Theorem prdsbas 13275

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.) (Revised by Zhi Wang, 18-Aug-2024.)

Hypotheses

Ref	Expression
prdsbas.p	⊢ 𝑃 = (𝑆Xs𝑅)
prdsbas.s	⊢ (𝜑 → 𝑆 ∈ 𝑉)
prdsbas.r	⊢ (𝜑 → 𝑅 ∈ 𝑊)
prdsbas.b	⊢ 𝐵 = (Base‘𝑃)
prdsbas.i	⊢ (𝜑 → dom 𝑅 = 𝐼)

Assertion

Ref	Expression
prdsbas	⊢ (𝜑 → 𝐵 = X𝑥 ∈ 𝐼 (Base‘(𝑅‘𝑥)))

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.

Step	Hyp	Ref	Expression
1		prdsbas.p	. 2 ⊢ 𝑃 = (𝑆Xs𝑅)
2		prdsbas.s	. 2 ⊢ (𝜑 → 𝑆 ∈ 𝑉)
3		prdsbas.r	. 2 ⊢ (𝜑 → 𝑅 ∈ 𝑊)
4		prdsbas.b	. 2 ⊢ 𝐵 = (Base‘𝑃)
5		baseid 13052	. 2 ⊢ Base = Slot (Base‘ndx)
6		basendxnn 13054	. 2 ⊢ (Base‘ndx) ∈ ℕ
7		prdsbas.i	. . . 4 ⊢ (𝜑 → dom 𝑅 = 𝐼)
8		dmexg 4964	. . . . 5 ⊢ (𝑅 ∈ 𝑊 → dom 𝑅 ∈ V)
9	3, 8	syl 14	. . . 4 ⊢ (𝜑 → dom 𝑅 ∈ V)
10	7, 9	eqeltrrd 2287	. . 3 ⊢ (𝜑 → 𝐼 ∈ V)
11		basfn 13057	. . . . 5 ⊢ Base Fn V
12		vex 2782	. . . . . 6 ⊢ 𝑥 ∈ V
13		fvexg 5622	. . . . . 6 ⊢ ((𝑅 ∈ 𝑊 ∧ 𝑥 ∈ V) → (𝑅‘𝑥) ∈ V)
14	3, 12, 13	sylancl 413	. . . . 5 ⊢ (𝜑 → (𝑅‘𝑥) ∈ V)
15		funfvex 5620	. . . . . 6 ⊢ ((Fun Base ∧ (𝑅‘𝑥) ∈ dom Base) → (Base‘(𝑅‘𝑥)) ∈ V)
16	15	funfni 5399	. . . . 5 ⊢ ((Base Fn V ∧ (𝑅‘𝑥) ∈ V) → (Base‘(𝑅‘𝑥)) ∈ V)
17	11, 14, 16	sylancr 414	. . . 4 ⊢ (𝜑 → (Base‘(𝑅‘𝑥)) ∈ V)
18	17	ralrimivw 2584	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝐼 (Base‘(𝑅‘𝑥)) ∈ V)
19		ixpexgg 6839	. . 3 ⊢ ((𝐼 ∈ V ∧ ∀𝑥 ∈ 𝐼 (Base‘(𝑅‘𝑥)) ∈ V) → X𝑥 ∈ 𝐼 (Base‘(𝑅‘𝑥)) ∈ V)
20	10, 18, 19	syl2anc 411	. 2 ⊢ (𝜑 → X𝑥 ∈ 𝐼 (Base‘(𝑅‘𝑥)) ∈ V)
21		snsstp1 3797	. . . . 5 ⊢ {⟨(Base‘ndx), X𝑥 ∈ 𝐼 (Base‘(𝑅‘𝑥))⟩} ⊆ {⟨(Base‘ndx), X𝑥 ∈ 𝐼 (Base‘(𝑅‘𝑥))⟩, ⟨(+g‘ndx), (𝑓 ∈ X𝑥 ∈ 𝐼 (Base‘(𝑅‘𝑥)), 𝑔 ∈ X𝑥 ∈ 𝐼 (Base‘(𝑅‘𝑥)) ↦ (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(+g‘(𝑅‘𝑥))(𝑔‘𝑥))))⟩, ⟨(.r‘ndx), (𝑓 ∈ X𝑥 ∈ 𝐼 (Base‘(𝑅‘𝑥)), 𝑔 ∈ X𝑥 ∈ 𝐼 (Base‘(𝑅‘𝑥)) ↦ (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(.r‘(𝑅‘𝑥))(𝑔‘𝑥))))⟩}
