Theorem stoweidlem21 44382

Description: Once the Stone Weierstrass theorem has been proven for approximating nonnegative functions, then this lemma is used to extend the result to functions with (possibly) negative values. (Contributed by Glauco Siliprandi, 20-Apr-2017.)

Hypotheses

Ref	Expression
stoweidlem21.1	⊢ Ⅎ𝑡𝐺
stoweidlem21.2	⊢ Ⅎ𝑡𝐻
stoweidlem21.3	⊢ Ⅎ𝑡𝑆
stoweidlem21.4	⊢ Ⅎ𝑡𝜑
stoweidlem21.5	⊢ 𝐺 = (𝑡 ∈ 𝑇 ↦ ((𝐻‘𝑡) + 𝑆))
stoweidlem21.6	⊢ (𝜑 → 𝐹:𝑇⟶ℝ)
stoweidlem21.7	⊢ (𝜑 → 𝑆 ∈ ℝ)
stoweidlem21.8	⊢ ((𝜑 ∧ 𝑓 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) + (𝑔‘𝑡))) ∈ 𝐴)
stoweidlem21.9	⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝑡 ∈ 𝑇 ↦ 𝑥) ∈ 𝐴)
stoweidlem21.10	⊢ (𝜑 → ∀𝑓 ∈ 𝐴 𝑓:𝑇⟶ℝ)
stoweidlem21.11	⊢ (𝜑 → 𝐻 ∈ 𝐴)
stoweidlem21.12	⊢ (𝜑 → ∀𝑡 ∈ 𝑇 (abs‘((𝐻‘𝑡) − ((𝐹‘𝑡) − 𝑆))) < 𝐸)

Assertion

Ref	Expression
stoweidlem21	⊢ (𝜑 → ∃𝑓 ∈ 𝐴 ∀𝑡 ∈ 𝑇 (abs‘((𝑓‘𝑡) − (𝐹‘𝑡))) < 𝐸)

Distinct variable groups:   𝑓,𝑔,𝑡,𝑇   𝐴,𝑓,𝑔   𝑓,𝐸,𝑔   𝑓,𝐹,𝑔   𝑓,𝐺,𝑔   𝑓,𝐻,𝑔   𝜑,𝑓,𝑔   𝑆,𝑔   𝑥,𝑡,𝑇   𝑥,𝐴   𝑥,𝑆   𝜑,𝑥

Allowed substitution hints:   𝜑(𝑡)   𝐴(𝑡)   𝑆(𝑡,𝑓)   𝐸(𝑥,𝑡)   𝐹(𝑥,𝑡)   𝐺(𝑥,𝑡)   𝐻(𝑥,𝑡)

