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Theorem stoweidlem2 44708
Description: lemma for stoweid 44769: 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 β„²π‘‘πœ‘
stoweidlem2.2 ((πœ‘ ∧ 𝑓 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴) β†’ (𝑑 ∈ 𝑇 ↦ ((π‘“β€˜π‘‘) Β· (π‘”β€˜π‘‘))) ∈ 𝐴)
stoweidlem2.3 ((πœ‘ ∧ π‘₯ ∈ ℝ) β†’ (𝑑 ∈ 𝑇 ↦ π‘₯) ∈ 𝐴)
stoweidlem2.4 ((πœ‘ ∧ 𝑓 ∈ 𝐴) β†’ 𝑓:π‘‡βŸΆβ„)
stoweidlem2.5 (πœ‘ β†’ 𝐸 ∈ ℝ)
stoweidlem2.6 (πœ‘ β†’ 𝐹 ∈ 𝐴)
Assertion
Ref Expression
stoweidlem2 (πœ‘ β†’ (𝑑 ∈ 𝑇 ↦ (𝐸 Β· (πΉβ€˜π‘‘))) ∈ 𝐴)
Distinct variable groups:   𝑓,𝑔,𝑑,𝐹   𝑓,𝐸,𝑑   𝐴,𝑓,𝑔   𝑇,𝑓,𝑔,𝑑   πœ‘,𝑓,𝑔   π‘₯,𝑑,𝐸   π‘₯,𝐴   π‘₯,𝑇   πœ‘,π‘₯
Allowed substitution hints:   πœ‘(𝑑)   𝐴(𝑑)   𝐸(𝑔)   𝐹(π‘₯)

Proof of Theorem stoweidlem2
Dummy variable 𝑠 is distinct from all other variables.
StepHypRef Expression
1 stoweidlem2.1 . . 3 β„²π‘‘πœ‘
2 simpr 485 . . . . . 6 ((πœ‘ ∧ 𝑑 ∈ 𝑇) β†’ 𝑑 ∈ 𝑇)
3 stoweidlem2.5 . . . . . . 7 (πœ‘ β†’ 𝐸 ∈ ℝ)
43adantr 481 . . . . . 6 ((πœ‘ ∧ 𝑑 ∈ 𝑇) β†’ 𝐸 ∈ ℝ)
5 eqidd 2733 . . . . . . . 8 (𝑠 = 𝑑 β†’ 𝐸 = 𝐸)
65cbvmptv 5261 . . . . . . 7 (𝑠 ∈ 𝑇 ↦ 𝐸) = (𝑑 ∈ 𝑇 ↦ 𝐸)
76fvmpt2 7009 . . . . . 6 ((𝑑 ∈ 𝑇 ∧ 𝐸 ∈ ℝ) β†’ ((𝑠 ∈ 𝑇 ↦ 𝐸)β€˜π‘‘) = 𝐸)
82, 4, 7syl2anc 584 . . . . 5 ((πœ‘ ∧ 𝑑 ∈ 𝑇) β†’ ((𝑠 ∈ 𝑇 ↦ 𝐸)β€˜π‘‘) = 𝐸)
98eqcomd 2738 . . . 4 ((πœ‘ ∧ 𝑑 ∈ 𝑇) β†’ 𝐸 = ((𝑠 ∈ 𝑇 ↦ 𝐸)β€˜π‘‘))
109oveq1d 7423 . . 3 ((πœ‘ ∧ 𝑑 ∈ 𝑇) β†’ (𝐸 Β· (πΉβ€˜π‘‘)) = (((𝑠 ∈ 𝑇 ↦ 𝐸)β€˜π‘‘) Β· (πΉβ€˜π‘‘)))
111, 10mpteq2da 5246 . 2 (πœ‘ β†’ (𝑑 ∈ 𝑇 ↦ (𝐸 Β· (πΉβ€˜π‘‘))) = (𝑑 ∈ 𝑇 ↦ (((𝑠 ∈ 𝑇 ↦ 𝐸)β€˜π‘‘) Β· (πΉβ€˜π‘‘))))
12 id 22 . . . . . . . . 9 (π‘₯ = 𝐸 β†’ π‘₯ = 𝐸)
