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Theorem prdsbas 17409
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 (πœ‘ β†’ 𝑆 ∈ 𝑉)
prdsbas.r (πœ‘ β†’ 𝑅 ∈ π‘Š)
prdsbas.b 𝐡 = (Baseβ€˜π‘ƒ)
prdsbas.i (πœ‘ β†’ dom 𝑅 = 𝐼)
Assertion
Ref Expression
prdsbas (πœ‘ β†’ 𝐡 = Xπ‘₯ ∈ 𝐼 (Baseβ€˜(π‘…β€˜π‘₯)))
Distinct variable groups:   π‘₯,𝐡   πœ‘,π‘₯   π‘₯,𝐼   π‘₯,𝑃   π‘₯,𝑅   π‘₯,𝑆
Allowed substitution hints:   𝑉(π‘₯)   π‘Š(π‘₯)

Proof of Theorem prdsbas
Dummy variables π‘Ž 𝑐 𝑑 𝑒 𝑓 𝑔 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 prdsbas.p . . 3 𝑃 = (𝑆Xs𝑅)
2 eqid 2726 . . 3 (Baseβ€˜π‘†) = (Baseβ€˜π‘†)
3 prdsbas.i . . 3 (πœ‘ β†’ dom 𝑅 = 𝐼)
4 eqidd 2727 . . 3 (πœ‘ β†’ Xπ‘₯ ∈ 𝐼 (Baseβ€˜(π‘…β€˜π‘₯)) = Xπ‘₯ ∈ 𝐼 (Baseβ€˜(π‘…β€˜π‘₯)))
5 eqidd 2727 . . 3 (πœ‘ β†’ (𝑓 ∈ Xπ‘₯ ∈ 𝐼 (Baseβ€˜(π‘…β€˜π‘₯)), 𝑔 ∈ Xπ‘₯ ∈ 𝐼 (Baseβ€˜(π‘…β€˜π‘₯)) ↦ (π‘₯ ∈ 𝐼 ↦ ((π‘“β€˜π‘₯)(+gβ€˜(π‘…β€˜π‘₯))(π‘”β€˜π‘₯)))) = (𝑓 ∈ Xπ‘₯ ∈ 𝐼 (Baseβ€˜(π‘…β€˜π‘₯)), 𝑔 ∈ Xπ‘₯ ∈ 𝐼 (Baseβ€˜(π‘…β€˜π‘₯)) ↦ (π‘₯ ∈ 𝐼 ↦ ((π‘“β€˜π‘₯)(+gβ€˜(π‘…β€˜π‘₯))(π‘”β€˜π‘₯)))))
6 eqidd 2727 . . 3 (πœ‘ β†’ (𝑓 ∈ Xπ‘₯ ∈ 𝐼 (Baseβ€˜(π‘…β€˜π‘₯)), 𝑔 ∈ Xπ‘₯ ∈ 𝐼 (Baseβ€˜(π‘…β€˜π‘₯)) ↦ (π‘₯ ∈ 𝐼 ↦ ((π‘“β€˜π‘₯)(.rβ€˜(π‘…β€˜π‘₯))(π‘”β€˜π‘₯)))) = (𝑓 ∈ Xπ‘₯ ∈ 𝐼 (Baseβ€˜(π‘…β€˜π‘₯)), 𝑔 ∈ Xπ‘₯ ∈ 𝐼 (Baseβ€˜(π‘…β€˜π‘₯)) ↦ (π‘₯ ∈ 𝐼 ↦ ((π‘“β€˜π‘₯)(.rβ€˜(π‘…β€˜π‘₯))(π‘”β€˜π‘₯)))))
7 eqidd 2727 . . 3 (πœ‘ β†’ (𝑓 ∈ (Baseβ€˜π‘†), 𝑔 ∈ Xπ‘₯ ∈ 𝐼 (Baseβ€˜(π‘…β€˜π‘₯)) ↦ (π‘₯ ∈ 𝐼 ↦ (𝑓( ·𝑠 β€˜(π‘…β€˜π‘₯))(π‘”β€˜π‘₯)))) = (𝑓 ∈ (Baseβ€˜π‘†), 𝑔 ∈ Xπ‘₯ ∈ 𝐼 (Baseβ€˜(π‘…β€˜π‘₯)) ↦ (π‘₯ ∈ 𝐼 ↦ (𝑓( ·𝑠 β€˜(π‘…β€˜π‘₯))(π‘”β€˜π‘₯)))))
8 eqidd 2727 . . 3 (πœ‘ β†’ (𝑓 ∈ Xπ‘₯ ∈ 𝐼 (Baseβ€˜(π‘…β€˜π‘₯)), 𝑔 ∈ Xπ‘₯ ∈ 𝐼 (Baseβ€˜(π‘…β€˜π‘₯)) ↦ (𝑆 Ξ£g (π‘₯ ∈ 𝐼 ↦ ((π‘“β€˜π‘₯)(Β·π‘–β€˜(π‘…β€˜π‘₯))(π‘”β€˜π‘₯))))) = (𝑓 ∈ Xπ‘₯ ∈ 𝐼 (Baseβ€˜(π‘…β€˜π‘₯)), 𝑔 ∈ Xπ‘₯ ∈ 𝐼 (Baseβ€˜(π‘…β€˜π‘₯)) ↦ (𝑆 Ξ£g (π‘₯ ∈ 𝐼 ↦ ((π‘“β€˜π‘₯)(Β·π‘–β€˜(π‘…β€˜π‘₯))(π‘”β€˜π‘₯))))))
9 eqidd 2727 . . 3 (πœ‘ β†’ (∏tβ€˜(TopOpen ∘ 𝑅)) = (∏tβ€˜(TopOpen ∘ 𝑅)))
