MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  flval Unicode version

Theorem flval 11158
Description: Value of the floor (greatest integer) function. The floor of  A is the (unique) integer less than or equal to  A whose successor is strictly greater than  A. (Contributed by NM, 14-Nov-2004.) (Revised by Mario Carneiro, 2-Nov-2013.)
Assertion
Ref Expression
flval  |-  ( A  e.  RR  ->  ( |_ `  A )  =  ( iota_ x  e.  ZZ ( x  <_  A  /\  A  <  ( x  + 
1 ) ) ) )
Distinct variable group:    x, A

Proof of Theorem flval
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 breq2 4176 . . . 4  |-  ( y  =  A  ->  (
x  <_  y  <->  x  <_  A ) )
2 breq1 4175 . . . 4  |-  ( y  =  A  ->  (
y  <  ( x  +  1 )  <->  A  <  ( x  +  1 ) ) )
31, 2anbi12d 692 . . 3  |-  ( y  =  A  ->  (
( x  <_  y  /\  y  <  ( x  +  1 ) )  <-> 
( x  <_  A  /\  A  <  ( x  +  1 ) ) ) )
43riotabidv 6510 . 2  |-  ( y  =  A  ->  ( iota_ x  e.  ZZ ( x  <_  y  /\  y  <  ( x  + 
1 ) ) )  =  ( iota_ x  e.  ZZ ( x  <_  A  /\  A  <  (
x  +  1 ) ) ) )
5 df-fl 11157 . 2  |-  |_  =  ( y  e.  RR  |->  ( iota_ x  e.  ZZ ( x  <_  y  /\  y  <  ( x  + 
1 ) ) ) )
6 riotaex 6512 . 2  |-  ( iota_ x  e.  ZZ ( x  <_  A  /\  A  <  ( x  +  1 ) ) )  e. 
_V
74, 5, 6fvmpt 5765 1  |-  ( A  e.  RR  ->  ( |_ `  A )  =  ( iota_ x  e.  ZZ ( x  <_  A  /\  A  <  ( x  + 
1 ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1721   class class class wbr 4172   ` cfv 5413  (class class class)co 6040   iota_crio 6501   RRcr 8945   1c1 8947    + caddc 8949    < clt 9076    <_ cle 9077   ZZcz 10238   |_cfl 11156
This theorem is referenced by:  flcl  11159  fllelt  11161  flbi  11178  ltflcei  26140  lxflflp1  26142
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pr 4363
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-iota 5377  df-fun 5415  df-fv 5421  df-riota 6508  df-fl 11157
  Copyright terms: Public domain W3C validator