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Mirrors > Home > MPE Home > Th. List > ex-ceil | Structured version Visualization version GIF version |
Description: Example for df-ceil 12634. (Contributed by AV, 4-Sep-2021.) |
Ref | Expression |
---|---|
ex-ceil | ⊢ ((⌈‘(3 / 2)) = 2 ∧ (⌈‘-(3 / 2)) = -1) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ex-fl 27434 | . 2 ⊢ ((⌊‘(3 / 2)) = 1 ∧ (⌊‘-(3 / 2)) = -2) | |
2 | 3re 11132 | . . . . . . 7 ⊢ 3 ∈ ℝ | |
3 | 2 | rehalfcli 11319 | . . . . . 6 ⊢ (3 / 2) ∈ ℝ |
4 | 3 | renegcli 10380 | . . . . 5 ⊢ -(3 / 2) ∈ ℝ |
5 | ceilval 12679 | . . . . 5 ⊢ (-(3 / 2) ∈ ℝ → (⌈‘-(3 / 2)) = -(⌊‘--(3 / 2))) | |
6 | 4, 5 | ax-mp 5 | . . . 4 ⊢ (⌈‘-(3 / 2)) = -(⌊‘--(3 / 2)) |
7 | 3 | recni 10090 | . . . . . . . . . 10 ⊢ (3 / 2) ∈ ℂ |
8 | 7 | negnegi 10389 | . . . . . . . . 9 ⊢ --(3 / 2) = (3 / 2) |
9 | 8 | eqcomi 2660 | . . . . . . . 8 ⊢ (3 / 2) = --(3 / 2) |
10 | 9 | fveq2i 6232 | . . . . . . 7 ⊢ (⌊‘(3 / 2)) = (⌊‘--(3 / 2)) |
11 | 10 | eqeq1i 2656 | . . . . . 6 ⊢ ((⌊‘(3 / 2)) = 1 ↔ (⌊‘--(3 / 2)) = 1) |
12 | 11 | biimpi 206 | . . . . 5 ⊢ ((⌊‘(3 / 2)) = 1 → (⌊‘--(3 / 2)) = 1) |
13 | 12 | negeqd 10313 | . . . 4 ⊢ ((⌊‘(3 / 2)) = 1 → -(⌊‘--(3 / 2)) = -1) |
14 | 6, 13 | syl5eq 2697 | . . 3 ⊢ ((⌊‘(3 / 2)) = 1 → (⌈‘-(3 / 2)) = -1) |
15 | ceilval 12679 | . . . . 5 ⊢ ((3 / 2) ∈ ℝ → (⌈‘(3 / 2)) = -(⌊‘-(3 / 2))) | |
16 | 3, 15 | ax-mp 5 | . . . 4 ⊢ (⌈‘(3 / 2)) = -(⌊‘-(3 / 2)) |
17 | negeq 10311 | . . . . 5 ⊢ ((⌊‘-(3 / 2)) = -2 → -(⌊‘-(3 / 2)) = --2) | |
18 | 2cn 11129 | . . . . . 6 ⊢ 2 ∈ ℂ | |
19 | 18 | negnegi 10389 | . . . . 5 ⊢ --2 = 2 |
20 | 17, 19 | syl6eq 2701 | . . . 4 ⊢ ((⌊‘-(3 / 2)) = -2 → -(⌊‘-(3 / 2)) = 2) |
21 | 16, 20 | syl5eq 2697 | . . 3 ⊢ ((⌊‘-(3 / 2)) = -2 → (⌈‘(3 / 2)) = 2) |
22 | 14, 21 | anim12ci 590 | . 2 ⊢ (((⌊‘(3 / 2)) = 1 ∧ (⌊‘-(3 / 2)) = -2) → ((⌈‘(3 / 2)) = 2 ∧ (⌈‘-(3 / 2)) = -1)) |
23 | 1, 22 | ax-mp 5 | 1 ⊢ ((⌈‘(3 / 2)) = 2 ∧ (⌈‘-(3 / 2)) = -1) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 383 = wceq 1523 ∈ wcel 2030 ‘cfv 5926 (class class class)co 6690 ℝcr 9973 1c1 9975 -cneg 10305 / cdiv 10722 2c2 11108 3c3 11109 ⌊cfl 12631 ⌈cceil 12632 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 ax-un 6991 ax-cnex 10030 ax-resscn 10031 ax-1cn 10032 ax-icn 10033 ax-addcl 10034 ax-addrcl 10035 ax-mulcl 10036 ax-mulrcl 10037 ax-mulcom 10038 ax-addass 10039 ax-mulass 10040 ax-distr 10041 ax-i2m1 10042 ax-1ne0 10043 ax-1rid 10044 ax-rnegex 10045 ax-rrecex 10046 ax-cnre 10047 ax-pre-lttri 10048 ax-pre-lttrn 10049 ax-pre-ltadd 10050 ax-pre-mulgt0 10051 ax-pre-sup 10052 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1055 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-nel 2927 df-ral 2946 df-rex 2947 df-reu 2948 df-rmo 2949 df-rab 2950 df-v 3233 df-sbc 3469 df-csb 3567 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-pss 3623 df-nul 3949 df-if 4120 df-pw 4193 df-sn 4211 df-pr 4213 df-tp 4215 df-op 4217 df-uni 4469 df-iun 4554 df-br 4686 df-opab 4746 df-mpt 4763 df-tr 4786 df-id 5053 df-eprel 5058 df-po 5064 df-so 5065 df-fr 5102 df-we 5104 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 df-pred 5718 df-ord 5764 df-on 5765 df-lim 5766 df-suc 5767 df-iota 5889 df-fun 5928 df-fn 5929 df-f 5930 df-f1 5931 df-fo 5932 df-f1o 5933 df-fv 5934 df-riota 6651 df-ov 6693 df-oprab 6694 df-mpt2 6695 df-om 7108 df-wrecs 7452 df-recs 7513 df-rdg 7551 df-er 7787 df-en 7998 df-dom 7999 df-sdom 8000 df-sup 8389 df-inf 8390 df-pnf 10114 df-mnf 10115 df-xr 10116 df-ltxr 10117 df-le 10118 df-sub 10306 df-neg 10307 df-div 10723 df-nn 11059 df-2 11117 df-3 11118 df-4 11119 df-n0 11331 df-z 11416 df-uz 11726 df-fl 12633 df-ceil 12634 |
This theorem is referenced by: (None) |
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