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Mirrors > Home > MPE Home > Th. List > ex-fl | Structured version Visualization version GIF version |
Description: Example for df-fl 13829. 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 11259 | . . . 4 ⊢ 1 ∈ ℝ | |
2 | 3re 12344 | . . . . 5 ⊢ 3 ∈ ℝ | |
3 | 2 | rehalfcli 12513 | . . . 4 ⊢ (3 / 2) ∈ ℝ |
4 | 2cn 12339 | . . . . . . 7 ⊢ 2 ∈ ℂ | |
5 | 4 | mullidi 11264 | . . . . . 6 ⊢ (1 · 2) = 2 |
6 | 2lt3 12436 | . . . . . 6 ⊢ 2 < 3 | |
7 | 5, 6 | eqbrtri 5169 | . . . . 5 ⊢ (1 · 2) < 3 |
8 | 2pos 12367 | . . . . . 6 ⊢ 0 < 2 | |
9 | 2re 12338 | . . . . . . 7 ⊢ 2 ∈ ℝ | |
10 | 1, 2, 9 | ltmuldivi 12186 | . . . . . 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 230 | . . . 4 ⊢ 1 < (3 / 2) |
13 | 1, 3, 12 | ltleii 11382 | . . 3 ⊢ 1 ≤ (3 / 2) |
14 | 3lt4 12438 | . . . . . 6 ⊢ 3 < 4 | |
15 | 2t2e4 12428 | . . . . . 6 ⊢ (2 · 2) = 4 | |
16 | 14, 15 | breqtrri 5175 | . . . . 5 ⊢ 3 < (2 · 2) |
17 | 9, 8 | pm3.2i 470 | . . . . . 6 ⊢ (2 ∈ ℝ ∧ 0 < 2) |
18 | ltdivmul 12141 | . . . . . 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 231 | . . . 4 ⊢ (3 / 2) < 2 |
21 | df-2 12327 | . . . 4 ⊢ 2 = (1 + 1) | |
22 | 20, 21 | breqtri 5173 | . . 3 ⊢ (3 / 2) < (1 + 1) |
23 | 1z 12645 | . . . 4 ⊢ 1 ∈ ℤ | |
24 | flbi 13853 | . . . 4 ⊢ (((3 / 2) ∈ ℝ ∧ 1 ∈ ℤ) → ((⌊‘(3 / 2)) = 1 ↔ (1 ≤ (3 / 2) ∧ (3 / 2) < (1 + 1)))) | |
25 | 3, 23, 24 | mp2an 692 | . . 3 ⊢ ((⌊‘(3 / 2)) = 1 ↔ (1 ≤ (3 / 2) ∧ (3 / 2) < (1 + 1))) |
26 | 13, 22, 25 | mpbir2an 711 | . 2 ⊢ (⌊‘(3 / 2)) = 1 |
27 | 9 | renegcli 11568 | . . . 4 ⊢ -2 ∈ ℝ |
28 | 3 | renegcli 11568 | . . . 4 ⊢ -(3 / 2) ∈ ℝ |
29 | 3, 9 | ltnegi 11805 | . . . . 5 ⊢ ((3 / 2) < 2 ↔ -2 < -(3 / 2)) |
30 | 20, 29 | mpbi 230 | . . . 4 ⊢ -2 < -(3 / 2) |
31 | 27, 28, 30 | ltleii 11382 | . . 3 ⊢ -2 ≤ -(3 / 2) |
32 | 4 | negcli 11575 | . . . . . . 7 ⊢ -2 ∈ ℂ |
33 | ax-1cn 11211 | . . . . . . 7 ⊢ 1 ∈ ℂ | |
34 | negdi2 11565 | . . . . . . 7 ⊢ ((-2 ∈ ℂ ∧ 1 ∈ ℂ) → -(-2 + 1) = (--2 − 1)) | |
35 | 32, 33, 34 | mp2an 692 | . . . . . 6 ⊢ -(-2 + 1) = (--2 − 1) |
36 | 4 | negnegi 11577 | . . . . . . 7 ⊢ --2 = 2 |
37 | 36 | oveq1i 7441 | . . . . . 6 ⊢ (--2 − 1) = (2 − 1) |
38 | 35, 37 | eqtri 2763 | . . . . 5 ⊢ -(-2 + 1) = (2 − 1) |
39 | 2m1e1 12390 | . . . . . 6 ⊢ (2 − 1) = 1 | |
40 | 39, 12 | eqbrtri 5169 | . . . . 5 ⊢ (2 − 1) < (3 / 2) |
41 | 38, 40 | eqbrtri 5169 | . . . 4 ⊢ -(-2 + 1) < (3 / 2) |
42 | 27, 1 | readdcli 11274 | . . . . 5 ⊢ (-2 + 1) ∈ ℝ |
43 | 42, 3 | ltnegcon1i 11812 | . . . 4 ⊢ (-(-2 + 1) < (3 / 2) ↔ -(3 / 2) < (-2 + 1)) |
44 | 41, 43 | mpbi 230 | . . 3 ⊢ -(3 / 2) < (-2 + 1) |
45 | 2z 12647 | . . . . 5 ⊢ 2 ∈ ℤ | |
46 | znegcl 12650 | . . . . 5 ⊢ (2 ∈ ℤ → -2 ∈ ℤ) | |
47 | 45, 46 | ax-mp 5 | . . . 4 ⊢ -2 ∈ ℤ |
48 | flbi 13853 | . . . 4 ⊢ ((-(3 / 2) ∈ ℝ ∧ -2 ∈ ℤ) → ((⌊‘-(3 / 2)) = -2 ↔ (-2 ≤ -(3 / 2) ∧ -(3 / 2) < (-2 + 1)))) | |
49 | 28, 47, 48 | mp2an 692 | . . 3 ⊢ ((⌊‘-(3 / 2)) = -2 ↔ (-2 ≤ -(3 / 2) ∧ -(3 / 2) < (-2 + 1))) |
50 | 31, 44, 49 | mpbir2an 711 | . 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 206 ∧ wa 395 = wceq 1537 ∈ wcel 2106 class class class wbr 5148 ‘cfv 6563 (class class class)co 7431 ℂcc 11151 ℝcr 11152 0cc0 11153 1c1 11154 + caddc 11156 · cmul 11158 < clt 11293 ≤ cle 11294 − cmin 11490 -cneg 11491 / cdiv 11918 2c2 12319 3c3 12320 4c4 12321 ℤcz 12611 ⌊cfl 13827 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 ax-pre-sup 11231 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-sup 9480 df-inf 9481 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-div 11919 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-n0 12525 df-z 12612 df-uz 12877 df-fl 13829 |
This theorem is referenced by: ex-ceil 30477 |
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