Theorem stoweidlem21 44252

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 7151	. . . . . . . 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 2742	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → 𝑆 = ((𝑇 × {𝑆})‘𝑡))
7	6	oveq2d 7373	. . . . 5 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → ((𝐻‘𝑡) + 𝑆) = ((𝐻‘𝑡) + ((𝑇 × {𝑆})‘𝑡)))
8	2, 7	mpteq2da 5203	. . . 4 ⊢ (𝜑 → (𝑡 ∈ 𝑇 ↦ ((𝐻‘𝑡) + 𝑆)) = (𝑡 ∈ 𝑇 ↦ ((𝐻‘𝑡) + ((𝑇 × {𝑆})‘𝑡))))
9	1, 8	eqtrid 2788	. . 3 ⊢ (𝜑 → 𝐺 = (𝑡 ∈ 𝑇 ↦ ((𝐻‘𝑡) + ((𝑇 × {𝑆})‘𝑡))))
10		stoweidlem21.11	. . . 4 ⊢ (𝜑 → 𝐻 ∈ 𝐴)
11		fconstmpt 5694	. . . . . 6 ⊢ (𝑇 × {𝑆}) = (𝑠 ∈ 𝑇 ↦ 𝑆)
12		stoweidlem21.3	. . . . . . 7 ⊢ Ⅎ𝑡𝑆
13		nfcv 2907	. . . . . . 7 ⊢ Ⅎ𝑠𝑆
14		eqidd 2737	. . . . . . 7 ⊢ (𝑠 = 𝑡 → 𝑆 = 𝑆)
15	12, 13, 14	cbvmpt 5216	. . . . . 6 ⊢ (𝑠 ∈ 𝑇 ↦ 𝑆) = (𝑡 ∈ 𝑇 ↦ 𝑆)
16	11, 15	eqtri 2764	. . . . 5 ⊢ (𝑇 × {𝑆}) = (𝑡 ∈ 𝑇 ↦ 𝑆)
17	12	nfeq2 2924	. . . . . . . . . 10 ⊢ Ⅎ𝑡 𝑥 = 𝑆
18		simpl 483	. . . . . . . . . 10 ⊢ ((𝑥 = 𝑆 ∧ 𝑡 ∈ 𝑇) → 𝑥 = 𝑆)
19	17, 18	mpteq2da 5203	. . . . . . . . 9 ⊢ (𝑥 = 𝑆 → (𝑡 ∈ 𝑇 ↦ 𝑥) = (𝑡 ∈ 𝑇 ↦ 𝑆))
20	19	eleq1d 2822	. . . . . . . 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 3534	. . . . . 6 ⊢ (𝑆 ∈ ℝ → (𝜑 → (𝑡 ∈ 𝑇 ↦ 𝑆) ∈ 𝐴))
25	3, 24	mpcom 38	. . . . 5 ⊢ (𝜑 → (𝑡 ∈ 𝑇 ↦ 𝑆) ∈ 𝐴)
26	16, 25	eqeltrid 2842	. . . 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 2907	. . . . . 6 ⊢ Ⅎ𝑡𝑇
30	12	nfsn 4668	. . . . . 6 ⊢ Ⅎ𝑡{𝑆}
31	29, 30	nfxp 5666	. . . . 5 ⊢ Ⅎ𝑡(𝑇 × {𝑆})
32	27, 28, 31	stoweidlem8 44239	. . . 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 2838	. 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 6649	. . . . . . . . . . . . 13 ⊢ (𝑓 = 𝐻 → (𝑓:𝑇⟶ℝ ↔ 𝐻:𝑇⟶ℝ))
38	37	rspccva 3580	. . . . . . . . . . . 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 7035	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (𝐻‘𝑡) ∈ ℝ)
41	3	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → 𝑆 ∈ ℝ)
42	40, 41	readdcld 11184	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → ((𝐻‘𝑡) + 𝑆) ∈ ℝ)
43	1	fvmpt2 6959	. . . . . . . . 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 7372	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → ((𝐺‘𝑡) − (𝐹‘𝑡)) = (((𝐻‘𝑡) + 𝑆) − (𝐹‘𝑡)))
46	40	recnd 11183	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (𝐻‘𝑡) ∈ ℂ)
