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 11259 | . . . . . . 7 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℂ) | |
2 | npcan1 10334 | . . . . . . . 8 ⊢ (𝑁 ∈ ℂ → ((𝑁 − 1) + 1) = 𝑁) | |
3 | 2 | eqcomd 2616 | . . . . . . 7 ⊢ (𝑁 ∈ ℂ → 𝑁 = ((𝑁 − 1) + 1)) |
4 | 1, 3 | syl 17 | . . . . . 6 ⊢ (𝑁 ∈ ℤ → 𝑁 = ((𝑁 − 1) + 1)) |
5 | 4 | oveq1d 6564 | . . . . 5 ⊢ (𝑁 ∈ ℤ → (𝑁 / 2) = (((𝑁 − 1) + 1) / 2)) |
6 | peano2zm 11297 | . . . . . . 7 ⊢ (𝑁 ∈ ℤ → (𝑁 − 1) ∈ ℤ) | |
7 | 6 | zcnd 11359 | . . . . . 6 ⊢ (𝑁 ∈ ℤ → (𝑁 − 1) ∈ ℂ) |
8 | 1cnd 9935 | . . . . . 6 ⊢ (𝑁 ∈ ℤ → 1 ∈ ℂ) | |
9 | 2cnne0 11119 | . . . . . . 7 ⊢ (2 ∈ ℂ ∧ 2 ≠ 0) | |
10 | 9 | a1i 11 | . . . . . 6 ⊢ (𝑁 ∈ ℤ → (2 ∈ ℂ ∧ 2 ≠ 0)) |
11 | divdir 10589 | . . . . . 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 1318 | . . . . 5 ⊢ (𝑁 ∈ ℤ → (((𝑁 − 1) + 1) / 2) = (((𝑁 − 1) / 2) + (1 / 2))) |
13 | 5, 12 | eqtrd 2644 | . . . 4 ⊢ (𝑁 ∈ ℤ → (𝑁 / 2) = (((𝑁 − 1) / 2) + (1 / 2))) |
14 | 13 | fveq2d 6107 | . . 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 11126 | . . . 4 ⊢ 0 ≤ (1 / 2) | |
17 | halflt1 11127 | . . . 4 ⊢ (1 / 2) < 1 | |
18 | 16, 17 | pm3.2i 470 | . . 3 ⊢ (0 ≤ (1 / 2) ∧ (1 / 2) < 1) |
19 | zob 14921 | . . . . 5 ⊢ (𝑁 ∈ ℤ → (((𝑁 + 1) / 2) ∈ ℤ ↔ ((𝑁 − 1) / 2) ∈ ℤ)) | |
20 | 19 | biimpa 500 | . . . 4 ⊢ ((𝑁 ∈ ℤ ∧ ((𝑁 + 1) / 2) ∈ ℤ) → ((𝑁 − 1) / 2) ∈ ℤ) |
21 | halfre 11123 | . . . 4 ⊢ (1 / 2) ∈ ℝ | |
22 | flbi2 12480 | . . . 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 693 | . . 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 247 | . 2 ⊢ ((𝑁 ∈ ℤ ∧ ((𝑁 + 1) / 2) ∈ ℤ) → (⌊‘(((𝑁 − 1) / 2) + (1 / 2))) = ((𝑁 − 1) / 2)) |
25 | 15, 24 | eqtrd 2644 | 1 ⊢ ((𝑁 ∈ ℤ ∧ ((𝑁 + 1) / 2) ∈ ℤ) → (⌊‘(𝑁 / 2)) = ((𝑁 − 1) / 2)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 class class class wbr 4583 ‘cfv 5804 (class class class)co 6549 ℂcc 9813 ℝcr 9814 0cc0 9815 1c1 9816 + caddc 9818 < clt 9953 ≤ cle 9954 − cmin 10145 / cdiv 10563 2c2 10947 ℤcz 11254 ⌊cfl 12453 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-sup 8231 df-inf 8232 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-n0 11170 df-z 11255 df-uz 11564 df-fl 12455 |
This theorem is referenced by: nn0ofldiv2 42120 dignn0flhalflem2 42208 nn0sumshdiglemB 42212 |
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