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Theorem ceilval 12501
Description: The value of the ceiling function. (Contributed by David A. Wheeler, 19-May-2015.)
Assertion
Ref Expression
ceilval (𝐴 ∈ ℝ → (⌈‘𝐴) = -(⌊‘-𝐴))

Proof of Theorem ceilval
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 negeq 10152 . . . 4 (𝑥 = 𝐴 → -𝑥 = -𝐴)
21fveq2d 6107 . . 3 (𝑥 = 𝐴 → (⌊‘-𝑥) = (⌊‘-𝐴))
32negeqd 10154 . 2 (𝑥 = 𝐴 → -(⌊‘-𝑥) = -(⌊‘-𝐴))
4 df-ceil 12456 . 2 ⌈ = (𝑥 ∈ ℝ ↦ -(⌊‘-𝑥))
5 negex 10158 . 2 -(⌊‘-𝐴) ∈ V
63, 4, 5fvmpt 6191 1 (𝐴 ∈ ℝ → (⌈‘𝐴) = -(⌊‘-𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1475  wcel 1977  cfv 5804  cr 9814  -cneg 10146  cfl 12453  cceil 12454
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-iota 5768  df-fun 5806  df-fv 5812  df-ov 6552  df-neg 10148  df-ceil 12456
This theorem is referenced by:  ceilcl  12505  ceilge  12507  ceilm1lt  12509  ceille  12511  ceilid  12512  ex-ceil  26697
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