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Mirrors > Home > ILE Home > Th. List > ex-ceil | GIF version |
Description: Example for df-ceil 10044. (Contributed by AV, 4-Sep-2021.) |
Ref | Expression |
---|---|
ex-ceil | ⊢ ((⌈‘(3 / 2)) = 2 ∧ (⌈‘-(3 / 2)) = -1) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ex-fl 12937 | . 2 ⊢ ((⌊‘(3 / 2)) = 1 ∧ (⌊‘-(3 / 2)) = -2) | |
2 | 3z 9083 | . . . . . . 7 ⊢ 3 ∈ ℤ | |
3 | 2nn 8881 | . . . . . . 7 ⊢ 2 ∈ ℕ | |
4 | znq 9416 | . . . . . . 7 ⊢ ((3 ∈ ℤ ∧ 2 ∈ ℕ) → (3 / 2) ∈ ℚ) | |
5 | 2, 3, 4 | mp2an 422 | . . . . . 6 ⊢ (3 / 2) ∈ ℚ |
6 | qnegcl 9428 | . . . . . 6 ⊢ ((3 / 2) ∈ ℚ → -(3 / 2) ∈ ℚ) | |
7 | 5, 6 | ax-mp 5 | . . . . 5 ⊢ -(3 / 2) ∈ ℚ |
8 | ceilqval 10079 | . . . . 5 ⊢ (-(3 / 2) ∈ ℚ → (⌈‘-(3 / 2)) = -(⌊‘--(3 / 2))) | |
9 | 7, 8 | ax-mp 5 | . . . 4 ⊢ (⌈‘-(3 / 2)) = -(⌊‘--(3 / 2)) |
10 | qcn 9426 | . . . . . . . . . . 11 ⊢ ((3 / 2) ∈ ℚ → (3 / 2) ∈ ℂ) | |
11 | 5, 10 | ax-mp 5 | . . . . . . . . . 10 ⊢ (3 / 2) ∈ ℂ |
12 | 11 | negnegi 8032 | . . . . . . . . 9 ⊢ --(3 / 2) = (3 / 2) |
13 | 12 | eqcomi 2143 | . . . . . . . 8 ⊢ (3 / 2) = --(3 / 2) |
14 | 13 | fveq2i 5424 | . . . . . . 7 ⊢ (⌊‘(3 / 2)) = (⌊‘--(3 / 2)) |
15 | 14 | eqeq1i 2147 | . . . . . 6 ⊢ ((⌊‘(3 / 2)) = 1 ↔ (⌊‘--(3 / 2)) = 1) |
16 | 15 | biimpi 119 | . . . . 5 ⊢ ((⌊‘(3 / 2)) = 1 → (⌊‘--(3 / 2)) = 1) |
17 | 16 | negeqd 7957 | . . . 4 ⊢ ((⌊‘(3 / 2)) = 1 → -(⌊‘--(3 / 2)) = -1) |
18 | 9, 17 | syl5eq 2184 | . . 3 ⊢ ((⌊‘(3 / 2)) = 1 → (⌈‘-(3 / 2)) = -1) |
19 | ceilqval 10079 | . . . . 5 ⊢ ((3 / 2) ∈ ℚ → (⌈‘(3 / 2)) = -(⌊‘-(3 / 2))) | |
20 | 5, 19 | ax-mp 5 | . . . 4 ⊢ (⌈‘(3 / 2)) = -(⌊‘-(3 / 2)) |
21 | negeq 7955 | . . . . 5 ⊢ ((⌊‘-(3 / 2)) = -2 → -(⌊‘-(3 / 2)) = --2) | |
22 | 2cn 8791 | . . . . . 6 ⊢ 2 ∈ ℂ | |
23 | 22 | negnegi 8032 | . . . . 5 ⊢ --2 = 2 |
24 | 21, 23 | syl6eq 2188 | . . . 4 ⊢ ((⌊‘-(3 / 2)) = -2 → -(⌊‘-(3 / 2)) = 2) |
25 | 20, 24 | syl5eq 2184 | . . 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 1331 ∈ wcel 1480 ‘cfv 5123 (class class class)co 5774 ℂcc 7618 1c1 7621 -cneg 7934 / cdiv 8432 ℕcn 8720 2c2 8771 3c3 8772 ℤcz 9054 ℚcq 9411 ⌊cfl 10041 ⌈cceil 10042 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-cnex 7711 ax-resscn 7712 ax-1cn 7713 ax-1re 7714 ax-icn 7715 ax-addcl 7716 ax-addrcl 7717 ax-mulcl 7718 ax-mulrcl 7719 ax-addcom 7720 ax-mulcom 7721 ax-addass 7722 ax-mulass 7723 ax-distr 7724 ax-i2m1 7725 ax-0lt1 7726 ax-1rid 7727 ax-0id 7728 ax-rnegex 7729 ax-precex 7730 ax-cnre 7731 ax-pre-ltirr 7732 ax-pre-ltwlin 7733 ax-pre-lttrn 7734 ax-pre-apti 7735 ax-pre-ltadd 7736 ax-pre-mulgt0 7737 ax-pre-mulext 7738 ax-arch 7739 |
This theorem depends on definitions: df-bi 116 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-nel 2404 df-ral 2421 df-rex 2422 df-reu 2423 df-rmo 2424 df-rab 2425 df-v 2688 df-sbc 2910 df-csb 3004 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-int 3772 df-iun 3815 df-br 3930 df-opab 3990 df-mpt 3991 df-id 4215 df-po 4218 df-iso 4219 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-fv 5131 df-riota 5730 df-ov 5777 df-oprab 5778 df-mpo 5779 df-1st 6038 df-2nd 6039 df-pnf 7802 df-mnf 7803 df-xr 7804 df-ltxr 7805 df-le 7806 df-sub 7935 df-neg 7936 df-reap 8337 df-ap 8344 df-div 8433 df-inn 8721 df-2 8779 df-3 8780 df-4 8781 df-n0 8978 df-z 9055 df-q 9412 df-rp 9442 df-fl 10043 df-ceil 10044 |
This theorem is referenced by: (None) |
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