MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  prdshom Structured version   Visualization version   GIF version

Theorem prdshom 17456
Description: Structure product hom-sets. (Contributed by Mario Carneiro, 7-Jan-2017.) (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)
prdshom.h 𝐻 = (Hom β€˜π‘ƒ)
Assertion
Ref Expression
prdshom (πœ‘ β†’ 𝐻 = (𝑓 ∈ 𝐡, 𝑔 ∈ 𝐡 ↦ Xπ‘₯ ∈ 𝐼 ((π‘“β€˜π‘₯)(Hom β€˜(π‘…β€˜π‘₯))(π‘”β€˜π‘₯))))
Distinct variable groups:   𝑓,𝑔,π‘₯,𝐡   πœ‘,𝑓,𝑔,π‘₯   𝑓,𝐼,𝑔,π‘₯   𝑃,𝑓,𝑔,π‘₯   𝑅,𝑓,𝑔,π‘₯   𝑆,𝑓,𝑔,π‘₯
Allowed substitution hints:   𝐻(π‘₯,𝑓,𝑔)   𝑉(π‘₯,𝑓,𝑔)   π‘Š(π‘₯,𝑓,𝑔)

Proof of Theorem prdshom
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 eqid 2728 . . 3 (Baseβ€˜π‘†) = (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 17446 . . 3 (πœ‘ β†’ 𝐡 = Xπ‘₯ ∈ 𝐼 (Baseβ€˜(π‘…β€˜π‘₯)))
8 eqid 2728 . . . 4 (+gβ€˜π‘ƒ) = (+gβ€˜π‘ƒ)
91, 4, 5, 6, 3, 8prdsplusg 17447 . . 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 2728 . . . 4 (.rβ€˜π‘ƒ) = (.rβ€˜π‘ƒ)
111, 4, 5, 6, 3, 10prdsmulr 17448 . . 3 (πœ‘ β†’ (.rβ€˜π‘ƒ) = (𝑓 ∈ 𝐡, 𝑔 ∈ 𝐡 ↦ (π‘₯ ∈ 𝐼 ↦ ((π‘“β€˜π‘₯)(.rβ€˜(π‘…β€˜π‘₯))(π‘”β€˜π‘₯)))))
12 eqid 2728 . . . 4 ( ·𝑠 β€˜π‘ƒ) = ( ·𝑠 β€˜π‘ƒ)
131, 4, 5, 6, 3, 2, 12prdsvsca 17449 . . 3 (πœ‘ β†’ ( ·𝑠 β€˜π‘ƒ) = (𝑓 ∈ (Baseβ€˜π‘†), 𝑔 ∈ 𝐡 ↦ (π‘₯ ∈ 𝐼 ↦ (𝑓( ·𝑠 β€˜(π‘…β€˜π‘₯))(π‘”β€˜π‘₯)))))
14 eqidd 2729 . . 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‘₯))))))
15 eqid 2728 . . . 4 (TopSetβ€˜π‘ƒ) = (TopSetβ€˜π‘ƒ)
161, 4, 5, 6, 3, 15prdstset 17455 . . 3 (πœ‘ β†’ (TopSetβ€˜π‘ƒ) = (∏tβ€˜(TopOpen ∘ 𝑅)))
17 eqid 2728 . . . 4 (leβ€˜π‘ƒ) = (leβ€˜π‘ƒ)
181, 4, 5, 6, 3, 17prdsle 17451 . . 3 (πœ‘ β†’ (leβ€˜π‘ƒ) = {βŸ¨π‘“, π‘”βŸ© ∣ ({𝑓, 𝑔} βŠ† 𝐡 ∧ βˆ€π‘₯ ∈ 𝐼 (π‘“β€˜π‘₯)(leβ€˜(π‘…β€˜π‘₯))(π‘”β€˜π‘₯))})
19 eqid 2728 . . . 4 (distβ€˜π‘ƒ) = (distβ€˜π‘ƒ)
201, 4, 5, 6, 3, 19prdsds 17453 . . 3 (πœ‘ β†’ (distβ€˜π‘ƒ) = (𝑓 ∈ 𝐡, 𝑔 ∈ 𝐡 ↦ sup((ran (π‘₯ ∈ 𝐼 ↦ ((π‘“β€˜π‘₯)(distβ€˜(π‘…β€˜π‘₯))(π‘”β€˜π‘₯))) βˆͺ {0}), ℝ*, < )))
