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Theorem ceilval 12068
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.
StepHypRef Expression
1 negeq 9869 . . . 4  |-  ( x  =  A  ->  -u x  =  -u A )
21fveq2d 5883 . . 3  |-  ( x  =  A  ->  ( |_ `  -u x )  =  ( |_ `  -u A
) )
32negeqd 9871 . 2  |-  ( x  =  A  ->  -u ( |_ `  -u x )  = 
-u ( |_ `  -u A ) )
4 df-ceil 12030 . 2  |- =  ( x  e.  RR  |->  -u ( |_ `  -u x
) )
5 negex 9875 . 2  |-  -u ( |_ `  -u A )  e. 
_V
63, 4, 5fvmpt 5962 1  |-  ( A  e.  RR  ->  ( `  A )  =  -u ( |_ `  -u A
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1438    e. wcel 1869   ` cfv 5599   RRcr 9540   -ucneg 9863   |_cfl 12027  ⌈cceil 12028
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1666  ax-4 1679  ax-5 1749  ax-6 1795  ax-7 1840  ax-9 1873  ax-10 1888  ax-11 1893  ax-12 1906  ax-13 2054  ax-ext 2401  ax-sep 4544  ax-nul 4553  ax-pr 4658
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 985  df-tru 1441  df-ex 1661  df-nf 1665  df-sb 1788  df-eu 2270  df-mo 2271  df-clab 2409  df-cleq 2415  df-clel 2418  df-nfc 2573  df-ne 2621  df-ral 2781  df-rex 2782  df-rab 2785  df-v 3084  df-sbc 3301  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3763  df-if 3911  df-sn 3998  df-pr 4000  df-op 4004  df-uni 4218  df-br 4422  df-opab 4481  df-mpt 4482  df-id 4766  df-xp 4857  df-rel 4858  df-cnv 4859  df-co 4860  df-dm 4861  df-iota 5563  df-fun 5601  df-fv 5607  df-ov 6306  df-neg 9865  df-ceil 12030
This theorem is referenced by:  ceilcl  12072  ceilge  12074  ceilm1lt  12076  ceille  12078  ceilid  12079
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