Theorem stoweidlem2 46793

Description: lemma for stoweid 46854: here we prove that the subalgebra of continuous functions, which contains constant functions, is closed under scaling. (Contributed by Glauco Siliprandi, 20-Apr-2017.)

Hypotheses

Ref	Expression
stoweidlem2.1	⊢ Ⅎ𝑡𝜑
stoweidlem2.2	⊢ ((𝜑 ∧ 𝑓 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) · (𝑔‘𝑡))) ∈ 𝐴)
stoweidlem2.3	⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝑡 ∈ 𝑇 ↦ 𝑥) ∈ 𝐴)
stoweidlem2.4	⊢ ((𝜑 ∧ 𝑓 ∈ 𝐴) → 𝑓:𝑇⟶ℝ)
stoweidlem2.5	⊢ (𝜑 → 𝐸 ∈ ℝ)
stoweidlem2.6	⊢ (𝜑 → 𝐹 ∈ 𝐴)

Assertion

Ref	Expression
stoweidlem2	⊢ (𝜑 → (𝑡 ∈ 𝑇 ↦ (𝐸 · (𝐹‘𝑡))) ∈ 𝐴)

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

Allowed substitution hints:   𝜑(𝑡)   𝐴(𝑡)   𝐸(𝑔)   𝐹(𝑥)