10 eqidd 2727 . . 3 (πœ‘ β†’ {βŸ¨π‘“, π‘”βŸ© ∣ ({𝑓, 𝑔} βŠ† Xπ‘₯ ∈ 𝐼 (Baseβ€˜(π‘…β€˜π‘₯)) ∧ βˆ€π‘₯ ∈ 𝐼 (π‘“β€˜π‘₯)(leβ€˜(π‘…β€˜π‘₯))(π‘”β€˜π‘₯))} = {βŸ¨π‘“, π‘”βŸ© ∣ ({𝑓, 𝑔} βŠ† Xπ‘₯ ∈ 𝐼 (Baseβ€˜(π‘…β€˜π‘₯)) ∧ βˆ€π‘₯ ∈ 𝐼 (π‘“β€˜π‘₯)(leβ€˜(π‘…β€˜π‘₯))(π‘”β€˜π‘₯))})
11 eqidd 2727 . . 3 (πœ‘ β†’ (𝑓 ∈ Xπ‘₯ ∈ 𝐼 (Baseβ€˜(π‘…β€˜π‘₯)), 𝑔 ∈ Xπ‘₯ ∈ 𝐼 (Baseβ€˜(π‘…β€˜π‘₯)) ↦ sup((ran (π‘₯ ∈ 𝐼 ↦ ((π‘“β€˜π‘₯)(distβ€˜(π‘…β€˜π‘₯))(π‘”β€˜π‘₯))) βˆͺ {0}), ℝ*, < )) = (𝑓 ∈ Xπ‘₯ ∈ 𝐼 (Baseβ€˜(π‘…β€˜π‘₯)), 𝑔 ∈ Xπ‘₯ ∈ 𝐼 (Baseβ€˜(π‘…β€˜π‘₯)) ↦ sup((ran (π‘₯ ∈ 𝐼 ↦ ((π‘“β€˜π‘₯)(distβ€˜(π‘…β€˜π‘₯))(π‘”β€˜π‘₯))) βˆͺ {0}), ℝ*, < )))
12 eqidd 2727 . . 3 (πœ‘ β†’ (𝑓 ∈ Xπ‘₯ ∈ 𝐼 (Baseβ€˜(π‘…β€˜π‘₯)), 𝑔 ∈ Xπ‘₯ ∈ 𝐼 (Baseβ€˜(π‘…β€˜π‘₯)) ↦ Xπ‘₯ ∈ 𝐼 ((π‘“β€˜π‘₯)(Hom β€˜(π‘…β€˜π‘₯))(π‘”β€˜π‘₯))) = (𝑓 ∈ Xπ‘₯ ∈ 𝐼 (Baseβ€˜(π‘…β€˜π‘₯)), 𝑔 ∈ Xπ‘₯ ∈ 𝐼 (Baseβ€˜(π‘…β€˜π‘₯)) ↦ Xπ‘₯ ∈ 𝐼 ((π‘“β€˜π‘₯)(Hom β€˜(π‘…β€˜π‘₯))(π‘”β€˜π‘₯))))
13 eqidd 2727 . . 3 (πœ‘ β†’ (π‘Ž ∈ (Xπ‘₯ ∈ 𝐼 (Baseβ€˜(π‘…β€˜π‘₯)) Γ— Xπ‘₯ ∈ 𝐼 (Baseβ€˜(π‘…β€˜π‘₯))), 𝑐 ∈ Xπ‘₯ ∈ 𝐼 (Baseβ€˜(π‘…β€˜π‘₯)) ↦ (𝑑 ∈ ((2nd β€˜π‘Ž)(𝑓 ∈ Xπ‘₯ ∈ 𝐼 (Baseβ€˜(π‘…β€˜π‘₯)), 𝑔 ∈ Xπ‘₯ ∈ 𝐼 (Baseβ€˜(π‘…β€˜π‘₯)) ↦ Xπ‘₯ ∈ 𝐼 ((π‘“β€˜π‘₯)(Hom β€˜(π‘…β€˜π‘₯))(π‘”β€˜π‘₯)))𝑐), 𝑒 ∈ ((𝑓 ∈ Xπ‘₯ ∈ 𝐼 (Baseβ€˜(π‘…β€˜π‘₯)), 𝑔 ∈ Xπ‘₯ ∈ 𝐼 (Baseβ€˜(π‘…β€˜π‘₯)) ↦ Xπ‘₯ ∈ 𝐼 ((π‘“β€˜π‘₯)(Hom β€˜(π‘…β€˜π‘₯))(π‘”β€˜π‘₯)))β€˜π‘Ž) ↦ (π‘₯ ∈ 𝐼 ↦ ((π‘‘β€˜π‘₯)(⟨((1st β€˜π‘Ž)β€˜π‘₯), ((2nd β€˜π‘Ž)β€˜π‘₯)⟩(compβ€˜(π‘…β€˜π‘₯))(π‘β€˜π‘₯))(π‘’β€˜π‘₯))))) = (π‘Ž ∈ (Xπ‘₯ ∈ 𝐼 (Baseβ€˜(π‘…β€˜π‘₯)) Γ— Xπ‘₯ ∈ 𝐼 (Baseβ€˜(π‘…β€˜π‘₯))), 𝑐 ∈ Xπ‘₯ ∈ 𝐼 (Baseβ€˜(π‘…β€˜π‘₯)) ↦ (𝑑 ∈ ((2nd β€˜π‘Ž)(𝑓 ∈ Xπ‘₯ ∈ 𝐼 (Baseβ€˜(π‘…β€˜π‘₯)), 𝑔 ∈ Xπ‘₯ ∈ 𝐼 (Baseβ€˜(π‘…β€˜π‘₯)) ↦ Xπ‘₯ ∈ 𝐼 ((π‘“β€˜π‘₯)(Hom β€˜(π‘…β€˜π‘₯))(π‘”β€˜π‘₯)))𝑐), 𝑒 ∈ ((𝑓 ∈ Xπ‘₯ ∈ 𝐼 (Baseβ€˜(π‘…β€˜π‘₯)), 𝑔 ∈ Xπ‘₯ ∈ 𝐼 (Baseβ€˜(π‘…β€˜π‘₯)) ↦ Xπ‘₯ ∈ 𝐼 ((π‘“β€˜π‘₯)(Hom β€˜(π‘…β€˜π‘₯))(π‘”β€˜π‘₯)))β€˜π‘Ž) ↦ (π‘₯ ∈ 𝐼 ↦ ((π‘‘β€˜π‘₯)(⟨((1st β€˜π‘Ž)β€˜π‘₯), ((2nd β€˜π‘Ž)β€˜π‘₯)⟩(compβ€˜(π‘…β€˜π‘₯))(π‘β€˜π‘₯))(π‘’β€˜π‘₯))))))
14 prdsbas.s . . 3 (πœ‘ β†’ 𝑆 ∈ 𝑉)
15 prdsbas.r . . 3 (πœ‘ β†’ 𝑅 ∈ π‘Š)
161, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15prdsval 17407 . 2 (πœ‘ β†’ 𝑃 = (({⟨(Baseβ€˜ndx), Xπ‘₯ ∈ 𝐼 (Baseβ€˜(π‘…β€˜π‘₯))⟩, ⟨(+gβ€˜ndx), (𝑓 ∈ Xπ‘₯ ∈ 𝐼 (Baseβ€˜(π‘…β€˜π‘₯)), 𝑔 ∈ Xπ‘₯ ∈ 𝐼 (Baseβ€˜(π‘…β€˜π‘₯)) ↦ (π‘₯ ∈ 𝐼 ↦ ((π‘“β€˜π‘₯)(+gβ€˜(π‘…β€˜π‘₯))(π‘”β€˜π‘₯))))⟩, ⟨(.rβ€˜ndx), (𝑓 ∈ Xπ‘₯ ∈ 𝐼 (Baseβ€˜(π‘…β€˜π‘₯)), 𝑔 ∈ Xπ‘₯ ∈ 𝐼 (Baseβ€˜(π‘…β€˜π‘₯)) ↦ (π‘₯ ∈ 𝐼 ↦ ((π‘“β€˜π‘₯)(.rβ€˜(π‘…β€˜π‘₯))(π‘”β€˜π‘₯))))⟩} βˆͺ {⟨(Scalarβ€˜ndx), π‘†βŸ©, ⟨( ·𝑠 β€˜ndx), (𝑓 ∈ (Baseβ€˜π‘†), 𝑔 ∈ Xπ‘₯ ∈ 𝐼 (Baseβ€˜(π‘…β€˜π‘₯)) ↦ (π‘₯ ∈ 𝐼 ↦ (𝑓( ·𝑠 β€˜(π‘…β€˜π‘₯))(π‘”β€˜π‘₯))))⟩, ⟨(Β·π‘–β€˜ndx), (𝑓 ∈ Xπ‘₯ ∈ 𝐼 (Baseβ€˜(π‘…β€˜π‘₯)), 𝑔 ∈ Xπ‘₯ ∈ 𝐼 (Baseβ€˜(π‘…β€˜π‘₯)) ↦ (𝑆 Ξ£g (π‘₯ ∈ 𝐼 ↦ ((π‘“β€˜π‘₯)(Β·π‘–β€˜(π‘…β€˜π‘₯))(π‘”β€˜π‘₯)))))⟩}) βˆͺ ({⟨(TopSetβ€˜ndx), (∏tβ€˜(TopOpen ∘ 𝑅))⟩, ⟨(leβ€˜ndx), {βŸ¨π‘“, π‘”βŸ© ∣ ({𝑓, 𝑔} βŠ† Xπ‘₯ ∈ 𝐼 (Baseβ€˜(π‘…β€˜π‘₯)) ∧ βˆ€π‘₯ ∈ 𝐼 (π‘“β€˜π‘₯)(leβ€˜(π‘…β€˜π‘₯))(π‘”β€˜π‘₯))}⟩, ⟨(distβ€˜ndx), (𝑓 ∈ Xπ‘₯ ∈ 𝐼 (Baseβ€˜(π‘…β€˜π‘₯)), 𝑔 ∈ Xπ‘₯ ∈ 𝐼 (Baseβ€˜(π‘…β€˜π‘₯)) ↦ sup((ran (π‘₯ ∈ 𝐼 ↦ ((π‘“β€˜π‘₯)(distβ€˜(π‘…β€˜π‘₯))(π‘”β€˜π‘₯))) βˆͺ {0}), ℝ*, < ))⟩} βˆͺ {⟨(Hom β€˜ndx), (𝑓 ∈ Xπ‘₯ ∈ 𝐼 (Baseβ€˜(π‘…β€˜π‘₯)), 𝑔 ∈ Xπ‘₯ ∈ 𝐼 (Baseβ€˜(π‘…β€˜π‘₯)) ↦ Xπ‘₯ ∈ 𝐼 ((π‘“β€˜π‘₯)(Hom β€˜(π‘…β€˜π‘₯))(π‘”β€˜π‘₯)))⟩, ⟨(compβ€˜ndx), (π‘Ž ∈ (Xπ‘₯ ∈ 𝐼 (Baseβ€˜(π‘…β€˜π‘₯)) Γ— Xπ‘₯ ∈ 𝐼 (Baseβ€˜(π‘…β€˜π‘₯))), 𝑐 ∈ Xπ‘₯ ∈ 𝐼 (Baseβ€˜(π‘…β€˜π‘₯)) ↦ (𝑑 ∈ ((2nd β€˜π‘Ž)(𝑓 ∈ Xπ‘₯ ∈ 𝐼 (Baseβ€˜(π‘…β€˜π‘₯)), 𝑔 ∈ Xπ‘₯ ∈ 𝐼 (Baseβ€˜(π‘…β€˜π‘₯)) ↦ Xπ‘₯ ∈ 𝐼 ((π‘“β€˜π‘₯)(Hom β€˜(π‘…β€˜π‘₯))(π‘”β€˜π‘₯)))𝑐), 𝑒 ∈ ((𝑓 ∈ Xπ‘₯ ∈ 𝐼 (Baseβ€˜(π‘…β€˜π‘₯)), 𝑔 ∈ Xπ‘₯ ∈ 𝐼 (Baseβ€˜(π‘…β€˜π‘₯)) ↦ Xπ‘₯ ∈ 𝐼 ((π‘“β€˜π‘₯)(Hom β€˜(π‘…β€˜π‘₯))(π‘”β€˜π‘₯)))β€˜π‘Ž) ↦ (π‘₯ ∈ 𝐼 ↦ ((π‘‘β€˜π‘₯)(⟨((1st β€˜π‘Ž)β€˜π‘₯), ((2nd β€˜π‘Ž)β€˜π‘₯)⟩(compβ€˜(π‘…β€˜π‘₯))(π‘β€˜π‘₯))(π‘’β€˜π‘₯)))))⟩})))
17 prdsbas.b . 2 𝐡 = (Baseβ€˜π‘ƒ)
18 baseid 17153 . 2 Base = Slot (Baseβ€˜ndx)
1918strfvss 17126 . . . . . . 7 (Baseβ€˜(π‘…β€˜π‘₯)) βŠ† βˆͺ ran (π‘…β€˜π‘₯)
20 fvssunirn 6917 . . . . . . . 8 (π‘…β€˜π‘₯) βŠ† βˆͺ ran 𝑅
21 rnss 5931 . . . . . . . 8 ((π‘…β€˜π‘₯) βŠ† βˆͺ ran 𝑅 β†’ ran (π‘…β€˜π‘₯) βŠ† ran βˆͺ ran 𝑅)
