Theorem prdsbas 14282

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 13463	. 2 ⊢ Base = Slot (Base‘ndx)
6		basendxnn 13465	. 2 ⊢ (Base‘ndx) ∈ ℕ
7		prdsbas.i	. . . 4 ⊢ (𝜑 → dom 𝑅 = 𝐼)
8		dmexg 5048	. . . . 5 ⊢ (𝑅 ∈ 𝑊 → dom 𝑅 ∈ V)
9	3, 8	syl 14	. . . 4 ⊢ (𝜑 → dom 𝑅 ∈ V)
10	7, 9	eqeltrrd 2316	. . 3 ⊢ (𝜑 → 𝐼 ∈ V)
11		basfn 13468	. . . . 5 ⊢ Base Fn V
12		vex 2824	. . . . . 6 ⊢ 𝑥 ∈ V
13		fvexg 5717	. . . . . 6 ⊢ ((𝑅 ∈ 𝑊 ∧ 𝑥 ∈ V) → (𝑅‘𝑥) ∈ V)
14	3, 12, 13	sylancl 417	. . . . 5 ⊢ (𝜑 → (𝑅‘𝑥) ∈ V)
15		funfvex 5715	. . . . . 6 ⊢ ((Fun Base ∧ (𝑅‘𝑥) ∈ dom Base) → (Base‘(𝑅‘𝑥)) ∈ V)
16	15	funfni 5486	. . . . 5 ⊢ ((Base Fn V ∧ (𝑅‘𝑥) ∈ V) → (Base‘(𝑅‘𝑥)) ∈ V)
17	11, 14, 16	sylancr 418	. . . 4 ⊢ (𝜑 → (Base‘(𝑅‘𝑥)) ∈ V)
18	17	ralrimivw 2624	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝐼 (Base‘(𝑅‘𝑥)) ∈ V)
19		ixpexgg 7010	. . 3 ⊢ ((𝐼 ∈ V ∧ ∀𝑥 ∈ 𝐼 (Base‘(𝑅‘𝑥)) ∈ V) → X𝑥 ∈ 𝐼 (Base‘(𝑅‘𝑥)) ∈ V)
20	10, 18, 19	syl2anc 415	. 2 ⊢ (𝜑 → X𝑥 ∈ 𝐼 (Base‘(𝑅‘𝑥)) ∈ V)
21		snsstp1 3865	. . . . 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 3392	. . . . 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 3257	. . . 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 3392	. . . 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 3257	. . 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 2238	. . . 4 ⊢ (Base‘𝑆) = (Base‘𝑆)
27		eqidd 2239	. . . 4 ⊢ (𝜑 → X𝑥 ∈ 𝐼 (Base‘(𝑅‘𝑥)) = X𝑥 ∈ 𝐼 (Base‘(𝑅‘𝑥)))
28		eqidd 2239	. . . 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 2239	. . . 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 2239	. . . 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 2239	. . . 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 2239	. . . 4 ⊢ (𝜑 → (∏t‘(TopOpen ∘ 𝑅)) = (∏t‘(TopOpen ∘ 𝑅)))
33		eqidd 2239	. . . 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 2239	. . . 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 2239	. . . 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 2239	. . . 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 14279	. . 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 3299	. 2 ⊢ (𝜑 → {⟨(Base‘ndx), X𝑥 ∈ 𝐼 (Base‘(𝑅‘𝑥))⟩} ⊆ 𝑃)
39	1, 2, 3, 4, 5, 6, 20, 38	prdsbaslemss 14280	1 ⊢ (𝜑 → 𝐵 = X𝑥 ∈ 𝐼 (Base‘(𝑅‘𝑥)))

This proof depends on syntax axioms:   → wi 4   ∧ wa 104   = wceq 1402   ∈ wcel 2209  ∀wral 2528  Vcvv 2821   ∪ cun 3218   ⊆ wss 3220  {csn 3709  {cpr 3710  {ctp 3711  ⟨cop 3712   class class class wbr 4131  {copab 4192   ↦ cmpt 4193   × cxp 4774  dom cdm 4776  ran crn 4777   ∘ ccom 4780   Fn wfn 5375  ‘cfv 5380  (class class class)co 6088   ∈ cmpo 6090  1^st c1st 6376  2^nd c2nd 6377  Xcixp 6986  supcsup 7328  0cc0 8185  ℝ^*cxr 8365   < clt 8366  ndxcnx 13406  Basecbs 13409  +_gcplusg 13489  ._rcmulr 13490  Scalarcsca 13492   ·_𝑠 cvsca 13493  ·_𝑖cip 13494  TopSetcts 13495  lecple 13496  distcds 13498  Hom chom 13500  compcco 13501  TopOpenctopn 13652  ∏_tcpt 13667   Σ_g cgsu 14239  X_scprds 14275

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4248  ax-sep 4251  ax-pow 4313  ax-pr 4348  ax-un 4580  ax-setind 4686  ax-cnex 8276  ax-resscn 8277  ax-1cn 8278  ax-1re 8279  ax-icn 8280  ax-addcl 8281  ax-addrcl 8282  ax-mulcl 8283  ax-addcom 8285  ax-mulcom 8286  ax-addass 8287  ax-mulass 8288  ax-distr 8289  ax-i2m1 8290  ax-0lt1 8291  ax-1rid 8292  ax-0id 8293  ax-rnegex 8294  ax-cnre 8296  ax-pre-ltirr 8297  ax-pre-ltwlin 8298  ax-pre-lttrn 8299  ax-pre-apti 8300  ax-pre-ltadd 8301

This proof depends on definitions:  df-bi 117  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3690  df-sn 3715  df-pr 3716  df-tp 3717  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4132  df-opab 4194  df-mpt 4195  df-id 4440  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-rn 4787  df-res 4788  df-ima 4789  df-iota 5340  df-fun 5382  df-fn 5383  df-f 5384  df-f1 5385  df-fo 5386  df-f1o 5387  df-fv 5388  df-riota 6041  df-ov 6091  df-oprab 6092  df-mpo 6093  df-1st 6378  df-2nd 6379  df-map 6929  df-ixp 6987  df-sup 7330  df-pnf 8368  df-mnf 8369  df-xr 8370  df-ltxr 8371  df-le 8372  df-sub 8506  df-neg 8507  df-inn 9314  df-2 9372  df-3 9373  df-4 9374  df-5 9375  df-6 9376  df-7 9377  df-8 9378  df-9 9379  df-n0 9575  df-z 9654  df-dec 9788  df-uz 9937  df-fz 10428  df-struct 13411  df-ndx 13412  df-slot 13413  df-base 13415  df-plusg 13502  df-mulr 13503  df-sca 13505  df-vsca 13506  df-ip 13507  df-tset 13508  df-ple 13509  df-ds 13511  df-hom 13513  df-cco 13514  df-rest 13653  df-topn 13654  df-topgen 13672  df-pt 13673  df-prds 14276

This theorem is used by:  prdsplusg  14283  prdsmulr  14284  prdsbas2  14285  pwsbas  14311