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Theorem flval 11890
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 4446 . . . 4  |-  ( y  =  A  ->  (
x  <_  y  <->  x  <_  A ) )
2 breq1 4445 . . . 4  |-  ( y  =  A  ->  (
y  <  ( x  +  1 )  <->  A  <  ( x  +  1 ) ) )
31, 2anbi12d 710 . . 3  |-  ( y  =  A  ->  (
( x  <_  y  /\  y  <  ( x  +  1 ) )  <-> 
( x  <_  A  /\  A  <  ( x  +  1 ) ) ) )
43riotabidv 6240 . 2  |-  ( y  =  A  ->  ( iota_ x  e.  ZZ  (
x  <_  y  /\  y  <  ( x  + 
1 ) ) )  =  ( iota_ x  e.  ZZ  ( x  <_  A  /\  A  <  (
x  +  1 ) ) ) )
5 df-fl 11888 . 2  |-  |_  =  ( y  e.  RR  |->  ( iota_ x  e.  ZZ  ( x  <_  y  /\  y  <  ( x  + 
1 ) ) ) )
6 riotaex 6242 . 2  |-  ( iota_ x  e.  ZZ  ( x  <_  A  /\  A  <  ( x  +  1 ) ) )  e. 
_V
74, 5, 6fvmpt 5943 1  |-  ( A  e.  RR  ->  ( |_ `  A )  =  ( iota_ x  e.  ZZ  ( x  <_  A  /\  A  <  ( x  + 
1 ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1374    e. wcel 1762   class class class wbr 4442   ` cfv 5581   iota_crio 6237  (class class class)co 6277   RRcr 9482   1c1 9484    + caddc 9486    < clt 9619    <_ cle 9620   ZZcz 10855   |_cfl 11886
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-sep 4563  ax-nul 4571  ax-pr 4681
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-ral 2814  df-rex 2815  df-rab 2818  df-v 3110  df-sbc 3327  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3781  df-if 3935  df-sn 4023  df-pr 4025  df-op 4029  df-uni 4241  df-br 4443  df-opab 4501  df-mpt 4502  df-id 4790  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-iota 5544  df-fun 5583  df-fv 5589  df-riota 6238  df-fl 11888
This theorem is referenced by:  flcl  11891  fllelt  11893  flbi  11910  dfceil2  11926  ltflcei  29608  lxflflp1  29610
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