Theorem stoweidlem9 40226

Description: Lemma for stoweid 40280: here the Stone Weierstrass theorem is proven for the trivial case, T is the empty set. (Contributed by Glauco Siliprandi, 20-Apr-2017.)

Hypotheses

Ref	Expression
stoweidlem9.1	|- ( ph -> T = (/) )
stoweidlem9.2	|- ( ph -> ( t e. T |-> 1 ) e. A )

Assertion

Ref	Expression
stoweidlem9	|- ( ph -> E. g e. A A. t e. T ( abs ` ( ( g ` t ) - ( F ` t ) ) ) < E )

Distinct variable groups:   A, g   g, E   g, F   t, g, T

Allowed substitution hints:   ph( t, g)   A( t)   E( t)   F( t)

Proof of Theorem stoweidlem9

Step	Hyp	Ref	Expression
1		stoweidlem9.1	. . . 4 |- ( ph -> T = (/) )
2		mpteq1 4737	. . . . 5 |- ( T = (/) -> ( t e. T |-> 1 ) = ( t e. (/) |-> 1 ) )
3		mpt0 6021	. . . . 5 |- ( t e. (/) |-> 1 ) = (/)
4	2, 3	syl6eq 2672	. . . 4 |- ( T = (/) -> ( t e. T |-> 1 ) = (/) )
5	1, 4	syl 17	. . 3 |- ( ph -> ( t e. T |-> 1 ) = (/) )
6		stoweidlem9.2	. . 3 |- ( ph -> ( t e. T |-> 1 ) e. A )
7	5, 6	eqeltrrd 2702	. 2 |- ( ph -> (/) e. A )
8		rzal 4073	. . 3 |- ( T = (/) -> A. t e. T ( abs ` ( ( (/) ` t ) - ( F ` t ) ) ) < E )
9	1, 8	syl 17	. 2 |- ( ph -> A. t e. T ( abs ` ( ( (/) ` t ) - ( F ` t ) ) ) < E )
10		fveq1 6190	. . . . . . 7 |- ( g = (/) -> ( g ` t ) = ( (/) ` t ) )
11	10	oveq1d 6665	. . . . . 6 |- ( g = (/) -> ( ( g ` t ) - ( F ` t ) ) = ( ( (/) ` t ) - ( F ` t ) ) )
12	11	fveq2d 6195	. . . . 5 |- ( g = (/) -> ( abs ` ( ( g ` t ) - ( F ` t ) ) ) = ( abs ` ( ( (/) ` t ) - ( F ` t ) ) ) )
13	12	breq1d 4663	. . . 4 |- ( g = (/) -> ( ( abs ` ( ( g ` t ) - ( F ` t ) ) ) < E <-> ( abs ` ( ( (/) ` t ) - ( F ` t ) ) ) < E ) )
14	13	ralbidv 2986	. . 3 |- ( g = (/) -> ( A. t e. T ( abs ` ( ( g ` t ) - ( F ` t ) ) ) < E <-> A. t e. T ( abs ` ( ( (/) ` t ) - ( F ` t ) ) ) < E ) )
15	14	rspcev 3309	. 2 |- ( ( (/) e. A /\ A. t e. T ( abs ` ( ( (/) ` t ) - ( F ` t ) ) ) < E ) -> E. g e. A A. t e. T ( abs ` ( ( g ` t ) - ( F ` t ) ) ) < E )
16	7, 9, 15	syl2anc 693	1 |- ( ph -> E. g e. A A. t e. T ( abs ` ( ( g ` t ) - ( F ` t ) ) ) < E )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   A.wral 2912   E.wrex 2913   (/)c0 3915   class class class wbr 4653   |-> cmpt 4729   ` cfv 5888  (class class class)co 6650   1c1 9937   < clt 10074   - cmin 10266   abscabs 13974

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-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

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-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fn 5891  df-fv 5896  df-ov 6653

This theorem is referenced by:  stoweid  40280