22 uniss 4910 . . . . . . . 8 (ran (π‘…β€˜π‘₯) βŠ† ran βˆͺ ran 𝑅 β†’ βˆͺ ran (π‘…β€˜π‘₯) βŠ† βˆͺ ran βˆͺ ran 𝑅)
2320, 21, 22mp2b 10 . . . . . . 7 βˆͺ ran (π‘…β€˜π‘₯) βŠ† βˆͺ ran βˆͺ ran 𝑅
2419, 23sstri 3986 . . . . . 6 (Baseβ€˜(π‘…β€˜π‘₯)) βŠ† βˆͺ ran βˆͺ ran 𝑅
2524rgenw 3059 . . . . 5 βˆ€π‘₯ ∈ 𝐼 (Baseβ€˜(π‘…β€˜π‘₯)) βŠ† βˆͺ ran βˆͺ ran 𝑅
26 iunss 5041 . . . . 5 (βˆͺ π‘₯ ∈ 𝐼 (Baseβ€˜(π‘…β€˜π‘₯)) βŠ† βˆͺ ran βˆͺ ran 𝑅 ↔ βˆ€π‘₯ ∈ 𝐼 (Baseβ€˜(π‘…β€˜π‘₯)) βŠ† βˆͺ ran βˆͺ ran 𝑅)
2725, 26mpbir 230 . . . 4 βˆͺ π‘₯ ∈ 𝐼 (Baseβ€˜(π‘…β€˜π‘₯)) βŠ† βˆͺ ran βˆͺ ran 𝑅
28 rnexg 7891 . . . . . 6 (𝑅 ∈ π‘Š β†’ ran 𝑅 ∈ V)
29 uniexg 7726 . . . . . 6 (ran 𝑅 ∈ V β†’ βˆͺ ran 𝑅 ∈ V)
3015, 28, 293syl 18 . . . . 5 (πœ‘ β†’ βˆͺ ran 𝑅 ∈ V)
31 rnexg 7891 . . . . 5 (βˆͺ ran 𝑅 ∈ V β†’ ran βˆͺ ran 𝑅 ∈ V)
32 uniexg 7726 . . . . 5 (ran βˆͺ ran 𝑅 ∈ V β†’ βˆͺ ran βˆͺ ran 𝑅 ∈ V)
3330, 31, 323syl 18 . . . 4 (πœ‘ β†’ βˆͺ ran βˆͺ ran 𝑅 ∈ V)
34 ssexg 5316 . . . 4 ((βˆͺ π‘₯ ∈ 𝐼 (Baseβ€˜(π‘…β€˜π‘₯)) βŠ† βˆͺ ran βˆͺ ran 𝑅 ∧ βˆͺ ran βˆͺ ran 𝑅 ∈ V) β†’ βˆͺ π‘₯ ∈ 𝐼 (Baseβ€˜(π‘…β€˜π‘₯)) ∈ V)
3527, 33, 34sylancr 586 . . 3 (πœ‘ β†’ βˆͺ π‘₯ ∈ 𝐼 (Baseβ€˜(π‘…β€˜π‘₯)) ∈ V)
36 ixpssmap2g 8920 . . 3 (βˆͺ π‘₯ ∈ 𝐼 (Baseβ€˜(π‘…β€˜π‘₯)) ∈ V β†’ Xπ‘₯ ∈ 𝐼 (Baseβ€˜(π‘…β€˜π‘₯)) βŠ† (βˆͺ π‘₯ ∈ 𝐼 (Baseβ€˜(π‘…β€˜π‘₯)) ↑m 𝐼))
37 ovex 7437 . . . 4 (βˆͺ π‘₯ ∈ 𝐼 (Baseβ€˜(π‘…β€˜π‘₯)) ↑m 𝐼) ∈ V
3837ssex 5314 . . 3 (Xπ‘₯ ∈ 𝐼 (Baseβ€˜(π‘…β€˜π‘₯)) βŠ† (βˆͺ π‘₯ ∈ 𝐼 (Baseβ€˜(π‘…β€˜π‘₯)) ↑m 𝐼) β†’ Xπ‘₯ ∈ 𝐼 (Baseβ€˜(π‘…β€˜π‘₯)) ∈ V)
3935, 36, 383syl 18 . 2 (πœ‘ β†’ Xπ‘₯ ∈ 𝐼 (Baseβ€˜(π‘…β€˜π‘₯)) ∈ V)
40 snsstp1 4814 . . . 4 {⟨(Baseβ€˜ndx), Xπ‘₯ ∈ 𝐼 (Baseβ€˜(π‘…β€˜π‘₯))⟩} βŠ† {⟨(Baseβ€˜ndx), Xπ‘₯ ∈ 𝐼 (Baseβ€˜(π‘…β€˜π‘₯))⟩, ⟨(+gβ€˜ndx), (𝑓 ∈ Xπ‘₯ ∈ 𝐼 (Baseβ€˜(π‘…β€˜π‘₯)), 𝑔 ∈ Xπ‘₯ ∈ 𝐼 (Baseβ€˜(π‘…β€˜π‘₯)) ↦ (π‘₯ ∈ 𝐼 ↦ ((π‘“β€˜π‘₯)(+gβ€˜(π‘…β€˜π‘₯))(π‘”β€˜π‘₯))))⟩, ⟨(.rβ€˜ndx), (𝑓 ∈ Xπ‘₯ ∈ 𝐼 (Baseβ€˜(π‘…β€˜π‘₯)), 𝑔 ∈ Xπ‘₯ ∈ 𝐼 (Baseβ€˜(π‘…β€˜π‘₯)) ↦ (π‘₯ ∈ 𝐼 ↦ ((π‘“β€˜π‘₯)(.rβ€˜(π‘…β€˜π‘₯))(π‘”β€˜π‘₯))))⟩}
