Theorem prdsbas 17402

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 2732	. . 3 ⊢ (Base‘𝑆) = (Base‘𝑆)
3		prdsbas.i	. . 3 ⊢ (𝜑 → dom 𝑅 = 𝐼)
4		eqidd 2733	. . 3 ⊢ (𝜑 → X𝑥 ∈ 𝐼 (Base‘(𝑅‘𝑥)) = X𝑥 ∈ 𝐼 (Base‘(𝑅‘𝑥)))
5		eqidd 2733	. . 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 2733	. . 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 2733	. . 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 2733	. . 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 2733	. . 3 ⊢ (𝜑 → (∏t‘(TopOpen ∘ 𝑅)) = (∏t‘(TopOpen ∘ 𝑅)))
10		eqidd 2733	. . 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 2733	. . 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 2733	. . 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 2733	. . 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 17400	. 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 17146	. 2 ⊢ Base = Slot (Base‘ndx)
19	18	strfvss 17119	. . . . . . 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 3065	. . . . 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 7894	. . . . . 6 ⊢ (𝑅 ∈ 𝑊 → ran 𝑅 ∈ V)
29		uniexg 7729	. . . . . 6 ⊢ (ran 𝑅 ∈ V → ∪ ran 𝑅 ∈ V)
30	15, 28, 29	3syl 18	. . . . 5 ⊢ (𝜑 → ∪ ran 𝑅 ∈ V)
31		rnexg 7894	. . . . 5 ⊢ (∪ ran 𝑅 ∈ V → ran ∪ ran 𝑅 ∈ V)
32		uniexg 7729	. . . . 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 587	. . 3 ⊢ (𝜑 → ∪ 𝑥 ∈ 𝐼 (Base‘(𝑅‘𝑥)) ∈ V)
36		ixpssmap2g 8920	. . 3 ⊢ (∪ 𝑥 ∈ 𝐼 (Base‘(𝑅‘𝑥)) ∈ V → X𝑥 ∈ 𝐼 (Base‘(𝑅‘𝑥)) ⊆ (∪ 𝑥 ∈ 𝐼 (Base‘(𝑅‘𝑥)) ↑m 𝐼))
37		ovex 7441	. . . 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 17398	1 ⊢ (𝜑 → 𝐵 = X𝑥 ∈ 𝐼 (Base‘(𝑅‘𝑥)))

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1541   ∈ wcel 2106  ∀wral 3061  Vcvv 3474   ∪ 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 7408   ∈ cmpo 7410  1^st c1st 7972  2^nd c2nd 7973   ↑_m cmap 8819  Xcixp 8890  supcsup 9434  0cc0 11109  ℝ^*cxr 11246   < clt 11247  ndxcnx 17125  Basecbs 17143  +_gcplusg 17196  ._rcmulr 17197  Scalarcsca 17199   ·_𝑠 cvsca 17200  ·_𝑖cip 17201  TopSetcts 17202  lecple 17203  distcds 17205  Hom chom 17207  compcco 17208  TopOpenctopn 17366  ∏_tcpt 17383   Σ_g cgsu 17385  X_scprds 17390

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7724  ax-cnex 11165  ax-resscn 11166  ax-1cn 11167  ax-icn 11168  ax-addcl 11169  ax-addrcl 11170  ax-mulcl 11171  ax-mulrcl 11172  ax-mulcom 11173  ax-addass 11174  ax-mulass 11175  ax-distr 11176  ax-i2m1 11177  ax-1ne0 11178  ax-1rid 11179  ax-rnegex 11180  ax-rrecex 11181  ax-cnre 11182  ax-pre-lttri 11183  ax-pre-lttrn 11184  ax-pre-ltadd 11185  ax-pre-mulgt0 11186

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  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 7364  df-ov 7411  df-oprab 7412  df-mpo 7413  df-om 7855  df-1st 7974  df-2nd 7975  df-frecs 8265  df-wrecs 8296  df-recs 8370  df-rdg 8409  df-1o 8465  df-er 8702  df-map 8821  df-ixp 8891  df-en 8939  df-dom 8940  df-sdom 8941  df-fin 8942  df-sup 9436  df-pnf 11249  df-mnf 11250  df-xr 11251  df-ltxr 11252  df-le 11253  df-sub 11445  df-neg 11446  df-nn 12212  df-2 12274  df-3 12275  df-4 12276  df-5 12277  df-6 12278  df-7 12279  df-8 12280  df-9 12281  df-n0 12472  df-z 12558  df-dec 12677  df-uz 12822  df-fz 13484  df-struct 17079  df-slot 17114  df-ndx 17126  df-base 17144  df-plusg 17209  df-mulr 17210  df-sca 17212  df-vsca 17213  df-ip 17214  df-tset 17215  df-ple 17216  df-ds 17218  df-hom 17220  df-cco 17221  df-prds 17392

This theorem is referenced by:  prdsplusg  17403  prdsmulr  17404  prdsvsca  17405  prdsip  17406  prdsle  17407  prdsds  17409  prdstset  17411  prdshom  17412  prdsco  17413  prdsbas2  17414  pwsbas  17432  dsmmval  21288  frlmip  21332  prdstps  23132  rrxip  24906  prdstotbnd  36657