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Metamath Proof Explorer

Theorem vciOLD

Description: Obsolete version of cvsi . The properties of a complex vector space, which is an Abelian group (i.e. the vectors, with the operation of vector addition) accompanied by a scalar multiplication operation on the field of complex numbers. The variable W was chosen because _V is already used for the universal class. (Contributed by NM, 3-Nov-2006) (New usage is discouraged.) (Proof modification is discouraged.)

Ref Expression
Hypotheses vciOLD.1 ⊢ 𝐺 = ( 1st ‘ 𝑊 )
vciOLD.2 ⊢ 𝑆 = ( 2nd ‘ 𝑊 )
vciOLD.3 ⊢ 𝑋 = ran 𝐺
Assertion vciOLD ( 𝑊 ∈ CVecOLD → ( 𝐺 ∈ AbelOp ∧ 𝑆 : ( ℂ × 𝑋 ) ⟶ 𝑋 ∧ ∀ 𝑥 ∈ 𝑋 ( ( 1 𝑆 𝑥 ) = 𝑥 ∧ ∀ 𝑦 ∈ ℂ ( ∀ 𝑧 ∈ 𝑋 ( 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 ) ) ) ) ) ) )

Proof

Step Hyp Ref Expression
1 vciOLD.1 ⊢ 𝐺 = ( 1st ‘ 𝑊 )
2 vciOLD.2 ⊢ 𝑆 = ( 2nd ‘ 𝑊 )
3 vciOLD.3 ⊢ 𝑋 = ran 𝐺
4 1 eqeq2i ⊢ ( 𝑔 = 𝐺 ↔ 𝑔 = ( 1st ‘ 𝑊 ) )
5 eleq1 ⊢ ( 𝑔 = 𝐺 → ( 𝑔 ∈ AbelOp ↔ 𝐺 ∈ AbelOp ) )
6 rneq ⊢ ( 𝑔 = 𝐺 → ran 𝑔 = ran 𝐺 )
7 6 3 eqtr4di ⊢ ( 𝑔 = 𝐺 → ran 𝑔 = 𝑋 )
8 xpeq2 ⊢ ( ran 𝑔 = 𝑋 → ( ℂ × ran 𝑔 ) = ( ℂ × 𝑋 ) )
9 8 feq2d ⊢ ( ran 𝑔 = 𝑋 → ( 𝑠 : ( ℂ × ran 𝑔 ) ⟶ ran 𝑔 ↔ 𝑠 : ( ℂ × 𝑋 ) ⟶ ran 𝑔 ) )
10 feq3 ⊢ ( ran 𝑔 = 𝑋 → ( 𝑠 : ( ℂ × 𝑋 ) ⟶ ran 𝑔 ↔ 𝑠 : ( ℂ × 𝑋 ) ⟶ 𝑋 ) )
11 9 10 bitrd ⊢ ( ran 𝑔 = 𝑋 → ( 𝑠 : ( ℂ × ran 𝑔 ) ⟶ ran 𝑔 ↔ 𝑠 : ( ℂ × 𝑋 ) ⟶ 𝑋 ) )
12 7 11 syl ⊢ ( 𝑔 = 𝐺 → ( 𝑠 : ( ℂ × ran 𝑔 ) ⟶ ran 𝑔 ↔ 𝑠 : ( ℂ × 𝑋 ) ⟶ 𝑋 ) )
13 oveq ⊢ ( 𝑔 = 𝐺 → ( 𝑥 𝑔 𝑧 ) = ( 𝑥 𝐺 𝑧 ) )
14 13 oveq2d ⊢ ( 𝑔 = 𝐺 → ( 𝑦 𝑠 ( 𝑥 𝑔 𝑧 ) ) = ( 𝑦 𝑠 ( 𝑥 𝐺 𝑧 ) ) )
