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Theorem stoweidlem2 42164
Description: lemma for stoweid 42225: 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 485 . . . . . 6 ((𝜑𝑡𝑇) → 𝑡𝑇)
3 stoweidlem2.5 . . . . . . 7 (𝜑𝐸 ∈ ℝ)
43adantr 481 . . . . . 6 ((𝜑𝑡𝑇) → 𝐸 ∈ ℝ)
5 eqidd 2819 . . . . . . . 8 (𝑠 = 𝑡𝐸 = 𝐸)
65cbvmptv 5160 . . . . . . 7 (𝑠𝑇𝐸) = (𝑡𝑇𝐸)
76fvmpt2 6771 . . . . . 6 ((𝑡𝑇𝐸 ∈ ℝ) → ((𝑠𝑇𝐸)‘𝑡) = 𝐸)
82, 4, 7syl2anc 584 . . . . 5 ((𝜑𝑡𝑇) → ((𝑠𝑇𝐸)‘𝑡) = 𝐸)
98eqcomd 2824 . . . 4 ((𝜑𝑡𝑇) → 𝐸 = ((𝑠𝑇𝐸)‘𝑡))
109oveq1d 7160 . . 3 ((𝜑𝑡𝑇) → (𝐸 · (𝐹𝑡)) = (((𝑠𝑇𝐸)‘𝑡) · (𝐹𝑡)))
111, 10mpteq2da 5151 . 2 (𝜑 → (𝑡𝑇 ↦ (𝐸 · (𝐹𝑡))) = (𝑡𝑇 ↦ (((𝑠𝑇𝐸)‘𝑡) · (𝐹𝑡))))
12 id 22 . . . . . . . . 9 (𝑥 = 𝐸𝑥 = 𝐸)
1312mpteq2dv 5153 . . . . . . . 8 (𝑥 = 𝐸 → (𝑡𝑇𝑥) = (𝑡𝑇𝐸))
1413eleq1d 2894 . . . . . . 7 (𝑥 = 𝐸 → ((𝑡𝑇𝑥) ∈ 𝐴 ↔ (𝑡𝑇𝐸) ∈ 𝐴))
1514imbi2d 342 . . . . . 6 (𝑥 = 𝐸 → ((𝜑 → (𝑡𝑇𝑥) ∈ 𝐴) ↔ (𝜑 → (𝑡𝑇𝐸) ∈ 𝐴)))
16 stoweidlem2.3 . . . . . . 7 ((𝜑𝑥 ∈ ℝ) → (𝑡𝑇𝑥) ∈ 𝐴)
1716expcom 414 . . . . . 6 (𝑥 ∈ ℝ → (𝜑 → (𝑡𝑇𝑥) ∈ 𝐴))
1815, 17vtoclga 3571 . . . . 5 (𝐸 ∈ ℝ → (𝜑 → (𝑡𝑇𝐸) ∈ 𝐴))
193, 18mpcom 38 . . . 4 (𝜑 → (𝑡𝑇𝐸) ∈ 𝐴)
206, 19eqeltrid 2914 . . 3 (𝜑 → (𝑠𝑇𝐸) ∈ 𝐴)
21 fveq1 6662 . . . . . . . 8 (𝑓 = (𝑠𝑇𝐸) → (𝑓𝑡) = ((𝑠𝑇𝐸)‘𝑡))
2221oveq1d 7160 . . . . . . 7 (𝑓 = (𝑠𝑇𝐸) → ((𝑓𝑡) · (𝐹𝑡)) = (((𝑠𝑇𝐸)‘𝑡) · (𝐹𝑡)))
2322mpteq2dv 5153 . . . . . 6 (𝑓 = (𝑠𝑇𝐸) → (𝑡𝑇 ↦ ((𝑓𝑡) · (𝐹𝑡))) = (𝑡𝑇 ↦ (((𝑠𝑇𝐸)‘𝑡) · (𝐹𝑡))))
2423eleq1d 2894 . . . . 5 (𝑓 = (𝑠𝑇𝐸) → ((𝑡𝑇 ↦ ((𝑓𝑡) · (𝐹𝑡))) ∈ 𝐴 ↔ (𝑡𝑇 ↦ (((𝑠𝑇𝐸)‘𝑡) · (𝐹𝑡))) ∈ 𝐴))
