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Mirrors > Home > MPE Home > Th. List > Mathboxes > zofldiv2ALTV | Structured version Visualization version GIF version |
Description: The floor of an odd numer divided by 2 is equal to the odd number first decreased by 1 and then divided by 2. (Contributed by AV, 7-Jun-2020.) (Revised by AV, 18-Jun-2020.) |
Ref | Expression |
---|---|
zofldiv2ALTV | ⊢ (𝑁 ∈ Odd → (⌊‘(𝑁 / 2)) = ((𝑁 − 1) / 2)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oddz 43673 | . . . . 5 ⊢ (𝑁 ∈ Odd → 𝑁 ∈ ℤ) | |
2 | 1 | zcnd 12076 | . . . 4 ⊢ (𝑁 ∈ Odd → 𝑁 ∈ ℂ) |
3 | npcan1 11053 | . . . . . . 7 ⊢ (𝑁 ∈ ℂ → ((𝑁 − 1) + 1) = 𝑁) | |
4 | 3 | eqcomd 2824 | . . . . . 6 ⊢ (𝑁 ∈ ℂ → 𝑁 = ((𝑁 − 1) + 1)) |
5 | 4 | oveq1d 7160 | . . . . 5 ⊢ (𝑁 ∈ ℂ → (𝑁 / 2) = (((𝑁 − 1) + 1) / 2)) |
6 | peano2cnm 10940 | . . . . . 6 ⊢ (𝑁 ∈ ℂ → (𝑁 − 1) ∈ ℂ) | |
7 | 1cnd 10624 | . . . . . 6 ⊢ (𝑁 ∈ ℂ → 1 ∈ ℂ) | |
8 | 2cnne0 11835 | . . . . . . 7 ⊢ (2 ∈ ℂ ∧ 2 ≠ 0) | |
9 | 8 | a1i 11 | . . . . . 6 ⊢ (𝑁 ∈ ℂ → (2 ∈ ℂ ∧ 2 ≠ 0)) |
10 | divdir 11311 | . . . . . 6 ⊢ (((𝑁 − 1) ∈ ℂ ∧ 1 ∈ ℂ ∧ (2 ∈ ℂ ∧ 2 ≠ 0)) → (((𝑁 − 1) + 1) / 2) = (((𝑁 − 1) / 2) + (1 / 2))) | |
11 | 6, 7, 9, 10 | syl3anc 1363 | . . . . 5 ⊢ (𝑁 ∈ ℂ → (((𝑁 − 1) + 1) / 2) = (((𝑁 − 1) / 2) + (1 / 2))) |
12 | 5, 11 | eqtrd 2853 | . . . 4 ⊢ (𝑁 ∈ ℂ → (𝑁 / 2) = (((𝑁 − 1) / 2) + (1 / 2))) |
13 | 2, 12 | syl 17 | . . 3 ⊢ (𝑁 ∈ Odd → (𝑁 / 2) = (((𝑁 − 1) / 2) + (1 / 2))) |
14 | 13 | fveq2d 6667 | . 2 ⊢ (𝑁 ∈ Odd → (⌊‘(𝑁 / 2)) = (⌊‘(((𝑁 − 1) / 2) + (1 / 2)))) |
15 | halfge0 11842 | . . . 4 ⊢ 0 ≤ (1 / 2) | |
16 | halflt1 11843 | . . . 4 ⊢ (1 / 2) < 1 | |
17 | 15, 16 | pm3.2i 471 | . . 3 ⊢ (0 ≤ (1 / 2) ∧ (1 / 2) < 1) |
18 | oddm1div2z 43676 | . . . 4 ⊢ (𝑁 ∈ Odd → ((𝑁 − 1) / 2) ∈ ℤ) | |
19 | halfre 11839 | . . . 4 ⊢ (1 / 2) ∈ ℝ | |
20 | flbi2 13175 | . . . 4 ⊢ ((((𝑁 − 1) / 2) ∈ ℤ ∧ (1 / 2) ∈ ℝ) → ((⌊‘(((𝑁 − 1) / 2) + (1 / 2))) = ((𝑁 − 1) / 2) ↔ (0 ≤ (1 / 2) ∧ (1 / 2) < 1))) | |
21 | 18, 19, 20 | sylancl 586 | . . 3 ⊢ (𝑁 ∈ Odd → ((⌊‘(((𝑁 − 1) / 2) + (1 / 2))) = ((𝑁 − 1) / 2) ↔ (0 ≤ (1 / 2) ∧ (1 / 2) < 1))) |
22 | 17, 21 | mpbiri 259 | . 2 ⊢ (𝑁 ∈ Odd → (⌊‘(((𝑁 − 1) / 2) + (1 / 2))) = ((𝑁 − 1) / 2)) |
23 | 14, 22 | eqtrd 2853 | 1 ⊢ (𝑁 ∈ Odd → (⌊‘(𝑁 / 2)) = ((𝑁 − 1) / 2)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 = wceq 1528 ∈ wcel 2105 ≠ wne 3013 class class class wbr 5057 ‘cfv 6348 (class class class)co 7145 ℂcc 10523 ℝcr 10524 0cc0 10525 1c1 10526 + caddc 10528 < clt 10663 ≤ cle 10664 − cmin 10858 / cdiv 11285 2c2 11680 ℤcz 11969 ⌊cfl 13148 Odd codd 43667 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 ax-pre-sup 10603 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-er 8278 df-en 8498 df-dom 8499 df-sdom 8500 df-sup 8894 df-inf 8895 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-div 11286 df-nn 11627 df-2 11688 df-n0 11886 df-z 11970 df-uz 12232 df-fl 13150 df-odd 43669 |
This theorem is referenced by: oddflALTV 43705 |
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