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Theorem prdsvsca 17406
Description: Scalar multiplication in a structure product. (Contributed by Stefan O'Rear, 5-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)
prdsvsca.k 𝐾 = (Baseβ€˜π‘†)
prdsvsca.m Β· = ( ·𝑠 β€˜π‘ƒ)
Assertion
Ref Expression
prdsvsca (πœ‘ β†’ Β· = (𝑓 ∈ 𝐾, 𝑔 ∈ 𝐡 ↦ (π‘₯ ∈ 𝐼 ↦ (𝑓( ·𝑠 β€˜(π‘…β€˜π‘₯))(π‘”β€˜π‘₯)))))
Distinct variable groups:   𝑓,𝑔,π‘₯,𝐡   𝑓,𝐾,𝑔   πœ‘,𝑓,𝑔,π‘₯   𝑓,𝐼,𝑔,π‘₯   𝑃,𝑓,𝑔,π‘₯   𝑅,𝑓,𝑔,π‘₯   𝑆,𝑓,𝑔,π‘₯
Allowed substitution hints:   Β· (π‘₯,𝑓,𝑔)   𝐾(π‘₯)   𝑉(π‘₯,𝑓,𝑔)   π‘Š(π‘₯,𝑓,𝑔)

Proof of Theorem prdsvsca
Dummy variables π‘Ž 𝑐 𝑑 𝑒 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 prdsbas.p . . 3 𝑃 = (𝑆Xs𝑅)
2 prdsvsca.k . . 3 𝐾 = (Baseβ€˜π‘†)
3 prdsbas.i . . 3 (πœ‘ β†’ dom 𝑅 = 𝐼)
4 prdsbas.s . . . 4 (πœ‘ β†’ 𝑆 ∈ 𝑉)
5 prdsbas.r . . . 4 (πœ‘ β†’ 𝑅 ∈ π‘Š)
6 prdsbas.b . . . 4 𝐡 = (Baseβ€˜π‘ƒ)
71, 4, 5, 6, 3prdsbas 17403 . . 3 (πœ‘ β†’ 𝐡 = Xπ‘₯ ∈ 𝐼 (Baseβ€˜(π‘…β€˜π‘₯)))
8 eqid 2733 . . . 4 (+gβ€˜π‘ƒ) = (+gβ€˜π‘ƒ)
91, 4, 5, 6, 3, 8prdsplusg 17404 . . 3 (πœ‘ β†’ (+gβ€˜π‘ƒ) = (𝑓 ∈ 𝐡, 𝑔 ∈ 𝐡 ↦ (π‘₯ ∈ 𝐼 ↦ ((π‘“β€˜π‘₯)(+gβ€˜(π‘…β€˜π‘₯))(π‘”β€˜π‘₯)))))
10 eqid 2733 . . . 4 (.rβ€˜π‘ƒ) = (.rβ€˜π‘ƒ)
111, 4, 5, 6, 3, 10prdsmulr 17405 . . 3 (πœ‘ β†’ (.rβ€˜π‘ƒ) = (𝑓 ∈ 𝐡, 𝑔 ∈ 𝐡 ↦ (π‘₯ ∈ 𝐼 ↦ ((π‘“β€˜π‘₯)(.rβ€˜(π‘…β€˜π‘₯))(π‘”β€˜π‘₯)))))
12 eqidd 2734 . . 3 (πœ‘ β†’ (𝑓 ∈ 𝐾, 𝑔 ∈ 𝐡 ↦ (π‘₯ ∈ 𝐼 ↦ (𝑓( ·𝑠 β€˜(π‘…β€˜π‘₯))(π‘”β€˜π‘₯)))) = (𝑓 ∈ 𝐾, 𝑔 ∈ 𝐡 ↦ (π‘₯ ∈ 𝐼 ↦ (𝑓( ·𝑠 β€˜(π‘…β€˜π‘₯))(π‘”β€˜π‘₯)))))
13 eqidd 2734 . . 3 (πœ‘ β†’ (𝑓 ∈ 𝐡, 𝑔 ∈ 𝐡 ↦ (𝑆 Ξ£g (π‘₯ ∈ 𝐼 ↦ ((π‘“β€˜π‘₯)(Β·π‘–β€˜(π‘…β€˜π‘₯))(π‘”β€˜π‘₯))))) = (𝑓 ∈ 𝐡, 𝑔 ∈ 𝐡 ↦ (𝑆 Ξ£g (π‘₯ ∈ 𝐼 ↦ ((π‘“β€˜π‘₯)(Β·π‘–β€˜(π‘…β€˜π‘₯))(π‘”β€˜π‘₯))))))
14 eqidd 2734 . . 3 (πœ‘ β†’ (∏tβ€˜(TopOpen ∘ 𝑅)) = (∏tβ€˜(TopOpen ∘ 𝑅)))