Proof of Theorem stoweidlem21

Dummy variable 𝑠 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		stoweidlem21.5	. . . 4 ⊢ 𝐺 = (𝑡 ∈ 𝑇 ↦ ((𝐻‘𝑡) + 𝑆))
2		stoweidlem21.4	. . . . 5 ⊢ Ⅎ𝑡𝜑
3		stoweidlem21.7	. . . . . . . 8 ⊢ (𝜑 → 𝑆 ∈ ℝ)
4		fvconst2g 7156	. . . . . . . 8 ⊢ ((𝑆 ∈ ℝ ∧ 𝑡 ∈ 𝑇) → ((𝑇 × {𝑆})‘𝑡) = 𝑆)
5	3, 4	sylan 580	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → ((𝑇 × {𝑆})‘𝑡) = 𝑆)
6	5	eqcomd 2737	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → 𝑆 = ((𝑇 × {𝑆})‘𝑡))
7	6	oveq2d 7378	. . . . 5 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → ((𝐻‘𝑡) + 𝑆) = ((𝐻‘𝑡) + ((𝑇 × {𝑆})‘𝑡)))
8	2, 7	mpteq2da 5208	. . . 4 ⊢ (𝜑 → (𝑡 ∈ 𝑇 ↦ ((𝐻‘𝑡) + 𝑆)) = (𝑡 ∈ 𝑇 ↦ ((𝐻‘𝑡) + ((𝑇 × {𝑆})‘𝑡))))
9	1, 8	eqtrid 2783	. . 3 ⊢ (𝜑 → 𝐺 = (𝑡 ∈ 𝑇 ↦ ((𝐻‘𝑡) + ((𝑇 × {𝑆})‘𝑡))))
10		stoweidlem21.11	. . . 4 ⊢ (𝜑 → 𝐻 ∈ 𝐴)
11		fconstmpt 5699	. . . . . 6 ⊢ (𝑇 × {𝑆}) = (𝑠 ∈ 𝑇 ↦ 𝑆)
12		stoweidlem21.3	. . . . . . 7 ⊢ Ⅎ𝑡𝑆
13		nfcv 2902	. . . . . . 7 ⊢ Ⅎ𝑠𝑆
14		eqidd 2732	. . . . . . 7 ⊢ (𝑠 = 𝑡 → 𝑆 = 𝑆)
15	12, 13, 14	cbvmpt 5221	. . . . . 6 ⊢ (𝑠 ∈ 𝑇 ↦ 𝑆) = (𝑡 ∈ 𝑇 ↦ 𝑆)
16	11, 15	eqtri 2759	. . . . 5 ⊢ (𝑇 × {𝑆}) = (𝑡 ∈ 𝑇 ↦ 𝑆)
17	12	nfeq2 2919	. . . . . . . . . 10 ⊢ Ⅎ𝑡 𝑥 = 𝑆
18		simpl 483	. . . . . . . . . 10 ⊢ ((𝑥 = 𝑆 ∧ 𝑡 ∈ 𝑇) → 𝑥 = 𝑆)
19	17, 18	mpteq2da 5208	. . . . . . . . 9 ⊢ (𝑥 = 𝑆 → (𝑡 ∈ 𝑇 ↦ 𝑥) = (𝑡 ∈ 𝑇 ↦ 𝑆))
20	19	eleq1d 2817	. . . . . . . 8 ⊢ (𝑥 = 𝑆 → ((𝑡 ∈ 𝑇 ↦ 𝑥) ∈ 𝐴 ↔ (𝑡 ∈ 𝑇 ↦ 𝑆) ∈ 𝐴))
21	20	imbi2d 340	. . . . . . 7 ⊢ (𝑥 = 𝑆 → ((𝜑 → (𝑡 ∈ 𝑇 ↦ 𝑥) ∈ 𝐴) ↔ (𝜑 → (𝑡 ∈ 𝑇 ↦ 𝑆) ∈ 𝐴)))
22		stoweidlem21.9	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝑡 ∈ 𝑇 ↦ 𝑥) ∈ 𝐴)
23	22	expcom 414	. . . . . . 7 ⊢ (𝑥 ∈ ℝ → (𝜑 → (𝑡 ∈ 𝑇 ↦ 𝑥) ∈ 𝐴))
24	21, 23	vtoclga 3535	. . . . . 6 ⊢ (𝑆 ∈ ℝ → (𝜑 → (𝑡 ∈ 𝑇 ↦ 𝑆) ∈ 𝐴))
25	3, 24	mpcom 38	. . . . 5 ⊢ (𝜑 → (𝑡 ∈ 𝑇 ↦ 𝑆) ∈ 𝐴)
26	16, 25	eqeltrid 2836	. . . 4 ⊢ (𝜑 → (𝑇 × {𝑆}) ∈ 𝐴)
27		stoweidlem21.8	. . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) + (𝑔‘𝑡))) ∈ 𝐴)
28		stoweidlem21.2	. . . . 5 ⊢ Ⅎ𝑡𝐻
29		nfcv 2902	. . . . . 6 ⊢ Ⅎ𝑡𝑇
30	12	nfsn 4673	. . . . . 6 ⊢ Ⅎ𝑡{𝑆}
31	29, 30	nfxp 5671	. . . . 5 ⊢ Ⅎ𝑡(𝑇 × {𝑆})
32	27, 28, 31	stoweidlem8 44369	. . . 4 ⊢ ((𝜑 ∧ 𝐻 ∈ 𝐴 ∧ (𝑇 × {𝑆}) ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝐻‘𝑡) + ((𝑇 × {𝑆})‘𝑡))) ∈ 𝐴)
33	10, 26, 32	mpd3an23 1463	. . 3 ⊢ (𝜑 → (𝑡 ∈ 𝑇 ↦ ((𝐻‘𝑡) + ((𝑇 × {𝑆})‘𝑡))) ∈ 𝐴)
34	9, 33	eqeltrd 2832	. 2 ⊢ (𝜑 → 𝐺 ∈ 𝐴)
35		simpr 485	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → 𝑡 ∈ 𝑇)
36		stoweidlem21.10	. . . . . . . . . . . 12 ⊢ (𝜑 → ∀𝑓 ∈ 𝐴 𝑓:𝑇⟶ℝ)
37		feq1 6654	. . . . . . . . . . . . 13 ⊢ (𝑓 = 𝐻 → (𝑓:𝑇⟶ℝ ↔ 𝐻:𝑇⟶ℝ))
38	37	rspccva 3581	. . . . . . . . . . . 12 ⊢ ((∀𝑓 ∈ 𝐴 𝑓:𝑇⟶ℝ ∧ 𝐻 ∈ 𝐴) → 𝐻:𝑇⟶ℝ)
