Metamath Proof Explorer

Theorem isvclem

Description: Lemma for isvcOLD . (Contributed by NM, 31-May-2008) (New usage is discouraged.)

Ref Expression
Hypothesis isvclem.1 ⊢ 𝑋 = ran 𝐺
Assertion isvclem ( ( 𝐺 ∈ V ∧ 𝑆 ∈ V ) → ( ⟨ 𝐺 , 𝑆 ⟩ ∈ 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 isvclem.1 ⊢ 𝑋 = ran 𝐺
2 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 ) ) ) ) ) ) }
3 2 eleq2i ⊢ ( ⟨ 𝐺 , 𝑆 ⟩ ∈ CVecOLD ↔ ⟨ 𝐺 , 𝑆 ⟩ ∈ { ⟨ 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 ) ) ) ) ) ) } )
4 eleq1 ⊢ ( 𝑔 = 𝐺 → ( 𝑔 ∈ AbelOp ↔ 𝐺 ∈ AbelOp ) )
5 rneq ⊢ ( 𝑔 = 𝐺 → ran 𝑔 = ran 𝐺 )
6 5 1 eqtr4di ⊢ ( 𝑔 = 𝐺 → ran 𝑔 = 𝑋 )
7 xpeq2 ⊢ ( ran 𝑔 = 𝑋 → ( ℂ × ran 𝑔 ) = ( ℂ × 𝑋 ) )
8 7 feq2d ⊢ ( ran 𝑔 = 𝑋 → ( 𝑠 : ( ℂ × ran 𝑔 ) ⟶ ran 𝑔 ↔ 𝑠 : ( ℂ × 𝑋 ) ⟶ ran 𝑔 ) )
9 feq3 ⊢ ( ran 𝑔 = 𝑋 → ( 𝑠 : ( ℂ × 𝑋 ) ⟶ ran 𝑔 ↔ 𝑠 : ( ℂ × 𝑋 ) ⟶ 𝑋 ) )
10 8 9 bitrd ⊢ ( ran 𝑔 = 𝑋 → ( 𝑠 : ( ℂ × ran 𝑔 ) ⟶ ran 𝑔 ↔ 𝑠 : ( ℂ × 𝑋 ) ⟶ 𝑋 ) )
11 6 10 syl ⊢ ( 𝑔 = 𝐺 → ( 𝑠 : ( ℂ × ran 𝑔 ) ⟶ ran 𝑔 ↔ 𝑠 : ( ℂ × 𝑋 ) ⟶ 𝑋 ) )
12 oveq ⊢ ( 𝑔 = 𝐺 → ( 𝑥 𝑔 𝑧 ) = ( 𝑥 𝐺 𝑧 ) )
13 12 oveq2d ⊢ ( 𝑔 = 𝐺 → ( 𝑦 𝑠 ( 𝑥 𝑔 𝑧 ) ) = ( 𝑦 𝑠 ( 𝑥 𝐺 𝑧 ) ) )
14 oveq ⊢ ( 𝑔 = 𝐺 → ( ( 𝑦 𝑠 𝑥 ) 𝑔 ( 𝑦 𝑠 𝑧 ) ) = ( ( 𝑦 𝑠 𝑥 ) 𝐺 ( 𝑦 𝑠 𝑧 ) ) )
15 13 14 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 ) ) ) )
16 6 15 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 ) ) ) )
17 oveq ⊢ ( 𝑔 = 𝐺 → ( ( 𝑦 𝑠 𝑥 ) 𝑔 ( 𝑧 𝑠 𝑥 ) ) = ( ( 𝑦 𝑠 𝑥 ) 𝐺 ( 𝑧 𝑠 𝑥 ) ) )
18 17 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 ) ) ) )
19 18 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 ) ) ) ) )
20 19 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 ) ) ) ) )
21 16 20 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 ) ) ) ) ) )
22 21 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 ) ) ) ) ) )
23 22 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 ) ) ) ) ) ) )
24 6 23 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 ) ) ) ) ) ) )
25 4 11 24 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 ) ) ) ) ) ) ) )
