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Mirrors > Home > MPE Home > Th. List > ex-fl | Structured version Visualization version GIF version |
Description: Example for df-fl 13367. Example by David A. Wheeler. (Contributed by Mario Carneiro, 18-Jun-2015.) |
Ref | Expression |
---|---|
ex-fl | ⊢ ((⌊‘(3 / 2)) = 1 ∧ (⌊‘-(3 / 2)) = -2) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1re 10833 | . . . 4 ⊢ 1 ∈ ℝ | |
2 | 3re 11910 | . . . . 5 ⊢ 3 ∈ ℝ | |
3 | 2 | rehalfcli 12079 | . . . 4 ⊢ (3 / 2) ∈ ℝ |
4 | 2cn 11905 | . . . . . . 7 ⊢ 2 ∈ ℂ | |
5 | 4 | mulid2i 10838 | . . . . . 6 ⊢ (1 · 2) = 2 |
6 | 2lt3 12002 | . . . . . 6 ⊢ 2 < 3 | |
7 | 5, 6 | eqbrtri 5074 | . . . . 5 ⊢ (1 · 2) < 3 |
8 | 2pos 11933 | . . . . . 6 ⊢ 0 < 2 | |
9 | 2re 11904 | . . . . . . 7 ⊢ 2 ∈ ℝ | |
10 | 1, 2, 9 | ltmuldivi 11752 | . . . . . 6 ⊢ (0 < 2 → ((1 · 2) < 3 ↔ 1 < (3 / 2))) |
11 | 8, 10 | ax-mp 5 | . . . . 5 ⊢ ((1 · 2) < 3 ↔ 1 < (3 / 2)) |
12 | 7, 11 | mpbi 233 | . . . 4 ⊢ 1 < (3 / 2) |
13 | 1, 3, 12 | ltleii 10955 | . . 3 ⊢ 1 ≤ (3 / 2) |
14 | 3lt4 12004 | . . . . . 6 ⊢ 3 < 4 | |
15 | 2t2e4 11994 | . . . . . 6 ⊢ (2 · 2) = 4 | |
16 | 14, 15 | breqtrri 5080 | . . . . 5 ⊢ 3 < (2 · 2) |
17 | 9, 8 | pm3.2i 474 | . . . . . 6 ⊢ (2 ∈ ℝ ∧ 0 < 2) |
18 | ltdivmul 11707 | . . . . . 6 ⊢ ((3 ∈ ℝ ∧ 2 ∈ ℝ ∧ (2 ∈ ℝ ∧ 0 < 2)) → ((3 / 2) < 2 ↔ 3 < (2 · 2))) | |
19 | 2, 9, 17, 18 | mp3an 1463 | . . . . 5 ⊢ ((3 / 2) < 2 ↔ 3 < (2 · 2)) |
20 | 16, 19 | mpbir 234 | . . . 4 ⊢ (3 / 2) < 2 |
21 | df-2 11893 | . . . 4 ⊢ 2 = (1 + 1) | |
22 | 20, 21 | breqtri 5078 | . . 3 ⊢ (3 / 2) < (1 + 1) |
23 | 1z 12207 | . . . 4 ⊢ 1 ∈ ℤ | |
24 | flbi 13391 | . . . 4 ⊢ (((3 / 2) ∈ ℝ ∧ 1 ∈ ℤ) → ((⌊‘(3 / 2)) = 1 ↔ (1 ≤ (3 / 2) ∧ (3 / 2) < (1 + 1)))) | |
25 | 3, 23, 24 | mp2an 692 | . . 3 ⊢ ((⌊‘(3 / 2)) = 1 ↔ (1 ≤ (3 / 2) ∧ (3 / 2) < (1 + 1))) |
26 | 13, 22, 25 | mpbir2an 711 | . 2 ⊢ (⌊‘(3 / 2)) = 1 |
27 | 9 | renegcli 11139 | . . . 4 ⊢ -2 ∈ ℝ |
28 | 3 | renegcli 11139 | . . . 4 ⊢ -(3 / 2) ∈ ℝ |
29 | 3, 9 | ltnegi 11376 | . . . . 5 ⊢ ((3 / 2) < 2 ↔ -2 < -(3 / 2)) |
30 | 20, 29 | mpbi 233 | . . . 4 ⊢ -2 < -(3 / 2) |
31 | 27, 28, 30 | ltleii 10955 | . . 3 ⊢ -2 ≤ -(3 / 2) |
32 | 4 | negcli 11146 | . . . . . . 7 ⊢ -2 ∈ ℂ |
33 | ax-1cn 10787 | . . . . . . 7 ⊢ 1 ∈ ℂ | |
34 | negdi2 11136 | . . . . . . 7 ⊢ ((-2 ∈ ℂ ∧ 1 ∈ ℂ) → -(-2 + 1) = (--2 − 1)) | |
35 | 32, 33, 34 | mp2an 692 | . . . . . 6 ⊢ -(-2 + 1) = (--2 − 1) |
36 | 4 | negnegi 11148 | . . . . . . 7 ⊢ --2 = 2 |
37 | 36 | oveq1i 7223 | . . . . . 6 ⊢ (--2 − 1) = (2 − 1) |
38 | 35, 37 | eqtri 2765 | . . . . 5 ⊢ -(-2 + 1) = (2 − 1) |
39 | 2m1e1 11956 | . . . . . 6 ⊢ (2 − 1) = 1 | |
