Theorem stoweidlem21 46036

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 7222	. . . . . . . 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 2743	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → 𝑆 = ((𝑇 × {𝑆})‘𝑡))
7	6	oveq2d 7447	. . . . 5 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → ((𝐻‘𝑡) + 𝑆) = ((𝐻‘𝑡) + ((𝑇 × {𝑆})‘𝑡)))
8	2, 7	mpteq2da 5240	. . . 4 ⊢ (𝜑 → (𝑡 ∈ 𝑇 ↦ ((𝐻‘𝑡) + 𝑆)) = (𝑡 ∈ 𝑇 ↦ ((𝐻‘𝑡) + ((𝑇 × {𝑆})‘𝑡))))
9	1, 8	eqtrid 2789	. . 3 ⊢ (𝜑 → 𝐺 = (𝑡 ∈ 𝑇 ↦ ((𝐻‘𝑡) + ((𝑇 × {𝑆})‘𝑡))))
10		stoweidlem21.11	. . . 4 ⊢ (𝜑 → 𝐻 ∈ 𝐴)
11		fconstmpt 5747	. . . . . 6 ⊢ (𝑇 × {𝑆}) = (𝑠 ∈ 𝑇 ↦ 𝑆)
12		stoweidlem21.3	. . . . . . 7 ⊢ Ⅎ𝑡𝑆
13		nfcv 2905	. . . . . . 7 ⊢ Ⅎ𝑠𝑆
14		eqidd 2738	. . . . . . 7 ⊢ (𝑠 = 𝑡 → 𝑆 = 𝑆)
15	12, 13, 14	cbvmpt 5253	. . . . . 6 ⊢ (𝑠 ∈ 𝑇 ↦ 𝑆) = (𝑡 ∈ 𝑇 ↦ 𝑆)
16	11, 15	eqtri 2765	. . . . 5 ⊢ (𝑇 × {𝑆}) = (𝑡 ∈ 𝑇 ↦ 𝑆)
17	12	nfeq2 2923	. . . . . . . . . 10 ⊢ Ⅎ𝑡 𝑥 = 𝑆
18		simpl 482	. . . . . . . . . 10 ⊢ ((𝑥 = 𝑆 ∧ 𝑡 ∈ 𝑇) → 𝑥 = 𝑆)
19	17, 18	mpteq2da 5240	. . . . . . . . 9 ⊢ (𝑥 = 𝑆 → (𝑡 ∈ 𝑇 ↦ 𝑥) = (𝑡 ∈ 𝑇 ↦ 𝑆))
20	19	eleq1d 2826	. . . . . . . 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 413	. . . . . . 7 ⊢ (𝑥 ∈ ℝ → (𝜑 → (𝑡 ∈ 𝑇 ↦ 𝑥) ∈ 𝐴))
24	21, 23	vtoclga 3577	. . . . . 6 ⊢ (𝑆 ∈ ℝ → (𝜑 → (𝑡 ∈ 𝑇 ↦ 𝑆) ∈ 𝐴))
25	3, 24	mpcom 38	. . . . 5 ⊢ (𝜑 → (𝑡 ∈ 𝑇 ↦ 𝑆) ∈ 𝐴)
26	16, 25	eqeltrid 2845	. . . 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 2905	. . . . . 6 ⊢ Ⅎ𝑡𝑇
30	12	nfsn 4707	. . . . . 6 ⊢ Ⅎ𝑡{𝑆}
31	29, 30	nfxp 5718	. . . . 5 ⊢ Ⅎ𝑡(𝑇 × {𝑆})
32	27, 28, 31	stoweidlem8 46023	. . . 4 ⊢ ((𝜑 ∧ 𝐻 ∈ 𝐴 ∧ (𝑇 × {𝑆}) ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝐻‘𝑡) + ((𝑇 × {𝑆})‘𝑡))) ∈ 𝐴)
33	10, 26, 32	mpd3an23 1465	. . 3 ⊢ (𝜑 → (𝑡 ∈ 𝑇 ↦ ((𝐻‘𝑡) + ((𝑇 × {𝑆})‘𝑡))) ∈ 𝐴)
34	9, 33	eqeltrd 2841	. 2 ⊢ (𝜑 → 𝐺 ∈ 𝐴)
35		simpr 484	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → 𝑡 ∈ 𝑇)
36		stoweidlem21.10	. . . . . . . . . . . 12 ⊢ (𝜑 → ∀𝑓 ∈ 𝐴 𝑓:𝑇⟶ℝ)
37		feq1 6716	. . . . . . . . . . . . 13 ⊢ (𝑓 = 𝐻 → (𝑓:𝑇⟶ℝ ↔ 𝐻:𝑇⟶ℝ))
38	37	rspccva 3621	. . . . . . . . . . . 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 7104	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (𝐻‘𝑡) ∈ ℝ)
41	3	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → 𝑆 ∈ ℝ)
42	40, 41	readdcld 11290	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → ((𝐻‘𝑡) + 𝑆) ∈ ℝ)
