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Theorem flval 10245
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 4004 . . . 4  |-  ( y  =  A  ->  (
x  <_  y  <->  x  <_  A ) )
2 breq1 4003 . . . 4  |-  ( y  =  A  ->  (
y  <  ( x  +  1 )  <->  A  <  ( x  +  1 ) ) )
31, 2anbi12d 473 . . 3  |-  ( y  =  A  ->  (
( x  <_  y  /\  y  <  ( x  +  1 ) )  <-> 
( x  <_  A  /\  A  <  ( x  +  1 ) ) ) )
43riotabidv 5826 . 2  |-  ( y  =  A  ->  ( iota_ x  e.  ZZ  (
x  <_  y  /\  y  <  ( x  + 
1 ) ) )  =  ( iota_ x  e.  ZZ  ( x  <_  A  /\  A  <  (
x  +  1 ) ) ) )
5 df-fl 10243 . 2  |-  |_  =  ( y  e.  RR  |->  ( iota_ x  e.  ZZ  ( x  <_  y  /\  y  <  ( x  + 
1 ) ) ) )
6 zex 9238 . . 3  |-  ZZ  e.  _V
7 riotaexg 5828 . . 3  |-  ( ZZ  e.  _V  ->  ( iota_ x  e.  ZZ  (
x  <_  y  /\  y  <  ( x  + 
1 ) ) )  e.  _V )
86, 7ax-mp 5 . 2  |-  ( iota_ x  e.  ZZ  ( x  <_  y  /\  y  <  ( x  +  1 ) ) )  e. 
_V
94, 5, 8fvmpt3i 5591 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 104    = wceq 1353    e. wcel 2148   _Vcvv 2737   class class class wbr 4000   ` cfv 5211   iota_crio 5823  (class class class)co 5868   RRcr 7788   1c1 7790    + caddc 7792    < clt 7969    <_ cle 7970   ZZcz 9229   |_cfl 10241
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-pow 4171  ax-pr 4205  ax-un 4429  ax-cnex 7880  ax-resscn 7881
This theorem depends on definitions:  df-bi 117  df-3or 979  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2739  df-sbc 2963  df-un 3133  df-in 3135  df-ss 3142  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-br 4001  df-opab 4062  df-mpt 4063  df-id 4289  df-xp 4628  df-rel 4629  df-cnv 4630  df-co 4631  df-dm 4632  df-iota 5173  df-fun 5213  df-fv 5219  df-riota 5824  df-ov 5871  df-neg 8108  df-z 9230  df-fl 10243
This theorem is referenced by:  flqcl  10246  flapcl  10248  flqlelt  10249  flqbi  10263
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