Theorem flval 10636

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 4115	. . . 4 |- ( y = A -> ( x <_ y <-> x <_ A ) )
2		breq1 4114	. . . 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 6007	. 2 |- ( y = A -> ( iota_ x e. ZZ ( x <_ y /\ y < ( x + 1 ) ) ) = ( iota_ x e. ZZ ( x <_ A /\ A < ( x + 1 ) ) ) )
5		df-fl 10634	. 2 |- |_ = ( y e. RR |-> ( iota_ x e. ZZ ( x <_ y /\ y < ( x + 1 ) ) ) )
6		zex 9588	. . 3 |- ZZ e. _V
7		riotaexg 6009	. . 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 5759	1 |- ( A e. RR -> ( |_ ` A ) = ( iota_ x e. ZZ ( x <_ A /\ A < ( x + 1 ) ) ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1398   e. wcel 2205   _Vcvv 2815   class class class wbr 4111   ` cfv 5354   iota_crio 6004  (class class class)co 6052   RRcr 8128   1c1 8130   + caddc 8132   < clt 8310   <_ cle 8311   ZZcz 9579   |_cfl 10632

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 2207  ax-14 2208  ax-ext 2216  ax-sep 4230  ax-pow 4289  ax-pr 4324  ax-un 4556  ax-cnex 8220  ax-resscn 8221

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 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-rab 2531  df-v 2817  df-sbc 3045  df-un 3217  df-in 3219  df-ss 3226  df-pw 3673  df-sn 3697  df-pr 3698  df-op 3700  df-uni 3917  df-br 4112  df-opab 4174  df-mpt 4175  df-id 4416  df-xp 4757  df-rel 4758  df-cnv 4759  df-co 4760  df-dm 4761  df-iota 5314  df-fun 5356  df-fv 5362  df-riota 6005  df-ov 6055  df-neg 8449  df-z 9580  df-fl 10634

This theorem is referenced by:  flqcl  10637  flapcl  10639  flqlelt  10640  flqbi  10654