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Mirrors > Home > MPE Home > Th. List > ex-fl | Structured version Visualization version GIF version |
Description: Example for df-fl 13522. 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 10985 | . . . 4 ⊢ 1 ∈ ℝ | |
2 | 3re 12063 | . . . . 5 ⊢ 3 ∈ ℝ | |
3 | 2 | rehalfcli 12232 | . . . 4 ⊢ (3 / 2) ∈ ℝ |
4 | 2cn 12058 | . . . . . . 7 ⊢ 2 ∈ ℂ | |
5 | 4 | mulid2i 10990 | . . . . . 6 ⊢ (1 · 2) = 2 |
6 | 2lt3 12155 | . . . . . 6 ⊢ 2 < 3 | |
7 | 5, 6 | eqbrtri 5094 | . . . . 5 ⊢ (1 · 2) < 3 |
8 | 2pos 12086 | . . . . . 6 ⊢ 0 < 2 | |
9 | 2re 12057 | . . . . . . 7 ⊢ 2 ∈ ℝ | |
10 | 1, 2, 9 | ltmuldivi 11905 | . . . . . 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 11108 | . . 3 ⊢ 1 ≤ (3 / 2) |
14 | 3lt4 12157 | . . . . . 6 ⊢ 3 < 4 | |
15 | 2t2e4 12147 | . . . . . 6 ⊢ (2 · 2) = 4 | |
16 | 14, 15 | breqtrri 5100 | . . . . 5 ⊢ 3 < (2 · 2) |
17 | 9, 8 | pm3.2i 471 | . . . . . 6 ⊢ (2 ∈ ℝ ∧ 0 < 2) |
18 | ltdivmul 11860 | . . . . . 6 ⊢ ((3 ∈ ℝ ∧ 2 ∈ ℝ ∧ (2 ∈ ℝ ∧ 0 < 2)) → ((3 / 2) < 2 ↔ 3 < (2 · 2))) | |
19 | 2, 9, 17, 18 | mp3an 1460 | . . . . 5 ⊢ ((3 / 2) < 2 ↔ 3 < (2 · 2)) |
20 | 16, 19 | mpbir 230 | . . . 4 ⊢ (3 / 2) < 2 |
21 | df-2 12046 | . . . 4 ⊢ 2 = (1 + 1) | |
22 | 20, 21 | breqtri 5098 | . . 3 ⊢ (3 / 2) < (1 + 1) |
23 | 1z 12360 | . . . 4 ⊢ 1 ∈ ℤ | |
24 | flbi 13546 | . . . 4 ⊢ (((3 / 2) ∈ ℝ ∧ 1 ∈ ℤ) → ((⌊‘(3 / 2)) = 1 ↔ (1 ≤ (3 / 2) ∧ (3 / 2) < (1 + 1)))) | |
25 | 3, 23, 24 | mp2an 689 | . . 3 ⊢ ((⌊‘(3 / 2)) = 1 ↔ (1 ≤ (3 / 2) ∧ (3 / 2) < (1 + 1))) |
26 | 13, 22, 25 | mpbir2an 708 | . 2 ⊢ (⌊‘(3 / 2)) = 1 |
27 | 9 | renegcli 11292 | . . . 4 ⊢ -2 ∈ ℝ |
28 | 3 | renegcli 11292 | . . . 4 ⊢ -(3 / 2) ∈ ℝ |
29 | 3, 9 | ltnegi 11529 | . . . . 5 ⊢ ((3 / 2) < 2 ↔ -2 < -(3 / 2)) |
30 | 20, 29 | mpbi 229 | . . . 4 ⊢ -2 < -(3 / 2) |
31 | 27, 28, 30 | ltleii 11108 | . . 3 ⊢ -2 ≤ -(3 / 2) |
32 | 4 | negcli 11299 | . . . . . . 7 ⊢ -2 ∈ ℂ |
33 | ax-1cn 10939 | . . . . . . 7 ⊢ 1 ∈ ℂ | |
34 | negdi2 11289 | . . . . . . 7 ⊢ ((-2 ∈ ℂ ∧ 1 ∈ ℂ) → -(-2 + 1) = (--2 − 1)) | |
35 | 32, 33, 34 | mp2an 689 | . . . . . 6 ⊢ -(-2 + 1) = (--2 − 1) |
36 | 4 | negnegi 11301 | . . . . . . 7 ⊢ --2 = 2 |
37 | 36 | oveq1i 7277 | . . . . . 6 ⊢ (--2 − 1) = (2 − 1) |
38 | 35, 37 | eqtri 2766 | . . . . 5 ⊢ -(-2 + 1) = (2 − 1) |
39 | 2m1e1 12109 | . . . . . 6 ⊢ (2 − 1) = 1 | |
40 | 39, 12 | eqbrtri 5094 | . . . . 5 ⊢ (2 − 1) < (3 / 2) |
41 | 38, 40 | eqbrtri 5094 | . . . 4 ⊢ -(-2 + 1) < (3 / 2) |
42 | 27, 1 | readdcli 11000 | . . . . 5 ⊢ (-2 + 1) ∈ ℝ |