22		ssun1 3347	. . . . 5 ⊢ {⟨(Base‘ndx), X𝑥 ∈ 𝐼 (Base‘(𝑅‘𝑥))⟩, ⟨(+g‘ndx), (𝑓 ∈ X𝑥 ∈ 𝐼 (Base‘(𝑅‘𝑥)), 𝑔 ∈ X𝑥 ∈ 𝐼 (Base‘(𝑅‘𝑥)) ↦ (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(+g‘(𝑅‘𝑥))(𝑔‘𝑥))))⟩, ⟨(.r‘ndx), (𝑓 ∈ X𝑥 ∈ 𝐼 (Base‘(𝑅‘𝑥)), 𝑔 ∈ X𝑥 ∈ 𝐼 (Base‘(𝑅‘𝑥)) ↦ (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(.r‘(𝑅‘𝑥))(𝑔‘𝑥))))⟩} ⊆ ({⟨(Base‘ndx), X𝑥 ∈ 𝐼 (Base‘(𝑅‘𝑥))⟩, ⟨(+g‘ndx), (𝑓 ∈ X𝑥 ∈ 𝐼 (Base‘(𝑅‘𝑥)), 𝑔 ∈ X𝑥 ∈ 𝐼 (Base‘(𝑅‘𝑥)) ↦ (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(+g‘(𝑅‘𝑥))(𝑔‘𝑥))))⟩, ⟨(.r‘ndx), (𝑓 ∈ X𝑥 ∈ 𝐼 (Base‘(𝑅‘𝑥)), 𝑔 ∈ X𝑥 ∈ 𝐼 (Base‘(𝑅‘𝑥)) ↦ (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(.r‘(𝑅‘𝑥))(𝑔‘𝑥))))⟩} ∪ {⟨(Scalar‘ndx), 𝑆⟩, ⟨( ·𝑠 ‘ndx), (𝑓 ∈ (Base‘𝑆), 𝑔 ∈ X𝑥 ∈ 𝐼 (Base‘(𝑅‘𝑥)) ↦ (𝑥 ∈ 𝐼 ↦ (𝑓( ·𝑠 ‘(𝑅‘𝑥))(𝑔‘𝑥))))⟩, ⟨(·𝑖‘ndx), (𝑓 ∈ X𝑥 ∈ 𝐼 (Base‘(𝑅‘𝑥)), 𝑔 ∈ X𝑥 ∈ 𝐼 (Base‘(𝑅‘𝑥)) ↦ (𝑆 Σg (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(·𝑖‘(𝑅‘𝑥))(𝑔‘𝑥)))))⟩})
23	21, 22	sstri 3213	. . . 4 ⊢ {⟨(Base‘ndx), X𝑥 ∈ 𝐼 (Base‘(𝑅‘𝑥))⟩} ⊆ ({⟨(Base‘ndx), X𝑥 ∈ 𝐼 (Base‘(𝑅‘𝑥))⟩, ⟨(+g‘ndx), (𝑓 ∈ X𝑥 ∈ 𝐼 (Base‘(𝑅‘𝑥)), 𝑔 ∈ X𝑥 ∈ 𝐼 (Base‘(𝑅‘𝑥)) ↦ (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(+g‘(𝑅‘𝑥))(𝑔‘𝑥))))⟩, ⟨(.r‘ndx), (𝑓 ∈ X𝑥 ∈ 𝐼 (Base‘(𝑅‘𝑥)), 𝑔 ∈ X𝑥 ∈ 𝐼 (Base‘(𝑅‘𝑥)) ↦ (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(.r‘(𝑅‘𝑥))(𝑔‘𝑥))))⟩} ∪ {⟨(Scalar‘ndx), 𝑆⟩, ⟨( ·𝑠 ‘ndx), (𝑓 ∈ (Base‘𝑆), 𝑔 ∈ X𝑥 ∈ 𝐼 (Base‘(𝑅‘𝑥)) ↦ (𝑥 ∈ 𝐼 ↦ (𝑓( ·𝑠 ‘(𝑅‘𝑥))(𝑔‘𝑥))))⟩, ⟨(·𝑖‘ndx), (𝑓 ∈ X𝑥 ∈ 𝐼 (Base‘(𝑅‘𝑥)), 𝑔 ∈ X𝑥 ∈ 𝐼 (Base‘(𝑅‘𝑥)) ↦ (𝑆 Σg (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(·𝑖‘(𝑅‘𝑥))(𝑔‘𝑥)))))⟩})
