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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 12137 | . 2 ⊢ (𝑁 ∈ ℕ ↔ (𝑁 = 1 ∨ 𝑁 ∈ (ℤ≥‘2))) | |
2 | 1lt4 11621 | . . . . . 6 ⊢ 1 < 4 | |
3 | 1nn0 11723 | . . . . . . 7 ⊢ 1 ∈ ℕ0 | |
4 | 4nn 11522 | . . . . . . 7 ⊢ 4 ∈ ℕ | |
5 | divfl0 13007 | . . . . . . 7 ⊢ ((1 ∈ ℕ0 ∧ 4 ∈ ℕ) → (1 < 4 ↔ (⌊‘(1 / 4)) = 0)) | |
6 | 3, 4, 5 | mp2an 680 | . . . . . 6 ⊢ (1 < 4 ↔ (⌊‘(1 / 4)) = 0) |
7 | 2, 6 | mpbi 222 | . . . . 5 ⊢ (⌊‘(1 / 4)) = 0 |
8 | 1re 10437 | . . . . . . 7 ⊢ 1 ∈ ℝ | |
9 | 4re 11523 | . . . . . . 7 ⊢ 4 ∈ ℝ | |
10 | 4ne0 11553 | . . . . . . 7 ⊢ 4 ≠ 0 | |
11 | redivcl 11158 | . . . . . . . . 9 ⊢ ((1 ∈ ℝ ∧ 4 ∈ ℝ ∧ 4 ≠ 0) → (1 / 4) ∈ ℝ) | |
12 | 11 | flcld 12981 | . . . . . . . 8 ⊢ ((1 ∈ ℝ ∧ 4 ∈ ℝ ∧ 4 ≠ 0) → (⌊‘(1 / 4)) ∈ ℤ) |
13 | 12 | zred 11898 | . . . . . . 7 ⊢ ((1 ∈ ℝ ∧ 4 ∈ ℝ ∧ 4 ≠ 0) → (⌊‘(1 / 4)) ∈ ℝ) |
14 | 8, 9, 10, 13 | mp3an 1441 | . . . . . 6 ⊢ (⌊‘(1 / 4)) ∈ ℝ |
15 | 14 | eqlei 10548 | . . . . 5 ⊢ ((⌊‘(1 / 4)) = 0 → (⌊‘(1 / 4)) ≤ 0) |
16 | 7, 15 | mp1i 13 | . . . 4 ⊢ (𝑁 = 1 → (⌊‘(1 / 4)) ≤ 0) |
17 | fvoveq1 6997 | . . . 4 ⊢ (𝑁 = 1 → (⌊‘(𝑁 / 4)) = (⌊‘(1 / 4))) | |
18 | oveq1 6981 | . . . . . . 7 ⊢ (𝑁 = 1 → (𝑁 − 1) = (1 − 1)) | |
19 | 1m1e0 11510 | . . . . . . 7 ⊢ (1 − 1) = 0 | |
20 | 18, 19 | syl6eq 2823 | . . . . . 6 ⊢ (𝑁 = 1 → (𝑁 − 1) = 0) |
21 | 20 | oveq1d 6989 | . . . . 5 ⊢ (𝑁 = 1 → ((𝑁 − 1) / 2) = (0 / 2)) |
22 | 2cnne0 11655 | . . . . . 6 ⊢ (2 ∈ ℂ ∧ 2 ≠ 0) | |
23 | div0 11127 | . . . . . 6 ⊢ ((2 ∈ ℂ ∧ 2 ≠ 0) → (0 / 2) = 0) | |
24 | 22, 23 | ax-mp 5 | . . . . 5 ⊢ (0 / 2) = 0 |
25 | 21, 24 | syl6eq 2823 | . . . 4 ⊢ (𝑁 = 1 → ((𝑁 − 1) / 2) = 0) |
26 | 16, 17, 25 | 3brtr4d 4957 | . . 3 ⊢ (𝑁 = 1 → (⌊‘(𝑁 / 4)) ≤ ((𝑁 − 1) / 2)) |
27 | fldiv4lem1div2uz2 13019 | . . 3 ⊢ (𝑁 ∈ (ℤ≥‘2) → (⌊‘(𝑁 / 4)) ≤ ((𝑁 − 1) / 2)) | |
28 | 26, 27 | jaoi 844 | . 2 ⊢ ((𝑁 = 1 ∨ 𝑁 ∈ (ℤ≥‘2)) → (⌊‘(𝑁 / 4)) ≤ ((𝑁 − 1) / 2)) |
