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Theorem ceilval 13799
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 11448 . . . 4 (𝑥 = 𝐴 → -𝑥 = -𝐴)
21fveq2d 6892 . . 3 (𝑥 = 𝐴 → (⌊‘-𝑥) = (⌊‘-𝐴))
32negeqd 11450 . 2 (𝑥 = 𝐴 → -(⌊‘-𝑥) = -(⌊‘-𝐴))
4 df-ceil 13754 . 2 ⌈ = (𝑥 ∈ ℝ ↦ -(⌊‘-𝑥))
5 negex 11454 . 2 -(⌊‘-𝐴) ∈ V
63, 4, 5fvmpt 6994 1 (𝐴 ∈ ℝ → (⌈‘𝐴) = -(⌊‘-𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1542  wcel 2107  cfv 6540  cr 11105  -cneg 11441  cfl 13751  cceil 13752
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 5298  ax-nul 5305  ax-pr 5426
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 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-iota 6492  df-fun 6542  df-fv 6548  df-ov 7407  df-neg 11443  df-ceil 13754
This theorem is referenced by:  ceilcl  13803  ceilge  13806  ceilm1lt  13809  ceille  13811  ceilid  13812  ex-ceil  29681
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