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Theorem prdsbas 16098
Description: Base set of a structure product. (Contributed by Stefan O'Rear, 3-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)
prdsbas.b 𝐵 = (Base‘𝑃)
prdsbas.i (𝜑 → dom 𝑅 = 𝐼)
Assertion
Ref Expression
prdsbas (𝜑𝐵 = X𝑥𝐼 (Base‘(𝑅𝑥)))
Distinct variable groups:   𝑥,𝐵   𝜑,𝑥   𝑥,𝐼   𝑥,𝑃   𝑥,𝑅   𝑥,𝑆
Allowed substitution hints:   𝑉(𝑥)   𝑊(𝑥)

Proof of Theorem prdsbas
Dummy variables 𝑎 𝑐 𝑑 𝑒 𝑓 𝑔 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 prdsbas.p . . 3 𝑃 = (𝑆Xs𝑅)
2 eqid 2620 . . 3 (Base‘𝑆) = (Base‘𝑆)
3 prdsbas.i . . 3 (𝜑 → dom 𝑅 = 𝐼)
4 eqidd 2621 . . 3 (𝜑X𝑥𝐼 (Base‘(𝑅𝑥)) = X𝑥𝐼 (Base‘(𝑅𝑥)))
5 eqidd 2621 . . 3 (𝜑 → (𝑓X𝑥𝐼 (Base‘(𝑅𝑥)), 𝑔X𝑥𝐼 (Base‘(𝑅𝑥)) ↦ (𝑥𝐼 ↦ ((𝑓𝑥)(+g‘(𝑅𝑥))(𝑔𝑥)))) = (𝑓X𝑥𝐼 (Base‘(𝑅𝑥)), 𝑔X𝑥𝐼 (Base‘(𝑅𝑥)) ↦ (𝑥𝐼 ↦ ((𝑓𝑥)(+g‘(𝑅𝑥))(𝑔𝑥)))))
6 eqidd 2621 . . 3 (𝜑 → (𝑓X𝑥𝐼 (Base‘(𝑅𝑥)), 𝑔X𝑥𝐼 (Base‘(𝑅𝑥)) ↦ (𝑥𝐼 ↦ ((𝑓𝑥)(.r‘(𝑅𝑥))(𝑔𝑥)))) = (𝑓X𝑥𝐼 (Base‘(𝑅𝑥)), 𝑔X𝑥𝐼 (Base‘(𝑅𝑥)) ↦ (𝑥𝐼 ↦ ((𝑓𝑥)(.r‘(𝑅𝑥))(𝑔𝑥)))))
7 eqidd 2621 . . 3 (𝜑 → (𝑓 ∈ (Base‘𝑆), 𝑔X𝑥𝐼 (Base‘(𝑅𝑥)) ↦ (𝑥𝐼 ↦ (𝑓( ·𝑠 ‘(𝑅𝑥))(𝑔𝑥)))) = (𝑓 ∈ (Base‘𝑆), 𝑔X𝑥𝐼 (Base‘(𝑅𝑥)) ↦ (𝑥𝐼 ↦ (𝑓( ·𝑠 ‘(𝑅𝑥))(𝑔𝑥)))))
8 eqidd 2621 . . 3 (𝜑 → (𝑓X𝑥𝐼 (Base‘(𝑅𝑥)), 𝑔X𝑥𝐼 (Base‘(𝑅𝑥)) ↦ (𝑆 Σg (𝑥𝐼 ↦ ((𝑓𝑥)(·𝑖‘(𝑅𝑥))(𝑔𝑥))))) = (𝑓X𝑥𝐼 (Base‘(𝑅𝑥)), 𝑔X𝑥𝐼 (Base‘(𝑅𝑥)) ↦ (𝑆 Σg (𝑥𝐼 ↦ ((𝑓𝑥)(·𝑖‘(𝑅𝑥))(𝑔𝑥))))))
9 eqidd 2621 . . 3 (𝜑 → (∏t‘(TopOpen ∘ 𝑅)) = (∏t‘(TopOpen ∘ 𝑅)))