41 ssun1 4167 . . . 4 {⟨(Baseβ€˜ndx), Xπ‘₯ ∈ 𝐼 (Baseβ€˜(π‘…β€˜π‘₯))⟩, ⟨(+gβ€˜ndx), (𝑓 ∈ Xπ‘₯ ∈ 𝐼 (Baseβ€˜(π‘…β€˜π‘₯)), 𝑔 ∈ Xπ‘₯ ∈ 𝐼 (Baseβ€˜(π‘…β€˜π‘₯)) ↦ (π‘₯ ∈ 𝐼 ↦ ((π‘“β€˜π‘₯)(+gβ€˜(π‘…β€˜π‘₯))(π‘”β€˜π‘₯))))⟩, ⟨(.rβ€˜ndx), (𝑓 ∈ Xπ‘₯ ∈ 𝐼 (Baseβ€˜(π‘…β€˜π‘₯)), 𝑔 ∈ Xπ‘₯ ∈ 𝐼 (Baseβ€˜(π‘…β€˜π‘₯)) ↦ (π‘₯ ∈ 𝐼 ↦ ((π‘“β€˜π‘₯)(.rβ€˜(π‘…β€˜π‘₯))(π‘”β€˜π‘₯))))⟩} βŠ† ({⟨(Baseβ€˜ndx), Xπ‘₯ ∈ 𝐼 (Baseβ€˜(π‘…β€˜π‘₯))⟩, ⟨(+gβ€˜ndx), (𝑓 ∈ Xπ‘₯ ∈ 𝐼 (Baseβ€˜(π‘…β€˜π‘₯)), 𝑔 ∈ Xπ‘₯ ∈ 𝐼 (Baseβ€˜(π‘…β€˜π‘₯)) ↦ (π‘₯ ∈ 𝐼 ↦ ((π‘“β€˜π‘₯)(+gβ€˜(π‘…β€˜π‘₯))(π‘”β€˜π‘₯))))⟩, ⟨(.rβ€˜ndx), (𝑓 ∈ Xπ‘₯ ∈ 𝐼 (Baseβ€˜(π‘…β€˜π‘₯)), 𝑔 ∈ Xπ‘₯ ∈ 𝐼 (Baseβ€˜(π‘…β€˜π‘₯)) ↦ (π‘₯ ∈ 𝐼 ↦ ((π‘“β€˜π‘₯)(.rβ€˜(π‘…β€˜π‘₯))(π‘”β€˜π‘₯))))⟩} βˆͺ {⟨(Scalarβ€˜ndx), π‘†βŸ©, ⟨( ·𝑠 β€˜ndx), (𝑓 ∈ (Baseβ€˜π‘†), 𝑔 ∈ Xπ‘₯ ∈ 𝐼 (Baseβ€˜(π‘…β€˜π‘₯)) ↦ (π‘₯ ∈ 𝐼 ↦ (𝑓( ·𝑠 β€˜(π‘…β€˜π‘₯))(π‘”β€˜π‘₯))))⟩, ⟨(Β·π‘–β€˜ndx), (𝑓 ∈ Xπ‘₯ ∈ 𝐼 (Baseβ€˜(π‘…β€˜π‘₯)), 𝑔 ∈ Xπ‘₯ ∈ 𝐼 (Baseβ€˜(π‘…β€˜π‘₯)) ↦ (𝑆 Ξ£g (π‘₯ ∈ 𝐼 ↦ ((π‘“β€˜π‘₯)(Β·π‘–β€˜(π‘…β€˜π‘₯))(π‘”β€˜π‘₯)))))⟩})
4240, 41sstri 3986 . . 3 {⟨(Baseβ€˜ndx), Xπ‘₯ ∈ 𝐼 (Baseβ€˜(π‘…β€˜π‘₯))⟩} βŠ† ({⟨(Baseβ€˜ndx), Xπ‘₯ ∈ 𝐼 (Baseβ€˜(π‘…β€˜π‘₯))⟩, ⟨(+gβ€˜ndx), (𝑓 ∈ Xπ‘₯ ∈ 𝐼 (Baseβ€˜(π‘…β€˜π‘₯)), 𝑔 ∈ Xπ‘₯ ∈ 𝐼 (Baseβ€˜(π‘…β€˜π‘₯)) ↦ (π‘₯ ∈ 𝐼 ↦ ((π‘“β€˜π‘₯)(+gβ€˜(π‘…β€˜π‘₯))(π‘”β€˜π‘₯))))⟩, ⟨(.rβ€˜ndx), (𝑓 ∈ Xπ‘₯ ∈ 𝐼 (Baseβ€˜(π‘…β€˜π‘₯)), 𝑔 ∈ Xπ‘₯ ∈ 𝐼 (Baseβ€˜(π‘…β€˜π‘₯)) ↦ (π‘₯ ∈ 𝐼 ↦ ((π‘“β€˜π‘₯)(.rβ€˜(π‘…β€˜π‘₯))(π‘”β€˜π‘₯))))⟩} βˆͺ {⟨(Scalarβ€˜ndx), π‘†βŸ©, ⟨( ·𝑠 β€˜ndx), (𝑓 ∈ (Baseβ€˜π‘†), 𝑔 ∈ Xπ‘₯ ∈ 𝐼 (Baseβ€˜(π‘…β€˜π‘₯)) ↦ (π‘₯ ∈ 𝐼 ↦ (𝑓( ·𝑠 β€˜(π‘…β€˜π‘₯))(π‘”β€˜π‘₯))))⟩, ⟨(Β·π‘–β€˜ndx), (𝑓 ∈ Xπ‘₯ ∈ 𝐼 (Baseβ€˜(π‘…β€˜π‘₯)), 𝑔 ∈ Xπ‘₯ ∈ 𝐼 (Baseβ€˜(π‘…β€˜π‘₯)) ↦ (𝑆 Ξ£g (π‘₯ ∈ 𝐼 ↦ ((π‘“β€˜π‘₯)(Β·π‘–β€˜(π‘…β€˜π‘₯))(π‘”β€˜π‘₯)))))⟩})
43 ssun1 4167 . . 3 ({⟨(Baseβ€˜ndx), Xπ‘₯ ∈ 𝐼 (Baseβ€˜(π‘…β€˜π‘₯))⟩, ⟨(+gβ€˜ndx), (𝑓 ∈ Xπ‘₯ ∈ 𝐼 (Baseβ€˜(π‘…β€˜π‘₯)), 𝑔 ∈ Xπ‘₯ ∈ 𝐼 (Baseβ€˜(π‘…β€˜π‘₯)) ↦ (π‘₯ ∈ 𝐼 ↦ ((π‘“β€˜π‘₯)(+gβ€˜(π‘…β€˜π‘₯))(π‘”β€˜π‘₯))))⟩, ⟨(.rβ€˜ndx), (𝑓 ∈ Xπ‘₯ ∈ 𝐼 (Baseβ€˜(π‘…β€˜π‘₯)), 𝑔 ∈ Xπ‘₯ ∈ 𝐼 (Baseβ€˜(π‘…β€˜π‘₯)) ↦ (π‘₯ ∈ 𝐼 ↦ ((π‘“β€˜π‘₯)(.rβ€˜(π‘…β€˜π‘₯))(π‘”β€˜π‘₯))))⟩} βˆͺ {⟨(Scalarβ€˜ndx), π‘†βŸ©, ⟨( ·𝑠 