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Definition df-prds 12722
Description: Define a structure product. This can be a product of groups, rings, modules, or ordered topological fields; any unused components will have garbage in them but this is usually not relevant for the purpose of inheriting the structures present in the factors. (Contributed by Stefan O'Rear, 3-Jan-2015.) (Revised by Thierry Arnoux, 15-Jun-2019.)
Assertion
Ref Expression
df-prds Xs = (𝑠 ∈ V, π‘Ÿ ∈ V ↦ ⦋Xπ‘₯ ∈ dom π‘Ÿ(Baseβ€˜(π‘Ÿβ€˜π‘₯)) / π‘£β¦Œβ¦‹(𝑓 ∈ 𝑣, 𝑔 ∈ 𝑣 ↦ Xπ‘₯ ∈ dom π‘Ÿ((π‘“β€˜π‘₯)(Hom β€˜(π‘Ÿβ€˜π‘₯))(π‘”β€˜π‘₯))) / β„Žβ¦Œ(({⟨(Baseβ€˜ndx), π‘£βŸ©, ⟨(+gβ€˜ndx), (𝑓 ∈ 𝑣, 𝑔 ∈ 𝑣 ↦ (π‘₯ ∈ dom π‘Ÿ ↦ ((π‘“β€˜π‘₯)(+gβ€˜(π‘Ÿβ€˜π‘₯))(π‘”β€˜π‘₯))))⟩, ⟨(.rβ€˜ndx), (𝑓 ∈ 𝑣, 𝑔 ∈ 𝑣 ↦ (π‘₯ ∈ dom π‘Ÿ ↦ ((π‘“β€˜π‘₯)(.rβ€˜(π‘Ÿβ€˜π‘₯))(π‘”β€˜π‘₯))))⟩} βˆͺ {⟨(Scalarβ€˜ndx), π‘ βŸ©, ⟨( ·𝑠 β€˜ndx), (𝑓 ∈ (Baseβ€˜π‘ ), 𝑔 ∈ 𝑣 ↦ (π‘₯ ∈ dom π‘Ÿ ↦ (𝑓( ·𝑠 β€˜(π‘Ÿβ€˜π‘₯))(π‘”β€˜π‘₯))))⟩, ⟨(Β·π‘–β€˜ndx), (𝑓 ∈ 𝑣, 𝑔 ∈ 𝑣 ↦ (𝑠 Ξ£g (π‘₯ ∈ dom π‘Ÿ ↦ ((π‘“β€˜π‘₯)(Β·π‘–β€˜(π‘Ÿβ€˜π‘₯))(π‘”β€˜π‘₯)))))⟩}) βˆͺ ({⟨(TopSetβ€˜ndx), (∏tβ€˜(TopOpen ∘ π‘Ÿ))⟩, ⟨(leβ€˜ndx), {βŸ¨π‘“, π‘”βŸ© ∣ ({𝑓, 𝑔} βŠ† 𝑣 ∧ βˆ€π‘₯ ∈ dom π‘Ÿ(π‘“β€˜π‘₯)(leβ€˜(π‘Ÿβ€˜π‘₯))(π‘”β€˜π‘₯))}⟩, ⟨(distβ€˜ndx), (𝑓 ∈ 𝑣, 𝑔 ∈ 𝑣 ↦ sup((ran (π‘₯ ∈ dom π‘Ÿ ↦ ((π‘“β€˜π‘₯)(distβ€˜(π‘Ÿβ€˜π‘₯))(π‘”β€˜π‘₯))) βˆͺ {0}), ℝ*, < ))⟩} βˆͺ {⟨(Hom β€˜ndx), β„ŽβŸ©, ⟨(compβ€˜ndx), (π‘Ž ∈ (𝑣 Γ— 𝑣), 𝑐 ∈ 𝑣 ↦ (𝑑 ∈ (π‘β„Ž(2nd β€˜π‘Ž)), 𝑒 ∈ (β„Žβ€˜π‘Ž) ↦ (π‘₯ ∈ dom π‘Ÿ ↦ ((π‘‘β€˜π‘₯)(⟨((1st β€˜π‘Ž)β€˜π‘₯), ((2nd β€˜π‘Ž)β€˜π‘₯)⟩(compβ€˜(π‘Ÿβ€˜π‘₯))(π‘β€˜π‘₯))(π‘’β€˜π‘₯)))))⟩})))
Distinct variable group:   π‘Ž,𝑐,𝑑,𝑒,𝑓,𝑔,β„Ž,𝑠,π‘Ÿ,π‘₯,𝑣

Detailed syntax breakdown of Definition df-prds
StepHypRef Expression
1 cprds 12720 . 2 class Xs
2 vs . . 3 setvar 𝑠
3 vr . . 3 setvar π‘Ÿ
4 cvv 2739 . . 3 class V
5 vv . . . 4 setvar 𝑣
6 vx . . . . 5 setvar π‘₯
73cv 1352 . . . . . 6 class π‘Ÿ
87cdm 4628 . . . . 5 class dom π‘Ÿ
96cv 1352 . . . . . . 7 class π‘₯
109, 7cfv 5218 . . . . . 6 class (π‘Ÿβ€˜π‘₯)
11 cbs 12465 . . . . . 6 class Base
1210, 11cfv 5218 . . . . 5 class (Baseβ€˜(π‘Ÿβ€˜π‘₯))
136, 8, 12cixp 6701 . . . 4 class Xπ‘₯ ∈ dom π‘Ÿ(Baseβ€˜(π‘Ÿβ€˜π‘₯))
14 vh . . . . 5 setvar β„Ž
15 vf . . . . . 6 setvar 𝑓
16 vg . . . . . 6 setvar 𝑔
175cv 1352 . . . . . 6 class 𝑣
1815cv 1352 . . . . . . . . 9 class 𝑓
199, 18cfv 5218 . . . . . . . 8 class (π‘“β€˜π‘₯)
2016cv 1352 . . . . . . . . 9 class 𝑔
