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Mirrors > Home > ILE Home > Th. List > ex-fl | GIF version |
Description: Example for df-fl 9114. 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 7026 | . . . 4 ⊢ 1 ∈ ℝ | |
2 | 3re 7989 | . . . . 5 ⊢ 3 ∈ ℝ | |
3 | 2 | rehalfcli 8173 | . . . 4 ⊢ (3 / 2) ∈ ℝ |
4 | 2cn 7986 | . . . . . . 7 ⊢ 2 ∈ ℂ | |
5 | 4 | mulid2i 7030 | . . . . . 6 ⊢ (1 · 2) = 2 |
6 | 2lt3 8087 | . . . . . 6 ⊢ 2 < 3 | |
7 | 5, 6 | eqbrtri 3783 | . . . . 5 ⊢ (1 · 2) < 3 |
8 | 2pos 8007 | . . . . . 6 ⊢ 0 < 2 | |
9 | 2re 7985 | . . . . . . 7 ⊢ 2 ∈ ℝ | |
10 | 1, 2, 9 | ltmuldivi 7888 | . . . . . 6 ⊢ (0 < 2 → ((1 · 2) < 3 ↔ 1 < (3 / 2))) |
11 | 8, 10 | ax-mp 7 | . . . . 5 ⊢ ((1 · 2) < 3 ↔ 1 < (3 / 2)) |
12 | 7, 11 | mpbi 133 | . . . 4 ⊢ 1 < (3 / 2) |
13 | 1, 3, 12 | ltleii 7120 | . . 3 ⊢ 1 ≤ (3 / 2) |
14 | 3lt4 8089 | . . . . . 6 ⊢ 3 < 4 | |
15 | 2t2e4 8069 | . . . . . 6 ⊢ (2 · 2) = 4 | |
16 | 14, 15 | breqtrri 3789 | . . . . 5 ⊢ 3 < (2 · 2) |
17 | 9, 8 | pm3.2i 257 | . . . . . 6 ⊢ (2 ∈ ℝ ∧ 0 < 2) |
18 | ltdivmul 7842 | . . . . . 6 ⊢ ((3 ∈ ℝ ∧ 2 ∈ ℝ ∧ (2 ∈ ℝ ∧ 0 < 2)) → ((3 / 2) < 2 ↔ 3 < (2 · 2))) | |
19 | 2, 9, 17, 18 | mp3an 1232 | . . . . 5 ⊢ ((3 / 2) < 2 ↔ 3 < (2 · 2)) |
20 | 16, 19 | mpbir 134 | . . . 4 ⊢ (3 / 2) < 2 |
21 | df-2 7973 | . . . 4 ⊢ 2 = (1 + 1) | |
22 | 20, 21 | breqtri 3787 | . . 3 ⊢ (3 / 2) < (1 + 1) |
23 | 3z 8274 | . . . . 5 ⊢ 3 ∈ ℤ | |
24 | 2nn 8077 | . . . . 5 ⊢ 2 ∈ ℕ | |
25 | znq 8559 | . . . . 5 ⊢ ((3 ∈ ℤ ∧ 2 ∈ ℕ) → (3 / 2) ∈ ℚ) | |
26 | 23, 24, 25 | mp2an 402 | . . . 4 ⊢ (3 / 2) ∈ ℚ |
27 | 1z 8271 | . . . 4 ⊢ 1 ∈ ℤ | |
28 | flqbi 9132 | . . . 4 ⊢ (((3 / 2) ∈ ℚ ∧ 1 ∈ ℤ) → ((⌊‘(3 / 2)) = 1 ↔ (1 ≤ (3 / 2) ∧ (3 / 2) < (1 + 1)))) | |
29 | 26, 27, 28 | mp2an 402 | . . 3 ⊢ ((⌊‘(3 / 2)) = 1 ↔ (1 ≤ (3 / 2) ∧ (3 / 2) < (1 + 1))) |
30 | 13, 22, 29 | mpbir2an 849 | . 2 ⊢ (⌊‘(3 / 2)) = 1 |
31 | 9 | renegcli 7273 | . . . 4 ⊢ -2 ∈ ℝ |
32 | 3 | renegcli 7273 | . . . 4 ⊢ -(3 / 2) ∈ ℝ |
33 | 3, 9 | ltnegi 7485 | . . . . 5 ⊢ ((3 / 2) < 2 ↔ -2 < -(3 / 2)) |
