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Theorem flval 9644
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 3841 . . . 4  |-  ( y  =  A  ->  (
x  <_  y  <->  x  <_  A ) )
2 breq1 3840 . . . 4  |-  ( y  =  A  ->  (
y  <  ( x  +  1 )  <->  A  <  ( x  +  1 ) ) )
31, 2anbi12d 457 . . 3  |-  ( y  =  A  ->  (
( x  <_  y  /\  y  <  ( x  +  1 ) )  <-> 
( x  <_  A  /\  A  <  ( x  +  1 ) ) ) )
43riotabidv 5592 . 2  |-  ( y  =  A  ->  ( iota_ x  e.  ZZ  (
x  <_  y  /\  y  <  ( x  + 
1 ) ) )  =  ( iota_ x  e.  ZZ  ( x  <_  A  /\  A  <  (
x  +  1 ) ) ) )
5 df-fl 9642 . 2  |-  |_  =  ( y  e.  RR  |->  ( iota_ x  e.  ZZ  ( x  <_  y  /\  y  <  ( x  + 
1 ) ) ) )
6 zex 8729 . . 3  |-  ZZ  e.  _V
7 riotaexg 5594 . . 3  |-  ( ZZ  e.  _V  ->  ( iota_ x  e.  ZZ  (
x  <_  y  /\  y  <  ( x  + 
1 ) ) )  e.  _V )
86, 7ax-mp 7 . 2  |-  ( iota_ x  e.  ZZ  ( x  <_  y  /\  y  <  ( x  +  1 ) ) )  e. 
_V
94, 5, 8fvmpt3i 5368 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 102    = wceq 1289    e. wcel 1438   _Vcvv 2619   class class class wbr 3837   ` cfv 5002   iota_crio 5589  (class class class)co 5634   RRcr 7328   1c1 7330    + caddc 7332    < clt 7501    <_ cle 7502   ZZcz 8720   |_cfl 9640
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-pow 4001  ax-pr 4027  ax-un 4251  ax-cnex 7415  ax-resscn 7416
This theorem depends on definitions:  df-bi 115  df-3or 925  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-rab 2368  df-v 2621  df-sbc 2839  df-un 3001  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-br 3838  df-opab 3892  df-mpt 3893  df-id 4111  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-iota 4967  df-fun 5004  df-fv 5010  df-riota 5590  df-ov 5637  df-neg 7635  df-z 8721  df-fl 9642
This theorem is referenced by:  flqcl  9645  flapcl  9647  flqlelt  9648  flqbi  9662
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