26 feq1 ⊢ ( 𝑠 = 𝑆 → ( 𝑠 : ( ℂ × 𝑋 ) ⟶ 𝑋 ↔ 𝑆 : ( ℂ × 𝑋 ) ⟶ 𝑋 ) )
27 oveq ⊢ ( 𝑠 = 𝑆 → ( 1 𝑠 𝑥 ) = ( 1 𝑆 𝑥 ) )
28 27 eqeq1d ⊢ ( 𝑠 = 𝑆 → ( ( 1 𝑠 𝑥 ) = 𝑥 ↔ ( 1 𝑆 𝑥 ) = 𝑥 ) )
29 oveq ⊢ ( 𝑠 = 𝑆 → ( 𝑦 𝑠 ( 𝑥 𝐺 𝑧 ) ) = ( 𝑦 𝑆 ( 𝑥 𝐺 𝑧 ) ) )
30 oveq ⊢ ( 𝑠 = 𝑆 → ( 𝑦 𝑠 𝑥 ) = ( 𝑦 𝑆 𝑥 ) )
31 oveq ⊢ ( 𝑠 = 𝑆 → ( 𝑦 𝑠 𝑧 ) = ( 𝑦 𝑆 𝑧 ) )
32 30 31 oveq12d ⊢ ( 𝑠 = 𝑆 → ( ( 𝑦 𝑠 𝑥 ) 𝐺 ( 𝑦 𝑠 𝑧 ) ) = ( ( 𝑦 𝑆 𝑥 ) 𝐺 ( 𝑦 𝑆 𝑧 ) ) )
33 29 32 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 ) ) ) )
34 33 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 ) ) ) )
35 oveq ⊢ ( 𝑠 = 𝑆 → ( ( 𝑦 + 𝑧 ) 𝑠 𝑥 ) = ( ( 𝑦 + 𝑧 ) 𝑆 𝑥 ) )
36 oveq ⊢ ( 𝑠 = 𝑆 → ( 𝑧 𝑠 𝑥 ) = ( 𝑧 𝑆 𝑥 ) )
37 30 36 oveq12d ⊢ ( 𝑠 = 𝑆 → ( ( 𝑦 𝑠 𝑥 ) 𝐺 ( 𝑧 𝑠 𝑥 ) ) = ( ( 𝑦 𝑆 𝑥 ) 𝐺 ( 𝑧 𝑆 𝑥 ) ) )
38 35 37 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 ) ) ) )
39 oveq ⊢ ( 𝑠 = 𝑆 → ( ( 𝑦 · 𝑧 ) 𝑠 𝑥 ) = ( ( 𝑦 · 𝑧 ) 𝑆 𝑥 ) )
40 oveq ⊢ ( 𝑠 = 𝑆 → ( 𝑦 𝑠 ( 𝑧 𝑠 𝑥 ) ) = ( 𝑦 𝑆 ( 𝑧 𝑠 𝑥 ) ) )
41 36 oveq2d ⊢ ( 𝑠 = 𝑆 → ( 𝑦 𝑆 ( 𝑧 𝑠 𝑥 ) ) = ( 𝑦 𝑆 ( 𝑧 𝑆 𝑥 ) ) )
42 40 41 eqtrd ⊢ ( 𝑠 = 𝑆 → ( 𝑦 𝑠 ( 𝑧 𝑠 𝑥 ) ) = ( 𝑦 𝑆 ( 𝑧 𝑆 𝑥 ) ) )
43 39 42 eqeq12d ⊢ ( 𝑠 = 𝑆 → ( ( ( 𝑦 · 𝑧 ) 𝑠 𝑥 ) = ( 𝑦 𝑠 ( 𝑧 𝑠 𝑥 ) ) ↔ ( ( 𝑦 · 𝑧 ) 𝑆 𝑥 ) = ( 𝑦 𝑆 ( 𝑧 𝑆 𝑥 ) ) ) )
44 38 43 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 ) ) ) ) )
45 44 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 ) ) ) ) )
46 34 45 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 ) ) ) ) ) )
47 46 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 ) ) ) ) ) )
48 28 47 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 ) ) ) ) ) ) )
49 48 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 ) ) ) ) ) ) )
50 26 49 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 ) ) ) ) ) ) ) )
51 25 50 opelopabg ⊢ ( ( 𝐺 ∈ V ∧ 𝑆 ∈ V ) → ( ⟨ 𝐺 , 𝑆 ⟩ ∈ { ⟨ 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 ) ) ) ) ) ) ) )
52 3 51 bitrid ⊢ ( ( 𝐺 ∈ V ∧ 𝑆 ∈ V ) → ( ⟨ 𝐺 , 𝑆 ⟩ ∈ 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 ) ) ) ) ) ) ) )