43 | 42, 3 | ltnegcon1i 11536 | . . . 4 ⊢ (-(-2 + 1) < (3 / 2) ↔ -(3 / 2) < (-2 + 1)) |
44 | 41, 43 | mpbi 229 | . . 3 ⊢ -(3 / 2) < (-2 + 1) |
45 | 2z 12362 | . . . . 5 ⊢ 2 ∈ ℤ | |
46 | znegcl 12365 | . . . . 5 ⊢ (2 ∈ ℤ → -2 ∈ ℤ) | |
47 | 45, 46 | ax-mp 5 | . . . 4 ⊢ -2 ∈ ℤ |
48 | flbi 13546 | . . . 4 ⊢ ((-(3 / 2) ∈ ℝ ∧ -2 ∈ ℤ) → ((⌊‘-(3 / 2)) = -2 ↔ (-2 ≤ -(3 / 2) ∧ -(3 / 2) < (-2 + 1)))) | |
49 | 28, 47, 48 | mp2an 689 | . . 3 ⊢ ((⌊‘-(3 / 2)) = -2 ↔ (-2 ≤ -(3 / 2) ∧ -(3 / 2) < (-2 + 1))) |
50 | 31, 44, 49 | mpbir2an 708 | . 2 ⊢ (⌊‘-(3 / 2)) = -2 |
51 | 26, 50 | pm3.2i 471 | 1 ⊢ ((⌊‘(3 / 2)) = 1 ∧ (⌊‘-(3 / 2)) = -2) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 396 = wceq 1539 ∈ wcel 2106 class class class wbr 5073 ‘cfv 6426 (class class class)co 7267 ℂcc 10879 ℝcr 10880 0cc0 10881 1c1 10882 + caddc 10884 · cmul 10886 < clt 11019 ≤ cle 11020 − cmin 11215 -cneg 11216 / cdiv 11642 2c2 12038 3c3 12039 4c4 12040 ℤcz 12329 ⌊cfl 13520 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-sep 5221 ax-nul 5228 ax-pow 5286 ax-pr 5350 ax-un 7578 ax-cnex 10937 ax-resscn 10938 ax-1cn 10939 ax-icn 10940 ax-addcl 10941 ax-addrcl 10942 ax-mulcl 10943 ax-mulrcl 10944 ax-mulcom 10945 ax-addass 10946 ax-mulass 10947 ax-distr 10948 ax-i2m1 10949 ax-1ne0 10950 ax-1rid 10951 ax-rnegex 10952 ax-rrecex 10953 ax-cnre 10954 ax-pre-lttri 10955 ax-pre-lttrn 10956 ax-pre-ltadd 10957 ax-pre-mulgt0 10958 ax-pre-sup 10959 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-reu 3071 df-rmo 3072 df-rab 3073 df-v 3431 df-sbc 3716 df-csb 3832 df-dif 3889 df-un 3891 df-in 3893 df-ss 3903 df-pss 3905 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-iun 4926 df-br 5074 df-opab 5136 df-mpt 5157 df-tr 5191 df-id 5484 df-eprel 5490 df-po 5498 df-so 5499 df-fr 5539 df-we 5541 df-xp 5590 df-rel 5591 df-cnv 5592 df-co 5593 df-dm 5594 df-rn 5595 df-res 5596 df-ima 5597 df-pred 6195 df-ord 6262 df-on 6263 df-lim 6264 df-suc 6265 df-iota 6384 df-fun 6428 df-fn 6429 df-f 6430 df-f1 6431 df-fo 6432 df-f1o 6433 df-fv 6434 df-riota 7224 df-ov 7270 df-oprab 7271 df-mpo 7272 df-om 7703 df-2nd 7821 df-frecs 8084 df-wrecs 8115 df-recs 8189 df-rdg 8228 df-er 8485 df-en 8721 df-dom 8722 df-sdom 8723 df-sup 9188 df-inf 9189 df-pnf 11021 df-mnf 11022 df-xr 11023 df-ltxr 11024 df-le 11025 df-sub 11217 df-neg 11218 df-div 11643 df-nn 11984 df-2 12046 df-3 12047 df-4 12048 df-n0 12244 df-z 12330 df-uz 12593 df-fl 13522 |
This theorem is referenced by: ex-ceil 28820 |
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