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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 43876 | . . . . 5 ⊢ (𝑁 ∈ Odd → 𝑁 ∈ ℤ) | |
2 | 1 | zcnd 12070 | . . . 4 ⊢ (𝑁 ∈ Odd → 𝑁 ∈ ℂ) |
3 | npcan1 11046 | . . . . . . 7 ⊢ (𝑁 ∈ ℂ → ((𝑁 − 1) + 1) = 𝑁) | |
4 | 3 | eqcomd 2826 | . . . . . 6 ⊢ (𝑁 ∈ ℂ → 𝑁 = ((𝑁 − 1) + 1)) |
5 | 4 | oveq1d 7152 | . . . . 5 ⊢ (𝑁 ∈ ℂ → (𝑁 / 2) = (((𝑁 − 1) + 1) / 2)) |
6 | peano2cnm 10933 | . . . . . 6 ⊢ (𝑁 ∈ ℂ → (𝑁 − 1) ∈ ℂ) | |
7 | 1cnd 10617 | . . . . . 6 ⊢ (𝑁 ∈ ℂ → 1 ∈ ℂ) | |
8 | 2cnne0 11829 | . . . . . . 7 ⊢ (2 ∈ ℂ ∧ 2 ≠ 0) | |
9 | 8 | a1i 11 | . . . . . 6 ⊢ (𝑁 ∈ ℂ → (2 ∈ ℂ ∧ 2 ≠ 0)) |
10 | divdir 11304 | . . . . . 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 1367 | . . . . 5 ⊢ (𝑁 ∈ ℂ → (((𝑁 − 1) + 1) / 2) = (((𝑁 − 1) / 2) + (1 / 2))) |
12 | 5, 11 | eqtrd 2855 | . . . 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 6655 | . 2 ⊢ (𝑁 ∈ Odd → (⌊‘(𝑁 / 2)) = (⌊‘(((𝑁 − 1) / 2) + (1 / 2)))) |
15 | halfge0 11836 | . . . 4 ⊢ 0 ≤ (1 / 2) | |
16 | halflt1 11837 | . . . 4 ⊢ (1 / 2) < 1 | |
17 | 15, 16 | pm3.2i 473 | . . 3 ⊢ (0 ≤ (1 / 2) ∧ (1 / 2) < 1) |
18 | oddm1div2z 43879 | . . . 4 ⊢ (𝑁 ∈ Odd → ((𝑁 − 1) / 2) ∈ ℤ) | |
19 | halfre 11833 | . . . 4 ⊢ (1 / 2) ∈ ℝ | |
20 | flbi2 13172 | . . . 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 588 | . . 3 ⊢ (𝑁 ∈ Odd → ((⌊‘(((𝑁 − 1) / 2) + (1 / 2))) = ((𝑁 − 1) / 2) ↔ (0 ≤ (1 / 2) ∧ (1 / 2) < 1))) |
22 | 17, 21 | mpbiri 260 | . 2 ⊢ (𝑁 ∈ Odd → (⌊‘(((𝑁 − 1) / 2) + (1 / 2))) = ((𝑁 − 1) / 2)) |
23 | 14, 22 | eqtrd 2855 | 1 ⊢ (𝑁 ∈ Odd → (⌊‘(𝑁 / 2)) = ((𝑁 − 1) / 2)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ≠ wne 3011 class class class wbr 5047 ‘cfv 6336 (class class class)co 7137 ℂcc 10516 ℝcr 10517 0cc0 10518 1c1 10519 + caddc 10521 < clt 10656 ≤ cle 10657 − cmin 10851 / cdiv 11278 2c2 11674 ℤcz 11963 ⌊cfl 13145 Odd codd 43870 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2792 ax-sep 5184 ax-nul 5191 ax-pow 5247 ax-pr 5311 ax-un 7442 ax-cnex 10574 ax-resscn 10575 ax-1cn 10576 ax-icn 10577 ax-addcl 10578 ax-addrcl 10579 ax-mulcl 10580 ax-mulrcl 10581 ax-mulcom 10582 ax-addass 10583 ax-mulass 10584 ax-distr 10585 ax-i2m1 10586 ax-1ne0 10587 ax-1rid 10588 ax-rnegex 10589 ax-rrecex 10590 ax-cnre 10591 ax-pre-lttri 10592 ax-pre-lttrn 10593 ax-pre-ltadd 10594 ax-pre-mulgt0 10595 ax-pre-sup 10596 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2891 df-nfc 2959 df-ne 3012 df-nel 3119 df-ral 3138 df-rex 3139 df-reu 3140 df-rmo 3141 df-rab 3142 df-v 3483 df-sbc 3759 df-csb 3867 df-dif 3922 df-un 3924 df-in 3926 df-ss 3935 df-pss 3937 df-nul 4275 df-if 4449 df-pw 4522 df-sn 4549 df-pr 4551 df-tp 4553 df-op 4555 df-uni 4820 df-iun 4902 df-br 5048 df-opab 5110 df-mpt 5128 df-tr 5154 df-id 5441 df-eprel 5446 df-po 5455 df-so 5456 df-fr 5495 df-we 5497 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-pred 6129 df-ord 6175 df-on 6176 df-lim 6177 df-suc 6178 df-iota 6295 df-fun 6338 df-fn 6339 df-f 6340 df-f1 6341 df-fo 6342 df-f1o 6343 df-fv 6344 df-riota 7095 df-ov 7140 df-oprab 7141 df-mpo 7142 df-om 7562 df-wrecs 7928 df-recs 7989 df-rdg 8027 df-er 8270 df-en 8491 df-dom 8492 df-sdom 8493 df-sup 8887 df-inf 8888 df-pnf 10658 df-mnf 10659 df-xr 10660 df-ltxr 10661 df-le 10662 df-sub 10853 df-neg 10854 df-div 11279 df-nn 11620 df-2 11682 df-n0 11880 df-z 11964 df-uz 12226 df-fl 13147 df-odd 43872 |
This theorem is referenced by: oddflALTV 43908 |
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