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Mirrors > Home > MPE Home > Th. List > fldiv4lem1div2 | Structured version Visualization version GIF version |
Description: The floor of a positive integer divided by 4 is less than or equal to the half of the integer minus 1. (Contributed by AV, 9-Jul-2021.) |
Ref | Expression |
---|---|
fldiv4lem1div2 | ⊢ (𝑁 ∈ ℕ → (⌊‘(𝑁 / 4)) ≤ ((𝑁 − 1) / 2)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elnn1uz2 11641 | . 2 ⊢ (𝑁 ∈ ℕ ↔ (𝑁 = 1 ∨ 𝑁 ∈ (ℤ≥‘2))) | |
2 | 1lt4 11076 | . . . . . 6 ⊢ 1 < 4 | |
3 | 1nn0 11185 | . . . . . . 7 ⊢ 1 ∈ ℕ0 | |
4 | 4nn 11064 | . . . . . . 7 ⊢ 4 ∈ ℕ | |
5 | divfl0 12487 | . . . . . . 7 ⊢ ((1 ∈ ℕ0 ∧ 4 ∈ ℕ) → (1 < 4 ↔ (⌊‘(1 / 4)) = 0)) | |
6 | 3, 4, 5 | mp2an 704 | . . . . . 6 ⊢ (1 < 4 ↔ (⌊‘(1 / 4)) = 0) |
7 | 2, 6 | mpbi 219 | . . . . 5 ⊢ (⌊‘(1 / 4)) = 0 |
8 | 1re 9918 | . . . . . . 7 ⊢ 1 ∈ ℝ | |
9 | 4re 10974 | . . . . . . 7 ⊢ 4 ∈ ℝ | |
10 | 4ne0 10994 | . . . . . . 7 ⊢ 4 ≠ 0 | |
11 | redivcl 10623 | . . . . . . . . 9 ⊢ ((1 ∈ ℝ ∧ 4 ∈ ℝ ∧ 4 ≠ 0) → (1 / 4) ∈ ℝ) | |
12 | 11 | flcld 12461 | . . . . . . . 8 ⊢ ((1 ∈ ℝ ∧ 4 ∈ ℝ ∧ 4 ≠ 0) → (⌊‘(1 / 4)) ∈ ℤ) |
13 | 12 | zred 11358 | . . . . . . 7 ⊢ ((1 ∈ ℝ ∧ 4 ∈ ℝ ∧ 4 ≠ 0) → (⌊‘(1 / 4)) ∈ ℝ) |
14 | 8, 9, 10, 13 | mp3an 1416 | . . . . . 6 ⊢ (⌊‘(1 / 4)) ∈ ℝ |
15 | 14 | eqlei 10026 | . . . . 5 ⊢ ((⌊‘(1 / 4)) = 0 → (⌊‘(1 / 4)) ≤ 0) |
16 | 7, 15 | mp1i 13 | . . . 4 ⊢ (𝑁 = 1 → (⌊‘(1 / 4)) ≤ 0) |
17 | oveq1 6556 | . . . . 5 ⊢ (𝑁 = 1 → (𝑁 / 4) = (1 / 4)) | |
18 | 17 | fveq2d 6107 | . . . 4 ⊢ (𝑁 = 1 → (⌊‘(𝑁 / 4)) = (⌊‘(1 / 4))) |
19 | oveq1 6556 | . . . . . . 7 ⊢ (𝑁 = 1 → (𝑁 − 1) = (1 − 1)) | |
20 | 1m1e0 10966 | . . . . . . 7 ⊢ (1 − 1) = 0 | |
21 | 19, 20 | syl6eq 2660 | . . . . . 6 ⊢ (𝑁 = 1 → (𝑁 − 1) = 0) |
22 | 21 | oveq1d 6564 | . . . . 5 ⊢ (𝑁 = 1 → ((𝑁 − 1) / 2) = (0 / 2)) |
23 | 2cnne0 11119 | . . . . . 6 ⊢ (2 ∈ ℂ ∧ 2 ≠ 0) | |
24 | div0 10594 | . . . . . 6 ⊢ ((2 ∈ ℂ ∧ 2 ≠ 0) → (0 / 2) = 0) | |
25 | 23, 24 | ax-mp 5 | . . . . 5 ⊢ (0 / 2) = 0 |
26 | 22, 25 | syl6eq 2660 | . . . 4 ⊢ (𝑁 = 1 → ((𝑁 − 1) / 2) = 0) |
27 | 16, 18, 26 | 3brtr4d 4615 | . . 3 ⊢ (𝑁 = 1 → (⌊‘(𝑁 / 4)) ≤ ((𝑁 − 1) / 2)) |
28 | fldiv4lem1div2uz2 12499 | . . 3 ⊢ (𝑁 ∈ (ℤ≥‘2) → (⌊‘(𝑁 / 4)) ≤ ((𝑁 − 1) / 2)) | |
29 | 27, 28 | jaoi 393 | . 2 ⊢ ((𝑁 = 1 ∨ 𝑁 ∈ (ℤ≥‘2)) → (⌊‘(𝑁 / 4)) ≤ ((𝑁 − 1) / 2)) |
30 | 1, 29 | sylbi 206 | 1 ⊢ (𝑁 ∈ ℕ → (⌊‘(𝑁 / 4)) ≤ ((𝑁 − 1) / 2)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∨ wo 382 ∧ wa 383 ∧ w3a 1031 = 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 < clt 9953 ≤ cle 9954 − cmin 10145 / cdiv 10563 ℕcn 10897 2c2 10947 4c4 10949 ℕ0cn0 11169 ℤ≥cuz 11563 ⌊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-3 10957 df-4 10958 df-n0 11170 df-z 11255 df-uz 11564 df-rp 11709 df-fl 12455 |
This theorem is referenced by: gausslemma2dlem0g 24887 |
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