43	1	fvmpt2 7027	. . . . . . . . 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 7446	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → ((𝐺‘𝑡) − (𝐹‘𝑡)) = (((𝐻‘𝑡) + 𝑆) − (𝐹‘𝑡)))
46	40	recnd 11289	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (𝐻‘𝑡) ∈ ℂ)
47		stoweidlem21.6	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹:𝑇⟶ℝ)
48	47	ffvelcdmda 7104	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (𝐹‘𝑡) ∈ ℝ)
49	48	recnd 11289	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (𝐹‘𝑡) ∈ ℂ)
50	3	recnd 11289	. . . . . . . . 9 ⊢ (𝜑 → 𝑆 ∈ ℂ)
51	50	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → 𝑆 ∈ ℂ)
52	46, 49, 51	subsub3d 11650	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → ((𝐻‘𝑡) − ((𝐹‘𝑡) − 𝑆)) = (((𝐻‘𝑡) + 𝑆) − (𝐹‘𝑡)))
53	45, 52	eqtr4d 2780	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → ((𝐺‘𝑡) − (𝐹‘𝑡)) = ((𝐻‘𝑡) − ((𝐹‘𝑡) − 𝑆)))
54	53	fveq2d 6910	. . . . 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 3251	. . . . 5 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (abs‘((𝐻‘𝑡) − ((𝐹‘𝑡) − 𝑆))) < 𝐸)
57	54, 56	eqbrtrd 5165	. . . 4 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (abs‘((𝐺‘𝑡) − (𝐹‘𝑡))) < 𝐸)
58	57	ex 412	. . 3 ⊢ (𝜑 → (𝑡 ∈ 𝑇 → (abs‘((𝐺‘𝑡) − (𝐹‘𝑡))) < 𝐸))
59	2, 58	ralrimi 3257	. 2 ⊢ (𝜑 → ∀𝑡 ∈ 𝑇 (abs‘((𝐺‘𝑡) − (𝐹‘𝑡))) < 𝐸)
60		stoweidlem21.1	. . . . 5 ⊢ Ⅎ𝑡𝐺
61	60	nfeq2 2923	. . . 4 ⊢ Ⅎ𝑡 𝑓 = 𝐺
62		fveq1 6905	. . . . . . 7 ⊢ (𝑓 = 𝐺 → (𝑓‘𝑡) = (𝐺‘𝑡))
63	62	oveq1d 7446	. . . . . 6 ⊢ (𝑓 = 𝐺 → ((𝑓‘𝑡) − (𝐹‘𝑡)) = ((𝐺‘𝑡) − (𝐹‘𝑡)))
64	63	fveq2d 6910	. . . . 5 ⊢ (𝑓 = 𝐺 → (abs‘((𝑓‘𝑡) − (𝐹‘𝑡))) = (abs‘((𝐺‘𝑡) − (𝐹‘𝑡))))
65	64	breq1d 5153	. . . 4 ⊢ (𝑓 = 𝐺 → ((abs‘((𝑓‘𝑡) − (𝐹‘𝑡))) < 𝐸 ↔ (abs‘((𝐺‘𝑡) − (𝐹‘𝑡))) < 𝐸))
66	61, 65	ralbid 3273	. . 3 ⊢ (𝑓 = 𝐺 → (∀𝑡 ∈ 𝑇 (abs‘((𝑓‘𝑡) − (𝐹‘𝑡))) < 𝐸 ↔ ∀𝑡 ∈ 𝑇 (abs‘((𝐺‘𝑡) − (𝐹‘𝑡))) < 𝐸))
67	66	rspcev 3622	. 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 395   ∧ w3a 1087   = wceq 1540  Ⅎwnf 1783   ∈ wcel 2108  Ⅎwnfc 2890  ∀wral 3061  ∃wrex 3070  {csn 4626   class class class wbr 5143   ↦ cmpt 5225   × cxp 5683  ⟶wf 6557  ‘cfv 6561  (class class class)co 7431  ℂcc 11153  ℝcr 11154   + caddc 11158   < clt 11295   − cmin 11492  abscabs 15273

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-po 5592  df-so 5593  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-er 8745  df-en 8986  df-dom 8987  df-sdom 8988  df-pnf 11297  df-mnf 11298  df-ltxr 11300  df-sub 11494

This theorem is referenced by:  stoweidlem62  46077