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Mirrors > Home > MPE Home > Th. List > ex-fl | Structured version Visualization version GIF version |
Description: Example for df-fl 13012. 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 10490 | . . . 4 ⊢ 1 ∈ ℝ | |
2 | 3re 11567 | . . . . 5 ⊢ 3 ∈ ℝ | |
3 | 2 | rehalfcli 11736 | . . . 4 ⊢ (3 / 2) ∈ ℝ |
4 | 2cn 11562 | . . . . . . 7 ⊢ 2 ∈ ℂ | |
5 | 4 | mulid2i 10495 | . . . . . 6 ⊢ (1 · 2) = 2 |
6 | 2lt3 11659 | . . . . . 6 ⊢ 2 < 3 | |
7 | 5, 6 | eqbrtri 4985 | . . . . 5 ⊢ (1 · 2) < 3 |
8 | 2pos 11590 | . . . . . 6 ⊢ 0 < 2 | |
9 | 2re 11561 | . . . . . . 7 ⊢ 2 ∈ ℝ | |
10 | 1, 2, 9 | ltmuldivi 11410 | . . . . . 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 231 | . . . 4 ⊢ 1 < (3 / 2) |
13 | 1, 3, 12 | ltleii 10612 | . . 3 ⊢ 1 ≤ (3 / 2) |
14 | 3lt4 11661 | . . . . . 6 ⊢ 3 < 4 | |
15 | 2t2e4 11651 | . . . . . 6 ⊢ (2 · 2) = 4 | |
16 | 14, 15 | breqtrri 4991 | . . . . 5 ⊢ 3 < (2 · 2) |
17 | 9, 8 | pm3.2i 471 | . . . . . 6 ⊢ (2 ∈ ℝ ∧ 0 < 2) |
18 | ltdivmul 11365 | . . . . . 6 ⊢ ((3 ∈ ℝ ∧ 2 ∈ ℝ ∧ (2 ∈ ℝ ∧ 0 < 2)) → ((3 / 2) < 2 ↔ 3 < (2 · 2))) | |
19 | 2, 9, 17, 18 | mp3an 1453 | . . . . 5 ⊢ ((3 / 2) < 2 ↔ 3 < (2 · 2)) |
20 | 16, 19 | mpbir 232 | . . . 4 ⊢ (3 / 2) < 2 |
21 | df-2 11550 | . . . 4 ⊢ 2 = (1 + 1) | |
22 | 20, 21 | breqtri 4989 | . . 3 ⊢ (3 / 2) < (1 + 1) |
23 | 1z 11862 | . . . 4 ⊢ 1 ∈ ℤ | |
24 | flbi 13036 | . . . 4 ⊢ (((3 / 2) ∈ ℝ ∧ 1 ∈ ℤ) → ((⌊‘(3 / 2)) = 1 ↔ (1 ≤ (3 / 2) ∧ (3 / 2) < (1 + 1)))) | |
25 | 3, 23, 24 | mp2an 688 | . . 3 ⊢ ((⌊‘(3 / 2)) = 1 ↔ (1 ≤ (3 / 2) ∧ (3 / 2) < (1 + 1))) |
26 | 13, 22, 25 | mpbir2an 707 | . 2 ⊢ (⌊‘(3 / 2)) = 1 |
27 | 9 | renegcli 10797 | . . . 4 ⊢ -2 ∈ ℝ |
28 | 3 | renegcli 10797 | . . . 4 ⊢ -(3 / 2) ∈ ℝ |
29 | 3, 9 | ltnegi 11034 | . . . . 5 ⊢ ((3 / 2) < 2 ↔ -2 < -(3 / 2)) |
30 | 20, 29 | mpbi 231 | . . . 4 ⊢ -2 < -(3 / 2) |
31 | 27, 28, 30 | ltleii 10612 | . . 3 ⊢ -2 ≤ -(3 / 2) |
32 | 4 | negcli 10804 | . . . . . . 7 ⊢ -2 ∈ ℂ |
33 | ax-1cn 10444 | . . . . . . 7 ⊢ 1 ∈ ℂ | |
34 | negdi2 10794 | . . . . . . 7 ⊢ ((-2 ∈ ℂ ∧ 1 ∈ ℂ) → -(-2 + 1) = (--2 − 1)) | |
35 | 32, 33, 34 | mp2an 688 | . . . . . 6 ⊢ -(-2 + 1) = (--2 − 1) |
36 | 4 | negnegi 10806 | . . . . . . 7 ⊢ --2 = 2 |
37 | 36 | oveq1i 7029 | . . . . . 6 ⊢ (--2 − 1) = (2 − 1) |
38 | 35, 37 | eqtri 2818 | . . . . 5 ⊢ -(-2 + 1) = (2 − 1) |
39 | 2m1e1 11613 | . . . . . 6 ⊢ (2 − 1) = 1 | |
40 | 39, 12 | eqbrtri 4985 | . . . . 5 ⊢ (2 − 1) < (3 / 2) |
41 | 38, 40 | eqbrtri 4985 | . . . 4 ⊢ -(-2 + 1) < (3 / 2) |
42 | 27, 1 | readdcli 10505 | . . . . 5 ⊢ (-2 + 1) ∈ ℝ |
43 | 42, 3 | ltnegcon1i 11041 | . . . 4 ⊢ (-(-2 + 1) < (3 / 2) ↔ -(3 / 2) < (-2 + 1)) |
