Theorem ceilval 13878

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.

Step	Hyp	Ref	Expression
1		negeq 11500	. . . 4 ⊢ (𝑥 = 𝐴 → -𝑥 = -𝐴)
2	1	fveq2d 6910	. . 3 ⊢ (𝑥 = 𝐴 → (⌊‘-𝑥) = (⌊‘-𝐴))
3	2	negeqd 11502	. 2 ⊢ (𝑥 = 𝐴 → -(⌊‘-𝑥) = -(⌊‘-𝐴))
4		df-ceil 13833	. 2 ⊢ ⌈ = (𝑥 ∈ ℝ ↦ -(⌊‘-𝑥))
5		negex 11506	. 2 ⊢ -(⌊‘-𝐴) ∈ V
6	3, 4, 5	fvmpt 7016	1 ⊢ (𝐴 ∈ ℝ → (⌈‘𝐴) = -(⌊‘-𝐴))

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2108  ‘cfv 6561  ℝcr 11154  -cneg 11493  ⌊cfl 13830  ⌈cceil 13831

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-iota 6514  df-fun 6563  df-fv 6569  df-ov 7434  df-neg 11495  df-ceil 13833

This theorem is referenced by:  ceilcl  13882  ceilge  13885  ceilm1lt  13888  ceille  13890  ceilid  13891  ex-ceil  30467  ceildivmod  47341