34 | 20, 33 | mpbi 133 | . . . 4 ⊢ -2 < -(3 / 2) |
35 | 31, 32, 34 | ltleii 7120 | . . 3 ⊢ -2 ≤ -(3 / 2) |
36 | 4 | negcli 7279 | . . . . . . 7 ⊢ -2 ∈ ℂ |
37 | ax-1cn 6977 | . . . . . . 7 ⊢ 1 ∈ ℂ | |
38 | negdi2 7269 | . . . . . . 7 ⊢ ((-2 ∈ ℂ ∧ 1 ∈ ℂ) → -(-2 + 1) = (--2 − 1)) | |
39 | 36, 37, 38 | mp2an 402 | . . . . . 6 ⊢ -(-2 + 1) = (--2 − 1) |
40 | 4 | negnegi 7281 | . . . . . . 7 ⊢ --2 = 2 |
41 | 40 | oveq1i 5522 | . . . . . 6 ⊢ (--2 − 1) = (2 − 1) |
42 | 39, 41 | eqtri 2060 | . . . . 5 ⊢ -(-2 + 1) = (2 − 1) |
43 | 2m1e1 8034 | . . . . . 6 ⊢ (2 − 1) = 1 | |
44 | 43, 12 | eqbrtri 3783 | . . . . 5 ⊢ (2 − 1) < (3 / 2) |
45 | 42, 44 | eqbrtri 3783 | . . . 4 ⊢ -(-2 + 1) < (3 / 2) |
46 | 31, 1 | readdcli 7040 | . . . . 5 ⊢ (-2 + 1) ∈ ℝ |
47 | 46, 3 | ltnegcon1i 7491 | . . . 4 ⊢ (-(-2 + 1) < (3 / 2) ↔ -(3 / 2) < (-2 + 1)) |
48 | 45, 47 | mpbi 133 | . . 3 ⊢ -(3 / 2) < (-2 + 1) |
49 | qnegcl 8571 | . . . . 5 ⊢ ((3 / 2) ∈ ℚ → -(3 / 2) ∈ ℚ) | |
50 | 26, 49 | ax-mp 7 | . . . 4 ⊢ -(3 / 2) ∈ ℚ |
51 | 2z 8273 | . . . . 5 ⊢ 2 ∈ ℤ | |
52 | znegcl 8276 | . . . . 5 ⊢ (2 ∈ ℤ → -2 ∈ ℤ) | |
53 | 51, 52 | ax-mp 7 | . . . 4 ⊢ -2 ∈ ℤ |
54 | flqbi 9132 | . . . 4 ⊢ ((-(3 / 2) ∈ ℚ ∧ -2 ∈ ℤ) → ((⌊‘-(3 / 2)) = -2 ↔ (-2 ≤ -(3 / 2) ∧ -(3 / 2) < (-2 + 1)))) | |
55 | 50, 53, 54 | mp2an 402 | . . 3 ⊢ ((⌊‘-(3 / 2)) = -2 ↔ (-2 ≤ -(3 / 2) ∧ -(3 / 2) < (-2 + 1))) |
56 | 35, 48, 55 | mpbir2an 849 | . 2 ⊢ (⌊‘-(3 / 2)) = -2 |
57 | 30, 56 | pm3.2i 257 | 1 ⊢ ((⌊‘(3 / 2)) = 1 ∧ (⌊‘-(3 / 2)) = -2) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 97 ↔ wb 98 = wceq 1243 ∈ wcel 1393 class class class wbr 3764 ‘cfv 4902 (class class class)co 5512 ℂcc 6887 ℝcr 6888 0cc0 6889 1c1 6890 + caddc 6892 · cmul 6894 < clt 7060 ≤ cle 7061 − cmin 7182 -cneg 7183 / cdiv 7651 ℕcn 7914 2c2 7964 3c3 7965 4c4 7966 ℤcz 8245 ℚcq 8554 ⌊cfl 9112 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-13 1404 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-coll 3872 ax-sep 3875 ax-nul 3883 ax-pow 3927 ax-pr 3944 ax-un 4170 ax-setind 4262 ax-iinf 