24		ssun1 3347	. . . 4 ⊢ ({⟨(Base‘ndx), X𝑥 ∈ 𝐼 (Base‘(𝑅‘𝑥))⟩, ⟨(+g‘ndx), (𝑓 ∈ X𝑥 ∈ 𝐼 (Base‘(𝑅‘𝑥)), 𝑔 ∈ X𝑥 ∈ 𝐼 (Base‘(𝑅‘𝑥)) ↦ (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(+g‘(𝑅‘𝑥))(𝑔‘𝑥))))⟩, ⟨(.r‘ndx), (𝑓 ∈ X𝑥 ∈ 𝐼 (Base‘(𝑅‘𝑥)), 𝑔 ∈ X𝑥 ∈ 𝐼 (Base‘(𝑅‘𝑥)) ↦ (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(.r‘(𝑅‘𝑥))(𝑔‘𝑥))))⟩} ∪ {⟨(Scalar‘ndx), 𝑆⟩, ⟨( ·𝑠 ‘ndx), (𝑓 ∈ (Base‘𝑆), 𝑔 ∈ X𝑥 ∈ 𝐼 (Base‘(𝑅‘𝑥)) ↦ (𝑥 ∈ 𝐼 ↦ (𝑓( ·𝑠 ‘(𝑅‘𝑥))(𝑔‘𝑥))))⟩, ⟨(·𝑖‘ndx), (𝑓 ∈ X𝑥 ∈ 𝐼 (Base‘(𝑅‘𝑥)), 𝑔 ∈ X𝑥 ∈ 𝐼 (Base‘(𝑅‘𝑥)) ↦ (𝑆 Σg (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(·𝑖‘(𝑅‘𝑥))(𝑔‘𝑥)))))⟩}) ⊆ (({⟨(Base‘ndx), X𝑥 ∈ 𝐼 (Base‘(𝑅‘𝑥))⟩, ⟨(+g‘ndx), (𝑓 ∈ X𝑥 ∈ 𝐼 (Base‘(𝑅‘𝑥)), 𝑔 ∈ X𝑥 ∈ 𝐼 (Base‘(𝑅‘𝑥)) ↦ (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(+g‘(𝑅‘𝑥))(𝑔‘𝑥))))⟩, ⟨(.r‘ndx), (𝑓 ∈ X𝑥 ∈ 𝐼 (Base‘(𝑅‘𝑥)), 𝑔 ∈ X𝑥 ∈ 𝐼 (Base‘(𝑅‘𝑥)) ↦ (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(.r‘(𝑅‘𝑥))(𝑔‘𝑥))))⟩} ∪ {⟨(Scalar‘ndx), 𝑆⟩, ⟨( ·𝑠 ‘ndx), (𝑓 ∈ (Base‘𝑆), 𝑔 ∈ X𝑥 ∈ 𝐼 (Base‘(𝑅‘𝑥)) ↦ (𝑥 ∈ 𝐼 ↦ (𝑓( ·𝑠 ‘(𝑅‘𝑥))(𝑔‘𝑥))))⟩, ⟨(·𝑖‘ndx), (𝑓 ∈ X𝑥 ∈ 𝐼 (Base‘(𝑅‘𝑥)), 𝑔 ∈ X𝑥 ∈ 𝐼 (Base‘(𝑅‘𝑥)) ↦ (𝑆 Σg (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(·𝑖‘(𝑅‘𝑥))(𝑔‘𝑥)))))⟩}) ∪ ({⟨(TopSet‘ndx), (∏t‘(TopOpen ∘ 𝑅))⟩, ⟨(le‘ndx), {⟨𝑓, 𝑔⟩ ∣ ({𝑓, 𝑔} ⊆ X𝑥 ∈ 𝐼 (Base‘(𝑅‘𝑥)) ∧ ∀𝑥 ∈ 𝐼 (𝑓‘𝑥)(le‘(𝑅‘𝑥))(𝑔‘𝑥))}⟩, ⟨(dist‘ndx), (𝑓 ∈ X𝑥 ∈ 𝐼 (Base‘(𝑅‘𝑥)), 𝑔 ∈ X𝑥 ∈ 𝐼 (Base‘(𝑅‘𝑥)) ↦ sup((ran (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(dist‘(𝑅‘𝑥))(𝑔‘𝑥))) ∪ {0}), ℝ*, < ))⟩} ∪ {⟨(Hom ‘ndx), (𝑓 ∈ X𝑥 ∈ 𝐼 (Base‘(𝑅‘𝑥)), 𝑔 ∈ X𝑥 ∈ 𝐼 (Base‘(𝑅‘𝑥)) ↦ X𝑥 ∈ 𝐼 ((𝑓‘𝑥)(Hom ‘(𝑅‘𝑥))(𝑔‘𝑥)))⟩, ⟨(comp‘ndx), (𝑎 ∈ (X𝑥 ∈ 𝐼 (Base‘(𝑅‘𝑥)) × X𝑥 ∈ 𝐼 (Base‘(𝑅‘𝑥))), 𝑐 ∈ X𝑥 ∈ 𝐼 (Base‘(𝑅‘𝑥)) ↦ (𝑑 ∈ ((2nd ‘𝑎)(𝑓 ∈ X𝑥 ∈ 𝐼 (Base‘(𝑅‘𝑥)), 𝑔 ∈ X𝑥 ∈ 𝐼 (Base‘(𝑅‘𝑥)) ↦ X𝑥 ∈ 𝐼 ((𝑓‘𝑥)(Hom ‘(𝑅‘𝑥))(𝑔‘𝑥)))𝑐), 𝑒 ∈ ((𝑓 ∈ X𝑥 ∈ 𝐼 (Base‘(𝑅‘𝑥)), 𝑔 ∈ X𝑥 ∈ 𝐼 (Base‘(𝑅‘𝑥)) ↦ X𝑥 ∈ 𝐼 ((𝑓‘𝑥)(Hom ‘(𝑅‘𝑥))(𝑔‘𝑥)))‘𝑎) ↦ (𝑥 ∈ 𝐼 ↦ ((𝑑‘𝑥)(⟨((1st ‘𝑎)‘𝑥), ((2nd ‘𝑎)‘𝑥)⟩(comp‘(𝑅‘𝑥))(𝑐‘𝑥))(𝑒‘𝑥)))))⟩}))
