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Theorem prdsvsca 17408
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 17405 . . 3 (πœ‘ β†’ 𝐡 = Xπ‘₯ ∈ 𝐼 (Baseβ€˜(π‘…β€˜π‘₯)))
8 eqid 2732 . . . 4 (+gβ€˜π‘ƒ) = (+gβ€˜π‘ƒ)
91, 4, 5, 6, 3, 8prdsplusg 17406 . . 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 2732 . . . 4 (.rβ€˜π‘ƒ) = (.rβ€˜π‘ƒ)
111, 4, 5, 6, 3, 10prdsmulr 17407 . . 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 2733 . . 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 2733 . . 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 2733 . . 3 (πœ‘ β†’ (∏tβ€˜(TopOpen ∘ 𝑅)) = (∏tβ€˜(TopOpen ∘ 𝑅)))
15 eqidd 2733 . . 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 2733 . . 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 2733 . . 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 2733 . . 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 17403 . 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 17267 . 2 ·𝑠 = Slot ( ·𝑠 β€˜ndx)
22 ovssunirn 7447 . . . . . . . . . . 11 (𝑓( ·𝑠 β€˜(π‘…β€˜π‘₯))(π‘”β€˜π‘₯)) βŠ† βˆͺ ran ( ·𝑠 β€˜(π‘…β€˜π‘₯))
2321strfvss 17122 . . . . . . . . . . . . 13 ( ·𝑠 β€˜(π‘…β€˜π‘₯)) βŠ† βˆͺ ran (π‘…β€˜π‘₯)
24 fvssunirn 6924 . . . . . . . . . . . . . 14 (π‘…β€˜π‘₯) βŠ† βˆͺ ran 𝑅
25 rnss 5938 . . . . . . . . . . . . . 14 ((π‘…β€˜π‘₯) βŠ† βˆͺ ran 𝑅 β†’ ran (π‘…β€˜π‘₯) βŠ† ran βˆͺ ran 𝑅)
26 uniss 4916 . . . . . . . . . . . . . 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 3991 . . . . . . . . . . . 12 ( ·𝑠 β€˜(π‘…β€˜π‘₯)) βŠ† βˆͺ ran βˆͺ ran 𝑅
29 rnss 5938 . . . . . . . . . . . 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 4916 . . . . . . . . . . . 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 3991 . . . . . . . . . 10 (𝑓( ·𝑠 β€˜(π‘…β€˜π‘₯))(π‘”β€˜π‘₯)) βŠ† βˆͺ ran βˆͺ ran βˆͺ ran 𝑅
33 ovex 7444 . . . . . . . . . . 11 (𝑓( ·𝑠 β€˜(π‘…β€˜π‘₯))(π‘”β€˜π‘₯)) ∈ V
3433elpw 4606 . . . . . . . . . 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 7116 . . . . . . 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 7897 . . . . . . . . . . 11 (𝑅 ∈ π‘Š β†’ ran 𝑅 ∈ V)
39 uniexg 7732 . . . . . . . . . . 11 (ran 𝑅 ∈ V β†’ βˆͺ ran 𝑅 ∈ V)
405, 38, 393syl 18 . . . . . . . . . 10 (πœ‘ β†’ βˆͺ ran 𝑅 ∈ V)
41 rnexg 7897 . . . . . . . . . 10 (βˆͺ ran 𝑅 ∈ V β†’ ran βˆͺ ran 𝑅 ∈ V)
42 uniexg 7732 . . . . . . . . . 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 7897 . . . . . . . . 9 (βˆͺ ran βˆͺ ran 𝑅 ∈ V β†’ ran βˆͺ ran βˆͺ ran 𝑅 ∈ V)
45 uniexg 7732 . . . . . . . . 9 (ran βˆͺ ran βˆͺ ran 𝑅 ∈ V β†’ βˆͺ ran βˆͺ ran βˆͺ ran 𝑅 ∈ V)
46 pwexg 5376 . . . . . . . . 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 7898 . . . . . . . . 9 (πœ‘ β†’ dom 𝑅 ∈ V)
493, 48eqeltrrd 2834 . . . . . . . 8 (πœ‘ β†’ 𝐼 ∈ V)
5047, 49elmapd 8836 . . . . . . 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 256 . . . . . 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 3150 . . . . 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 3150 . . . 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 2732 . . . . 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 8056 . . . 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 6905 . . . . 5 𝐾 ∈ V