219, 20cfv 5218 . . . . . . . 8 class (π‘”β€˜π‘₯)
22 chom 12550 . . . . . . . . 9 class Hom
2310, 22cfv 5218 . . . . . . . 8 class (Hom β€˜(π‘Ÿβ€˜π‘₯))
2419, 21, 23co 5878 . . . . . . 7 class ((π‘“β€˜π‘₯)(Hom β€˜(π‘Ÿβ€˜π‘₯))(π‘”β€˜π‘₯))
256, 8, 24cixp 6701 . . . . . 6 class Xπ‘₯ ∈ dom π‘Ÿ((π‘“β€˜π‘₯)(Hom β€˜(π‘Ÿβ€˜π‘₯))(π‘”β€˜π‘₯))
2615, 16, 17, 17, 25cmpo 5880 . . . . 5 class (𝑓 ∈ 𝑣, 𝑔 ∈ 𝑣 ↦ Xπ‘₯ ∈ dom π‘Ÿ((π‘“β€˜π‘₯)(Hom β€˜(π‘Ÿβ€˜π‘₯))(π‘”β€˜π‘₯)))
27 cnx 12462 . . . . . . . . . 10 class ndx
2827, 11cfv 5218 . . . . . . . . 9 class (Baseβ€˜ndx)
2928, 17cop 3597 . . . . . . . 8 class ⟨(Baseβ€˜ndx), π‘£βŸ©
30 cplusg 12539 . . . . . . . . . 10 class +g
3127, 30cfv 5218 . . . . . . . . 9 class (+gβ€˜ndx)
3210, 30cfv 5218 . . . . . . . . . . . 12 class (+gβ€˜(π‘Ÿβ€˜π‘₯))
3319, 21, 32co 5878 . . . . . . . . . . 11 class ((π‘“β€˜π‘₯)(+gβ€˜(π‘Ÿβ€˜π‘₯))(π‘”β€˜π‘₯))
346, 8, 33cmpt 4066 . . . . . . . . . 10 class (π‘₯ ∈ dom π‘Ÿ ↦ ((π‘“β€˜π‘₯)(+gβ€˜(π‘Ÿβ€˜π‘₯))(π‘”β€˜π‘₯)))
3515, 16, 17, 17, 34cmpo 5880 . . . . . . . . 9 class (𝑓 ∈ 𝑣, 𝑔 ∈ 𝑣 ↦ (π‘₯ ∈ dom π‘Ÿ ↦ ((π‘“β€˜π‘₯)(+gβ€˜(π‘Ÿβ€˜π‘₯))(π‘”β€˜π‘₯))))
3631, 35cop 3597 . . . . . . . 8 class ⟨(+gβ€˜ndx), (𝑓 ∈ 𝑣, 𝑔 ∈ 𝑣 ↦ (π‘₯ ∈ dom π‘Ÿ ↦ ((π‘“β€˜π‘₯)(+gβ€˜(π‘Ÿβ€˜π‘₯))(π‘”β€˜π‘₯))))⟩
37 cmulr 12540 . . . . . . . . . 10 class .r
3827, 37cfv 5218 . . . . . . . . 9 class (.rβ€˜ndx)
3910, 37cfv 5218 . . . . . . . . . . . 12 class (.rβ€˜(π‘Ÿβ€˜π‘₯))
4019, 21, 39co 5878 . . . . . . . . . . 11 class ((π‘“β€˜π‘₯)(.rβ€˜(π‘Ÿβ€˜π‘₯))(π‘”β€˜π‘₯))
416, 8, 40cmpt 4066 . . . . . . . . . 10 class (π‘₯ ∈ dom π‘Ÿ ↦ ((π‘“β€˜π‘₯)(.rβ€˜(π‘Ÿβ€˜π‘₯))(π‘”β€˜π‘₯)))
4215, 16, 17, 17, 41cmpo 5880 . . . . . . . . 9 class (𝑓 ∈ 𝑣, 𝑔 ∈ 𝑣 ↦ (π‘₯ ∈ dom π‘Ÿ ↦ ((π‘“β€˜π‘₯)(.rβ€˜(π‘Ÿβ€˜π‘₯))(π‘”β€˜π‘₯))))
4338, 42cop 3597 . . . . . . . 8 class ⟨(.rβ€˜ndx), (𝑓 ∈ 𝑣, 𝑔 ∈ 𝑣 ↦ (π‘₯ ∈ dom π‘Ÿ ↦ ((π‘“β€˜π‘₯)(.rβ€˜(π‘Ÿβ€˜π‘₯))(π‘”β€˜π‘₯))))⟩
4429, 36, 43ctp 3596 . . . . . . 7 class {⟨(Baseβ€˜ndx), π‘£βŸ©, ⟨(+gβ€˜ndx), (𝑓 ∈ 𝑣, 𝑔 ∈ 𝑣 ↦ (π‘₯ ∈ dom π‘Ÿ ↦ ((π‘“β€˜π‘₯)(+gβ€˜(π‘Ÿβ€˜π‘₯))(π‘”β€˜π‘₯))))⟩, ⟨(.rβ€˜ndx), (𝑓 ∈ 𝑣, 𝑔 ∈ 𝑣 ↦ (π‘₯ ∈ dom π‘Ÿ ↦ ((π‘“β€˜π‘₯)(.rβ€˜(π‘Ÿβ€˜π‘₯))(π‘”β€˜π‘₯))))⟩}
45 csca 12542 . . . . . . . . . 10 class Scalar
4627, 45cfv 5218 . . . . . . . . 9 class (Scalarβ€˜ndx)
472cv 1352 . . . . . . . . 9 class 𝑠
4846, 47cop 3597 . . . . . . . 8 class ⟨(Scalarβ€˜ndx), π‘ βŸ©
49 cvsca 12543 . . . . . . . . . 10 class ·𝑠
