Theorem stoweidlem2 44718

Description: lemma for stoweid 44779: 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 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → 𝑡 ∈ 𝑇)
3		stoweidlem2.5	. . . . . . 7 ⊢ (𝜑 → 𝐸 ∈ ℝ)
4	3	adantr 482	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → 𝐸 ∈ ℝ)
5		eqidd 2734	. . . . . . . 8 ⊢ (𝑠 = 𝑡 → 𝐸 = 𝐸)
6	5	cbvmptv 5262	. . . . . . 7 ⊢ (𝑠 ∈ 𝑇 ↦ 𝐸) = (𝑡 ∈ 𝑇 ↦ 𝐸)
7	6	fvmpt2 7010	. . . . . 6 ⊢ ((𝑡 ∈ 𝑇 ∧ 𝐸 ∈ ℝ) → ((𝑠 ∈ 𝑇 ↦ 𝐸)‘𝑡) = 𝐸)
8	2, 4, 7	syl2anc 585	. . . . 5 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → ((𝑠 ∈ 𝑇 ↦ 𝐸)‘𝑡) = 𝐸)
9	8	eqcomd 2739	. . . 4 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → 𝐸 = ((𝑠 ∈ 𝑇 ↦ 𝐸)‘𝑡))
10	9	oveq1d 7424	. . 3 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (𝐸 · (𝐹‘𝑡)) = (((𝑠 ∈ 𝑇 ↦ 𝐸)‘𝑡) · (𝐹‘𝑡)))
11	1, 10	mpteq2da 5247	. 2 ⊢ (𝜑 → (𝑡 ∈ 𝑇 ↦ (𝐸 · (𝐹‘𝑡))) = (𝑡 ∈ 𝑇 ↦ (((𝑠 ∈ 𝑇 ↦ 𝐸)‘𝑡) · (𝐹‘𝑡))))
12		id 22	. . . . . . . . 9 ⊢ (𝑥 = 𝐸 → 𝑥 = 𝐸)
13	12	mpteq2dv 5251	. . . . . . . 8 ⊢ (𝑥 = 𝐸 → (𝑡 ∈ 𝑇 ↦ 𝑥) = (𝑡 ∈ 𝑇 ↦ 𝐸))
14	13	eleq1d 2819	. . . . . . 7 ⊢ (𝑥 = 𝐸 → ((𝑡 ∈ 𝑇 ↦ 𝑥) ∈ 𝐴 ↔ (𝑡 ∈ 𝑇 ↦ 𝐸) ∈ 𝐴))
15	14	imbi2d 341	. . . . . 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 415	. . . . . 6 ⊢ (𝑥 ∈ ℝ → (𝜑 → (𝑡 ∈ 𝑇 ↦ 𝑥) ∈ 𝐴))
18	15, 17	vtoclga 3566	. . . . 5 ⊢ (𝐸 ∈ ℝ → (𝜑 → (𝑡 ∈ 𝑇 ↦ 𝐸) ∈ 𝐴))
19	3, 18	mpcom 38	. . . 4 ⊢ (𝜑 → (𝑡 ∈ 𝑇 ↦ 𝐸) ∈ 𝐴)
20	6, 19	eqeltrid 2838	. . 3 ⊢ (𝜑 → (𝑠 ∈ 𝑇 ↦ 𝐸) ∈ 𝐴)
21		fveq1 6891	. . . . . . . 8 ⊢ (𝑓 = (𝑠 ∈ 𝑇 ↦ 𝐸) → (𝑓‘𝑡) = ((𝑠 ∈ 𝑇 ↦ 𝐸)‘𝑡))
22	21	oveq1d 7424	. . . . . . 7 ⊢ (𝑓 = (𝑠 ∈ 𝑇 ↦ 𝐸) → ((𝑓‘𝑡) · (𝐹‘𝑡)) = (((𝑠 ∈ 𝑇 ↦ 𝐸)‘𝑡) · (𝐹‘𝑡)))
23	22	mpteq2dv 5251	. . . . . 6 ⊢ (𝑓 = (𝑠 ∈ 𝑇 ↦ 𝐸) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) · (𝐹‘𝑡))) = (𝑡 ∈ 𝑇 ↦ (((𝑠 ∈ 𝑇 ↦ 𝐸)‘𝑡) · (𝐹‘𝑡))))
24	23	eleq1d 2819	. . . . 5 ⊢ (𝑓 = (𝑠 ∈ 𝑇 ↦ 𝐸) → ((𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) · (𝐹‘𝑡))) ∈ 𝐴 ↔ (𝑡 ∈ 𝑇 ↦ (((𝑠 ∈ 𝑇 ↦ 𝐸)‘𝑡) · (𝐹‘𝑡))) ∈ 𝐴))