25	23, 24	sstri 3213	. . 3 ⊢ {⟨(Base‘ndx), X𝑥 ∈ 𝐼 (Base‘(𝑅‘𝑥))⟩} ⊆ (({⟨(Base‘ndx), X𝑥 ∈ 𝐼 (Base‘(𝑅‘𝑥))⟩, ⟨(+g‘ndx), (𝑓 ∈ X𝑥 ∈ 𝐼 (Base‘(𝑅‘𝑥)), 𝑔 ∈ X𝑥 ∈ 𝐼 (Base‘(𝑅‘𝑥)) ↦ (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(+g‘(𝑅‘𝑥))(𝑔‘𝑥))))⟩, ⟨(.r‘ndx), (𝑓 ∈ X𝑥 ∈ 𝐼 (Base‘(𝑅‘𝑥)), 𝑔 ∈ X𝑥 ∈ 𝐼 (Base‘(𝑅‘𝑥)) ↦ (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(.r‘(𝑅‘𝑥))(𝑔‘𝑥))))⟩} ∪ {⟨(Scalar‘ndx), 𝑆⟩, ⟨( ·𝑠 ‘ndx), (𝑓 ∈ (Base‘𝑆), 𝑔 ∈ X𝑥 ∈ 𝐼 (Base‘(𝑅‘𝑥)) ↦ (𝑥 ∈ 𝐼 ↦ (𝑓( ·𝑠 ‘(𝑅‘𝑥))(𝑔‘𝑥))))⟩, ⟨(·𝑖‘ndx), (𝑓 ∈ X𝑥 ∈ 𝐼 (Base‘(𝑅‘𝑥)), 𝑔 ∈ X𝑥 ∈ 𝐼 (Base‘(𝑅‘𝑥)) ↦ (𝑆 Σg (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(·𝑖‘(𝑅‘𝑥))(𝑔‘𝑥)))))⟩}) ∪ ({⟨(TopSet‘ndx), (∏t‘(TopOpen ∘ 𝑅))⟩, ⟨(le‘ndx), {⟨𝑓, 𝑔⟩ ∣ ({𝑓, 𝑔} ⊆ X𝑥 ∈ 𝐼 (Base‘(𝑅‘𝑥)) ∧ ∀𝑥 ∈ 𝐼 (𝑓‘𝑥)(le‘(𝑅‘𝑥))(𝑔‘𝑥))}⟩, ⟨(dist‘ndx), (𝑓 ∈ X𝑥 ∈ 𝐼 (Base‘(𝑅‘𝑥)), 𝑔 ∈ X𝑥 ∈ 𝐼 (Base‘(𝑅‘𝑥)) ↦ sup((ran (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(dist‘(𝑅‘𝑥))(𝑔‘𝑥))) ∪ {0}), ℝ*, < ))⟩} ∪ {⟨(Hom ‘ndx), (𝑓 ∈ X𝑥 ∈ 𝐼 (Base‘(𝑅‘𝑥)), 𝑔 ∈ X𝑥 ∈ 𝐼 (Base‘(𝑅‘𝑥)) ↦ X𝑥 ∈ 𝐼 ((𝑓‘𝑥)(Hom ‘(𝑅‘𝑥))(𝑔‘𝑥)))⟩, ⟨(comp‘ndx), (𝑎 ∈ (X𝑥 ∈ 𝐼 (Base‘(𝑅‘𝑥)) × X𝑥 ∈ 𝐼 (Base‘(𝑅‘𝑥))), 𝑐 ∈ X𝑥 ∈ 𝐼 (Base‘(𝑅‘𝑥)) ↦ (𝑑 ∈ ((2nd ‘𝑎)(𝑓 ∈ X𝑥 ∈ 𝐼 (Base‘(𝑅‘𝑥)), 𝑔 ∈ X𝑥 ∈ 𝐼 (Base‘(𝑅‘𝑥)) ↦ X𝑥 ∈ 𝐼 ((𝑓‘𝑥)(Hom ‘(𝑅‘𝑥))(𝑔‘𝑥)))𝑐), 𝑒 ∈ ((𝑓 ∈ X𝑥 ∈ 𝐼 (Base‘(𝑅‘𝑥)), 𝑔 ∈ X𝑥 ∈ 𝐼 (Base‘(𝑅‘𝑥)) ↦ X𝑥 ∈ 𝐼 ((𝑓‘𝑥)(Hom ‘(𝑅‘𝑥))(𝑔‘𝑥)))‘𝑎) ↦ (𝑥 ∈ 𝐼 ↦ ((𝑑‘𝑥)(⟨((1st ‘𝑎)‘𝑥), ((2nd ‘𝑎)‘𝑥)⟩(comp‘(𝑅‘𝑥))(𝑐‘𝑥))(𝑒‘𝑥)))))⟩}))
26		eqid 2209	. . . 4 ⊢ (Base‘𝑆) = (Base‘𝑆)
27		eqidd 2210	. . . 4 ⊢ (𝜑 → X𝑥 ∈ 𝐼 (Base‘(𝑅‘𝑥)) = X𝑥 ∈ 𝐼 (Base‘(𝑅‘𝑥)))
