Theorem ceilval 13855

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 11474	. . . 4 ⊢ (𝑥 = 𝐴 → -𝑥 = -𝐴)
2	1	fveq2d 6880	. . 3 ⊢ (𝑥 = 𝐴 → (⌊‘-𝑥) = (⌊‘-𝐴))
3	2	negeqd 11476	. 2 ⊢ (𝑥 = 𝐴 → -(⌊‘-𝑥) = -(⌊‘-𝐴))
4		df-ceil 13810	. 2 ⊢ ⌈ = (𝑥 ∈ ℝ ↦ -(⌊‘-𝑥))
5		negex 11480	. 2 ⊢ -(⌊‘-𝐴) ∈ V
6	3, 4, 5	fvmpt 6986	1 ⊢ (𝐴 ∈ ℝ → (⌈‘𝐴) = -(⌊‘-𝐴))

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2108  ‘cfv 6531  ℝcr 11128  -cneg 11467  ⌊cfl 13807  ⌈cceil 13808

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 2707  ax-sep 5266  ax-nul 5276  ax-pr 5402

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3416  df-v 3461  df-dif 3929  df-un 3931  df-ss 3943  df-nul 4309  df-if 4501  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-br 5120  df-opab 5182  df-mpt 5202  df-id 5548  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-iota 6484  df-fun 6533  df-fv 6539  df-ov 7408  df-neg 11469  df-ceil 13810

This theorem is referenced by:  ceilcl  13859  ceilge  13862  ceilm1lt  13865  ceille  13867  ceilid  13868  ex-ceil  30429  ceilbi  47362  ceildivmod  47368