Theorem flval 10341

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 4033	. . . 4 |- ( y = A -> ( x <_ y <-> x <_ A ) )
2		breq1 4032	. . . 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 5875	. 2 |- ( y = A -> ( iota_ x e. ZZ ( x <_ y /\ y < ( x + 1 ) ) ) = ( iota_ x e. ZZ ( x <_ A /\ A < ( x + 1 ) ) ) )
5		df-fl 10339	. 2 |- |_ = ( y e. RR |-> ( iota_ x e. ZZ ( x <_ y /\ y < ( x + 1 ) ) ) )
6		zex 9326	. . 3 |- ZZ e. _V
7		riotaexg 5877	. . 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 5637	1 |- ( A e. RR -> ( |_ ` A ) = ( iota_ x e. ZZ ( x <_ A /\ A < ( x + 1 ) ) ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1364   e. wcel 2164   _Vcvv 2760   class class class wbr 4029   ` cfv 5254   iota_crio 5872  (class class class)co 5918   RRcr 7871   1c1 7873   + caddc 7875   < clt 8054   <_ cle 8055   ZZcz 9317   |_cfl 10337

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-cnex 7963  ax-resscn 7964

This theorem depends on definitions:  df-bi 117  df-3or 981  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-rab 2481  df-v 2762  df-sbc 2986  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-br 4030  df-opab 4091  df-mpt 4092  df-id 4324  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-iota 5215  df-fun 5256  df-fv 5262  df-riota 5873  df-ov 5921  df-neg 8193  df-z 9318  df-fl 10339

This theorem is referenced by:  flqcl  10342  flapcl  10344  flqlelt  10345  flqbi  10359