1312mpteq2dv 5250 . . . . . . . 8 (π‘₯ = 𝐸 β†’ (𝑑 ∈ 𝑇 ↦ π‘₯) = (𝑑 ∈ 𝑇 ↦ 𝐸))
1413eleq1d 2818 . . . . . . 7 (π‘₯ = 𝐸 β†’ ((𝑑 ∈ 𝑇 ↦ π‘₯) ∈ 𝐴 ↔ (𝑑 ∈ 𝑇 ↦ 𝐸) ∈ 𝐴))
1514imbi2d 340 . . . . . 6 (π‘₯ = 𝐸 β†’ ((πœ‘ β†’ (𝑑 ∈ 𝑇 ↦ π‘₯) ∈ 𝐴) ↔ (πœ‘ β†’ (𝑑 ∈ 𝑇 ↦ 𝐸) ∈ 𝐴)))
16 stoweidlem2.3 . . . . . . 7 ((πœ‘ ∧ π‘₯ ∈ ℝ) β†’ (𝑑 ∈ 𝑇 ↦ π‘₯) ∈ 𝐴)
1716expcom 414 . . . . . 6 (π‘₯ ∈ ℝ β†’ (πœ‘ β†’ (𝑑 ∈ 𝑇 ↦ π‘₯) ∈ 𝐴))
1815, 17vtoclga 3565 . . . . 5 (𝐸 ∈ ℝ β†’ (πœ‘ β†’ (𝑑 ∈ 𝑇 ↦ 𝐸) ∈ 𝐴))
193, 18mpcom 38 . . . 4 (πœ‘ β†’ (𝑑 ∈ 𝑇 ↦ 𝐸) ∈ 𝐴)
206, 19eqeltrid 2837 . . 3 (πœ‘ β†’ (𝑠 ∈ 𝑇 ↦ 𝐸) ∈ 𝐴)
21 fveq1 6890 . . . . . . . 8 (𝑓 = (𝑠 ∈ 𝑇 ↦ 𝐸) β†’ (π‘“β€˜π‘‘) = ((𝑠 ∈ 𝑇 ↦ 𝐸)β€˜π‘‘))
2221oveq1d 7423 . . . . . . 7 (𝑓 = (𝑠 ∈ 𝑇 ↦ 𝐸) β†’ ((π‘“β€˜π‘‘) Β· (πΉβ€˜π‘‘)) = (((𝑠 ∈ 𝑇 ↦ 𝐸)β€˜π‘‘) Β· (πΉβ€˜π‘‘)))
2322mpteq2dv 5250 . . . . . 6 (𝑓 = (𝑠 ∈ 𝑇 ↦ 𝐸) β†’ (𝑑 ∈ 𝑇 ↦ ((π‘“β€˜π‘‘) Β· (πΉβ€˜π‘‘))) = (𝑑 ∈ 𝑇 ↦ (((𝑠 ∈ 𝑇 ↦ 𝐸)β€˜π‘‘) Β· (πΉβ€˜π‘‘))))
2423eleq1d 2818 . . . . 5 (𝑓 = (𝑠 ∈ 𝑇 ↦ 𝐸) β†’ ((𝑑 ∈ 𝑇 ↦ ((π‘“β€˜π‘‘) Β· (πΉβ€˜π‘‘))) ∈ 𝐴 ↔ (𝑑 ∈ 𝑇 ↦ (((𝑠 ∈ 𝑇 ↦ 𝐸)β€˜π‘‘) Β· (πΉβ€˜π‘‘))) ∈ 𝐴))
2524imbi2d 340 . . . 4 (𝑓 = (𝑠 ∈ 𝑇 ↦ 𝐸) β†’ ((πœ‘ β†’ (𝑑 ∈ 𝑇 ↦ ((π‘“β€˜π‘‘) Β· (πΉβ€˜π‘‘))) ∈ 𝐴) ↔ (πœ‘ β†’ (𝑑 ∈ 𝑇 ↦ (((𝑠 ∈ 𝑇 ↦ 𝐸)β€˜π‘‘) Β· (πΉβ€˜π‘‘))) ∈ 𝐴)))
26 stoweidlem2.6 . . . . . . 7 (πœ‘ β†’ 𝐹 ∈ 𝐴)
2726adantr 481 . . . . . 6 ((πœ‘ ∧ 𝑓 ∈ 𝐴) β†’ 𝐹 ∈ 𝐴)
28 fveq1 6890 . . . . . . . . . . 11 (𝑔 = 𝐹 β†’ (π‘”β€˜π‘‘) = (πΉβ€˜π‘‘))
2928oveq2d 7424 . . . . . . . . . 10 (𝑔 = 𝐹 β†’ ((π‘“β€˜π‘‘) Β· (π‘”β€˜π‘‘)) = ((π‘“β€˜π‘‘) Β· (πΉβ€˜π‘‘)))
3029mpteq2dv 5250 . . . . . . . . 9 (𝑔 = 𝐹 β†’ (𝑑 ∈ 𝑇 ↦ ((π‘“β€˜π‘‘) Β· (π‘”β€˜π‘‘))) = (𝑑 ∈ 𝑇 ↦ ((π‘“β€˜π‘‘) Β· (πΉβ€˜π‘‘))))
