Theorem prdsbas 17427

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	. . 3 ⊢ 𝑃 = (𝑆Xs𝑅)
2		eqid 2730	. . 3 ⊢ (Base‘𝑆) = (Base‘𝑆)
3		prdsbas.i	. . 3 ⊢ (𝜑 → dom 𝑅 = 𝐼)
4		eqidd 2731	. . 3 ⊢ (𝜑 → X𝑥 ∈ 𝐼 (Base‘(𝑅‘𝑥)) = X𝑥 ∈ 𝐼 (Base‘(𝑅‘𝑥)))
5		eqidd 2731	. . 3 ⊢ (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)))))
6		eqidd 2731	. . 3 ⊢ (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)))))
7		eqidd 2731	. . 3 ⊢ (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)))))
8		eqidd 2731	. . 3 ⊢ (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))))))
9		eqidd 2731	. . 3 ⊢ (𝜑 → (∏t‘(TopOpen ∘ 𝑅)) = (∏t‘(TopOpen ∘ 𝑅)))
10		eqidd 2731	. . 3 ⊢ (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))})
11		eqidd 2731	. . 3 ⊢ (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*, < )))
12		eqidd 2731	. . 3 ⊢ (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))))
13		eqidd 2731	. . 3 ⊢ (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))))))
14		prdsbas.s	. . 3 ⊢ (𝜑 → 𝑆 ∈ 𝑉)
15		prdsbas.r	. . 3 ⊢ (𝜑 → 𝑅 ∈ 𝑊)
16	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15	prdsval 17425	. 2 ⊢ (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)))))⟩})))
17		prdsbas.b	. 2 ⊢ 𝐵 = (Base‘𝑃)
18		baseid 17189	. 2 ⊢ Base = Slot (Base‘ndx)
19	18	strfvss 17164	. . . . . . 7 ⊢ (Base‘(𝑅‘𝑥)) ⊆ ∪ ran (𝑅‘𝑥)
20		fvssunirn 6894	. . . . . . . 8 ⊢ (𝑅‘𝑥) ⊆ ∪ ran 𝑅
21		rnss 5906	. . . . . . . 8 ⊢ ((𝑅‘𝑥) ⊆ ∪ ran 𝑅 → ran (𝑅‘𝑥) ⊆ ran ∪ ran 𝑅)
22		uniss 4882	. . . . . . . 8 ⊢ (ran (𝑅‘𝑥) ⊆ ran ∪ ran 𝑅 → ∪ ran (𝑅‘𝑥) ⊆ ∪ ran ∪ ran 𝑅)
23	20, 21, 22	mp2b 10	. . . . . . 7 ⊢ ∪ ran (𝑅‘𝑥) ⊆ ∪ ran ∪ ran 𝑅
24	19, 23	sstri 3959	. . . . . 6 ⊢ (Base‘(𝑅‘𝑥)) ⊆ ∪ ran ∪ ran 𝑅
25	24	rgenw 3049	. . . . 5 ⊢ ∀𝑥 ∈ 𝐼 (Base‘(𝑅‘𝑥)) ⊆ ∪ ran ∪ ran 𝑅
26		iunss 5012	. . . . 5 ⊢ (∪ 𝑥 ∈ 𝐼 (Base‘(𝑅‘𝑥)) ⊆ ∪ ran ∪ ran 𝑅 ↔ ∀𝑥 ∈ 𝐼 (Base‘(𝑅‘𝑥)) ⊆ ∪ ran ∪ ran 𝑅)
27	25, 26	mpbir 231	. . . 4 ⊢ ∪ 𝑥 ∈ 𝐼 (Base‘(𝑅‘𝑥)) ⊆ ∪ ran ∪ ran 𝑅
28		rnexg 7881	. . . . . 6 ⊢ (𝑅 ∈ 𝑊 → ran 𝑅 ∈ V)
29		uniexg 7719	. . . . . 6 ⊢ (ran 𝑅 ∈ V → ∪ ran 𝑅 ∈ V)
30	15, 28, 29	3syl 18	. . . . 5 ⊢ (𝜑 → ∪ ran 𝑅 ∈ V)
31		rnexg 7881	. . . . 5 ⊢ (∪ ran 𝑅 ∈ V → ran ∪ ran 𝑅 ∈ V)
32		uniexg 7719	. . . . 5 ⊢ (ran ∪ ran 𝑅 ∈ V → ∪ ran ∪ ran 𝑅 ∈ V)
33	30, 31, 32	3syl 18	. . . 4 ⊢ (𝜑 → ∪ ran ∪ ran 𝑅 ∈ V)
34		ssexg 5281	. . . 4 ⊢ ((∪ 𝑥 ∈ 𝐼 (Base‘(𝑅‘𝑥)) ⊆ ∪ ran ∪ ran 𝑅 ∧ ∪ ran ∪ ran 𝑅 ∈ V) → ∪ 𝑥 ∈ 𝐼 (Base‘(𝑅‘𝑥)) ∈ V)
35	27, 33, 34	sylancr 587	. . 3 ⊢ (𝜑 → ∪ 𝑥 ∈ 𝐼 (Base‘(𝑅‘𝑥)) ∈ V)
36		ixpssmap2g 8903	. . 3 ⊢ (∪ 𝑥 ∈ 𝐼 (Base‘(𝑅‘𝑥)) ∈ V → X𝑥 ∈ 𝐼 (Base‘(𝑅‘𝑥)) ⊆ (∪ 𝑥 ∈ 𝐼 (Base‘(𝑅‘𝑥)) ↑m 𝐼))
37		ovex 7423	. . . 4 ⊢ (∪ 𝑥 ∈ 𝐼 (Base‘(𝑅‘𝑥)) ↑m 𝐼) ∈ V
38	37	ssex 5279	. . 3 ⊢ (X𝑥 ∈ 𝐼 (Base‘(𝑅‘𝑥)) ⊆ (∪ 𝑥 ∈ 𝐼 (Base‘(𝑅‘𝑥)) ↑m 𝐼) → X𝑥 ∈ 𝐼 (Base‘(𝑅‘𝑥)) ∈ V)
39	35, 36, 38	3syl 18	. 2 ⊢ (𝜑 → X𝑥 ∈ 𝐼 (Base‘(𝑅‘𝑥)) ∈ V)
40		snsstp1 4783	. . . 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))))⟩}
41		ssun1 4144	. . . 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))))⟩} ⊆ ({⟨(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)))))⟩})
42	40, 41	sstri 3959	. . 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)))))⟩})
43		ssun1 4144	. . 3 ⊢ ({⟨(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)))))⟩}))
44	42, 43	sstri 3959	. 2 ⊢ {⟨(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)))))⟩}))
45	16, 17, 18, 39, 44	prdsbaslem 17423	1 ⊢ (𝜑 → 𝐵 = X𝑥 ∈ 𝐼 (Base‘(𝑅‘𝑥)))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∀wral 3045  Vcvv 3450   ∪ cun 3915   ⊆ wss 3917  {csn 4592  {cpr 4594  {ctp 4596  ⟨cop 4598  ∪ cuni 4874  ∪ ciun 4958   class class class wbr 5110  {copab 5172   ↦ cmpt 5191   × cxp 5639  dom cdm 5641  ran crn 5642   ∘ ccom 5645  ‘cfv 6514  (class class class)co 7390   ∈ cmpo 7392  1^st c1st 7969  2^nd c2nd 7970   ↑_m cmap 8802  Xcixp 8873  supcsup 9398  0cc0 11075  ℝ^*cxr 11214   < clt 11215  ndxcnx 17170  Basecbs 17186  +_gcplusg 17227  ._rcmulr 17228  Scalarcsca 17230   ·_𝑠 cvsca 17231  ·_𝑖cip 17232  TopSetcts 17233  lecple 17234  distcds 17236  Hom chom 17238  compcco 17239  TopOpenctopn 17391  ∏_tcpt 17408   Σ_g cgsu 17410  X_scprds 17415

