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| Description: Example for df-fl 13832. 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 11261 | . . . 4 ⊢ 1 ∈ ℝ | |
| 2 | 3re 12346 | . . . . 5 ⊢ 3 ∈ ℝ | |
| 3 | 2 | rehalfcli 12515 | . . . 4 ⊢ (3 / 2) ∈ ℝ | 
| 4 | 2cn 12341 | . . . . . . 7 ⊢ 2 ∈ ℂ | |
| 5 | 4 | mullidi 11266 | . . . . . 6 ⊢ (1 · 2) = 2 | 
| 6 | 2lt3 12438 | . . . . . 6 ⊢ 2 < 3 | |
| 7 | 5, 6 | eqbrtri 5164 | . . . . 5 ⊢ (1 · 2) < 3 | 
| 8 | 2pos 12369 | . . . . . 6 ⊢ 0 < 2 | |
| 9 | 2re 12340 | . . . . . . 7 ⊢ 2 ∈ ℝ | |
| 10 | 1, 2, 9 | ltmuldivi 12188 | . . . . . 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 11384 | . . 3 ⊢ 1 ≤ (3 / 2) | 
| 14 | 3lt4 12440 | . . . . . 6 ⊢ 3 < 4 | |
| 15 | 2t2e4 12430 | . . . . . 6 ⊢ (2 · 2) = 4 | |
| 16 | 14, 15 | breqtrri 5170 | . . . . 5 ⊢ 3 < (2 · 2) | 
| 17 | 9, 8 | pm3.2i 470 | . . . . . 6 ⊢ (2 ∈ ℝ ∧ 0 < 2) | 
| 18 | ltdivmul 12143 | . . . . . 6 ⊢ ((3 ∈ ℝ ∧ 2 ∈ ℝ ∧ (2 ∈ ℝ ∧ 0 < 2)) → ((3 / 2) < 2 ↔ 3 < (2 · 2))) | |
| 19 | 2, 9, 17, 18 | mp3an 1463 | . . . . 5 ⊢ ((3 / 2) < 2 ↔ 3 < (2 · 2)) | 
| 20 | 16, 19 | mpbir 231 | . . . 4 ⊢ (3 / 2) < 2 | 
| 21 | df-2 12329 | . . . 4 ⊢ 2 = (1 + 1) | |
| 22 | 20, 21 | breqtri 5168 | . . 3 ⊢ (3 / 2) < (1 + 1) | 
| 23 | 1z 12647 | . . . 4 ⊢ 1 ∈ ℤ | |
| 24 | flbi 13856 | . . . 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 11570 | . . . 4 ⊢ -2 ∈ ℝ | 
| 28 | 3 | renegcli 11570 | . . . 4 ⊢ -(3 / 2) ∈ ℝ | 
| 29 | 3, 9 | ltnegi 11807 | . . . . 5 ⊢ ((3 / 2) < 2 ↔ -2 < -(3 / 2)) | 
| 30 | 20, 29 | mpbi 230 | . . . 4 ⊢ -2 < -(3 / 2) | 
| 31 | 27, 28, 30 | ltleii 11384 | . . 3 ⊢ -2 ≤ -(3 / 2) | 
| 32 | 4 | negcli 11577 | . . . . . . 7 ⊢ -2 ∈ ℂ | 
| 33 | ax-1cn 11213 | . . . . . . 7 ⊢ 1 ∈ ℂ | |
| 34 | negdi2 11567 | . . . . . . 7 ⊢ ((-2 ∈ ℂ ∧ 1 ∈ ℂ) → -(-2 + 1) = (--2 − 1)) | |
| 35 | 32, 33, 34 | mp2an 692 | . . . . . 6 ⊢ -(-2 + 1) = (--2 − 1) | 
| 36 | 4 | negnegi 11579 | . . . . . . 7 ⊢ --2 = 2 | 
| 37 | 36 | oveq1i 7441 | . . . . . 6 ⊢ (--2 − 1) = (2 − 1) | 
| 38 | 35, 37 | eqtri 2765 | . . . . 5 ⊢ -(-2 + 1) = (2 − 1) | 
| 39 | 2m1e1 12392 | . . . . . 6 ⊢ (2 − 1) = 1 | |
| 40 | 39, 12 | eqbrtri 5164 | . . . . 5 ⊢ (2 − 1) < (3 / 2) | 
| 41 | 38, 40 | eqbrtri 5164 | . . . 4 ⊢ -(-2 + 1) < (3 / 2) | 
| 42 | 27, 1 | readdcli 11276 | . . . . 5 ⊢ (-2 + 1) ∈ ℝ | 
| 43 | 42, 3 | ltnegcon1i 11814 | . . . 4 ⊢ (-(-2 + 1) < (3 / 2) ↔ -(3 / 2) < (-2 + 1)) | 
| 44 | 41, 43 | mpbi 230 | . . 3 ⊢ -(3 / 2) < (-2 + 1) | 
| 45 | 2z 12649 | . . . . 5 ⊢ 2 ∈ ℤ | |
| 46 | znegcl 12652 | . . . . 5 ⊢ (2 ∈ ℤ → -2 ∈ ℤ) | |
| 47 | 45, 46 | ax-mp 5 | . . . 4 ⊢ -2 ∈ ℤ | 
| 48 | flbi 13856 | . . . 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 1540 ∈ wcel 2108 class class class wbr 5143 ‘cfv 6561 (class class class)co 7431 ℂcc 11153 ℝcr 11154 0cc0 11155 1c1 11156 + caddc 11158 · cmul 11160 < clt 11295 ≤ cle 11296 − cmin 11492 -cneg 11493 / cdiv 11920 2c2 12321 3c3 12322 4c4 12323 ℤcz 12613 ⌊cfl 13830 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-pre-sup 11233 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-sup 9482 df-inf 9483 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-n0 12527 df-z 12614 df-uz 12879 df-fl 13832 | 
| This theorem is referenced by: ex-ceil 30467 | 
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