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Mirrors > Home > MPE Home > Th. List > ex-fl | Structured version Visualization version GIF version |
Description: Example for df-fl 13704. 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 11162 | . . . 4 ⊢ 1 ∈ ℝ | |
2 | 3re 12240 | . . . . 5 ⊢ 3 ∈ ℝ | |
3 | 2 | rehalfcli 12409 | . . . 4 ⊢ (3 / 2) ∈ ℝ |
4 | 2cn 12235 | . . . . . . 7 ⊢ 2 ∈ ℂ | |
5 | 4 | mulid2i 11167 | . . . . . 6 ⊢ (1 · 2) = 2 |
6 | 2lt3 12332 | . . . . . 6 ⊢ 2 < 3 | |
7 | 5, 6 | eqbrtri 5131 | . . . . 5 ⊢ (1 · 2) < 3 |
8 | 2pos 12263 | . . . . . 6 ⊢ 0 < 2 | |
9 | 2re 12234 | . . . . . . 7 ⊢ 2 ∈ ℝ | |
10 | 1, 2, 9 | ltmuldivi 12082 | . . . . . 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 229 | . . . 4 ⊢ 1 < (3 / 2) |
13 | 1, 3, 12 | ltleii 11285 | . . 3 ⊢ 1 ≤ (3 / 2) |
14 | 3lt4 12334 | . . . . . 6 ⊢ 3 < 4 | |
15 | 2t2e4 12324 | . . . . . 6 ⊢ (2 · 2) = 4 | |
16 | 14, 15 | breqtrri 5137 | . . . . 5 ⊢ 3 < (2 · 2) |
17 | 9, 8 | pm3.2i 472 | . . . . . 6 ⊢ (2 ∈ ℝ ∧ 0 < 2) |
18 | ltdivmul 12037 | . . . . . 6 ⊢ ((3 ∈ ℝ ∧ 2 ∈ ℝ ∧ (2 ∈ ℝ ∧ 0 < 2)) → ((3 / 2) < 2 ↔ 3 < (2 · 2))) | |
19 | 2, 9, 17, 18 | mp3an 1462 | . . . . 5 ⊢ ((3 / 2) < 2 ↔ 3 < (2 · 2)) |
20 | 16, 19 | mpbir 230 | . . . 4 ⊢ (3 / 2) < 2 |
21 | df-2 12223 | . . . 4 ⊢ 2 = (1 + 1) | |
22 | 20, 21 | breqtri 5135 | . . 3 ⊢ (3 / 2) < (1 + 1) |
23 | 1z 12540 | . . . 4 ⊢ 1 ∈ ℤ | |
24 | flbi 13728 | . . . 4 ⊢ (((3 / 2) ∈ ℝ ∧ 1 ∈ ℤ) → ((⌊‘(3 / 2)) = 1 ↔ (1 ≤ (3 / 2) ∧ (3 / 2) < (1 + 1)))) | |
25 | 3, 23, 24 | mp2an 691 | . . 3 ⊢ ((⌊‘(3 / 2)) = 1 ↔ (1 ≤ (3 / 2) ∧ (3 / 2) < (1 + 1))) |
26 | 13, 22, 25 | mpbir2an 710 | . 2 ⊢ (⌊‘(3 / 2)) = 1 |
27 | 9 | renegcli 11469 | . . . 4 ⊢ -2 ∈ ℝ |
28 | 3 | renegcli 11469 | . . . 4 ⊢ -(3 / 2) ∈ ℝ |
29 | 3, 9 | ltnegi 11706 | . . . . 5 ⊢ ((3 / 2) < 2 ↔ -2 < -(3 / 2)) |
30 | 20, 29 | mpbi 229 | . . . 4 ⊢ -2 < -(3 / 2) |
31 | 27, 28, 30 | ltleii 11285 | . . 3 ⊢ -2 ≤ -(3 / 2) |
32 | 4 | negcli 11476 | . . . . . . 7 ⊢ -2 ∈ ℂ |
33 | ax-1cn 11116 | . . . . . . 7 ⊢ 1 ∈ ℂ | |
34 | negdi2 11466 | . . . . . . 7 ⊢ ((-2 ∈ ℂ ∧ 1 ∈ ℂ) → -(-2 + 1) = (--2 − 1)) | |
35 | 32, 33, 34 | mp2an 691 | . . . . . 6 ⊢ -(-2 + 1) = (--2 − 1) |
36 | 4 | negnegi 11478 | . . . . . . 7 ⊢ --2 = 2 |
37 | 36 | oveq1i 7372 | . . . . . 6 ⊢ (--2 − 1) = (2 − 1) |
38 | 35, 37 | eqtri 2765 | . . . . 5 ⊢ -(-2 + 1) = (2 − 1) |
39 | 2m1e1 12286 | . . . . . 6 ⊢ (2 − 1) = 1 | |
40 | 39, 12 | eqbrtri 5131 | . . . . 5 ⊢ (2 − 1) < (3 / 2) |
41 | 38, 40 | eqbrtri 5131 | . . . 4 ⊢ -(-2 + 1) < (3 / 2) |
42 | 27, 1 | readdcli 11177 | . . . . 5 ⊢ (-2 + 1) ∈ ℝ |
