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Theorem prdsbas 16324
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 2770 . . 3 (Base‘𝑆) = (Base‘𝑆)
3 prdsbas.i . . 3 (𝜑 → dom 𝑅 = 𝐼)
4 eqidd 2771 . . 3 (𝜑X𝑥𝐼 (Base‘(𝑅𝑥)) = X𝑥𝐼 (Base‘(𝑅𝑥)))
5 eqidd 2771 . . 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 2771 . . 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 2771 . . 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 2771 . . 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 2771 . . 3 (𝜑 → (∏t‘(TopOpen ∘ 𝑅)) = (∏t‘(TopOpen ∘ 𝑅)))
10 eqidd 2771 . . 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 2771 . . 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 2771 . . 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 2771 . . 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 16322 . 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 16125 . 2 Base = Slot (Base‘ndx)
1918strfvss 16086 . . . . . . 7 (Base‘(𝑅𝑥)) ⊆ ran (𝑅𝑥)
20 fvssunirn 6358 . . . . . . . 8 (𝑅𝑥) ⊆ ran 𝑅
21 rnss 5492 . . . . . . . 8 ((𝑅𝑥) ⊆ ran 𝑅 → ran (𝑅𝑥) ⊆ ran ran 𝑅)
22 uniss 4593 . . . . . . . 8 (ran (𝑅𝑥) ⊆ ran ran 𝑅 ran (𝑅𝑥) ⊆ ran ran 𝑅)
2320, 21, 22mp2b 10 . . . . . . 7 ran (𝑅𝑥) ⊆ ran ran 𝑅
2419, 23sstri 3759 . . . . . 6 (Base‘(𝑅𝑥)) ⊆ ran ran 𝑅
2524rgenw 3072 . . . . 5 𝑥𝐼 (Base‘(𝑅𝑥)) ⊆ ran ran 𝑅
26 iunss 4693 . . . . 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 7244 . . . . . 6 (𝑅𝑊 → ran 𝑅 ∈ V)
29 uniexg 7101 . . . . . 6 (ran 𝑅 ∈ V → ran 𝑅 ∈ V)
3015, 28, 293syl 18 . . . . 5 (𝜑 ran 𝑅 ∈ V)
31 rnexg 7244 . . . . 5 ( ran 𝑅 ∈ V → ran ran 𝑅 ∈ V)
32 uniexg 7101 . . . . 5 (ran ran 𝑅 ∈ V → ran ran 𝑅 ∈ V)
3330, 31, 323syl 18 . . . 4 (𝜑 ran ran 𝑅 ∈ V)
34 ssexg 4935 . . . 4 (( 𝑥𝐼 (Base‘(𝑅𝑥)) ⊆ ran ran 𝑅 ran ran 𝑅 ∈ V) → 𝑥𝐼 (Base‘(𝑅𝑥)) ∈ V)
3527, 33, 34sylancr 567 . . 3 (𝜑 𝑥𝐼 (Base‘(𝑅𝑥)) ∈ V)
36 ixpssmap2g 8090 . . 3 ( 𝑥𝐼 (Base‘(𝑅𝑥)) ∈ V → X𝑥𝐼 (Base‘(𝑅𝑥)) ⊆ ( 𝑥𝐼 (Base‘(𝑅𝑥)) ↑𝑚 𝐼))
37 ovex 6822 . . . 4 ( 𝑥𝐼 (Base‘(𝑅𝑥)) ↑𝑚 𝐼) ∈ V
3837ssex 4933 . . 3 (X𝑥𝐼 (Base‘(𝑅𝑥)) ⊆ ( 𝑥𝐼 (Base‘(𝑅𝑥)) ↑𝑚 𝐼) → X𝑥𝐼 (Base‘(𝑅𝑥)) ∈ V)
3935, 36, 383syl 18 . 2 (𝜑X𝑥𝐼 (Base‘(𝑅𝑥)) ∈ V)
40 snsstp1 4480 . . . 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 3925 . . . 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 3759 . . 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 3925 . . 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 3759 . 