21 eqidd 2729 . . 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‘₯))))
22 eqidd 2729 . . 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‘₯))))))
231, 2, 3, 7, 9, 11, 13, 14, 16, 18, 20, 21, 22, 4, 5prdsval 17444 . 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), ( ·𝑠 β€˜π‘ƒ)⟩, ⟨(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), (TopSetβ€˜π‘ƒ)⟩, ⟨(leβ€˜ndx), (leβ€˜π‘ƒ)⟩, ⟨(distβ€˜ndx), (distβ€˜π‘ƒ)⟩} 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‘₯)))))⟩})))
24 prdshom.h . 2 𝐻 = (Hom β€˜π‘ƒ)
25 homid 17400 . 2 Hom = Slot (Hom β€˜ndx)
26 ovssunirn 7462 . . . . . . . . . . 11 ((π‘“β€˜π‘₯)(Hom β€˜(π‘…β€˜π‘₯))(π‘”β€˜π‘₯)) βŠ† βˆͺ ran (Hom β€˜(π‘…β€˜π‘₯))
2725strfvss 17163 . . . . . . . . . . . . 13 (Hom β€˜(π‘…β€˜π‘₯)) βŠ† βˆͺ ran (π‘…β€˜π‘₯)
28 fvssunirn 6935 . . . . . . . . . . . . . 14 (π‘…β€˜π‘₯) βŠ† βˆͺ ran 𝑅
29 rnss 5945 . . . . . . . . . . . . . 14 ((π‘…β€˜π‘₯) βŠ† βˆͺ ran 𝑅 β†’ ran (π‘…β€˜π‘₯) βŠ† ran βˆͺ ran 𝑅)
30 uniss 4920 . . . . . . . . . . . . . 14 (ran (π‘…β€˜π‘₯) βŠ† ran βˆͺ ran 𝑅 β†’ βˆͺ ran (π‘…β€˜π‘₯) βŠ† βˆͺ ran βˆͺ ran 𝑅)
3128, 29, 30mp2b 10 . . . . . . . . . . . . 13 βˆͺ ran (π‘…β€˜π‘₯) βŠ† βˆͺ ran βˆͺ ran 𝑅
3227, 31sstri 3991 . . . . . . . . . . . 12 (Hom β€˜(π‘…β€˜π‘₯)) βŠ† βˆͺ ran βˆͺ ran 𝑅
33 rnss 5945 . . . . . . . . . . . 12 ((Hom β€˜(π‘…β€˜π‘₯)) βŠ† βˆͺ ran βˆͺ ran 𝑅 β†’ ran (Hom β€˜(π‘…β€˜π‘₯)) βŠ† ran βˆͺ ran βˆͺ ran 𝑅)
34 uniss 4920 . . . . . . . . . . . 12 (ran (Hom β€˜(π‘…β€˜π‘₯)) βŠ† ran βˆͺ ran βˆͺ ran 𝑅 β†’ βˆͺ ran (Hom β€˜(π‘…β€˜π‘₯)) βŠ† βˆͺ ran βˆͺ ran βˆͺ ran 𝑅)
3532, 33, 34mp2b 10 . . . . . . . . . . 11 βˆͺ ran (Hom β€˜(π‘…β€˜π‘₯)) βŠ† βˆͺ ran βˆͺ ran βˆͺ ran 𝑅
3626, 35sstri 3991 . . . . . . . . . 10 ((π‘“β€˜π‘₯)(Hom β€˜(π‘…β€˜π‘₯))(π‘”β€˜π‘₯)) βŠ† βˆͺ ran βˆͺ ran βˆͺ ran 𝑅
3736rgenw 3062 . . . . . . . . 9 βˆ€π‘₯ ∈ 𝐼 ((π‘“β€˜π‘₯)(Hom β€˜(π‘…β€˜π‘₯))(π‘”β€˜π‘₯)) βŠ† βˆͺ ran βˆͺ ran βˆͺ ran 𝑅