β€˜ndx), (𝑓 ∈ (Baseβ€˜π‘†), 𝑔 ∈ Xπ‘₯ ∈ 𝐼 (Baseβ€˜(π‘…β€˜π‘₯)) ↦ (π‘₯ ∈ 𝐼 ↦ (𝑓( ·𝑠 β€˜(π‘…β€˜π‘₯))(π‘”β€˜π‘₯))))⟩, ⟨(Β·π‘–β€˜ndx), (𝑓 ∈ Xπ‘₯ ∈ 𝐼 (Baseβ€˜(π‘…β€˜π‘₯)), 𝑔 ∈ Xπ‘₯ ∈ 𝐼 (Baseβ€˜(π‘…β€˜π‘₯)) ↦ (𝑆 Ξ£g (π‘₯ ∈ 𝐼 ↦ ((π‘“β€˜π‘₯)(Β·π‘–β€˜(π‘…β€˜π‘₯))(π‘”β€˜π‘₯)))))⟩}) βŠ† (({⟨(Baseβ€˜ndx), Xπ‘₯ ∈ 𝐼 (Baseβ€˜(π‘…β€˜π‘₯))⟩, ⟨(+gβ€˜ndx), (𝑓 ∈ Xπ‘₯ ∈ 𝐼 (Baseβ€˜(π‘…β€˜π‘₯)), 𝑔 ∈ Xπ‘₯ ∈ 𝐼 (Baseβ€˜(π‘…β€˜π‘₯)) ↦ (π‘₯ ∈ 𝐼 ↦ ((π‘“β€˜π‘₯)(+gβ€˜(π‘…β€˜π‘₯))(π‘”β€˜π‘₯))))⟩, ⟨(.rβ€˜ndx), (𝑓 ∈ Xπ‘₯ ∈ 𝐼 (Baseβ€˜(π‘…β€˜π‘₯)), 𝑔 ∈ Xπ‘₯ ∈ 𝐼 (Baseβ€˜(π‘…β€˜π‘₯)) ↦ (π‘₯ ∈ 𝐼 ↦ ((π‘“β€˜π‘₯)(.rβ€˜(π‘…β€˜π‘₯))(π‘”β€˜π‘₯))))⟩} βˆͺ {⟨(Scalarβ€˜ndx), π‘†βŸ©, ⟨( ·𝑠 β€˜ndx), (𝑓 ∈ (Baseβ€˜π‘†), 𝑔 ∈ Xπ‘₯ ∈ 𝐼 (Baseβ€˜(π‘…β€˜π‘₯)) ↦ (π‘₯ ∈ 𝐼 ↦ (𝑓( ·𝑠 β€˜(π‘…β€˜π‘₯))(π‘”β€˜π‘₯))))⟩, ⟨(Β·π‘–β€˜ndx), (𝑓 ∈ Xπ‘₯ ∈ 𝐼 (Baseβ€˜(π‘…β€˜π‘₯)), 𝑔 ∈ Xπ‘₯ ∈ 𝐼 (Baseβ€˜(π‘…β€˜π‘₯)) ↦ (𝑆 Ξ£g (π‘₯ ∈ 𝐼 ↦ ((π‘“β€˜π‘₯)(Β·π‘–β€˜(π‘…β€˜π‘₯))(π‘”β€˜π‘₯)))))⟩}) βˆͺ ({⟨(TopSetβ€˜ndx), (∏tβ€˜(TopOpen ∘ 𝑅))⟩, ⟨(leβ€˜ndx), {βŸ¨π‘“, π‘”βŸ© ∣ ({𝑓, 𝑔} βŠ† Xπ‘₯ ∈ 𝐼 (Baseβ€˜(π‘…β€˜π‘₯)) ∧ βˆ€π‘₯ ∈ 𝐼 (π‘“β€˜π‘₯)(leβ€˜(π‘…β€˜π‘₯))(π‘”β€˜π‘₯))}⟩, ⟨(distβ€˜ndx), (𝑓 ∈ Xπ‘₯ ∈ 𝐼 (Baseβ€˜(π‘…β€˜π‘₯)), 𝑔 ∈ Xπ‘₯ ∈ 𝐼 (Baseβ€˜(π‘…β€˜π‘₯)) ↦ sup((ran (π‘₯ ∈ 𝐼 ↦ ((π‘“β€˜π‘₯)(distβ€˜(π‘…β€˜π‘₯))(π‘”β€˜π‘₯))) βˆͺ {0}), ℝ*, < ))⟩} βˆͺ {⟨(Hom β€˜ndx), (𝑓 ∈ Xπ‘₯ ∈ 𝐼 (Baseβ€˜(π‘…β€˜π‘₯)), 𝑔 ∈ Xπ‘₯ ∈ 𝐼 (Baseβ€˜(π‘…β€˜π‘₯)) ↦ Xπ‘₯ ∈ 𝐼 ((π‘“β€˜π‘₯)(Hom β€˜(π‘…β€˜π‘₯))(π‘”β€˜π‘₯)))⟩, ⟨(compβ€˜ndx), (π‘Ž ∈ (Xπ‘₯ ∈ 𝐼 (Baseβ€˜(π‘…β€˜π‘₯)) Γ— Xπ‘₯ ∈ 𝐼 (Baseβ€˜(π‘…β€˜π‘₯))), 𝑐 ∈ Xπ‘₯ ∈ 𝐼 (Baseβ€˜(π‘…β€˜π‘₯)) ↦ (𝑑 ∈ ((2nd β€˜π‘Ž)(𝑓 ∈ Xπ‘₯ ∈ 𝐼 (Baseβ€˜(π‘…β€˜π‘₯)), 𝑔 ∈ Xπ‘₯ ∈ 𝐼 (Baseβ€˜(π‘…β€˜π‘₯)) ↦ Xπ‘₯ ∈ 𝐼 ((π‘“β€˜π‘₯)(Hom β€˜(π‘…β€˜π‘₯))(π‘”β€˜π‘₯)))𝑐), 𝑒 ∈ ((𝑓 ∈ Xπ‘₯ ∈ 𝐼 (Baseβ€˜(π‘…β€˜π‘₯)), 𝑔 ∈ Xπ‘₯ ∈ 𝐼 (Baseβ€˜(π‘…β€˜π‘₯)) ↦ Xπ‘₯ ∈ 𝐼 ((π‘“β€˜π‘₯)(Hom β€˜(π‘…β€˜π‘₯))(π‘”β€˜π‘₯)))β€˜π‘Ž) ↦ (π‘₯ ∈ 𝐼 ↦ ((π‘‘β€˜π‘₯)(⟨((1st β€˜π‘Ž)β€˜π‘₯), ((2nd β€˜π‘Ž)β€˜π‘₯)⟩(compβ€˜(π‘…β€˜π‘₯))(π‘β€˜π‘₯))(π‘’β€˜π‘₯)))))⟩}))
