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Mirrors > Home > ILE Home > Th. List > flqlt | GIF version |
Description: The floor function value is less than the next integer. (Contributed by Jim Kingdon, 8-Oct-2021.) |
Ref | Expression |
---|---|
flqlt | ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℤ) → (𝐴 < 𝐵 ↔ (⌊‘𝐴) < 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | flqge 10225 | . . 3 ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℤ) → (𝐵 ≤ 𝐴 ↔ 𝐵 ≤ (⌊‘𝐴))) | |
2 | 1 | notbid 662 | . 2 ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℤ) → (¬ 𝐵 ≤ 𝐴 ↔ ¬ 𝐵 ≤ (⌊‘𝐴))) |
3 | zq 9572 | . . 3 ⊢ (𝐵 ∈ ℤ → 𝐵 ∈ ℚ) | |
4 | qltnle 10189 | . . 3 ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) → (𝐴 < 𝐵 ↔ ¬ 𝐵 ≤ 𝐴)) | |
5 | 3, 4 | sylan2 284 | . 2 ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℤ) → (𝐴 < 𝐵 ↔ ¬ 𝐵 ≤ 𝐴)) |
6 | simpl 108 | . . . . 5 ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℤ) → 𝐴 ∈ ℚ) | |
7 | 6 | flqcld 10220 | . . . 4 ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℤ) → (⌊‘𝐴) ∈ ℤ) |
8 | zq 9572 | . . . 4 ⊢ ((⌊‘𝐴) ∈ ℤ → (⌊‘𝐴) ∈ ℚ) | |
9 | 7, 8 | syl 14 | . . 3 ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℤ) → (⌊‘𝐴) ∈ ℚ) |
10 | 3 | adantl 275 | . . 3 ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℤ) → 𝐵 ∈ ℚ) |
11 | qltnle 10189 | . . 3 ⊢ (((⌊‘𝐴) ∈ ℚ ∧ 𝐵 ∈ ℚ) → ((⌊‘𝐴) < 𝐵 ↔ ¬ 𝐵 ≤ (⌊‘𝐴))) | |
12 | 9, 10, 11 | syl2anc 409 | . 2 ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℤ) → ((⌊‘𝐴) < 𝐵 ↔ ¬ 𝐵 ≤ (⌊‘𝐴))) |
13 | 2, 5, 12 | 3bitr4d 219 | 1 ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℤ) → (𝐴 < 𝐵 ↔ (⌊‘𝐴) < 𝐵)) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 103 ↔ wb 104 ∈ wcel 2141 class class class wbr 3987 ‘cfv 5196 < clt 7941 ≤ cle 7942 ℤcz 9199 ℚcq 9565 ⌊cfl 10211 |
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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-sep 4105 ax-pow 4158 ax-pr 4192 ax-un 4416 ax-setind 4519 ax-cnex 7852 ax-resscn 7853 ax-1cn 7854 ax-1re 7855 ax-icn 7856 ax-addcl 7857 ax-addrcl 7858 ax-mulcl 7859 ax-mulrcl 7860 ax-addcom 7861 ax-mulcom 7862 ax-addass 7863 ax-mulass 7864 ax-distr 7865 ax-i2m1 7866 ax-0lt1 7867 ax-1rid 7868 ax-0id 7869 ax-rnegex 7870 ax-precex 7871 ax-cnre 7872 ax-pre-ltirr 7873 ax-pre-ltwlin 7874 ax-pre-lttrn 7875 ax-pre-apti 7876 ax-pre-ltadd 7877 ax-pre-mulgt0 7878 ax-pre-mulext 7879 ax-arch 7880 |
This theorem depends on definitions: df-bi 116 df-3or 974 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-nel 2436 df-ral 2453 df-rex 2454 df-reu 2455 df-rmo 2456 df-rab 2457 df-v 2732 df-sbc 2956 df-csb 3050 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-pw 3566 df-sn 3587 df-pr 3588 df-op 3590 df-uni 3795 df-int 3830 df-iun 3873 df-br 3988 df-opab 4049 df-mpt 4050 df-id 4276 df-po 4279 df-iso 4280 df-xp 4615 df-rel 4616 df-cnv 4617 df-co 4618 df-dm 4619 df-rn 4620 df-res 4621 df-ima 4622 df-iota 5158 df-fun 5198 df-fn 5199 df-f 5200 df-fv 5204 df-riota 5806 df-ov 5853 df-oprab 5854 df-mpo 5855 df-1st 6116 df-2nd 6117 df-pnf 7943 df-mnf 7944 df-xr 7945 df-ltxr 7946 df-le 7947 df-sub 8079 df-neg 8080 df-reap 8481 df-ap 8488 df-div 8577 df-inn 8866 df-n0 9123 df-z 9200 df-q 9566 df-rp 9598 df-fl 10213 |
This theorem is referenced by: hashdvds 12162 |
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