10 eqidd 2621 . . 3 (𝜑 → {⟨𝑓, 𝑔⟩ ∣ ({𝑓, 𝑔} ⊆ X𝑥𝐼 (Base‘(𝑅𝑥)) ∧ ∀𝑥𝐼 (𝑓𝑥)(le‘(𝑅𝑥))(𝑔𝑥))} = {⟨𝑓, 𝑔⟩ ∣ ({𝑓, 𝑔} ⊆ X𝑥𝐼 (Base‘(𝑅𝑥)) ∧ ∀𝑥𝐼 (𝑓𝑥)(le‘(𝑅𝑥))(𝑔𝑥))})
11 eqidd 2621 . . 3 (𝜑 → (𝑓X𝑥𝐼 (Base‘(𝑅𝑥)), 𝑔X𝑥𝐼 (Base‘(𝑅𝑥)) ↦ sup((ran (𝑥𝐼 ↦ ((𝑓𝑥)(dist‘(𝑅𝑥))(𝑔𝑥))) ∪ {0}), ℝ*, < )) = (𝑓X𝑥𝐼 (Base‘(𝑅𝑥)), 𝑔X𝑥𝐼 (Base‘(𝑅𝑥)) ↦ sup((ran (𝑥𝐼 ↦ ((𝑓𝑥)(dist‘(𝑅𝑥))(𝑔𝑥))) ∪ {0}), ℝ*, < )))
12 eqidd 2621 . . 3 (𝜑 → (𝑓X𝑥𝐼 (Base‘(𝑅𝑥)), 𝑔X𝑥𝐼 (Base‘(𝑅𝑥)) ↦ X𝑥𝐼 ((𝑓𝑥)(Hom ‘(𝑅𝑥))(𝑔𝑥))) = (𝑓X𝑥𝐼 (Base‘(𝑅𝑥)), 𝑔X𝑥𝐼 (Base‘(𝑅𝑥)) ↦ X𝑥𝐼 ((𝑓𝑥)(Hom ‘(𝑅𝑥))(𝑔𝑥))))
13 eqidd 2621 . . 3 (𝜑 → (𝑎 ∈ (X𝑥𝐼 (Base‘(𝑅𝑥)) × X𝑥𝐼 (Base‘(𝑅𝑥))), 𝑐X𝑥𝐼 (Base‘(𝑅𝑥)) ↦ (𝑑 ∈ (𝑐(𝑓X𝑥𝐼 (Base‘(𝑅𝑥)), 𝑔X𝑥𝐼 (Base‘(𝑅𝑥)) ↦ X𝑥𝐼 ((𝑓𝑥)(Hom ‘(𝑅𝑥))(𝑔𝑥)))(2nd𝑎)), 𝑒 ∈ ((𝑓X𝑥𝐼 (Base‘(𝑅𝑥)), 𝑔X𝑥𝐼 (Base‘(𝑅𝑥)) ↦ X𝑥𝐼 ((𝑓𝑥)(Hom ‘(𝑅𝑥))(𝑔𝑥)))‘𝑎) ↦ (𝑥𝐼 ↦ ((𝑑𝑥)(⟨((1st𝑎)‘𝑥), ((2nd𝑎)‘𝑥)⟩(comp‘(𝑅𝑥))(𝑐𝑥))(𝑒𝑥))))) = (𝑎 ∈ (X𝑥𝐼 (Base‘(𝑅𝑥)) × X𝑥𝐼 (Base‘(𝑅𝑥))), 𝑐X𝑥𝐼 (Base‘(𝑅𝑥)) ↦ (𝑑 ∈ (𝑐(𝑓X𝑥𝐼 (Base‘(𝑅𝑥)), 𝑔X𝑥𝐼 (Base‘(𝑅𝑥)) ↦ X𝑥𝐼 ((𝑓𝑥)(Hom ‘(𝑅𝑥))(𝑔𝑥)))(2nd𝑎)), 𝑒 ∈ ((𝑓X𝑥𝐼 (Base‘(𝑅𝑥)), 𝑔X𝑥𝐼 (Base‘(𝑅𝑥)) ↦ X𝑥𝐼 ((𝑓𝑥)(Hom ‘(𝑅𝑥))(𝑔𝑥)))‘𝑎) ↦ (𝑥𝐼 ↦ ((𝑑𝑥)(⟨((1st𝑎)‘𝑥), ((2nd𝑎)‘𝑥)⟩(comp‘(𝑅𝑥))(𝑐𝑥))(𝑒𝑥))))))
14 prdsbas.s . . 3 (𝜑𝑆𝑉)
15 prdsbas.r . . 3 (𝜑𝑅𝑊)
161, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15prdsval 16096 . 2 (𝜑𝑃 = (({⟨(Base‘ndx), X𝑥𝐼 (Base‘(𝑅𝑥))⟩, ⟨(+g‘ndx), (𝑓X𝑥𝐼 (Base‘(𝑅𝑥)), 𝑔X𝑥𝐼 (Base‘(𝑅𝑥)) ↦ (𝑥𝐼 ↦ ((𝑓𝑥)(+g‘(𝑅𝑥))(𝑔𝑥))))⟩, ⟨(.r‘ndx), (𝑓X𝑥𝐼 (Base‘(𝑅𝑥)), 𝑔X𝑥𝐼 (Base‘(𝑅𝑥)) ↦ (𝑥𝐼 ↦ ((𝑓𝑥)(.r‘(𝑅𝑥))(𝑔𝑥))))⟩} ∪ {⟨(Scalar‘ndx), 𝑆⟩, ⟨( ·𝑠 ‘ndx), (𝑓 ∈ (Base‘𝑆), 𝑔X𝑥𝐼 (Base‘(𝑅𝑥)) ↦ (𝑥𝐼 ↦ (𝑓( ·𝑠 ‘(𝑅𝑥))(𝑔𝑥))))⟩, ⟨(·𝑖‘ndx), (𝑓X𝑥𝐼 (Base‘(𝑅𝑥)), 𝑔X𝑥𝐼 (Base‘(𝑅𝑥)) ↦ (𝑆 Σg (𝑥𝐼 ↦ ((𝑓𝑥)(·𝑖‘(𝑅𝑥))(𝑔𝑥)))))⟩}) ∪ ({⟨(TopSet‘ndx), (∏t‘(TopOpen ∘ 𝑅))⟩, ⟨(le‘ndx), {⟨𝑓, 𝑔⟩ ∣ ({𝑓, 𝑔} ⊆ X𝑥𝐼 (Base‘(𝑅𝑥)) ∧ ∀𝑥𝐼 (𝑓𝑥)(le‘(𝑅𝑥))(𝑔𝑥))}⟩, ⟨(dist‘ndx), (𝑓X𝑥𝐼 (Base‘(𝑅𝑥)), 𝑔X𝑥𝐼 (Base‘(𝑅𝑥)) ↦ sup((ran (𝑥𝐼 ↦ ((𝑓𝑥)(dist‘(𝑅𝑥))(𝑔𝑥))) ∪ {0}), ℝ*, < ))⟩} ∪ {⟨(Hom ‘ndx), (𝑓X𝑥𝐼 (Base‘(𝑅𝑥)), 𝑔X𝑥𝐼 (Base‘(𝑅𝑥)) ↦ X𝑥𝐼 ((𝑓𝑥)(Hom ‘(𝑅𝑥))(𝑔𝑥)))⟩, ⟨(comp‘ndx), (𝑎 ∈ (X𝑥𝐼 (Base‘(𝑅𝑥)) × X𝑥𝐼 (Base‘(𝑅𝑥))), 𝑐X𝑥𝐼 (Base‘(𝑅𝑥)) ↦ (𝑑 ∈ (𝑐(𝑓X𝑥𝐼 (Base‘(𝑅𝑥)), 𝑔X𝑥𝐼 (Base‘(𝑅𝑥)) ↦ X𝑥𝐼 ((𝑓𝑥)(Hom ‘(𝑅𝑥))(𝑔𝑥)))(2nd𝑎)), 𝑒 ∈ ((𝑓X𝑥𝐼 (Base‘(𝑅𝑥)), 𝑔X𝑥𝐼 (Base‘(𝑅𝑥)) ↦ X𝑥𝐼 ((𝑓𝑥)(Hom ‘(𝑅𝑥))(𝑔𝑥)))‘𝑎) ↦ (𝑥𝐼 ↦ ((𝑑𝑥)(⟨((1st𝑎)‘𝑥), ((2nd𝑎)‘𝑥)⟩(comp‘(𝑅𝑥))(𝑐𝑥))(𝑒𝑥)))))⟩})))
17 prdsbas.b . 2 𝐵 = (Base‘𝑃)
18 baseid 15900 . 2 Base = Slot (Base‘ndx)
1918strfvss 15861 . . . . . . 7 (Base‘(𝑅𝑥)) ⊆ ran (𝑅𝑥)
20 fvssunirn 6204 . . . . . . . 8 (𝑅𝑥) ⊆ ran 𝑅
21 rnss 5343 . . . . . . . 8 ((𝑅𝑥) ⊆ ran 𝑅 → ran (𝑅𝑥) ⊆ ran ran 𝑅)
22 uniss 4449 . . . . . . . 8 (ran (𝑅𝑥) ⊆ ran ran 𝑅 ran (𝑅𝑥) ⊆ ran ran 𝑅)
2320, 21, 22mp2b 10 . . . . . . 7 ran (𝑅𝑥) ⊆ ran ran 𝑅
2419, 23sstri 3604 . . . . . 6 (Base‘(𝑅𝑥)) ⊆ ran ran 𝑅
2524rgenw 2921 . . . . 5 𝑥𝐼 (Base‘(𝑅𝑥)) ⊆ ran ran 𝑅
26 iunss 4552 . . . . 5 ( 𝑥𝐼 (Base‘(𝑅𝑥)) ⊆ ran ran 𝑅 ↔ ∀𝑥𝐼 (Base‘(𝑅𝑥)) ⊆ ran ran 𝑅)
2725, 26mpbir 221 . . . 4 𝑥𝐼 (Base‘(𝑅𝑥)) ⊆ ran ran 𝑅
28 rnexg 7083 . . . . . 6 (𝑅𝑊 → ran 𝑅 ∈ V)
29 uniexg 6940 . . . . . 6 (ran 𝑅 ∈ V → ran 𝑅 ∈ V)
