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Theorem flval 9116
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 3768 . . . 4  |-  ( y  =  A  ->  (
x  <_  y  <->  x  <_  A ) )
2 breq1 3767 . . . 4  |-  ( y  =  A  ->  (
y  <  ( x  +  1 )  <->  A  <  ( x  +  1 ) ) )
31, 2anbi12d 442 . . 3  |-  ( y  =  A  ->  (
( x  <_  y  /\  y  <  ( x  +  1 ) )  <-> 
( x  <_  A  /\  A  <  ( x  +  1 ) ) ) )
43riotabidv 5470 . 2  |-  ( y  =  A  ->  ( iota_ x  e.  ZZ  (
x  <_  y  /\  y  <  ( x  + 
1 ) ) )  =  ( iota_ x  e.  ZZ  ( x  <_  A  /\  A  <  (
x  +  1 ) ) ) )
5 df-fl 9114 . 2  |-  |_  =  ( y  e.  RR  |->  ( iota_ x  e.  ZZ  ( x  <_  y  /\  y  <  ( x  + 
1 ) ) ) )
6 zex 8254 . . 3  |-  ZZ  e.  _V
7 riotaexg 5472 . . 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 5252 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 97    = wceq 1243    e. wcel 1393   _Vcvv 2557   class class class wbr 3764   ` cfv 4902   iota_crio 5467  (class class class)co 5512   RRcr 6888   1c1 6890    + caddc 6892    < clt 7060    <_ cle 7061   ZZcz 8245   |_cfl 9112
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944  ax-un 4170  ax-cnex 6975  ax-resscn 6976
This theorem depends on definitions:  df-bi 110  df-3or 886  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-rab 2315  df-v 2559  df-sbc 2765  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-br 3765  df-opab 3819  df-mpt 3820  df-id 4030  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-iota 4867  df-fun 4904  df-fv 4910  df-riota 5468  df-ov 5515  df-neg 7185  df-z 8246  df-fl 9114
This theorem is referenced by:  flqcl  9117  flqlelt  9118  flqbi  9132
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