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Mirrors > Home > MPE Home > Th. List > ico01fl0 | Structured version Visualization version GIF version |
Description: The floor of a real number in [0, 1) is 0. Remark: may shorten the proof of modid 13867 or a version of it where the antecedent is membership in an interval. (Contributed by BJ, 29-Jun-2019.) |
Ref | Expression |
---|---|
ico01fl0 | ⊢ (𝐴 ∈ (0[,)1) → (⌊‘𝐴) = 0) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0re 11220 | . . . 4 ⊢ 0 ∈ ℝ | |
2 | 1xr 11277 | . . . 4 ⊢ 1 ∈ ℝ* | |
3 | icossre 13411 | . . . 4 ⊢ ((0 ∈ ℝ ∧ 1 ∈ ℝ*) → (0[,)1) ⊆ ℝ) | |
4 | 1, 2, 3 | mp2an 689 | . . 3 ⊢ (0[,)1) ⊆ ℝ |
5 | 4 | sseli 3973 | . 2 ⊢ (𝐴 ∈ (0[,)1) → 𝐴 ∈ ℝ) |
6 | 0xr 11265 | . . . 4 ⊢ 0 ∈ ℝ* | |
7 | elico1 13373 | . . . 4 ⊢ ((0 ∈ ℝ* ∧ 1 ∈ ℝ*) → (𝐴 ∈ (0[,)1) ↔ (𝐴 ∈ ℝ* ∧ 0 ≤ 𝐴 ∧ 𝐴 < 1))) | |
8 | 6, 2, 7 | mp2an 689 | . . 3 ⊢ (𝐴 ∈ (0[,)1) ↔ (𝐴 ∈ ℝ* ∧ 0 ≤ 𝐴 ∧ 𝐴 < 1)) |
9 | 8 | simp2bi 1143 | . 2 ⊢ (𝐴 ∈ (0[,)1) → 0 ≤ 𝐴) |
10 | 8 | simp3bi 1144 | . 2 ⊢ (𝐴 ∈ (0[,)1) → 𝐴 < 1) |
11 | recn 11202 | . . . . . 6 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℂ) | |
12 | 11 | addlidd 11419 | . . . . 5 ⊢ (𝐴 ∈ ℝ → (0 + 𝐴) = 𝐴) |
13 | 12 | fveqeq2d 6893 | . . . 4 ⊢ (𝐴 ∈ ℝ → ((⌊‘(0 + 𝐴)) = 0 ↔ (⌊‘𝐴) = 0)) |
14 | 0z 12573 | . . . . 5 ⊢ 0 ∈ ℤ | |
15 | flbi2 13788 | . . . . 5 ⊢ ((0 ∈ ℤ ∧ 𝐴 ∈ ℝ) → ((⌊‘(0 + 𝐴)) = 0 ↔ (0 ≤ 𝐴 ∧ 𝐴 < 1))) | |
16 | 14, 15 | mpan 687 | . . . 4 ⊢ (𝐴 ∈ ℝ → ((⌊‘(0 + 𝐴)) = 0 ↔ (0 ≤ 𝐴 ∧ 𝐴 < 1))) |
17 | 13, 16 | bitr3d 281 | . . 3 ⊢ (𝐴 ∈ ℝ → ((⌊‘𝐴) = 0 ↔ (0 ≤ 𝐴 ∧ 𝐴 < 1))) |
18 | 17 | biimpar 477 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ (0 ≤ 𝐴 ∧ 𝐴 < 1)) → (⌊‘𝐴) = 0) |
19 | 5, 9, 10, 18 | syl12anc 834 | 1 ⊢ (𝐴 ∈ (0[,)1) → (⌊‘𝐴) = 0) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ⊆ wss 3943 class class class wbr 5141 ‘cfv 6537 (class class class)co 7405 ℝcr 11111 0cc0 11112 1c1 11113 + caddc 11115 ℝ*cxr 11251 < clt 11252 ≤ cle 11253 ℤcz 12562 [,)cico 13332 ⌊cfl 13761 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 ax-pre-sup 11190 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6294 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7853 df-2nd 7975 df-frecs 8267 df-wrecs 8298 df-recs 8372 df-rdg 8411 df-er 8705 df-en 8942 df-dom 8943 df-sdom 8944 df-sup 9439 df-inf 9440 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-nn 12217 df-n0 12477 df-z 12563 df-uz 12827 df-ico 13336 df-fl 13763 |
This theorem is referenced by: dnizeq0 35859 dignnld 47561 digexp 47565 |
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