Theorem prdsbas 17410

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 2731	. . 3 ⊢ (Base‘𝑆) = (Base‘𝑆)
3		prdsbas.i	. . 3 ⊢ (𝜑 → dom 𝑅 = 𝐼)
4		eqidd 2732	. . 3 ⊢ (𝜑 → X𝑥 ∈ 𝐼 (Base‘(𝑅‘𝑥)) = X𝑥 ∈ 𝐼 (Base‘(𝑅‘𝑥)))
5		eqidd 2732	. . 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 2732	. . 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 2732	. . 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 2732	. . 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 2732	. . 3 ⊢ (𝜑 → (∏t‘(TopOpen ∘ 𝑅)) = (∏t‘(TopOpen ∘ 𝑅)))
10		eqidd 2732	. . 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 2732	. . 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 2732	. . 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 2732	. . 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 17408	. 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 17154	. 2 ⊢ Base = Slot (Base‘ndx)
19	18	strfvss 17127	. . . . . . 7 ⊢ (Base‘(𝑅‘𝑥)) ⊆ ∪ ran (𝑅‘𝑥)
20		fvssunirn 6924	. . . . . . . 8 ⊢ (𝑅‘𝑥) ⊆ ∪ ran 𝑅
21		rnss 5938	. . . . . . . 8 ⊢ ((𝑅‘𝑥) ⊆ ∪ ran 𝑅 → ran (𝑅‘𝑥) ⊆ ran ∪ ran 𝑅)
22		uniss 4916	. . . . . . . 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 3991	. . . . . 6 ⊢ (Base‘(𝑅‘𝑥)) ⊆ ∪ ran ∪ ran 𝑅
25	24	rgenw 3064	. . . . 5 ⊢ ∀𝑥 ∈ 𝐼 (Base‘(𝑅‘𝑥)) ⊆ ∪ ran ∪ ran 𝑅
26		iunss 5048	. . . . 5 ⊢ (∪ 𝑥 ∈ 𝐼 (Base‘(𝑅‘𝑥)) ⊆ ∪ ran ∪ ran 𝑅 ↔ ∀𝑥 ∈ 𝐼 (Base‘(𝑅‘𝑥)) ⊆ ∪ ran ∪ ran 𝑅)
27	25, 26	mpbir 230	. . . 4 ⊢ ∪ 𝑥 ∈ 𝐼 (Base‘(𝑅‘𝑥)) ⊆ ∪ ran ∪ ran 𝑅
28		rnexg 7899	. . . . . 6 ⊢ (𝑅 ∈ 𝑊 → ran 𝑅 ∈ V)
29		uniexg 7734	. . . . . 6 ⊢ (ran 𝑅 ∈ V → ∪ ran 𝑅 ∈ V)
30	15, 28, 29	3syl 18	. . . . 5 ⊢ (𝜑 → ∪ ran 𝑅 ∈ V)
31		rnexg 7899	. . . . 5 ⊢ (∪ ran 𝑅 ∈ V → ran ∪ ran 𝑅 ∈ V)
32		uniexg 7734	. . . . 5 ⊢ (ran ∪ ran 𝑅 ∈ V → ∪ ran ∪ ran 𝑅 ∈ V)
33	30, 31, 32	3syl 18	. . . 4 ⊢ (𝜑 → ∪ ran ∪ ran 𝑅 ∈ V)
34		ssexg 5323	. . . 4 ⊢ ((∪ 𝑥 ∈ 𝐼 (Base‘(𝑅‘𝑥)) ⊆ ∪ ran ∪ ran 𝑅 ∧ ∪ ran ∪ ran 𝑅 ∈ V) → ∪ 𝑥 ∈ 𝐼 (Base‘(𝑅‘𝑥)) ∈ V)
35	27, 33, 34	sylancr 586	. . 3 ⊢ (𝜑 → ∪ 𝑥 ∈ 𝐼 (Base‘(𝑅‘𝑥)) ∈ V)
36		ixpssmap2g 8927	. . 3 ⊢ (∪ 𝑥 ∈ 𝐼 (Base‘(𝑅‘𝑥)) ∈ V → X𝑥 ∈ 𝐼 (Base‘(𝑅‘𝑥)) ⊆ (∪ 𝑥 ∈ 𝐼 (Base‘(𝑅‘𝑥)) ↑m 𝐼))
37		ovex 7445	. . . 4 ⊢ (∪ 𝑥 ∈ 𝐼 (Base‘(𝑅‘𝑥)) ↑m 𝐼) ∈ V
38	37	ssex 5321	. . 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 4819	. . . 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 4172	. . . 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 3991	. . 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 4172	. . 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 3991	. 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 17406	1 ⊢ (𝜑 → 𝐵 = X𝑥 ∈ 𝐼 (Base‘(𝑅‘𝑥)))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2105  ∀wral 3060  Vcvv 3473   ∪ cun 3946   ⊆ wss 3948  {csn 4628  {cpr 4630  {ctp 4632  ⟨cop 4634  ∪ cuni 4908  ∪ ciun 4997   class class class wbr 5148  {copab 5210   ↦ cmpt 5231   × cxp 5674  dom cdm 5676  ran crn 5677   ∘ ccom 5680  ‘cfv 6543  (class class class)co 7412   ∈ cmpo 7414  1^st c1st 7977  2^nd c2nd 7978   ↑_m cmap 8826  Xcixp 8897  supcsup 9441  0cc0 11116  ℝ^*cxr 11254   < clt 11255  ndxcnx 17133  Basecbs 17151  +_gcplusg 17204  ._rcmulr 17205  Scalarcsca 17207   ·_𝑠 cvsca 17208  ·_𝑖cip 17209  TopSetcts 17210  lecple 17211  distcds 17213  Hom chom 17215  compcco 17216  TopOpenctopn 17374  ∏_tcpt 17391   Σ_g cgsu 17393  X_scprds 17398