5027, 49cfv 5218 . . . . . . . . 9 class ( ·𝑠 β€˜ndx)
5147, 11cfv 5218 . . . . . . . . . 10 class (Baseβ€˜π‘ )
5210, 49cfv 5218 . . . . . . . . . . . 12 class ( ·𝑠 β€˜(π‘Ÿβ€˜π‘₯))
5318, 21, 52co 5878 . . . . . . . . . . 11 class (𝑓( ·𝑠 β€˜(π‘Ÿβ€˜π‘₯))(π‘”β€˜π‘₯))
546, 8, 53cmpt 4066 . . . . . . . . . 10 class (π‘₯ ∈ dom π‘Ÿ ↦ (𝑓( ·𝑠 β€˜(π‘Ÿβ€˜π‘₯))(π‘”β€˜π‘₯)))
5515, 16, 51, 17, 54cmpo 5880 . . . . . . . . 9 class (𝑓 ∈ (Baseβ€˜π‘ ), 𝑔 ∈ 𝑣 ↦ (π‘₯ ∈ dom π‘Ÿ ↦ (𝑓( ·𝑠 β€˜(π‘Ÿβ€˜π‘₯))(π‘”β€˜π‘₯))))
5650, 55cop 3597 . . . . . . . 8 class ⟨( ·𝑠 β€˜ndx), (𝑓 ∈ (Baseβ€˜π‘ ), 𝑔 ∈ 𝑣 ↦ (π‘₯ ∈ dom π‘Ÿ ↦ (𝑓( ·𝑠 β€˜(π‘Ÿβ€˜π‘₯))(π‘”β€˜π‘₯))))⟩
57 cip 12544 . . . . . . . . . 10 class ·𝑖
5827, 57cfv 5218 . . . . . . . . 9 class (Β·π‘–β€˜ndx)
5910, 57cfv 5218 . . . . . . . . . . . . 13 class (Β·π‘–β€˜(π‘Ÿβ€˜π‘₯))
6019, 21, 59co 5878 . . . . . . . . . . . 12 class ((π‘“β€˜π‘₯)(Β·π‘–β€˜(π‘Ÿβ€˜π‘₯))(π‘”β€˜π‘₯))
616, 8, 60cmpt 4066 . . . . . . . . . . 11 class (π‘₯ ∈ dom π‘Ÿ ↦ ((π‘“β€˜π‘₯)(Β·π‘–β€˜(π‘Ÿβ€˜π‘₯))(π‘”β€˜π‘₯)))
62 cgsu 12712 . . . . . . . . . . 11 class Ξ£g
6347, 61, 62co 5878 . . . . . . . . . 10 class (𝑠 Ξ£g (π‘₯ ∈ dom π‘Ÿ ↦ ((π‘“β€˜π‘₯)(Β·π‘–β€˜(π‘Ÿβ€˜π‘₯))(π‘”β€˜π‘₯))))
6415, 16, 17, 17, 63cmpo 5880 . . . . . . . . 9 class (𝑓 ∈ 𝑣, 𝑔 ∈ 𝑣 ↦ (𝑠 Ξ£g (π‘₯ ∈ dom π‘Ÿ ↦ ((π‘“β€˜π‘₯)(Β·π‘–β€˜(π‘Ÿβ€˜π‘₯))(π‘”β€˜π‘₯)))))
6558, 64cop 3597 . . . . . . . 8 class ⟨(Β·π‘–β€˜ndx), (𝑓 ∈ 𝑣, 𝑔 ∈ 𝑣 ↦ (𝑠 Ξ£g (π‘₯ ∈ dom π‘Ÿ ↦ ((π‘“β€˜π‘₯)(Β·π‘–β€˜(π‘Ÿβ€˜π‘₯))(π‘”β€˜π‘₯)))))⟩
6648, 56, 65ctp 3596 . . . . . . 7 class {⟨(Scalarβ€˜ndx), π‘ βŸ©, ⟨( ·𝑠 β€˜ndx), (𝑓 ∈ (Baseβ€˜π‘ ), 𝑔 ∈ 𝑣 ↦ (π‘₯ ∈ dom π‘Ÿ ↦ (𝑓( ·𝑠 β€˜(π‘Ÿβ€˜π‘₯))(π‘”β€˜π‘₯))))⟩, ⟨(Β·π‘–β€˜ndx), (𝑓 ∈ 𝑣, 𝑔 ∈ 𝑣 ↦ (𝑠 Ξ£g (π‘₯ ∈ dom π‘Ÿ ↦ ((π‘“β€˜π‘₯)(Β·π‘–β€˜(π‘Ÿβ€˜π‘₯))(π‘”β€˜π‘₯)))))⟩}
6744, 66cun 3129 . . . . . 6 class ({⟨(Baseβ€˜ndx), π‘£βŸ©, ⟨(+gβ€˜ndx), (𝑓 ∈ 𝑣, 𝑔 ∈ 𝑣 ↦ (π‘₯ ∈ dom π‘Ÿ ↦ ((π‘“β€˜π‘₯)(+gβ€˜(π‘Ÿβ€˜π‘₯))(π‘”β€˜π‘₯))))⟩, ⟨(.rβ€˜ndx), (𝑓 ∈ 𝑣, 𝑔 ∈ 𝑣 ↦ (π‘₯ ∈ dom π‘Ÿ ↦ ((π‘“β€˜π‘₯)(.rβ€˜(π‘Ÿβ€˜π‘₯))(π‘”β€˜π‘₯))))⟩} βˆͺ {⟨(Scalarβ€˜ndx), π‘ βŸ©, ⟨( ·𝑠 β€˜ndx), (𝑓 ∈ (Baseβ€˜π‘ ), 𝑔 ∈ 𝑣 ↦ (π‘₯ ∈ dom π‘Ÿ ↦ (𝑓( ·𝑠 β€˜(π‘Ÿβ€˜π‘₯))(π‘”β€˜π‘₯))))⟩, ⟨(Β·π‘–β€˜ndx), (𝑓 ∈ 𝑣, 𝑔 ∈ 𝑣 ↦ (𝑠 Ξ£g (π‘₯ ∈ dom π‘Ÿ ↦ ((π‘“β€˜π‘₯)(Β·π‘–β€˜(π‘Ÿβ€˜π‘₯))(π‘”β€˜π‘₯)))))⟩})
68 cts 12545 . . . . . . . . . 10 class TopSet
6927, 68cfv 5218 . . . . . . . . 9 class (TopSetβ€˜ndx)
70 ctopn 12695 . . . . . . . . . . 11 class TopOpen
