Theorem ceilval 12639

Description: The value of the ceiling function. (Contributed by David A. Wheeler, 19-May-2015.)

Assertion

Ref	Expression
ceilval	|- ( A e. RR -> (⌈ ` A ) = -u ( |_ ` -u A ) )

Proof of Theorem ceilval

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		negeq 10273	. . . 4 |- ( x = A -> -u x = -u A )
2	1	fveq2d 6195	. . 3 |- ( x = A -> ( |_ ` -u x ) = ( |_ ` -u A ) )
3	2	negeqd 10275	. 2 |- ( x = A -> -u ( |_ ` -u x ) = -u ( |_ ` -u A ) )
4		df-ceil 12594	. 2 |- ⌈ = ( x e. RR |-> -u ( |_ ` -u x ) )
5		negex 10279	. 2 |- -u ( |_ ` -u A ) e. _V
6	3, 4, 5	fvmpt 6282	1 |- ( A e. RR -> (⌈ ` A ) = -u ( |_ ` -u A ) )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   ` cfv 5888   RRcr 9935   -ucneg 10267   |_cfl 12591  ⌈cceil 12592

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fv 5896  df-ov 6653  df-neg 10269  df-ceil 12594

This theorem is referenced by:  ceilcl  12643  ceilge  12645  ceilm1lt  12647  ceille  12649  ceilid  12650  ex-ceil  27305