15 eqidd 2734 . . 3 (πœ‘ β†’ {βŸ¨π‘“, π‘”βŸ© ∣ ({𝑓, 𝑔} βŠ† 𝐡 ∧ βˆ€π‘₯ ∈ 𝐼 (π‘“β€˜π‘₯)(leβ€˜(π‘…β€˜π‘₯))(π‘”β€˜π‘₯))} = {βŸ¨π‘“, π‘”βŸ© ∣ ({𝑓, 𝑔} βŠ† 𝐡 ∧ βˆ€π‘₯ ∈ 𝐼 (π‘“β€˜π‘₯)(leβ€˜(π‘…β€˜π‘₯))(π‘”β€˜π‘₯))})
16 eqidd 2734 . . 3 (πœ‘ β†’ (𝑓 ∈ 𝐡, 𝑔 ∈ 𝐡 ↦ sup((ran (π‘₯ ∈ 𝐼 ↦ ((π‘“β€˜π‘₯)(distβ€˜(π‘…β€˜π‘₯))(π‘”β€˜π‘₯))) βˆͺ {0}), ℝ*, < )) = (𝑓 ∈ 𝐡, 𝑔 ∈ 𝐡 ↦ sup((ran (π‘₯ ∈ 𝐼 ↦ ((π‘“β€˜π‘₯)(distβ€˜(π‘…β€˜π‘₯))(π‘”β€˜π‘₯))) βˆͺ {0}), ℝ*, < )))
17 eqidd 2734 . . 3 (πœ‘ β†’ (𝑓 ∈ 𝐡, 𝑔 ∈ 𝐡 ↦ Xπ‘₯ ∈ 𝐼 ((π‘“β€˜π‘₯)(Hom β€˜(π‘…β€˜π‘₯))(π‘”β€˜π‘₯))) = (𝑓 ∈ 𝐡, 𝑔 ∈ 𝐡 ↦ Xπ‘₯ ∈ 𝐼 ((π‘“β€˜π‘₯)(Hom β€˜(π‘…β€˜π‘₯))(π‘”β€˜π‘₯))))
18 eqidd 2734 . . 3 (πœ‘ β†’ (π‘Ž ∈ (𝐡 Γ— 𝐡), 𝑐 ∈ 𝐡 ↦ (𝑑 ∈ ((2nd β€˜π‘Ž)(𝑓 ∈ 𝐡, 𝑔 ∈ 𝐡 ↦ Xπ‘₯ ∈ 𝐼 ((π‘“β€˜π‘₯)(Hom β€˜(π‘…β€˜π‘₯))(π‘”β€˜π‘₯)))𝑐), 𝑒 ∈ ((𝑓 ∈ 𝐡, 𝑔 ∈ 𝐡 ↦ 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 ∈ (𝐡 Γ— 𝐡), 𝑐 ∈ 𝐡 ↦ (𝑑 ∈ ((2nd β€˜π‘Ž)(𝑓 ∈ 𝐡, 𝑔 ∈ 𝐡 ↦ Xπ‘₯ ∈ 𝐼 ((π‘“β€˜π‘₯)(Hom β€˜(π‘…β€˜π‘₯))(π‘”β€˜π‘₯)))𝑐), 𝑒 ∈ ((𝑓 ∈ 𝐡, 𝑔 ∈ 𝐡 ↦ 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‘₯))))))
191, 2, 3, 7, 9, 11, 12, 13, 14, 15, 16, 17, 18, 4, 5prdsval 17401 . 2 (πœ‘ β†’ 𝑃 = (({⟨(Baseβ€˜ndx), 𝐡⟩, ⟨(+gβ€˜ndx), (+gβ€˜π‘ƒ)⟩, ⟨(.rβ€˜ndx), (.rβ€˜π‘ƒ)⟩} βˆͺ {⟨(Scalarβ€˜ndx), π‘†βŸ©, ⟨( ·𝑠 β€˜ndx), (𝑓 ∈ 𝐾, 𝑔 ∈ 𝐡 ↦ (π‘₯ ∈ 𝐼 ↦ (𝑓( ·𝑠 β€˜(π‘…β€˜π‘₯))(π‘”β€˜π‘₯))))⟩, ⟨(Β·π‘–β€˜ndx), (𝑓 ∈ 𝐡, 𝑔 ∈ 𝐡 ↦ (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† 𝐡 ∧ βˆ€π‘₯ ∈ 𝐼 (π‘“β€˜π‘₯)(leβ€˜(π‘…β€˜π‘₯))(π‘”β€˜π‘₯))}⟩, ⟨(distβ€˜ndx), (𝑓 ∈ 𝐡, 𝑔 ∈ 𝐡 ↦ sup((ran (π‘₯ ∈ 𝐼 ↦ ((π‘“β€˜π‘₯)(distβ€˜(π‘…β€˜π‘₯))(π‘”β€˜π‘₯))) βˆͺ {0}), ℝ*, < ))⟩} βˆͺ {⟨(Hom β€˜ndx), (𝑓 ∈ 𝐡, 𝑔 ∈ 𝐡 ↦ Xπ‘₯ ∈ 𝐼 ((π‘“β€˜π‘₯)(Hom β€˜(π‘…β€˜π‘₯))(π‘”β€˜π‘₯)))⟩, ⟨(compβ€˜ndx), (π‘Ž ∈ (𝐡 Γ— 𝐡), 𝑐 ∈ 𝐡 ↦ (𝑑 ∈ ((2nd β€˜π‘Ž)(𝑓 ∈ 𝐡, 𝑔 ∈ 𝐡 ↦ Xπ‘₯ ∈ 𝐼 ((π‘“β€˜π‘₯)(Hom β€˜(π‘…β€˜π‘₯))(π‘”β€˜π‘₯)))𝑐), 𝑒 ∈ ((𝑓 ∈ 𝐡, 𝑔 ∈ 𝐡 ↦ 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‘₯)))))⟩})))