7170, 7ccom 4632 . . . . . . . . . 10 class (TopOpen ∘ π‘Ÿ)
72 cpt 12710 . . . . . . . . . 10 class ∏t
7371, 72cfv 5218 . . . . . . . . 9 class (∏tβ€˜(TopOpen ∘ π‘Ÿ))
7469, 73cop 3597 . . . . . . . 8 class ⟨(TopSetβ€˜ndx), (∏tβ€˜(TopOpen ∘ π‘Ÿ))⟩
75 cple 12546 . . . . . . . . . 10 class le
7627, 75cfv 5218 . . . . . . . . 9 class (leβ€˜ndx)
7718, 20cpr 3595 . . . . . . . . . . . 12 class {𝑓, 𝑔}
7877, 17wss 3131 . . . . . . . . . . 11 wff {𝑓, 𝑔} βŠ† 𝑣
7910, 75cfv 5218 . . . . . . . . . . . . 13 class (leβ€˜(π‘Ÿβ€˜π‘₯))
8019, 21, 79wbr 4005 . . . . . . . . . . . 12 wff (π‘“β€˜π‘₯)(leβ€˜(π‘Ÿβ€˜π‘₯))(π‘”β€˜π‘₯)
8180, 6, 8wral 2455 . . . . . . . . . . 11 wff βˆ€π‘₯ ∈ dom π‘Ÿ(π‘“β€˜π‘₯)(leβ€˜(π‘Ÿβ€˜π‘₯))(π‘”β€˜π‘₯)
8278, 81wa 104 . . . . . . . . . 10 wff ({𝑓, 𝑔} βŠ† 𝑣 ∧ βˆ€π‘₯ ∈ dom π‘Ÿ(π‘“β€˜π‘₯)(leβ€˜(π‘Ÿβ€˜π‘₯))(π‘”β€˜π‘₯))
8382, 15, 16copab 4065 . . . . . . . . 9 class {βŸ¨π‘“, π‘”βŸ© ∣ ({𝑓, 𝑔} βŠ† 𝑣 ∧ βˆ€π‘₯ ∈ dom π‘Ÿ(π‘“β€˜π‘₯)(leβ€˜(π‘Ÿβ€˜π‘₯))(π‘”β€˜π‘₯))}
8476, 83cop 3597 . . . . . . . 8 class ⟨(leβ€˜ndx), {βŸ¨π‘“, π‘”βŸ© ∣ ({𝑓, 𝑔} βŠ† 𝑣 ∧ βˆ€π‘₯ ∈ dom π‘Ÿ(π‘“β€˜π‘₯)(leβ€˜(π‘Ÿβ€˜π‘₯))(π‘”β€˜π‘₯))}⟩
85 cds 12548 . . . . . . . . . 10 class dist
8627, 85cfv 5218 . . . . . . . . 9 class (distβ€˜ndx)
8710, 85cfv 5218 . . . . . . . . . . . . . . 15 class (distβ€˜(π‘Ÿβ€˜π‘₯))
8819, 21, 87co 5878 . . . . . . . . . . . . . 14 class ((π‘“β€˜π‘₯)(distβ€˜(π‘Ÿβ€˜π‘₯))(π‘”β€˜π‘₯))
896, 8, 88cmpt 4066 . . . . . . . . . . . . 13 class (π‘₯ ∈ dom π‘Ÿ ↦ ((π‘“β€˜π‘₯)(distβ€˜(π‘Ÿβ€˜π‘₯))(π‘”β€˜π‘₯)))
9089crn 4629 . . . . . . . . . . . 12 class ran (π‘₯ ∈ dom π‘Ÿ ↦ ((π‘“β€˜π‘₯)(distβ€˜(π‘Ÿβ€˜π‘₯))(π‘”β€˜π‘₯)))
91 cc0 7814 . . . . . . . . . . . . 13 class 0
9291csn 3594 . . . . . . . . . . . 12 class {0}
9390, 92cun 3129 . . . . . . . . . . 11 class (ran (π‘₯ ∈ dom π‘Ÿ ↦ ((π‘“β€˜π‘₯)(distβ€˜(π‘Ÿβ€˜π‘₯))(π‘”β€˜π‘₯))) βˆͺ {0})
94 cxr 7994 . . . . . . . . . . 11 class ℝ*
95 clt 7995 . . . . . . . . . . 11 class <
9693, 94, 95csup 6984 . . . . . . . . . 10 class sup((ran (π‘₯ ∈ dom π‘Ÿ ↦ ((π‘“β€˜π‘₯)(distβ€˜(π‘Ÿβ€˜π‘₯))(π‘”β€˜π‘₯))) βˆͺ {0}), ℝ*, < )
9715, 16, 17, 17, 96cmpo 5880 . . . . . . . . 9 class (𝑓 ∈ 𝑣, 𝑔 ∈ 𝑣 ↦ sup((ran (π‘₯ ∈ dom π‘Ÿ ↦ ((π‘“β€˜π‘₯)(distβ€˜(π‘Ÿβ€˜π‘₯))(π‘”β€˜π‘₯))) βˆͺ {0}), ℝ*, < ))
9886, 97cop 3597 . . . . . . . 8 class ⟨(distβ€˜ndx), (𝑓 ∈ 𝑣, 𝑔 ∈ 𝑣 ↦ sup((ran (π‘₯ ∈ dom π‘Ÿ ↦ ((π‘“β€˜π‘₯)(distβ€˜(π‘Ÿβ€˜π‘₯))(π‘”β€˜π‘₯))) βˆͺ {0}), ℝ*, < ))⟩
