Theorem stoweidlem21 46012

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 7158	. . . . . . . 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 2735	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → 𝑆 = ((𝑇 × {𝑆})‘𝑡))
7	6	oveq2d 7385	. . . . 5 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → ((𝐻‘𝑡) + 𝑆) = ((𝐻‘𝑡) + ((𝑇 × {𝑆})‘𝑡)))
8	2, 7	mpteq2da 5194	. . . 4 ⊢ (𝜑 → (𝑡 ∈ 𝑇 ↦ ((𝐻‘𝑡) + 𝑆)) = (𝑡 ∈ 𝑇 ↦ ((𝐻‘𝑡) + ((𝑇 × {𝑆})‘𝑡))))
9	1, 8	eqtrid 2776	. . 3 ⊢ (𝜑 → 𝐺 = (𝑡 ∈ 𝑇 ↦ ((𝐻‘𝑡) + ((𝑇 × {𝑆})‘𝑡))))
10		stoweidlem21.11	. . . 4 ⊢ (𝜑 → 𝐻 ∈ 𝐴)
11		fconstmpt 5693	. . . . . 6 ⊢ (𝑇 × {𝑆}) = (𝑠 ∈ 𝑇 ↦ 𝑆)
12		stoweidlem21.3	. . . . . . 7 ⊢ Ⅎ𝑡𝑆
13		nfcv 2891	. . . . . . 7 ⊢ Ⅎ𝑠𝑆
14		eqidd 2730	. . . . . . 7 ⊢ (𝑠 = 𝑡 → 𝑆 = 𝑆)
15	12, 13, 14	cbvmpt 5204	. . . . . 6 ⊢ (𝑠 ∈ 𝑇 ↦ 𝑆) = (𝑡 ∈ 𝑇 ↦ 𝑆)
16	11, 15	eqtri 2752	. . . . 5 ⊢ (𝑇 × {𝑆}) = (𝑡 ∈ 𝑇 ↦ 𝑆)
17	12	nfeq2 2909	. . . . . . . . . 10 ⊢ Ⅎ𝑡 𝑥 = 𝑆
18		simpl 482	. . . . . . . . . 10 ⊢ ((𝑥 = 𝑆 ∧ 𝑡 ∈ 𝑇) → 𝑥 = 𝑆)
19	17, 18	mpteq2da 5194	. . . . . . . . 9 ⊢ (𝑥 = 𝑆 → (𝑡 ∈ 𝑇 ↦ 𝑥) = (𝑡 ∈ 𝑇 ↦ 𝑆))
20	19	eleq1d 2813	. . . . . . . 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 3540	. . . . . 6 ⊢ (𝑆 ∈ ℝ → (𝜑 → (𝑡 ∈ 𝑇 ↦ 𝑆) ∈ 𝐴))
25	3, 24	mpcom 38	. . . . 5 ⊢ (𝜑 → (𝑡 ∈ 𝑇 ↦ 𝑆) ∈ 𝐴)
26	16, 25	eqeltrid 2832	. . . 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 2891	. . . . . 6 ⊢ Ⅎ𝑡𝑇
30	12	nfsn 4667	. . . . . 6 ⊢ Ⅎ𝑡{𝑆}
31	29, 30	nfxp 5664	. . . . 5 ⊢ Ⅎ𝑡(𝑇 × {𝑆})
32	27, 28, 31	stoweidlem8 45999	. . . 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 2828	. 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 6648	. . . . . . . . . . . . 13 ⊢ (𝑓 = 𝐻 → (𝑓:𝑇⟶ℝ ↔ 𝐻:𝑇⟶ℝ))
38	37	rspccva 3584	. . . . . . . . . . . 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 7038	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (𝐻‘𝑡) ∈ ℝ)
41	3	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → 𝑆 ∈ ℝ)
42	40, 41	readdcld 11179	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → ((𝐻‘𝑡) + 𝑆) ∈ ℝ)
43	1	fvmpt2 6961	. . . . . . . . 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 7384	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → ((𝐺‘𝑡) − (𝐹‘𝑡)) = (((𝐻‘𝑡) + 𝑆) − (𝐹‘𝑡)))
46	40	recnd 11178	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (𝐻‘𝑡) ∈ ℂ)
47		stoweidlem21.6	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹:𝑇⟶ℝ)
48	47	ffvelcdmda 7038	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (𝐹‘𝑡) ∈ ℝ)
49	48	recnd 11178	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (𝐹‘𝑡) ∈ ℂ)
