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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 12567 | . . . . . . 7 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℂ) | |
2 | npcan1 11643 | . . . . . . . 8 ⊢ (𝑁 ∈ ℂ → ((𝑁 − 1) + 1) = 𝑁) | |
3 | 2 | eqcomd 2732 | . . . . . . 7 ⊢ (𝑁 ∈ ℂ → 𝑁 = ((𝑁 − 1) + 1)) |
4 | 1, 3 | syl 17 | . . . . . 6 ⊢ (𝑁 ∈ ℤ → 𝑁 = ((𝑁 − 1) + 1)) |
5 | 4 | oveq1d 7420 | . . . . 5 ⊢ (𝑁 ∈ ℤ → (𝑁 / 2) = (((𝑁 − 1) + 1) / 2)) |
6 | peano2zm 12609 | . . . . . . 7 ⊢ (𝑁 ∈ ℤ → (𝑁 − 1) ∈ ℤ) | |
7 | 6 | zcnd 12671 | . . . . . 6 ⊢ (𝑁 ∈ ℤ → (𝑁 − 1) ∈ ℂ) |
8 | 1cnd 11213 | . . . . . 6 ⊢ (𝑁 ∈ ℤ → 1 ∈ ℂ) | |
9 | 2cnne0 12426 | . . . . . . 7 ⊢ (2 ∈ ℂ ∧ 2 ≠ 0) | |
10 | 9 | a1i 11 | . . . . . 6 ⊢ (𝑁 ∈ ℤ → (2 ∈ ℂ ∧ 2 ≠ 0)) |
11 | divdir 11901 | . . . . . 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 1368 | . . . . 5 ⊢ (𝑁 ∈ ℤ → (((𝑁 − 1) + 1) / 2) = (((𝑁 − 1) / 2) + (1 / 2))) |
13 | 5, 12 | eqtrd 2766 | . . . 4 ⊢ (𝑁 ∈ ℤ → (𝑁 / 2) = (((𝑁 − 1) / 2) + (1 / 2))) |
14 | 13 | fveq2d 6889 | . . 3 ⊢ (𝑁 ∈ ℤ → (⌊‘(𝑁 / 2)) = (⌊‘(((𝑁 − 1) / 2) + (1 / 2)))) |
15 | 14 | adantr 480 | . 2 ⊢ ((𝑁 ∈ ℤ ∧ ((𝑁 + 1) / 2) ∈ ℤ) → (⌊‘(𝑁 / 2)) = (⌊‘(((𝑁 − 1) / 2) + (1 / 2)))) |
16 | halfge0 12433 | . . . 4 ⊢ 0 ≤ (1 / 2) | |
17 | halflt1 12434 | . . . 4 ⊢ (1 / 2) < 1 | |
18 | 16, 17 | pm3.2i 470 | . . 3 ⊢ (0 ≤ (1 / 2) ∧ (1 / 2) < 1) |
19 | zob 16309 | . . . . 5 ⊢ (𝑁 ∈ ℤ → (((𝑁 + 1) / 2) ∈ ℤ ↔ ((𝑁 − 1) / 2) ∈ ℤ)) | |
20 | 19 | biimpa 476 | . . . 4 ⊢ ((𝑁 ∈ ℤ ∧ ((𝑁 + 1) / 2) ∈ ℤ) → ((𝑁 − 1) / 2) ∈ ℤ) |
21 | halfre 12430 | . . . 4 ⊢ (1 / 2) ∈ ℝ | |
22 | flbi2 13788 | . . . 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 585 | . . 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 258 | . 2 ⊢ ((𝑁 ∈ ℤ ∧ ((𝑁 + 1) / 2) ∈ ℤ) → (⌊‘(((𝑁 − 1) / 2) + (1 / 2))) = ((𝑁 − 1) / 2)) |
25 | 15, 24 | eqtrd 2766 | 1 ⊢ ((𝑁 ∈ ℤ ∧ ((𝑁 + 1) / 2) ∈ ℤ) → (⌊‘(𝑁 / 2)) = ((𝑁 − 1) / 2)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1533 ∈ wcel 2098 ≠ wne 2934 class class class wbr 5141 ‘cfv 6537 (class class class)co 7405 ℂcc 11110 ℝcr 11111 0cc0 11112 1c1 11113 + caddc 11115 < clt 11252 ≤ cle 11253 − cmin 11448 / cdiv 11875 2c2 12271 ℤcz 12562 ⌊cfl 13761 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 ax-pre-sup 11190 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6294 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7853 df-2nd 7975 df-frecs 8267 df-wrecs 8298 df-recs 8372 df-rdg 8411 df-er 8705 df-en 8942 df-dom 8943 df-sdom 8944 df-sup 9439 df-inf 9440 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-div 11876 df-nn 12217 df-2 12279 df-n0 12477 df-z 12563 df-uz 12827 df-fl 13763 |
This theorem is referenced by: nn0ofldiv2 47490 dignn0flhalflem2 47574 nn0sumshdiglemB 47578 |
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