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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 12617 | . . . . . . . 8 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℝ) | |
| 2 | peano2re 11434 | . . . . . . . 8 ⊢ (𝑁 ∈ ℝ → (𝑁 + 1) ∈ ℝ) | |
| 3 | 1, 2 | syl 17 | . . . . . . 7 ⊢ (𝑁 ∈ ℤ → (𝑁 + 1) ∈ ℝ) |
| 4 | 3 | rehalfcld 12513 | . . . . . 6 ⊢ (𝑁 ∈ ℤ → ((𝑁 + 1) / 2) ∈ ℝ) |
| 5 | flltp1 13840 | . . . . . 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 13838 | . . . . . . . 8 ⊢ (𝑁 ∈ ℤ → (⌊‘((𝑁 + 1) / 2)) ∈ ℤ) |
| 8 | 7 | zred 12722 | . . . . . . 7 ⊢ (𝑁 ∈ ℤ → (⌊‘((𝑁 + 1) / 2)) ∈ ℝ) |
| 9 | 1red 11262 | . . . . . . 7 ⊢ (𝑁 ∈ ℤ → 1 ∈ ℝ) | |
| 10 | 8, 9 | readdcld 11290 | . . . . . 6 ⊢ (𝑁 ∈ ℤ → ((⌊‘((𝑁 + 1) / 2)) + 1) ∈ ℝ) |
| 11 | 2rp 13039 | . . . . . . 7 ⊢ 2 ∈ ℝ+ | |
| 12 | 11 | a1i 11 | . . . . . 6 ⊢ (𝑁 ∈ ℤ → 2 ∈ ℝ+) |
| 13 | 3, 10, 12 | ltdivmuld 13128 | . . . . 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 11289 | . . . . . . 7 ⊢ (𝑁 ∈ ℤ → 1 ∈ ℂ) |
| 16 | 15 | 2timesd 12509 | . . . . . 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 12344 | . . . . . 6 ⊢ (𝑁 ∈ ℤ → 2 ∈ ℂ) | |
| 19 | 8 | recnd 11289 | . . . . . 6 ⊢ (𝑁 ∈ ℤ → (⌊‘((𝑁 + 1) / 2)) ∈ ℂ) |
| 20 | 18, 19, 15 | adddid 11285 | . . . . 5 ⊢ (𝑁 ∈ ℤ → (2 · ((⌊‘((𝑁 + 1) / 2)) + 1)) = ((2 · (⌊‘((𝑁 + 1) / 2))) + (2 · 1))) |
| 21 | 2re 12340 | . . . . . . . . 9 ⊢ 2 ∈ ℝ | |
| 22 | 21 | a1i 11 | . . . . . . . 8 ⊢ (𝑁 ∈ ℤ → 2 ∈ ℝ) |
| 23 | 22, 8 | remulcld 11291 | . . . . . . 7 ⊢ (𝑁 ∈ ℤ → (2 · (⌊‘((𝑁 + 1) / 2))) ∈ ℝ) |
| 24 | 23 | recnd 11289 | . . . . . 6 ⊢ (𝑁 ∈ ℤ → (2 · (⌊‘((𝑁 + 1) / 2))) ∈ ℂ) |
| 25 | 24, 15, 15 | addassd 11283 | . . . . 5 ⊢ (𝑁 ∈ ℤ → (((2 · (⌊‘((𝑁 + 1) / 2))) + 1) + 1) = ((2 · (⌊‘((𝑁 + 1) / 2))) + (1 + 1))) |
| 26 | 17, 20, 25 | 3eqtr4d 2787 | . . . 4 ⊢ (𝑁 ∈ ℤ → (2 · ((⌊‘((𝑁 + 1) / 2)) + 1)) = (((2 · (⌊‘((𝑁 + 1) / 2))) + 1) + 1)) |
| 27 | 14, 26 | breqtrd 5169 | . . 3 ⊢ (𝑁 ∈ ℤ → (𝑁 + 1) < (((2 · (⌊‘((𝑁 + 1) / 2))) + 1) + 1)) |
| 28 | 23, 9 | readdcld 11290 | . . . 4 ⊢ (𝑁 ∈ ℤ → ((2 · (⌊‘((𝑁 + 1) / 2))) + 1) ∈ ℝ) |
| 29 | 1, 28, 9 | ltadd1d 11856 | . . 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 12649 | . . . . 5 ⊢ 2 ∈ ℤ | |
| 32 | 31 | a1i 11 | . . . 4 ⊢ (𝑁 ∈ ℤ → 2 ∈ ℤ) |
| 33 | 32, 7 | zmulcld 12728 | . . 3 ⊢ (𝑁 ∈ ℤ → (2 · (⌊‘((𝑁 + 1) / 2))) ∈ ℤ) |
| 34 | zleltp1 12668 | . . 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 2108 class class class wbr 5143 ‘cfv 6561 (class class class)co 7431 ℝcr 11154 1c1 11156 + caddc 11158 · cmul 11160 < clt 11295 ≤ cle 11296 / cdiv 11920 2c2 12321 ℤcz 12613 ℝ+crp 13034 ⌊cfl 13830 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-pre-sup 11233 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-sup 9482 df-inf 9483 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-2 12329 df-n0 12527 df-z 12614 df-uz 12879 df-rp 13035 df-fl 13832 |
| This theorem is referenced by: ovolunlem1a 25531 |
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