28		eqidd 2210	. . . 4 ⊢ (𝜑 → (𝑓 ∈ X𝑥 ∈ 𝐼 (Base‘(𝑅‘𝑥)), 𝑔 ∈ X𝑥 ∈ 𝐼 (Base‘(𝑅‘𝑥)) ↦ (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(+g‘(𝑅‘𝑥))(𝑔‘𝑥)))) = (𝑓 ∈ X𝑥 ∈ 𝐼 (Base‘(𝑅‘𝑥)), 𝑔 ∈ X𝑥 ∈ 𝐼 (Base‘(𝑅‘𝑥)) ↦ (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(+g‘(𝑅‘𝑥))(𝑔‘𝑥)))))
29		eqidd 2210	. . . 4 ⊢ (𝜑 → (𝑓 ∈ X𝑥 ∈ 𝐼 (Base‘(𝑅‘𝑥)), 𝑔 ∈ X𝑥 ∈ 𝐼 (Base‘(𝑅‘𝑥)) ↦ (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(.r‘(𝑅‘𝑥))(𝑔‘𝑥)))) = (𝑓 ∈ X𝑥 ∈ 𝐼 (Base‘(𝑅‘𝑥)), 𝑔 ∈ X𝑥 ∈ 𝐼 (Base‘(𝑅‘𝑥)) ↦ (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(.r‘(𝑅‘𝑥))(𝑔‘𝑥)))))
30		eqidd 2210	. . . 4 ⊢ (𝜑 → (𝑓 ∈ (Base‘𝑆), 𝑔 ∈ X𝑥 ∈ 𝐼 (Base‘(𝑅‘𝑥)) ↦ (𝑥 ∈ 𝐼 ↦ (𝑓( ·𝑠 ‘(𝑅‘𝑥))(𝑔‘𝑥)))) = (𝑓 ∈ (Base‘𝑆), 𝑔 ∈ X𝑥 ∈ 𝐼 (Base‘(𝑅‘𝑥)) ↦ (𝑥 ∈ 𝐼 ↦ (𝑓( ·𝑠 ‘(𝑅‘𝑥))(𝑔‘𝑥)))))
31		eqidd 2210	. . . 4 ⊢ (𝜑 → (𝑓 ∈ X𝑥 ∈ 𝐼 (Base‘(𝑅‘𝑥)), 𝑔 ∈ X𝑥 ∈ 𝐼 (Base‘(𝑅‘𝑥)) ↦ (𝑆 Σg (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(·𝑖‘(𝑅‘𝑥))(𝑔‘𝑥))))) = (𝑓 ∈ X𝑥 ∈ 𝐼 (Base‘(𝑅‘𝑥)), 𝑔 ∈ X𝑥 ∈ 𝐼 (Base‘(𝑅‘𝑥)) ↦ (𝑆 Σg (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(·𝑖‘(𝑅‘𝑥))(𝑔‘𝑥))))))
32		eqidd 2210	. . . 4 ⊢ (𝜑 → (∏t‘(TopOpen ∘ 𝑅)) = (∏t‘(TopOpen ∘ 𝑅)))
33		eqidd 2210	. . . 4 ⊢ (𝜑 → {⟨𝑓, 𝑔⟩ ∣ ({𝑓, 𝑔} ⊆ X𝑥 ∈ 𝐼 (Base‘(𝑅‘𝑥)) ∧ ∀𝑥 ∈ 𝐼 (𝑓‘𝑥)(le‘(𝑅‘𝑥))(𝑔‘𝑥))} = {⟨𝑓, 𝑔⟩ ∣ ({𝑓, 𝑔} ⊆ X𝑥 ∈ 𝐼 (Base‘(𝑅‘𝑥)) ∧ ∀𝑥 ∈ 𝐼 (𝑓‘𝑥)(le‘(𝑅‘𝑥))(𝑔‘𝑥))})
34		eqidd 2210	. . . 4 ⊢ (𝜑 → (𝑓 ∈ X𝑥 ∈ 𝐼 (Base‘(𝑅‘𝑥)), 𝑔 ∈ X𝑥 ∈ 𝐼 (Base‘(𝑅‘𝑥)) ↦ sup((ran (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(dist‘(𝑅‘𝑥))(𝑔‘𝑥))) ∪ {0}), ℝ*, < )) = (𝑓 ∈ X𝑥 ∈ 𝐼 (Base‘(𝑅‘𝑥)), 𝑔 ∈ X𝑥 ∈ 𝐼 (Base‘(𝑅‘𝑥)) ↦ sup((ran (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(dist‘(𝑅‘𝑥))(𝑔‘𝑥))) ∪ {0}), ℝ*, < )))
35		eqidd 2210	. . . 4 ⊢ (𝜑 → (𝑓 ∈ X𝑥 ∈ 𝐼 (Base‘(𝑅‘𝑥)), 𝑔 ∈ X𝑥 ∈ 𝐼 (Base‘(𝑅‘𝑥)) ↦ X𝑥 ∈ 𝐼 ((𝑓‘𝑥)(Hom ‘(𝑅‘𝑥))(𝑔‘𝑥))) = (𝑓 ∈ X𝑥 ∈ 𝐼 (Base‘(𝑅‘𝑥)), 𝑔 ∈ X𝑥 ∈ 𝐼 (Base‘(𝑅‘𝑥)) ↦ X𝑥 ∈ 𝐼 ((𝑓‘𝑥)(Hom ‘(𝑅‘𝑥))(𝑔‘𝑥))))