20 prdsvsca.m . 2 Β· = ( ·𝑠 β€˜π‘ƒ)
21 vscaid 17265 . 2 ·𝑠 = Slot ( ·𝑠 β€˜ndx)
22 ovssunirn 7445 . . . . . . . . . . 11 (𝑓( ·𝑠 β€˜(π‘…β€˜π‘₯))(π‘”β€˜π‘₯)) βŠ† βˆͺ ran ( ·𝑠 β€˜(π‘…β€˜π‘₯))
2321strfvss 17120 . . . . . . . . . . . . 13 ( ·𝑠 β€˜(π‘…β€˜π‘₯)) βŠ† βˆͺ ran (π‘…β€˜π‘₯)
24 fvssunirn 6925 . . . . . . . . . . . . . 14 (π‘…β€˜π‘₯) βŠ† βˆͺ ran 𝑅
25 rnss 5939 . . . . . . . . . . . . . 14 ((π‘…β€˜π‘₯) βŠ† βˆͺ ran 𝑅 β†’ ran (π‘…β€˜π‘₯) βŠ† ran βˆͺ ran 𝑅)
26 uniss 4917 . . . . . . . . . . . . . 14 (ran (π‘…β€˜π‘₯) βŠ† ran βˆͺ ran 𝑅 β†’ βˆͺ ran (π‘…β€˜π‘₯) βŠ† βˆͺ ran βˆͺ ran 𝑅)
2724, 25, 26mp2b 10 . . . . . . . . . . . . 13 βˆͺ ran (π‘…β€˜π‘₯) βŠ† βˆͺ ran βˆͺ ran 𝑅
2823, 27sstri 3992 . . . . . . . . . . . 12 ( ·𝑠 β€˜(π‘…β€˜π‘₯)) βŠ† βˆͺ ran βˆͺ ran 𝑅
29 rnss 5939 . . . . . . . . . . . 12 (( ·𝑠 β€˜(π‘…β€˜π‘₯)) βŠ† βˆͺ ran βˆͺ ran 𝑅 β†’ ran ( ·𝑠 β€˜(π‘…β€˜π‘₯)) βŠ† ran βˆͺ ran βˆͺ ran 𝑅)
30 uniss 4917 . . . . . . . . . . . 12 (ran ( ·𝑠 β€˜(π‘…β€˜π‘₯)) βŠ† ran βˆͺ ran βˆͺ ran 𝑅 β†’ βˆͺ ran ( ·𝑠 β€˜(π‘…β€˜π‘₯)) βŠ† βˆͺ ran βˆͺ ran βˆͺ ran 𝑅)
3128, 29, 30mp2b 10 . . . . . . . . . . 11 βˆͺ ran ( ·𝑠 β€˜(π‘…β€˜π‘₯)) βŠ† βˆͺ ran βˆͺ ran βˆͺ ran 𝑅
3222, 31sstri 3992 . . . . . . . . . 10 (𝑓( ·𝑠 β€˜(π‘…β€˜π‘₯))(π‘”β€˜π‘₯)) βŠ† βˆͺ ran βˆͺ ran βˆͺ ran 𝑅
33 ovex 7442 . . . . . . . . . . 11 (𝑓( ·𝑠 β€˜(π‘…β€˜π‘₯))(π‘”β€˜π‘₯)) ∈ V
3433elpw 4607 . . . . . . . . . 10 ((𝑓( ·𝑠 β€˜(π‘…β€˜π‘₯))(π‘”β€˜π‘₯)) ∈ 𝒫 βˆͺ ran βˆͺ ran βˆͺ ran 𝑅 ↔ (𝑓( ·𝑠 β€˜(π‘…β€˜π‘₯))(π‘”β€˜π‘₯)) βŠ† βˆͺ ran βˆͺ ran βˆͺ ran 𝑅)
3532, 34mpbir 230 . . . . . . . . 9 (𝑓( ·𝑠 β€˜(π‘…β€˜π‘₯))(π‘”β€˜π‘₯)) ∈ 𝒫 βˆͺ ran βˆͺ ran βˆͺ ran 𝑅
