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Theorem stoweidlem2 39523
Description: lemma for stoweid 39584: 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 477 . . . . . 6 ((𝜑𝑡𝑇) → 𝑡𝑇)
3 stoweidlem2.5 . . . . . . 7 (𝜑𝐸 ∈ ℝ)
43adantr 481 . . . . . 6 ((𝜑𝑡𝑇) → 𝐸 ∈ ℝ)
5 eqidd 2622 . . . . . . . 8 (𝑠 = 𝑡𝐸 = 𝐸)
65cbvmptv 4710 . . . . . . 7 (𝑠𝑇𝐸) = (𝑡𝑇𝐸)
76fvmpt2 6248 . . . . . 6 ((𝑡𝑇𝐸 ∈ ℝ) → ((𝑠𝑇𝐸)‘𝑡) = 𝐸)
82, 4, 7syl2anc 692 . . . . 5 ((𝜑𝑡𝑇) → ((𝑠𝑇𝐸)‘𝑡) = 𝐸)
98eqcomd 2627 . . . 4 ((𝜑𝑡𝑇) → 𝐸 = ((𝑠𝑇𝐸)‘𝑡))
109oveq1d 6619 . . 3 ((𝜑𝑡𝑇) → (𝐸 · (𝐹𝑡)) = (((𝑠𝑇𝐸)‘𝑡) · (𝐹𝑡)))
111, 10mpteq2da 4703 . 2 (𝜑 → (𝑡𝑇 ↦ (𝐸 · (𝐹𝑡))) = (𝑡𝑇 ↦ (((𝑠𝑇𝐸)‘𝑡) · (𝐹𝑡))))
12 id 22 . . . . . . . . 9 (𝑥 = 𝐸𝑥 = 𝐸)
1312mpteq2dv 4705 . . . . . . . 8 (𝑥 = 𝐸 → (𝑡𝑇𝑥) = (𝑡𝑇𝐸))
1413eleq1d 2683 . . . . . . 7 (𝑥 = 𝐸 → ((𝑡𝑇𝑥) ∈ 𝐴 ↔ (𝑡𝑇𝐸) ∈ 𝐴))
1514imbi2d 330 . . . . . 6 (𝑥 = 𝐸 → ((𝜑 → (𝑡𝑇𝑥) ∈ 𝐴) ↔ (𝜑 → (𝑡𝑇𝐸) ∈ 𝐴)))
16 stoweidlem2.3 . . . . . . 7 ((𝜑𝑥 ∈ ℝ) → (𝑡𝑇𝑥) ∈ 𝐴)
1716expcom 451 . . . . . 6 (𝑥 ∈ ℝ → (𝜑 → (𝑡𝑇𝑥) ∈ 𝐴))
1815, 17vtoclga 3258 . . . . 5 (𝐸 ∈ ℝ → (𝜑 → (𝑡𝑇𝐸) ∈ 𝐴))
193, 18mpcom 38 . . . 4 (𝜑 → (𝑡𝑇𝐸) ∈ 𝐴)
206, 19syl5eqel 2702 . . 3 (𝜑 → (𝑠𝑇𝐸) ∈ 𝐴)
21 fveq1 6147 . . . . . . . 8 (𝑓 = (𝑠𝑇𝐸) → (𝑓𝑡) = ((𝑠𝑇𝐸)‘𝑡))
2221oveq1d 6619 . . . . . . 7 (𝑓 = (𝑠𝑇𝐸) → ((𝑓𝑡) · (𝐹𝑡)) = (((𝑠𝑇𝐸)‘𝑡) · (𝐹𝑡)))
2322mpteq2dv 4705 . . . . . 6 (𝑓 = (𝑠𝑇𝐸) → (𝑡𝑇 ↦ ((𝑓𝑡) · (𝐹𝑡))) = (𝑡𝑇 ↦ (((𝑠𝑇𝐸)‘𝑡) · (𝐹𝑡))))
2423eleq1d 2683 . . . . 5 (𝑓 = (𝑠𝑇𝐸) → ((𝑡𝑇 ↦ ((𝑓𝑡) · (𝐹𝑡))) ∈ 𝐴 ↔ (𝑡𝑇 ↦ (((𝑠𝑇𝐸)‘𝑡) · (𝐹𝑡))) ∈ 𝐴))
