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| Mirrors > Home > MPE Home > Th. List > Mathboxes > zofldiv2ALTV | Structured version Visualization version GIF version | ||
| Description: The floor of an odd number 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 47618 | . . . . 5 ⊢ (𝑁 ∈ Odd → 𝑁 ∈ ℤ) | |
| 2 | 1 | zcnd 12723 | . . . 4 ⊢ (𝑁 ∈ Odd → 𝑁 ∈ ℂ) |
| 3 | npcan1 11688 | . . . . . . 7 ⊢ (𝑁 ∈ ℂ → ((𝑁 − 1) + 1) = 𝑁) | |
| 4 | 3 | eqcomd 2743 | . . . . . 6 ⊢ (𝑁 ∈ ℂ → 𝑁 = ((𝑁 − 1) + 1)) |
| 5 | 4 | oveq1d 7446 | . . . . 5 ⊢ (𝑁 ∈ ℂ → (𝑁 / 2) = (((𝑁 − 1) + 1) / 2)) |
| 6 | peano2cnm 11575 | . . . . . 6 ⊢ (𝑁 ∈ ℂ → (𝑁 − 1) ∈ ℂ) | |
| 7 | 1cnd 11256 | . . . . . 6 ⊢ (𝑁 ∈ ℂ → 1 ∈ ℂ) | |
| 8 | 2cnne0 12476 | . . . . . . 7 ⊢ (2 ∈ ℂ ∧ 2 ≠ 0) | |
| 9 | 8 | a1i 11 | . . . . . 6 ⊢ (𝑁 ∈ ℂ → (2 ∈ ℂ ∧ 2 ≠ 0)) |
| 10 | divdir 11947 | . . . . . 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 1373 | . . . . 5 ⊢ (𝑁 ∈ ℂ → (((𝑁 − 1) + 1) / 2) = (((𝑁 − 1) / 2) + (1 / 2))) |
| 12 | 5, 11 | eqtrd 2777 | . . . 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 6910 | . 2 ⊢ (𝑁 ∈ Odd → (⌊‘(𝑁 / 2)) = (⌊‘(((𝑁 − 1) / 2) + (1 / 2)))) |
| 15 | halfge0 12483 | . . . 4 ⊢ 0 ≤ (1 / 2) | |
| 16 | halflt1 12484 | . . . 4 ⊢ (1 / 2) < 1 | |
| 17 | 15, 16 | pm3.2i 470 | . . 3 ⊢ (0 ≤ (1 / 2) ∧ (1 / 2) < 1) |
| 18 | oddm1div2z 47621 | . . . 4 ⊢ (𝑁 ∈ Odd → ((𝑁 − 1) / 2) ∈ ℤ) | |
| 19 | halfre 12480 | . . . 4 ⊢ (1 / 2) ∈ ℝ | |
| 20 | flbi2 13857 | . . . 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 258 | . 2 ⊢ (𝑁 ∈ Odd → (⌊‘(((𝑁 − 1) / 2) + (1 / 2))) = ((𝑁 − 1) / 2)) |
| 23 | 14, 22 | eqtrd 2777 | 1 ⊢ (𝑁 ∈ Odd → (⌊‘(𝑁 / 2)) = ((𝑁 − 1) / 2)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ≠ wne 2940 class class class wbr 5143 ‘cfv 6561 (class class class)co 7431 ℂcc 11153 ℝcr 11154 0cc0 11155 1c1 11156 + caddc 11158 < clt 11295 ≤ cle 11296 − cmin 11492 / cdiv 11920 2c2 12321 ℤcz 12613 ⌊cfl 13830 Odd codd 47612 |
| 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-fl 13832 df-odd 47614 |
| This theorem is referenced by: oddflALTV 47650 |
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