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Theorem prdsbas 17446
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 2728 . . 3 (Baseβ€˜π‘†) = (Baseβ€˜π‘†)
3 prdsbas.i . . 3 (πœ‘ β†’ dom 𝑅 = 𝐼)
4 eqidd 2729 . . 3 (πœ‘ β†’ Xπ‘₯ ∈ 𝐼 (Baseβ€˜(π‘…β€˜π‘₯)) = Xπ‘₯ ∈ 𝐼 (Baseβ€˜(π‘…β€˜π‘₯)))
5 eqidd 2729 . . 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 2729 . . 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 2729 . . 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 2729 . . 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 2729 . . 3 (πœ‘ β†’ (∏tβ€˜(TopOpen ∘ 𝑅)) = (∏tβ€˜(TopOpen ∘ 𝑅)))
10 eqidd 2729 . . 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 2729 . . 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 2729 . . 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 2729 . . 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 17444 . 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 17190 . 2 Base = Slot (Baseβ€˜ndx)
1918strfvss 17163 . . . . . . 7 (Baseβ€˜(π‘…β€˜π‘₯)) βŠ† βˆͺ ran (π‘…β€˜π‘₯)
20 fvssunirn 6935 . . . . . . . 8 (π‘…β€˜π‘₯) βŠ† βˆͺ ran 𝑅
21 rnss 5945 . . . . . . . 8 ((π‘…β€˜π‘₯) βŠ† βˆͺ ran 𝑅 β†’ ran (π‘…β€˜π‘₯) βŠ† ran βˆͺ ran 𝑅)
22 uniss 4920 . . . . . . . 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 3991 . . . . . 6 (Baseβ€˜(π‘…β€˜π‘₯)) βŠ† βˆͺ ran βˆͺ ran 𝑅
2524rgenw 3062 . . . . 5 βˆ€π‘₯ ∈ 𝐼 (Baseβ€˜(π‘…β€˜π‘₯)) βŠ† βˆͺ ran βˆͺ ran 𝑅
26 iunss 5052 . . . . 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 7916 . . . . . 6 (𝑅 ∈ π‘Š β†’ ran 𝑅 ∈ V)
29 uniexg 7751 . . . . . 6 (ran 𝑅 ∈ V β†’ βˆͺ ran 𝑅 ∈ V)
3015, 28, 293syl 18 . . . . 5 (πœ‘ β†’ βˆͺ ran 𝑅 ∈ V)
31 rnexg 7916 . . . . 5 (βˆͺ ran 𝑅 ∈ V β†’ ran βˆͺ ran 𝑅 ∈ V)
32 uniexg 7751 . . . . 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 5327 . . . 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 585 . . 3 (πœ‘ β†’ βˆͺ π‘₯ ∈ 𝐼 (Baseβ€˜(π‘…β€˜π‘₯)) ∈ V)
36 ixpssmap2g 8952 . . 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 7459 . . . 4 (βˆͺ π‘₯ ∈ 𝐼 (Baseβ€˜(π‘…β€˜π‘₯)) ↑m 𝐼) ∈ V
3837ssex 5325 . . 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 4824 . . . 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 4174 . . . 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 3991 . . 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 4174 . . 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 3991 . 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 17442 1 (πœ‘ β†’ 𝐡 = Xπ‘₯ ∈ 𝐼 (Baseβ€˜(π‘…β€˜π‘₯)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 394   = wceq 1533   ∈ wcel 2098  βˆ€wral 3058  Vcvv 3473   βˆͺ cun 3947   βŠ† wss 3949  {csn 4632  {cpr 4634  {ctp 4636  βŸ¨cop 4638  βˆͺ cuni 4912  βˆͺ ciun 5000   class class class wbr 5152  {copab 5214   ↦ cmpt 5235   Γ— cxp 5680  dom cdm 5682  ran crn 5683   ∘ ccom 5686  β€˜cfv 6553  (class class class)co 7426   ∈ cmpo 7428  1st c1st 7997  2nd c2nd 7998   ↑m cmap 8851  Xcixp 8922  supcsup 9471  0cc0 11146  β„*cxr 11285   < clt 11286  ndxcnx 17169  Basecbs 17187  +gcplusg 17240  .rcmulr 17241  Scalarcsca 17243   ·𝑠 cvsca 17244  Β·π‘–cip 17245  TopSetcts 17246  lecple 17247  distcds 17249  Hom chom 17251  compcco 17252  TopOpenctopn 17410  βˆtcpt 17427   Ξ£g cgsu 17429  Xscprds 17434
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 2166  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433  ax-un 7746  ax-cnex 11202  ax-resscn 11203  ax-1cn 11204  ax-icn 11205  ax-addcl 11206  ax-addrcl 11207  ax-mulcl 11208  ax-mulrcl 11209  ax-mulcom 11210  ax-addass 11211  ax-mulass 11212  ax-distr 11213  ax-i2m1 11214  ax-1ne0 11215  ax-1rid 11216  ax-rnegex 11217  ax-rrecex 11218  ax-cnre 11219  ax-pre-lttri 11220  ax-pre-lttrn 11221  ax-pre-ltadd 11222  ax-pre-mulgt0 11223
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-reu 3375  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-tp 4637  df-op 4639  df-uni 4913  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-tr 5270  df-id 5580  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-we 5639  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-pred 6310  df-ord 6377  df-on 6378  df-lim 6379  df-suc 6380  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-riota 7382  df-ov 7429  df-oprab 7430  df-mpo 7431  df-om 7877  df-1st 7999  df-2nd 8000  df-frecs 8293  df-wrecs 8324  df-recs 8398  df-rdg 8437  df-1o 8493  df-er 8731  df-map 8853  df-ixp 8923  df-en 8971  df-dom 8972  df-sdom 8973  df-fin 8974  df-sup 9473  df-pnf 11288  df-mnf 11289  df-xr 11290  df-ltxr 11291  df-le 11292  df-sub 11484  df-neg 11485  df-nn 12251  df-2 12313  df-3 12314  df-4 12315  df-5 12316  df-6 12317  df-7 12318  df-8 12319  df-9 12320  df-n0 12511  df-z 12597  df-dec 12716  df-uz 12861  df-fz 13525  df-struct 17123  df-slot 17158  df-ndx 17170  df-base 17188  df-plusg 17253  df-mulr 17254  df-sca 17256  df-vsca 17257  df-ip 17258  df-tset 17259  df-ple 17260  df-ds 17262  df-hom 17264  df-cco 17265  df-prds 17436
This theorem is referenced by:  prdsplusg  17447  prdsmulr  17448  prdsvsca  17449  prdsip  17450  prdsle  17451  prdsds  17453  prdstset  17455  prdshom  17456  prdsco  17457  prdsbas2  17458  pwsbas  17476  dsmmval  21675  frlmip  21719  prdstps  23553  rrxip  25338  prdstotbnd  37300
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