Theorem elcnfn 28741

Description: Property defining a continuous functional. (Contributed by NM, 11-Feb-2006.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)

Assertion

Ref	Expression
elcnfn	|- ( T e. ContFn <-> ( T : ~H --> CC /\ A. x e. ~H A. y e. RR+ E. z e. RR+ A. w e. ~H ( ( normh ` ( w -h x ) ) < z -> ( abs ` ( ( T ` w ) - ( T ` x ) ) ) < y ) ) )

Distinct variable group:   x, w, y, z, T

Proof of Theorem elcnfn

Dummy variable t is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq1 6190	. . . . . . . . 9 |- ( t = T -> ( t ` w ) = ( T ` w ) )
2		fveq1 6190	. . . . . . . . 9 |- ( t = T -> ( t ` x ) = ( T ` x ) )
3	1, 2	oveq12d 6668	. . . . . . . 8 |- ( t = T -> ( ( t ` w ) - ( t ` x ) ) = ( ( T ` w ) - ( T ` x ) ) )
4	3	fveq2d 6195	. . . . . . 7 |- ( t = T -> ( abs ` ( ( t ` w ) - ( t ` x ) ) ) = ( abs ` ( ( T ` w ) - ( T ` x ) ) ) )
5	4	breq1d 4663	. . . . . 6 |- ( t = T -> ( ( abs ` ( ( t ` w ) - ( t ` x ) ) ) < y <-> ( abs ` ( ( T ` w ) - ( T ` x ) ) ) < y ) )
6	5	imbi2d 330	. . . . 5 |- ( t = T -> ( ( ( normh ` ( w -h x ) ) < z -> ( abs ` ( ( t ` w ) - ( t ` x ) ) ) < y ) <-> ( ( normh ` ( w -h x ) ) < z -> ( abs ` ( ( T ` w ) - ( T ` x ) ) ) < y ) ) )
7	6	rexralbidv 3058	. . . 4 |- ( t = T -> ( E. z e. RR+ A. w e. ~H ( ( normh ` ( w -h x ) ) < z -> ( abs ` ( ( t ` w ) - ( t ` x ) ) ) < y ) <-> E. z e. RR+ A. w e. ~H ( ( normh ` ( w -h x ) ) < z -> ( abs ` ( ( T ` w ) - ( T ` x ) ) ) < y ) ) )
8	7	2ralbidv 2989	. . 3 |- ( t = T -> ( A. x e. ~H A. y e. RR+ E. z e. RR+ A. w e. ~H ( ( normh ` ( w -h x ) ) < z -> ( abs ` ( ( t ` w ) - ( t ` x ) ) ) < y ) <-> A. x e. ~H A. y e. RR+ E. z e. RR+ A. w e. ~H ( ( normh ` ( w -h x ) ) < z -> ( abs ` ( ( T ` w ) - ( T ` x ) ) ) < y ) ) )
9		df-cnfn 28706	. . 3 |- ContFn = { t e. ( CC ^m ~H ) | A. x e. ~H A. y e. RR+ E. z e. RR+ A. w e. ~H ( ( normh ` ( w -h x ) ) < z -> ( abs ` ( ( t ` w ) - ( t ` x ) ) ) < y ) }
10	8, 9	elrab2 3366	. 2 |- ( T e. ContFn <-> ( T e. ( CC ^m ~H ) /\ A. x e. ~H A. y e. RR+ E. z e. RR+ A. w e. ~H ( ( normh ` ( w -h x ) ) < z -> ( abs ` ( ( T ` w ) - ( T ` x ) ) ) < y ) ) )
11		cnex 10017	. . . 4 |- CC e. _V
12		ax-hilex 27856	. . . 4 |- ~H e. _V
13	11, 12	elmap 7886	. . 3 |- ( T e. ( CC ^m ~H ) <-> T : ~H --> CC )
14	13	anbi1i 731	. 2 |- ( ( T e. ( CC ^m ~H ) /\ A. x e. ~H A. y e. RR+ E. z e. RR+ A. w e. ~H ( ( normh ` ( w -h x ) ) < z -> ( abs ` ( ( T ` w ) - ( T ` x ) ) ) < y ) ) <-> ( T : ~H --> CC /\ A. x e. ~H A. y e. RR+ E. z e. RR+ A. w e. ~H ( ( normh ` ( w -h x ) ) < z -> ( abs ` ( ( T ` w ) - ( T ` x ) ) ) < y ) ) )
15	10, 14	bitri 264	1 |- ( T e. ContFn <-> ( T : ~H --> CC /\ A. x e. ~H A. y e. RR+ E. z e. RR+ A. w e. ~H ( ( normh ` ( w -h x ) ) < z -> ( abs ` ( ( T ` w ) - ( T ` x ) ) ) < y ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   E.wrex 2913   class class class wbr 4653   -->wf 5884   ` cfv 5888  (class class class)co 6650   ^m cmap 7857   CCcc 9934   < clt 10074   - cmin 10266   RR+crp 11832   abscabs 13974   ~Hchil 27776   normhcno 27780   -h cmv 27782   ContFnccnfn 27810

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-hilex 27856

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-map 7859  df-cnfn 28706

This theorem is referenced by:  cnfnc  28789  0cnfn  28839  lnfnconi  28914