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 16321 1 (𝜑𝐵 = X𝑥𝐼 (Base‘(𝑅𝑥)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1630  wcel 2144  wral 3060  Vcvv 3349  cun 3719  wss 3721  {csn 4314  {cpr 4316  {ctp 4318  cop 4320   cuni 4572   ciun 4652   class class class wbr 4784  {copab 4844  cmpt 4861   × cxp 5247  dom cdm 5249  ran crn 5250  ccom 5253  cfv 6031  (class class class)co 6792  cmpt2 6794  1st c1st 7312  2nd c2nd 7313  𝑚 cmap 8008  Xcixp 8061  supcsup 8501  0cc0 10137  *cxr 10274   < clt 10275  ndxcnx 16060  Basecbs 16063  +gcplusg 16148  .rcmulr 16149  Scalarcsca 16151   ·𝑠 cvsca 16152  ·𝑖cip 16153  TopSetcts 16154  lecple 16155  distcds 16157  Hom chom 16159  compcco 16160  TopOpenctopn 16289  tcpt 16306   Σg cgsu 16308  Xscprds 16313
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-sep 4912  ax-nul 4920  ax-pow 4971  ax-pr 5034  ax-un 7095  ax-cnex 10193  ax-resscn 10194  ax-1cn 10195  ax-icn 10196  ax-addcl 10197  ax-addrcl 10198  ax-mulcl 10199  ax-mulrcl 10200  ax-mulcom 10201  ax-addass 10202  ax-mulass 10203  ax-distr 10204  ax-i2m1 10205  ax-1ne0 10206  ax-1rid 10207  ax-rnegex 10208  ax-rrecex 10209  ax-cnre 10210  ax-pre-lttri 10211  ax-pre-lttrn 10212  ax-pre-ltadd 10213  ax-pre-mulgt0 10214
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3or 1071  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-nel 3046  df-ral 3065  df-rex 3066  df-reu 3067  df-rab 3069  df-v 3351  df-sbc 3586  df-csb 3681  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-pss 3737  df-nul 4062  df-if 4224  df-pw 4297  df-sn 4315  df-pr 4317  df-tp 4319  df-op 4321  df-uni 4573  df-int 4610  df-iun 4654  df-br 4785  df-opab 4845  df-mpt 4862  df-tr 4885  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-riota 6753  df-ov 6795  df-oprab 6796  df-mpt2 6797  df-om 7212  df-1st 7314  df-2nd 7315  df-wrecs 7558  df-recs 7620  df-rdg 7658  df-1o 7712  df-oadd 7716  df-er 7895  df-map 8010  df-ixp 8062  df-en 8109  df-dom 8110  df-sdom 8111  df-fin 8112  df-sup 8503  df-pnf 10277  df-mnf 10278  df-xr 10279  df-ltxr 10280  df-le 10281  df-sub 10469  df-neg 10470  df-nn 11222  df-2 11280  df-3 11281  df-4 11282  df-5 11283  df-6 11284  df-7 11285  df-8 11286  df-9 11287  df-n0 11494  df-z 11579  df-dec 11695  df-uz 11888  df-fz 12533  df-struct 16065  df-ndx 16066  df-slot 16067  df-base 16069  df-plusg 16161  df-mulr 16162  df-sca 16164  df-vsca 16165  df-ip 16166  df-tset 16167  df-ple 16168  df-ds 16171  df-hom 16173  df-cco 16174  df-prds 16315
This theorem is referenced by:  prdsplusg  16325  prdsmulr  16326  prdsvsca  16327  prdsip  16328  prdsle  16329  prdsds  16331  prdstset  16333  prdshom  16334  prdsco  16335  prdsbas2  16336  pwsbas  16354  dsmmval  20294  frlmip  20333  prdstps  21652  rrxip  23396  prdstotbnd  33918
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