3130eleq1d 2818 . . . . . . . 8 (𝑔 = 𝐹 β†’ ((𝑑 ∈ 𝑇 ↦ ((π‘“β€˜π‘‘) Β· (π‘”β€˜π‘‘))) ∈ 𝐴 ↔ (𝑑 ∈ 𝑇 ↦ ((π‘“β€˜π‘‘) Β· (πΉβ€˜π‘‘))) ∈ 𝐴))
3231imbi2d 340 . . . . . . 7 (𝑔 = 𝐹 β†’ (((πœ‘ ∧ 𝑓 ∈ 𝐴) β†’ (𝑑 ∈ 𝑇 ↦ ((π‘“β€˜π‘‘) Β· (π‘”β€˜π‘‘))) ∈ 𝐴) ↔ ((πœ‘ ∧ 𝑓 ∈ 𝐴) β†’ (𝑑 ∈ 𝑇 ↦ ((π‘“β€˜π‘‘) Β· (πΉβ€˜π‘‘))) ∈ 𝐴)))
33 stoweidlem2.2 . . . . . . . . 9 ((πœ‘ ∧ 𝑓 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴) β†’ (𝑑 ∈ 𝑇 ↦ ((π‘“β€˜π‘‘) Β· (π‘”β€˜π‘‘))) ∈ 𝐴)
34333comr 1125 . . . . . . . 8 ((𝑔 ∈ 𝐴 ∧ πœ‘ ∧ 𝑓 ∈ 𝐴) β†’ (𝑑 ∈ 𝑇 ↦ ((π‘“β€˜π‘‘) Β· (π‘”β€˜π‘‘))) ∈ 𝐴)
35343expib 1122 . . . . . . 7 (𝑔 ∈ 𝐴 β†’ ((πœ‘ ∧ 𝑓 ∈ 𝐴) β†’ (𝑑 ∈ 𝑇 ↦ ((π‘“β€˜π‘‘) Β· (π‘”β€˜π‘‘))) ∈ 𝐴))
3632, 35vtoclga 3565 . . . . . 6 (𝐹 ∈ 𝐴 β†’ ((πœ‘ ∧ 𝑓 ∈ 𝐴) β†’ (𝑑 ∈ 𝑇 ↦ ((π‘“β€˜π‘‘) Β· (πΉβ€˜π‘‘))) ∈ 𝐴))
3727, 36mpcom 38 . . . . 5 ((πœ‘ ∧ 𝑓 ∈ 𝐴) β†’ (𝑑 ∈ 𝑇 ↦ ((π‘“β€˜π‘‘) Β· (πΉβ€˜π‘‘))) ∈ 𝐴)
3837expcom 414 . . . 4 (𝑓 ∈ 𝐴 β†’ (πœ‘ β†’ (𝑑 ∈ 𝑇 ↦ ((π‘“β€˜π‘‘) Β· (πΉβ€˜π‘‘))) ∈ 𝐴))
3925, 38vtoclga 3565 . . 3 ((𝑠 ∈ 𝑇 ↦ 𝐸) ∈ 𝐴 β†’ (πœ‘ β†’ (𝑑 ∈ 𝑇 ↦ (((𝑠 ∈ 𝑇 ↦ 𝐸)β€˜π‘‘) Β· (πΉβ€˜π‘‘))) ∈ 𝐴))
4020, 39mpcom 38 . 2 (πœ‘ β†’ (𝑑 ∈ 𝑇 ↦ (((𝑠 ∈ 𝑇 ↦ 𝐸)β€˜π‘‘) Β· (πΉβ€˜π‘‘))) ∈ 𝐴)
4111, 40eqeltrd 2833 1 (πœ‘ β†’ (𝑑 ∈ 𝑇 ↦ (𝐸 Β· (πΉβ€˜π‘‘))) ∈ 𝐴)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 396   ∧ w3a 1087   = wceq 1541  β„²wnf 1785   ∈ wcel 2106   ↦ cmpt 5231  βŸΆwf 6539  β€˜cfv 6543  (class class class)co 7408  β„cr 11108   Β· cmul 11114
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fv 6551  df-ov 7411
This theorem is referenced by:  stoweidlem17  44723
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