9974, 84, 98ctp 3596 . . . . . . 7 class {⟨(TopSetβ€˜ndx), (∏tβ€˜(TopOpen ∘ π‘Ÿ))⟩, ⟨(leβ€˜ndx), {βŸ¨π‘“, π‘”βŸ© ∣ ({𝑓, 𝑔} βŠ† 𝑣 ∧ βˆ€π‘₯ ∈ dom π‘Ÿ(π‘“β€˜π‘₯)(leβ€˜(π‘Ÿβ€˜π‘₯))(π‘”β€˜π‘₯))}⟩, ⟨(distβ€˜ndx), (𝑓 ∈ 𝑣, 𝑔 ∈ 𝑣 ↦ sup((ran (π‘₯ ∈ dom π‘Ÿ ↦ ((π‘“β€˜π‘₯)(distβ€˜(π‘Ÿβ€˜π‘₯))(π‘”β€˜π‘₯))) βˆͺ {0}), ℝ*, < ))⟩}
10027, 22cfv 5218 . . . . . . . . 9 class (Hom β€˜ndx)
10114cv 1352 . . . . . . . . 9 class β„Ž
102100, 101cop 3597 . . . . . . . 8 class ⟨(Hom β€˜ndx), β„ŽβŸ©
103 cco 12551 . . . . . . . . . 10 class comp
10427, 103cfv 5218 . . . . . . . . 9 class (compβ€˜ndx)
105 va . . . . . . . . . 10 setvar π‘Ž
106 vc . . . . . . . . . 10 setvar 𝑐
10717, 17cxp 4626 . . . . . . . . . 10 class (𝑣 Γ— 𝑣)
108 vd . . . . . . . . . . 11 setvar 𝑑
109 ve . . . . . . . . . . 11 setvar 𝑒
110106cv 1352 . . . . . . . . . . . 12 class 𝑐
111105cv 1352 . . . . . . . . . . . . 13 class π‘Ž
112 c2nd 6143 . . . . . . . . . . . . 13 class 2nd
113111, 112cfv 5218 . . . . . . . . . . . 12 class (2nd β€˜π‘Ž)
114110, 113, 101co 5878 . . . . . . . . . . 11 class (π‘β„Ž(2nd β€˜π‘Ž))
115111, 101cfv 5218 . . . . . . . . . . 11 class (β„Žβ€˜π‘Ž)
116108cv 1352 . . . . . . . . . . . . . 14 class 𝑑
1179, 116cfv 5218 . . . . . . . . . . . . 13 class (π‘‘β€˜π‘₯)
118109cv 1352 . . . . . . . . . . . . . 14 class 𝑒
1199, 118cfv 5218 . . . . . . . . . . . . 13 class (π‘’β€˜π‘₯)
120 c1st 6142 . . . . . . . . . . . . . . . . 17 class 1st
121111, 120cfv 5218 . . . . . . . . . . . . . . . 16 class (1st β€˜π‘Ž)
1229, 121cfv 5218 . . . . . . . . . . . . . . 15 class ((1st β€˜π‘Ž)β€˜π‘₯)
1239, 113cfv 5218 . . . . . . . . . . . . . . 15 class ((2nd β€˜π‘Ž)β€˜π‘₯)
124122, 123cop 3597 . . . . . . . . . . . . . 14 class ⟨((1st β€˜π‘Ž)β€˜π‘₯), ((2nd β€˜π‘Ž)β€˜π‘₯)⟩
1259, 110cfv 5218 . . . . . . . . . . . . . 14 class (π‘β€˜π‘₯)
12610, 103cfv 5218 . . . . . . . . . . . . . 14 class (compβ€˜(π‘Ÿβ€˜π‘₯))
127124, 125, 126co 5878 . . . . . . . . . . . . 13 class (⟨((1st β€˜π‘Ž)β€˜π‘₯), ((2nd β€˜π‘Ž)β€˜π‘₯)⟩(compβ€˜(π‘Ÿβ€˜π‘₯))(π‘β€˜π‘₯))
128117, 119, 127co 5878 . . . . . . . . . . . 12 class ((π‘‘β€˜π‘₯)(⟨((1st β€˜π‘Ž)β€˜π‘₯), ((2nd β€˜π‘Ž)β€˜π‘₯)⟩(compβ€˜(π‘Ÿβ€˜π‘₯))(π‘β€˜π‘₯))(π‘’β€˜π‘₯))
1296, 8, 128cmpt 4066 . . . . . . . . . . 11 class (π‘₯ ∈ dom π‘Ÿ ↦ ((π‘‘β€˜π‘₯)(⟨((1st β€˜π‘Ž)β€˜π‘₯), ((2nd β€˜π‘Ž)β€˜π‘₯)⟩(compβ€˜(π‘Ÿβ€˜π‘₯))(π‘β€˜π‘₯))(π‘’β€˜π‘₯)))
