MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  prdssca Structured version   Visualization version   GIF version

Theorem prdssca 16037
Description: Scalar ring of a structure product. (Contributed by Stefan O'Rear, 5-Jan-2015.) (Revised by Mario Carneiro, 15-Aug-2015.) (Revised by Thierry Arnoux, 16-Jun-2019.)
Hypotheses
Ref Expression
prdsbas.p 𝑃 = (𝑆Xs𝑅)
prdsbas.s (𝜑𝑆𝑉)
prdsbas.r (𝜑𝑅𝑊)
Assertion
Ref Expression
prdssca (𝜑𝑆 = (Scalar‘𝑃))

Proof of Theorem prdssca
Dummy variables 𝑎 𝑐 𝑑 𝑒 𝑓 𝑔 𝑥 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 prdsbas.p . . . 4 𝑃 = (𝑆Xs𝑅)
2 eqid 2621 . . . 4 (Base‘𝑆) = (Base‘𝑆)
3 eqidd 2622 . . . 4 (𝜑 → dom 𝑅 = dom 𝑅)
4 eqidd 2622 . . . 4 (𝜑X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)) = X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)))
5 eqidd 2622 . . . 4 (𝜑 → (𝑓X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)), 𝑔X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)) ↦ (𝑥 ∈ dom 𝑅 ↦ ((𝑓𝑥)(+g‘(𝑅𝑥))(𝑔𝑥)))) = (𝑓X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)), 𝑔X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)) ↦ (𝑥 ∈ dom 𝑅 ↦ ((𝑓𝑥)(+g‘(𝑅𝑥))(𝑔𝑥)))))
6 eqidd 2622 . . . 4 (𝜑 → (𝑓X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)), 𝑔X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)) ↦ (𝑥 ∈ dom 𝑅 ↦ ((𝑓𝑥)(.r‘(𝑅𝑥))(𝑔𝑥)))) = (𝑓X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)), 𝑔X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)) ↦ (𝑥 ∈ dom 𝑅 ↦ ((𝑓𝑥)(.r‘(𝑅𝑥))(𝑔𝑥)))))
7 eqidd 2622 . . . 4 (𝜑 → (𝑓 ∈ (Base‘𝑆), 𝑔X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)) ↦ (𝑥 ∈ dom 𝑅 ↦ (𝑓( ·𝑠 ‘(𝑅𝑥))(𝑔𝑥)))) = (𝑓 ∈ (Base‘𝑆), 𝑔X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)) ↦ (𝑥 ∈ dom 𝑅 ↦ (𝑓( ·𝑠 ‘(𝑅𝑥))(𝑔𝑥)))))
8 eqidd 2622 . . . 4 (𝜑 → (𝑓X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)), 𝑔X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)) ↦ (𝑆 Σg (𝑥 ∈ dom 𝑅 ↦ ((𝑓𝑥)(·𝑖‘(𝑅𝑥))(𝑔𝑥))))) = (𝑓X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)), 𝑔X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)) ↦ (𝑆 Σg (𝑥 ∈ dom 𝑅 ↦ ((𝑓𝑥)(·𝑖‘(𝑅𝑥))(𝑔𝑥))))))
9 eqidd 2622 . . . 4 (𝜑 → (∏t‘(TopOpen ∘ 𝑅)) = (∏t‘(TopOpen ∘ 𝑅)))
10 eqidd 2622 . . . 4 (𝜑 → {⟨𝑓, 𝑔⟩ ∣ ({𝑓, 𝑔} ⊆ X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)) ∧ ∀𝑥 ∈ dom 𝑅(𝑓𝑥)(le‘(𝑅𝑥))(𝑔𝑥))} = {⟨𝑓, 𝑔⟩ ∣ ({𝑓, 𝑔} ⊆ X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)) ∧ ∀𝑥 ∈ dom 𝑅(𝑓𝑥)(le‘(𝑅𝑥))(𝑔𝑥))})
11 eqidd 2622 . . . 4 (𝜑 → (𝑓X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)), 𝑔X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)) ↦ sup((ran (𝑥 ∈ dom 𝑅 ↦ ((𝑓𝑥)(dist‘(𝑅𝑥))(𝑔𝑥))) ∪ {0}), ℝ*, < )) = (𝑓X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)), 𝑔X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)) ↦ sup((ran (𝑥 ∈ dom 𝑅 ↦ ((𝑓𝑥)(dist‘(𝑅𝑥))(𝑔𝑥))) ∪ {0}), ℝ*, < )))