38 ss2ixp 8935 . . . . . . . . 9 (βˆ€π‘₯ ∈ 𝐼 ((π‘“β€˜π‘₯)(Hom β€˜(π‘…β€˜π‘₯))(π‘”β€˜π‘₯)) βŠ† βˆͺ ran βˆͺ ran βˆͺ ran 𝑅 β†’ Xπ‘₯ ∈ 𝐼 ((π‘“β€˜π‘₯)(Hom β€˜(π‘…β€˜π‘₯))(π‘”β€˜π‘₯)) βŠ† Xπ‘₯ ∈ 𝐼 βˆͺ ran βˆͺ ran βˆͺ ran 𝑅)
3937, 38ax-mp 5 . . . . . . . 8 Xπ‘₯ ∈ 𝐼 ((π‘“β€˜π‘₯)(Hom β€˜(π‘…β€˜π‘₯))(π‘”β€˜π‘₯)) βŠ† Xπ‘₯ ∈ 𝐼 βˆͺ ran βˆͺ ran βˆͺ ran 𝑅
405dmexd 7917 . . . . . . . . . 10 (πœ‘ β†’ dom 𝑅 ∈ V)
413, 40eqeltrrd 2830 . . . . . . . . 9 (πœ‘ β†’ 𝐼 ∈ V)
42 rnexg 7916 . . . . . . . . . . . 12 (𝑅 ∈ π‘Š β†’ ran 𝑅 ∈ V)
43 uniexg 7751 . . . . . . . . . . . 12 (ran 𝑅 ∈ V β†’ βˆͺ ran 𝑅 ∈ V)
445, 42, 433syl 18 . . . . . . . . . . 11 (πœ‘ β†’ βˆͺ ran 𝑅 ∈ V)
45 rnexg 7916 . . . . . . . . . . 11 (βˆͺ ran 𝑅 ∈ V β†’ ran βˆͺ ran 𝑅 ∈ V)
46 uniexg 7751 . . . . . . . . . . 11 (ran βˆͺ ran 𝑅 ∈ V β†’ βˆͺ ran βˆͺ ran 𝑅 ∈ V)
4744, 45, 463syl 18 . . . . . . . . . 10 (πœ‘ β†’ βˆͺ ran βˆͺ ran 𝑅 ∈ V)
48 rnexg 7916 . . . . . . . . . 10 (βˆͺ ran βˆͺ ran 𝑅 ∈ V β†’ ran βˆͺ ran βˆͺ ran 𝑅 ∈ V)
49 uniexg 7751 . . . . . . . . . 10 (ran βˆͺ ran βˆͺ ran 𝑅 ∈ V β†’ βˆͺ ran βˆͺ ran βˆͺ ran 𝑅 ∈ V)
5047, 48, 493syl 18 . . . . . . . . 9 (πœ‘ β†’ βˆͺ ran βˆͺ ran βˆͺ ran 𝑅 ∈ V)
51 ixpconstg 8931 . . . . . . . . 9 ((𝐼 ∈ V ∧ βˆͺ ran βˆͺ ran βˆͺ ran 𝑅 ∈ V) β†’ Xπ‘₯ ∈ 𝐼 βˆͺ ran βˆͺ ran βˆͺ ran 𝑅 = (βˆͺ ran βˆͺ ran βˆͺ ran 𝑅 ↑m 𝐼))
5241, 50, 51syl2anc 582 . . . . . . . 8 (πœ‘ β†’ Xπ‘₯ ∈ 𝐼 βˆͺ ran βˆͺ ran βˆͺ ran 𝑅 = (βˆͺ ran βˆͺ ran βˆͺ ran 𝑅 ↑m 𝐼))
5339, 52sseqtrid 4034 . . . . . . 7 (πœ‘ β†’ Xπ‘₯ ∈ 𝐼 ((π‘“β€˜π‘₯)(Hom β€˜(π‘…β€˜π‘₯))(π‘”β€˜π‘₯)) βŠ† (βˆͺ ran βˆͺ ran βˆͺ ran 𝑅 ↑m 𝐼))
54 ovex 7459 . . . . . . . 8 (βˆͺ ran βˆͺ ran βˆͺ ran 𝑅 ↑m 𝐼) ∈ V
5554elpw2 5351 . . . . . . 7 (Xπ‘₯ ∈ 𝐼 ((π‘“β€˜π‘₯)(Hom β€˜(π‘…β€˜π‘₯))(π‘”β€˜π‘₯)) ∈ 𝒫 (βˆͺ ran βˆͺ ran βˆͺ ran 𝑅 ↑m 𝐼) ↔ Xπ‘₯ ∈ 𝐼 ((π‘“β€˜π‘₯)(Hom β€˜(π‘…β€˜π‘₯))(π‘”β€˜π‘₯)) βŠ† (βˆͺ ran βˆͺ ran βˆͺ ran 𝑅 ↑m 𝐼))
