Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > flqmulnn0 | GIF version |
Description: Move a nonnegative integer in and out of a floor. (Contributed by Jim Kingdon, 10-Oct-2021.) |
Ref | Expression |
---|---|
flqmulnn0 | ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ ℚ) → (𝑁 · (⌊‘𝐴)) ≤ (⌊‘(𝑁 · 𝐴))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | flqcl 10198 | . . . . 5 ⊢ (𝐴 ∈ ℚ → (⌊‘𝐴) ∈ ℤ) | |
2 | 1 | adantl 275 | . . . 4 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ ℚ) → (⌊‘𝐴) ∈ ℤ) |
3 | 2 | zred 9304 | . . 3 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ ℚ) → (⌊‘𝐴) ∈ ℝ) |
4 | qre 9554 | . . . 4 ⊢ (𝐴 ∈ ℚ → 𝐴 ∈ ℝ) | |
5 | 4 | adantl 275 | . . 3 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ ℚ) → 𝐴 ∈ ℝ) |
6 | simpl 108 | . . . 4 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ ℚ) → 𝑁 ∈ ℕ0) | |
7 | 6 | nn0red 9159 | . . 3 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ ℚ) → 𝑁 ∈ ℝ) |
8 | 6 | nn0ge0d 9161 | . . 3 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ ℚ) → 0 ≤ 𝑁) |
9 | flqle 10203 | . . . 4 ⊢ (𝐴 ∈ ℚ → (⌊‘𝐴) ≤ 𝐴) | |
10 | 9 | adantl 275 | . . 3 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ ℚ) → (⌊‘𝐴) ≤ 𝐴) |
11 | 3, 5, 7, 8, 10 | lemul2ad 8826 | . 2 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ ℚ) → (𝑁 · (⌊‘𝐴)) ≤ (𝑁 · 𝐴)) |
12 | nn0z 9202 | . . . . 5 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ ℤ) | |
13 | zq 9555 | . . . . 5 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℚ) | |
14 | 12, 13 | syl 14 | . . . 4 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ ℚ) |
15 | qmulcl 9566 | . . . 4 ⊢ ((𝑁 ∈ ℚ ∧ 𝐴 ∈ ℚ) → (𝑁 · 𝐴) ∈ ℚ) | |
16 | 14, 15 | sylan 281 | . . 3 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ ℚ) → (𝑁 · 𝐴) ∈ ℚ) |
17 | zmulcl 9235 | . . . 4 ⊢ ((𝑁 ∈ ℤ ∧ (⌊‘𝐴) ∈ ℤ) → (𝑁 · (⌊‘𝐴)) ∈ ℤ) | |
18 | 12, 1, 17 | syl2an 287 | . . 3 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ ℚ) → (𝑁 · (⌊‘𝐴)) ∈ ℤ) |
19 | flqge 10207 | . . 3 ⊢ (((𝑁 · 𝐴) ∈ ℚ ∧ (𝑁 · (⌊‘𝐴)) ∈ ℤ) → ((𝑁 · (⌊‘𝐴)) ≤ (𝑁 · 𝐴) ↔ (𝑁 · (⌊‘𝐴)) ≤ (⌊‘(𝑁 · 𝐴)))) | |
20 | 16, 18, 19 | syl2anc 409 | . 2 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ ℚ) → ((𝑁 · (⌊‘𝐴)) ≤ (𝑁 · 𝐴) ↔ (𝑁 · (⌊‘𝐴)) ≤ (⌊‘(𝑁 · 𝐴)))) |
21 | 11, 20 | mpbid 146 | 1 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ ℚ) → (𝑁 · (⌊‘𝐴)) ≤ (⌊‘(𝑁 · 𝐴))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 ∈ wcel 2135 class class class wbr 3976 ‘cfv 5182 (class class class)co 5836 ℝcr 7743 · cmul 7749 ≤ cle 7925 ℕ0cn0 9105 ℤcz 9182 ℚcq 9548 ⌊cfl 10193 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-pow 4147 ax-pr 4181 ax-un 4405 ax-setind 4508 ax-cnex 7835 ax-resscn 7836 ax-1cn 7837 ax-1re 7838 ax-icn 7839 ax-addcl 7840 ax-addrcl 7841 ax-mulcl 7842 ax-mulrcl 7843 ax-addcom 7844 ax-mulcom 7845 ax-addass 7846 ax-mulass 7847 ax-distr 7848 ax-i2m1 7849 ax-0lt1 7850 ax-1rid 7851 ax-0id 7852 ax-rnegex 7853 ax-precex 7854 ax-cnre 7855 ax-pre-ltirr 7856 ax-pre-ltwlin 7857 ax-pre-lttrn 7858 ax-pre-apti 7859 ax-pre-ltadd 7860 ax-pre-mulgt0 7861 ax-pre-mulext 7862 ax-arch 7863 |
This theorem depends on definitions: df-bi 116 df-3or 968 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-nel 2430 df-ral 2447 df-rex 2448 df-reu 2449 df-rmo 2450 df-rab 2451 df-v 2723 df-sbc 2947 df-csb 3041 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-int 3819 df-iun 3862 df-br 3977 df-opab 4038 df-mpt 4039 df-id 4265 df-po 4268 df-iso 4269 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-rn 4609 df-res 4610 df-ima 4611 df-iota 5147 df-fun 5184 df-fn 5185 df-f 5186 df-fv 5190 df-riota 5792 df-ov 5839 df-oprab 5840 df-mpo 5841 df-1st 6100 df-2nd 6101 df-pnf 7926 df-mnf 7927 df-xr 7928 df-ltxr 7929 df-le 7930 df-sub 8062 df-neg 8063 df-reap 8464 df-ap 8471 df-div 8560 df-inn 8849 df-n0 9106 df-z 9183 df-q 9549 df-rp 9581 df-fl 10195 |
This theorem is referenced by: modqmulnn 10267 |
Copyright terms: Public domain | W3C validator |