36		eqidd 2210	. . . 4 ⊢ (𝜑 → (𝑎 ∈ (X𝑥 ∈ 𝐼 (Base‘(𝑅‘𝑥)) × X𝑥 ∈ 𝐼 (Base‘(𝑅‘𝑥))), 𝑐 ∈ X𝑥 ∈ 𝐼 (Base‘(𝑅‘𝑥)) ↦ (𝑑 ∈ ((2nd ‘𝑎)(𝑓 ∈ X𝑥 ∈ 𝐼 (Base‘(𝑅‘𝑥)), 𝑔 ∈ X𝑥 ∈ 𝐼 (Base‘(𝑅‘𝑥)) ↦ X𝑥 ∈ 𝐼 ((𝑓‘𝑥)(Hom ‘(𝑅‘𝑥))(𝑔‘𝑥)))𝑐), 𝑒 ∈ ((𝑓 ∈ X𝑥 ∈ 𝐼 (Base‘(𝑅‘𝑥)), 𝑔 ∈ X𝑥 ∈ 𝐼 (Base‘(𝑅‘𝑥)) ↦ X𝑥 ∈ 𝐼 ((𝑓‘𝑥)(Hom ‘(𝑅‘𝑥))(𝑔‘𝑥)))‘𝑎) ↦ (𝑥 ∈ 𝐼 ↦ ((𝑑‘𝑥)(⟨((1st ‘𝑎)‘𝑥), ((2nd ‘𝑎)‘𝑥)⟩(comp‘(𝑅‘𝑥))(𝑐‘𝑥))(𝑒‘𝑥))))) = (𝑎 ∈ (X𝑥 ∈ 𝐼 (Base‘(𝑅‘𝑥)) × X𝑥 ∈ 𝐼 (Base‘(𝑅‘𝑥))), 𝑐 ∈ X𝑥 ∈ 𝐼 (Base‘(𝑅‘𝑥)) ↦ (𝑑 ∈ ((2nd ‘𝑎)(𝑓 ∈ X𝑥 ∈ 𝐼 (Base‘(𝑅‘𝑥)), 𝑔 ∈ X𝑥 ∈ 𝐼 (Base‘(𝑅‘𝑥)) ↦ X𝑥 ∈ 𝐼 ((𝑓‘𝑥)(Hom ‘(𝑅‘𝑥))(𝑔‘𝑥)))𝑐), 𝑒 ∈ ((𝑓 ∈ X𝑥 ∈ 𝐼 (Base‘(𝑅‘𝑥)), 𝑔 ∈ X𝑥 ∈ 𝐼 (Base‘(𝑅‘𝑥)) ↦ X𝑥 ∈ 𝐼 ((𝑓‘𝑥)(Hom ‘(𝑅‘𝑥))(𝑔‘𝑥)))‘𝑎) ↦ (𝑥 ∈ 𝐼 ↦ ((𝑑‘𝑥)(⟨((1st ‘𝑎)‘𝑥), ((2nd ‘𝑎)‘𝑥)⟩(comp‘(𝑅‘𝑥))(𝑐‘𝑥))(𝑒‘𝑥))))))
37	1, 26, 7, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 2, 3	prdsval 13272	. . 3 ⊢ (𝜑 → 𝑃 = (({⟨(Base‘ndx), X𝑥 ∈ 𝐼 (Base‘(𝑅‘𝑥))⟩, ⟨(+g‘ndx), (𝑓 ∈ X𝑥 ∈ 𝐼 (Base‘(𝑅‘𝑥)), 𝑔 ∈ X𝑥 ∈ 𝐼 (Base‘(𝑅‘𝑥)) ↦ (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(+g‘(𝑅‘𝑥))(𝑔‘𝑥))))⟩, ⟨(.r‘ndx), (𝑓 ∈ X𝑥 ∈ 𝐼 (Base‘(𝑅‘𝑥)), 𝑔 ∈ X𝑥 ∈ 𝐼 (Base‘(𝑅‘𝑥)) ↦ (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(.r‘(𝑅‘𝑥))(𝑔‘𝑥))))⟩} ∪ {⟨(Scalar‘ndx), 𝑆⟩, ⟨( ·𝑠 ‘ndx), (𝑓 ∈ (Base‘𝑆), 𝑔 ∈ X𝑥 ∈ 𝐼 (Base‘(𝑅‘𝑥)) ↦ (𝑥 ∈ 𝐼 ↦ (𝑓( ·𝑠 ‘(𝑅‘𝑥))(𝑔‘𝑥))))⟩, ⟨(·𝑖‘ndx), (𝑓 ∈ X𝑥 ∈ 𝐼 (Base‘(𝑅‘𝑥)), 𝑔 ∈ X𝑥 ∈ 𝐼 (Base‘(𝑅‘𝑥)) ↦ (𝑆 Σg (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(·𝑖‘(𝑅‘𝑥))(𝑔‘𝑥)))))⟩}) ∪ ({⟨(TopSet‘ndx), (∏t‘(TopOpen ∘ 𝑅))⟩, ⟨(le‘ndx), {⟨𝑓, 𝑔⟩ ∣ ({𝑓, 𝑔} ⊆ X𝑥 ∈ 𝐼 (Base‘(𝑅‘𝑥)) ∧ ∀𝑥 ∈ 𝐼 (𝑓‘𝑥)(le‘(𝑅‘𝑥))(𝑔‘𝑥))}⟩, ⟨(dist‘ndx), (𝑓 ∈ X𝑥 ∈ 𝐼 (Base‘(𝑅‘𝑥)), 𝑔 ∈ X𝑥 ∈ 𝐼 (Base‘(𝑅‘𝑥)) ↦ sup((ran (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(dist‘(𝑅‘𝑥))(𝑔‘𝑥))) ∪ {0}), ℝ*, < ))⟩} ∪ {⟨(Hom ‘ndx), (𝑓 ∈ X𝑥 ∈ 𝐼 (Base‘(𝑅‘𝑥)), 𝑔 ∈ X𝑥 ∈ 𝐼 (Base‘(𝑅‘𝑥)) ↦ X𝑥 ∈ 𝐼 ((𝑓‘𝑥)(Hom ‘(𝑅‘𝑥))(𝑔‘𝑥)))⟩, ⟨(comp‘ndx), (𝑎 ∈ (X𝑥 ∈ 𝐼 (Base‘(𝑅‘𝑥)) × X𝑥 ∈ 𝐼 (Base‘(𝑅‘𝑥))), 𝑐 ∈ X𝑥 ∈ 𝐼 (Base‘(𝑅‘𝑥)) ↦ (𝑑 ∈ ((2nd ‘𝑎)(𝑓 ∈ X𝑥 ∈ 𝐼 (Base‘(𝑅‘𝑥)), 𝑔 ∈ X𝑥 ∈ 𝐼 (Base‘(𝑅‘𝑥)) ↦ X𝑥 ∈ 𝐼 ((𝑓‘𝑥)(Hom ‘(𝑅‘𝑥))(𝑔‘𝑥)))𝑐), 𝑒 ∈ ((𝑓 ∈ X𝑥 ∈ 𝐼 (Base‘(𝑅‘𝑥)), 𝑔 ∈ X𝑥 ∈ 𝐼 (Base‘(𝑅‘𝑥)) ↦ X𝑥 ∈ 𝐼 ((𝑓‘𝑥)(Hom ‘(𝑅‘𝑥))(𝑔‘𝑥)))‘𝑎) ↦ (𝑥 ∈ 𝐼 ↦ ((𝑑‘𝑥)(⟨((1st ‘𝑎)‘𝑥), ((2nd ‘𝑎)‘𝑥)⟩(comp‘(𝑅‘𝑥))(𝑐‘𝑥))(𝑒‘𝑥)))))⟩})))
38	25, 37	sseqtrrid 3255	. 2 ⊢ (𝜑 → {⟨(Base‘ndx), X𝑥 ∈ 𝐼 (Base‘(𝑅‘𝑥))⟩} ⊆ 𝑃)
39	1, 2, 3, 4, 5, 6, 20, 38	prdsbaslemss 13273	1 ⊢ (𝜑 → 𝐵 = X𝑥 ∈ 𝐼 (Base‘(𝑅‘𝑥)))

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1375   ∈ wcel 2180  ∀wral 2488  Vcvv 2779   ∪ cun 3175   ⊆ wss 3177  {csn 3646  {cpr 3647  {ctp 3648  ⟨cop 3649   class class class wbr 4062  {copab 4123   ↦ cmpt 4124   × cxp 4694  dom cdm 4696  ran crn 4697   ∘ ccom 4700   Fn wfn 5289  ‘cfv 5294  (class class class)co 5974   ∈ cmpo 5976  1^st c1st 6254  2^nd c2nd 6255  Xcixp 6815  supcsup 7117  0cc0 7967  ℝ^*cxr 8148   < clt 8149  ndxcnx 12995  Basecbs 12998  +_gcplusg 13076  ._rcmulr 13077  Scalarcsca 13079   ·_𝑠 cvsca 13080  ·_𝑖cip 13081  TopSetcts 13082  lecple 13083  distcds 13085  Hom chom 13087  compcco 13088  TopOpenctopn 13239  ∏_tcpt 13254   Σ_g cgsu 13256  X_scprds 13264

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 713  ax-5 1473  ax-7 1474  ax-gen 1475  ax-ie1 1519  ax-ie2 1520  ax-8 1530  ax-10 1531  ax-11 1532  ax-i12 1533  ax-bndl 1535  ax-4 1536  ax-17 1552  ax-i9 1556  ax-ial 1560  ax-i5r 1561  ax-13 2182  ax-14 2183  ax-ext 2191  ax-coll 4178  ax-sep 4181  ax-pow 4237  ax-pr 4272  ax-un 4501  ax-setind 4606  ax-cnex 8058  ax-resscn 8059  ax-1cn 8060  ax-1re 8061  ax-icn 8062  ax-addcl 8063  ax-addrcl 8064  ax-mulcl 8065  ax-addcom 8067  ax-mulcom 8068  ax-addass 8069  ax-mulass 8070  ax-distr 8071  ax-i2m1 8072  ax-0lt1 8073  ax-1rid 8074  ax-0id 8075  ax-rnegex 8076  ax-cnre 8078  ax-pre-ltirr 8079  ax-pre-ltwlin 8080  ax-pre-lttrn 8081  ax-pre-apti 8082  ax-pre-ltadd 8083