586fvexi 6905 . . . . 5 𝐡 ∈ V
5957, 58xpex 7742 . . . 4 (𝐾 Γ— 𝐡) ∈ V
60 ovex 7444 . . . 4 (𝒫 βˆͺ ran βˆͺ ran βˆͺ ran 𝑅 ↑m 𝐼) ∈ V
61 fex2 7926 . . . 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 1453 . . 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 4820 . . . 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 4173 . . . 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 3991 . . 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 4172 . . 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 3991 . 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 17401 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 396   = wceq 1541   ∈ wcel 2106  βˆ€wral 3061  Vcvv 3474   βˆͺ cun 3946   βŠ† wss 3948  π’« cpw 4602  {csn 4628  {cpr 4630  {ctp 4632  βŸ¨cop 4634  βˆͺ cuni 4908   class class class wbr 5148  {copab 5210   ↦ cmpt 5231   Γ— cxp 5674  dom cdm 5676  ran crn 5677   ∘ ccom 5680  βŸΆwf 6539  β€˜cfv 6543  (class class class)co 7411   ∈ cmpo 7413  1st c1st 7975  2nd c2nd 7976   ↑m cmap 8822  Xcixp 8893  supcsup 9437  0cc0 11112  β„*cxr 11249   < clt 11250  ndxcnx 17128  Basecbs 17146  +gcplusg 17199  .rcmulr 17200  Scalarcsca 17202   ·𝑠 cvsca 17203  Β·π‘–cip 17204  TopSetcts 17205  lecple 17206  distcds 17208  Hom chom 17210  compcco 17211  TopOpenctopn 17369  βˆtcpt 17386   Ξ£g cgsu 17388  Xscprds 17393
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 7727  ax-cnex 11168  ax-resscn 11169  ax-1cn 11170  ax-icn 11171  ax-addcl 11172  ax-addrcl 11173  ax-mulcl 11174  ax-mulrcl 11175  ax-mulcom 11176  ax-addass 11177  ax-mulass 11178  ax-distr 11179  ax-i2m1 11180  ax-1ne0 11181  ax-1rid 11182  ax-rnegex 11183  ax-rrecex 11184  ax-cnre 11185  ax-pre-lttri 11186  ax-pre-lttrn 11187  ax-pre-ltadd 11188  ax-pre-mulgt0 11189
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 7367  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7858  df-1st 7977  df-2nd 7978  df-frecs 8268  df-wrecs 8299  df-recs 8373  df-rdg 8412  df-1o 8468  df-er 8705  df-map 8824  df-ixp 8894  df-en 8942  df-dom 8943  df-sdom 8944  df-fin 8945  df-sup 9439  df-pnf 11252  df-mnf 11253  df-xr 11254  df-ltxr 11255  df-le 11256  df-sub 11448  df-neg 11449  df-nn 12215  df-2 12277  df-3 12278  df-4 12279  df-5 12280  df-6 12281  df-7 12282  df-8 12283  df-9 12284  df-n0 12475  df-z 12561  df-dec 12680  df-uz 12825  df-fz 13487  df-struct 17082  df-slot 17117  df-ndx 17129  df-base 17147  df-plusg 17212  df-mulr 17213  df-sca 17215  df-vsca 17216  df-ip 17217  df-tset 17218  df-ple 17219  df-ds 17221  df-hom 17223  df-cco 17224  df-prds 17395
This theorem is referenced by:  prdsle  17410  prdsds  17412  prdstset  17414  prdshom  17415  prdsco  17416  prdsvscaval  17427
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