Theorem 0cnfn 28839

Description: The identically zero function is a continuous Hilbert space functional. (Contributed by NM, 7-Feb-2006.) (New usage is discouraged.)

Assertion

Ref	Expression
0cnfn	|- ( ~H X. { 0 } ) e. ContFn

Proof of Theorem 0cnfn

Dummy variables x y z w are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		0cn 10032	. . 3 |- 0 e. CC
2	1	fconst6 6095	. 2 |- ( ~H X. { 0 } ) : ~H --> CC
3		1rp 11836	. . . 4 |- 1 e. RR+
4		c0ex 10034	. . . . . . . . . . . . 13 |- 0 e. _V
5	4	fvconst2 6469	. . . . . . . . . . . 12 |- ( w e. ~H -> ( ( ~H X. { 0 } ) ` w ) = 0 )
6	4	fvconst2 6469	. . . . . . . . . . . 12 |- ( x e. ~H -> ( ( ~H X. { 0 } ) ` x ) = 0 )
7	5, 6	oveqan12rd 6670	. . . . . . . . . . 11 |- ( ( x e. ~H /\ w e. ~H ) -> ( ( ( ~H X. { 0 } ) ` w ) - ( ( ~H X. { 0 } ) ` x ) ) = ( 0 - 0 ) )
8	7	adantlr 751	. . . . . . . . . 10 |- ( ( ( x e. ~H /\ y e. RR+ ) /\ w e. ~H ) -> ( ( ( ~H X. { 0 } ) ` w ) - ( ( ~H X. { 0 } ) ` x ) ) = ( 0 - 0 ) )
9		0m0e0 11130	. . . . . . . . . 10 |- ( 0 - 0 ) = 0
10	8, 9	syl6eq 2672	. . . . . . . . 9 |- ( ( ( x e. ~H /\ y e. RR+ ) /\ w e. ~H ) -> ( ( ( ~H X. { 0 } ) ` w ) - ( ( ~H X. { 0 } ) ` x ) ) = 0 )
11	10	fveq2d 6195	. . . . . . . 8 |- ( ( ( x e. ~H /\ y e. RR+ ) /\ w e. ~H ) -> ( abs ` ( ( ( ~H X. { 0 } ) ` w ) - ( ( ~H X. { 0 } ) ` x ) ) ) = ( abs ` 0 ) )
12		abs0 14025	. . . . . . . 8 |- ( abs ` 0 ) = 0
13	11, 12	syl6eq 2672	. . . . . . 7 |- ( ( ( x e. ~H /\ y e. RR+ ) /\ w e. ~H ) -> ( abs ` ( ( ( ~H X. { 0 } ) ` w ) - ( ( ~H X. { 0 } ) ` x ) ) ) = 0 )
14		rpgt0 11844	. . . . . . . 8 |- ( y e. RR+ -> 0 < y )
15	14	ad2antlr 763	. . . . . . 7 |- ( ( ( x e. ~H /\ y e. RR+ ) /\ w e. ~H ) -> 0 < y )
16	13, 15	eqbrtrd 4675	. . . . . 6 |- ( ( ( x e. ~H /\ y e. RR+ ) /\ w e. ~H ) -> ( abs ` ( ( ( ~H X. { 0 } ) ` w ) - ( ( ~H X. { 0 } ) ` x ) ) ) < y )
17	16	a1d 25	. . . . 5 |- ( ( ( x e. ~H /\ y e. RR+ ) /\ w e. ~H ) -> ( ( normh ` ( w -h x ) ) < 1 -> ( abs ` ( ( ( ~H X. { 0 } ) ` w ) - ( ( ~H X. { 0 } ) ` x ) ) ) < y ) )
18	17	ralrimiva 2966	. . . 4 |- ( ( x e. ~H /\ y e. RR+ ) -> A. w e. ~H ( ( normh ` ( w -h x ) ) < 1 -> ( abs ` ( ( ( ~H X. { 0 } ) ` w ) - ( ( ~H X. { 0 } ) ` x ) ) ) < y ) )
19		breq2 4657	. . . . . . 7 |- ( z = 1 -> ( ( normh ` ( w -h x ) ) < z <-> ( normh ` ( w -h x ) ) < 1 ) )
20	19	imbi1d 331	. . . . . 6 |- ( z = 1 -> ( ( ( normh ` ( w -h x ) ) < z -> ( abs ` ( ( ( ~H X. { 0 } ) ` w ) - ( ( ~H X. { 0 } ) ` x ) ) ) < y ) <-> ( ( normh ` ( w -h x ) ) < 1 -> ( abs ` ( ( ( ~H X. { 0 } ) ` w ) - ( ( ~H X. { 0 } ) ` x ) ) ) < y ) ) )
21	20	ralbidv 2986	. . . . 5 |- ( z = 1 -> ( A. w e. ~H ( ( normh ` ( w -h x ) ) < z -> ( abs ` ( ( ( ~H X. { 0 } ) ` w ) - ( ( ~H X. { 0 } ) ` x ) ) ) < y ) <-> A. w e. ~H ( ( normh ` ( w -h x ) ) < 1 -> ( abs ` ( ( ( ~H X. { 0 } ) ` w ) - ( ( ~H X. { 0 } ) ` x ) ) ) < y ) ) )
22	21	rspcev 3309	. . . 4 |- ( ( 1 e. RR+ /\ A. w e. ~H ( ( normh ` ( w -h x ) ) < 1 -> ( abs ` ( ( ( ~H X. { 0 } ) ` w ) - ( ( ~H X. { 0 } ) ` x ) ) ) < y ) ) -> E. z e. RR+ A. w e. ~H ( ( normh ` ( w -h x ) ) < z -> ( abs ` ( ( ( ~H X. { 0 } ) ` w ) - ( ( ~H X. { 0 } ) ` x ) ) ) < y ) )
23	3, 18, 22	sylancr 695	. . 3 |- ( ( x e. ~H /\ y e. RR+ ) -> E. z e. RR+ A. w e. ~H ( ( normh ` ( w -h x ) ) < z -> ( abs ` ( ( ( ~H X. { 0 } ) ` w ) - ( ( ~H X. { 0 } ) ` x ) ) ) < y ) )
24	23	rgen2 2975	. 2 |- A. x e. ~H A. y e. RR+ E. z e. RR+ A. w e. ~H ( ( normh ` ( w -h x ) ) < z -> ( abs ` ( ( ( ~H X. { 0 } ) ` w ) - ( ( ~H X. { 0 } ) ` x ) ) ) < y )
25		elcnfn 28741	. 2 |- ( ( ~H X. { 0 } ) e. ContFn <-> ( ( ~H X. { 0 } ) : ~H --> CC /\ A. x e. ~H A. y e. RR+ E. z e. RR+ A. w e. ~H ( ( normh ` ( w -h x ) ) < z -> ( abs ` ( ( ( ~H X. { 0 } ) ` w ) - ( ( ~H X. { 0 } ) ` x ) ) ) < y ) ) )
26	2, 24, 25	mpbir2an 955	1 |- ( ~H X. { 0 } ) e. ContFn

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   E.wrex 2913   {csn 4177   class class class wbr 4653   X. cxp 5112   -->wf 5884   ` cfv 5888  (class class class)co 6650   CCcc 9934   0cc0 9936   1c1 9937   < 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-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-hilex 27856

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  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-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-n0 11293  df-z 11378  df-uz 11688  df-rp 11833  df-seq 12802  df-exp 12861  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-cnfn 28706

This theorem is referenced by:  nmcfnex  28912  nmcfnlb  28913  riesz4  28923  riesz1  28924