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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 12881 | . 2 ⊢ (𝑁 ∈ ℕ ↔ (𝑁 = 1 ∨ 𝑁 ∈ (ℤ≥‘2))) | |
2 | 1lt4 12360 | . . . . . 6 ⊢ 1 < 4 | |
3 | 1nn0 12460 | . . . . . . 7 ⊢ 1 ∈ ℕ0 | |
4 | 4nn 12267 | . . . . . . 7 ⊢ 4 ∈ ℕ | |
5 | divfl0 13761 | . . . . . . 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 11186 | . . . . . . 7 ⊢ 1 ∈ ℝ | |
9 | 4re 12268 | . . . . . . 7 ⊢ 4 ∈ ℝ | |
10 | 4ne0 12292 | . . . . . . 7 ⊢ 4 ≠ 0 | |
11 | redivcl 11905 | . . . . . . . . 9 ⊢ ((1 ∈ ℝ ∧ 4 ∈ ℝ ∧ 4 ≠ 0) → (1 / 4) ∈ ℝ) | |
12 | 11 | flcld 13735 | . . . . . . . 8 ⊢ ((1 ∈ ℝ ∧ 4 ∈ ℝ ∧ 4 ≠ 0) → (⌊‘(1 / 4)) ∈ ℤ) |
13 | 12 | zred 12638 | . . . . . . 7 ⊢ ((1 ∈ ℝ ∧ 4 ∈ ℝ ∧ 4 ≠ 0) → (⌊‘(1 / 4)) ∈ ℝ) |
14 | 8, 9, 10, 13 | mp3an 1461 | . . . . . 6 ⊢ (⌊‘(1 / 4)) ∈ ℝ |
15 | 14 | eqlei 11296 | . . . . 5 ⊢ ((⌊‘(1 / 4)) = 0 → (⌊‘(1 / 4)) ≤ 0) |
16 | 7, 15 | mp1i 13 | . . . 4 ⊢ (𝑁 = 1 → (⌊‘(1 / 4)) ≤ 0) |
17 | fvoveq1 7407 | . . . 4 ⊢ (𝑁 = 1 → (⌊‘(𝑁 / 4)) = (⌊‘(1 / 4))) | |
18 | oveq1 7391 | . . . . . . 7 ⊢ (𝑁 = 1 → (𝑁 − 1) = (1 − 1)) | |
19 | 1m1e0 12256 | . . . . . . 7 ⊢ (1 − 1) = 0 | |
20 | 18, 19 | eqtrdi 2787 | . . . . . 6 ⊢ (𝑁 = 1 → (𝑁 − 1) = 0) |
21 | 20 | oveq1d 7399 | . . . . 5 ⊢ (𝑁 = 1 → ((𝑁 − 1) / 2) = (0 / 2)) |
22 | 2cnne0 12394 | . . . . . 6 ⊢ (2 ∈ ℂ ∧ 2 ≠ 0) | |
23 | div0 11874 | . . . . . 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 5164 | . . 3 ⊢ (𝑁 = 1 → (⌊‘(𝑁 / 4)) ≤ ((𝑁 − 1) / 2)) |
27 | fldiv4lem1div2uz2 13773 | . . 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 5132 ‘cfv 6523 (class class class)co 7384 ℂcc 11080 ℝcr 11081 0cc0 11082 1c1 11083 < clt 11220 ≤ cle 11221 − cmin 11416 / cdiv 11843 ℕcn 12184 2c2 12239 4c4 12241 ℕ0cn0 12444 ℤ≥cuz 12794 ⌊cfl 13727 |
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 5283 ax-nul 5290 ax-pow 5347 ax-pr 5411 ax-un 7699 ax-cnex 11138 ax-resscn 11139 ax-1cn 11140 ax-icn 11141 ax-addcl 11142 ax-addrcl 11143 ax-mulcl 11144 ax-mulrcl 11145 ax-mulcom 11146 ax-addass 11147 ax-mulass 11148 ax-distr 11149 ax-i2m1 11150 ax-1ne0 11151 ax-1rid 11152 ax-rnegex 11153 ax-rrecex 11154 ax-cnre 11155 ax-pre-lttri 11156 ax-pre-lttrn 11157 ax-pre-ltadd 11158 ax-pre-mulgt0 11159 ax-pre-sup 11160 |
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 3371 df-reu 3372 df-rab 3426 df-v 3468 df-sbc 3765 df-csb 3881 df-dif 3938 df-un 3940 df-in 3942 df-ss 3952 df-pss 3954 df-nul 4310 df-if 4514 df-pw 4589 df-sn 4614 df-pr 4616 df-op 4620 df-uni 4893 df-iun 4983 df-br 5133 df-opab 5195 df-mpt 5216 df-tr 5250 df-id 5558 df-eprel 5564 df-po 5572 df-so 5573 df-fr 5615 df-we 5617 df-xp 5666 df-rel 5667 df-cnv 5668 df-co 5669 df-dm 5670 df-rn 5671 df-res 5672 df-ima 5673 df-pred 6280 df-ord 6347 df-on 6348 df-lim 6349 df-suc 6350 df-iota 6475 df-fun 6525 df-fn 6526 df-f 6527 df-f1 6528 df-fo 6529 df-f1o 6530 df-fv 6531 df-riota 7340 df-ov 7387 df-oprab 7388 df-mpo 7389 df-om 7830 df-2nd 7949 df-frecs 8239 df-wrecs 8270 df-recs 8344 df-rdg 8383 df-er 8677 df-en 8913 df-dom 8914 df-sdom 8915 df-sup 9409 df-inf 9410 df-pnf 11222 df-mnf 11223 df-xr 11224 df-ltxr 11225 df-le 11226 df-sub 11418 df-neg 11419 df-div 11844 df-nn 12185 df-2 12247 df-3 12248 df-4 12249 df-n0 12445 df-z 12531 df-uz 12795 df-rp 12947 df-fl 13729 |
This theorem is referenced by: gausslemma2dlem0g 26769 |
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