12 eqidd 2622 . . . 4 (𝜑 → (𝑓X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)), 𝑔X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)) ↦ X𝑥 ∈ dom 𝑅((𝑓𝑥)(Hom ‘(𝑅𝑥))(𝑔𝑥))) = (𝑓X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)), 𝑔X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)) ↦ X𝑥 ∈ dom 𝑅((𝑓𝑥)(Hom ‘(𝑅𝑥))(𝑔𝑥))))
13 eqidd 2622 . . . 4 (𝜑 → (𝑎 ∈ (X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)) × X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥))), 𝑐X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)) ↦ (𝑑 ∈ (𝑐(𝑓X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)), 𝑔X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)) ↦ X𝑥 ∈ dom 𝑅((𝑓𝑥)(Hom ‘(𝑅𝑥))(𝑔𝑥)))(2nd𝑎)), 𝑒 ∈ ((𝑓X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)), 𝑔X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)) ↦ X𝑥 ∈ dom 𝑅((𝑓𝑥)(Hom ‘(𝑅𝑥))(𝑔𝑥)))‘𝑎) ↦ (𝑥 ∈ dom 𝑅 ↦ ((𝑑𝑥)(⟨((1st𝑎)‘𝑥), ((2nd𝑎)‘𝑥)⟩(comp‘(𝑅𝑥))(𝑐𝑥))(𝑒𝑥))))) = (𝑎 ∈ (X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)) × X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥))), 𝑐X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)) ↦ (𝑑 ∈ (𝑐(𝑓X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)), 𝑔X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)) ↦ X𝑥 ∈ dom 𝑅((𝑓𝑥)(Hom ‘(𝑅𝑥))(𝑔𝑥)))(2nd𝑎)), 𝑒 ∈ ((𝑓X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)), 𝑔X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)) ↦ X𝑥 ∈ dom 𝑅((𝑓𝑥)(Hom ‘(𝑅𝑥))(𝑔𝑥)))‘𝑎) ↦ (𝑥 ∈ dom 𝑅 ↦ ((𝑑𝑥)(⟨((1st𝑎)‘𝑥), ((2nd𝑎)‘𝑥)⟩(comp‘(𝑅𝑥))(𝑐𝑥))(𝑒𝑥))))))
14 prdsbas.s . . . 4 (𝜑𝑆𝑉)
15 prdsbas.r . . . 4 (𝜑𝑅𝑊)
161, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15prdsval 16036 . . 3 (𝜑𝑃 = (({⟨(Base‘ndx), X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥))⟩, ⟨(+g‘ndx), (𝑓X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)), 𝑔X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)) ↦ (𝑥 ∈ dom 𝑅 ↦ ((𝑓𝑥)(+g‘(𝑅𝑥))(𝑔𝑥))))⟩, ⟨(.r‘ndx), (𝑓X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)), 𝑔X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)) ↦ (𝑥 ∈ dom 𝑅 ↦ ((𝑓𝑥)(.r‘(𝑅𝑥))(𝑔𝑥))))⟩} ∪ {⟨(Scalar‘ndx), 𝑆⟩, ⟨( ·𝑠 ‘ndx), (𝑓 ∈ (Base‘𝑆), 𝑔X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)) ↦ (𝑥 ∈ dom 𝑅 ↦ (𝑓( ·𝑠 ‘(𝑅𝑥))(𝑔𝑥))))⟩, ⟨(·𝑖‘ndx), (𝑓X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)), 𝑔X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)) ↦ (𝑆 Σg (𝑥 ∈ dom 𝑅 ↦ ((𝑓𝑥)(·𝑖‘(𝑅𝑥))(𝑔𝑥)))))⟩}) ∪ ({⟨(TopSet‘ndx), (∏t‘(TopOpen ∘ 𝑅))⟩, ⟨(le‘ndx), {⟨𝑓, 