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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 13102 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 10478 | . . . 4 ⊢ 0 ∈ ℝ | |
2 | 1xr 10536 | . . . 4 ⊢ 1 ∈ ℝ* | |
3 | icossre 12656 | . . . 4 ⊢ ((0 ∈ ℝ ∧ 1 ∈ ℝ*) → (0[,)1) ⊆ ℝ) | |
4 | 1, 2, 3 | mp2an 688 | . . 3 ⊢ (0[,)1) ⊆ ℝ |
5 | 4 | sseli 3880 | . 2 ⊢ (𝐴 ∈ (0[,)1) → 𝐴 ∈ ℝ) |
6 | 0xr 10523 | . . . 4 ⊢ 0 ∈ ℝ* | |
7 | elico1 12620 | . . . 4 ⊢ ((0 ∈ ℝ* ∧ 1 ∈ ℝ*) → (𝐴 ∈ (0[,)1) ↔ (𝐴 ∈ ℝ* ∧ 0 ≤ 𝐴 ∧ 𝐴 < 1))) | |
8 | 6, 2, 7 | mp2an 688 | . . 3 ⊢ (𝐴 ∈ (0[,)1) ↔ (𝐴 ∈ ℝ* ∧ 0 ≤ 𝐴 ∧ 𝐴 < 1)) |
9 | 8 | simp2bi 1137 | . 2 ⊢ (𝐴 ∈ (0[,)1) → 0 ≤ 𝐴) |
10 | 8 | simp3bi 1138 | . 2 ⊢ (𝐴 ∈ (0[,)1) → 𝐴 < 1) |
11 | recn 10462 | . . . . . 6 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℂ) | |
12 | 11 | addid2d 10677 | . . . . 5 ⊢ (𝐴 ∈ ℝ → (0 + 𝐴) = 𝐴) |
13 | 12 | fveqeq2d 6538 | . . . 4 ⊢ (𝐴 ∈ ℝ → ((⌊‘(0 + 𝐴)) = 0 ↔ (⌊‘𝐴) = 0)) |
14 | 0z 11829 | . . . . 5 ⊢ 0 ∈ ℤ | |
15 | flbi2 13025 | . . . . 5 ⊢ ((0 ∈ ℤ ∧ 𝐴 ∈ ℝ) → ((⌊‘(0 + 𝐴)) = 0 ↔ (0 ≤ 𝐴 ∧ 𝐴 < 1))) | |
16 | 14, 15 | mpan 686 | . . . 4 ⊢ (𝐴 ∈ ℝ → ((⌊‘(0 + 𝐴)) = 0 ↔ (0 ≤ 𝐴 ∧ 𝐴 < 1))) |
17 | 13, 16 | bitr3d 282 | . . 3 ⊢ (𝐴 ∈ ℝ → ((⌊‘𝐴) = 0 ↔ (0 ≤ 𝐴 ∧ 𝐴 < 1))) |
18 | 17 | biimpar 478 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ (0 ≤ 𝐴 ∧ 𝐴 < 1)) → (⌊‘𝐴) = 0) |
19 | 5, 9, 10, 18 | syl12anc 833 | 1 ⊢ (𝐴 ∈ (0[,)1) → (⌊‘𝐴) = 0) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 ∧ w3a 1078 = wceq 1520 ∈ wcel 2079 ⊆ wss 3854 class class class wbr 4956 ‘cfv 6217 (class class class)co 7007 ℝcr 10371 0cc0 10372 1c1 10373 + caddc 10375 ℝ*cxr 10509 < clt 10510 ≤ cle 10511 ℤcz 11818 [,)cico 12579 ⌊cfl 12998 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1775 ax-4 1789 ax-5 1886 ax-6 1945 ax-7 1990 ax-8 2081 ax-9 2089 ax-10 2110 ax-11 2124 ax-12 2139 ax-13 2342 ax-ext 2767 ax-sep 5088 ax-nul 5095 ax-pow 5150 ax-pr 5214 ax-un 7310 ax-cnex 10428 ax-resscn 10429 ax-1cn 10430 ax-icn 10431 ax-addcl 10432 ax-addrcl 10433 ax-mulcl 10434 ax-mulrcl 10435 ax-mulcom 10436 ax-addass 10437 ax-mulass 10438 ax-distr 10439 ax-i2m1 10440 ax-1ne0 10441 ax-1rid 10442 ax-rnegex 10443 ax-rrecex 10444 ax-cnre 10445 ax-pre-lttri 10446 ax-pre-lttrn 10447 ax-pre-ltadd 10448 ax-pre-mulgt0 10449 ax-pre-sup 10450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1079 df-3an 1080 df-tru 1523 df-ex 1760 df-nf 1764 df-sb 2041 df-mo 2574 df-eu 2610 df-clab 2774 df-cleq 2786 df-clel 2861 df-nfc 2933 df-ne 2983 df-nel 3089 df-ral 3108 df-rex 3109 df-reu 3110 df-rmo 3111 df-rab 3112 df-v 3434 df-sbc 3702 df-csb 3807 df-dif 3857 df-un 3859 df-in 3861 df-ss 3869 df-pss 3871 df-nul 4207 df-if 4376 df-pw 4449 df-sn 4467 df-pr 4469 df-tp 4471 df-op 4473 df-uni 4740 df-iun 4821 df-br 4957 df-opab 5019 df-mpt 5036 df-tr 5058 df-id 5340 df-eprel 5345 df-po 5354 df-so 5355 df-fr 5394 df-we 5396 df-xp 5441 df-rel 5442 df-cnv 5443 df-co 5444 df-dm 5445 df-rn 5446 df-res 5447 df-ima 5448 df-pred 6015 df-ord 6061 df-on 6062 df-lim 6063 df-suc 6064 df-iota 6181 df-fun 6219 df-fn 6220 df-f 6221 df-f1 6222 df-fo 6223 df-f1o 6224 df-fv 6225 df-riota 6968 df-ov 7010 df-oprab 7011 df-mpo 7012 df-om 7428 df-wrecs 7789 df-recs 7851 df-rdg 7889 df-er 8130 df-en 8348 df-dom 8349 df-sdom 8350 df-sup 8742 df-inf 8743 df-pnf 10512 df-mnf 10513 df-xr 10514 df-ltxr 10515 df-le 10516 df-sub 10708 df-neg 10709 df-nn 11476 df-n0 11735 df-z 11819 df-uz 12083 df-ico 12583 df-fl 13000 |
This theorem is referenced by: dnizeq0 33367 dignnld 44098 digexp 44102 |
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