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Theorem ceilval 13749
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 11398 . . . 4 (𝑥 = 𝐴 → -𝑥 = -𝐴)
21fveq2d 6847 . . 3 (𝑥 = 𝐴 → (⌊‘-𝑥) = (⌊‘-𝐴))
32negeqd 11400 . 2 (𝑥 = 𝐴 → -(⌊‘-𝑥) = -(⌊‘-𝐴))
4 df-ceil 13704 . 2 ⌈ = (𝑥 ∈ ℝ ↦ -(⌊‘-𝑥))
5 negex 11404 . 2 -(⌊‘-𝐴) ∈ V
63, 4, 5fvmpt 6949 1 (𝐴 ∈ ℝ → (⌈‘𝐴) = -(⌊‘-𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1542  wcel 2107  cfv 6497  cr 11055  -cneg 11391  cfl 13701  cceil 13702
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pr 5385
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-iota 6449  df-fun 6499  df-fv 6505  df-ov 7361  df-neg 11393  df-ceil 13704
This theorem is referenced by:  ceilcl  13753  ceilge  13756  ceilm1lt  13759  ceille  13761  ceilid  13762  ex-ceil  29434
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