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Mathbox for Alexander van der Vekens |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > zofldiv2 | Structured version Visualization version GIF version |
Description: The floor of an odd integer divided by 2 is equal to the integer first decreased by 1 and then divided by 2. (Contributed by AV, 7-Jun-2020.) |
Ref | Expression |
---|---|
zofldiv2 | ⊢ ((𝑁 ∈ ℤ ∧ ((𝑁 + 1) / 2) ∈ ℤ) → (⌊‘(𝑁 / 2)) = ((𝑁 − 1) / 2)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | zcn 11709 | . . . . . . 7 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℂ) | |
2 | npcan1 10779 | . . . . . . . 8 ⊢ (𝑁 ∈ ℂ → ((𝑁 − 1) + 1) = 𝑁) | |
3 | 2 | eqcomd 2831 | . . . . . . 7 ⊢ (𝑁 ∈ ℂ → 𝑁 = ((𝑁 − 1) + 1)) |
4 | 1, 3 | syl 17 | . . . . . 6 ⊢ (𝑁 ∈ ℤ → 𝑁 = ((𝑁 − 1) + 1)) |
5 | 4 | oveq1d 6920 | . . . . 5 ⊢ (𝑁 ∈ ℤ → (𝑁 / 2) = (((𝑁 − 1) + 1) / 2)) |
6 | peano2zm 11748 | . . . . . . 7 ⊢ (𝑁 ∈ ℤ → (𝑁 − 1) ∈ ℤ) | |
7 | 6 | zcnd 11811 | . . . . . 6 ⊢ (𝑁 ∈ ℤ → (𝑁 − 1) ∈ ℂ) |
8 | 1cnd 10351 | . . . . . 6 ⊢ (𝑁 ∈ ℤ → 1 ∈ ℂ) | |
9 | 2cnne0 11568 | . . . . . . 7 ⊢ (2 ∈ ℂ ∧ 2 ≠ 0) | |
10 | 9 | a1i 11 | . . . . . 6 ⊢ (𝑁 ∈ ℤ → (2 ∈ ℂ ∧ 2 ≠ 0)) |
11 | divdir 11035 | . . . . . 6 ⊢ (((𝑁 − 1) ∈ ℂ ∧ 1 ∈ ℂ ∧ (2 ∈ ℂ ∧ 2 ≠ 0)) → (((𝑁 − 1) + 1) / 2) = (((𝑁 − 1) / 2) + (1 / 2))) | |
12 | 7, 8, 10, 11 | syl3anc 1494 | . . . . 5 ⊢ (𝑁 ∈ ℤ → (((𝑁 − 1) + 1) / 2) = (((𝑁 − 1) / 2) + (1 / 2))) |
13 | 5, 12 | eqtrd 2861 | . . . 4 ⊢ (𝑁 ∈ ℤ → (𝑁 / 2) = (((𝑁 − 1) / 2) + (1 / 2))) |
14 | 13 | fveq2d 6437 | . . 3 ⊢ (𝑁 ∈ ℤ → (⌊‘(𝑁 / 2)) = (⌊‘(((𝑁 − 1) / 2) + (1 / 2)))) |
15 | 14 | adantr 474 | . 2 ⊢ ((𝑁 ∈ ℤ ∧ ((𝑁 + 1) / 2) ∈ ℤ) → (⌊‘(𝑁 / 2)) = (⌊‘(((𝑁 − 1) / 2) + (1 / 2)))) |
16 | halfge0 11575 | . . . 4 ⊢ 0 ≤ (1 / 2) | |
17 | halflt1 11576 | . . . 4 ⊢ (1 / 2) < 1 | |
18 | 16, 17 | pm3.2i 464 | . . 3 ⊢ (0 ≤ (1 / 2) ∧ (1 / 2) < 1) |
19 | zob 15457 | . . . . 5 ⊢ (𝑁 ∈ ℤ → (((𝑁 + 1) / 2) ∈ ℤ ↔ ((𝑁 − 1) / 2) ∈ ℤ)) | |
20 | 19 | biimpa 470 | . . . 4 ⊢ ((𝑁 ∈ ℤ ∧ ((𝑁 + 1) / 2) ∈ ℤ) → ((𝑁 − 1) / 2) ∈ ℤ) |
21 | halfre 11572 | . . . 4 ⊢ (1 / 2) ∈ ℝ | |
22 | flbi2 12913 | . . . 4 ⊢ ((((𝑁 − 1) / 2) ∈ ℤ ∧ (1 / 2) ∈ ℝ) → ((⌊‘(((𝑁 − 1) / 2) + (1 / 2))) = ((𝑁 − 1) / 2) ↔ (0 ≤ (1 / 2) ∧ (1 / 2) < 1))) | |