130108, 109, 114, 115, 129cmpo 5880 . . . . . . . . . 10 class (𝑑 ∈ (π‘β„Ž(2nd β€˜π‘Ž)), 𝑒 ∈ (β„Žβ€˜π‘Ž) ↦ (π‘₯ ∈ dom π‘Ÿ ↦ ((π‘‘β€˜π‘₯)(⟨((1st β€˜π‘Ž)β€˜π‘₯), ((2nd β€˜π‘Ž)β€˜π‘₯)⟩(compβ€˜(π‘Ÿβ€˜π‘₯))(π‘β€˜π‘₯))(π‘’β€˜π‘₯))))
131105, 106, 107, 17, 130cmpo 5880 . . . . . . . . 9 class (π‘Ž ∈ (𝑣 Γ— 𝑣), 𝑐 ∈ 𝑣 ↦ (𝑑 ∈ (π‘β„Ž(2nd β€˜π‘Ž)), 𝑒 ∈ (β„Žβ€˜π‘Ž) ↦ (π‘₯ ∈ dom π‘Ÿ ↦ ((π‘‘β€˜π‘₯)(⟨((1st β€˜π‘Ž)β€˜π‘₯), ((2nd β€˜π‘Ž)β€˜π‘₯)⟩(compβ€˜(π‘Ÿβ€˜π‘₯))(π‘β€˜π‘₯))(π‘’β€˜π‘₯)))))
132104, 131cop 3597 . . . . . . . 8 class ⟨(compβ€˜ndx), (π‘Ž ∈ (𝑣 Γ— 𝑣), 𝑐 ∈ 𝑣 ↦ (𝑑 ∈ (π‘β„Ž(2nd β€˜π‘Ž)), 𝑒 ∈ (β„Žβ€˜π‘Ž) ↦ (π‘₯ ∈ dom π‘Ÿ ↦ ((π‘‘β€˜π‘₯)(⟨((1st β€˜π‘Ž)β€˜π‘₯), ((2nd β€˜π‘Ž)β€˜π‘₯)⟩(compβ€˜(π‘Ÿβ€˜π‘₯))(π‘β€˜π‘₯))(π‘’β€˜π‘₯)))))⟩
133102, 132cpr 3595 . . . . . . 7 class {⟨(Hom β€˜ndx), β„ŽβŸ©, ⟨(compβ€˜ndx), (π‘Ž ∈ (𝑣 Γ— 𝑣), 𝑐 ∈ 𝑣 ↦ (𝑑 ∈ (π‘β„Ž(2nd β€˜π‘Ž)), 𝑒 ∈ (β„Žβ€˜π‘Ž) ↦ (π‘₯ ∈ dom π‘Ÿ ↦ ((π‘‘β€˜π‘₯)(⟨((1st β€˜π‘Ž)β€˜π‘₯), ((2nd β€˜π‘Ž)β€˜π‘₯)⟩(compβ€˜(π‘Ÿβ€˜π‘₯))(π‘β€˜π‘₯))(π‘’β€˜π‘₯)))))⟩}
13499, 133cun 3129 . . . . . 6 class ({⟨(TopSetβ€˜ndx), (∏tβ€˜(TopOpen ∘ π‘Ÿ))⟩, ⟨(leβ€˜ndx), {βŸ¨π‘“, π‘”βŸ© ∣ ({𝑓, 𝑔} βŠ† 𝑣 ∧ βˆ€π‘₯ ∈ dom π‘Ÿ(π‘“β€˜π‘₯)(leβ€˜(π‘Ÿβ€˜π‘₯))(π‘”β€˜π‘₯))}⟩, ⟨(distβ€˜ndx), (𝑓 ∈ 𝑣, 𝑔 ∈ 𝑣 ↦ sup((ran (π‘₯ ∈ dom π‘Ÿ ↦ ((π‘“β€˜π‘₯)(distβ€˜(π‘Ÿβ€˜π‘₯))(π‘”β€˜π‘₯))) βˆͺ {0}), ℝ*, < ))⟩} βˆͺ {⟨(Hom β€˜ndx), β„ŽβŸ©, ⟨(compβ€˜ndx), (π‘Ž ∈ (𝑣 Γ— 𝑣), 𝑐 ∈ 𝑣 ↦ (𝑑 ∈ (π‘β„Ž(2nd β€˜π‘Ž)), 𝑒 ∈ (β„Žβ€˜π‘Ž) ↦ (π‘₯ ∈ dom π‘Ÿ ↦ ((π‘‘β€˜π‘₯)(⟨((1st β€˜π‘Ž)β€˜π‘₯), ((2nd β€˜π‘Ž)β€˜π‘₯)⟩(compβ€˜(π‘Ÿβ€˜π‘₯))(π‘β€˜π‘₯))(π‘’β€˜π‘₯)))))⟩})
13567, 134cun 3129 . . . . 5 class (({⟨(Baseβ€˜ndx), π‘£βŸ©, ⟨(+gβ€˜ndx), (𝑓 ∈ 𝑣, 𝑔 ∈ 𝑣 ↦ (π‘₯ ∈ dom π‘Ÿ ↦ ((π‘“β€˜π‘₯)(+gβ€˜(π‘Ÿβ€˜π‘₯))(π‘”β€˜π‘₯))))⟩, ⟨(.rβ€˜ndx), (𝑓 ∈ 𝑣, 𝑔 ∈ 𝑣 ↦ (π‘₯ ∈ dom π‘Ÿ ↦ ((π‘“β€˜π‘₯)(.rβ€˜(π‘Ÿβ€˜π‘₯))(π‘”β€˜π‘₯))))⟩} βˆͺ {⟨(Scalarβ€˜ndx), π‘ βŸ©, ⟨( ·𝑠 β€˜ndx), (𝑓 ∈ (Baseβ€˜π‘ ), 𝑔 ∈ 𝑣 ↦ (π‘₯ ∈ dom π‘Ÿ ↦ (𝑓( ·𝑠 β€˜(π‘Ÿβ€˜π‘₯))(π‘”β€˜π‘₯))))⟩, ⟨(Β·π‘–β€˜ndx), (𝑓 ∈ 𝑣, 𝑔 ∈ 𝑣 ↦ (𝑠 Ξ£g (π‘₯ ∈ dom π‘Ÿ ↦ ((π‘“β€˜π‘₯)(Β·π‘–β€˜(π‘Ÿβ€˜π‘₯))(π‘”β€˜π‘₯)))))⟩}) βˆͺ ({⟨(TopSetβ€˜ndx), (∏tβ€˜(TopOpen ∘ π‘Ÿ))⟩, ⟨(leβ€˜ndx), {βŸ¨π‘“, π‘”βŸ© ∣ ({𝑓, 𝑔} βŠ† 𝑣 ∧ βˆ€π‘₯ ∈ dom π‘Ÿ(π‘“β€˜π‘₯)(leβ€˜(π‘Ÿβ€˜π‘₯))(π‘”β€˜π‘₯))}⟩, ⟨(distβ€˜ndx), (𝑓 ∈ 𝑣, 𝑔 ∈ 𝑣 ↦ sup((ran (π‘₯ ∈ dom π‘Ÿ ↦ ((π‘“β€˜π‘₯)(distβ€˜(π‘Ÿβ€˜π‘₯))(π‘”β€˜π‘₯))) βˆͺ {0}), ℝ*, < ))⟩} βˆͺ {⟨(Hom β€˜ndx), β„ŽβŸ©, ⟨(compβ€˜ndx), (π‘Ž ∈ (𝑣 Γ— 𝑣), 𝑐 ∈ 𝑣 ↦ (𝑑 ∈ (π‘β„Ž(2nd β€˜π‘Ž)), 𝑒 ∈ (β„Žβ€˜π‘Ž) ↦ (π‘₯ ∈ dom π‘Ÿ ↦ ((π‘‘β€˜π‘₯)(⟨((1st β€˜π‘Ž)β€˜π‘₯), ((2nd β€˜π‘Ž)β€˜π‘₯)⟩(compβ€˜(π‘Ÿβ€˜π‘₯))(π‘β€˜π‘₯))(π‘’β€˜π‘₯)))))⟩}))