25	24	imbi2d 341	. . . 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 482	. . . . . 6 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐴) → 𝐹 ∈ 𝐴)
28		fveq1 6891	. . . . . . . . . . 11 ⊢ (𝑔 = 𝐹 → (𝑔‘𝑡) = (𝐹‘𝑡))
29	28	oveq2d 7425	. . . . . . . . . 10 ⊢ (𝑔 = 𝐹 → ((𝑓‘𝑡) · (𝑔‘𝑡)) = ((𝑓‘𝑡) · (𝐹‘𝑡)))
30	29	mpteq2dv 5251	. . . . . . . . 9 ⊢ (𝑔 = 𝐹 → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) · (𝑔‘𝑡))) = (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) · (𝐹‘𝑡))))
31	30	eleq1d 2819	. . . . . . . 8 ⊢ (𝑔 = 𝐹 → ((𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) · (𝑔‘𝑡))) ∈ 𝐴 ↔ (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) · (𝐹‘𝑡))) ∈ 𝐴))
32	31	imbi2d 341	. . . . . . 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 1126	. . . . . . . 8 ⊢ ((𝑔 ∈ 𝐴 ∧ 𝜑 ∧ 𝑓 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) · (𝑔‘𝑡))) ∈ 𝐴)
35	34	3expib 1123	. . . . . . 7 ⊢ (𝑔 ∈ 𝐴 → ((𝜑 ∧ 𝑓 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) · (𝑔‘𝑡))) ∈ 𝐴))
36	32, 35	vtoclga 3566	. . . . . 6 ⊢ (𝐹 ∈ 𝐴 → ((𝜑 ∧ 𝑓 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) · (𝐹‘𝑡))) ∈ 𝐴))
37	27, 36	mpcom 38	. . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) · (𝐹‘𝑡))) ∈ 𝐴)
38	37	expcom 415	. . . 4 ⊢ (𝑓 ∈ 𝐴 → (𝜑 → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) · (𝐹‘𝑡))) ∈ 𝐴))
39	25, 38	vtoclga 3566	. . 3 ⊢ ((𝑠 ∈ 𝑇 ↦ 𝐸) ∈ 𝐴 → (𝜑 → (𝑡 ∈ 𝑇 ↦ (((𝑠 ∈ 𝑇 ↦ 𝐸)‘𝑡) · (𝐹‘𝑡))) ∈ 𝐴))
40	20, 39	mpcom 38	. 2 ⊢ (𝜑 → (𝑡 ∈ 𝑇 ↦ (((𝑠 ∈ 𝑇 ↦ 𝐸)‘𝑡) · (𝐹‘𝑡))) ∈ 𝐴)
41	11, 40	eqeltrd 2834	1 ⊢ (𝜑 → (𝑡 ∈ 𝑇 ↦ (𝐸 · (𝐹‘𝑡))) ∈ 𝐴)

Syntax hints:   → wi 4   ∧ wa 397   ∧ w3a 1088   = wceq 1542  Ⅎwnf 1786   ∈ wcel 2107   ↦ cmpt 5232  ⟶wf 6540  ‘cfv 6544  (class class class)co 7409  ℝcr 11109   · cmul 11115

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fv 6552  df-ov 7412

This theorem is referenced by:  stoweidlem17  44733