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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 13807 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 11162 | . . . 4 ⊢ 0 ∈ ℝ | |
2 | 1xr 11219 | . . . 4 ⊢ 1 ∈ ℝ* | |
3 | icossre 13351 | . . . 4 ⊢ ((0 ∈ ℝ ∧ 1 ∈ ℝ*) → (0[,)1) ⊆ ℝ) | |
4 | 1, 2, 3 | mp2an 691 | . . 3 ⊢ (0[,)1) ⊆ ℝ |
5 | 4 | sseli 3941 | . 2 ⊢ (𝐴 ∈ (0[,)1) → 𝐴 ∈ ℝ) |
6 | 0xr 11207 | . . . 4 ⊢ 0 ∈ ℝ* | |
7 | elico1 13313 | . . . 4 ⊢ ((0 ∈ ℝ* ∧ 1 ∈ ℝ*) → (𝐴 ∈ (0[,)1) ↔ (𝐴 ∈ ℝ* ∧ 0 ≤ 𝐴 ∧ 𝐴 < 1))) | |
8 | 6, 2, 7 | mp2an 691 | . . 3 ⊢ (𝐴 ∈ (0[,)1) ↔ (𝐴 ∈ ℝ* ∧ 0 ≤ 𝐴 ∧ 𝐴 < 1)) |
9 | 8 | simp2bi 1147 | . 2 ⊢ (𝐴 ∈ (0[,)1) → 0 ≤ 𝐴) |
10 | 8 | simp3bi 1148 | . 2 ⊢ (𝐴 ∈ (0[,)1) → 𝐴 < 1) |
11 | recn 11146 | . . . . . 6 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℂ) | |
12 | 11 | addid2d 11361 | . . . . 5 ⊢ (𝐴 ∈ ℝ → (0 + 𝐴) = 𝐴) |
13 | 12 | fveqeq2d 6851 | . . . 4 ⊢ (𝐴 ∈ ℝ → ((⌊‘(0 + 𝐴)) = 0 ↔ (⌊‘𝐴) = 0)) |
14 | 0z 12515 | . . . . 5 ⊢ 0 ∈ ℤ | |
15 | flbi2 13728 | . . . . 5 ⊢ ((0 ∈ ℤ ∧ 𝐴 ∈ ℝ) → ((⌊‘(0 + 𝐴)) = 0 ↔ (0 ≤ 𝐴 ∧ 𝐴 < 1))) | |
16 | 14, 15 | mpan 689 | . . . 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 479 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ (0 ≤ 𝐴 ∧ 𝐴 < 1)) → (⌊‘𝐴) = 0) |
19 | 5, 9, 10, 18 | syl12anc 836 | 1 ⊢ (𝐴 ∈ (0[,)1) → (⌊‘𝐴) = 0) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 ∧ w3a 1088 = wceq 1542 ∈ wcel 2107 ⊆ wss 3911 class class class wbr 5106 ‘cfv 6497 (class class class)co 7358 ℝcr 11055 0cc0 11056 1c1 11057 + caddc 11059 ℝ*cxr 11193 < clt 11194 ≤ cle 11195 ℤcz 12504 [,)cico 13272 ⌊cfl 13701 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-cnex 11112 ax-resscn 11113 ax-1cn 11114 ax-icn 11115 ax-addcl 11116 ax-addrcl 11117 ax-mulcl 11118 ax-mulrcl 11119 ax-mulcom 11120 ax-addass 11121 ax-mulass 11122 ax-distr 11123 ax-i2m1 11124 ax-1ne0 11125 ax-1rid 11126 ax-rnegex 11127 ax-rrecex 11128 ax-cnre 11129 ax-pre-lttri 11130 ax-pre-lttrn 11131 ax-pre-ltadd 11132 ax-pre-mulgt0 11133 ax-pre-sup 11134 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6254 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-riota 7314 df-ov 7361 df-oprab 7362 df-mpo 7363 df-om 7804 df-2nd 7923 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-rdg 8357 df-er 8651 df-en 8887 df-dom 8888 df-sdom 8889 df-sup 9383 df-inf 9384 df-pnf 11196 df-mnf 11197 df-xr 11198 df-ltxr 11199 df-le 11200 df-sub 11392 df-neg 11393 df-nn 12159 df-n0 12419 df-z 12505 df-uz 12769 df-ico 13276 df-fl 13703 |
This theorem is referenced by: dnizeq0 34984 dignnld 46775 digexp 46779 |
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