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Theorem ceilingval 28255
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 9044 . . . 4  |-  ( x  =  A  ->  -u x  =  -u A )
21fveq2d 5529 . . 3  |-  ( x  =  A  ->  ( |_ `  -u x )  =  ( |_ `  -u A
) )
32negeqd 9046 . 2  |-  ( x  =  A  ->  -u ( |_ `  -u x )  = 
-u ( |_ `  -u A ) )
4 df-ceiling 28254 . 2  |- =  ( x  e.  RR  |->  -u ( |_ `  -u x
) )
5 negex 9050 . 2  |-  -u ( |_ `  -u A )  e. 
_V
63, 4, 5fvmpt 5602 1  |-  ( A  e.  RR  ->  ( `  A )  =  -u ( |_ `  -u A
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623    e. wcel 1684   ` cfv 5255   RRcr 8736   -ucneg 9038   |_cfl 10924  ⌈ccei 28253
This theorem is referenced by:  ceilingcl  28256
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-iota 5219  df-fun 5257  df-fv 5263  df-ov 5861  df-neg 9040  df-ceiling 28254
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