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Mirrors > Home > MPE Home > Th. List > flodddiv4lt | Structured version Visualization version GIF version |
Description: The floor of an odd number divided by 4 is less than the odd number divided by 4. (Contributed by AV, 4-Jul-2021.) |
Ref | Expression |
---|---|
flodddiv4lt | ⊢ ((𝑁 ∈ ℤ ∧ ¬ 2 ∥ 𝑁) → (⌊‘(𝑁 / 4)) < (𝑁 / 4)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl 481 | . 2 ⊢ ((𝑁 ∈ ℤ ∧ ¬ 2 ∥ 𝑁) → 𝑁 ∈ ℤ) | |
2 | 4z 12624 | . . . 4 ⊢ 4 ∈ ℤ | |
3 | 4ne0 12348 | . . . 4 ⊢ 4 ≠ 0 | |
4 | 2, 3 | pm3.2i 469 | . . 3 ⊢ (4 ∈ ℤ ∧ 4 ≠ 0) |
5 | 4 | a1i 11 | . 2 ⊢ ((𝑁 ∈ ℤ ∧ ¬ 2 ∥ 𝑁) → (4 ∈ ℤ ∧ 4 ≠ 0)) |
6 | 4dvdseven 16347 | . . . 4 ⊢ (4 ∥ 𝑁 → 2 ∥ 𝑁) | |
7 | 6 | con3i 154 | . . 3 ⊢ (¬ 2 ∥ 𝑁 → ¬ 4 ∥ 𝑁) |
8 | 7 | adantl 480 | . 2 ⊢ ((𝑁 ∈ ℤ ∧ ¬ 2 ∥ 𝑁) → ¬ 4 ∥ 𝑁) |
9 | fldivndvdslt 16388 | . 2 ⊢ ((𝑁 ∈ ℤ ∧ (4 ∈ ℤ ∧ 4 ≠ 0) ∧ ¬ 4 ∥ 𝑁) → (⌊‘(𝑁 / 4)) < (𝑁 / 4)) | |
10 | 1, 5, 8, 9 | syl3anc 1368 | 1 ⊢ ((𝑁 ∈ ℤ ∧ ¬ 2 ∥ 𝑁) → (⌊‘(𝑁 / 4)) < (𝑁 / 4)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 394 ∈ wcel 2098 ≠ wne 2930 class class class wbr 5143 ‘cfv 6542 (class class class)co 7415 0cc0 11136 < clt 11276 / cdiv 11899 2c2 12295 4c4 12297 ℤcz 12586 ⌊cfl 13785 ∥ cdvds 16228 |
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 2166 ax-ext 2696 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7737 ax-cnex 11192 ax-resscn 11193 ax-1cn 11194 ax-icn 11195 ax-addcl 11196 ax-addrcl 11197 ax-mulcl 11198 ax-mulrcl 11199 ax-mulcom 11200 ax-addass 11201 ax-mulass 11202 ax-distr 11203 ax-i2m1 11204 ax-1ne0 11205 ax-1rid 11206 ax-rnegex 11207 ax-rrecex 11208 ax-cnre 11209 ax-pre-lttri 11210 ax-pre-lttrn 11211 ax-pre-ltadd 11212 ax-pre-mulgt0 11213 ax-pre-sup 11214 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 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 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3960 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7371 df-ov 7418 df-oprab 7419 df-mpo 7420 df-om 7868 df-2nd 7990 df-frecs 8283 df-wrecs 8314 df-recs 8388 df-rdg 8427 df-er 8721 df-en 8961 df-dom 8962 df-sdom 8963 df-sup 9463 df-inf 9464 df-pnf 11278 df-mnf 11279 df-xr 11280 df-ltxr 11281 df-le 11282 df-sub 11474 df-neg 11475 df-div 11900 df-nn 12241 df-2 12303 df-3 12304 df-4 12305 df-n0 12501 df-z 12587 df-uz 12851 df-fl 13787 df-dvds 16229 |
This theorem is referenced by: flodddiv4t2lthalf 16390 |
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