Metamath Proof Explorer


Definition df-mvr

Description: Define the generating elements of the power series algebra. (Contributed by Mario Carneiro, 7-Jan-2015)

Ref Expression
Assertion df-mvr mVar = ( 𝑖 ∈ V , 𝑟 ∈ V ↦ ( 𝑥𝑖 ↦ ( 𝑓 ∈ { ∈ ( ℕ0m 𝑖 ) ∣ ( “ ℕ ) ∈ Fin } ↦ if ( 𝑓 = ( 𝑦𝑖 ↦ if ( 𝑦 = 𝑥 , 1 , 0 ) ) , ( 1r𝑟 ) , ( 0g𝑟 ) ) ) ) )

Detailed syntax breakdown

Step Hyp Ref Expression
0 cmvr mVar
1 vi 𝑖
2 cvv V
3 vr 𝑟
4 vx 𝑥
5 1 cv 𝑖
6 vf 𝑓
7 vh
8 cn0 0
9 cmap m
10 8 5 9 co ( ℕ0m 𝑖 )
11 7 cv
12 11 ccnv
13 cn
14 12 13 cima ( “ ℕ )
15 cfn Fin
16 14 15 wcel ( “ ℕ ) ∈ Fin
17 16 7 10 crab { ∈ ( ℕ0m 𝑖 ) ∣ ( “ ℕ ) ∈ Fin }
18 6 cv 𝑓
19 vy 𝑦
20 19 cv 𝑦
21 4 cv 𝑥
22 20 21 wceq 𝑦 = 𝑥
23 c1 1
24 cc0 0
25 22 23 24 cif if ( 𝑦 = 𝑥 , 1 , 0 )
26 19 5 25 cmpt ( 𝑦𝑖 ↦ if ( 𝑦 = 𝑥 , 1 , 0 ) )
27 18 26 wceq 𝑓 = ( 𝑦𝑖 ↦ if ( 𝑦 = 𝑥 , 1 , 0 ) )
28 cur 1r
29 3 cv 𝑟
30 29 28 cfv ( 1r𝑟 )
31 c0g 0g
32 29 31 cfv ( 0g𝑟 )
33 27 30 32 cif if ( 𝑓 = ( 𝑦𝑖 ↦ if ( 𝑦 = 𝑥 , 1 , 0 ) ) , ( 1r𝑟 ) , ( 0g𝑟 ) )
34 6 17 33 cmpt ( 𝑓 ∈ { ∈ ( ℕ0m 𝑖 ) ∣ ( “ ℕ ) ∈ Fin } ↦ if ( 𝑓 = ( 𝑦𝑖 ↦ if ( 𝑦 = 𝑥 , 1 , 0 ) ) , ( 1r𝑟 ) , ( 0g𝑟 ) ) )
35 4 5 34 cmpt ( 𝑥𝑖 ↦ ( 𝑓 ∈ { ∈ ( ℕ0m 𝑖 ) ∣ ( “ ℕ ) ∈ Fin } ↦ if ( 𝑓 = ( 𝑦𝑖 ↦ if ( 𝑦 = 𝑥 , 1 , 0 ) ) , ( 1r𝑟 ) , ( 0g𝑟 ) ) ) )
36 1 3 2 2 35 cmpo ( 𝑖 ∈ V , 𝑟 ∈ V ↦ ( 𝑥𝑖 ↦ ( 𝑓 ∈ { ∈ ( ℕ0m 𝑖 ) ∣ ( “ ℕ ) ∈ Fin } ↦ if ( 𝑓 = ( 𝑦𝑖 ↦ if ( 𝑦 = 𝑥 , 1 , 0 ) ) , ( 1r𝑟 ) , ( 0g𝑟 ) ) ) ) )
37 0 36 wceq mVar = ( 𝑖 ∈ V , 𝑟 ∈ V ↦ ( 𝑥𝑖 ↦ ( 𝑓 ∈ { ∈ ( ℕ0m 𝑖 ) ∣ ( “ ℕ ) ∈ Fin } ↦ if ( 𝑓 = ( 𝑦𝑖 ↦ if ( 𝑦 = 𝑥 , 1 , 0 ) ) , ( 1r𝑟 ) , ( 0g𝑟 ) ) ) ) )