5653, 55sylibr 233 . . . . . 6 (πœ‘ β†’ Xπ‘₯ ∈ 𝐼 ((π‘“β€˜π‘₯)(Hom β€˜(π‘…β€˜π‘₯))(π‘”β€˜π‘₯)) ∈ 𝒫 (βˆͺ ran βˆͺ ran βˆͺ ran 𝑅 ↑m 𝐼))
5756ralrimivw 3147 . . . . 5 (πœ‘ β†’ βˆ€π‘” ∈ 𝐡 Xπ‘₯ ∈ 𝐼 ((π‘“β€˜π‘₯)(Hom β€˜(π‘…β€˜π‘₯))(π‘”β€˜π‘₯)) ∈ 𝒫 (βˆͺ ran βˆͺ ran βˆͺ ran 𝑅 ↑m 𝐼))
5857ralrimivw 3147 . . . 4 (πœ‘ β†’ βˆ€π‘“ ∈ 𝐡 βˆ€π‘” ∈ 𝐡 Xπ‘₯ ∈ 𝐼 ((π‘“β€˜π‘₯)(Hom β€˜(π‘…β€˜π‘₯))(π‘”β€˜π‘₯)) ∈ 𝒫 (βˆͺ ran βˆͺ ran βˆͺ ran 𝑅 ↑m 𝐼))
59 eqid 2728 . . . . 5 (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‘₯)))
6059fmpo 8078 . . . 4 (βˆ€π‘“ ∈ 𝐡 βˆ€π‘” ∈ 𝐡 Xπ‘₯ ∈ 𝐼 ((π‘“β€˜π‘₯)(Hom β€˜(π‘…β€˜π‘₯))(π‘”β€˜π‘₯)) ∈ 𝒫 (βˆͺ ran βˆͺ ran βˆͺ ran 𝑅 ↑m 𝐼) ↔ (𝑓 ∈ 𝐡, 𝑔 ∈ 𝐡 ↦ Xπ‘₯ ∈ 𝐼 ((π‘“β€˜π‘₯)(Hom β€˜(π‘…β€˜π‘₯))(π‘”β€˜π‘₯))):(𝐡 Γ— 𝐡)βŸΆπ’« (βˆͺ ran βˆͺ ran βˆͺ ran 𝑅 ↑m 𝐼))
6158, 60sylib 217 . . 3 (πœ‘ β†’ (𝑓 ∈ 𝐡, 𝑔 ∈ 𝐡 ↦ Xπ‘₯ ∈ 𝐼 ((π‘“β€˜π‘₯)(Hom β€˜(π‘…β€˜π‘₯))(π‘”β€˜π‘₯))):(𝐡 Γ— 𝐡)βŸΆπ’« (βˆͺ ran βˆͺ ran βˆͺ ran 𝑅 ↑m 𝐼))
626fvexi 6916 . . . . 5 𝐡 ∈ V
6362, 62xpex 7761 . . . 4 (𝐡 Γ— 𝐡) ∈ V
6463a1i 11 . . 3 (πœ‘ β†’ (𝐡 Γ— 𝐡) ∈ V)
6554pwex 5384 . . . 4 𝒫 (βˆͺ ran βˆͺ ran βˆͺ ran 𝑅 ↑m 𝐼) ∈ V
6665a1i 11 . . 3 (πœ‘ β†’ 𝒫 (βˆͺ ran βˆͺ ran βˆͺ ran 𝑅 ↑m 𝐼) ∈ V)
67 fex2 7947 . . 3 (((𝑓 ∈ 𝐡, 𝑔 ∈ 𝐡 ↦ Xπ‘₯ ∈ 𝐼 ((π‘“β€˜π‘₯)(Hom β€˜(π‘…β€˜π‘₯))(π‘”β€˜π‘₯))):(𝐡 Γ— 𝐡)βŸΆπ’« (βˆͺ ran βˆͺ ran βˆͺ ran 𝑅 ↑m 𝐼) ∧ (𝐡 Γ— 𝐡) ∈ V ∧ 𝒫 (βˆͺ ran βˆͺ ran βˆͺ ran 𝑅 ↑m 𝐼) ∈ V) β†’ (𝑓 ∈ 𝐡, 𝑔 ∈ 𝐡 ↦ Xπ‘₯ ∈ 𝐼 ((π‘“β€˜π‘₯)(Hom β€˜(π‘…β€˜π‘₯))(π‘”β€˜π‘₯))) ∈ V)
6861, 64, 66, 67syl3anc 1368 . 2 (πœ‘ β†’ (𝑓 ∈ 𝐡, 𝑔 ∈ 𝐡 ↦ Xπ‘₯ ∈ 𝐼 ((π‘“β€˜π‘₯)(Hom β€˜(π‘…β€˜π‘₯))(π‘”β€˜π‘₯))) ∈ V)
69 snsspr1 4822 . . . 4 {⟨(Hom β€˜ndx), (𝑓 ∈ 𝐡, 𝑔 ∈ 𝐡 ↦ Xπ‘₯ ∈ 𝐼 ((π‘“β€˜π‘₯)(Hom β€˜(π‘…β€˜π‘₯))(π‘”β€˜π‘₯)))⟩} βŠ† {⟨(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‘₯)))))⟩}
70 ssun2 4175 . . . 4 {⟨(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‘₯)))))⟩} βŠ† ({⟨(TopSetβ€˜ndx), (TopSetβ€˜π‘ƒ)⟩, ⟨(leβ€˜ndx), (leβ€˜π‘ƒ)⟩, ⟨(distβ€˜ndx), (distβ€˜π‘ƒ)⟩} 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‘₯)))))⟩})