2524imbi2d 330 . . . 4 (𝑓 = (𝑠𝑇𝐸) → ((𝜑 → (𝑡𝑇 ↦ ((𝑓𝑡) · (𝐹𝑡))) ∈ 𝐴) ↔ (𝜑 → (𝑡𝑇 ↦ (((𝑠𝑇𝐸)‘𝑡) · (𝐹𝑡))) ∈ 𝐴)))
26 stoweidlem2.6 . . . . . . 7 (𝜑𝐹𝐴)
2726adantr 481 . . . . . 6 ((𝜑𝑓𝐴) → 𝐹𝐴)
28 fveq1 6147 . . . . . . . . . . 11 (𝑔 = 𝐹 → (𝑔𝑡) = (𝐹𝑡))
2928oveq2d 6620 . . . . . . . . . 10 (𝑔 = 𝐹 → ((𝑓𝑡) · (𝑔𝑡)) = ((𝑓𝑡) · (𝐹𝑡)))
3029mpteq2dv 4705 . . . . . . . . 9 (𝑔 = 𝐹 → (𝑡𝑇 ↦ ((𝑓𝑡) · (𝑔𝑡))) = (𝑡𝑇 ↦ ((𝑓𝑡) · (𝐹𝑡))))
3130eleq1d 2683 . . . . . . . 8 (𝑔 = 𝐹 → ((𝑡𝑇 ↦ ((𝑓𝑡) · (𝑔𝑡))) ∈ 𝐴 ↔ (𝑡𝑇 ↦ ((𝑓𝑡) · (𝐹𝑡))) ∈ 𝐴))
3231imbi2d 330 . . . . . . 7 (𝑔 = 𝐹 → (((𝜑𝑓𝐴) → (𝑡𝑇 ↦ ((𝑓𝑡) · (𝑔𝑡))) ∈ 𝐴) ↔ ((𝜑𝑓𝐴) → (𝑡𝑇 ↦ ((𝑓𝑡) · (𝐹𝑡))) ∈ 𝐴)))
33 stoweidlem2.2 . . . . . . . . 9 ((𝜑𝑓𝐴𝑔𝐴) → (𝑡𝑇 ↦ ((𝑓𝑡) · (𝑔𝑡))) ∈ 𝐴)
34333comr 1270 . . . . . . . 8 ((𝑔𝐴𝜑𝑓𝐴) → (𝑡𝑇 ↦ ((𝑓𝑡) · (𝑔𝑡))) ∈ 𝐴)
35343expib 1265 . . . . . . 7 (𝑔𝐴 → ((𝜑𝑓𝐴) → (𝑡𝑇 ↦ ((𝑓𝑡) · (𝑔𝑡))) ∈ 𝐴))
3632, 35vtoclga 3258 . . . . . 6 (𝐹𝐴 → ((𝜑𝑓𝐴) → (𝑡𝑇 ↦ ((𝑓𝑡) · (𝐹𝑡))) ∈ 𝐴))
3727, 36mpcom 38 . . . . 5 ((𝜑𝑓𝐴) → (𝑡𝑇 ↦ ((𝑓𝑡) · (𝐹𝑡))) ∈ 𝐴)
3837expcom 451 . . . 4 (𝑓𝐴 → (𝜑 → (𝑡𝑇 ↦ ((𝑓𝑡) · (𝐹𝑡))) ∈ 𝐴))
3925, 38vtoclga 3258 . . 3 ((𝑠𝑇𝐸) ∈ 𝐴 → (𝜑 → (𝑡𝑇 ↦ (((𝑠𝑇𝐸)‘𝑡) · (𝐹𝑡))) ∈ 𝐴))
4020, 39mpcom 38 . 2 (𝜑 → (𝑡𝑇 ↦ (((𝑠𝑇𝐸)‘𝑡) · (𝐹𝑡))) ∈ 𝐴)
4111, 40eqeltrd 2698 1 (𝜑 → (𝑡𝑇 ↦ (𝐸 · (𝐹𝑡))) ∈ 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  w3a 1036   = wceq 1480  wnf 1705  wcel 1987  cmpt 4673  wf 5843  cfv 5847  (class class class)co 6604  cr 9879   · cmul 9885
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fv 5855  df-ov 6607
This theorem is referenced by:  stoweidlem17  39538
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