3635a1i 11 . . . . . . . 8 ((πœ‘ ∧ π‘₯ ∈ 𝐼) β†’ (𝑓( ·𝑠 β€˜(π‘…β€˜π‘₯))(π‘”β€˜π‘₯)) ∈ 𝒫 βˆͺ ran βˆͺ ran βˆͺ ran 𝑅)
3736fmpttd 7115 . . . . . . 7 (πœ‘ β†’ (π‘₯ ∈ 𝐼 ↦ (𝑓( ·𝑠 β€˜(π‘…β€˜π‘₯))(π‘”β€˜π‘₯))):πΌβŸΆπ’« βˆͺ ran βˆͺ ran βˆͺ ran 𝑅)
38 rnexg 7895 . . . . . . . . . . 11 (𝑅 ∈ π‘Š β†’ ran 𝑅 ∈ V)
39 uniexg 7730 . . . . . . . . . . 11 (ran 𝑅 ∈ V β†’ βˆͺ ran 𝑅 ∈ V)
405, 38, 393syl 18 . . . . . . . . . 10 (πœ‘ β†’ βˆͺ ran 𝑅 ∈ V)
41 rnexg 7895 . . . . . . . . . 10 (βˆͺ ran 𝑅 ∈ V β†’ ran βˆͺ ran 𝑅 ∈ V)
42 uniexg 7730 . . . . . . . . . 10 (ran βˆͺ ran 𝑅 ∈ V β†’ βˆͺ ran βˆͺ ran 𝑅 ∈ V)
4340, 41, 423syl 18 . . . . . . . . 9 (πœ‘ β†’ βˆͺ ran βˆͺ ran 𝑅 ∈ V)
44 rnexg 7895 . . . . . . . . 9 (βˆͺ ran βˆͺ ran 𝑅 ∈ V β†’ ran βˆͺ ran βˆͺ ran 𝑅 ∈ V)
45 uniexg 7730 . . . . . . . . 9 (ran βˆͺ ran βˆͺ ran 𝑅 ∈ V β†’ βˆͺ ran βˆͺ ran βˆͺ ran 𝑅 ∈ V)
46 pwexg 5377 . . . . . . . . 9 (βˆͺ ran βˆͺ ran βˆͺ ran 𝑅 ∈ V β†’ 𝒫 βˆͺ ran βˆͺ ran βˆͺ ran 𝑅 ∈ V)
4743, 44, 45, 464syl 19 . . . . . . . 8 (πœ‘ β†’ 𝒫 βˆͺ ran βˆͺ ran βˆͺ ran 𝑅 ∈ V)
485dmexd 7896 . . . . . . . . 9 (πœ‘ β†’ dom 𝑅 ∈ V)
493, 48eqeltrrd 2835 . . . . . . . 8 (πœ‘ β†’ 𝐼 ∈ V)
5047, 49elmapd 8834 . . . . . . 7 (πœ‘ β†’ ((π‘₯ ∈ 𝐼 ↦ (𝑓( ·𝑠 β€˜(π‘…β€˜π‘₯))(π‘”β€˜π‘₯))) ∈ (𝒫 βˆͺ ran βˆͺ ran βˆͺ ran 𝑅 ↑m 𝐼) ↔ (π‘₯ ∈ 𝐼 ↦ (𝑓( ·𝑠 β€˜(π‘…β€˜π‘₯))(π‘”β€˜π‘₯))):πΌβŸΆπ’« βˆͺ ran βˆͺ ran βˆͺ ran 𝑅))
5137, 50mpbird 257 . . . . . 6 (πœ‘ β†’ (π‘₯ ∈ 𝐼 ↦ (𝑓( ·𝑠 β€˜(π‘…β€˜π‘₯))(π‘”β€˜π‘₯))) ∈ (𝒫 βˆͺ ran βˆͺ ran βˆͺ ran 𝑅 ↑m 𝐼))
5251ralrimivw 3151 . . . . 5 (πœ‘ β†’ βˆ€π‘” ∈ 𝐡 (π‘₯ ∈ 𝐼 ↦ (𝑓( ·𝑠 β€˜(π‘…β€˜π‘₯))(π‘”β€˜π‘₯))) ∈ (𝒫 βˆͺ ran βˆͺ ran βˆͺ ran 𝑅 ↑m 𝐼))
5352ralrimivw 3151 . . . 4 (πœ‘ β†’ βˆ€π‘“ ∈ 𝐾 βˆ€π‘” ∈ 𝐡 (π‘₯ ∈ 𝐼 ↦ (𝑓( ·𝑠 β€˜(π‘…β€˜π‘₯))(π‘”β€˜π‘₯))) ∈ (𝒫 βˆͺ ran βˆͺ ran βˆͺ ran 𝑅 ↑m 𝐼))