This theorem depends on definitions:  df-bi 117  df-3or 984  df-3an 985  df-tru 1378  df-fal 1381  df-nf 1487  df-sb 1789  df-eu 2060  df-mo 2061  df-clab 2196  df-cleq 2202  df-clel 2205  df-nfc 2341  df-ne 2381  df-nel 2476  df-ral 2493  df-rex 2494  df-reu 2495  df-rab 2497  df-v 2781  df-sbc 3009  df-csb 3105  df-dif 3179  df-un 3181  df-in 3183  df-ss 3190  df-nul 3472  df-pw 3631  df-sn 3652  df-pr 3653  df-tp 3654  df-op 3655  df-uni 3868  df-int 3903  df-iun 3946  df-br 4063  df-opab 4125  df-mpt 4126  df-id 4361  df-xp 4702  df-rel 4703  df-cnv 4704  df-co 4705  df-dm 4706  df-rn 4707  df-res 4708  df-ima 4709  df-iota 5254  df-fun 5296  df-fn 5297  df-f 5298  df-f1 5299  df-fo 5300  df-f1o 5301  df-fv 5302  df-riota 5927  df-ov 5977  df-oprab 5978  df-mpo 5979  df-1st 6256  df-2nd 6257  df-map 6767  df-ixp 6816  df-sup 7119  df-pnf 8151  df-mnf 8152  df-xr 8153  df-ltxr 8154  df-le 8155  df-sub 8287  df-neg 8288  df-inn 9079  df-2 9137  df-3 9138  df-4 9139  df-5 9140  df-6 9141  df-7 9142  df-8 9143  df-9 9144  df-n0 9338  df-z 9415  df-dec 9547  df-uz 9691  df-fz 10173  df-struct 13000  df-ndx 13001  df-slot 13002  df-base 13004  df-plusg 13089  df-mulr 13090  df-sca 13092  df-vsca 13093  df-ip 13094  df-tset 13095  df-ple 13096  df-ds 13098  df-hom 13100  df-cco 13101  df-rest 13240  df-topn 13241  df-topgen 13259  df-pt 13260  df-prds 13266

This theorem is referenced by:  prdsplusg  13276  prdsmulr  13277  prdsbas2  13278  pwsbas  13291