39	36, 10, 38	syl2anc 584	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐻:𝑇⟶ℝ)
40	39	ffvelcdmda 7040	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (𝐻‘𝑡) ∈ ℝ)
41	3	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → 𝑆 ∈ ℝ)
42	40, 41	readdcld 11193	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → ((𝐻‘𝑡) + 𝑆) ∈ ℝ)
43	1	fvmpt2 6964	. . . . . . . . 9 ⊢ ((𝑡 ∈ 𝑇 ∧ ((𝐻‘𝑡) + 𝑆) ∈ ℝ) → (𝐺‘𝑡) = ((𝐻‘𝑡) + 𝑆))
44	35, 42, 43	syl2anc 584	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (𝐺‘𝑡) = ((𝐻‘𝑡) + 𝑆))
45	44	oveq1d 7377	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → ((𝐺‘𝑡) − (𝐹‘𝑡)) = (((𝐻‘𝑡) + 𝑆) − (𝐹‘𝑡)))
46	40	recnd 11192	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (𝐻‘𝑡) ∈ ℂ)
47		stoweidlem21.6	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹:𝑇⟶ℝ)
48	47	ffvelcdmda 7040	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (𝐹‘𝑡) ∈ ℝ)
49	48	recnd 11192	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (𝐹‘𝑡) ∈ ℂ)
50	3	recnd 11192	. . . . . . . . 9 ⊢ (𝜑 → 𝑆 ∈ ℂ)
51	50	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → 𝑆 ∈ ℂ)
52	46, 49, 51	subsub3d 11551	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → ((𝐻‘𝑡) − ((𝐹‘𝑡) − 𝑆)) = (((𝐻‘𝑡) + 𝑆) − (𝐹‘𝑡)))
53	45, 52	eqtr4d 2774	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → ((𝐺‘𝑡) − (𝐹‘𝑡)) = ((𝐻‘𝑡) − ((𝐹‘𝑡) − 𝑆)))
54	53	fveq2d 6851	. . . . 5 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (abs‘((𝐺‘𝑡) − (𝐹‘𝑡))) = (abs‘((𝐻‘𝑡) − ((𝐹‘𝑡) − 𝑆))))
55		stoweidlem21.12	. . . . . 6 ⊢ (𝜑 → ∀𝑡 ∈ 𝑇 (abs‘((𝐻‘𝑡) − ((𝐹‘𝑡) − 𝑆))) < 𝐸)
56	55	r19.21bi 3232	. . . . 5 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (abs‘((𝐻‘𝑡) − ((𝐹‘𝑡) − 𝑆))) < 𝐸)
57	54, 56	eqbrtrd 5132	. . . 4 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (abs‘((𝐺‘𝑡) − (𝐹‘𝑡))) < 𝐸)
58	57	ex 413	. . 3 ⊢ (𝜑 → (𝑡 ∈ 𝑇 → (abs‘((𝐺‘𝑡) − (𝐹‘𝑡))) < 𝐸))
59	2, 58	ralrimi 3238	. 2 ⊢ (𝜑 → ∀𝑡 ∈ 𝑇 (abs‘((𝐺‘𝑡) − (𝐹‘𝑡))) < 𝐸)
60		stoweidlem21.1	. . . . 5 ⊢ Ⅎ𝑡𝐺
61	60	nfeq2 2919	. . . 4 ⊢ Ⅎ𝑡 𝑓 = 𝐺
62		fveq1 6846	. . . . . . 7 ⊢ (𝑓 = 𝐺 → (𝑓‘𝑡) = (𝐺‘𝑡))
63	62	oveq1d 7377	. . . . . 6 ⊢ (𝑓 = 𝐺 → ((𝑓‘𝑡) − (𝐹‘𝑡)) = ((𝐺‘𝑡) − (𝐹‘𝑡)))
64	63	fveq2d 6851	. . . . 5 ⊢ (𝑓 = 𝐺 → (abs‘((𝑓‘𝑡) − (𝐹‘𝑡))) = (abs‘((𝐺‘𝑡) − (𝐹‘𝑡))))
65	64	breq1d 5120	. . . 4 ⊢ (𝑓 = 𝐺 → ((abs‘((𝑓‘𝑡) − (𝐹‘𝑡))) < 𝐸 ↔ (abs‘((𝐺‘𝑡) − (𝐹‘𝑡))) < 𝐸))
66	61, 65	ralbid 3254	. . 3 ⊢ (𝑓 = 𝐺 → (∀𝑡 ∈ 𝑇 (abs‘((𝑓‘𝑡) − (𝐹‘𝑡))) < 𝐸 ↔ ∀𝑡 ∈ 𝑇 (abs‘((𝐺‘𝑡) − (𝐹‘𝑡))) < 𝐸))
67	66	rspcev 3582	. 2 ⊢ ((𝐺 ∈ 𝐴 ∧ ∀𝑡 ∈ 𝑇 (abs‘((𝐺‘𝑡) − (𝐹‘𝑡))) < 𝐸) → ∃𝑓 ∈ 𝐴 ∀𝑡 ∈ 𝑇 (abs‘((𝑓‘𝑡) − (𝐹‘𝑡))) < 𝐸)
68	34, 59, 67	syl2anc 584	1 ⊢ (𝜑 → ∃𝑓 ∈ 𝐴 ∀𝑡 ∈ 𝑇 (abs‘((𝑓‘𝑡) − (𝐹‘𝑡))) < 𝐸)