3015, 28, 293syl 18 . . . . 5 (𝜑 ran 𝑅 ∈ V)
31 rnexg 7083 . . . . 5 ( ran 𝑅 ∈ V → ran ran 𝑅 ∈ V)
32 uniexg 6940 . . . . 5 (ran ran 𝑅 ∈ V → ran ran 𝑅 ∈ V)
3330, 31, 323syl 18 . . . 4 (𝜑 ran ran 𝑅 ∈ V)
34 ssexg 4795 . . . 4 (( 𝑥𝐼 (Base‘(𝑅𝑥)) ⊆ ran ran 𝑅 ran ran 𝑅 ∈ V) → 𝑥𝐼 (Base‘(𝑅𝑥)) ∈ V)
3527, 33, 34sylancr 694 . . 3 (𝜑 𝑥𝐼 (Base‘(𝑅𝑥)) ∈ V)
36 ixpssmap2g 7922 . . 3 ( 𝑥𝐼 (Base‘(𝑅𝑥)) ∈ V → X𝑥𝐼 (Base‘(𝑅𝑥)) ⊆ ( 𝑥𝐼 (Base‘(𝑅𝑥)) ↑𝑚 𝐼))
37 ovex 6663 . . . 4 ( 𝑥𝐼 (Base‘(𝑅𝑥)) ↑𝑚 𝐼) ∈ V
3837ssex 4793 . . 3 (X𝑥𝐼 (Base‘(𝑅𝑥)) ⊆ ( 𝑥𝐼 (Base‘(𝑅𝑥)) ↑𝑚 𝐼) → X𝑥𝐼 (Base‘(𝑅𝑥)) ∈ V)
3935, 36, 383syl 18 . 2 (𝜑X𝑥𝐼 (Base‘(𝑅𝑥)) ∈ V)
40 snsstp1 4338 . . . 4 {⟨(Base‘ndx), X𝑥𝐼 (Base‘(𝑅𝑥))⟩} ⊆ {⟨(Base‘ndx), X𝑥𝐼 (Base‘(𝑅𝑥))⟩, ⟨(+g‘ndx), (𝑓X𝑥𝐼 (Base‘(𝑅𝑥)), 𝑔X𝑥𝐼 (Base‘(𝑅𝑥)) ↦ (𝑥𝐼 ↦ ((𝑓𝑥)(+g‘(𝑅𝑥))(𝑔𝑥))))⟩, ⟨(.r‘ndx), (𝑓X𝑥𝐼 (Base‘(𝑅𝑥)), 𝑔X𝑥𝐼 (Base‘(𝑅𝑥)) ↦ (𝑥𝐼 ↦ ((𝑓𝑥)(.r‘(𝑅𝑥))(𝑔𝑥))))⟩}
41 ssun1 3768 . . . 4 {⟨(Base‘ndx), X𝑥𝐼 (Base‘(𝑅𝑥))⟩, ⟨(+g‘ndx), (𝑓X𝑥𝐼 (Base‘(𝑅𝑥)), 𝑔X𝑥𝐼 (Base‘(𝑅𝑥)) ↦ (𝑥𝐼 ↦ ((𝑓𝑥)(+g‘(𝑅𝑥))(𝑔𝑥))))⟩, ⟨(.r‘ndx), (𝑓X𝑥𝐼 (Base‘(𝑅𝑥)), 𝑔X𝑥𝐼 (Base‘(𝑅𝑥)) ↦ (𝑥𝐼 ↦ ((𝑓𝑥)(.r‘(𝑅𝑥))(𝑔𝑥))))⟩} ⊆ ({⟨(Base‘ndx), X𝑥𝐼 (Base‘(𝑅𝑥))⟩, ⟨(+g‘ndx), (𝑓X𝑥𝐼 (Base‘(𝑅𝑥)), 𝑔X𝑥𝐼 (Base‘(𝑅𝑥)) ↦ (𝑥𝐼 ↦ ((𝑓𝑥)(+g‘(𝑅𝑥))(𝑔𝑥))))⟩, ⟨(.r‘ndx), (𝑓X𝑥𝐼 (Base‘(𝑅𝑥)), 𝑔X𝑥𝐼 (Base‘(𝑅𝑥)) ↦ (𝑥𝐼 ↦ ((𝑓𝑥)(.r‘(𝑅𝑥))(𝑔𝑥))))⟩} ∪ {⟨(Scalar‘ndx), 𝑆⟩, ⟨( ·𝑠 ‘ndx), (𝑓 ∈ (Base‘𝑆), 𝑔X𝑥𝐼 (Base‘(𝑅𝑥)) ↦ (𝑥𝐼 ↦ (𝑓( ·𝑠 ‘(𝑅𝑥))(𝑔𝑥))))⟩, ⟨(·𝑖‘ndx), (𝑓X𝑥𝐼 (Base‘(𝑅𝑥)), 𝑔X𝑥𝐼 (Base‘(𝑅𝑥)) ↦ (𝑆 Σg (𝑥𝐼 ↦ ((𝑓𝑥)(·𝑖‘(𝑅𝑥))(𝑔𝑥)))))⟩})
