Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > ex-fl | Structured version Visualization version GIF version |
Description: Example for df-fl 12455. 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 9918 | . . . 4 ⊢ 1 ∈ ℝ | |
2 | 3re 10971 | . . . . 5 ⊢ 3 ∈ ℝ | |
3 | 2 | rehalfcli 11158 | . . . 4 ⊢ (3 / 2) ∈ ℝ |
4 | 2cn 10968 | . . . . . . 7 ⊢ 2 ∈ ℂ | |
5 | 4 | mulid2i 9922 | . . . . . 6 ⊢ (1 · 2) = 2 |
6 | 2lt3 11072 | . . . . . 6 ⊢ 2 < 3 | |
7 | 5, 6 | eqbrtri 4604 | . . . . 5 ⊢ (1 · 2) < 3 |
8 | 2pos 10989 | . . . . . 6 ⊢ 0 < 2 | |
9 | 2re 10967 | . . . . . . 7 ⊢ 2 ∈ ℝ | |
10 | 1, 2, 9 | ltmuldivi 10823 | . . . . . 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 219 | . . . 4 ⊢ 1 < (3 / 2) |
13 | 1, 3, 12 | ltleii 10039 | . . 3 ⊢ 1 ≤ (3 / 2) |
14 | 3lt4 11074 | . . . . . 6 ⊢ 3 < 4 | |
15 | 2t2e4 11054 | . . . . . 6 ⊢ (2 · 2) = 4 | |
16 | 14, 15 | breqtrri 4610 | . . . . 5 ⊢ 3 < (2 · 2) |
17 | 9, 8 | pm3.2i 470 | . . . . . 6 ⊢ (2 ∈ ℝ ∧ 0 < 2) |
18 | ltdivmul 10777 | . . . . . 6 ⊢ ((3 ∈ ℝ ∧ 2 ∈ ℝ ∧ (2 ∈ ℝ ∧ 0 < 2)) → ((3 / 2) < 2 ↔ 3 < (2 · 2))) | |
19 | 2, 9, 17, 18 | mp3an 1416 | . . . . 5 ⊢ ((3 / 2) < 2 ↔ 3 < (2 · 2)) |
20 | 16, 19 | mpbir 220 | . . . 4 ⊢ (3 / 2) < 2 |
21 | df-2 10956 | . . . 4 ⊢ 2 = (1 + 1) | |
22 | 20, 21 | breqtri 4608 | . . 3 ⊢ (3 / 2) < (1 + 1) |
23 | 1z 11284 | . . . 4 ⊢ 1 ∈ ℤ | |
24 | flbi 12479 | . . . 4 ⊢ (((3 / 2) ∈ ℝ ∧ 1 ∈ ℤ) → ((⌊‘(3 / 2)) = 1 ↔ (1 ≤ (3 / 2) ∧ (3 / 2) < (1 + 1)))) | |
25 | 3, 23, 24 | mp2an 704 | . . 3 ⊢ ((⌊‘(3 / 2)) = 1 ↔ (1 ≤ (3 / 2) ∧ (3 / 2) < (1 + 1))) |
26 | 13, 22, 25 | mpbir2an 957 | . 2 ⊢ (⌊‘(3 / 2)) = 1 |
27 | 9 | renegcli 10221 | . . . 4 ⊢ -2 ∈ ℝ |
28 | 3 | renegcli 10221 | . . . 4 ⊢ -(3 / 2) ∈ ℝ |
29 | 3, 9 | ltnegi 10451 | . . . . 5 ⊢ ((3 / 2) < 2 ↔ -2 < -(3 / 2)) |
30 | 20, 29 | mpbi 219 | . . . 4 ⊢ -2 < -(3 / 2) |
31 | 27, 28, 30 | ltleii 10039 | . . 3 ⊢ -2 ≤ -(3 / 2) |
32 | 4 | negcli 10228 | . . . . . . 7 ⊢ -2 ∈ ℂ |
33 | ax-1cn 9873 | . . . . . . 7 ⊢ 1 ∈ ℂ | |
34 | negdi2 10218 | . . . . . . 7 ⊢ ((-2 ∈ ℂ ∧ 1 ∈ ℂ) → -(-2 + 1) = (--2 − 1)) | |
35 | 32, 33, 34 | mp2an 704 | . . . . . 6 ⊢ -(-2 + 1) = (--2 − 1) |
36 | 4 | negnegi 10230 | . . . . . . 7 ⊢ --2 = 2 |
37 | 36 | oveq1i 6559 | . . . . . 6 ⊢ (--2 − 1) = (2 − 1) |
38 | 35, 37 | eqtri 2632 | . . . . 5 ⊢ -(-2 + 1) = (2 − 1) |
39 | 2m1e1 11012 | . . . . . 6 ⊢ (2 − 1) = 1 | |
40 | 39, 12 | eqbrtri 4604 | . . . . 5 ⊢ (2 − 1) < (3 / 2) |
41 | 38, 40 | eqbrtri 4604 | . . . 4 ⊢ -(-2 + 1) < (3 / 2) |
42 | 27, 1 | readdcli 9932 | . . . . 5 ⊢ (-2 + 1) ∈ ℝ |
43 | 42, 3 | ltnegcon1i 10458 | . . . 4 ⊢ (-(-2 + 1) < (3 / 2) ↔ -(3 / 2) < (-2 + 1)) |
44 | 41, 43 | mpbi 219 | . . 3 ⊢ -(3 / 2) < (-2 + 1) |
45 | 2z 11286 | . . . . 5 ⊢ 2 ∈ ℤ | |
46 | znegcl 11289 | . . . . 5 ⊢ (2 ∈ ℤ → -2 ∈ ℤ) | |
47 | 45, 46 | ax-mp 5 | . . . 4 ⊢ -2 ∈ ℤ |
48 | flbi 12479 | . . . 4 ⊢ ((-(3 / 2) ∈ ℝ ∧ -2 ∈ ℤ) → ((⌊‘-(3 / 2)) = -2 ↔ (-2 ≤ -(3 / 2) ∧ -(3 / 2) < (-2 + 1)))) | |
49 | 28, 47, 48 | mp2an 704 | . . 3 ⊢ ((⌊‘-(3 / 2)) = -2 ↔ (-2 ≤ -(3 / 2) ∧ -(3 / 2) < (-2 + 1))) |
50 | 31, 44, 49 | mpbir2an 957 | . 2 ⊢ (⌊‘-(3 / 2)) = -2 |
51 | 26, 50 | pm3.2i 470 | 1 ⊢ ((⌊‘(3 / 2)) = 1 ∧ (⌊‘-(3 / 2)) = -2) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 class class class wbr 4583 ‘cfv 5804 (class class class)co 6549 ℂcc 9813 ℝcr 9814 0cc0 9815 1c1 9816 + caddc 9818 · cmul 9820 < clt 9953 ≤ cle 9954 − cmin 10145 -cneg 10146 / cdiv 10563 2c2 10947 3c3 10948 4c4 10949 ℤcz 11254 ⌊cfl 12453 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-sup 8231 df-inf 8232 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-3 10957 df-4 10958 df-n0 11170 df-z 11255 df-uz 11564 df-fl 12455 |
This theorem is referenced by: ex-ceil 26697 |
Copyright terms: Public domain | W3C validator |