Mathbox for Glauco Siliprandi < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  stoweidlem2 Structured version   Visualization version   GIF version

Theorem stoweidlem2 38895
 Description: lemma for stoweid 38956: 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 𝑡𝜑
stoweidlem2.2 ((𝜑𝑓𝐴𝑔𝐴) → (𝑡𝑇 ↦ ((𝑓𝑡) · (𝑔𝑡))) ∈ 𝐴)
stoweidlem2.3 ((𝜑𝑥 ∈ ℝ) → (𝑡𝑇𝑥) ∈ 𝐴)
stoweidlem2.4 ((𝜑𝑓𝐴) → 𝑓:𝑇⟶ℝ)
stoweidlem2.5 (𝜑𝐸 ∈ ℝ)
stoweidlem2.6 (𝜑𝐹𝐴)
Assertion
Ref Expression
stoweidlem2 (𝜑 → (𝑡𝑇 ↦ (𝐸 · (𝐹𝑡))) ∈ 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.
StepHypRef Expression
1 stoweidlem2.1 . . 3 𝑡𝜑
2 simpr 476 . . . . . 6 ((𝜑𝑡𝑇) → 𝑡𝑇)
3 stoweidlem2.5 . . . . . . 7 (𝜑𝐸 ∈ ℝ)
43adantr 480 . . . . . 6 ((𝜑𝑡𝑇) → 𝐸 ∈ ℝ)
5 eqidd 2611 . . . . . . . 8 (𝑠 = 𝑡𝐸 = 𝐸)
65cbvmptv 4678 . . . . . . 7 (𝑠𝑇𝐸) = (𝑡𝑇𝐸)
76fvmpt2 6200 . . . . . 6 ((𝑡𝑇𝐸 ∈ ℝ) → ((𝑠𝑇𝐸)‘𝑡) = 𝐸)
82, 4, 7syl2anc 691 . . . . 5 ((𝜑𝑡𝑇) → ((𝑠𝑇𝐸)‘𝑡) = 𝐸)
98eqcomd 2616 . . . 4 ((𝜑𝑡𝑇) → 𝐸 = ((𝑠𝑇𝐸)‘𝑡))
109oveq1d 6564 . . 3 ((𝜑𝑡𝑇) → (𝐸 · (𝐹𝑡)) = (((𝑠𝑇𝐸)‘𝑡) · (𝐹𝑡)))
111, 10mpteq2da 4671 . 2 (𝜑 → (𝑡𝑇 ↦ (𝐸 · (𝐹𝑡))) = (𝑡𝑇 ↦ (((𝑠𝑇𝐸)‘𝑡) · (𝐹𝑡))))
12 id 22 . . . . . . . . 9 (𝑥 = 𝐸𝑥 = 𝐸)
1312mpteq2dv 4673 . . . . . . . 8 (𝑥 = 𝐸 → (𝑡𝑇𝑥) = (𝑡𝑇𝐸))
1413eleq1d 2672 . . . . . . 7 (𝑥 = 𝐸 → ((𝑡𝑇𝑥) ∈ 𝐴 ↔ (𝑡𝑇𝐸) ∈ 𝐴))
1514imbi2d 329 . . . . . 6 (𝑥 = 𝐸 → ((𝜑 → (𝑡𝑇𝑥) ∈ 𝐴) ↔ (𝜑 → (𝑡𝑇𝐸) ∈ 𝐴)))
16 stoweidlem2.3 . . . . . . 7 ((𝜑𝑥 ∈ ℝ) → (𝑡𝑇𝑥) ∈ 𝐴)
1716expcom 450 . . . . . 6 (𝑥 ∈ ℝ → (𝜑 → (𝑡𝑇𝑥) ∈ 𝐴))
1815, 17vtoclga 3245 . . . . 5 (𝐸 ∈ ℝ → (𝜑 → (𝑡𝑇𝐸) ∈ 𝐴))
193, 18mpcom 37 . . . 4 (𝜑 → (𝑡𝑇𝐸) ∈ 𝐴)
206, 19syl5eqel 2692 . . 3 (𝜑 → (𝑠𝑇𝐸) ∈ 𝐴)
21 fveq1 6102 . . . . . . . 8 (𝑓 = (𝑠𝑇𝐸) → (𝑓𝑡) = ((𝑠𝑇𝐸)‘𝑡))
2221oveq1d 6564 . . . . . . 7 (𝑓 = (𝑠𝑇𝐸) → ((𝑓𝑡) · (𝐹𝑡)) = (((𝑠𝑇𝐸)‘𝑡) · (𝐹𝑡)))