54 eqid 2733 . . . . 5 (𝑓 ∈ 𝐾, 𝑔 ∈ 𝐡 ↦ (π‘₯ ∈ 𝐼 ↦ (𝑓( ·𝑠 β€˜(π‘…β€˜π‘₯))(π‘”β€˜π‘₯)))) = (𝑓 ∈ 𝐾, 𝑔 ∈ 𝐡 ↦ (π‘₯ ∈ 𝐼 ↦ (𝑓( ·𝑠 β€˜(π‘…β€˜π‘₯))(π‘”β€˜π‘₯))))
5554fmpo 8054 . . . 4 (βˆ€π‘“ ∈ 𝐾 βˆ€π‘” ∈ 𝐡 (π‘₯ ∈ 𝐼 ↦ (𝑓( ·𝑠 β€˜(π‘…β€˜π‘₯))(π‘”β€˜π‘₯))) ∈ (𝒫 βˆͺ ran βˆͺ ran βˆͺ ran 𝑅 ↑m 𝐼) ↔ (𝑓 ∈ 𝐾, 𝑔 ∈ 𝐡 ↦ (π‘₯ ∈ 𝐼 ↦ (𝑓( ·𝑠 β€˜(π‘…β€˜π‘₯))(π‘”β€˜π‘₯)))):(𝐾 Γ— 𝐡)⟢(𝒫 βˆͺ ran βˆͺ ran βˆͺ ran 𝑅 ↑m 𝐼))
5653, 55sylib 217 . . 3 (πœ‘ β†’ (𝑓 ∈ 𝐾, 𝑔 ∈ 𝐡 ↦ (π‘₯ ∈ 𝐼 ↦ (𝑓( ·𝑠 β€˜(π‘…β€˜π‘₯))(π‘”β€˜π‘₯)))):(𝐾 Γ— 𝐡)⟢(𝒫 βˆͺ ran βˆͺ ran βˆͺ ran 𝑅 ↑m 𝐼))
572fvexi 6906 . . . . 5 𝐾 ∈ V
586fvexi 6906 . . . . 5 𝐡 ∈ V
5957, 58xpex 7740 . . . 4 (𝐾 Γ— 𝐡) ∈ V
60 ovex 7442 . . . 4 (𝒫 βˆͺ ran βˆͺ ran βˆͺ ran 𝑅 ↑m 𝐼) ∈ V
61 fex2 7924 . . . 4 (((𝑓 ∈ 𝐾, 𝑔 ∈ 𝐡 ↦ (π‘₯ ∈ 𝐼 ↦ (𝑓( ·𝑠 β€˜(π‘…β€˜π‘₯))(π‘”β€˜π‘₯)))):(𝐾 Γ— 𝐡)⟢(𝒫 βˆͺ ran βˆͺ ran βˆͺ ran 𝑅 ↑m 𝐼) ∧ (𝐾 Γ— 𝐡) ∈ V ∧ (𝒫 βˆͺ ran βˆͺ ran βˆͺ ran 𝑅 ↑m 𝐼) ∈ V) β†’ (𝑓 ∈ 𝐾, 𝑔 ∈ 𝐡 ↦ (π‘₯ ∈ 𝐼 ↦ (𝑓( ·𝑠 β€˜(π‘…β€˜π‘₯))(π‘”β€˜π‘₯)))) ∈ V)
6259, 60, 61mp3an23 1454 . . 3 ((𝑓 ∈ 𝐾, 𝑔 ∈ 𝐡 ↦ (π‘₯ ∈ 𝐼 ↦ (𝑓( ·𝑠 β€˜(π‘…β€˜π‘₯))(π‘”β€˜π‘₯)))):(𝐾 Γ— 𝐡)⟢(𝒫 βˆͺ ran βˆͺ ran βˆͺ ran 𝑅 ↑m 𝐼) β†’ (𝑓 ∈ 𝐾, 𝑔 ∈ 𝐡 ↦ (π‘₯ ∈ 𝐼 ↦ (𝑓( ·𝑠 β€˜(π‘…β€˜π‘₯))(π‘”β€˜π‘₯)))) ∈ V)
6356, 62syl 17 . 2 (πœ‘ β†’ (𝑓 ∈ 𝐾, 𝑔 ∈ 𝐡 ↦ (π‘₯ ∈ 𝐼 ↦ (𝑓( ·𝑠 β€˜(π‘…β€˜π‘₯))(π‘”β€˜π‘₯)))) ∈ V)
64 snsstp2 4821 . . . 4 {⟨( ·𝑠 β€˜ndx), (𝑓 ∈ 𝐾, 𝑔 ∈ 𝐡 ↦ (π‘₯ ∈ 𝐼 ↦ (𝑓( ·𝑠 β€˜(π‘…β€˜π‘₯))(π‘”β€˜π‘₯))))⟩} βŠ† {⟨(Scalarβ€˜ndx), π‘†βŸ©, ⟨( ·𝑠 β€˜ndx), (𝑓 ∈ 𝐾, 𝑔 ∈ 𝐡 ↦ (π‘₯ ∈ 𝐼 ↦ (𝑓( ·𝑠 β€˜(π‘…β€˜π‘₯))(π‘”β€˜π‘₯))))⟩, ⟨(Β·π‘–β€˜ndx), (𝑓 ∈ 𝐡, 𝑔 ∈ 𝐡 ↦ (𝑆 Ξ£g (π‘₯ ∈ 𝐼 ↦ ((π‘“β€˜π‘₯)(Β·π‘–β€˜(π‘…β€˜π‘₯))(π‘”β€˜π‘₯)))))⟩}
