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Theorem flval 11208
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 4219 . . . 4  |-  ( y  =  A  ->  (
x  <_  y  <->  x  <_  A ) )
2 breq1 4218 . . . 4  |-  ( y  =  A  ->  (
y  <  ( x  +  1 )  <->  A  <  ( x  +  1 ) ) )
31, 2anbi12d 693 . . 3  |-  ( y  =  A  ->  (
( x  <_  y  /\  y  <  ( x  +  1 ) )  <-> 
( x  <_  A  /\  A  <  ( x  +  1 ) ) ) )
43riotabidv 6554 . 2  |-  ( y  =  A  ->  ( iota_ x  e.  ZZ ( x  <_  y  /\  y  <  ( x  + 
1 ) ) )  =  ( iota_ x  e.  ZZ ( x  <_  A  /\  A  <  (
x  +  1 ) ) ) )
5 df-fl 11207 . 2  |-  |_  =  ( y  e.  RR  |->  ( iota_ x  e.  ZZ ( x  <_  y  /\  y  <  ( x  + 
1 ) ) ) )
6 riotaex 6556 . 2  |-  ( iota_ x  e.  ZZ ( x  <_  A  /\  A  <  ( x  +  1 ) ) )  e. 
_V
74, 5, 6fvmpt 5809 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 360    = wceq 1653    e. wcel 1726   class class class wbr 4215   ` cfv 5457  (class class class)co 6084   iota_crio 6545   RRcr 8994   1c1 8996    + caddc 8998    < clt 9125    <_ cle 9126   ZZcz 10287   |_cfl 11206
This theorem is referenced by:  flcl  11209  fllelt  11211  flbi  11228  ltflcei  26247  lxflflp1  26249
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pr 4406
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-iota 5421  df-fun 5459  df-fv 5465  df-riota 6552  df-fl 11207
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