Theorem flval 10379

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 4038	. . . 4 |- ( y = A -> ( x <_ y <-> x <_ A ) )
2		breq1 4037	. . . 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 5882	. 2 |- ( y = A -> ( iota_ x e. ZZ ( x <_ y /\ y < ( x + 1 ) ) ) = ( iota_ x e. ZZ ( x <_ A /\ A < ( x + 1 ) ) ) )
5		df-fl 10377	. 2 |- |_ = ( y e. RR |-> ( iota_ x e. ZZ ( x <_ y /\ y < ( x + 1 ) ) ) )
6		zex 9352	. . 3 |- ZZ e. _V
7		riotaexg 5884	. . 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 5644	1 |- ( A e. RR -> ( |_ ` A ) = ( iota_ x e. ZZ ( x <_ A /\ A < ( x + 1 ) ) ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1364   e. wcel 2167   _Vcvv 2763   class class class wbr 4034   ` cfv 5259   iota_crio 5879  (class class class)co 5925   RRcr 7895   1c1 7897   + caddc 7899   < clt 8078   <_ cle 8079   ZZcz 9343   |_cfl 10375

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4152  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-cnex 7987  ax-resscn 7988

This theorem depends on definitions:  df-bi 117  df-3or 981  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-rab 2484  df-v 2765  df-sbc 2990  df-un 3161  df-in 3163  df-ss 3170  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-br 4035  df-opab 4096  df-mpt 4097  df-id 4329  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-iota 5220  df-fun 5261  df-fv 5267  df-riota 5880  df-ov 5928  df-neg 8217  df-z 9344  df-fl 10377

This theorem is referenced by:  flqcl  10380  flapcl  10382  flqlelt  10383  flqbi  10397