𝑔⟩ ∣ ({𝑓, 𝑔} ⊆ X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)) ∧ ∀𝑥 ∈ dom 𝑅(𝑓𝑥)(le‘(𝑅𝑥))(𝑔𝑥))}⟩, ⟨(dist‘ndx), (𝑓X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)), 𝑔X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)) ↦ sup((ran (𝑥 ∈ dom 𝑅 ↦ ((𝑓𝑥)(dist‘(𝑅𝑥))(𝑔𝑥))) ∪ {0}), ℝ*, < ))⟩} ∪ {⟨(Hom ‘ndx), (𝑓X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)), 𝑔X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)) ↦ X𝑥 ∈ dom 𝑅((𝑓𝑥)(Hom ‘(𝑅𝑥))(𝑔𝑥)))⟩, ⟨(comp‘ndx), (𝑎 ∈ (X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)) × X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥))), 𝑐X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)) ↦ (𝑑 ∈ (𝑐(𝑓X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)), 𝑔X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)) ↦ X𝑥 ∈ dom 𝑅((𝑓𝑥)(Hom ‘(𝑅𝑥))(𝑔𝑥)))(2nd𝑎)), 𝑒 ∈ ((𝑓X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)), 𝑔X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)) ↦ X𝑥 ∈ dom 𝑅((𝑓𝑥)(Hom ‘(𝑅𝑥))(𝑔𝑥)))‘𝑎) ↦ (𝑥 ∈ dom 𝑅 ↦ ((𝑑𝑥)(⟨((1st𝑎)‘𝑥), ((2nd𝑎)‘𝑥)⟩(comp‘(𝑅𝑥))(𝑐𝑥))(𝑒𝑥)))))⟩})))
17 eqid 2621 . . 3 (Scalar‘𝑃) = (Scalar‘𝑃)
18 scaid 15935 . . 3 Scalar = Slot (Scalar‘ndx)
19 elex 3198 . . . 4 (𝑆𝑉𝑆 ∈ V)
2014, 19syl 17 . . 3 (𝜑𝑆 ∈ V)
21 snsstp1 4315 . . . . 5 {⟨(Scalar‘ndx), 𝑆⟩} ⊆ {⟨(Scalar‘ndx), 𝑆⟩, ⟨( ·𝑠 ‘ndx), (𝑓 ∈ (Base‘𝑆), 𝑔X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)) ↦ (𝑥 ∈ dom 𝑅 ↦ (𝑓( ·𝑠 ‘(𝑅𝑥))(𝑔𝑥))))⟩, ⟨(·𝑖‘ndx), (𝑓X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)), 𝑔X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)) ↦ (𝑆 Σg (𝑥 ∈ dom 𝑅 ↦ ((𝑓𝑥)(·𝑖‘(𝑅𝑥))(𝑔𝑥)))))⟩}
22 ssun2 3755 . . . . 5 {⟨(Scalar‘ndx), 𝑆⟩, ⟨( ·𝑠 ‘ndx), (𝑓 ∈ (Base‘𝑆), 𝑔X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)) ↦ (𝑥 ∈ dom 𝑅 ↦ (𝑓( ·𝑠 ‘(𝑅𝑥))(𝑔𝑥))))⟩, ⟨(·𝑖‘ndx), (𝑓X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)), 𝑔X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)) ↦ (𝑆 Σg (𝑥 ∈ dom 𝑅 ↦ ((𝑓𝑥)(·𝑖‘(𝑅𝑥))(𝑔𝑥)))))⟩} ⊆ ({⟨(Base‘ndx), X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥))⟩, ⟨(+g‘ndx), (𝑓X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)), 𝑔X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)) ↦ (𝑥 ∈ dom 𝑅 ↦ ((𝑓𝑥)(+g‘(𝑅𝑥))(𝑔𝑥))))⟩, ⟨(.r‘ndx), (𝑓X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)), 𝑔X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)) ↦ (𝑥 ∈ dom 𝑅 ↦ ((𝑓𝑥)(.r‘(𝑅𝑥))(𝑔𝑥))))⟩} ∪ {⟨(Scalar‘ndx), 𝑆⟩, ⟨( ·𝑠 ‘ndx), (𝑓 ∈ (Base‘𝑆), 𝑔X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)) ↦ (𝑥 ∈ dom 𝑅 ↦ (𝑓( ·𝑠 ‘(𝑅𝑥))(𝑔𝑥))))⟩, ⟨(·𝑖‘ndx), (𝑓X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)), 𝑔X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)) ↦ (𝑆 Σg (𝑥 ∈ dom 𝑅 ↦ ((𝑓𝑥)(·𝑖‘(𝑅𝑥))(𝑔𝑥)))))⟩})
