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Mirrors > Home > ILE Home > Th. List > ex-ceil | GIF version |
Description: Example for df-ceil 10012. (Contributed by AV, 4-Sep-2021.) |
Ref | Expression |
---|---|
ex-ceil | ⊢ ((⌈‘(3 / 2)) = 2 ∧ (⌈‘-(3 / 2)) = -1) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ex-fl 12864 | . 2 ⊢ ((⌊‘(3 / 2)) = 1 ∧ (⌊‘-(3 / 2)) = -2) | |
2 | 3z 9051 | . . . . . . 7 ⊢ 3 ∈ ℤ | |
3 | 2nn 8849 | . . . . . . 7 ⊢ 2 ∈ ℕ | |
4 | znq 9384 | . . . . . . 7 ⊢ ((3 ∈ ℤ ∧ 2 ∈ ℕ) → (3 / 2) ∈ ℚ) | |
5 | 2, 3, 4 | mp2an 422 | . . . . . 6 ⊢ (3 / 2) ∈ ℚ |
6 | qnegcl 9396 | . . . . . 6 ⊢ ((3 / 2) ∈ ℚ → -(3 / 2) ∈ ℚ) | |
7 | 5, 6 | ax-mp 5 | . . . . 5 ⊢ -(3 / 2) ∈ ℚ |
8 | ceilqval 10047 | . . . . 5 ⊢ (-(3 / 2) ∈ ℚ → (⌈‘-(3 / 2)) = -(⌊‘--(3 / 2))) | |
9 | 7, 8 | ax-mp 5 | . . . 4 ⊢ (⌈‘-(3 / 2)) = -(⌊‘--(3 / 2)) |
10 | qcn 9394 | . . . . . . . . . . 11 ⊢ ((3 / 2) ∈ ℚ → (3 / 2) ∈ ℂ) | |
11 | 5, 10 | ax-mp 5 | . . . . . . . . . 10 ⊢ (3 / 2) ∈ ℂ |
12 | 11 | negnegi 8000 | . . . . . . . . 9 ⊢ --(3 / 2) = (3 / 2) |
13 | 12 | eqcomi 2121 | . . . . . . . 8 ⊢ (3 / 2) = --(3 / 2) |
14 | 13 | fveq2i 5392 | . . . . . . 7 ⊢ (⌊‘(3 / 2)) = (⌊‘--(3 / 2)) |
15 | 14 | eqeq1i 2125 | . . . . . 6 ⊢ ((⌊‘(3 / 2)) = 1 ↔ (⌊‘--(3 / 2)) = 1) |
16 | 15 | biimpi 119 | . . . . 5 ⊢ ((⌊‘(3 / 2)) = 1 → (⌊‘--(3 / 2)) = 1) |
17 | 16 | negeqd 7925 | . . . 4 ⊢ ((⌊‘(3 / 2)) = 1 → -(⌊‘--(3 / 2)) = -1) |
18 | 9, 17 | syl5eq 2162 | . . 3 ⊢ ((⌊‘(3 / 2)) = 1 → (⌈‘-(3 / 2)) = -1) |
19 | ceilqval 10047 | . . . . 5 ⊢ ((3 / 2) ∈ ℚ → (⌈‘(3 / 2)) = -(⌊‘-(3 / 2))) | |
20 | 5, 19 | ax-mp 5 | . . . 4 ⊢ (⌈‘(3 / 2)) = -(⌊‘-(3 / 2)) |
21 | negeq 7923 | . . . . 5 ⊢ ((⌊‘-(3 / 2)) = -2 → -(⌊‘-(3 / 2)) = --2) | |
22 | 2cn 8759 | . . . . . 6 ⊢ 2 ∈ ℂ | |
23 | 22 | negnegi 8000 | . . . . 5 ⊢ --2 = 2 |
24 | 21, 23 | syl6eq 2166 | . . . 4 ⊢ ((⌊‘-(3 / 2)) = -2 → -(⌊‘-(3 / 2)) = 2) |
25 | 20, 24 | syl5eq 2162 | . . 3 ⊢ ((⌊‘-(3 / 2)) = -2 → (⌈‘(3 / 2)) = 2) |
26 | 18, 25 | anim12ci 337 | . 2 ⊢ (((⌊‘(3 / 2)) = 1 ∧ (⌊‘-(3 / 2)) = -2) → ((⌈‘(3 / 2)) = 2 ∧ (⌈‘-(3 / 2)) = -1)) |
27 | 1, 26 | ax-mp 5 | 1 ⊢ ((⌈‘(3 / 2)) = 2 ∧ (⌈‘-(3 / 2)) = -1) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 103 = wceq 1316 ∈ wcel 1465 ‘cfv 5093 (class class class)co 5742 ℂcc 7586 1c1 7589 -cneg 7902 / cdiv 8400 ℕcn 8688 2c2 8739 3c3 8740 ℤcz 9022 ℚcq 9379 ⌊cfl 10009 ⌈cceil 10010 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 588 ax-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-13 1476 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-pow 4068 ax-pr 4101 ax-un 4325 ax-setind 4422 ax-cnex 7679 ax-resscn 7680 ax-1cn 7681 ax-1re 7682 ax-icn 7683 ax-addcl 7684 ax-addrcl 7685 ax-mulcl 7686 ax-mulrcl 7687 ax-addcom 7688 ax-mulcom 7689 ax-addass 7690 ax-mulass 7691 ax-distr 7692 ax-i2m1 7693 ax-0lt1 7694 ax-1rid 7695 ax-0id 7696 ax-rnegex 7697 ax-precex 7698 ax-cnre 7699 ax-pre-ltirr 7700 ax-pre-ltwlin 7701 ax-pre-lttrn 7702 ax-pre-apti 7703 ax-pre-ltadd 7704 ax-pre-mulgt0 7705 ax-pre-mulext 7706 ax-arch 7707 |
This theorem depends on definitions: df-bi 116 df-3or 948 df-3an 949 df-tru 1319 df-fal 1322 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ne 2286 df-nel 2381 df-ral 2398 df-rex 2399 df-reu 2400 df-rmo 2401 df-rab 2402 df-v 2662 df-sbc 2883 df-csb 2976 df-dif 3043 df-un 3045 df-in 3047 df-ss 3054 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-int 3742 df-iun 3785 df-br 3900 df-opab 3960 df-mpt 3961 df-id 4185 df-po 4188 df-iso 4189 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-rn 4520 df-res 4521 df-ima 4522 df-iota 5058 df-fun 5095 df-fn 5096 df-f 5097 df-fv 5101 df-riota 5698 df-ov 5745 df-oprab 5746 df-mpo 5747 df-1st 6006 df-2nd 6007 df-pnf 7770 df-mnf 7771 df-xr 7772 df-ltxr 7773 df-le 7774 df-sub 7903 df-neg 7904 df-reap 8305 df-ap 8312 df-div 8401 df-inn 8689 df-2 8747 df-3 8748 df-4 8749 df-n0 8946 df-z 9023 df-q 9380 df-rp 9410 df-fl 10011 df-ceil 10012 |
This theorem is referenced by: (None) |
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