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Theorem ceilingval 28528
Description: The value of the ceiling function. (Contributed by David A. Wheeler, 19-May-2015.)
Assertion
Ref Expression
ceilingval  |-  ( A  e.  RR  ->  ( `  A )  =  -u ( |_ `  -u A
) )

Proof of Theorem ceilingval
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 negeq 9298 . . . 4  |-  ( x  =  A  ->  -u x  =  -u A )
21fveq2d 5732 . . 3  |-  ( x  =  A  ->  ( |_ `  -u x )  =  ( |_ `  -u A
) )
32negeqd 9300 . 2  |-  ( x  =  A  ->  -u ( |_ `  -u x )  = 
-u ( |_ `  -u A ) )
4 df-ceiling 28527 . 2  |- =  ( x  e.  RR  |->  -u ( |_ `  -u x
) )
5 negex 9304 . 2  |-  -u ( |_ `  -u A )  e. 
_V
63, 4, 5fvmpt 5806 1  |-  ( A  e.  RR  ->  ( `  A )  =  -u ( |_ `  -u A
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1652    e. wcel 1725   ` cfv 5454   RRcr 8989   -ucneg 9292   |_cfl 11201  ⌈ccei 28526
This theorem is referenced by:  ceilingcl  28529
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pr 4403
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-iota 5418  df-fun 5456  df-fv 5462  df-ov 6084  df-neg 9294  df-ceiling 28527
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