4240, 41sstri 3604 . . 3 {⟨(Base‘ndx), X𝑥𝐼 (Base‘(𝑅𝑥))⟩} ⊆ ({⟨(Base‘ndx), X𝑥𝐼 (Base‘(𝑅𝑥))⟩, ⟨(+g‘ndx), (𝑓X𝑥𝐼 (Base‘(𝑅𝑥)), 𝑔X𝑥𝐼 (Base‘(𝑅𝑥)) ↦ (𝑥𝐼 ↦ ((𝑓𝑥)(+g‘(𝑅𝑥))(𝑔𝑥))))⟩, ⟨(.r‘ndx), (𝑓X𝑥𝐼 (Base‘(𝑅𝑥)), 𝑔X𝑥𝐼 (Base‘(𝑅𝑥)) ↦ (𝑥𝐼 ↦ ((𝑓𝑥)(.r‘(𝑅𝑥))(𝑔𝑥))))⟩} ∪ {⟨(Scalar‘ndx), 𝑆⟩, ⟨( ·𝑠 ‘ndx), (𝑓 ∈ (Base‘𝑆), 𝑔X𝑥𝐼 (Base‘(𝑅𝑥)) ↦ (𝑥𝐼 ↦ (𝑓( ·𝑠 ‘(𝑅𝑥))(𝑔𝑥))))⟩, ⟨(·𝑖‘ndx), (𝑓X𝑥𝐼 (Base‘(𝑅𝑥)), 𝑔X𝑥𝐼 (Base‘(𝑅𝑥)) ↦ (𝑆 Σg (𝑥𝐼 ↦ ((𝑓𝑥)(·𝑖‘(𝑅𝑥))(𝑔𝑥)))))⟩})
43 ssun1 3768 . . 3 ({⟨(Base‘ndx), X𝑥𝐼 (Base‘(𝑅𝑥))⟩, ⟨(+g‘ndx), (𝑓X𝑥𝐼 (Base‘(𝑅𝑥)), 𝑔X𝑥𝐼 (Base‘(𝑅𝑥)) ↦ (𝑥𝐼 ↦ ((𝑓𝑥)(+g‘(𝑅𝑥))(𝑔𝑥))))⟩, ⟨(.r‘ndx), (𝑓X𝑥𝐼 (Base‘(𝑅𝑥)), 𝑔X𝑥𝐼 (Base‘(𝑅𝑥)) ↦ (𝑥𝐼 ↦ ((𝑓𝑥)(.r‘(𝑅𝑥))(𝑔𝑥))))⟩} ∪ {⟨(Scalar‘ndx), 𝑆⟩, ⟨( ·𝑠 ‘ndx), (𝑓 ∈ (Base‘𝑆), 𝑔X𝑥𝐼 (Base‘(𝑅𝑥)) ↦ (𝑥𝐼 ↦ (𝑓( ·𝑠 ‘(𝑅𝑥))(𝑔𝑥))))⟩, ⟨(·𝑖‘ndx), (𝑓X𝑥𝐼 (Base‘(𝑅𝑥)), 𝑔X𝑥𝐼 (Base‘(𝑅𝑥)) ↦ (𝑆 Σg (𝑥𝐼 ↦ ((𝑓𝑥)(·𝑖‘(𝑅𝑥))(𝑔𝑥)))))⟩}) ⊆ (({⟨(Base‘ndx), X𝑥𝐼 (Base‘(𝑅𝑥))⟩, ⟨(+g‘ndx), (𝑓X𝑥𝐼 (Base‘(𝑅𝑥)), 𝑔X𝑥𝐼 (Base‘(𝑅𝑥)) ↦ (𝑥𝐼 ↦ ((𝑓𝑥)(+g‘(𝑅𝑥))(𝑔𝑥))))⟩, ⟨(.r‘ndx), (𝑓X𝑥𝐼 (Base‘(𝑅𝑥)), 𝑔X𝑥𝐼 (Base‘(𝑅𝑥)) ↦ (𝑥𝐼 ↦ ((𝑓𝑥)(.r‘(𝑅𝑥))(𝑔𝑥))))⟩} ∪ {⟨(Scalar‘ndx), 𝑆⟩, ⟨( ·𝑠 ‘ndx), (𝑓 ∈ (Base‘𝑆), 𝑔X𝑥𝐼 (Base‘(𝑅𝑥)) ↦ (𝑥𝐼 ↦ (𝑓( ·𝑠 ‘(𝑅𝑥))(𝑔𝑥))))⟩, ⟨(·𝑖‘ndx), (𝑓X𝑥𝐼 (Base‘(𝑅𝑥)), 𝑔X𝑥𝐼 (Base‘(𝑅𝑥)) ↦ (𝑆 Σg (𝑥𝐼 ↦ ((𝑓𝑥)(·𝑖‘(𝑅𝑥))(𝑔𝑥)))))⟩}) ∪ ({⟨(TopSet‘ndx), (∏t‘(TopOpen ∘ 𝑅))⟩, ⟨(le‘ndx), {⟨𝑓, 𝑔⟩ ∣ ({𝑓, 𝑔} ⊆ X𝑥𝐼 (Base‘(𝑅𝑥)) ∧ ∀𝑥𝐼 (𝑓𝑥)(le‘(𝑅𝑥))(𝑔𝑥))}⟩, ⟨(dist‘ndx), (𝑓X𝑥𝐼 (Base‘(𝑅𝑥)), 𝑔X𝑥𝐼 (Base‘(𝑅𝑥)) ↦ sup((ran (𝑥𝐼 ↦ ((𝑓𝑥)(dist‘(𝑅𝑥))(𝑔𝑥))) ∪ {0}), ℝ*, < ))⟩} ∪ {⟨(Hom ‘ndx), (𝑓X𝑥𝐼 (Base‘(𝑅𝑥)), 𝑔X𝑥𝐼 (Base‘(𝑅𝑥)) ↦ X𝑥𝐼 ((𝑓𝑥)(Hom ‘(𝑅𝑥))(𝑔𝑥)))⟩, ⟨(comp‘ndx), (𝑎 ∈ (X𝑥𝐼 (Base‘(𝑅𝑥)) × X𝑥𝐼 (Base‘(𝑅𝑥))), 𝑐X𝑥𝐼 (Base‘(𝑅𝑥)) ↦ (𝑑 ∈ (𝑐(𝑓X𝑥𝐼 (Base‘(𝑅𝑥)), 𝑔X𝑥𝐼 (Base‘(𝑅𝑥)) ↦ X𝑥𝐼 ((𝑓𝑥)(Hom ‘(𝑅𝑥))(𝑔𝑥)))(2nd𝑎)), 𝑒 ∈ ((𝑓X𝑥𝐼 (Base‘(𝑅𝑥)), 𝑔X𝑥𝐼 (Base‘(𝑅𝑥)) ↦ X𝑥𝐼 ((𝑓𝑥)(Hom ‘(𝑅𝑥))(𝑔𝑥)))‘𝑎) ↦ (𝑥𝐼 ↦ ((𝑑𝑥)(⟨((1st𝑎)‘𝑥), ((2nd𝑎)‘𝑥)⟩(comp‘(𝑅𝑥))(𝑐𝑥))(𝑒𝑥)))))⟩}))
4442, 43sstri 3604 . 2 {⟨(Base‘ndx), X𝑥𝐼 (Base‘(𝑅𝑥))⟩} ⊆ (({⟨(Base‘ndx), X𝑥𝐼 (Base‘(𝑅𝑥))⟩, ⟨(+g‘ndx), (𝑓X𝑥𝐼 (Base‘(𝑅𝑥)), 𝑔X𝑥𝐼 (Base‘(𝑅𝑥)) ↦ (𝑥𝐼 ↦ ((𝑓𝑥)(+g‘(𝑅𝑥))(𝑔𝑥))))⟩, ⟨(.r‘ndx), (𝑓X𝑥𝐼 (Base‘(𝑅𝑥)), 𝑔X𝑥𝐼 (Base‘(𝑅𝑥)) ↦ (𝑥𝐼 ↦ ((𝑓𝑥)(.r‘(𝑅𝑥))(𝑔𝑥))))⟩} ∪ {⟨(Scalar‘ndx), 𝑆⟩, ⟨( ·𝑠 ‘ndx), (𝑓 ∈ (Base‘𝑆), 𝑔X𝑥𝐼 (Base‘(𝑅𝑥)) ↦ (𝑥𝐼 ↦ (𝑓( ·𝑠 ‘(𝑅𝑥))(𝑔𝑥))))⟩, ⟨(·𝑖‘ndx), (𝑓X𝑥𝐼 (Base‘(𝑅𝑥)), 𝑔X𝑥𝐼 (Base‘(𝑅𝑥)) ↦ (𝑆 Σg (𝑥𝐼 ↦ ((𝑓𝑥)(·𝑖‘(𝑅𝑥))(𝑔𝑥)))))⟩}) ∪ ({⟨(TopSet‘ndx), (∏t‘(TopOpen ∘ 𝑅))⟩, ⟨(le‘ndx), {⟨𝑓, 𝑔⟩ ∣ ({𝑓, 𝑔} ⊆ X𝑥𝐼 (Base‘(𝑅𝑥)) ∧ ∀𝑥𝐼 (𝑓𝑥)(le‘(𝑅𝑥))(𝑔𝑥))}⟩, ⟨(dist‘ndx), (𝑓X𝑥𝐼 (Base‘(𝑅𝑥)), 𝑔X𝑥𝐼 (Base‘(𝑅𝑥)) ↦ sup((ran (𝑥𝐼 ↦ ((𝑓𝑥)(dist‘(𝑅𝑥))(𝑔𝑥))) ∪ {0}), ℝ*, < ))⟩} ∪ {⟨(Hom ‘ndx), (𝑓X𝑥𝐼 (Base‘(𝑅𝑥)), 𝑔X𝑥𝐼 (Base‘(𝑅𝑥)) ↦ X𝑥𝐼 ((𝑓𝑥)(Hom ‘(𝑅𝑥))(𝑔𝑥)))⟩, ⟨(comp‘ndx), (𝑎 ∈ (X𝑥𝐼 (Base‘(𝑅𝑥)) × X𝑥𝐼 (Base‘(𝑅𝑥))), 𝑐X𝑥𝐼 (Base‘(𝑅𝑥)) ↦ (𝑑 ∈ (𝑐(𝑓X𝑥𝐼 (Base‘(𝑅𝑥)), 𝑔X𝑥𝐼 (Base‘(𝑅𝑥)) ↦ X𝑥𝐼 ((𝑓𝑥)(Hom ‘(𝑅𝑥))(𝑔𝑥)))(2nd𝑎)), 𝑒 ∈ ((𝑓X𝑥𝐼 (Base‘(𝑅𝑥)), 𝑔X𝑥𝐼 (Base‘(𝑅𝑥)) ↦ X𝑥𝐼 ((𝑓𝑥)(Hom ‘(𝑅𝑥))(𝑔𝑥)))‘𝑎) ↦ (𝑥𝐼 ↦ ((𝑑𝑥)(⟨((1st𝑎)‘𝑥), ((2nd𝑎)‘𝑥)⟩(comp‘(𝑅𝑥))(𝑐𝑥))(𝑒𝑥)))))⟩}))