2524imbi2d 342 . . . 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 6662 . . . . . . . . . . 11 (𝑔 = 𝐹 → (𝑔𝑡) = (𝐹𝑡))
2928oveq2d 7161 . . . . . . . . . 10 (𝑔 = 𝐹 → ((𝑓𝑡) · (𝑔𝑡)) = ((𝑓𝑡) · (𝐹𝑡)))
3029mpteq2dv 5153 . . . . . . . . 9 (𝑔 = 𝐹 → (𝑡𝑇 ↦ ((𝑓𝑡) · (𝑔𝑡))) = (𝑡𝑇 ↦ ((𝑓𝑡) · (𝐹𝑡))))
3130eleq1d 2894 . . . . . . . 8 (𝑔 = 𝐹 → ((𝑡𝑇 ↦ ((𝑓𝑡) · (𝑔𝑡))) ∈ 𝐴 ↔ (𝑡𝑇 ↦ ((𝑓𝑡) · (𝐹𝑡))) ∈ 𝐴))
3231imbi2d 342 . . . . . . 7 (𝑔 = 𝐹 → (((𝜑𝑓𝐴) → (𝑡𝑇 ↦ ((𝑓𝑡) · (𝑔𝑡))) ∈ 𝐴) ↔ ((𝜑𝑓𝐴) → (𝑡𝑇 ↦ ((𝑓𝑡) · (𝐹𝑡))) ∈ 𝐴)))
33 stoweidlem2.2 . . . . . . . . 9 ((𝜑𝑓𝐴𝑔𝐴) → (𝑡𝑇 ↦ ((𝑓𝑡) · (𝑔𝑡))) ∈ 𝐴)
34333comr 1117 . . . . . . . 8 ((𝑔𝐴𝜑𝑓𝐴) → (𝑡𝑇 ↦ ((𝑓𝑡) · (𝑔𝑡))) ∈ 𝐴)
35343expib 1114 . . . . . . 7 (𝑔𝐴 → ((𝜑𝑓𝐴) → (𝑡𝑇 ↦ ((𝑓𝑡) · (𝑔𝑡))) ∈ 𝐴))
3632, 35vtoclga 3571 . . . . . 6 (𝐹𝐴 → ((𝜑𝑓𝐴) → (𝑡𝑇 ↦ ((𝑓𝑡) · (𝐹𝑡))) ∈ 𝐴))
3727, 36mpcom 38 . . . . 5 ((𝜑𝑓𝐴) → (𝑡𝑇 ↦ ((𝑓𝑡) · (𝐹𝑡))) ∈ 𝐴)
3837expcom 414 . . . 4 (𝑓𝐴 → (𝜑 → (𝑡𝑇 ↦ ((𝑓𝑡) · (𝐹𝑡))) ∈ 𝐴))
3925, 38vtoclga 3571 . . 3 ((𝑠𝑇𝐸) ∈ 𝐴 → (𝜑 → (𝑡𝑇 ↦ (((𝑠𝑇𝐸)‘𝑡) · (𝐹𝑡))) ∈ 𝐴))
4020, 39mpcom 38 . 2 (𝜑 → (𝑡𝑇 ↦ (((𝑠𝑇𝐸)‘𝑡) · (𝐹𝑡))) ∈ 𝐴)
4111, 40eqeltrd 2910 1 (𝜑 → (𝑡𝑇 ↦ (𝐸 · (𝐹𝑡))) ∈ 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1079   = wceq 1528  wnf 1775  wcel 2105  cmpt 5137  wf 6344  cfv 6348  (class class class)co 7145  cr 10524   · cmul 10530
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fv 6356  df-ov 7148
This theorem is referenced by:  stoweidlem17  42179
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