2321, 22sstri 3592 . . . 4 {⟨(Scalar‘ndx), 𝑆⟩} ⊆ ({⟨(Base‘ndx), X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥))⟩, ⟨(+g‘ndx), (𝑓X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)), 𝑔X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)) ↦ (𝑥 ∈ dom 𝑅 ↦ ((𝑓𝑥)(+g‘(𝑅𝑥))(𝑔𝑥))))⟩, ⟨(.r‘ndx), (𝑓X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)), 𝑔X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)) ↦ (𝑥 ∈ dom 𝑅 ↦ ((𝑓𝑥)(.r‘(𝑅𝑥))(𝑔𝑥))))⟩} ∪ {⟨(Scalar‘ndx), 𝑆⟩, ⟨( ·𝑠 ‘ndx), (𝑓 ∈ (Base‘𝑆), 𝑔X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)) ↦ (𝑥 ∈ dom 𝑅 ↦ (𝑓( ·𝑠 ‘(𝑅𝑥))(𝑔𝑥))))⟩, ⟨(·𝑖‘ndx), (𝑓X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)), 𝑔X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)) ↦ (𝑆 Σg (𝑥 ∈ dom 𝑅 ↦ ((𝑓𝑥)(·𝑖‘(𝑅𝑥))(𝑔𝑥)))))⟩})
24 ssun1 3754 . . . 4 ({⟨(Base‘ndx), X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥))⟩, ⟨(+g‘ndx), (𝑓X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)), 𝑔X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)) ↦ (𝑥 ∈ dom 𝑅 ↦ ((𝑓𝑥)(+g‘(𝑅𝑥))(𝑔𝑥))))⟩, ⟨(.r‘ndx), (𝑓X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)), 𝑔X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)) ↦ (𝑥 ∈ dom 𝑅 ↦ ((𝑓𝑥)(.r‘(𝑅𝑥))(𝑔𝑥))))⟩} ∪ {⟨(Scalar‘ndx), 𝑆⟩, ⟨( ·𝑠 ‘ndx), (𝑓 ∈ (Base‘𝑆), 𝑔X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)) ↦ (𝑥 ∈ dom 𝑅 ↦ (𝑓( ·𝑠 ‘(𝑅𝑥))(𝑔𝑥))))⟩, ⟨(·𝑖‘ndx), (𝑓X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)), 𝑔X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)) ↦ (𝑆 Σg (𝑥 ∈ dom 𝑅 ↦ ((𝑓𝑥)(·𝑖‘(𝑅𝑥))(𝑔𝑥)))))⟩}) ⊆ (({⟨(Base‘ndx), X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥))⟩, ⟨(+g‘ndx), (𝑓X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)), 𝑔X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)) ↦ (𝑥 ∈ dom 𝑅 ↦ ((𝑓𝑥)(+g‘(𝑅𝑥))(𝑔𝑥))))⟩, ⟨(.r‘ndx), (𝑓X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)), 𝑔X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)) ↦ (𝑥 ∈ dom 𝑅 ↦ ((𝑓𝑥)(.r‘(𝑅𝑥))(𝑔𝑥))))⟩} ∪ {⟨(Scalar‘ndx), 𝑆⟩, ⟨( ·𝑠 ‘ndx), (𝑓 ∈ (Base‘𝑆), 𝑔X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)) ↦ (𝑥 ∈ dom 𝑅 ↦ (𝑓( ·𝑠 ‘(𝑅𝑥))(𝑔𝑥))))⟩, ⟨(·𝑖‘ndx), (𝑓X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)), 𝑔X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)) ↦ (𝑆 Σg (𝑥 ∈ dom 𝑅 ↦ ((𝑓𝑥)(·𝑖‘(𝑅𝑥))(𝑔𝑥)))))⟩}) ∪ ({⟨(TopSet‘ndx), (∏t‘(TopOpen ∘ 𝑅))⟩, ⟨(le‘ndx), {⟨𝑓, 𝑔⟩ ∣ ({𝑓, 𝑔} ⊆ X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)) ∧ ∀𝑥 ∈ dom 𝑅(𝑓𝑥)(le‘(𝑅𝑥))(𝑔𝑥))}⟩, ⟨(dist‘ndx), (𝑓X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)), 𝑔X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)) ↦ sup((ran (𝑥 ∈ dom 𝑅 ↦ ((𝑓𝑥)(dist‘(𝑅𝑥))(𝑔𝑥))) ∪ {0}), ℝ*, < ))⟩} ∪ {⟨(Hom ‘ndx), (𝑓X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)), 𝑔X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)) ↦ X𝑥 ∈ dom 𝑅((𝑓𝑥)(Hom ‘(𝑅𝑥))(𝑔𝑥)))⟩, ⟨(comp‘ndx), (𝑎 ∈ (X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)) × X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥))), 𝑐X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)) ↦ (𝑑 ∈ (𝑐(𝑓X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)), 𝑔X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)) ↦ X𝑥 ∈ dom 𝑅((𝑓𝑥)(Hom ‘(𝑅𝑥))(𝑔𝑥)))(2nd𝑎)), 𝑒 ∈ ((𝑓X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)), 𝑔X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)) ↦ X𝑥 ∈ dom 𝑅((𝑓𝑥)(Hom ‘(𝑅𝑥))(𝑔𝑥)))‘𝑎) ↦ (𝑥 ∈ dom 𝑅 ↦ ((𝑑𝑥)(⟨((1st𝑎)‘𝑥), ((2nd𝑎)‘𝑥)⟩(comp‘(𝑅𝑥))(𝑐𝑥))(𝑒𝑥)))))⟩}))
2523, 24sstri 3592 . . 3 {⟨(Scalar‘ndx), 𝑆⟩} ⊆ (({⟨(Base‘ndx), X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥))⟩, ⟨(+g‘ndx), (𝑓X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)), 𝑔X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)) ↦ (𝑥 ∈ dom 𝑅 ↦ ((𝑓𝑥)(+g‘(𝑅𝑥))(𝑔𝑥))))⟩, ⟨(.r‘ndx), (𝑓X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)), 𝑔X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)) ↦ (𝑥 ∈ dom 𝑅 ↦ ((𝑓𝑥)(.r‘(𝑅𝑥))(𝑔𝑥))))⟩} ∪ {⟨(Scalar‘ndx), 𝑆⟩, ⟨( ·𝑠 ‘ndx), (𝑓 ∈ (Base‘𝑆), 𝑔X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)) ↦ (𝑥 ∈ dom 𝑅 ↦ (𝑓( ·𝑠 ‘(𝑅𝑥))(𝑔𝑥))))⟩, ⟨(·𝑖‘ndx), (𝑓X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)), 𝑔X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)) ↦ (𝑆 Σg (𝑥 ∈ dom 𝑅 ↦ ((𝑓𝑥)(·𝑖‘(𝑅𝑥))(𝑔𝑥)))))⟩}) ∪ ({⟨(TopSet‘ndx), (∏t‘(TopOpen ∘ 𝑅))⟩, ⟨(le‘ndx), {⟨𝑓, 𝑔⟩ ∣ ({𝑓, 𝑔} ⊆ X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)) ∧ ∀𝑥 ∈ dom 𝑅(𝑓𝑥)(le‘(𝑅𝑥))(𝑔𝑥))}⟩, ⟨(dist‘ndx), (𝑓X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)), 𝑔X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)) ↦ sup((ran (𝑥 ∈ dom 𝑅 ↦ ((𝑓𝑥)(dist‘(𝑅𝑥))(𝑔𝑥))) ∪ {0}), ℝ*, < ))⟩} ∪ {⟨(Hom ‘ndx), (𝑓X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)), 𝑔X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)) ↦ X𝑥 ∈ dom 𝑅((𝑓𝑥)(Hom ‘(𝑅𝑥))(𝑔𝑥)))⟩, ⟨(comp‘ndx), (𝑎 ∈ (X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)) × X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥))), 𝑐X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)) ↦ (𝑑 ∈ (𝑐(𝑓X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)), 𝑔X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)) ↦ X𝑥 ∈ dom 𝑅((𝑓𝑥)(Hom ‘(𝑅𝑥))(𝑔𝑥)))(2nd𝑎)), 𝑒 ∈ ((𝑓X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)), 𝑔X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)) ↦ X𝑥 ∈ dom 𝑅((𝑓𝑥)(Hom ‘(𝑅𝑥))(𝑔𝑥)))‘𝑎) ↦ (𝑥 ∈ dom 𝑅 ↦ ((𝑑𝑥)(⟨((1st𝑎)‘𝑥), ((2nd𝑎)‘𝑥)⟩(comp‘(𝑅𝑥))(𝑐𝑥))(𝑒𝑥)))))⟩}))