13614, 26, 135csb 3059 . . . 4 class ⦋(𝑓 ∈ 𝑣, 𝑔 ∈ 𝑣 ↦ Xπ‘₯ ∈ dom π‘Ÿ((π‘“β€˜π‘₯)(Hom β€˜(π‘Ÿβ€˜π‘₯))(π‘”β€˜π‘₯))) / β„Žβ¦Œ(({⟨(Baseβ€˜ndx), π‘£βŸ©, ⟨(+gβ€˜ndx), (𝑓 ∈ 𝑣, 𝑔 ∈ 𝑣 ↦ (π‘₯ ∈ dom π‘Ÿ ↦ ((π‘“β€˜π‘₯)(+gβ€˜(π‘Ÿβ€˜π‘₯))(π‘”β€˜π‘₯))))⟩, ⟨(.rβ€˜ndx), (𝑓 ∈ 𝑣, 𝑔 ∈ 𝑣 ↦ (π‘₯ ∈ dom π‘Ÿ ↦ ((π‘“β€˜π‘₯)(.rβ€˜(π‘Ÿβ€˜π‘₯))(π‘”β€˜π‘₯))))⟩} βˆͺ {⟨(Scalarβ€˜ndx), π‘ βŸ©, ⟨( ·𝑠 β€˜ndx), (𝑓 ∈ (Baseβ€˜π‘ ), 𝑔 ∈ 𝑣 ↦ (π‘₯ ∈ dom π‘Ÿ ↦ (𝑓( ·𝑠 β€˜(π‘Ÿβ€˜π‘₯))(π‘”β€˜π‘₯))))⟩, ⟨(Β·π‘–β€˜ndx), (𝑓 ∈ 𝑣, 𝑔 ∈ 𝑣 ↦ (𝑠 Ξ£g (π‘₯ ∈ dom π‘Ÿ ↦ ((π‘“β€˜π‘₯)(Β·π‘–β€˜(π‘Ÿβ€˜π‘₯))(π‘”β€˜π‘₯)))))⟩}) βˆͺ ({⟨(TopSetβ€˜ndx), (∏tβ€˜(TopOpen ∘ π‘Ÿ))⟩, ⟨(leβ€˜ndx), {βŸ¨π‘“, π‘”βŸ© ∣ ({𝑓, 𝑔} βŠ† 𝑣 ∧ βˆ€π‘₯ ∈ dom π‘Ÿ(π‘“β€˜π‘₯)(leβ€˜(π‘Ÿβ€˜π‘₯))(π‘”β€˜π‘₯))}⟩, ⟨(distβ€˜ndx), (𝑓 ∈ 𝑣, 𝑔 ∈ 𝑣 ↦ sup((ran (π‘₯ ∈ dom π‘Ÿ ↦ ((π‘“β€˜π‘₯)(distβ€˜(π‘Ÿβ€˜π‘₯))(π‘”β€˜π‘₯))) βˆͺ {0}), ℝ*, < ))⟩} βˆͺ {⟨(Hom β€˜ndx), β„ŽβŸ©, ⟨(compβ€˜ndx), (π‘Ž ∈ (𝑣 Γ— 𝑣), 𝑐 ∈ 𝑣 ↦ (𝑑 ∈ (π‘β„Ž(2nd β€˜π‘Ž)), 𝑒 ∈ (β„Žβ€˜π‘Ž) ↦ (π‘₯ ∈ dom π‘Ÿ ↦ ((π‘‘β€˜π‘₯)(⟨((1st β€˜π‘Ž)β€˜π‘₯), ((2nd β€˜π‘Ž)β€˜π‘₯)⟩(compβ€˜(π‘Ÿβ€˜π‘₯))(π‘β€˜π‘₯))(π‘’β€˜π‘₯)))))⟩}))
1375, 13, 136csb 3059 . . 3 class ⦋Xπ‘₯ ∈ dom π‘Ÿ(Baseβ€˜(π‘Ÿβ€˜π‘₯)) / π‘£β¦Œβ¦‹(𝑓 ∈ 𝑣, 𝑔 ∈ 𝑣 ↦ Xπ‘₯ ∈ dom π‘Ÿ((π‘“β€˜π‘₯)(Hom β€˜(π‘Ÿβ€˜π‘₯))(π‘”β€˜π‘₯))) / β„Žβ¦Œ(({⟨(Baseβ€˜ndx), π‘£βŸ©, ⟨(+gβ€˜ndx), (𝑓 ∈ 𝑣, 𝑔 ∈ 𝑣 ↦ (π‘₯ ∈ dom π‘Ÿ ↦ ((π‘“β€˜π‘₯)(+gβ€˜(π‘Ÿβ€˜π‘₯))(π‘”β€˜π‘₯))))⟩, ⟨(.rβ€˜ndx), (𝑓 ∈ 𝑣, 𝑔 ∈ 𝑣 ↦ (π‘₯ ∈ dom π‘Ÿ ↦ ((π‘“β€˜π‘₯)(.rβ€˜(π‘Ÿβ€˜π‘₯))(π‘”β€˜π‘₯))))⟩} βˆͺ {⟨(Scalarβ€˜ndx), π‘ βŸ©, ⟨( ·𝑠 β€˜ndx), (𝑓 ∈ (Baseβ€˜π‘ ), 𝑔 ∈ 𝑣 ↦ (π‘₯ ∈ dom π‘Ÿ ↦ (𝑓( ·𝑠 β€˜(π‘Ÿβ€˜π‘₯))(π‘”β€˜π‘₯))))⟩, ⟨(Β·π‘–β€˜ndx), (𝑓 ∈ 𝑣, 𝑔 ∈ 𝑣 ↦ (𝑠 Ξ£g (π‘₯ ∈ dom π‘Ÿ ↦ ((π‘“β€˜π‘₯)(Β·π‘–β€˜(π‘Ÿβ€˜π‘₯))(π‘”β€˜π‘₯)))))⟩}) βˆͺ ({⟨(TopSetβ€˜ndx), (∏tβ€˜(TopOpen ∘ π‘Ÿ))⟩, ⟨(leβ€˜ndx), {βŸ¨π‘“, π‘”βŸ© ∣ ({𝑓, 𝑔} βŠ† 𝑣 ∧ βˆ€π‘₯ ∈ dom π‘Ÿ(π‘“β€˜π‘₯)(leβ€˜(π‘Ÿβ€˜π‘₯))(π‘”β€˜π‘₯))}⟩, ⟨(distβ€˜ndx), (𝑓 ∈ 𝑣, 𝑔 ∈ 𝑣 ↦ sup((ran (π‘₯ ∈ dom π‘Ÿ ↦ ((π‘“β€˜π‘₯)(distβ€˜(π‘Ÿβ€˜π‘₯))(π‘”β€˜π‘₯))) βˆͺ {0}), ℝ*, < ))⟩} βˆͺ {⟨(Hom β€˜ndx), β„ŽβŸ©, ⟨(compβ€˜ndx), (π‘Ž ∈ (𝑣 Γ— 𝑣), 𝑐 ∈ 𝑣 ↦ (𝑑 ∈ (π‘β„Ž(2nd β€˜π‘Ž)), 𝑒 ∈ (β„Žβ€˜π‘Ž) ↦ (π‘₯ ∈ dom π‘Ÿ ↦ ((π‘‘β€˜π‘₯)(⟨((1st β€˜π‘Ž)β€˜π‘₯), ((2nd β€˜π‘Ž)β€˜π‘₯)⟩(compβ€˜(π‘Ÿβ€˜π‘₯))(π‘β€˜π‘₯))(π‘’β€˜π‘₯)))))⟩}))