65 ssun2 4174 . . . 4 {⟨(Scalarβ€˜ndx), π‘†βŸ©, ⟨( ·𝑠 β€˜ndx), (𝑓 ∈ 𝐾, 𝑔 ∈ 𝐡 ↦ (π‘₯ ∈ 𝐼 ↦ (𝑓( ·𝑠 β€˜(π‘…β€˜π‘₯))(π‘”β€˜π‘₯))))⟩, ⟨(Β·π‘–β€˜ndx), (𝑓 ∈ 𝐡, 𝑔 ∈ 𝐡 ↦ (𝑆 Ξ£g (π‘₯ ∈ 𝐼 ↦ ((π‘“β€˜π‘₯)(Β·π‘–β€˜(π‘…β€˜π‘₯))(π‘”β€˜π‘₯)))))⟩} βŠ† ({⟨(Baseβ€˜ndx), 𝐡⟩, ⟨(+gβ€˜ndx), (+gβ€˜π‘ƒ)⟩, ⟨(.rβ€˜ndx), (.rβ€˜π‘ƒ)⟩} βˆͺ {⟨(Scalarβ€˜ndx), π‘†βŸ©, ⟨( ·𝑠 β€˜ndx), (𝑓 ∈ 𝐾, 𝑔 ∈ 𝐡 ↦ (π‘₯ ∈ 𝐼 ↦ (𝑓( ·𝑠 β€˜(π‘…β€˜π‘₯))(π‘”β€˜π‘₯))))⟩, ⟨(Β·π‘–β€˜ndx), (𝑓 ∈ 𝐡, 𝑔 ∈ 𝐡 ↦ (𝑆 Ξ£g (π‘₯ ∈ 𝐼 ↦ ((π‘“β€˜π‘₯)(Β·π‘–β€˜(π‘…β€˜π‘₯))(π‘”β€˜π‘₯)))))⟩})
6664, 65sstri 3992 . . 3 {⟨( ·𝑠 β€˜ndx), (𝑓 ∈ 𝐾, 𝑔 ∈ 𝐡 ↦ (π‘₯ ∈ 𝐼 ↦ (𝑓( ·𝑠 β€˜(π‘…β€˜π‘₯))(π‘”β€˜π‘₯))))⟩} βŠ† ({⟨(Baseβ€˜ndx), 𝐡⟩, ⟨(+gβ€˜ndx), (+gβ€˜π‘ƒ)⟩, ⟨(.rβ€˜ndx), (.rβ€˜π‘ƒ)⟩} βˆͺ {⟨(Scalarβ€˜ndx), π‘†βŸ©, ⟨( ·𝑠 β€˜ndx), (𝑓 ∈ 𝐾, 𝑔 ∈ 𝐡 ↦ (π‘₯ ∈ 𝐼 ↦ (𝑓( ·𝑠 β€˜(π‘…β€˜π‘₯))(π‘”β€˜π‘₯))))⟩, ⟨(Β·π‘–β€˜ndx), (𝑓 ∈ 𝐡, 𝑔 ∈ 𝐡 ↦ (𝑆 Ξ£g (π‘₯ ∈ 𝐼 ↦ ((π‘“β€˜π‘₯)(Β·π‘–β€˜(π‘…β€˜π‘₯))(π‘”β€˜π‘₯)))))⟩})
67 ssun1 4173 . . 3 ({⟨(Baseβ€˜ndx), 𝐡⟩, ⟨(+gβ€˜ndx), (+gβ€˜π‘ƒ)⟩, ⟨(.rβ€˜ndx), (.rβ€˜π‘ƒ)⟩} βˆͺ {⟨(Scalarβ€˜ndx), π‘†βŸ©, ⟨( ·𝑠 β€˜ndx), (𝑓 ∈ 𝐾, 𝑔 ∈ 𝐡 ↦ (π‘₯ ∈ 𝐼 ↦ (𝑓( ·𝑠 β€˜(π‘…β€˜π‘₯))(π‘”β€˜π‘₯))))⟩, ⟨(Β·π‘–β€˜ndx), (𝑓 ∈ 𝐡, 𝑔 ∈ 𝐡 ↦ (𝑆 Ξ£g (π‘₯ ∈ 𝐼 ↦ ((π‘“β€˜π‘₯)(Β·π‘–β€˜(π‘…β€˜π‘₯))(π‘”β€˜π‘₯)))))⟩}) βŠ† (({⟨(Baseβ€˜ndx), 𝐡⟩, ⟨(+gβ€˜ndx), (+gβ€˜π‘ƒ)⟩, ⟨(.rβ€˜ndx), (.rβ€˜π‘ƒ)⟩} βˆͺ {⟨(Scalarβ€˜ndx), π‘†βŸ©, ⟨( ·𝑠 β€˜ndx), (𝑓 ∈ 𝐾, 𝑔 ∈ 𝐡 ↦ (π‘₯ ∈ 𝐼 ↦ (𝑓( ·𝑠 β€˜(π‘…β€˜π‘₯))(π‘”β€˜π‘₯))))⟩, ⟨(Β·π‘–β€˜ndx), (𝑓 ∈ 𝐡, 𝑔 ∈ 𝐡 ↦ (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† 𝐡 ∧ βˆ€π‘₯ ∈ 𝐼 (π‘“β€˜π‘₯)(leβ€˜(π‘…β€˜π‘₯))(π‘”β€˜π‘₯))}⟩, ⟨(distβ€˜ndx), (𝑓 ∈ 𝐡, 𝑔 ∈ 𝐡 ↦ sup((ran (π‘₯ ∈ 𝐼 ↦ ((π‘“β€˜π‘₯)(distβ€˜(π‘…β€˜π‘₯))(π‘”β€˜π‘₯))) βˆͺ {0}), ℝ*, < ))⟩} βˆͺ {⟨(Hom β€˜ndx), (𝑓 ∈ 𝐡, 𝑔 ∈ 𝐡 ↦ Xπ‘₯ ∈ 𝐼 ((π‘“β€˜π‘₯)(Hom β€˜(π‘…β€˜π‘₯))(π‘”β€˜π‘₯)))⟩, ⟨(compβ€˜ndx), (π‘Ž ∈ (𝐡 Γ— 𝐡), 𝑐 ∈ 𝐡 ↦ (𝑑 ∈ ((2nd β€˜π‘Ž)(𝑓 ∈ 𝐡, 𝑔 ∈ 𝐡 ↦ Xπ‘₯ ∈ 𝐼 ((π‘“β€˜π‘₯)(Hom β€˜(π‘…β€˜π‘₯))(π‘”β€˜π‘₯)))𝑐), 𝑒 ∈ ((𝑓 ∈ 𝐡, 𝑔 ∈ 𝐡 ↦ 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‘₯)))))⟩}))