47		stoweidlem21.6	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹:𝑇⟶ℝ)
48	47	ffvelcdmda 7035	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (𝐹‘𝑡) ∈ ℝ)
49	48	recnd 11183	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (𝐹‘𝑡) ∈ ℂ)
50	3	recnd 11183	. . . . . . . . 9 ⊢ (𝜑 → 𝑆 ∈ ℂ)
51	50	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → 𝑆 ∈ ℂ)
52	46, 49, 51	subsub3d 11542	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → ((𝐻‘𝑡) − ((𝐹‘𝑡) − 𝑆)) = (((𝐻‘𝑡) + 𝑆) − (𝐹‘𝑡)))
53	45, 52	eqtr4d 2779	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → ((𝐺‘𝑡) − (𝐹‘𝑡)) = ((𝐻‘𝑡) − ((𝐹‘𝑡) − 𝑆)))
54	53	fveq2d 6846	. . . . 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 3234	. . . . 5 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (abs‘((𝐻‘𝑡) − ((𝐹‘𝑡) − 𝑆))) < 𝐸)
57	54, 56	eqbrtrd 5127	. . . 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 3240	. 2 ⊢ (𝜑 → ∀𝑡 ∈ 𝑇 (abs‘((𝐺‘𝑡) − (𝐹‘𝑡))) < 𝐸)
60		stoweidlem21.1	. . . . 5 ⊢ Ⅎ𝑡𝐺
61	60	nfeq2 2924	. . . 4 ⊢ Ⅎ𝑡 𝑓 = 𝐺
62		fveq1 6841	. . . . . . 7 ⊢ (𝑓 = 𝐺 → (𝑓‘𝑡) = (𝐺‘𝑡))
63	62	oveq1d 7372	. . . . . 6 ⊢ (𝑓 = 𝐺 → ((𝑓‘𝑡) − (𝐹‘𝑡)) = ((𝐺‘𝑡) − (𝐹‘𝑡)))
64	63	fveq2d 6846	. . . . 5 ⊢ (𝑓 = 𝐺 → (abs‘((𝑓‘𝑡) − (𝐹‘𝑡))) = (abs‘((𝐺‘𝑡) − (𝐹‘𝑡))))
65	64	breq1d 5115	. . . 4 ⊢ (𝑓 = 𝐺 → ((abs‘((𝑓‘𝑡) − (𝐹‘𝑡))) < 𝐸 ↔ (abs‘((𝐺‘𝑡) − (𝐹‘𝑡))) < 𝐸))
66	61, 65	ralbid 3256	. . 3 ⊢ (𝑓 = 𝐺 → (∀𝑡 ∈ 𝑇 (abs‘((𝑓‘𝑡) − (𝐹‘𝑡))) < 𝐸 ↔ ∀𝑡 ∈ 𝑇 (abs‘((𝐺‘𝑡) − (𝐹‘𝑡))) < 𝐸))
67	66	rspcev 3581	. 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 2887  ∀wral 3064  ∃wrex 3073  {csn 4586   class class class wbr 5105   ↦ cmpt 5188   × cxp 5631  ⟶wf 6492  ‘cfv 6496  (class class class)co 7357  ℂcc 11049  ℝcr 11050   + caddc 11054   < clt 11189   − cmin 11385  abscabs 15119

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 2707  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127

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 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-po 5545  df-so 5546  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-er 8648  df-en 8884  df-dom 8885  df-sdom 8886  df-pnf 11191  df-mnf 11192  df-ltxr 11194  df-sub 11387

This theorem is referenced by:  stoweidlem62  44293