15 oveq ⊢ ( 𝑔 = 𝐺 → ( ( 𝑦 𝑠 𝑥 ) 𝑔 ( 𝑦 𝑠 𝑧 ) ) = ( ( 𝑦 𝑠 𝑥 ) 𝐺 ( 𝑦 𝑠 𝑧 ) ) )
16 14 15 eqeq12d ⊢ ( 𝑔 = 𝐺 → ( ( 𝑦 𝑠 ( 𝑥 𝑔 𝑧 ) ) = ( ( 𝑦 𝑠 𝑥 ) 𝑔 ( 𝑦 𝑠 𝑧 ) ) ↔ ( 𝑦 𝑠 ( 𝑥 𝐺 𝑧 ) ) = ( ( 𝑦 𝑠 𝑥 ) 𝐺 ( 𝑦 𝑠 𝑧 ) ) ) )
17 7 16 raleqbidv ⊢ ( 𝑔 = 𝐺 → ( ∀ 𝑧 ∈ ran 𝑔 ( 𝑦 𝑠 ( 𝑥 𝑔 𝑧 ) ) = ( ( 𝑦 𝑠 𝑥 ) 𝑔 ( 𝑦 𝑠 𝑧 ) ) ↔ ∀ 𝑧 ∈ 𝑋 ( 𝑦 𝑠 ( 𝑥 𝐺 𝑧 ) ) = ( ( 𝑦 𝑠 𝑥 ) 𝐺 ( 𝑦 𝑠 𝑧 ) ) ) )
18 oveq ⊢ ( 𝑔 = 𝐺 → ( ( 𝑦 𝑠 𝑥 ) 𝑔 ( 𝑧 𝑠 𝑥 ) ) = ( ( 𝑦 𝑠 𝑥 ) 𝐺 ( 𝑧 𝑠 𝑥 ) ) )
19 18 eqeq2d ⊢ ( 𝑔 = 𝐺 → ( ( ( 𝑦 + 𝑧 ) 𝑠 𝑥 ) = ( ( 𝑦 𝑠 𝑥 ) 𝑔 ( 𝑧 𝑠 𝑥 ) ) ↔ ( ( 𝑦 + 𝑧 ) 𝑠 𝑥 ) = ( ( 𝑦 𝑠 𝑥 ) 𝐺 ( 𝑧 𝑠 𝑥 ) ) ) )
20 19 anbi1d ⊢ ( 𝑔 = 𝐺 → ( ( ( ( 𝑦 + 𝑧 ) 𝑠 𝑥 ) = ( ( 𝑦 𝑠 𝑥 ) 𝑔 ( 𝑧 𝑠 𝑥 ) ) ∧ ( ( 𝑦 · 𝑧 ) 𝑠 𝑥 ) = ( 𝑦 𝑠 ( 𝑧 𝑠 𝑥 ) ) ) ↔ ( ( ( 𝑦 + 𝑧 ) 𝑠 𝑥 ) = ( ( 𝑦 𝑠 𝑥 ) 𝐺 ( 𝑧 𝑠 𝑥 ) ) ∧ ( ( 𝑦 · 𝑧 ) 𝑠 𝑥 ) = ( 𝑦 𝑠 ( 𝑧 𝑠 𝑥 ) ) ) ) )
21 20 ralbidv ⊢ ( 𝑔 = 𝐺 → ( ∀ 𝑧 ∈ ℂ ( ( ( 𝑦 + 𝑧 ) 𝑠 𝑥 ) = ( ( 𝑦 𝑠 𝑥 ) 𝑔 ( 𝑧 𝑠 𝑥 ) ) ∧ ( ( 𝑦 · 𝑧 ) 𝑠 𝑥 ) = ( 𝑦 𝑠 ( 𝑧 𝑠 𝑥 ) ) ) ↔ ∀ 𝑧 ∈ ℂ ( ( ( 𝑦 + 𝑧 ) 𝑠 𝑥 ) = ( ( 𝑦 𝑠 𝑥 ) 𝐺 ( 𝑧 𝑠 𝑥 ) ) ∧ ( ( 𝑦 · 𝑧 ) 𝑠 𝑥 ) = ( 𝑦 𝑠 ( 𝑧 𝑠 𝑥 ) ) ) ) )
22 17 21 anbi12d ⊢ ( 𝑔 = 𝐺 → ( ( ∀ 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 ) ) ) ) ↔ ( ∀ 𝑧 ∈ 𝑋 ( 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 ) ) ) ) ) )
23 22 ralbidv ⊢ ( 𝑔 = 𝐺 → ( ∀ 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 ) ) ) ) ↔ ∀ 𝑦 ∈ ℂ ( ∀ 𝑧 ∈ 𝑋 ( 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 ) ) ) ) ) )
24 23 anbi2d ⊢ ( 𝑔 = 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 ) ) ) ) ) ↔ ( ( 1 𝑠 𝑥 ) = 𝑥 ∧ ∀ 𝑦 ∈ ℂ ( ∀ 𝑧 ∈ 𝑋 ( 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 ) ) ) ) ) ) )
