Theorem flval 10632

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.

Step	Hyp	Ref	Expression
1		breq2 4113	. . . 4 |- ( y = A -> ( x <_ y <-> x <_ A ) )
2		breq1 4112	. . . 4 |- ( y = A -> ( y < ( x + 1 ) <-> A < ( x + 1 ) ) )
3	1, 2	anbi12d 473	. . 3 |- ( y = A -> ( ( x <_ y /\ y < ( x + 1 ) ) <-> ( x <_ A /\ A < ( x + 1 ) ) ) )
4	3	riotabidv 6005	. 2 |- ( y = A -> ( iota_ x e. ZZ ( x <_ y /\ y < ( x + 1 ) ) ) = ( iota_ x e. ZZ ( x <_ A /\ A < ( x + 1 ) ) ) )
5		df-fl 10630	. 2 |- |_ = ( y e. RR |-> ( iota_ x e. ZZ ( x <_ y /\ y < ( x + 1 ) ) ) )
6		zex 9586	. . 3 |- ZZ e. _V
7		riotaexg 6007	. . 3 |- ( ZZ e. _V -> ( iota_ x e. ZZ ( x <_ y /\ y < ( x + 1 ) ) ) e. _V )
8	6, 7	ax-mp 5	. 2 |- ( iota_ x e. ZZ ( x <_ y /\ y < ( x + 1 ) ) ) e. _V
9	4, 5, 8	fvmpt3i 5757	1 |- ( A e. RR -> ( |_ ` A ) = ( iota_ x e. ZZ ( x <_ A /\ A < ( x + 1 ) ) ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1398   e. wcel 2203   _Vcvv 2813   class class class wbr 4109   ` cfv 5352   iota_crio 6002  (class class class)co 6050   RRcr 8126   1c1 8128   + caddc 8130   < clt 8308   <_ cle 8309   ZZcz 9577   |_cfl 10628

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-cnex 8218  ax-resscn 8219

This theorem depends on definitions:  df-bi 117  df-3or 1006  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-rab 2529  df-v 2815  df-sbc 3043  df-un 3215  df-in 3217  df-ss 3224  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-br 4110  df-opab 4172  df-mpt 4173  df-id 4414  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-iota 5312  df-fun 5354  df-fv 5360  df-riota 6003  df-ov 6053  df-neg 8447  df-z 9578  df-fl 10630

This theorem is referenced by:  flqcl  10633  flapcl  10635  flqlelt  10636  flqbi  10650