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Mirrors > Home > ILE Home > Th. List > ex-ceil | GIF version |
Description: Example for df-ceil 9275. (Contributed by AV, 4-Sep-2021.) |
Ref | Expression |
---|---|
ex-ceil | ⊢ ((⌈‘(3 / 2)) = 2 ∧ (⌈‘-(3 / 2)) = -1) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ex-fl 10563 | . 2 ⊢ ((⌊‘(3 / 2)) = 1 ∧ (⌊‘-(3 / 2)) = -2) | |
2 | 3z 8380 | . . . . . . 7 ⊢ 3 ∈ ℤ | |
3 | 2nn 8193 | . . . . . . 7 ⊢ 2 ∈ ℕ | |
4 | znq 8709 | . . . . . . 7 ⊢ ((3 ∈ ℤ ∧ 2 ∈ ℕ) → (3 / 2) ∈ ℚ) | |
5 | 2, 3, 4 | mp2an 416 | . . . . . 6 ⊢ (3 / 2) ∈ ℚ |
6 | qnegcl 8721 | . . . . . 6 ⊢ ((3 / 2) ∈ ℚ → -(3 / 2) ∈ ℚ) | |
7 | 5, 6 | ax-mp 7 | . . . . 5 ⊢ -(3 / 2) ∈ ℚ |
8 | ceilqval 9308 | . . . . 5 ⊢ (-(3 / 2) ∈ ℚ → (⌈‘-(3 / 2)) = -(⌊‘--(3 / 2))) | |
9 | 7, 8 | ax-mp 7 | . . . 4 ⊢ (⌈‘-(3 / 2)) = -(⌊‘--(3 / 2)) |
10 | qcn 8719 | . . . . . . . . . . 11 ⊢ ((3 / 2) ∈ ℚ → (3 / 2) ∈ ℂ) | |
11 | 5, 10 | ax-mp 7 | . . . . . . . . . 10 ⊢ (3 / 2) ∈ ℂ |
12 | 11 | negnegi 7378 | . . . . . . . . 9 ⊢ --(3 / 2) = (3 / 2) |
13 | 12 | eqcomi 2085 | . . . . . . . 8 ⊢ (3 / 2) = --(3 / 2) |
14 | 13 | fveq2i 5201 | . . . . . . 7 ⊢ (⌊‘(3 / 2)) = (⌊‘--(3 / 2)) |
15 | 14 | eqeq1i 2088 | . . . . . 6 ⊢ ((⌊‘(3 / 2)) = 1 ↔ (⌊‘--(3 / 2)) = 1) |
16 | 15 | biimpi 118 | . . . . 5 ⊢ ((⌊‘(3 / 2)) = 1 → (⌊‘--(3 / 2)) = 1) |
17 | 16 | negeqd 7303 | . . . 4 ⊢ ((⌊‘(3 / 2)) = 1 → -(⌊‘--(3 / 2)) = -1) |
18 | 9, 17 | syl5eq 2125 | . . 3 ⊢ ((⌊‘(3 / 2)) = 1 → (⌈‘-(3 / 2)) = -1) |
19 | ceilqval 9308 | . . . . 5 ⊢ ((3 / 2) ∈ ℚ → (⌈‘(3 / 2)) = -(⌊‘-(3 / 2))) | |
20 | 5, 19 | ax-mp 7 | . . . 4 ⊢ (⌈‘(3 / 2)) = -(⌊‘-(3 / 2)) |
21 | negeq 7301 | . . . . 5 ⊢ ((⌊‘-(3 / 2)) = -2 → -(⌊‘-(3 / 2)) = --2) | |
22 | 2cn 8110 | . . . . . 6 ⊢ 2 ∈ ℂ | |
23 | 22 | negnegi 7378 | . . . . 5 ⊢ --2 = 2 |
24 | 21, 23 | syl6eq 2129 | . . . 4 ⊢ ((⌊‘-(3 / 2)) = -2 → -(⌊‘-(3 / 2)) = 2) |
25 | 20, 24 | syl5eq 2125 | . . 3 ⊢ ((⌊‘-(3 / 2)) = -2 → (⌈‘(3 / 2)) = 2) |
26 | 18, 25 | anim12ci 332 | . 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 102 = wceq 1284 ∈ wcel 1433 ‘cfv 4922 (class class class)co 5532 ℂcc 6979 1c1 6982 -cneg 7280 / cdiv 7760 ℕcn 8039 2c2 8089 3c3 8090 ℤcz 8351 ℚcq 8704 ⌊cfl 9272 ⌈cceil 9273 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-13 1444 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-pow 3948 ax-pr 3964 ax-un 4188 ax-setind 4280 ax-cnex 7067 ax-resscn 7068 ax-1cn 7069 ax-1re 7070 ax-icn 7071 ax-addcl 7072 ax-addrcl 7073 ax-mulcl 7074 ax-mulrcl 7075 ax-addcom 7076 ax-mulcom 7077 ax-addass 7078 ax-mulass 7079 ax-distr 7080 ax-i2m1 7081 ax-0lt1 7082 ax-1rid 7083 ax-0id 7084 ax-rnegex 7085 ax-precex 7086 ax-cnre 7087 ax-pre-ltirr 7088 ax-pre-ltwlin 7089 ax-pre-lttrn 7090 ax-pre-apti 7091 ax-pre-ltadd 7092 ax-pre-mulgt0 7093 ax-pre-mulext 7094 ax-arch 7095 |
This theorem depends on definitions: df-bi 115 df-3or 920 df-3an 921 df-tru 1287 df-fal 1290 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ne 2246 df-nel 2340 df-ral 2353 df-rex 2354 df-reu 2355 df-rmo 2356 df-rab 2357 df-v 2603 df-sbc 2816 df-csb 2909 df-dif 2975 df-un 2977 df-in 2979 df-ss 2986 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-int 3637 df-iun 3680 df-br 3786 df-opab 3840 df-mpt 3841 df-id 4048 df-po 4051 df-iso 4052 df-xp 4369 df-rel 4370 df-cnv 4371 df-co 4372 df-dm 4373 df-rn 4374 df-res 4375 df-ima 4376 df-iota 4887 df-fun 4924 df-fn 4925 df-f 4926 df-fv 4930 df-riota 5488 df-ov 5535 df-oprab 5536 df-mpt2 5537 df-1st 5787 df-2nd 5788 df-pnf 7155 df-mnf 7156 df-xr 7157 df-ltxr 7158 df-le 7159 df-sub 7281 df-neg 7282 df-reap 7675 df-ap 7682 df-div 7761 df-inn 8040 df-2 8098 df-3 8099 df-4 8100 df-n0 8289 df-z 8352 df-q 8705 df-rp 8735 df-fl 9274 df-ceil 9275 |
This theorem is referenced by: (None) |
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