25 7 24 raleqbidv ⊢ ( 𝑔 = 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 ) ) ) ) ) ↔ ∀ 𝑥 ∈ 𝑋 ( ( 1 𝑠 𝑥 ) = 𝑥 ∧ ∀ 𝑦 ∈ ℂ ( ∀ 𝑧 ∈ 𝑋 ( 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 ) ) ) ) ) ) )
26 5 12 25 3anbi123d ⊢ ( 𝑔 = 𝐺 → ( ( 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 ) ) ) ) ) ) ↔ ( 𝐺 ∈ AbelOp ∧ 𝑠 : ( ℂ × 𝑋 ) ⟶ 𝑋 ∧ ∀ 𝑥 ∈ 𝑋 ( ( 1 𝑠 𝑥 ) = 𝑥 ∧ ∀ 𝑦 ∈ ℂ ( ∀ 𝑧 ∈ 𝑋 ( 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 ) ) ) ) ) ) ) )
27 4 26 sylbir ⊢ ( 𝑔 = ( 1st ‘ 𝑊 ) → ( ( 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 ) ) ) ) ) ) ↔ ( 𝐺 ∈ AbelOp ∧ 𝑠 : ( ℂ × 𝑋 ) ⟶ 𝑋 ∧ ∀ 𝑥 ∈ 𝑋 ( ( 1 𝑠 𝑥 ) = 𝑥 ∧ ∀ 𝑦 ∈ ℂ ( ∀ 𝑧 ∈ 𝑋 ( 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 ) ) ) ) ) ) ) )
28 2 eqeq2i ⊢ ( 𝑠 = 𝑆 ↔ 𝑠 = ( 2nd ‘ 𝑊 ) )
29 feq1 ⊢ ( 𝑠 = 𝑆 → ( 𝑠 : ( ℂ × 𝑋 ) ⟶ 𝑋 ↔ 𝑆 : ( ℂ × 𝑋 ) ⟶ 𝑋 ) )
30 oveq ⊢ ( 𝑠 = 𝑆 → ( 1 𝑠 𝑥 ) = ( 1 𝑆 𝑥 ) )
31 30 eqeq1d ⊢ ( 𝑠 = 𝑆 → ( ( 1 𝑠 𝑥 ) = 𝑥 ↔ ( 1 𝑆 𝑥 ) = 𝑥 ) )
32 oveq ⊢ ( 𝑠 = 𝑆 → ( 𝑦 𝑠 ( 𝑥 𝐺 𝑧 ) ) = ( 𝑦 𝑆 ( 𝑥 𝐺 𝑧 ) ) )
33 oveq ⊢ ( 𝑠 = 𝑆 → ( 𝑦 𝑠 𝑥 ) = ( 𝑦 𝑆 𝑥 ) )
34 oveq ⊢ ( 𝑠 = 𝑆 → ( 𝑦 𝑠 𝑧 ) = ( 𝑦 𝑆 𝑧 ) )
35 33 34 oveq12d ⊢ ( 𝑠 = 𝑆 → ( ( 𝑦 𝑠 𝑥 ) 𝐺 ( 𝑦 𝑠 𝑧 ) ) = ( ( 𝑦 𝑆 𝑥 ) 𝐺 ( 𝑦 𝑆 𝑧 ) ) )
36 32 35 eqeq12d ⊢ ( 𝑠 = 𝑆 → ( ( 𝑦 𝑠 ( 𝑥 𝐺 𝑧 ) ) = ( ( 𝑦 𝑠 𝑥 ) 𝐺 ( 𝑦 𝑠 𝑧 ) ) ↔ ( 𝑦 𝑆 ( 𝑥 𝐺 𝑧 ) ) = ( ( 𝑦 𝑆 𝑥 ) 𝐺 ( 𝑦 𝑆 𝑧 ) ) ) )
37 36 ralbidv ⊢ ( 𝑠 = 𝑆 → ( ∀ 𝑧 ∈ 𝑋 ( 𝑦 𝑠 ( 𝑥 𝐺 𝑧 ) ) = ( ( 𝑦 𝑠 𝑥 ) 𝐺 ( 𝑦 𝑠 𝑧 ) ) ↔ ∀ 𝑧 ∈ 𝑋 ( 𝑦 𝑆 ( 𝑥 𝐺 𝑧 ) ) = ( ( 𝑦 𝑆 𝑥 ) 𝐺 ( 𝑦 𝑆 𝑧 ) ) ) )
38 oveq ⊢ ( 𝑠 = 𝑆 → ( ( 𝑦 + 𝑧 ) 𝑠 𝑥 ) = ( ( 𝑦 + 𝑧 ) 𝑆 𝑥 ) )
39 oveq ⊢ ( 𝑠 = 𝑆 → ( 𝑧 𝑠 𝑥 ) = ( 𝑧 𝑆 𝑥 ) )
