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Mirrors > Home > MPE Home > Th. List > flhalf | Structured version Visualization version GIF version |
Description: Ordering relation for the floor of half of an integer. (Contributed by NM, 1-Jan-2006.) (Proof shortened by Mario Carneiro, 7-Jun-2016.) |
Ref | Expression |
---|---|
flhalf | ⊢ (𝑁 ∈ ℤ → 𝑁 ≤ (2 · (⌊‘((𝑁 + 1) / 2)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | zre 12615 | . . . . . . . 8 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℝ) | |
2 | peano2re 11432 | . . . . . . . 8 ⊢ (𝑁 ∈ ℝ → (𝑁 + 1) ∈ ℝ) | |
3 | 1, 2 | syl 17 | . . . . . . 7 ⊢ (𝑁 ∈ ℤ → (𝑁 + 1) ∈ ℝ) |
4 | 3 | rehalfcld 12511 | . . . . . 6 ⊢ (𝑁 ∈ ℤ → ((𝑁 + 1) / 2) ∈ ℝ) |
5 | flltp1 13837 | . . . . . 6 ⊢ (((𝑁 + 1) / 2) ∈ ℝ → ((𝑁 + 1) / 2) < ((⌊‘((𝑁 + 1) / 2)) + 1)) | |
6 | 4, 5 | syl 17 | . . . . 5 ⊢ (𝑁 ∈ ℤ → ((𝑁 + 1) / 2) < ((⌊‘((𝑁 + 1) / 2)) + 1)) |
7 | 4 | flcld 13835 | . . . . . . . 8 ⊢ (𝑁 ∈ ℤ → (⌊‘((𝑁 + 1) / 2)) ∈ ℤ) |
8 | 7 | zred 12720 | . . . . . . 7 ⊢ (𝑁 ∈ ℤ → (⌊‘((𝑁 + 1) / 2)) ∈ ℝ) |
9 | 1red 11260 | . . . . . . 7 ⊢ (𝑁 ∈ ℤ → 1 ∈ ℝ) | |
10 | 8, 9 | readdcld 11288 | . . . . . 6 ⊢ (𝑁 ∈ ℤ → ((⌊‘((𝑁 + 1) / 2)) + 1) ∈ ℝ) |
11 | 2rp 13037 | . . . . . . 7 ⊢ 2 ∈ ℝ+ | |
12 | 11 | a1i 11 | . . . . . 6 ⊢ (𝑁 ∈ ℤ → 2 ∈ ℝ+) |
13 | 3, 10, 12 | ltdivmuld 13126 | . . . . 5 ⊢ (𝑁 ∈ ℤ → (((𝑁 + 1) / 2) < ((⌊‘((𝑁 + 1) / 2)) + 1) ↔ (𝑁 + 1) < (2 · ((⌊‘((𝑁 + 1) / 2)) + 1)))) |
14 | 6, 13 | mpbid 232 | . . . 4 ⊢ (𝑁 ∈ ℤ → (𝑁 + 1) < (2 · ((⌊‘((𝑁 + 1) / 2)) + 1))) |
15 | 9 | recnd 11287 | . . . . . . 7 ⊢ (𝑁 ∈ ℤ → 1 ∈ ℂ) |
16 | 15 | 2timesd 12507 | . . . . . 6 ⊢ (𝑁 ∈ ℤ → (2 · 1) = (1 + 1)) |
17 | 16 | oveq2d 7447 | . . . . 5 ⊢ (𝑁 ∈ ℤ → ((2 · (⌊‘((𝑁 + 1) / 2))) + (2 · 1)) = ((2 · (⌊‘((𝑁 + 1) / 2))) + (1 + 1))) |
18 | 2cnd 12342 | . . . . . 6 ⊢ (𝑁 ∈ ℤ → 2 ∈ ℂ) | |
19 | 8 | recnd 11287 | . . . . . 6 ⊢ (𝑁 ∈ ℤ → (⌊‘((𝑁 + 1) / 2)) ∈ ℂ) |
20 | 18, 19, 15 | adddid 11283 | . . . . 5 ⊢ (𝑁 ∈ ℤ → (2 · ((⌊‘((𝑁 + 1) / 2)) + 1)) = ((2 · (⌊‘((𝑁 + 1) / 2))) + (2 · 1))) |