6866, 67sstri 3992 . 2 {⟨( ·𝑠 β€˜ndx), (𝑓 ∈ 𝐾, 𝑔 ∈ 𝐡 ↦ (π‘₯ ∈ 𝐼 ↦ (𝑓( ·𝑠 β€˜(π‘…β€˜π‘₯))(π‘”β€˜π‘₯))))⟩} βŠ† (({⟨(Baseβ€˜ndx), 𝐡⟩, ⟨(+gβ€˜ndx), (+gβ€˜π‘ƒ)⟩, ⟨(.rβ€˜ndx), (.rβ€˜π‘ƒ)⟩} βˆͺ {⟨(Scalarβ€˜ndx), π‘†βŸ©, ⟨( ·𝑠 β€˜ndx), (𝑓 ∈ 𝐾, 𝑔 ∈ 𝐡 ↦ (π‘₯ ∈ 𝐼 ↦ (𝑓( ·𝑠 β€˜(π‘…β€˜π‘₯))(π‘”β€˜π‘₯))))⟩, ⟨(Β·π‘–β€˜ndx), (𝑓 ∈ 𝐡, 𝑔 ∈ 𝐡 ↦ (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† 𝐡 ∧ βˆ€π‘₯ ∈ 𝐼 (π‘“β€˜π‘₯)(leβ€˜(π‘…β€˜π‘₯))(π‘”β€˜π‘₯))}⟩, ⟨(distβ€˜ndx), (𝑓 ∈ 𝐡, 𝑔 ∈ 𝐡 ↦ sup((ran (π‘₯ ∈ 𝐼 ↦ ((π‘“β€˜π‘₯)(distβ€˜(π‘…β€˜π‘₯))(π‘”β€˜π‘₯))) βˆͺ {0}), ℝ*, < ))⟩} βˆͺ {⟨(Hom β€˜ndx), (𝑓 ∈ 𝐡, 𝑔 ∈ 𝐡 ↦ Xπ‘₯ ∈ 𝐼 ((π‘“β€˜π‘₯)(Hom β€˜(π‘…β€˜π‘₯))(π‘”β€˜π‘₯)))⟩, ⟨(compβ€˜ndx), (π‘Ž ∈ (𝐡 Γ— 𝐡), 𝑐 ∈ 𝐡 ↦ (𝑑 ∈ ((2nd β€˜π‘Ž)(𝑓 ∈ 𝐡, 𝑔 ∈ 𝐡 ↦ Xπ‘₯ ∈ 𝐼 ((π‘“β€˜π‘₯)(Hom β€˜(π‘…β€˜π‘₯))(π‘”β€˜π‘₯)))𝑐), 𝑒 ∈ ((𝑓 ∈ 𝐡, 𝑔 ∈ 𝐡 ↦ 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‘₯)))))⟩}))
6919, 20, 21, 63, 68prdsbaslem 17399 1 (πœ‘ β†’ Β· = (𝑓 ∈ 𝐾, 𝑔 ∈ 𝐡 ↦ (π‘₯ ∈ 𝐼 ↦ (𝑓( ·𝑠 β€˜(π‘…β€˜π‘₯))(π‘”β€˜π‘₯)))))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 397   = wceq 1542   ∈ wcel 2107  βˆ€wral 3062  Vcvv 3475   βˆͺ cun 3947   βŠ† wss 3949  π’« cpw 4603  {csn 4629  {cpr 4631  {ctp 4633  βŸ¨cop 4635  βˆͺ cuni 4909   class class class wbr 5149  {copab 5211   ↦ cmpt 5232   Γ— cxp 5675  dom cdm 5677  ran crn 5678   ∘ ccom 5681  βŸΆwf 6540  β€˜cfv 6544  (class class class)co 7409   ∈ cmpo 7411  1st c1st 7973  2nd c2nd 7974   ↑m cmap 8820  Xcixp 8891  supcsup 9435  0cc0 11110  β„*cxr 11247   < clt 11248  ndxcnx 17126  Basecbs 17144  +gcplusg 17197  .rcmulr 17198  Scalarcsca 17200   ·𝑠 cvsca 17201  Β·π‘–cip 17202  TopSetcts 17203  lecple 17204  distcds 17206  Hom chom 17208  compcco 17209  TopOpenctopn 17367  βˆtcpt 17384   Ξ£g cgsu 17386  Xscprds 17391
