![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > ILE Home > Th. List > ceiqle | GIF version |
Description: The ceiling of a real number is the smallest integer greater than or equal to it. (Contributed by Jim Kingdon, 11-Oct-2021.) |
Ref | Expression |
---|---|
ceiqle | ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℤ ∧ 𝐴 ≤ 𝐵) → -(⌊‘-𝐴) ≤ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ceiqcl 10320 | . . . . . 6 ⊢ (𝐴 ∈ ℚ → -(⌊‘-𝐴) ∈ ℤ) | |
2 | 1 | zred 9388 | . . . . 5 ⊢ (𝐴 ∈ ℚ → -(⌊‘-𝐴) ∈ ℝ) |
3 | peano2rem 8237 | . . . . 5 ⊢ (-(⌊‘-𝐴) ∈ ℝ → (-(⌊‘-𝐴) − 1) ∈ ℝ) | |
4 | 2, 3 | syl 14 | . . . 4 ⊢ (𝐴 ∈ ℚ → (-(⌊‘-𝐴) − 1) ∈ ℝ) |
5 | 4 | 3ad2ant1 1019 | . . 3 ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℤ ∧ 𝐴 ≤ 𝐵) → (-(⌊‘-𝐴) − 1) ∈ ℝ) |
6 | qre 9638 | . . . 4 ⊢ (𝐴 ∈ ℚ → 𝐴 ∈ ℝ) | |
7 | 6 | 3ad2ant1 1019 | . . 3 ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℤ ∧ 𝐴 ≤ 𝐵) → 𝐴 ∈ ℝ) |
8 | zre 9270 | . . . 4 ⊢ (𝐵 ∈ ℤ → 𝐵 ∈ ℝ) | |
9 | 8 | 3ad2ant2 1020 | . . 3 ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℤ ∧ 𝐴 ≤ 𝐵) → 𝐵 ∈ ℝ) |
10 | ceiqm1l 10324 | . . . 4 ⊢ (𝐴 ∈ ℚ → (-(⌊‘-𝐴) − 1) < 𝐴) | |
11 | 10 | 3ad2ant1 1019 | . . 3 ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℤ ∧ 𝐴 ≤ 𝐵) → (-(⌊‘-𝐴) − 1) < 𝐴) |
12 | simp3 1000 | . . 3 ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℤ ∧ 𝐴 ≤ 𝐵) → 𝐴 ≤ 𝐵) | |
13 | 5, 7, 9, 11, 12 | ltletrd 8393 | . 2 ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℤ ∧ 𝐴 ≤ 𝐵) → (-(⌊‘-𝐴) − 1) < 𝐵) |
14 | zlem1lt 9322 | . . . 4 ⊢ ((-(⌊‘-𝐴) ∈ ℤ ∧ 𝐵 ∈ ℤ) → (-(⌊‘-𝐴) ≤ 𝐵 ↔ (-(⌊‘-𝐴) − 1) < 𝐵)) | |
15 | 1, 14 | sylan 283 | . . 3 ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℤ) → (-(⌊‘-𝐴) ≤ 𝐵 ↔ (-(⌊‘-𝐴) − 1) < 𝐵)) |
16 | 15 | 3adant3 1018 | . 2 ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℤ ∧ 𝐴 ≤ 𝐵) → (-(⌊‘-𝐴) ≤ 𝐵 ↔ (-(⌊‘-𝐴) − 1) < 𝐵)) |
17 | 13, 16 | mpbird 167 | 1 ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℤ ∧ 𝐴 ≤ 𝐵) → -(⌊‘-𝐴) ≤ 𝐵) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 105 ∧ w3a 979 ∈ wcel 2158 class class class wbr 4015 ‘cfv 5228 (class class class)co 5888 ℝcr 7823 1c1 7825 < clt 8005 ≤ cle 8006 − cmin 8141 -cneg 8142 ℤcz 9266 ℚcq 9632 ⌊cfl 10281 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1457 ax-7 1458 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-8 1514 ax-10 1515 ax-11 1516 ax-i12 1517 ax-bndl 1519 ax-4 1520 ax-17 1536 ax-i9 1540 ax-ial 1544 ax-i5r 1545 ax-13 2160 ax-14 2161 ax-ext 2169 ax-sep 4133 ax-pow 4186 ax-pr 4221 ax-un 4445 ax-setind 4548 ax-cnex 7915 ax-resscn 7916 ax-1cn 7917 ax-1re 7918 ax-icn 7919 ax-addcl 7920 ax-addrcl 7921 ax-mulcl 7922 ax-mulrcl 7923 ax-addcom 7924 ax-mulcom 7925 ax-addass 7926 ax-mulass 7927 ax-distr 7928 ax-i2m1 7929 ax-0lt1 7930 ax-1rid 7931 ax-0id 7932 ax-rnegex 7933 ax-precex 7934 ax-cnre 7935 ax-pre-ltirr 7936 ax-pre-ltwlin 7937 ax-pre-lttrn 7938 ax-pre-apti 7939 ax-pre-ltadd 7940 ax-pre-mulgt0 7941 ax-pre-mulext 7942 ax-arch 7943 |
This theorem depends on definitions: df-bi 117 df-3or 980 df-3an 981 df-tru 1366 df-fal 1369 df-nf 1471 df-sb 1773 df-eu 2039 df-mo 2040 df-clab 2174 df-cleq 2180 df-clel 2183 df-nfc 2318 df-ne 2358 df-nel 2453 df-ral 2470 df-rex 2471 df-reu 2472 df-rmo 2473 df-rab 2474 df-v 2751 df-sbc 2975 df-csb 3070 df-dif 3143 df-un 3145 df-in 3147 df-ss 3154 df-pw 3589 df-sn 3610 df-pr 3611 df-op 3613 df-uni 3822 df-int 3857 df-iun 3900 df-br 4016 df-opab 4077 df-mpt 4078 df-id 4305 df-po 4308 df-iso 4309 df-xp 4644 df-rel 4645 df-cnv 4646 df-co 4647 df-dm 4648 df-rn 4649 df-res 4650 df-ima 4651 df-iota 5190 df-fun 5230 df-fn 5231 df-f 5232 df-fv 5236 df-riota 5844 df-ov 5891 df-oprab 5892 df-mpo 5893 df-1st 6154 df-2nd 6155 df-pnf 8007 df-mnf 8008 df-xr 8009 df-ltxr 8010 df-le 8011 df-sub 8143 df-neg 8144 df-reap 8545 df-ap 8552 df-div 8643 df-inn 8933 df-n0 9190 df-z 9267 df-q 9633 df-rp 9667 df-fl 10283 |
This theorem is referenced by: ceilqle 10327 |
Copyright terms: Public domain | W3C validator |