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Mirrors > Home > ILE Home > Th. List > ex-ceil | GIF version |
Description: Example for df-ceil 9885. (Contributed by AV, 4-Sep-2021.) |
Ref | Expression |
---|---|
ex-ceil | ⊢ ((⌈‘(3 / 2)) = 2 ∧ (⌈‘-(3 / 2)) = -1) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ex-fl 12540 | . 2 ⊢ ((⌊‘(3 / 2)) = 1 ∧ (⌊‘-(3 / 2)) = -2) | |
2 | 3z 8935 | . . . . . . 7 ⊢ 3 ∈ ℤ | |
3 | 2nn 8733 | . . . . . . 7 ⊢ 2 ∈ ℕ | |
4 | znq 9266 | . . . . . . 7 ⊢ ((3 ∈ ℤ ∧ 2 ∈ ℕ) → (3 / 2) ∈ ℚ) | |
5 | 2, 3, 4 | mp2an 420 | . . . . . 6 ⊢ (3 / 2) ∈ ℚ |
6 | qnegcl 9278 | . . . . . 6 ⊢ ((3 / 2) ∈ ℚ → -(3 / 2) ∈ ℚ) | |
7 | 5, 6 | ax-mp 7 | . . . . 5 ⊢ -(3 / 2) ∈ ℚ |
8 | ceilqval 9920 | . . . . 5 ⊢ (-(3 / 2) ∈ ℚ → (⌈‘-(3 / 2)) = -(⌊‘--(3 / 2))) | |
9 | 7, 8 | ax-mp 7 | . . . 4 ⊢ (⌈‘-(3 / 2)) = -(⌊‘--(3 / 2)) |
10 | qcn 9276 | . . . . . . . . . . 11 ⊢ ((3 / 2) ∈ ℚ → (3 / 2) ∈ ℂ) | |
11 | 5, 10 | ax-mp 7 | . . . . . . . . . 10 ⊢ (3 / 2) ∈ ℂ |
12 | 11 | negnegi 7903 | . . . . . . . . 9 ⊢ --(3 / 2) = (3 / 2) |
13 | 12 | eqcomi 2104 | . . . . . . . 8 ⊢ (3 / 2) = --(3 / 2) |
14 | 13 | fveq2i 5356 | . . . . . . 7 ⊢ (⌊‘(3 / 2)) = (⌊‘--(3 / 2)) |
15 | 14 | eqeq1i 2107 | . . . . . 6 ⊢ ((⌊‘(3 / 2)) = 1 ↔ (⌊‘--(3 / 2)) = 1) |
16 | 15 | biimpi 119 | . . . . 5 ⊢ ((⌊‘(3 / 2)) = 1 → (⌊‘--(3 / 2)) = 1) |
17 | 16 | negeqd 7828 | . . . 4 ⊢ ((⌊‘(3 / 2)) = 1 → -(⌊‘--(3 / 2)) = -1) |
18 | 9, 17 | syl5eq 2144 | . . 3 ⊢ ((⌊‘(3 / 2)) = 1 → (⌈‘-(3 / 2)) = -1) |
19 | ceilqval 9920 | . . . . 5 ⊢ ((3 / 2) ∈ ℚ → (⌈‘(3 / 2)) = -(⌊‘-(3 / 2))) | |
20 | 5, 19 | ax-mp 7 | . . . 4 ⊢ (⌈‘(3 / 2)) = -(⌊‘-(3 / 2)) |
21 | negeq 7826 | . . . . 5 ⊢ ((⌊‘-(3 / 2)) = -2 → -(⌊‘-(3 / 2)) = --2) | |
22 | 2cn 8649 | . . . . . 6 ⊢ 2 ∈ ℂ | |
23 | 22 | negnegi 7903 | . . . . 5 ⊢ --2 = 2 |
24 | 21, 23 | syl6eq 2148 | . . . 4 ⊢ ((⌊‘-(3 / 2)) = -2 → -(⌊‘-(3 / 2)) = 2) |
25 | 20, 24 | syl5eq 2144 | . . 3 ⊢ ((⌊‘-(3 / 2)) = -2 → (⌈‘(3 / 2)) = 2) |
26 | 18, 25 | anim12ci 335 | . 2 ⊢ (((⌊‘(3 / 2)) = 1 ∧ (⌊‘-(3 / 2)) = -2) → ((⌈‘(3 / 2)) = 2 ∧ (⌈‘-(3 / 2)) = -1)) |