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 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725  ax-cnex 11166  ax-resscn 11167  ax-1cn 11168  ax-icn 11169  ax-addcl 11170  ax-addrcl 11171  ax-mulcl 11172  ax-mulrcl 11173  ax-mulcom 11174  ax-addass 11175  ax-mulass 11176  ax-distr 11177  ax-i2m1 11178  ax-1ne0 11179  ax-1rid 11180  ax-rnegex 11181  ax-rrecex 11182  ax-cnre 11183  ax-pre-lttri 11184  ax-pre-lttrn 11185  ax-pre-ltadd 11186  ax-pre-mulgt0 11187
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 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-tp 4634  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-riota 7365  df-ov 7412  df-oprab 7413  df-mpo 7414  df-om 7856  df-1st 7975  df-2nd 7976  df-frecs 8266  df-wrecs 8297  df-recs 8371  df-rdg 8410  df-1o 8466  df-er 8703  df-map 8822  df-ixp 8892  df-en 8940  df-dom 8941  df-sdom 8942  df-fin 8943  df-sup 9437  df-pnf 11250  df-mnf 11251  df-xr 11252  df-ltxr 11253  df-le 11254  df-sub 11446  df-neg 11447  df-nn 12213  df-2 12275  df-3 12276  df-4 12277  df-5 12278  df-6 12279  df-7 12280  df-8 12281  df-9 12282  df-n0 12473  df-z 12559  df-dec 12678  df-uz 12823  df-fz 13485  df-struct 17080  df-slot 17115  df-ndx 17127  df-base 17145  df-plusg 17210  df-mulr 17211  df-sca 17213  df-vsca 17214  df-ip 17215  df-tset 17216  df-ple 17217  df-ds 17219  df-hom 17221  df-cco 17222  df-prds 17393
This theorem is referenced by:  prdsle  17408  prdsds  17410  prdstset  17412  prdshom  17413  prdsco  17414  prdsvscaval  17425
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