Syntax hints:   → wi 4   ∧ wa 396   ∧ w3a 1087   = wceq 1541  Ⅎwnf 1785   ∈ wcel 2106  Ⅎwnfc 2882  ∀wral 3060  ∃wrex 3069  {csn 4591   class class class wbr 5110   ↦ cmpt 5193   × cxp 5636  ⟶wf 6497  ‘cfv 6501  (class class class)co 7362  ℂcc 11058  ℝcr 11059   + caddc 11063   < clt 11198   − cmin 11394  abscabs 15131

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677  ax-resscn 11117  ax-1cn 11118  ax-icn 11119  ax-addcl 11120  ax-addrcl 11121  ax-mulcl 11122  ax-mulrcl 11123  ax-mulcom 11124  ax-addass 11125  ax-mulass 11126  ax-distr 11127  ax-i2m1 11128  ax-1ne0 11129  ax-1rid 11130  ax-rnegex 11131  ax-rrecex 11132  ax-cnre 11133  ax-pre-lttri 11134  ax-pre-lttrn 11135  ax-pre-ltadd 11136

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-reu 3352  df-rab 3406  df-v 3448  df-sbc 3743  df-csb 3859  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-br 5111  df-opab 5173  df-mpt 5194  df-id 5536  df-po 5550  df-so 5551  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-riota 7318  df-ov 7365  df-oprab 7366  df-mpo 7367  df-er 8655  df-en 8891  df-dom 8892  df-sdom 8893  df-pnf 11200  df-mnf 11201  df-ltxr 11203  df-sub 11396

This theorem is referenced by:  stoweidlem62  44423