21 | 2re 12338 | . . . . . . . . 9 ⊢ 2 ∈ ℝ | |
22 | 21 | a1i 11 | . . . . . . . 8 ⊢ (𝑁 ∈ ℤ → 2 ∈ ℝ) |
23 | 22, 8 | remulcld 11289 | . . . . . . 7 ⊢ (𝑁 ∈ ℤ → (2 · (⌊‘((𝑁 + 1) / 2))) ∈ ℝ) |
24 | 23 | recnd 11287 | . . . . . 6 ⊢ (𝑁 ∈ ℤ → (2 · (⌊‘((𝑁 + 1) / 2))) ∈ ℂ) |
25 | 24, 15, 15 | addassd 11281 | . . . . 5 ⊢ (𝑁 ∈ ℤ → (((2 · (⌊‘((𝑁 + 1) / 2))) + 1) + 1) = ((2 · (⌊‘((𝑁 + 1) / 2))) + (1 + 1))) |
26 | 17, 20, 25 | 3eqtr4d 2785 | . . . 4 ⊢ (𝑁 ∈ ℤ → (2 · ((⌊‘((𝑁 + 1) / 2)) + 1)) = (((2 · (⌊‘((𝑁 + 1) / 2))) + 1) + 1)) |
27 | 14, 26 | breqtrd 5174 | . . 3 ⊢ (𝑁 ∈ ℤ → (𝑁 + 1) < (((2 · (⌊‘((𝑁 + 1) / 2))) + 1) + 1)) |
28 | 23, 9 | readdcld 11288 | . . . 4 ⊢ (𝑁 ∈ ℤ → ((2 · (⌊‘((𝑁 + 1) / 2))) + 1) ∈ ℝ) |
29 | 1, 28, 9 | ltadd1d 11854 | . . 3 ⊢ (𝑁 ∈ ℤ → (𝑁 < ((2 · (⌊‘((𝑁 + 1) / 2))) + 1) ↔ (𝑁 + 1) < (((2 · (⌊‘((𝑁 + 1) / 2))) + 1) + 1))) |
30 | 27, 29 | mpbird 257 | . 2 ⊢ (𝑁 ∈ ℤ → 𝑁 < ((2 · (⌊‘((𝑁 + 1) / 2))) + 1)) |
31 | 2z 12647 | . . . . 5 ⊢ 2 ∈ ℤ | |
32 | 31 | a1i 11 | . . . 4 ⊢ (𝑁 ∈ ℤ → 2 ∈ ℤ) |
33 | 32, 7 | zmulcld 12726 | . . 3 ⊢ (𝑁 ∈ ℤ → (2 · (⌊‘((𝑁 + 1) / 2))) ∈ ℤ) |
34 | zleltp1 12666 | . . 3 ⊢ ((𝑁 ∈ ℤ ∧ (2 · (⌊‘((𝑁 + 1) / 2))) ∈ ℤ) → (𝑁 ≤ (2 · (⌊‘((𝑁 + 1) / 2))) ↔ 𝑁 < ((2 · (⌊‘((𝑁 + 1) / 2))) + 1))) | |
35 | 33, 34 | mpdan 687 | . 2 ⊢ (𝑁 ∈ ℤ → (𝑁 ≤ (2 · (⌊‘((𝑁 + 1) / 2))) ↔ 𝑁 < ((2 · (⌊‘((𝑁 + 1) / 2))) + 1))) |
36 | 30, 35 | mpbird 257 | 1 ⊢ (𝑁 ∈ ℤ → 𝑁 ≤ (2 · (⌊‘((𝑁 + 1) / 2)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∈ wcel 2106 class class class wbr 5148 ‘cfv 6563 (class class class)co 7431 ℝcr 11152 1c1 11154 + caddc 11156 · cmul 11158 < clt 11293 ≤ cle 11294 / cdiv 11918 2c2 12319 ℤcz 12611 ℝ+crp 13032 ⌊cfl 13827 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 ax-pre-sup 11231 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-sup 9480 df-inf 9481 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-div 11919 df-nn 12265 df-2 12327 df-n0 12525 df-z 12612 df-uz 12877 df-rp 13033 df-fl 13829 |
This theorem is referenced by: ovolunlem1a 25545 |
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