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714  ax-cnex 11131  ax-resscn 11132  ax-1cn 11133  ax-icn 11134  ax-addcl 11135  ax-addrcl 11136  ax-mulcl 11137  ax-mulrcl 11138  ax-mulcom 11139  ax-addass 11140  ax-mulass 11141  ax-distr 11142  ax-i2m1 11143  ax-1ne0 11144  ax-1rid 11145  ax-rnegex 11146  ax-rrecex 11147  ax-cnre 11148  ax-pre-lttri 11149  ax-pre-lttrn 11150  ax-pre-ltadd 11151  ax-pre-mulgt0 11152

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-nel 3031  df-ral 3046  df-rex 3055  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-pss 3937  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-tp 4597  df-op 4599  df-uni 4875  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5536  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5594  df-we 5596  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-pred 6277  df-ord 6338  df-on 6339  df-lim 6340  df-suc 6341  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-riota 7347  df-ov 7393  df-oprab 7394  df-mpo 7395  df-om 7846  df-1st 7971  df-2nd 7972  df-frecs 8263  df-wrecs 8294  df-recs 8343  df-rdg 8381  df-1o 8437  df-er 8674  df-map 8804  df-ixp 8874  df-en 8922  df-dom 8923  df-sdom 8924  df-fin 8925  df-sup 9400  df-pnf 11217  df-mnf 11218  df-xr 11219  df-ltxr 11220  df-le 11221  df-sub 11414  df-neg 11415  df-nn 12194  df-2 12256  df-3 12257  df-4 12258  df-5 12259  df-6 12260  df-7 12261  df-8 12262  df-9 12263  df-n0 12450  df-z 12537  df-dec 12657  df-uz 12801  df-fz 13476  df-struct 17124  df-slot 17159  df-ndx 17171  df-base 17187  df-plusg 17240  df-mulr 17241  df-sca 17243  df-vsca 17244  df-ip 17245  df-tset 17246  df-ple 17247  df-ds 17249  df-hom 17251  df-cco 17252  df-prds 17417

This theorem is referenced by:  prdsplusg  17428  prdsmulr  17429  prdsvsca  17430  prdsip  17431  prdsle  17432  prdsds  17434  prdstset  17436  prdshom  17437  prdsco  17438  prdsbas2  17439  pwsbas  17457  dsmmval  21650  frlmip  21694  prdstps  23523  rrxip  25297  prdstotbnd  37795