4442, 43sstri 3986 . 2 {⟨(Baseβ€˜ndx), Xπ‘₯ ∈ 𝐼 (Baseβ€˜(π‘…β€˜π‘₯))⟩} βŠ† (({⟨(Baseβ€˜ndx), Xπ‘₯ ∈ 𝐼 (Baseβ€˜(π‘…β€˜π‘₯))⟩, ⟨(+gβ€˜ndx), (𝑓 ∈ Xπ‘₯ ∈ 𝐼 (Baseβ€˜(π‘…β€˜π‘₯)), 𝑔 ∈ Xπ‘₯ ∈ 𝐼 (Baseβ€˜(π‘…β€˜π‘₯)) ↦ (π‘₯ ∈ 𝐼 ↦ ((π‘“β€˜π‘₯)(+gβ€˜(π‘…β€˜π‘₯))(π‘”β€˜π‘₯))))⟩, ⟨(.rβ€˜ndx), (𝑓 ∈ Xπ‘₯ ∈ 𝐼 (Baseβ€˜(π‘…β€˜π‘₯)), 𝑔 ∈ Xπ‘₯ ∈ 𝐼 (Baseβ€˜(π‘…β€˜π‘₯)) ↦ (π‘₯ ∈ 𝐼 ↦ ((π‘“β€˜π‘₯)(.rβ€˜(π‘…β€˜π‘₯))(π‘”β€˜π‘₯))))⟩} βˆͺ {⟨(Scalarβ€˜ndx), π‘†βŸ©, ⟨( ·𝑠 β€˜ndx), (𝑓 ∈ (Baseβ€˜π‘†), 𝑔 ∈ Xπ‘₯ ∈ 𝐼 (Baseβ€˜(π‘…β€˜π‘₯)) ↦ (π‘₯ ∈ 𝐼 ↦ (𝑓( ·𝑠 β€˜(π‘…β€˜π‘₯))(π‘”β€˜π‘₯))))⟩, ⟨(Β·π‘–β€˜ndx), (𝑓 ∈ Xπ‘₯ ∈ 𝐼 (Baseβ€˜(π‘…β€˜π‘₯)), 𝑔 ∈ Xπ‘₯ ∈ 𝐼 (Baseβ€˜(π‘…β€˜π‘₯)) ↦ (𝑆 Ξ£g (π‘₯ ∈ 𝐼 ↦ ((π‘“β€˜π‘₯)(Β·π‘–β€˜(π‘…β€˜π‘₯))(π‘”β€˜π‘₯)))))⟩}) βˆͺ ({⟨(TopSetβ€˜ndx), (∏tβ€˜(TopOpen ∘ 𝑅))⟩, ⟨(leβ€˜ndx), {βŸ¨π‘“, π‘”βŸ© ∣ ({𝑓, 𝑔} βŠ† Xπ‘₯ ∈ 𝐼 (Baseβ€˜(π‘…β€˜π‘₯)) ∧ βˆ€π‘₯ ∈ 𝐼 (π‘“β€˜π‘₯)(leβ€˜(π‘…β€˜π‘₯))(π‘”β€˜π‘₯))}⟩, ⟨(distβ€˜ndx), (𝑓 ∈ Xπ‘₯ ∈ 𝐼 (Baseβ€˜(π‘…β€˜π‘₯)), 𝑔 ∈ Xπ‘₯ ∈ 𝐼 (Baseβ€˜(π‘…β€˜π‘₯)) ↦ sup((ran (π‘₯ ∈ 𝐼 ↦ ((π‘“β€˜π‘₯)(distβ€˜(π‘…β€˜π‘₯))(π‘”β€˜π‘₯))) βˆͺ {0}), ℝ*, < ))⟩} βˆͺ {⟨(Hom β€˜ndx), (𝑓 ∈ Xπ‘₯ ∈ 𝐼 (Baseβ€˜(π‘…β€˜π‘₯)), 𝑔 ∈ Xπ‘₯ ∈ 𝐼 (Baseβ€˜(π‘…β€˜π‘₯)) ↦ Xπ‘₯ ∈ 𝐼 ((π‘“β€˜π‘₯)(Hom β€˜(π‘…β€˜π‘₯))(π‘”β€˜π‘₯)))⟩, ⟨(compβ€˜ndx), (π‘Ž ∈ (Xπ‘₯ ∈ 𝐼 (Baseβ€˜(π‘…β€˜π‘₯)) Γ— Xπ‘₯ ∈ 𝐼 (Baseβ€˜(π‘…β€˜π‘₯))), 𝑐 ∈ Xπ‘₯ ∈ 𝐼 (Baseβ€˜(π‘…β€˜π‘₯)) ↦ (𝑑 ∈ ((2nd β€˜π‘Ž)(𝑓 ∈ Xπ‘₯ ∈ 𝐼 (Baseβ€˜(π‘…β€˜π‘₯)), 𝑔 ∈ Xπ‘₯ ∈ 𝐼 (Baseβ€˜(π‘…β€˜π‘₯)) ↦ Xπ‘₯ ∈ 𝐼 ((π‘“β€˜π‘₯)(Hom β€˜(π‘…β€˜π‘₯))(π‘”β€˜π‘₯)))𝑐), 𝑒 ∈ ((𝑓 ∈ Xπ‘₯ ∈ 𝐼 (Baseβ€˜(π‘…β€˜π‘₯)), 𝑔 ∈ Xπ‘₯ ∈ 𝐼 (Baseβ€˜(π‘…β€˜π‘₯)) ↦ Xπ‘₯ ∈ 𝐼 ((π‘“β€˜π‘₯)(Hom β€˜(π‘…β€˜π‘₯))(π‘”β€˜π‘₯)))β€˜π‘Ž) ↦ (π‘₯ ∈ 𝐼 ↦ ((π‘‘β€˜π‘₯)(⟨((1st β€˜π‘Ž)β€˜π‘₯), ((2nd β€˜π‘Ž)β€˜π‘₯)⟩(compβ€˜(π‘…β€˜π‘₯))(π‘β€˜π‘₯))(π‘’β€˜π‘₯)))))⟩}))
4516, 17, 18, 39, 44prdsbaslem 17405 1 (πœ‘ β†’ 𝐡 = Xπ‘₯ ∈ 𝐼 (Baseβ€˜(π‘…β€˜π‘₯)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 395   = wceq 1533   ∈ wcel 2098  βˆ€wral 3055  Vcvv 3468   βˆͺ cun 3941   βŠ† wss 3943  {csn 4623  {cpr 4625  {ctp 4627  βŸ¨cop 4629  βˆͺ cuni 4902  βˆͺ ciun 4990   class class class wbr 5141  {copab 5203   ↦ cmpt 5224   Γ— cxp 5667  dom cdm 5669  ran crn 5670   ∘ ccom 5673  β€˜cfv 6536  (class class class)co 7404   ∈ cmpo 7406  1st c1st 7969  2nd c2nd 7970   ↑m cmap 8819  Xcixp 8890  supcsup 9434  0cc0 11109  β„*cxr 11248   < clt 11249  ndxcnx 17132  Basecbs 17150  +gcplusg 17203  .rcmulr 17204  Scalarcsca 17206   ·𝑠 cvsca 17207  Β·π‘–cip 17208  TopSetcts 17209  lecple 17210  distcds 17212  Hom chom 17214  compcco 17215  TopOpenctopn 17373  βˆtcpt 17390   Ξ£g cgsu 17392  Xscprds 17397
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7721  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 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-nel 3041  df-ral 3056  df-rex 3065  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-pss 3962  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-tp 4628  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-pred 6293  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6488  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7360  df-ov 7407  df-oprab 7408  df-mpo 7409  df-om 7852  df-1st 7971  df-2nd 7972  df-frecs 8264  df-wrecs 8295  df-recs 8369  df-rdg 8408  df-1o 8464  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 11251  df-mnf 11252  df-xr 11253  df-ltxr 11254  df-le 11255  df-sub 11447  df-neg 11448  df-nn 12214  df-2 12276  df-3 12277  df-4 12278  df-5 12279  df-6 12280  df-7 12281  df-8 12282  df-9 12283  df-n0 12474  df-z 12560  df-dec 12679  df-uz 12824  df-fz 13488  df-struct 17086  df-slot 17121  df-ndx 17133  df-base 17151  df-plusg 17216  df-mulr 17217  df-sca 17219  df-vsca 17220  df-ip 17221  df-tset 17222  df-ple 17223  df-ds 17225  df-hom 17227  df-cco 17228  df-prds 17399
This theorem is referenced by:  prdsplusg  17410  prdsmulr  17411  prdsvsca  17412  prdsip  17413  prdsle  17414  prdsds  17416  prdstset  17418  prdshom  17419  prdsco  17420  prdsbas2  17421  pwsbas  17439  dsmmval  21624  frlmip  21668  prdstps  23483  rrxip  25268  prdstotbnd  37174
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