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Definition df-xps 17456
Description: Define a binary product on structures. (Contributed by Mario Carneiro, 14-Aug-2015.) (Revised by Jim Kingdon, 25-Sep-2023.)
Assertion
Ref Expression
df-xps Γ—s = (π‘Ÿ ∈ V, 𝑠 ∈ V ↦ (β—‘(π‘₯ ∈ (Baseβ€˜π‘Ÿ), 𝑦 ∈ (Baseβ€˜π‘ ) ↦ {βŸ¨βˆ…, π‘₯⟩, ⟨1o, π‘¦βŸ©}) β€œs ((Scalarβ€˜π‘Ÿ)Xs{βŸ¨βˆ…, π‘ŸβŸ©, ⟨1o, π‘ βŸ©})))
Distinct variable group:   𝑠,π‘Ÿ,π‘₯,𝑦

Detailed syntax breakdown of Definition df-xps
StepHypRef Expression
1 cxps 17452 . 2 class Γ—s
2 vr . . 3 setvar π‘Ÿ
3 vs . . 3 setvar 𝑠
4 cvv 3475 . . 3 class V
5 vx . . . . . 6 setvar π‘₯
6 vy . . . . . 6 setvar 𝑦
72cv 1541 . . . . . . 7 class π‘Ÿ
8 cbs 17144 . . . . . . 7 class Base
97, 8cfv 6544 . . . . . 6 class (Baseβ€˜π‘Ÿ)
103cv 1541 . . . . . . 7 class 𝑠
1110, 8cfv 6544 . . . . . 6 class (Baseβ€˜π‘ )
12 c0 4323 . . . . . . . 8 class βˆ…
135cv 1541 . . . . . . . 8 class π‘₯
1412, 13cop 4635 . . . . . . 7 class βŸ¨βˆ…, π‘₯⟩
15 c1o 8459 . . . . . . . 8 class 1o
166cv 1541 . . . . . . . 8 class 𝑦
1715, 16cop 4635 . . . . . . 7 class ⟨1o, π‘¦βŸ©
1814, 17cpr 4631 . . . . . 6 class {βŸ¨βˆ…, π‘₯⟩, ⟨1o, π‘¦βŸ©}
195, 6, 9, 11, 18cmpo 7411 . . . . 5 class (π‘₯ ∈ (Baseβ€˜π‘Ÿ), 𝑦 ∈ (Baseβ€˜π‘ ) ↦ {βŸ¨βˆ…, π‘₯⟩, ⟨1o, π‘¦βŸ©})
2019ccnv 5676 . . . 4 class β—‘(π‘₯ ∈ (Baseβ€˜π‘Ÿ), 𝑦 ∈ (Baseβ€˜π‘ ) ↦ {βŸ¨βˆ…, π‘₯⟩, ⟨1o, π‘¦βŸ©})
21 csca 17200 . . . . . 6 class Scalar
227, 21cfv 6544 . . . . 5 class (Scalarβ€˜π‘Ÿ)
2312, 7cop 4635 . . . . . 6 class βŸ¨βˆ…, π‘ŸβŸ©
2415, 10cop 4635 . . . . . 6 class ⟨1o, π‘ βŸ©
2523, 24cpr 4631 . . . . 5 class {βŸ¨βˆ…, π‘ŸβŸ©, ⟨1o, π‘ βŸ©}
26 cprds 17391 . . . . 5 class Xs
2722, 25, 26co 7409 . . . 4 class ((Scalarβ€˜π‘Ÿ)Xs{βŸ¨βˆ…, π‘ŸβŸ©, ⟨1o, π‘ βŸ©})
28 cimas 17450 . . . 4 class β€œs
2920, 27, 28co 7409 . . 3 class (β—‘(π‘₯ ∈ (Baseβ€˜π‘Ÿ), 𝑦 ∈ (Baseβ€˜π‘ ) ↦ {βŸ¨βˆ…, π‘₯⟩, ⟨1o, π‘¦βŸ©}) β€œs ((Scalarβ€˜π‘Ÿ)Xs{βŸ¨βˆ…, π‘ŸβŸ©, ⟨1o, π‘ βŸ©}))
302, 3, 4, 4, 29cmpo 7411 . 2 class (π‘Ÿ ∈ V, 𝑠 ∈ V ↦ (β—‘(π‘₯ ∈ (Baseβ€˜π‘Ÿ), 𝑦 ∈ (Baseβ€˜π‘ ) ↦ {βŸ¨βˆ…, π‘₯⟩, ⟨1o, π‘¦βŸ©}) β€œs ((Scalarβ€˜π‘Ÿ)Xs{βŸ¨βˆ…, π‘ŸβŸ©, ⟨1o, π‘ βŸ©})))
311, 30wceq 1542 1 wff Γ—s = (π‘Ÿ ∈ V, 𝑠 ∈ V ↦ (β—‘(π‘₯ ∈ (Baseβ€˜π‘Ÿ), 𝑦 ∈ (Baseβ€˜π‘ ) ↦ {βŸ¨βˆ…, π‘₯⟩, ⟨1o, π‘¦βŸ©}) β€œs ((Scalarβ€˜π‘Ÿ)Xs{βŸ¨βˆ…, π‘ŸβŸ©, ⟨1o, π‘ βŸ©})))
Colors of variables: wff setvar class
This definition is referenced by:  xpsval  17516
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