40 33 39 oveq12d ⊢ ( 𝑠 = 𝑆 → ( ( 𝑦 𝑠 𝑥 ) 𝐺 ( 𝑧 𝑠 𝑥 ) ) = ( ( 𝑦 𝑆 𝑥 ) 𝐺 ( 𝑧 𝑆 𝑥 ) ) )
41 38 40 eqeq12d ⊢ ( 𝑠 = 𝑆 → ( ( ( 𝑦 + 𝑧 ) 𝑠 𝑥 ) = ( ( 𝑦 𝑠 𝑥 ) 𝐺 ( 𝑧 𝑠 𝑥 ) ) ↔ ( ( 𝑦 + 𝑧 ) 𝑆 𝑥 ) = ( ( 𝑦 𝑆 𝑥 ) 𝐺 ( 𝑧 𝑆 𝑥 ) ) ) )
42 oveq ⊢ ( 𝑠 = 𝑆 → ( ( 𝑦 · 𝑧 ) 𝑠 𝑥 ) = ( ( 𝑦 · 𝑧 ) 𝑆 𝑥 ) )
43 39 oveq2d ⊢ ( 𝑠 = 𝑆 → ( 𝑦 𝑠 ( 𝑧 𝑠 𝑥 ) ) = ( 𝑦 𝑠 ( 𝑧 𝑆 𝑥 ) ) )
44 oveq ⊢ ( 𝑠 = 𝑆 → ( 𝑦 𝑠 ( 𝑧 𝑆 𝑥 ) ) = ( 𝑦 𝑆 ( 𝑧 𝑆 𝑥 ) ) )
45 43 44 eqtrd ⊢ ( 𝑠 = 𝑆 → ( 𝑦 𝑠 ( 𝑧 𝑠 𝑥 ) ) = ( 𝑦 𝑆 ( 𝑧 𝑆 𝑥 ) ) )
46 42 45 eqeq12d ⊢ ( 𝑠 = 𝑆 → ( ( ( 𝑦 · 𝑧 ) 𝑠 𝑥 ) = ( 𝑦 𝑠 ( 𝑧 𝑠 𝑥 ) ) ↔ ( ( 𝑦 · 𝑧 ) 𝑆 𝑥 ) = ( 𝑦 𝑆 ( 𝑧 𝑆 𝑥 ) ) ) )
47 41 46 anbi12d ⊢ ( 𝑠 = 𝑆 → ( ( ( ( 𝑦 + 𝑧 ) 𝑠 𝑥 ) = ( ( 𝑦 𝑠 𝑥 ) 𝐺 ( 𝑧 𝑠 𝑥 ) ) ∧ ( ( 𝑦 · 𝑧 ) 𝑠 𝑥 ) = ( 𝑦 𝑠 ( 𝑧 𝑠 𝑥 ) ) ) ↔ ( ( ( 𝑦 + 𝑧 ) 𝑆 𝑥 ) = ( ( 𝑦 𝑆 𝑥 ) 𝐺 ( 𝑧 𝑆 𝑥 ) ) ∧ ( ( 𝑦 · 𝑧 ) 𝑆 𝑥 ) = ( 𝑦 𝑆 ( 𝑧 𝑆 𝑥 ) ) ) ) )
48 47 ralbidv ⊢ ( 𝑠 = 𝑆 → ( ∀ 𝑧 ∈ ℂ ( ( ( 𝑦 + 𝑧 ) 𝑠 𝑥 ) = ( ( 𝑦 𝑠 𝑥 ) 𝐺 ( 𝑧 𝑠 𝑥 ) ) ∧ ( ( 𝑦 · 𝑧 ) 𝑠 𝑥 ) = ( 𝑦 𝑠 ( 𝑧 𝑠 𝑥 ) ) ) ↔ ∀ 𝑧 ∈ ℂ ( ( ( 𝑦 + 𝑧 ) 𝑆 𝑥 ) = ( ( 𝑦 𝑆 𝑥 ) 𝐺 ( 𝑧 𝑆 𝑥 ) ) ∧ ( ( 𝑦 · 𝑧 ) 𝑆 𝑥 ) = ( 𝑦 𝑆 ( 𝑧 𝑆 𝑥 ) ) ) ) )
49 37 48 anbi12d ⊢ ( 𝑠 = 𝑆 → ( ( ∀ 𝑧 ∈ 𝑋 ( 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 ) ) ) ) ↔ ( ∀ 𝑧 ∈ 𝑋 ( 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 ) ) ) ) ) )
50 49 ralbidv ⊢ ( 𝑠 = 𝑆 → ( ∀ 𝑦 ∈ ℂ ( ∀ 𝑧 ∈ 𝑋 ( 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 ) ) ) ) ↔ ∀ 𝑦 ∈ ℂ ( ∀ 𝑧 ∈ 𝑋 ( 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 ) ) ) ) ) )
51 31 50 anbi12d ⊢ ( 𝑠 = 𝑆 → ( ( ( 1 𝑠 𝑥 ) = 𝑥 ∧ ∀ 𝑦 ∈ ℂ ( ∀ 𝑧 ∈ 𝑋 ( 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 ) ) ) ) ) ↔ ( ( 1 𝑆 𝑥 ) = 𝑥 ∧ ∀ 𝑦 ∈ ℂ ( ∀ 𝑧 ∈ 𝑋 ( 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 ) ) ) ) ) ) )
