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Theorem ceilval 13875
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 11498 . . . 4 (𝑥 = 𝐴 → -𝑥 = -𝐴)
21fveq2d 6911 . . 3 (𝑥 = 𝐴 → (⌊‘-𝑥) = (⌊‘-𝐴))
32negeqd 11500 . 2 (𝑥 = 𝐴 → -(⌊‘-𝑥) = -(⌊‘-𝐴))
4 df-ceil 13830 . 2 ⌈ = (𝑥 ∈ ℝ ↦ -(⌊‘-𝑥))
5 negex 11504 . 2 -(⌊‘-𝐴) ∈ V
63, 4, 5fvmpt 7016 1 (𝐴 ∈ ℝ → (⌈‘𝐴) = -(⌊‘-𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1537  wcel 2106  cfv 6563  cr 11152  -cneg 11491  cfl 13827  cceil 13828
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-iota 6516  df-fun 6565  df-fv 6571  df-ov 7434  df-neg 11493  df-ceil 13830
This theorem is referenced by:  ceilcl  13879  ceilge  13882  ceilm1lt  13885  ceille  13887  ceilid  13888  ex-ceil  30477  ceildivmod  47279
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