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7729  ax-cnex 11172  ax-resscn 11173  ax-1cn 11174  ax-icn 11175  ax-addcl 11176  ax-addrcl 11177  ax-mulcl 11178  ax-mulrcl 11179  ax-mulcom 11180  ax-addass 11181  ax-mulass 11182  ax-distr 11183  ax-i2m1 11184  ax-1ne0 11185  ax-1rid 11186  ax-rnegex 11187  ax-rrecex 11188  ax-cnre 11189  ax-pre-lttri 11190  ax-pre-lttrn 11191  ax-pre-ltadd 11192  ax-pre-mulgt0 11193

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-tp 4633  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7368  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7860  df-1st 7979  df-2nd 7980  df-frecs 8272  df-wrecs 8303  df-recs 8377  df-rdg 8416  df-1o 8472  df-er 8709  df-map 8828  df-ixp 8898  df-en 8946  df-dom 8947  df-sdom 8948  df-fin 8949  df-sup 9443  df-pnf 11257  df-mnf 11258  df-xr 11259  df-ltxr 11260  df-le 11261  df-sub 11453  df-neg 11454  df-nn 12220  df-2 12282  df-3 12283  df-4 12284  df-5 12285  df-6 12286  df-7 12287  df-8 12288  df-9 12289  df-n0 12480  df-z 12566  df-dec 12685  df-uz 12830  df-fz 13492  df-struct 17087  df-slot 17122  df-ndx 17134  df-base 17152  df-plusg 17217  df-mulr 17218  df-sca 17220  df-vsca 17221  df-ip 17222  df-tset 17223  df-ple 17224  df-ds 17226  df-hom 17228  df-cco 17229  df-prds 17400

This theorem is referenced by:  prdsplusg  17411  prdsmulr  17412  prdsvsca  17413  prdsip  17414  prdsle  17415  prdsds  17417  prdstset  17419  prdshom  17420  prdsco  17421  prdsbas2  17422  pwsbas  17440  dsmmval  21598  frlmip  21642  prdstps  23452  rrxip  25237  prdstotbnd  37125