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Metamath Proof Explorer

Definition df-vc

Description: Define the class of all complex vector spaces. (Contributed by NM, 3-Nov-2006) (New usage is discouraged.)

Ref Expression
Assertion df-vc CVecOLD = { ⟨ 𝑔 , 𝑠 ⟩ ∣ ( 𝑔 ∈ AbelOp ∧ 𝑠 : ( ℂ × ran 𝑔 ) ⟶ ran 𝑔 ∧ ∀ 𝑥 ∈ ran 𝑔 ( ( 1 𝑠 𝑥 ) = 𝑥 ∧ ∀ 𝑦 ∈ ℂ ( ∀ 𝑧 ∈ ran 𝑔 ( 𝑦 𝑠 ( 𝑥 𝑔 𝑧 ) ) = ( ( 𝑦 𝑠 𝑥 ) 𝑔 ( 𝑦 𝑠 𝑧 ) ) ∧ ∀ 𝑧 ∈ ℂ ( ( ( 𝑦 + 𝑧 ) 𝑠 𝑥 ) = ( ( 𝑦 𝑠 𝑥 ) 𝑔 ( 𝑧 𝑠 𝑥 ) ) ∧ ( ( 𝑦 · 𝑧 ) 𝑠 𝑥 ) = ( 𝑦 𝑠 ( 𝑧 𝑠 𝑥 ) ) ) ) ) ) }