50	3	recnd 11178	. . . . . . . . 9 ⊢ (𝜑 → 𝑆 ∈ ℂ)
51	50	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → 𝑆 ∈ ℂ)
52	46, 49, 51	subsub3d 11539	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → ((𝐻‘𝑡) − ((𝐹‘𝑡) − 𝑆)) = (((𝐻‘𝑡) + 𝑆) − (𝐹‘𝑡)))
53	45, 52	eqtr4d 2767	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → ((𝐺‘𝑡) − (𝐹‘𝑡)) = ((𝐻‘𝑡) − ((𝐹‘𝑡) − 𝑆)))
54	53	fveq2d 6844	. . . . 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 3227	. . . . 5 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (abs‘((𝐻‘𝑡) − ((𝐹‘𝑡) − 𝑆))) < 𝐸)
57	54, 56	eqbrtrd 5124	. . . 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 3233	. 2 ⊢ (𝜑 → ∀𝑡 ∈ 𝑇 (abs‘((𝐺‘𝑡) − (𝐹‘𝑡))) < 𝐸)
60		stoweidlem21.1	. . . . 5 ⊢ Ⅎ𝑡𝐺
61	60	nfeq2 2909	. . . 4 ⊢ Ⅎ𝑡 𝑓 = 𝐺
62		fveq1 6839	. . . . . . 7 ⊢ (𝑓 = 𝐺 → (𝑓‘𝑡) = (𝐺‘𝑡))
63	62	oveq1d 7384	. . . . . 6 ⊢ (𝑓 = 𝐺 → ((𝑓‘𝑡) − (𝐹‘𝑡)) = ((𝐺‘𝑡) − (𝐹‘𝑡)))
64	63	fveq2d 6844	. . . . 5 ⊢ (𝑓 = 𝐺 → (abs‘((𝑓‘𝑡) − (𝐹‘𝑡))) = (abs‘((𝐺‘𝑡) − (𝐹‘𝑡))))
65	64	breq1d 5112	. . . 4 ⊢ (𝑓 = 𝐺 → ((abs‘((𝑓‘𝑡) − (𝐹‘𝑡))) < 𝐸 ↔ (abs‘((𝐺‘𝑡) − (𝐹‘𝑡))) < 𝐸))
66	61, 65	ralbid 3248	. . 3 ⊢ (𝑓 = 𝐺 → (∀𝑡 ∈ 𝑇 (abs‘((𝑓‘𝑡) − (𝐹‘𝑡))) < 𝐸 ↔ ∀𝑡 ∈ 𝑇 (abs‘((𝐺‘𝑡) − (𝐹‘𝑡))) < 𝐸))
67	66	rspcev 3585	. 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 1086   = wceq 1540  Ⅎwnf 1783   ∈ wcel 2109  Ⅎwnfc 2876  ∀wral 3044  ∃wrex 3053  {csn 4585   class class class wbr 5102   ↦ cmpt 5183   × cxp 5629  ⟶wf 6495  ‘cfv 6499  (class class class)co 7369  ℂcc 11042  ℝcr 11043   + caddc 11047   < clt 11184   − cmin 11381  abscabs 15176

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5246  ax-nul 5256  ax-pow 5315  ax-pr 5382  ax-un 7691  ax-resscn 11101  ax-1cn 11102  ax-icn 11103  ax-addcl 11104  ax-addrcl 11105  ax-mulcl 11106  ax-mulrcl 11107  ax-mulcom 11108  ax-addass 11109  ax-mulass 11110  ax-distr 11111  ax-i2m1 11112  ax-1ne0 11113  ax-1rid 11114  ax-rnegex 11115  ax-rrecex 11116  ax-cnre 11117  ax-pre-lttri 11118  ax-pre-lttrn 11119  ax-pre-ltadd 11120

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-reu 3352  df-rab 3403  df-v 3446  df-sbc 3751  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-br 5103  df-opab 5165  df-mpt 5184  df-id 5526  df-po 5539  df-so 5540  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-riota 7326  df-ov 7372  df-oprab 7373  df-mpo 7374  df-er 8648  df-en 8896  df-dom 8897  df-sdom 8898  df-pnf 11186  df-mnf 11187  df-ltxr 11189  df-sub 11383

This theorem is referenced by:  stoweidlem62  46053