2322mpteq2dv 4673 . . . . . 6 (𝑓 = (𝑠𝑇𝐸) → (𝑡𝑇 ↦ ((𝑓𝑡) · (𝐹𝑡))) = (𝑡𝑇 ↦ (((𝑠𝑇𝐸)‘𝑡) · (𝐹𝑡))))
2423eleq1d 2672 . . . . 5 (𝑓 = (𝑠𝑇𝐸) → ((𝑡𝑇 ↦ ((𝑓𝑡) · (𝐹𝑡))) ∈ 𝐴 ↔ (𝑡𝑇 ↦ (((𝑠𝑇𝐸)‘𝑡) · (𝐹𝑡))) ∈ 𝐴))
2524imbi2d 329 . . . 4 (𝑓 = (𝑠𝑇𝐸) → ((𝜑 → (𝑡𝑇 ↦ ((𝑓𝑡) · (𝐹𝑡))) ∈ 𝐴) ↔ (𝜑 → (𝑡𝑇 ↦ (((𝑠𝑇𝐸)‘𝑡) · (𝐹𝑡))) ∈ 𝐴)))
26 stoweidlem2.6 . . . . . . 7 (𝜑𝐹𝐴)
2726adantr 480 . . . . . 6 ((𝜑𝑓𝐴) → 𝐹𝐴)
28 fveq1 6102 . . . . . . . . . . 11 (𝑔 = 𝐹 → (𝑔𝑡) = (𝐹𝑡))
2928oveq2d 6565 . . . . . . . . . 10 (𝑔 = 𝐹 → ((𝑓𝑡) · (𝑔𝑡)) = ((𝑓𝑡) · (𝐹𝑡)))
3029mpteq2dv 4673 . . . . . . . . 9 (𝑔 = 𝐹 → (𝑡𝑇 ↦ ((𝑓𝑡) · (𝑔𝑡))) = (𝑡𝑇 ↦ ((𝑓𝑡) · (𝐹𝑡))))
3130eleq1d 2672 . . . . . . . 8 (𝑔 = 𝐹 → ((𝑡𝑇 ↦ ((𝑓𝑡) · (𝑔𝑡))) ∈ 𝐴 ↔ (𝑡𝑇 ↦ ((𝑓𝑡) · (𝐹𝑡))) ∈ 𝐴))
3231imbi2d 329 . . . . . . 7 (𝑔 = 𝐹 → (((𝜑𝑓𝐴) → (𝑡𝑇 ↦ ((𝑓𝑡) · (𝑔𝑡))) ∈ 𝐴) ↔ ((𝜑𝑓𝐴) → (𝑡𝑇 ↦ ((𝑓𝑡) · (𝐹𝑡))) ∈ 𝐴)))
33 stoweidlem2.2 . . . . . . . . 9 ((𝜑𝑓𝐴𝑔𝐴) → (𝑡𝑇 ↦ ((𝑓𝑡) · (𝑔𝑡))) ∈ 𝐴)
34333comr 1265 . . . . . . . 8 ((𝑔𝐴𝜑𝑓𝐴) → (𝑡𝑇 ↦ ((𝑓𝑡) · (𝑔𝑡))) ∈ 𝐴)
35343expib 1260 . . . . . . 7 (𝑔𝐴 → ((𝜑𝑓𝐴) → (𝑡𝑇 ↦ ((𝑓𝑡) · (𝑔𝑡))) ∈ 𝐴))
3632, 35vtoclga 3245 . . . . . 6 (𝐹𝐴 → ((𝜑𝑓𝐴) → (𝑡𝑇 ↦ ((𝑓𝑡) · (𝐹𝑡))) ∈ 𝐴))
3727, 36mpcom 37 . . . . 5 ((𝜑𝑓𝐴) → (𝑡𝑇 ↦ ((𝑓𝑡) · (𝐹𝑡))) ∈ 𝐴)
3837expcom 450 . . . 4 (𝑓𝐴 → (𝜑 → (𝑡𝑇 ↦ ((𝑓𝑡) · (𝐹𝑡))) ∈ 𝐴))
3925, 38vtoclga 3245 . . 3 ((𝑠𝑇𝐸) ∈ 𝐴 → (𝜑 → (𝑡𝑇 ↦ (((𝑠𝑇𝐸)‘𝑡) · (𝐹𝑡))) ∈ 𝐴))
4020, 39mpcom 37 . 2 (𝜑 → (𝑡𝑇 ↦ (((𝑠𝑇𝐸)‘𝑡) · (𝐹𝑡))) ∈ 𝐴)
4111, 40eqeltrd 2688 1 (𝜑 → (𝑡𝑇 ↦ (𝐸 · (𝐹𝑡))) ∈ 𝐴)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   ∧ w3a 1031   = wceq 1475  Ⅎwnf 1699   ∈ wcel 1977   ↦ cmpt 4643  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549  ℝcr 9814   · cmul 9820 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fv 5812  df-ov 6552 This theorem is referenced by:  stoweidlem17  38910
 Copyright terms: Public domain W3C validator