 ax-addass 10601 ax-mulass 10602 ax-distr 10603 ax-i2m1 10604 ax-1ne0 10605 ax-1rid 10606 ax-rnegex 10607 ax-rrecex 10608 ax-cnre 10609 ax-pre-lttri 10610 ax-pre-lttrn 10611 ax-pre-ltadd 10612 ax-pre-mulgt0 10613 ax-pre-sup 10614 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 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 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-tp 4571 df-op 4573 df-uni 4838 df-iun 4920 df-br 5066 df-opab 5128 df-mpt 5146 df-tr 5172 df-id 5459 df-eprel 5464 df-po 5473 df-so 5474 df-fr 5513 df-we 5515 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-pred 6147 df-ord 6193 df-on 6194 df-lim 6195 df-suc 6196 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 df-riota 7113 df-ov 7158 df-oprab 7159 df-mpo 7160 df-om 7580 df-wrecs 7946 df-recs 8007 df-rdg 8045 df-er 8288 df-en 8509 df-dom 8510 df-sdom 8511 df-sup 8905 df-inf 8906 df-pnf 10676 df-mnf 10677 df-xr 10678 df-ltxr 10679 df-le 10680 df-sub 10871 df-neg 10872 df-div 11297 df-nn 11638 df-2 11699 df-3 11700 df-4 11701 df-n0 11897 df-z 11981 df-uz 12243 df-rp 12389 df-fl 13161 |
This theorem is referenced by: gausslemma2dlem0g 25937 |
Copyright terms: Public domain | W3C validator |