Detailed syntax breakdown

Step Hyp Ref Expression
0 cvc ⊢ CVecOLD
1 vg ⊢ 𝑔
2 vs ⊢ 𝑠
3 1 cv ⊢ 𝑔
4 cablo ⊢ AbelOp
5 3 4 wcel ⊢ 𝑔 ∈ AbelOp
6 2 cv ⊢ 𝑠
7 cc ⊢ ℂ
8 3 crn ⊢ ran 𝑔
9 7 8 cxp ⊢ ( ℂ × ran 𝑔 )
10 9 8 6 wf ⊢ 𝑠 : ( ℂ × ran 𝑔 ) ⟶ ran 𝑔
11 vx ⊢ 𝑥
12 c1 ⊢ 1
13 11 cv ⊢ 𝑥
14 12 13 6 co ⊢ ( 1 𝑠 𝑥 )
15 14 13 wceq ⊢ ( 1 𝑠 𝑥 ) = 𝑥
16 vy ⊢ 𝑦
17 vz ⊢ 𝑧
18 16 cv ⊢ 𝑦
19 17 cv ⊢ 𝑧
20 13 19 3 co ⊢ ( 𝑥 𝑔 𝑧 )
21 18 20 6 co ⊢ ( 𝑦 𝑠 ( 𝑥 𝑔 𝑧 ) )
22 18 13 6 co ⊢ ( 𝑦 𝑠 𝑥 )
23 18 19 6 co ⊢ ( 𝑦 𝑠 𝑧 )
24 22 23 3 co ⊢ ( ( 𝑦 𝑠 𝑥 ) 𝑔 ( 𝑦 𝑠 𝑧 ) )
25 21 24 wceq ⊢ ( 𝑦 𝑠 ( 𝑥 𝑔 𝑧 ) ) = ( ( 𝑦 𝑠 𝑥 ) 𝑔 ( 𝑦 𝑠 𝑧 ) )
26 25 17 8 wral ⊢ ∀ 𝑧 ∈ ran 𝑔 ( 𝑦 𝑠 ( 𝑥 𝑔 𝑧 ) ) = ( ( 𝑦 𝑠 𝑥 ) 𝑔 ( 𝑦 𝑠 𝑧 ) )
27 caddc ⊢ +
28 18 19 27 co ⊢ ( 𝑦 + 𝑧 )
29 28 13 6 co ⊢ ( ( 𝑦 + 𝑧 ) 𝑠 𝑥 )
30 19 13 6 co ⊢ ( 𝑧 𝑠 𝑥 )
31 22 30 3 co ⊢ ( ( 𝑦 𝑠 𝑥 ) 𝑔 ( 𝑧 𝑠 𝑥 ) )
32 29 31 wceq ⊢ ( ( 𝑦 + 𝑧 ) 𝑠 𝑥 ) = ( ( 𝑦 𝑠 𝑥 ) 𝑔 ( 𝑧 𝑠 𝑥 ) )
33 cmul ⊢ ·
34 18 19 33 co ⊢ ( 𝑦 · 𝑧 )
35 34 13 6 co ⊢ ( ( 𝑦 · 𝑧 ) 𝑠 𝑥 )
36 18 30 6 co ⊢ ( 𝑦 𝑠 ( 𝑧 𝑠 𝑥 ) )
37 35 36 wceq ⊢ ( ( 𝑦 · 𝑧 ) 𝑠 𝑥 ) = ( 𝑦 𝑠 ( 𝑧 𝑠 𝑥 ) )
38 32 37 wa ⊢ ( ( ( 𝑦 + 𝑧 ) 𝑠 𝑥 ) = ( ( 𝑦 𝑠 𝑥 ) 𝑔 ( 𝑧 𝑠 𝑥 ) ) ∧ ( ( 𝑦 · 𝑧 ) 𝑠 𝑥 ) = ( 𝑦 𝑠 ( 𝑧 𝑠 𝑥 ) ) )
39 38 17 7 wral ⊢ ∀ 𝑧 ∈ ℂ ( ( ( 𝑦 + 𝑧 ) 𝑠 𝑥 ) = ( ( 𝑦 𝑠 𝑥 ) 𝑔 ( 𝑧 𝑠 𝑥 ) ) ∧ ( ( 𝑦 · 𝑧 ) 𝑠 𝑥 ) = ( 𝑦 𝑠 ( 𝑧 𝑠 𝑥 ) ) )
40 26 39 wa ⊢ ( ∀ 𝑧 ∈ ran 𝑔 ( 𝑦 𝑠 ( 𝑥 𝑔 𝑧 ) ) = ( ( 𝑦 𝑠 𝑥 ) 𝑔 ( 𝑦 𝑠 𝑧 ) ) ∧ ∀ 𝑧 ∈ ℂ ( ( ( 𝑦 + 𝑧 ) 𝑠 𝑥 ) = ( ( 𝑦 𝑠 𝑥 ) 𝑔 ( 𝑧 𝑠 𝑥 ) ) ∧ ( ( 𝑦 · 𝑧 ) 𝑠 𝑥 ) = ( 𝑦 𝑠 ( 𝑧 𝑠 𝑥 ) ) ) )
