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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 13903 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 11256 | . . . 4 ⊢ 0 ∈ ℝ | |
2 | 1xr 11313 | . . . 4 ⊢ 1 ∈ ℝ* | |
3 | icossre 13447 | . . . 4 ⊢ ((0 ∈ ℝ ∧ 1 ∈ ℝ*) → (0[,)1) ⊆ ℝ) | |
4 | 1, 2, 3 | mp2an 690 | . . 3 ⊢ (0[,)1) ⊆ ℝ |
5 | 4 | sseli 3978 | . 2 ⊢ (𝐴 ∈ (0[,)1) → 𝐴 ∈ ℝ) |
6 | 0xr 11301 | . . . 4 ⊢ 0 ∈ ℝ* | |
7 | elico1 13409 | . . . 4 ⊢ ((0 ∈ ℝ* ∧ 1 ∈ ℝ*) → (𝐴 ∈ (0[,)1) ↔ (𝐴 ∈ ℝ* ∧ 0 ≤ 𝐴 ∧ 𝐴 < 1))) | |
8 | 6, 2, 7 | mp2an 690 | . . 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 11238 | . . . . . 6 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℂ) | |
12 | 11 | addlidd 11455 | . . . . 5 ⊢ (𝐴 ∈ ℝ → (0 + 𝐴) = 𝐴) |
13 | 12 | fveqeq2d 6910 | . . . 4 ⊢ (𝐴 ∈ ℝ → ((⌊‘(0 + 𝐴)) = 0 ↔ (⌊‘𝐴) = 0)) |
14 | 0z 12609 | . . . . 5 ⊢ 0 ∈ ℤ | |
15 | flbi2 13824 | . . . . 5 ⊢ ((0 ∈ ℤ ∧ 𝐴 ∈ ℝ) → ((⌊‘(0 + 𝐴)) = 0 ↔ (0 ≤ 𝐴 ∧ 𝐴 < 1))) | |
16 | 14, 15 | mpan 688 | . . . 4 ⊢ (𝐴 ∈ ℝ → ((⌊‘(0 + 𝐴)) = 0 ↔ (0 ≤ 𝐴 ∧ 𝐴 < 1))) |
17 | 13, 16 | bitr3d 280 | . . 3 ⊢ (𝐴 ∈ ℝ → ((⌊‘𝐴) = 0 ↔ (0 ≤ 𝐴 ∧ 𝐴 < 1))) |
18 | 17 | biimpar 476 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ (0 ≤ 𝐴 ∧ 𝐴 < 1)) → (⌊‘𝐴) = 0) |
19 | 5, 9, 10, 18 | syl12anc 835 | 1 ⊢ (𝐴 ∈ (0[,)1) → (⌊‘𝐴) = 0) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ⊆ wss 3949 class class class wbr 5152 ‘cfv 6553 (class class class)co 7426 ℝcr 11147 0cc0 11148 1c1 11149 + caddc 11151 ℝ*cxr 11287 < clt 11288 ≤ cle 11289 ℤcz 12598 [,)cico 13368 ⌊cfl 13797 |
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 2166 ax-ext 2699 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7748 ax-cnex 11204 ax-resscn 11205 ax-1cn 11206 ax-icn 11207 ax-addcl 11208 ax-addrcl 11209 ax-mulcl 11210 ax-mulrcl 11211 ax-mulcom 11212 ax-addass 11213 ax-mulass 11214 ax-distr 11215 ax-i2m1 11216 ax-1ne0 11217 ax-1rid 11218 ax-rnegex 11219 ax-rrecex 11220 ax-cnre 11221 ax-pre-lttri 11222 ax-pre-lttrn 11223 ax-pre-ltadd 11224 ax-pre-mulgt0 11225 ax-pre-sup 11226 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-om 7879 df-2nd 8002 df-frecs 8295 df-wrecs 8326 df-recs 8400 df-rdg 8439 df-er 8733 df-en 8973 df-dom 8974 df-sdom 8975 df-sup 9475 df-inf 9476 df-pnf 11290 df-mnf 11291 df-xr 11292 df-ltxr 11293 df-le 11294 df-sub 11486 df-neg 11487 df-nn 12253 df-n0 12513 df-z 12599 df-uz 12863 df-ico 13372 df-fl 13799 |
This theorem is referenced by: dnizeq0 35991 dignnld 47772 digexp 47776 |
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