43 | 42, 3 | ltnegcon1i 11713 | . . . 4 ⊢ (-(-2 + 1) < (3 / 2) ↔ -(3 / 2) < (-2 + 1)) |
44 | 41, 43 | mpbi 229 | . . 3 ⊢ -(3 / 2) < (-2 + 1) |
45 | 2z 12542 | . . . . 5 ⊢ 2 ∈ ℤ | |
46 | znegcl 12545 | . . . . 5 ⊢ (2 ∈ ℤ → -2 ∈ ℤ) | |
47 | 45, 46 | ax-mp 5 | . . . 4 ⊢ -2 ∈ ℤ |
48 | flbi 13728 | . . . 4 ⊢ ((-(3 / 2) ∈ ℝ ∧ -2 ∈ ℤ) → ((⌊‘-(3 / 2)) = -2 ↔ (-2 ≤ -(3 / 2) ∧ -(3 / 2) < (-2 + 1)))) | |
49 | 28, 47, 48 | mp2an 691 | . . 3 ⊢ ((⌊‘-(3 / 2)) = -2 ↔ (-2 ≤ -(3 / 2) ∧ -(3 / 2) < (-2 + 1))) |
50 | 31, 44, 49 | mpbir2an 710 | . 2 ⊢ (⌊‘-(3 / 2)) = -2 |
51 | 26, 50 | pm3.2i 472 | 1 ⊢ ((⌊‘(3 / 2)) = 1 ∧ (⌊‘-(3 / 2)) = -2) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 397 = wceq 1542 ∈ wcel 2107 class class class wbr 5110 ‘cfv 6501 (class class class)co 7362 ℂcc 11056 ℝcr 11057 0cc0 11058 1c1 11059 + caddc 11061 · cmul 11063 < clt 11196 ≤ cle 11197 − cmin 11392 -cneg 11393 / cdiv 11819 2c2 12215 3c3 12216 4c4 12217 ℤcz 12506 ⌊cfl 13702 |
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 2708 ax-sep 5261 ax-nul 5268 ax-pow 5325 ax-pr 5389 ax-un 7677 ax-cnex 11114 ax-resscn 11115 ax-1cn 11116 ax-icn 11117 ax-addcl 11118 ax-addrcl 11119 ax-mulcl 11120 ax-mulrcl 11121 ax-mulcom 11122 ax-addass 11123 ax-mulass 11124 ax-distr 11125 ax-i2m1 11126 ax-1ne0 11127 ax-1rid 11128 ax-rnegex 11129 ax-rrecex 11130 ax-cnre 11131 ax-pre-lttri 11132 ax-pre-lttrn 11133 ax-pre-ltadd 11134 ax-pre-mulgt0 11135 ax-pre-sup 11136 |
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 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3066 df-rex 3075 df-rmo 3356 df-reu 3357 df-rab 3411 df-v 3450 df-sbc 3745 df-csb 3861 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-pss 3934 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-iun 4961 df-br 5111 df-opab 5173 df-mpt 5194 df-tr 5228 df-id 5536 df-eprel 5542 df-po 5550 df-so 5551 df-fr 5593 df-we 5595 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6258 df-ord 6325 df-on 6326 df-lim 6327 df-suc 6328 df-iota 6453 df-fun 6503 df-fn 6504 df-f 6505 df-f1 6506 df-fo 6507 df-f1o 6508 df-fv 6509 df-riota 7318 df-ov 7365 df-oprab 7366 df-mpo 7367 df-om 7808 df-2nd 7927 df-frecs 8217 df-wrecs 8248 df-recs 8322 df-rdg 8361 df-er 8655 df-en 8891 df-dom 8892 df-sdom 8893 df-sup 9385 df-inf 9386 df-pnf 11198 df-mnf 11199 df-xr 11200 df-ltxr 11201 df-le 11202 df-sub 11394 df-neg 11395 df-div 11820 df-nn 12161 df-2 12223 df-3 12224 df-4 12225 df-n0 12421 df-z 12507 df-uz 12771 df-fl 13704 |
This theorem is referenced by: ex-ceil 29434 |
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