40 | 39, 12 | eqbrtri 5074 | . . . . 5 ⊢ (2 − 1) < (3 / 2) |
41 | 38, 40 | eqbrtri 5074 | . . . 4 ⊢ -(-2 + 1) < (3 / 2) |
42 | 27, 1 | readdcli 10848 | . . . . 5 ⊢ (-2 + 1) ∈ ℝ |
43 | 42, 3 | ltnegcon1i 11383 | . . . 4 ⊢ (-(-2 + 1) < (3 / 2) ↔ -(3 / 2) < (-2 + 1)) |
44 | 41, 43 | mpbi 233 | . . 3 ⊢ -(3 / 2) < (-2 + 1) |
45 | 2z 12209 | . . . . 5 ⊢ 2 ∈ ℤ | |
46 | znegcl 12212 | . . . . 5 ⊢ (2 ∈ ℤ → -2 ∈ ℤ) | |
47 | 45, 46 | ax-mp 5 | . . . 4 ⊢ -2 ∈ ℤ |
48 | flbi 13391 | . . . 4 ⊢ ((-(3 / 2) ∈ ℝ ∧ -2 ∈ ℤ) → ((⌊‘-(3 / 2)) = -2 ↔ (-2 ≤ -(3 / 2) ∧ -(3 / 2) < (-2 + 1)))) | |
49 | 28, 47, 48 | mp2an 692 | . . 3 ⊢ ((⌊‘-(3 / 2)) = -2 ↔ (-2 ≤ -(3 / 2) ∧ -(3 / 2) < (-2 + 1))) |
50 | 31, 44, 49 | mpbir2an 711 | . 2 ⊢ (⌊‘-(3 / 2)) = -2 |
51 | 26, 50 | pm3.2i 474 | 1 ⊢ ((⌊‘(3 / 2)) = 1 ∧ (⌊‘-(3 / 2)) = -2) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 ∧ wa 399 = wceq 1543 ∈ wcel 2110 class class class wbr 5053 ‘cfv 6380 (class class class)co 7213 ℂcc 10727 ℝcr 10728 0cc0 10729 1c1 10730 + caddc 10732 · cmul 10734 < clt 10867 ≤ cle 10868 − cmin 11062 -cneg 11063 / cdiv 11489 2c2 11885 3c3 11886 4c4 11887 ℤcz 12176 ⌊cfl 13365 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-sep 5192 ax-nul 5199 ax-pow 5258 ax-pr 5322 ax-un 7523 ax-cnex 10785 ax-resscn 10786 ax-1cn 10787 ax-icn 10788 ax-addcl 10789 ax-addrcl 10790 ax-mulcl 10791 ax-mulrcl 10792 ax-mulcom 10793 ax-addass 10794 ax-mulass 10795 ax-distr 10796 ax-i2m1 10797 ax-1ne0 10798 ax-1rid 10799 ax-rnegex 10800 ax-rrecex 10801 ax-cnre 10802 ax-pre-lttri 10803 ax-pre-lttrn 10804 ax-pre-ltadd 10805 ax-pre-mulgt0 10806 ax-pre-sup 10807 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3410 df-sbc 3695 df-csb 3812 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-pss 3885 df-nul 4238 df-if 4440 df-pw 4515 df-sn 4542 df-pr 4544 df-tp 4546 df-op 4548 df-uni 4820 df-iun 4906 df-br 5054 df-opab 5116 df-mpt 5136 df-tr 5162 df-id 5455 df-eprel 5460 df-po 5468 df-so 5469 df-fr 5509 df-we 5511 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 df-pred 6160 df-ord 6216 df-on 6217 df-lim 6218 df-suc 6219 df-iota 6338 df-fun 6382 df-fn 6383 df-f 6384 df-f1 6385 df-fo 6386 df-f1o 6387 df-fv 6388 df-riota 7170 df-ov 7216 df-oprab 7217 df-mpo 7218 df-om 7645 df-wrecs 8047 df-recs 8108 df-rdg 8146 df-er 8391 df-en 8627 df-dom 8628 df-sdom 8629 df-sup 9058 df-inf 9059 df-pnf 10869 df-mnf 10870 df-xr 10871 df-ltxr 10872 df-le 10873 df-sub 11064 df-neg 11065 df-div 11490 df-nn 11831 df-2 11893 df-3 11894 df-4 11895 df-n0 12091 df-z 12177 df-uz 12439 df-fl 13367 |
This theorem is referenced by: ex-ceil 28531 |
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