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Mirrors > Home > ILE Home > Th. List > flid | GIF version |
Description: An integer is its own floor. (Contributed by NM, 15-Nov-2004.) |
Ref | Expression |
---|---|
flid | ⊢ (𝐴 ∈ ℤ → (⌊‘𝐴) = 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | zq 9655 | . . 3 ⊢ (𝐴 ∈ ℤ → 𝐴 ∈ ℚ) | |
2 | flqle 10308 | . . 3 ⊢ (𝐴 ∈ ℚ → (⌊‘𝐴) ≤ 𝐴) | |
3 | 1, 2 | syl 14 | . 2 ⊢ (𝐴 ∈ ℤ → (⌊‘𝐴) ≤ 𝐴) |
4 | zre 9286 | . . . 4 ⊢ (𝐴 ∈ ℤ → 𝐴 ∈ ℝ) | |
5 | 4 | leidd 8500 | . . 3 ⊢ (𝐴 ∈ ℤ → 𝐴 ≤ 𝐴) |
6 | flqge 10312 | . . . 4 ⊢ ((𝐴 ∈ ℚ ∧ 𝐴 ∈ ℤ) → (𝐴 ≤ 𝐴 ↔ 𝐴 ≤ (⌊‘𝐴))) | |
7 | 1, 6 | mpancom 422 | . . 3 ⊢ (𝐴 ∈ ℤ → (𝐴 ≤ 𝐴 ↔ 𝐴 ≤ (⌊‘𝐴))) |
8 | 5, 7 | mpbid 147 | . 2 ⊢ (𝐴 ∈ ℤ → 𝐴 ≤ (⌊‘𝐴)) |
9 | 1 | flqcld 10307 | . . . 4 ⊢ (𝐴 ∈ ℤ → (⌊‘𝐴) ∈ ℤ) |
10 | 9 | zred 9404 | . . 3 ⊢ (𝐴 ∈ ℤ → (⌊‘𝐴) ∈ ℝ) |
11 | 10, 4 | letri3d 8102 | . 2 ⊢ (𝐴 ∈ ℤ → ((⌊‘𝐴) = 𝐴 ↔ ((⌊‘𝐴) ≤ 𝐴 ∧ 𝐴 ≤ (⌊‘𝐴)))) |
12 | 3, 8, 11 | mpbir2and 946 | 1 ⊢ (𝐴 ∈ ℤ → (⌊‘𝐴) = 𝐴) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 105 = wceq 1364 ∈ wcel 2160 class class class wbr 4018 ‘cfv 5235 ≤ cle 8022 ℤcz 9282 ℚcq 9648 ⌊cfl 10298 |
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 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-sep 4136 ax-pow 4192 ax-pr 4227 ax-un 4451 ax-setind 4554 ax-cnex 7931 ax-resscn 7932 ax-1cn 7933 ax-1re 7934 ax-icn 7935 ax-addcl 7936 ax-addrcl 7937 ax-mulcl 7938 ax-mulrcl 7939 ax-addcom 7940 ax-mulcom 7941 ax-addass 7942 ax-mulass 7943 ax-distr 7944 ax-i2m1 7945 ax-0lt1 7946 ax-1rid 7947 ax-0id 7948 ax-rnegex 7949 ax-precex 7950 ax-cnre 7951 ax-pre-ltirr 7952 ax-pre-ltwlin 7953 ax-pre-lttrn 7954 ax-pre-apti 7955 ax-pre-ltadd 7956 ax-pre-mulgt0 7957 ax-pre-mulext 7958 ax-arch 7959 |
This theorem depends on definitions: df-bi 117 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-nel 2456 df-ral 2473 df-rex 2474 df-reu 2475 df-rmo 2476 df-rab 2477 df-v 2754 df-sbc 2978 df-csb 3073 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-int 3860 df-iun 3903 df-br 4019 df-opab 4080 df-mpt 4081 df-id 4311 df-po 4314 df-iso 4315 df-xp 4650 df-rel 4651 df-cnv 4652 df-co 4653 df-dm 4654 df-rn 4655 df-res 4656 df-ima 4657 df-iota 5196 df-fun 5237 df-fn 5238 df-f 5239 df-fv 5243 df-riota 5851 df-ov 5898 df-oprab 5899 df-mpo 5900 df-1st 6164 df-2nd 6165 df-pnf 8023 df-mnf 8024 df-xr 8025 df-ltxr 8026 df-le 8027 df-sub 8159 df-neg 8160 df-reap 8561 df-ap 8568 df-div 8659 df-inn 8949 df-n0 9206 df-z 9283 df-q 9649 df-rp 9683 df-fl 10300 |
This theorem is referenced by: flqidm 10315 flqidz 10316 ceilid 10345 flqeqceilz 10348 zmod10 10370 phiprmpw 12253 fldivp1 12379 |
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