4311 ax-cnex 6975 ax-resscn 6976 ax-1cn 6977 ax-1re 6978 ax-icn 6979 ax-addcl 6980 ax-addrcl 6981 ax-mulcl 6982 ax-mulrcl 6983 ax-addcom 6984 ax-mulcom 6985 ax-addass 6986 ax-mulass 6987 ax-distr 6988 ax-i2m1 6989 ax-1rid 6991 ax-0id 6992 ax-rnegex 6993 ax-precex 6994 ax-cnre 6995 ax-pre-ltirr 6996 ax-pre-ltwlin 6997 ax-pre-lttrn 6998 ax-pre-apti 6999 ax-pre-ltadd 7000 ax-pre-mulgt0 7001 ax-pre-mulext 7002 ax-arch 7003 |
This theorem depends on definitions: df-bi 110 df-dc 743 df-3or 886 df-3an 887 df-tru 1246 df-fal 1249 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ne 2206 df-nel 2207 df-ral 2311 df-rex 2312 df-reu 2313 df-rmo 2314 df-rab 2315 df-v 2559 df-sbc 2765 df-csb 2853 df-dif 2920 df-un 2922 df-in 2924 df-ss 2931 df-nul 3225 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-int 3616 df-iun 3659 df-br 3765 df-opab 3819 df-mpt 3820 df-tr 3855 df-eprel 4026 df-id 4030 df-po 4033 df-iso 4034 df-iord 4103 df-on 4105 df-suc 4108 df-iom 4314 df-xp 4351 df-rel 4352 df-cnv 4353 df-co 4354 df-dm 4355 df-rn 4356 df-res 4357 df-ima 4358 df-iota 4867 df-fun 4904 df-fn 4905 df-f 4906 df-f1 4907 df-fo 4908 df-f1o 4909 df-fv 4910 df-riota 5468 df-ov 5515 df-oprab 5516 df-mpt2 5517 df-1st 5767 df-2nd 5768 df-recs 5920 df-irdg 5957 df-1o 6001 df-2o 6002 df-oadd 6005 df-omul 6006 df-er 6106 df-ec 6108 df-qs 6112 df-ni 6402 df-pli 6403 df-mi 6404 df-lti 6405 df-plpq 6442 df-mpq 6443 df-enq 6445 df-nqqs 6446 df-plqqs 6447 df-mqqs 6448 df-1nqqs 6449 df-rq 6450 df-ltnqqs 6451 df-enq0 6522 df-nq0 6523 df-0nq0 6524 df-plq0 6525 df-mq0 6526 df-inp 6564 df-i1p 6565 df-iplp 6566 df-iltp 6568 df-enr 6811 df-nr 6812 df-ltr 6815 df-0r 6816 df-1r 6817 df-0 6896 df-1 6897 df-r 6899 df-lt 6902 df-pnf 7062 df-mnf 7063 df-xr 7064 df-ltxr 7065 df-le 7066 df-sub 7184 df-neg 7185 df-reap 7566 df-ap 7573 df-div 7652 df-inn 7915 df-2 7973 df-3 7974 df-4 7975 df-n0 8182 df-z 8246 df-q 8555 df-rp 8584 df-fl 9114 |
This theorem is referenced by: ex-ceil 9896 |
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