2616, 17, 18, 20, 25prdsvallem 16035 . 2 (𝜑 → (Scalar‘𝑃) = 𝑆)
2726eqcomd 2627 1 (𝜑𝑆 = (Scalar‘𝑃))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wcel 1987  wral 2907  Vcvv 3186  cun 3553  wss 3555  {csn 4148  {cpr 4150  {ctp 4152  cop 4154   class class class wbr 4613  {copab 4672  cmpt 4673   × cxp 5072  dom cdm 5074  ran crn 5075  ccom 5078  cfv 5847  (class class class)co 6604  cmpt2 6606  1st c1st 7111  2nd c2nd 7112  Xcixp 7852  supcsup 8290  0cc0 9880  *cxr 10017   < clt 10018  ndxcnx 15778  Basecbs 15781  +gcplusg 15862  .rcmulr 15863  Scalarcsca 15865   ·𝑠 cvsca 15866  ·𝑖cip 15867  TopSetcts 15868  lecple 15869  distcds 15871  Hom chom 15873  compcco 15874  TopOpenctopn 16003  tcpt 16020   Σg cgsu 16022  Xscprds 16027
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  ax-un 6902  ax-cnex 9936  ax-resscn 9937  ax-1cn 9938  ax-icn 9939  ax-addcl 9940  ax-addrcl 9941  ax-mulcl 9942  ax-mulrcl 9943  ax-mulcom 9944  ax-addass 9945  ax-mulass 9946  ax-distr 9947  ax-i2m1 9948  ax-1ne0 9949  ax-1rid 9950  ax-rnegex 9951  ax-rrecex 9952  ax-cnre 9953  ax-pre-lttri 9954  ax-pre-lttrn 9955  ax-pre-ltadd 9956  ax-pre-mulgt0 9957
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  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-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  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-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-int 4441  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  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-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-riota 6565  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-om 7013  df-1st 7113  df-2nd 7114  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-1o 7505  df-oadd 7509  df-er 7687  df-map 7804  df-ixp 7853  df-en 7900  df-dom 7901  df-sdom 7902  df-fin 7903  df-sup 8292  df-pnf 10020  df-mnf 10021  df-xr 10022  df-ltxr 10023  df-le 10024  df-sub 10212  df-neg 10213  df-nn 10965  df-2 11023  df-3 11024  df-4 11025  df-5 11026  df-6 11027  df-7 11028  df-8 11029  df-9 11030  df-n0 11237  df-z 11322  df-dec 11438  df-uz 11632  df-fz 12269  df-struct 15783  df-ndx 15784  df-slot 15785  df-base 15786  df-plusg 15875  df-mulr 15876  df-sca 15878  df-vsca 15879  df-ip 15880  df-tset 15881  df-ple 15882  df-ds 15885  df-hom 15887  df-cco 15888  df-prds 16029
This theorem is referenced by:  pwssca  16077  xpssca  16159  xpsvsca  16160  prdslmodd  18888  dsmmlss  20007
  Copyright terms: Public domain W3C validator