23 | 20, 21, 22 | sylancl 580 | . . 3 ⊢ ((𝑁 ∈ ℤ ∧ ((𝑁 + 1) / 2) ∈ ℤ) → ((⌊‘(((𝑁 − 1) / 2) + (1 / 2))) = ((𝑁 − 1) / 2) ↔ (0 ≤ (1 / 2) ∧ (1 / 2) < 1))) |
24 | 18, 23 | mpbiri 250 | . 2 ⊢ ((𝑁 ∈ ℤ ∧ ((𝑁 + 1) / 2) ∈ ℤ) → (⌊‘(((𝑁 − 1) / 2) + (1 / 2))) = ((𝑁 − 1) / 2)) |
25 | 15, 24 | eqtrd 2861 | 1 ⊢ ((𝑁 ∈ ℤ ∧ ((𝑁 + 1) / 2) ∈ ℤ) → (⌊‘(𝑁 / 2)) = ((𝑁 − 1) / 2)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 386 = wceq 1656 ∈ wcel 2164 ≠ wne 2999 class class class wbr 4873 ‘cfv 6123 (class class class)co 6905 ℂcc 10250 ℝcr 10251 0cc0 10252 1c1 10253 + caddc 10255 < clt 10391 ≤ cle 10392 − cmin 10585 / cdiv 11009 2c2 11406 ℤcz 11704 ⌊cfl 12886 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-sep 5005 ax-nul 5013 ax-pow 5065 ax-pr 5127 ax-un 7209 ax-cnex 10308 ax-resscn 10309 ax-1cn 10310 ax-icn 10311 ax-addcl 10312 ax-addrcl 10313 ax-mulcl 10314 ax-mulrcl 10315 ax-mulcom 10316 ax-addass 10317 ax-mulass 10318 ax-distr 10319 ax-i2m1 10320 ax-1ne0 10321 ax-1rid 10322 ax-rnegex 10323 ax-rrecex 10324 ax-cnre 10325 ax-pre-lttri 10326 ax-pre-lttrn 10327 ax-pre-ltadd 10328 ax-pre-mulgt0 10329 ax-pre-sup 10330 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3or 1112 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-nel 3103 df-ral 3122 df-rex 3123 df-reu 3124 df-rmo 3125 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-pss 3814 df-nul 4145 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-tp 4402 df-op 4404 df-uni 4659 df-iun 4742 df-br 4874 df-opab 4936 df-mpt 4953 df-tr 4976 df-id 5250 df-eprel 5255 df-po 5263 df-so 5264 df-fr 5301 df-we 5303 df-xp 5348 df-rel 5349 df-cnv 5350 df-co 5351 df-dm 5352 df-rn 5353 df-res 5354 df-ima 5355 df-pred 5920 df-ord 5966 df-on 5967 df-lim 5968 df-suc 5969 df-iota 6086 df-fun 6125 df-fn 6126 df-f 6127 df-f1 6128 df-fo 6129 df-f1o 6130 df-fv 6131 df-riota 6866 df-ov 6908 df-oprab 6909 df-mpt2 6910 df-om 7327 df-wrecs 7672 df-recs 7734 df-rdg 7772 df-er 8009 df-en 8223 df-dom 8224 df-sdom 8225 df-sup 8617 df-inf 8618 df-pnf 10393 df-mnf 10394 df-xr 10395 df-ltxr 10396 df-le 10397 df-sub 10587 df-neg 10588 df-div 11010 df-nn 11351 df-2 11414 df-n0 11619 df-z 11705 df-uz 11969 df-fl 12888 |
This theorem is referenced by: nn0ofldiv2 43166 dignn0flhalflem2 43250 nn0sumshdiglemB 43254 |
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