1382, 3, 4, 4, 137cmpo 5880 . 2 class (𝑠 ∈ V, π‘Ÿ ∈ V ↦ ⦋Xπ‘₯ ∈ dom π‘Ÿ(Baseβ€˜(π‘Ÿβ€˜π‘₯)) / π‘£β¦Œβ¦‹(𝑓 ∈ 𝑣, 𝑔 ∈ 𝑣 ↦ Xπ‘₯ ∈ dom π‘Ÿ((π‘“β€˜π‘₯)(Hom β€˜(π‘Ÿβ€˜π‘₯))(π‘”β€˜π‘₯))) / β„Žβ¦Œ(({⟨(Baseβ€˜ndx), π‘£βŸ©, ⟨(+gβ€˜ndx), (𝑓 ∈ 𝑣, 𝑔 ∈ 𝑣 ↦ (π‘₯ ∈ dom π‘Ÿ ↦ ((π‘“β€˜π‘₯)(+gβ€˜(π‘Ÿβ€˜π‘₯))(π‘”β€˜π‘₯))))⟩, ⟨(.rβ€˜ndx), (𝑓 ∈ 𝑣, 𝑔 ∈ 𝑣 ↦ (π‘₯ ∈ dom π‘Ÿ ↦ ((π‘“β€˜π‘₯)(.rβ€˜(π‘Ÿβ€˜π‘₯))(π‘”β€˜π‘₯))))⟩} βˆͺ {⟨(Scalarβ€˜ndx), π‘ βŸ©, ⟨( ·𝑠 β€˜ndx), (𝑓 ∈ (Baseβ€˜π‘ ), 𝑔 ∈ 𝑣 ↦ (π‘₯ ∈ dom π‘Ÿ ↦ (𝑓( ·𝑠 β€˜(π‘Ÿβ€˜π‘₯))(π‘”β€˜π‘₯))))⟩, ⟨(Β·π‘–β€˜ndx), (𝑓 ∈ 𝑣, 𝑔 ∈ 𝑣 ↦ (𝑠 Ξ£g (π‘₯ ∈ dom π‘Ÿ ↦ ((π‘“β€˜π‘₯)(Β·π‘–β€˜(π‘Ÿβ€˜π‘₯))(π‘”β€˜π‘₯)))))⟩}) βˆͺ ({⟨(TopSetβ€˜ndx), (∏tβ€˜(TopOpen ∘ π‘Ÿ))⟩, ⟨(leβ€˜ndx), {βŸ¨π‘“, π‘”βŸ© ∣ ({𝑓, 𝑔} βŠ† 𝑣 ∧ βˆ€π‘₯ ∈ dom π‘Ÿ(π‘“β€˜π‘₯)(leβ€˜(π‘Ÿβ€˜π‘₯))(π‘”β€˜π‘₯))}⟩, ⟨(distβ€˜ndx), (𝑓 ∈ 𝑣, 𝑔 ∈ 𝑣 ↦ sup((ran (π‘₯ ∈ dom π‘Ÿ ↦ ((π‘“β€˜π‘₯)(distβ€˜(π‘Ÿβ€˜π‘₯))(π‘”β€˜π‘₯))) βˆͺ {0}), ℝ*, < ))⟩} βˆͺ {⟨(Hom β€˜ndx), β„ŽβŸ©, ⟨(compβ€˜ndx), (π‘Ž ∈ (𝑣 Γ— 𝑣), 𝑐 ∈ 𝑣 ↦ (𝑑 ∈ (π‘β„Ž(2nd β€˜π‘Ž)), 𝑒 ∈ (β„Žβ€˜π‘Ž) ↦ (π‘₯ ∈ dom π‘Ÿ ↦ ((π‘‘β€˜π‘₯)(⟨((1st β€˜π‘Ž)β€˜π‘₯), ((2nd β€˜π‘Ž)β€˜π‘₯)⟩(compβ€˜(π‘Ÿβ€˜π‘₯))(π‘β€˜π‘₯))(π‘’β€˜π‘₯)))))⟩})))
1391, 138wceq 1353 1 wff Xs = (𝑠 ∈ V, π‘Ÿ ∈ V ↦ ⦋Xπ‘₯ ∈ dom π‘Ÿ(Baseβ€˜(π‘Ÿβ€˜π‘₯)) / π‘£β¦Œβ¦‹(𝑓 ∈ 𝑣, 𝑔 ∈ 𝑣 ↦ Xπ‘₯ ∈ dom π‘Ÿ((π‘“β€˜π‘₯)(Hom β€˜(π‘Ÿβ€˜π‘₯))(π‘”β€˜π‘₯))) / β„Žβ¦Œ(({⟨(Baseβ€˜ndx), π‘£βŸ©, ⟨(+gβ€˜ndx), (𝑓 ∈ 𝑣, 𝑔 ∈ 𝑣 ↦ (π‘₯ ∈ dom π‘Ÿ ↦ ((π‘“β€˜π‘₯)(+gβ€˜(π‘Ÿβ€˜π‘₯))(π‘”β€˜π‘₯))))⟩, ⟨(.rβ€˜ndx), (𝑓 ∈ 𝑣, 𝑔 ∈ 𝑣 ↦ (π‘₯ ∈ dom π‘Ÿ ↦ ((π‘“β€˜π‘₯)(.rβ€˜(π‘Ÿβ€˜π‘₯))(π‘”β€˜π‘₯))))⟩} βˆͺ {⟨(Scalarβ€˜ndx), π‘ βŸ©, ⟨( ·𝑠 β€˜ndx), (𝑓 ∈ (Baseβ€˜π‘ ), 𝑔 ∈ 𝑣 ↦ (π‘₯ ∈ dom π‘Ÿ ↦ (𝑓( ·𝑠 β€˜(π‘Ÿβ€˜π‘₯))(π‘”β€˜π‘₯))))⟩, ⟨(Β·π‘–β€˜ndx), (𝑓 ∈ 𝑣, 𝑔 ∈ 𝑣 ↦ (𝑠 Ξ£g (π‘₯ ∈ dom π‘Ÿ ↦ ((π‘“β€˜π‘₯)(Β·π‘–β€˜(π‘Ÿβ€˜π‘₯))(π‘”β€˜π‘₯)))))⟩}) βˆͺ ({⟨(TopSetβ€˜ndx), (∏tβ€˜(TopOpen ∘ π‘Ÿ))⟩, ⟨(leβ€˜ndx), {βŸ¨π‘“, π‘”βŸ© ∣ ({𝑓, 𝑔} βŠ† 𝑣 ∧ βˆ€π‘₯ ∈ dom π‘Ÿ(π‘“β€˜π‘₯)(leβ€˜(π‘Ÿβ€˜π‘₯))(π‘”β€˜π‘₯))}⟩, ⟨(distβ€˜ndx), (𝑓 ∈ 𝑣, 𝑔 ∈ 𝑣 ↦ sup((ran (π‘₯ ∈ dom π‘Ÿ ↦ ((π‘“β€˜π‘₯)(distβ€˜(π‘Ÿβ€˜π‘₯))(π‘”β€˜π‘₯))) βˆͺ {0}), ℝ*, < ))⟩} βˆͺ {⟨(Hom β€˜ndx), β„ŽβŸ©, ⟨(compβ€˜ndx), (π‘Ž ∈ (𝑣 Γ— 𝑣), 𝑐 ∈ 𝑣 ↦ (𝑑 ∈ (π‘β„Ž(2nd β€˜π‘Ž)), 𝑒 ∈ (β„Žβ€˜π‘Ž) ↦ (π‘₯ ∈ dom π‘Ÿ ↦ ((π‘‘β€˜π‘₯)(⟨((1st β€˜π‘Ž)β€˜π‘₯), ((2nd β€˜π‘Ž)β€˜π‘₯)⟩(compβ€˜(π‘Ÿβ€˜π‘₯))(π‘β€˜π‘₯))(π‘’β€˜π‘₯)))))⟩})))
Colors of variables: wff set class
This definition is referenced by:  reldmprds  12723  prdsex  12724
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