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| Mirrors > Home > MPE Home > Th. List > ceilval | Structured version Visualization version GIF version | ||
| Description: The value of the ceiling function. (Contributed by David A. Wheeler, 19-May-2015.) |
| Ref | Expression |
|---|---|
| ceilval | ⊢ (𝐴 ∈ ℝ → (⌈‘𝐴) = -(⌊‘-𝐴)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | negeq 11500 | . . . 4 ⊢ (𝑥 = 𝐴 → -𝑥 = -𝐴) | |
| 2 | 1 | fveq2d 6910 | . . 3 ⊢ (𝑥 = 𝐴 → (⌊‘-𝑥) = (⌊‘-𝐴)) |
| 3 | 2 | negeqd 11502 | . 2 ⊢ (𝑥 = 𝐴 → -(⌊‘-𝑥) = -(⌊‘-𝐴)) |
| 4 | df-ceil 13833 | . 2 ⊢ ⌈ = (𝑥 ∈ ℝ ↦ -(⌊‘-𝑥)) | |
| 5 | negex 11506 | . 2 ⊢ -(⌊‘-𝐴) ∈ V | |
| 6 | 3, 4, 5 | fvmpt 7016 | 1 ⊢ (𝐴 ∈ ℝ → (⌈‘𝐴) = -(⌊‘-𝐴)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 ‘cfv 6561 ℝcr 11154 -cneg 11493 ⌊cfl 13830 ⌈cceil 13831 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-iota 6514 df-fun 6563 df-fv 6569 df-ov 7434 df-neg 11495 df-ceil 13833 |
| This theorem is referenced by: ceilcl 13882 ceilge 13885 ceilm1lt 13888 ceille 13890 ceilid 13891 ex-ceil 30467 ceildivmod 47341 |
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