52 51 ralbidv ⊢ ( 𝑠 = 𝑆 → ( ∀ 𝑥 ∈ 𝑋 ( ( 1 𝑠 𝑥 ) = 𝑥 ∧ ∀ 𝑦 ∈ ℂ ( ∀ 𝑧 ∈ 𝑋 ( 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 ) ) ) ) ) ↔ ∀ 𝑥 ∈ 𝑋 ( ( 1 𝑆 𝑥 ) = 𝑥 ∧ ∀ 𝑦 ∈ ℂ ( ∀ 𝑧 ∈ 𝑋 ( 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 ) ) ) ) ) ) )
53 29 52 3anbi23d ⊢ ( 𝑠 = 𝑆 → ( ( 𝐺 ∈ AbelOp ∧ 𝑠 : ( ℂ × 𝑋 ) ⟶ 𝑋 ∧ ∀ 𝑥 ∈ 𝑋 ( ( 1 𝑠 𝑥 ) = 𝑥 ∧ ∀ 𝑦 ∈ ℂ ( ∀ 𝑧 ∈ 𝑋 ( 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 ) ) ) ) ) ) ↔ ( 𝐺 ∈ AbelOp ∧ 𝑆 : ( ℂ × 𝑋 ) ⟶ 𝑋 ∧ ∀ 𝑥 ∈ 𝑋 ( ( 1 𝑆 𝑥 ) = 𝑥 ∧ ∀ 𝑦 ∈ ℂ ( ∀ 𝑧 ∈ 𝑋 ( 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 ) ) ) ) ) ) ) )
54 28 53 sylbir ⊢ ( 𝑠 = ( 2nd ‘ 𝑊 ) → ( ( 𝐺 ∈ AbelOp ∧ 𝑠 : ( ℂ × 𝑋 ) ⟶ 𝑋 ∧ ∀ 𝑥 ∈ 𝑋 ( ( 1 𝑠 𝑥 ) = 𝑥 ∧ ∀ 𝑦 ∈ ℂ ( ∀ 𝑧 ∈ 𝑋 ( 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 ) ) ) ) ) ) ↔ ( 𝐺 ∈ AbelOp ∧ 𝑆 : ( ℂ × 𝑋 ) ⟶ 𝑋 ∧ ∀ 𝑥 ∈ 𝑋 ( ( 1 𝑆 𝑥 ) = 𝑥 ∧ ∀ 𝑦 ∈ ℂ ( ∀ 𝑧 ∈ 𝑋 ( 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 ) ) ) ) ) ) ) )
55 27 54 elopabi ⊢ ( 𝑊 ∈ { ⟨ 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 ) ) ) ) ) ) } → ( 𝐺 ∈ AbelOp ∧ 𝑆 : ( ℂ × 𝑋 ) ⟶ 𝑋 ∧ ∀ 𝑥 ∈ 𝑋 ( ( 1 𝑆 𝑥 ) = 𝑥 ∧ ∀ 𝑦 ∈ ℂ ( ∀ 𝑧 ∈ 𝑋 ( 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 ) ) ) ) ) ) )
56 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 ) ) ) ) ) ) }
57 55 56 eleq2s ⊢ ( 𝑊 ∈ CVecOLD → ( 𝐺 ∈ AbelOp ∧ 𝑆 : ( ℂ × 𝑋 ) ⟶ 𝑋 ∧ ∀ 𝑥 ∈ 𝑋 ( ( 1 𝑆 𝑥 ) = 𝑥 ∧ ∀ 𝑦 ∈ ℂ ( ∀ 𝑧 ∈ 𝑋 ( 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 ) ) ) ) ) ) )