27 | 1, 26 | ax-mp 7 | 1 ⊢ ((⌈‘(3 / 2)) = 2 ∧ (⌈‘-(3 / 2)) = -1) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 103 = wceq 1299 ∈ wcel 1448 ‘cfv 5059 (class class class)co 5706 ℂcc 7498 1c1 7501 -cneg 7805 / cdiv 8293 ℕcn 8578 2c2 8629 3c3 8630 ℤcz 8906 ℚcq 9261 ⌊cfl 9882 ⌈cceil 9883 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 584 ax-in2 585 ax-io 671 ax-5 1391 ax-7 1392 ax-gen 1393 ax-ie1 1437 ax-ie2 1438 ax-8 1450 ax-10 1451 ax-11 1452 ax-i12 1453 ax-bndl 1454 ax-4 1455 ax-13 1459 ax-14 1460 ax-17 1474 ax-i9 1478 ax-ial 1482 ax-i5r 1483 ax-ext 2082 ax-sep 3986 ax-pow 4038 ax-pr 4069 ax-un 4293 ax-setind 4390 ax-cnex 7586 ax-resscn 7587 ax-1cn 7588 ax-1re 7589 ax-icn 7590 ax-addcl 7591 ax-addrcl 7592 ax-mulcl 7593 ax-mulrcl 7594 ax-addcom 7595 ax-mulcom 7596 ax-addass 7597 ax-mulass 7598 ax-distr 7599 ax-i2m1 7600 ax-0lt1 7601 ax-1rid 7602 ax-0id 7603 ax-rnegex 7604 ax-precex 7605 ax-cnre 7606 ax-pre-ltirr 7607 ax-pre-ltwlin 7608 ax-pre-lttrn 7609 ax-pre-apti 7610 ax-pre-ltadd 7611 ax-pre-mulgt0 7612 ax-pre-mulext 7613 ax-arch 7614 |
This theorem depends on definitions: df-bi 116 df-3or 931 df-3an 932 df-tru 1302 df-fal 1305 df-nf 1405 df-sb 1704 df-eu 1963 df-mo 1964 df-clab 2087 df-cleq 2093 df-clel 2096 df-nfc 2229 df-ne 2268 df-nel 2363 df-ral 2380 df-rex 2381 df-reu 2382 df-rmo 2383 df-rab 2384 df-v 2643 df-sbc 2863 df-csb 2956 df-dif 3023 df-un 3025 df-in 3027 df-ss 3034 df-pw 3459 df-sn 3480 df-pr 3481 df-op 3483 df-uni 3684 df-int 3719 df-iun 3762 df-br 3876 df-opab 3930 df-mpt 3931 df-id 4153 df-po 4156 df-iso 4157 df-xp 4483 df-rel 4484 df-cnv 4485 df-co 4486 df-dm 4487 df-rn 4488 df-res 4489 df-ima 4490 df-iota 5024 df-fun 5061 df-fn 5062 df-f 5063 df-fv 5067 df-riota 5662 df-ov 5709 df-oprab 5710 df-mpo 5711 df-1st 5969 df-2nd 5970 df-pnf 7674 df-mnf 7675 df-xr 7676 df-ltxr 7677 df-le 7678 df-sub 7806 df-neg 7807 df-reap 8203 df-ap 8210 df-div 8294 df-inn 8579 df-2 8637 df-3 8638 df-4 8639 df-n0 8830 df-z 8907 df-q 9262 df-rp 9292 df-fl 9884 df-ceil 9885 |
This theorem is referenced by: (None) |
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