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Mirrors > Home > MPE Home > Th. List > flltnz | Structured version Visualization version GIF version |
Description: If A is not an integer, then the floor of A is less than A. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
Ref | Expression |
---|---|
flltnz | ⊢ ((𝐴 ∈ ℝ ∧ ¬ 𝐴 ∈ ℤ) → (⌊‘𝐴) < 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | reflcl 12980 | . . 3 ⊢ (𝐴 ∈ ℝ → (⌊‘𝐴) ∈ ℝ) | |
2 | 1 | adantr 473 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ ¬ 𝐴 ∈ ℤ) → (⌊‘𝐴) ∈ ℝ) |
3 | simpl 475 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ ¬ 𝐴 ∈ ℤ) → 𝐴 ∈ ℝ) | |
4 | fllelt 12981 | . . . 4 ⊢ (𝐴 ∈ ℝ → ((⌊‘𝐴) ≤ 𝐴 ∧ 𝐴 < ((⌊‘𝐴) + 1))) | |
5 | 4 | adantr 473 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ ¬ 𝐴 ∈ ℤ) → ((⌊‘𝐴) ≤ 𝐴 ∧ 𝐴 < ((⌊‘𝐴) + 1))) |
6 | 5 | simpld 487 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ ¬ 𝐴 ∈ ℤ) → (⌊‘𝐴) ≤ 𝐴) |
7 | simpr 477 | . . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ ¬ 𝐴 ∈ ℤ) → ¬ 𝐴 ∈ ℤ) | |
8 | flidz 12994 | . . . . . 6 ⊢ (𝐴 ∈ ℝ → ((⌊‘𝐴) = 𝐴 ↔ 𝐴 ∈ ℤ)) | |
9 | 8 | adantr 473 | . . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ ¬ 𝐴 ∈ ℤ) → ((⌊‘𝐴) = 𝐴 ↔ 𝐴 ∈ ℤ)) |
10 | 7, 9 | mtbird 317 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ ¬ 𝐴 ∈ ℤ) → ¬ (⌊‘𝐴) = 𝐴) |
11 | 10 | neqned 2969 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ ¬ 𝐴 ∈ ℤ) → (⌊‘𝐴) ≠ 𝐴) |
12 | 11 | necomd 3017 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ ¬ 𝐴 ∈ ℤ) → 𝐴 ≠ (⌊‘𝐴)) |
13 | 2, 3, 6, 12 | leneltd 10593 | 1 ⊢ ((𝐴 ∈ ℝ ∧ ¬ 𝐴 ∈ ℤ) → (⌊‘𝐴) < 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 198 ∧ wa 387 = wceq 1508 ∈ wcel 2051 class class class wbr 4926 ‘cfv 6186 (class class class)co 6975 ℝcr 10333 1c1 10335 + caddc 10337 < clt 10473 ≤ cle 10474 ℤcz 11792 ⌊cfl 12974 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1759 ax-4 1773 ax-5 1870 ax-6 1929 ax-7 1966 ax-8 2053 ax-9 2060 ax-10 2080 ax-11 2094 ax-12 2107 ax-13 2302 ax-ext 2745 ax-sep 5057 ax-nul 5064 ax-pow 5116 ax-pr 5183 ax-un 7278 ax-cnex 10390 ax-resscn 10391 ax-1cn 10392 ax-icn 10393 ax-addcl 10394 ax-addrcl 10395 ax-mulcl 10396 ax-mulrcl 10397 ax-mulcom 10398 ax-addass 10399 ax-mulass 10400 ax-distr 10401 ax-i2m1 10402 ax-1ne0 10403 ax-1rid 10404 ax-rnegex 10405 ax-rrecex 10406 ax-cnre 10407 ax-pre-lttri 10408 ax-pre-lttrn 10409 ax-pre-ltadd 10410 ax-pre-mulgt0 10411 ax-pre-sup 10412 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 835 df-3or 1070 df-3an 1071 df-tru 1511 df-ex 1744 df-nf 1748 df-sb 2017 df-mo 2548 df-eu 2585 df-clab 2754 df-cleq 2766 df-clel 2841 df-nfc 2913 df-ne 2963 df-nel 3069 df-ral 3088 df-rex 3089 df-reu 3090 df-rmo 3091 df-rab 3092 df-v 3412 df-sbc 3677 df-csb 3782 df-dif 3827 df-un 3829 df-in 3831 df-ss 3838 df-pss 3840 df-nul 4174 df-if 4346 df-pw 4419 df-sn 4437 df-pr 4439 df-tp 4441 df-op 4443 df-uni 4710 df-iun 4791 df-br 4927 df-opab 4989 df-mpt 5006 df-tr 5028 df-id 5309 df-eprel 5314 df-po 5323 df-so 5324 df-fr 5363 df-we 5365 df-xp 5410 df-rel 5411 df-cnv 5412 df-co 5413 df-dm 5414 df-rn 5415 df-res 5416 df-ima 5417 df-pred 5984 df-ord 6030 df-on 6031 df-lim 6032 df-suc 6033 df-iota 6150 df-fun 6188 df-fn 6189 df-f 6190 df-f1 6191 df-fo 6192 df-f1o 6193 df-fv 6194 df-riota 6936 df-ov 6978 df-oprab 6979 df-mpo 6980 df-om 7396 df-wrecs 7749 df-recs 7811 df-rdg 7849 df-er 8088 df-en 8306 df-dom 8307 df-sdom 8308 df-sup 8700 df-inf 8701 df-pnf 10475 df-mnf 10476 df-xr 10477 df-ltxr 10478 df-le 10479 df-sub 10671 df-neg 10672 df-nn 11439 df-n0 11707 df-z 11793 df-uz 12058 df-fl 12976 |
This theorem is referenced by: fldivndvdslt 15624 dirkercncflem4 41852 |
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