7169, 70sstri 3991 . . 3 {⟨(Hom β€˜ndx), (𝑓 ∈ 𝐡, 𝑔 ∈ 𝐡 ↦ Xπ‘₯ ∈ 𝐼 ((π‘“β€˜π‘₯)(Hom β€˜(π‘…β€˜π‘₯))(π‘”β€˜π‘₯)))⟩} βŠ† ({⟨(TopSetβ€˜ndx), (TopSetβ€˜π‘ƒ)⟩, ⟨(leβ€˜ndx), (leβ€˜π‘ƒ)⟩, ⟨(distβ€˜ndx), (distβ€˜π‘ƒ)⟩} 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‘₯)))))⟩})
72 ssun2 4175 . . 3 ({⟨(TopSetβ€˜ndx), (TopSetβ€˜π‘ƒ)⟩, ⟨(leβ€˜ndx), (leβ€˜π‘ƒ)⟩, ⟨(distβ€˜ndx), (distβ€˜π‘ƒ)⟩} 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‘₯)))))⟩}) βŠ† (({⟨(Baseβ€˜ndx), 𝐡⟩, ⟨(+gβ€˜ndx), (+gβ€˜π‘ƒ)⟩, ⟨(.rβ€˜ndx), (.rβ€˜π‘ƒ)⟩} βˆͺ {⟨(Scalarβ€˜ndx), π‘†βŸ©, ⟨( ·𝑠 β€˜ndx), ( ·𝑠 β€˜π‘ƒ)⟩, ⟨(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), (TopSetβ€˜π‘ƒ)⟩, ⟨(leβ€˜ndx), (leβ€˜π‘ƒ)⟩, ⟨(distβ€˜ndx), (distβ€˜π‘ƒ)⟩} 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‘₯)))))⟩}))
7371, 72sstri 3991 . 2 {⟨(Hom β€˜ndx), (𝑓 ∈ 𝐡, 𝑔 ∈ 𝐡 ↦ Xπ‘₯ ∈ 𝐼 ((π‘“β€˜π‘₯)(Hom 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), ( ·𝑠 β€˜π‘ƒ)⟩, ⟨(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), (TopSetβ€˜π‘ƒ)⟩, ⟨(leβ€˜ndx), (leβ€˜π‘ƒ)⟩, ⟨(distβ€˜ndx), (distβ€˜π‘ƒ)⟩} 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‘₯)))))⟩}))
7423, 24, 25, 68, 73prdsbaslem 17442 1 (πœ‘ β†’ 𝐻 = (𝑓 ∈ 𝐡, 𝑔 ∈ 𝐡 ↦ Xπ‘₯ ∈ 𝐼 ((π‘“β€˜π‘₯)(Hom β€˜(π‘…β€˜π‘₯))(π‘”β€˜π‘₯))))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   = wceq 1533   ∈ wcel 2098  βˆ€wral 3058  Vcvv 3473   βˆͺ cun 3947   βŠ† wss 3949  π’« cpw 4606  {csn 4632  {cpr 4634  {ctp 4636  βŸ¨cop 4638  βˆͺ cuni 4912   ↦ cmpt 5235   Γ— cxp 5680  dom cdm 5682  ran crn 5683  βŸΆwf 6549  β€˜cfv 6553  (class class class)co 7426   ∈ cmpo 7428  1st c1st 7997  2nd c2nd 7998   ↑m cmap 8851  Xcixp 8922  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   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:  prdsco  17457
  Copyright terms: Public domain W3C validator