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Theorem ceilval 12595
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 10233 . . . 4 (𝑥 = 𝐴 → -𝑥 = -𝐴)
21fveq2d 6162 . . 3 (𝑥 = 𝐴 → (⌊‘-𝑥) = (⌊‘-𝐴))
32negeqd 10235 . 2 (𝑥 = 𝐴 → -(⌊‘-𝑥) = -(⌊‘-𝐴))
4 df-ceil 12550 . 2 ⌈ = (𝑥 ∈ ℝ ↦ -(⌊‘-𝑥))
5 negex 10239 . 2 -(⌊‘-𝐴) ∈ V
63, 4, 5fvmpt 6249 1 (𝐴 ∈ ℝ → (⌈‘𝐴) = -(⌊‘-𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1480  wcel 1987  cfv 5857  cr 9895  -cneg 10227  cfl 12547  cceil 12548
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-nul 4759  ax-pr 4877
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2913  df-rex 2914  df-rab 2917  df-v 3192  df-sbc 3423  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-br 4624  df-opab 4684  df-mpt 4685  df-id 4999  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-iota 5820  df-fun 5859  df-fv 5865  df-ov 6618  df-neg 10229  df-ceil 12550
This theorem is referenced by:  ceilcl  12599  ceilge  12601  ceilm1lt  12603  ceille  12605  ceilid  12606  ex-ceil  27193
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