Proof of Theorem stoweidlem2

Dummy variable 𝑠 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		stoweidlem2.1	. . 3 ⊢ Ⅎ𝑡𝜑
2		simpr 490	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → 𝑡 ∈ 𝑇)
3		stoweidlem2.5	. . . . . . 7 ⊢ (𝜑 → 𝐸 ∈ ℝ)
4	3	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → 𝐸 ∈ ℝ)
5		eqidd 2766	. . . . . . . 8 ⊢ (𝑠 = 𝑡 → 𝐸 = 𝐸)
6	5	cbvmptv 5217	. . . . . . 7 ⊢ (𝑠 ∈ 𝑇 ↦ 𝐸) = (𝑡 ∈ 𝑇 ↦ 𝐸)
7	6	fvmpt2 7005	. . . . . 6 ⊢ ((𝑡 ∈ 𝑇 ∧ 𝐸 ∈ ℝ) → ((𝑠 ∈ 𝑇 ↦ 𝐸)‘𝑡) = 𝐸)
8	2, 4, 7	syl2anc 596	. . . . 5 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → ((𝑠 ∈ 𝑇 ↦ 𝐸)‘𝑡) = 𝐸)
9	8	eqcomd 2771	. . . 4 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → 𝐸 = ((𝑠 ∈ 𝑇 ↦ 𝐸)‘𝑡))
10	9	oveq1d 7434	. . 3 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (𝐸 · (𝐹‘𝑡)) = (((𝑠 ∈ 𝑇 ↦ 𝐸)‘𝑡) · (𝐹‘𝑡)))
11	1, 10	mpteq2da 5205	. 2 ⊢ (𝜑 → (𝑡 ∈ 𝑇 ↦ (𝐸 · (𝐹‘𝑡))) = (𝑡 ∈ 𝑇 ↦ (((𝑠 ∈ 𝑇 ↦ 𝐸)‘𝑡) · (𝐹‘𝑡))))
12		id 23	. . . . . . . . 9 ⊢ (𝑥 = 𝐸 → 𝑥 = 𝐸)
13	12	mpteq2dv 5207	. . . . . . . 8 ⊢ (𝑥 = 𝐸 → (𝑡 ∈ 𝑇 ↦ 𝑥) = (𝑡 ∈ 𝑇 ↦ 𝐸))
14	13	eleq1d 2850	. . . . . . 7 ⊢ (𝑥 = 𝐸 → ((𝑡 ∈ 𝑇 ↦ 𝑥) ∈ 𝐴 ↔ (𝑡 ∈ 𝑇 ↦ 𝐸) ∈ 𝐴))
15	14	imbi2d 343	. . . . . 6 ⊢ (𝑥 = 𝐸 → ((𝜑 → (𝑡 ∈ 𝑇 ↦ 𝑥) ∈ 𝐴) ↔ (𝜑 → (𝑡 ∈ 𝑇 ↦ 𝐸) ∈ 𝐴)))
16		stoweidlem2.3	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝑡 ∈ 𝑇 ↦ 𝑥) ∈ 𝐴)
17	16	expcom 419	. . . . . 6 ⊢ (𝑥 ∈ ℝ → (𝜑 → (𝑡 ∈ 𝑇 ↦ 𝑥) ∈ 𝐴))
18	15, 17	vtoclga 3543	. . . . 5 ⊢ (𝐸 ∈ ℝ → (𝜑 → (𝑡 ∈ 𝑇 ↦ 𝐸) ∈ 𝐴))
19	3, 18	mpcom 39	. . . 4 ⊢ (𝜑 → (𝑡 ∈ 𝑇 ↦ 𝐸) ∈ 𝐴)
20	6, 19	eqeltrid 2869	. . 3 ⊢ (𝜑 → (𝑠 ∈ 𝑇 ↦ 𝐸) ∈ 𝐴)
21		fveq1 6884	. . . . . . . 8 ⊢ (𝑓 = (𝑠 ∈ 𝑇 ↦ 𝐸) → (𝑓‘𝑡) = ((𝑠 ∈ 𝑇 ↦ 𝐸)‘𝑡))
22	21	oveq1d 7434	. . . . . . 7 ⊢ (𝑓 = (𝑠 ∈ 𝑇 ↦ 𝐸) → ((𝑓‘𝑡) · (𝐹‘𝑡)) = (((𝑠 ∈ 𝑇 ↦ 𝐸)‘𝑡) · (𝐹‘𝑡)))
23	22	mpteq2dv 5207	. . . . . 6 ⊢ (𝑓 = (𝑠 ∈ 𝑇 ↦ 𝐸) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) · (𝐹‘𝑡))) = (𝑡 ∈ 𝑇 ↦ (((𝑠 ∈ 𝑇 ↦ 𝐸)‘𝑡) · (𝐹‘𝑡))))
24	23	eleq1d 2850	. . . . 5 ⊢ (𝑓 = (𝑠 ∈ 𝑇 ↦ 𝐸) → ((𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) · (𝐹‘𝑡))) ∈ 𝐴 ↔ (𝑡 ∈ 𝑇 ↦ (((𝑠 ∈ 𝑇 ↦ 𝐸)‘𝑡) · (𝐹‘𝑡))) ∈ 𝐴))
25	24	imbi2d 343	. . . 4 ⊢ (𝑓 = (𝑠 ∈ 𝑇 ↦ 𝐸) → ((𝜑 → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) · (𝐹‘𝑡))) ∈ 𝐴) ↔ (𝜑 → (𝑡 ∈ 𝑇 ↦ (((𝑠 ∈ 𝑇 ↦ 𝐸)‘𝑡) · (𝐹‘𝑡))) ∈ 𝐴)))
26		stoweidlem2.6	. . . . . . 7 ⊢ (𝜑 → 𝐹 ∈ 𝐴)
27	26	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐴) → 𝐹 ∈ 𝐴)
28		fveq1 6884	. . . . . . . . . . 11 ⊢ (𝑔 = 𝐹 → (𝑔‘𝑡) = (𝐹‘𝑡))
29	28	oveq2d 7435	. . . . . . . . . 10 ⊢ (𝑔 = 𝐹 → ((𝑓‘𝑡) · (𝑔‘𝑡)) = ((𝑓‘𝑡) · (𝐹‘𝑡)))
30	29	mpteq2dv 5207	. . . . . . . . 9 ⊢ (𝑔 = 𝐹 → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) · (𝑔‘𝑡))) = (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) · (𝐹‘𝑡))))
31	30	eleq1d 2850	. . . . . . . 8 ⊢ (𝑔 = 𝐹 → ((𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) · (𝑔‘𝑡))) ∈ 𝐴 ↔ (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) · (𝐹‘𝑡))) ∈ 𝐴))
32	31	imbi2d 343	. . . . . . 7 ⊢ (𝑔 = 𝐹 → (((𝜑 ∧ 𝑓 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) · (𝑔‘𝑡))) ∈ 𝐴) ↔ ((𝜑 ∧ 𝑓 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) · (𝐹‘𝑡))) ∈ 𝐴)))
33		stoweidlem2.2	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) · (𝑔‘𝑡))) ∈ 𝐴)
34	33	3comr 1143	. . . . . . . 8 ⊢ ((𝑔 ∈ 𝐴 ∧ 𝜑 ∧ 𝑓 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) · (𝑔‘𝑡))) ∈ 𝐴)
35	34	3expib 1140	. . . . . . 7 ⊢ (𝑔 ∈ 𝐴 → ((𝜑 ∧ 𝑓 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) · (𝑔‘𝑡))) ∈ 𝐴))
36	32, 35	vtoclga 3543	. . . . . 6 ⊢ (𝐹 ∈ 𝐴 → ((𝜑 ∧ 𝑓 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) · (𝐹‘𝑡))) ∈ 𝐴))
37	27, 36	mpcom 39	. . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) · (𝐹‘𝑡))) ∈ 𝐴)
38	37	expcom 419	. . . 4 ⊢ (𝑓 ∈ 𝐴 → (𝜑 → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) · (𝐹‘𝑡))) ∈ 𝐴))
39	25, 38	vtoclga 3543	. . 3 ⊢ ((𝑠 ∈ 𝑇 ↦ 𝐸) ∈ 𝐴 → (𝜑 → (𝑡 ∈ 𝑇 ↦ (((𝑠 ∈ 𝑇 ↦ 𝐸)‘𝑡) · (𝐹‘𝑡))) ∈ 𝐴))
40	20, 39	mpcom 39	. 2 ⊢ (𝜑 → (𝑡 ∈ 𝑇 ↦ (((𝑠 ∈ 𝑇 ↦ 𝐸)‘𝑡) · (𝐹‘𝑡))) ∈ 𝐴)
41	11, 40	eqeltrd 2865	1 ⊢ (𝜑 → (𝑡 ∈ 𝑇 ↦ (𝐸 · (𝐹‘𝑡))) ∈ 𝐴)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1570  Ⅎwnf 1816   ∈ wcel 2146   ↦ cmpt 5194  ⟶wf 6536  ‘cfv 6540  (class class class)co 7419  ℝcr 11118   · cmul 11124

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-sep 5259  ax-nul 5271  ax-pr 5406

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3082  df-rex 3092  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-br 5112  df-opab 5176  df-mpt 5195  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-iota 6496  df-fun 6542  df-fv 6548  df-ov 7422

This theorem is used by:  stoweidlem17  46808