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Mirrors > Home > MPE Home > Th. List > ex-ceil | Structured version Visualization version GIF version |
Description: Example for df-ceil 12977. (Contributed by AV, 4-Sep-2021.) |
Ref | Expression |
---|---|
ex-ceil | ⊢ ((⌈‘(3 / 2)) = 2 ∧ (⌈‘-(3 / 2)) = -1) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ex-fl 28020 | . 2 ⊢ ((⌊‘(3 / 2)) = 1 ∧ (⌊‘-(3 / 2)) = -2) | |
2 | 3re 11519 | . . . . . . 7 ⊢ 3 ∈ ℝ | |
3 | 2 | rehalfcli 11695 | . . . . . 6 ⊢ (3 / 2) ∈ ℝ |
4 | 3 | renegcli 10747 | . . . . 5 ⊢ -(3 / 2) ∈ ℝ |
5 | ceilval 13022 | . . . . 5 ⊢ (-(3 / 2) ∈ ℝ → (⌈‘-(3 / 2)) = -(⌊‘--(3 / 2))) | |
6 | 4, 5 | ax-mp 5 | . . . 4 ⊢ (⌈‘-(3 / 2)) = -(⌊‘--(3 / 2)) |
7 | 3 | recni 10453 | . . . . . . . . . 10 ⊢ (3 / 2) ∈ ℂ |
8 | 7 | negnegi 10756 | . . . . . . . . 9 ⊢ --(3 / 2) = (3 / 2) |
9 | 8 | eqcomi 2782 | . . . . . . . 8 ⊢ (3 / 2) = --(3 / 2) |
10 | 9 | fveq2i 6500 | . . . . . . 7 ⊢ (⌊‘(3 / 2)) = (⌊‘--(3 / 2)) |
11 | 10 | eqeq1i 2778 | . . . . . 6 ⊢ ((⌊‘(3 / 2)) = 1 ↔ (⌊‘--(3 / 2)) = 1) |
12 | 11 | biimpi 208 | . . . . 5 ⊢ ((⌊‘(3 / 2)) = 1 → (⌊‘--(3 / 2)) = 1) |
13 | 12 | negeqd 10679 | . . . 4 ⊢ ((⌊‘(3 / 2)) = 1 → -(⌊‘--(3 / 2)) = -1) |
14 | 6, 13 | syl5eq 2821 | . . 3 ⊢ ((⌊‘(3 / 2)) = 1 → (⌈‘-(3 / 2)) = -1) |
15 | ceilval 13022 | . . . . 5 ⊢ ((3 / 2) ∈ ℝ → (⌈‘(3 / 2)) = -(⌊‘-(3 / 2))) | |
16 | 3, 15 | ax-mp 5 | . . . 4 ⊢ (⌈‘(3 / 2)) = -(⌊‘-(3 / 2)) |
17 | negeq 10677 | . . . . 5 ⊢ ((⌊‘-(3 / 2)) = -2 → -(⌊‘-(3 / 2)) = --2) | |
18 | 2cn 11514 | . . . . . 6 ⊢ 2 ∈ ℂ | |
19 | 18 | negnegi 10756 | . . . . 5 ⊢ --2 = 2 |
20 | 17, 19 | syl6eq 2825 | . . . 4 ⊢ ((⌊‘-(3 / 2)) = -2 → -(⌊‘-(3 / 2)) = 2) |
21 | 16, 20 | syl5eq 2821 | . . 3 ⊢ ((⌊‘-(3 / 2)) = -2 → (⌈‘(3 / 2)) = 2) |
22 | 14, 21 | anim12ci 605 | . 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 387 = wceq 1508 ∈ wcel 2051 ‘cfv 6186 (class class class)co 6975 ℝcr 10333 1c1 10335 -cneg 10670 / cdiv 11097 2c2 11494 3c3 11495 ⌊cfl 12974 ⌈cceil 12975 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1759 ax-4 1773 ax-5 1870 ax-6 1929 ax-7 1966 ax-8 2053 ax-9 2060 ax-10 2080 ax-11 2094 ax-12 2107 ax-13 2302 ax-ext 2745 ax-sep 5057 ax-nul 5064 ax-pow 5116 ax-pr 5183 ax-un 7278 ax-cnex 10390 ax-resscn 10391 ax-1cn 10392 ax-icn 10393 ax-addcl 10394 ax-addrcl 10395 ax-mulcl 10396 ax-mulrcl 10397 ax-mulcom 10398 ax-addass 10399 ax-mulass 10400 ax-distr 10401 ax-i2m1 10402 ax-1ne0 10403 ax-1rid 10404 ax-rnegex 10405 ax-rrecex 10406 ax-cnre 10407 ax-pre-lttri 10408 ax-pre-lttrn 10409 ax-pre-ltadd 10410 ax-pre-mulgt0 10411 ax-pre-sup 10412 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 835 df-3or 1070 df-3an 1071 df-tru 1511 df-ex 1744 df-nf 1748 df-sb 2017 df-mo 2548 df-eu 2585 df-clab 2754 df-cleq 2766 df-clel 2841 df-nfc 2913 df-ne 2963 df-nel 3069 df-ral 3088 df-rex 3089 df-reu 3090 df-rmo 3091 df-rab 3092 df-v 3412 df-sbc 3677 df-csb 3782 df-dif 3827 df-un 3829 df-in 3831 df-ss 3838 df-pss 3840 df-nul 4174 df-if 4346 df-pw 4419 df-sn 4437 df-pr 4439 df-tp 4441 df-op 4443 df-uni 4710 df-iun 4791 df-br 4927 df-opab 4989 df-mpt 5006 df-tr 5028 df-id 5309 df-eprel 5314 df-po 5323 df-so 5324 df-fr 5363 df-we 5365 df-xp 5410 df-rel 5411 df-cnv 5412 df-co 5413 df-dm 5414 df-rn 5415 df-res 5416 df-ima 5417 df-pred 5984 df-ord 6030 df-on 6031 df-lim 6032 df-suc 6033 df-iota 6150 df-fun 6188 df-fn 6189 df-f 6190 df-f1 6191 df-fo 6192 df-f1o 6193 df-fv 6194 df-riota 6936 df-ov 6978 df-oprab 6979 df-mpo 6980 df-om 7396 df-wrecs 7749 df-recs 7811 df-rdg 7849 df-er 8088 df-en 8306 df-dom 8307 df-sdom 8308 df-sup 8700 df-inf 8701 df-pnf 10475 df-mnf 10476 df-xr 10477 df-ltxr 10478 df-le 10479 df-sub 10671 df-neg 10672 df-div 11098 df-nn 11439 df-2 11502 df-3 11503 df-4 11504 df-n0 11707 df-z 11793 df-uz 12058 df-fl 12976 df-ceil 12977 |
This theorem is referenced by: (None) |
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