4516, 17, 18, 39, 44prdsvallem 16095 1 (𝜑𝐵 = X𝑥𝐼 (Base‘(𝑅𝑥)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1481  wcel 1988  wral 2909  Vcvv 3195  cun 3565  wss 3567  {csn 4168  {cpr 4170  {ctp 4172  cop 4174   cuni 4427   ciun 4511   class class class wbr 4644  {copab 4703  cmpt 4720   × cxp 5102  dom cdm 5104  ran crn 5105  ccom 5108  cfv 5876  (class class class)co 6635  cmpt2 6637  1st c1st 7151  2nd c2nd 7152  𝑚 cmap 7842  Xcixp 7893  supcsup 8331  0cc0 9921  *cxr 10058   < clt 10059  ndxcnx 15835  Basecbs 15838  +gcplusg 15922  .rcmulr 15923  Scalarcsca 15925   ·𝑠 cvsca 15926  ·𝑖cip 15927  TopSetcts 15928  lecple 15929  distcds 15931  Hom chom 15933  compcco 15934  TopOpenctopn 16063  tcpt 16080   Σg cgsu 16082  Xscprds 16087
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934  ax-cnex 9977  ax-resscn 9978  ax-1cn 9979  ax-icn 9980  ax-addcl 9981  ax-addrcl 9982  ax-mulcl 9983  ax-mulrcl 9984  ax-mulcom 9985  ax-addass 9986  ax-mulass 9987  ax-distr 9988  ax-i2m1 9989  ax-1ne0 9990  ax-1rid 9991  ax-rnegex 9992  ax-rrecex 9993  ax-cnre 9994  ax-pre-lttri 9995  ax-pre-lttrn 9996  ax-pre-ltadd 9997  ax-pre-mulgt0 9998
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-nel 2895  df-ral 2914  df-rex 2915  df-reu 2916  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-pss 3583  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-tp 4173  df-op 4175  df-uni 