29 | 1, 28 | sylbi 209 | 1 ⊢ (𝑁 ∈ ℕ → (⌊‘(𝑁 / 4)) ≤ ((𝑁 − 1) / 2)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 387 ∨ wo 834 ∧ w3a 1069 = wceq 1508 ∈ wcel 2051 ≠ wne 2960 class class class wbr 4925 ‘cfv 6185 (class class class)co 6974 ℂcc 10331 ℝcr 10332 0cc0 10333 1c1 10334 < clt 10472 ≤ cle 10473 − cmin 10668 / cdiv 11096 ℕcn 11437 2c2 11493 4c4 11495 ℕ0cn0 11705 ℤ≥cuz 12056 ⌊cfl 12973 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1759 ax-4 1773 ax-5 1870 ax-6 1929 ax-7 1966 ax-8 2053 ax-9 2060 ax-10 2080 ax-11 2094 ax-12 2107 ax-13 2302 ax-ext 2743 ax-sep 5056 ax-nul 5063 ax-pow 5115 ax-pr 5182 ax-un 7277 ax-cnex 10389 ax-resscn 10390 ax-1cn 10391 ax-icn 10392 ax-addcl 10393 ax-addrcl 10394 ax-mulcl 10395 ax-mulrcl 10396 ax-mulcom 10397 ax-addass 10398 ax-mulass 10399 ax-distr 10400 ax-i2m1 10401 ax-1ne0 10402 ax-1rid 10403 ax-rnegex 10404 ax-rrecex 10405 ax-cnre 10406 ax-pre-lttri 10407 ax-pre-lttrn 10408 ax-pre-ltadd 10409 ax-pre-mulgt0 10410 ax-pre-sup 10411 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 835 df-3or 1070 df-3an 1071 df-tru 1511 df-ex 1744 df-nf 1748 df-sb 2017 df-mo 2548 df-eu 2585 df-clab 2752 df-cleq 2764 df-clel 2839 df-nfc 2911 df-ne 2961 df-nel 3067 df-ral 3086 df-rex 3087 df-reu 3088 df-rmo 3089 df-rab 3090 df-v 3410 df-sbc 3675 df-csb 3780 df-dif 3825 df-un 3827 df-in 3829 df-ss 3836 df-pss 3838 df-nul 4173 df-if 4345 df-pw 4418 df-sn 4436 df-pr 4438 df-tp 4440 df-op 4442 df-uni 4709 df-iun 4790 df-br 4926 df-opab 4988 df-mpt 5005 df-tr 5027 df-id 5308 df-eprel 5313 df-po 5322 df-so 5323 df-fr 5362 df-we 5364 df-xp 5409 df-rel 5410 df-cnv 5411 df-co 5412 df-dm 5413 df-rn 5414 df-res 5415 df-ima 5416 df-pred 5983 df-ord 6029 df-on 6030 df-lim 6031 df-suc 6032 df-iota 6149 df-fun 6187 df-fn 6188 df-f 6189 df-f1 6190 df-fo 6191 df-f1o 6192 df-fv 6193 df-riota 6935 df-ov 6977 df-oprab 6978 df-mpo 6979 df-om 7395 df-wrecs 7748 df-recs 7810 df-rdg 7848 df-er 8087 df-en 8305 df-dom 8306 df-sdom 8307 df-sup 8699 df-inf 8700 df-pnf 10474 df-mnf 10475 df-xr 10476 df-ltxr 10477 df-le 10478 df-sub 10670 df-neg 10671 df-div 11097 df-nn 11438 df-2 11501 df-3 11502 df-4 11503 df-n0 11706 df-z 11792 df-uz 12057 df-rp 12203 df-fl 12975 |
This theorem is referenced by: gausslemma2dlem0g 25655 |
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