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Theorem prdsbas 17344
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 2733 . . 3 (Baseβ€˜π‘†) = (Baseβ€˜π‘†)
3 prdsbas.i . . 3 (πœ‘ β†’ dom 𝑅 = 𝐼)
4 eqidd 2734 . . 3 (πœ‘ β†’ Xπ‘₯ ∈ 𝐼 (Baseβ€˜(π‘…β€˜π‘₯)) = Xπ‘₯ ∈ 𝐼 (Baseβ€˜(π‘…β€˜π‘₯)))
5 eqidd 2734 . . 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 2734 . . 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 2734 . . 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 2734 . . 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 2734 . . 3 (πœ‘ β†’ (∏tβ€˜(TopOpen ∘ 𝑅)) = (∏tβ€˜(TopOpen ∘ 𝑅)))
10 eqidd 2734 . . 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 2734 . . 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 2734 . . 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 2734 . . 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 17342 . 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 17091 . 2 Base = Slot (Baseβ€˜ndx)
1918strfvss 17064 . . . . . . 7 (Baseβ€˜(π‘…β€˜π‘₯)) βŠ† βˆͺ ran (π‘…β€˜π‘₯)
20 fvssunirn 6876 . . . . . . . 8 (π‘…β€˜π‘₯) βŠ† βˆͺ ran 𝑅
21 rnss 5895 . . . . . . . 8 ((π‘…β€˜π‘₯) βŠ† βˆͺ ran 𝑅 β†’ ran (π‘…β€˜π‘₯) βŠ† ran βˆͺ ran 𝑅)
22 uniss 4874 . . . . . . . 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 3954 . . . . . 6 (Baseβ€˜(π‘…β€˜π‘₯)) βŠ† βˆͺ ran βˆͺ ran 𝑅
2524rgenw 3065 . . . . 5 βˆ€π‘₯ ∈ 𝐼 (Baseβ€˜(π‘…β€˜π‘₯)) βŠ† βˆͺ ran βˆͺ ran 𝑅
26 iunss 5006 . . . . 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 7842 . . . . . 6 (𝑅 ∈ π‘Š β†’ ran 𝑅 ∈ V)
29 uniexg 7678 . . . . . 6 (ran 𝑅 ∈ V β†’ βˆͺ ran 𝑅 ∈ V)
3015, 28, 293syl 18 . . . . 5 (πœ‘ β†’ βˆͺ ran 𝑅 ∈ V)
31 rnexg 7842 . . . . 5 (βˆͺ ran 𝑅 ∈ V β†’ ran βˆͺ ran 𝑅 ∈ V)
32 uniexg 7678 . . . . 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 5281 . . . 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 588 . . 3 (πœ‘ β†’ βˆͺ π‘₯ ∈ 𝐼 (Baseβ€˜(π‘…β€˜π‘₯)) ∈ V)
36 ixpssmap2g 8868 . . 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 7391 . . . 4 (βˆͺ π‘₯ ∈ 𝐼 (Baseβ€˜(π‘…β€˜π‘₯)) ↑m 𝐼) ∈ V
3837ssex 5279 . . 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 4777 . . . 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 4133 . . . 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 3954 . . 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 4133 . . 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 3954 . 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 17340 1 (πœ‘ β†’ 𝐡 = Xπ‘₯ ∈ 𝐼 (Baseβ€˜(π‘…β€˜π‘₯)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 397   = wceq 1542   ∈ wcel 2107  βˆ€wral 3061  Vcvv 3444   βˆͺ cun 3909   βŠ† wss 3911  {csn 4587  {cpr 4589  {ctp 4591  βŸ¨cop 4593  βˆͺ cuni 4866  βˆͺ ciun 4955   class class class wbr 5106  {copab 5168   ↦ cmpt 5189   Γ— cxp 5632  dom cdm 5634  ran crn 5635   ∘ ccom 5638  β€˜cfv 6497  (class class class)co 7358   ∈ cmpo 7360  1st c1st 7920  2nd c2nd 7921   ↑m cmap 8768  Xcixp 8838  supcsup 9381  0cc0 11056  β„*cxr 11193   < clt 11194  ndxcnx 17070  Basecbs 17088  +gcplusg 17138  .rcmulr 17139  Scalarcsca 17141   ·𝑠 cvsca 17142  Β·π‘–cip 17143  TopSetcts 17144  lecple 17145  distcds 17147  Hom chom 17149  compcco 17150  TopOpenctopn 17308  βˆtcpt 17325   Ξ£g cgsu 17327  Xscprds 17332
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673  ax-cnex 11112  ax-resscn 11113  ax-1cn 11114  ax-icn 11115  ax-addcl 11116  ax-addrcl 11117  ax-mulcl 11118  ax-mulrcl 11119  ax-mulcom 11120  ax-addass 11121  ax-mulass 11122  ax-distr 11123  ax-i2m1 11124  ax-1ne0 11125  ax-1rid 11126  ax-rnegex 11127  ax-rrecex 11128  ax-cnre 11129  ax-pre-lttri 11130  ax-pre-lttrn 11131  ax-pre-ltadd 11132  ax-pre-mulgt0 11133
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3353  df-rab 3407  df-v 3446  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3930  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-tp 4592  df-op 4594  df-uni 4867  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-tr 5224  df-id 5532  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5589  df-we 5591  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-pred 6254  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-riota 7314  df-ov 7361  df-oprab 7362  df-mpo 7363  df-om 7804  df-1st 7922  df-2nd 7923  df-frecs 8213  df-wrecs 8244  df-recs 8318  df-rdg 8357  df-1o 8413  df-er 8651  df-map 8770  df-ixp 8839  df-en 8887  df-dom 8888  df-sdom 8889  df-fin 8890  df-sup 9383  df-pnf 11196  df-mnf 11197  df-xr 11198  df-ltxr 11199  df-le 11200  df-sub 11392  df-neg 11393  df-nn 12159  df-2 12221  df-3 12222  df-4 12223  df-5 12224  df-6 12225  df-7 12226  df-8 12227  df-9 12228  df-n0 12419  df-z 12505  df-dec 12624  df-uz 12769  df-fz 13431  df-struct 17024  df-slot 17059  df-ndx 17071  df-base 17089  df-plusg 17151  df-mulr 17152  df-sca 17154  df-vsca 17155  df-ip 17156  df-tset 17157  df-ple 17158  df-ds 17160  df-hom 17162  df-cco 17163  df-prds 17334
This theorem is referenced by:  prdsplusg  17345  prdsmulr  17346  prdsvsca  17347  prdsip  17348  prdsle  17349  prdsds  17351  prdstset  17353  prdshom  17354  prdsco  17355  prdsbas2  17356  pwsbas  17374  dsmmval  21156  frlmip  21200  prdstps  22996  rrxip  24770  prdstotbnd  36299
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