4428  df-int 4467  df-iun 4513  df-br 4645  df-opab 4704  df-mpt 4721  df-tr 4744  df-id 5014  df-eprel 5019  df-po 5025  df-so 5026  df-fr 5063  df-we 5065  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-pred 5668  df-ord 5714  df-on 5715  df-lim 5716  df-suc 5717  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-fv 5884  df-riota 6596  df-ov 6638  df-oprab 6639  df-mpt2 6640  df-om 7051  df-1st 7153  df-2nd 7154  df-wrecs 7392  df-recs 7453  df-rdg 7491  df-1o 7545  df-oadd 7549  df-er 7727  df-map 7844  df-ixp 7894  df-en 7941  df-dom 7942  df-sdom 7943  df-fin 7944  df-sup 8333  df-pnf 10061  df-mnf 10062  df-xr 10063  df-ltxr 10064  df-le 10065  df-sub 10253  df-neg 10254  df-nn 11006  df-2 11064  df-3 11065  df-4 11066  df-5 11067  df-6 11068  df-7 11069  df-8 11070  df-9 11071  df-n0 11278  df-z 11363  df-dec 11479  df-uz 11673  df-fz 12312  df-struct 15840  df-ndx 15841  df-slot 15842  df-base 15844  df-plusg 15935  df-mulr 15936  df-sca 15938  df-vsca 15939  df-ip 15940  df-tset 15941  df-ple 15942  df-ds 15945  df-hom 15947  df-cco 15948  df-prds 16089
This theorem is referenced by:  prdsplusg  16099  prdsmulr  16100  prdsvsca  16101  prdsip  16102  prdsle  16103  prdsds  16105  prdstset  16107  prdshom  16108  prdsco  16109  prdsbas2  16110  pwsbas  16128  dsmmval  20059  frlmip  20098  prdstps  21413  rrxip  23159  prdstotbnd  33564
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