44 | 41, 43 | mpbi 231 | . . 3 ⊢ -(3 / 2) < (-2 + 1) |
45 | 2z 11864 | . . . . 5 ⊢ 2 ∈ ℤ | |
46 | znegcl 11867 | . . . . 5 ⊢ (2 ∈ ℤ → -2 ∈ ℤ) | |
47 | 45, 46 | ax-mp 5 | . . . 4 ⊢ -2 ∈ ℤ |
48 | flbi 13036 | . . . 4 ⊢ ((-(3 / 2) ∈ ℝ ∧ -2 ∈ ℤ) → ((⌊‘-(3 / 2)) = -2 ↔ (-2 ≤ -(3 / 2) ∧ -(3 / 2) < (-2 + 1)))) | |
49 | 28, 47, 48 | mp2an 688 | . . 3 ⊢ ((⌊‘-(3 / 2)) = -2 ↔ (-2 ≤ -(3 / 2) ∧ -(3 / 2) < (-2 + 1))) |
50 | 31, 44, 49 | mpbir2an 707 | . 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 207 ∧ wa 396 = wceq 1522 ∈ wcel 2080 class class class wbr 4964 ‘cfv 6228 (class class class)co 7019 ℂcc 10384 ℝcr 10385 0cc0 10386 1c1 10387 + caddc 10389 · cmul 10391 < clt 10524 ≤ cle 10525 − cmin 10719 -cneg 10720 / cdiv 11147 2c2 11542 3c3 11543 4c4 11544 ℤcz 11831 ⌊cfl 13010 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1778 ax-4 1792 ax-5 1889 ax-6 1948 ax-7 1993 ax-8 2082 ax-9 2090 ax-10 2111 ax-11 2125 ax-12 2140 ax-13 2343 ax-ext 2768 ax-sep 5097 ax-nul 5104 ax-pow 5160 ax-pr 5224 ax-un 7322 ax-cnex 10442 ax-resscn 10443 ax-1cn 10444 ax-icn 10445 ax-addcl 10446 ax-addrcl 10447 ax-mulcl 10448 ax-mulrcl 10449 ax-mulcom 10450 ax-addass 10451 ax-mulass 10452 ax-distr 10453 ax-i2m1 10454 ax-1ne0 10455 ax-1rid 10456 ax-rnegex 10457 ax-rrecex 10458 ax-cnre 10459 ax-pre-lttri 10460 ax-pre-lttrn 10461 ax-pre-ltadd 10462 ax-pre-mulgt0 10463 ax-pre-sup 10464 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1081 df-3an 1082 df-tru 1525 df-ex 1763 df-nf 1767 df-sb 2042 df-mo 2575 df-eu 2611 df-clab 2775 df-cleq 2787 df-clel 2862 df-nfc 2934 df-ne 2984 df-nel 3090 df-ral 3109 df-rex 3110 df-reu 3111 df-rmo 3112 df-rab 3113 df-v 3438 df-sbc 3708 df-csb 3814 df-dif 3864 df-un 3866 df-in 3868 df-ss 3876 df-pss 3878 df-nul 4214 df-if 4384 df-pw 4457 df-sn 4475 df-pr 4477 df-tp 4479 df-op 4481 df-uni 4748 df-iun 4829 df-br 4965 df-opab 5027 df-mpt 5044 df-tr 5067 df-id 5351 df-eprel 5356 df-po 5365 df-so 5366 df-fr 5405 df-we 5407 df-xp 5452 df-rel 5453 df-cnv 5454 df-co 5455 df-dm 5456 df-rn 5457 df-res 5458 df-ima 5459 df-pred 6026 df-ord 6072 df-on 6073 df-lim 6074 df-suc 6075 df-iota 6192 df-fun 6230 df-fn 6231 df-f 6232 df-f1 6233 df-fo 6234 df-f1o 6235 df-fv 6236 df-riota 6980 df-ov 7022 df-oprab 7023 df-mpo 7024 df-om 7440 df-wrecs 7801 df-recs 7863 df-rdg 7901 df-er 8142 df-en 8361 df-dom 8362 df-sdom 8363 df-sup 8755 df-inf 8756 df-pnf 10526 df-mnf 10527 df-xr 10528 df-ltxr 10529 df-le 10530 df-sub 10721 df-neg 10722 df-div 11148 df-nn 11489 df-2 11550 df-3 11551 df-4 11552 df-n0 11748 df-z 11832 df-uz 12094 df-fl 13012 |
This theorem is referenced by: ex-ceil 27911 |
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