Theorem flval 10707

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 4134	. . . 4 |- ( y = A -> ( x <_ y <-> x <_ A ) )
2		breq1 4133	. . . 4 |- ( y = A -> ( y < ( x + 1 ) <-> A < ( x + 1 ) ) )
3	1, 2	anbi12d 477	. . 3 |- ( y = A -> ( ( x <_ y /\ y < ( x + 1 ) ) <-> ( x <_ A /\ A < ( x + 1 ) ) ) )
4	3	riotabidv 6040	. 2 |- ( y = A -> ( iota_ x e. ZZ ( x <_ y /\ y < ( x + 1 ) ) ) = ( iota_ x e. ZZ ( x <_ A /\ A < ( x + 1 ) ) ) )
5		df-fl 10705	. 2 |- |_ = ( y e. RR |-> ( iota_ x e. ZZ ( x <_ y /\ y < ( x + 1 ) ) ) )
6		zex 9653	. . 3 |- ZZ e. _V
7		riotaexg 6042	. . 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 5785	1 |- ( A e. RR -> ( |_ ` A ) = ( iota_ x e. ZZ ( x <_ A /\ A < ( x + 1 ) ) ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   = wceq 1402   e. wcel 2209   _Vcvv 2821   class class class wbr 4130   ` cfv 5377   iota_crio 6037  (class class class)co 6085   RRcr 8178   1c1 8180   + caddc 8182   < clt 8360   <_ cle 8361   ZZcz 9644   |_cfl 10703

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4249  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-cnex 8270  ax-resscn 8271

This proof depends on definitions:  df-bi 117  df-3or 1010  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-rab 2537  df-v 2823  df-sbc 3052  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-br 4131  df-opab 4193  df-mpt 4194  df-id 4438  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-iota 5337  df-fun 5379  df-fv 5385  df-riota 6038  df-ov 6088  df-neg 8500  df-z 9645  df-fl 10705

This theorem is used by:  flqcl  10708  flapcl  10710  flqlelt  10711  flqbi  10725