41 40 16 7 wral ⊢ ∀ 𝑦 ∈ ℂ ( ∀ 𝑧 ∈ ran 𝑔 ( 𝑦 𝑠 ( 𝑥 𝑔 𝑧 ) ) = ( ( 𝑦 𝑠 𝑥 ) 𝑔 ( 𝑦 𝑠 𝑧 ) ) ∧ ∀ 𝑧 ∈ ℂ ( ( ( 𝑦 + 𝑧 ) 𝑠 𝑥 ) = ( ( 𝑦 𝑠 𝑥 ) 𝑔 ( 𝑧 𝑠 𝑥 ) ) ∧ ( ( 𝑦 · 𝑧 ) 𝑠 𝑥 ) = ( 𝑦 𝑠 ( 𝑧 𝑠 𝑥 ) ) ) )
42 15 41 wa ⊢ ( ( 1 𝑠 𝑥 ) = 𝑥 ∧ ∀ 𝑦 ∈ ℂ ( ∀ 𝑧 ∈ ran 𝑔 ( 𝑦 𝑠 ( 𝑥 𝑔 𝑧 ) ) = ( ( 𝑦 𝑠 𝑥 ) 𝑔 ( 𝑦 𝑠 𝑧 ) ) ∧ ∀ 𝑧 ∈ ℂ ( ( ( 𝑦 + 𝑧 ) 𝑠 𝑥 ) = ( ( 𝑦 𝑠 𝑥 ) 𝑔 ( 𝑧 𝑠 𝑥 ) ) ∧ ( ( 𝑦 · 𝑧 ) 𝑠 𝑥 ) = ( 𝑦 𝑠 ( 𝑧 𝑠 𝑥 ) ) ) ) )
43 42 11 8 wral ⊢ ∀ 𝑥 ∈ ran 𝑔 ( ( 1 𝑠 𝑥 ) = 𝑥 ∧ ∀ 𝑦 ∈ ℂ ( ∀ 𝑧 ∈ ran 𝑔 ( 𝑦 𝑠 ( 𝑥 𝑔 𝑧 ) ) = ( ( 𝑦 𝑠 𝑥 ) 𝑔 ( 𝑦 𝑠 𝑧 ) ) ∧ ∀ 𝑧 ∈ ℂ ( ( ( 𝑦 + 𝑧 ) 𝑠 𝑥 ) = ( ( 𝑦 𝑠 𝑥 ) 𝑔 ( 𝑧 𝑠 𝑥 ) ) ∧ ( ( 𝑦 · 𝑧 ) 𝑠 𝑥 ) = ( 𝑦 𝑠 ( 𝑧 𝑠 𝑥 ) ) ) ) )
44 5 10 43 w3a ⊢ ( 𝑔 ∈ AbelOp ∧ 𝑠 : ( ℂ × ran 𝑔 ) ⟶ ran 𝑔 ∧ ∀ 𝑥 ∈ ran 𝑔 ( ( 1 𝑠 𝑥 ) = 𝑥 ∧ ∀ 𝑦 ∈ ℂ ( ∀ 𝑧 ∈ ran 𝑔 ( 𝑦 𝑠 ( 𝑥 𝑔 𝑧 ) ) = ( ( 𝑦 𝑠 𝑥 ) 𝑔 ( 𝑦 𝑠 𝑧 ) ) ∧ ∀ 𝑧 ∈ ℂ ( ( ( 𝑦 + 𝑧 ) 𝑠 𝑥 ) = ( ( 𝑦 𝑠 𝑥 ) 𝑔 ( 𝑧 𝑠 𝑥 ) ) ∧ ( ( 𝑦 · 𝑧 ) 𝑠 𝑥 ) = ( 𝑦 𝑠 ( 𝑧 𝑠 𝑥 ) ) ) ) ) )
45 44 1 2 copab ⊢ { ⟨ 𝑔 , 𝑠 ⟩ ∣ ( 𝑔 ∈ AbelOp ∧ 𝑠 : ( ℂ × ran 𝑔 ) ⟶ ran 𝑔 ∧ ∀ 𝑥 ∈ ran 𝑔 ( ( 1 𝑠 𝑥 ) = 𝑥 ∧ ∀ 𝑦 ∈ ℂ ( ∀ 𝑧 ∈ ran 𝑔 ( 𝑦 𝑠 ( 𝑥 𝑔 𝑧 ) ) = ( ( 𝑦 𝑠 𝑥 ) 𝑔 ( 𝑦 𝑠 𝑧 ) ) ∧ ∀ 𝑧 ∈ ℂ ( ( ( 𝑦 + 𝑧 ) 𝑠 𝑥 ) = ( ( 𝑦 𝑠 𝑥 ) 𝑔 ( 𝑧 𝑠 𝑥 ) ) ∧ ( ( 𝑦 · 𝑧 ) 𝑠 𝑥 ) = ( 𝑦 𝑠 ( 𝑧 𝑠 𝑥 ) ) ) ) ) ) }
46 0 45 wceq ⊢ CVecOLD = { ⟨ 𝑔 , 𝑠 ⟩ ∣ ( 𝑔 ∈ AbelOp ∧ 𝑠 : ( ℂ × ran 𝑔 ) ⟶ ran 𝑔 ∧ ∀ 𝑥 ∈ ran 𝑔 ( ( 1 𝑠 𝑥 ) = 𝑥 ∧ ∀ 𝑦 ∈ ℂ ( ∀ 𝑧 ∈ ran 𝑔 ( 𝑦 𝑠 ( 𝑥 𝑔 𝑧 ) ) = ( ( 𝑦 𝑠 𝑥 ) 𝑔 ( 𝑦 𝑠 𝑧 ) ) ∧ ∀ 𝑧 ∈ ℂ ( ( ( 𝑦 + 𝑧 ) 𝑠 𝑥 ) = ( ( 𝑦 𝑠 𝑥 ) 𝑔 ( 𝑧 𝑠 𝑥 ) ) ∧ ( ( 𝑦 · 𝑧 ) 𝑠 𝑥 ) = ( 𝑦 𝑠 ( 𝑧 𝑠 𝑥 ) ) ) ) ) ) }