Metamath Proof Explorer


Theorem islfld

Description: Properties that determine a linear functional. TODO: use this in place of islfl when it shortens the proof. (Contributed by NM, 19-Oct-2014)

Ref Expression
Hypotheses islfld.v ( 𝜑𝑉 = ( Base ‘ 𝑊 ) )
islfld.a ( 𝜑+ = ( +g𝑊 ) )
islfld.d ( 𝜑𝐷 = ( Scalar ‘ 𝑊 ) )
islfld.s ( 𝜑· = ( ·𝑠𝑊 ) )
islfld.k ( 𝜑𝐾 = ( Base ‘ 𝐷 ) )
islfld.p ( 𝜑 = ( +g𝐷 ) )
islfld.t ( 𝜑× = ( .r𝐷 ) )
islfld.f ( 𝜑𝐹 = ( LFnl ‘ 𝑊 ) )
islfld.u ( 𝜑𝐺 : 𝑉𝐾 )
islfld.l ( ( 𝜑 ∧ ( 𝑟𝐾𝑥𝑉𝑦𝑉 ) ) → ( 𝐺 ‘ ( ( 𝑟 · 𝑥 ) + 𝑦 ) ) = ( ( 𝑟 × ( 𝐺𝑥 ) ) ( 𝐺𝑦 ) ) )
islfld.w ( 𝜑𝑊𝑋 )
Assertion islfld ( 𝜑𝐺𝐹 )

Proof

Step Hyp Ref Expression
1 islfld.v ( 𝜑𝑉 = ( Base ‘ 𝑊 ) )
2 islfld.a ( 𝜑+ = ( +g𝑊 ) )
3 islfld.d ( 𝜑𝐷 = ( Scalar ‘ 𝑊 ) )
4 islfld.s ( 𝜑· = ( ·𝑠𝑊 ) )
5 islfld.k ( 𝜑𝐾 = ( Base ‘ 𝐷 ) )
6 islfld.p ( 𝜑 = ( +g𝐷 ) )
7 islfld.t ( 𝜑× = ( .r𝐷 ) )
8 islfld.f ( 𝜑𝐹 = ( LFnl ‘ 𝑊 ) )
9 islfld.u ( 𝜑𝐺 : 𝑉𝐾 )
10 islfld.l ( ( 𝜑 ∧ ( 𝑟𝐾𝑥𝑉𝑦𝑉 ) ) → ( 𝐺 ‘ ( ( 𝑟 · 𝑥 ) + 𝑦 ) ) = ( ( 𝑟 × ( 𝐺𝑥 ) ) ( 𝐺𝑦 ) ) )
11 islfld.w ( 𝜑𝑊𝑋 )
12 3 fveq2d ( 𝜑 → ( Base ‘ 𝐷 ) = ( Base ‘ ( Scalar ‘ 𝑊 ) ) )
13 5 12 eqtrd ( 𝜑𝐾 = ( Base ‘ ( Scalar ‘ 𝑊 ) ) )
14 1 13 feq23d ( 𝜑 → ( 𝐺 : 𝑉𝐾𝐺 : ( Base ‘ 𝑊 ) ⟶ ( Base ‘ ( Scalar ‘ 𝑊 ) ) ) )
15 9 14 mpbid ( 𝜑𝐺 : ( Base ‘ 𝑊 ) ⟶ ( Base ‘ ( Scalar ‘ 𝑊 ) ) )
16 10 ralrimivvva ( 𝜑 → ∀ 𝑟𝐾𝑥𝑉𝑦𝑉 ( 𝐺 ‘ ( ( 𝑟 · 𝑥 ) + 𝑦 ) ) = ( ( 𝑟 × ( 𝐺𝑥 ) ) ( 𝐺𝑦 ) ) )
17 4 oveqd ( 𝜑 → ( 𝑟 · 𝑥 ) = ( 𝑟 ( ·𝑠𝑊 ) 𝑥 ) )
18 eqidd ( 𝜑𝑦 = 𝑦 )
19 2 17 18 oveq123d ( 𝜑 → ( ( 𝑟 · 𝑥 ) + 𝑦 ) = ( ( 𝑟 ( ·𝑠𝑊 ) 𝑥 ) ( +g𝑊 ) 𝑦 ) )
20 19 fveq2d ( 𝜑 → ( 𝐺 ‘ ( ( 𝑟 · 𝑥 ) + 𝑦 ) ) = ( 𝐺 ‘ ( ( 𝑟 ( ·𝑠𝑊 ) 𝑥 ) ( +g𝑊 ) 𝑦 ) ) )
21 3 fveq2d ( 𝜑 → ( +g𝐷 ) = ( +g ‘ ( Scalar ‘ 𝑊 ) ) )
22 6 21 eqtrd ( 𝜑 = ( +g ‘ ( Scalar ‘ 𝑊 ) ) )
23 3 fveq2d ( 𝜑 → ( .r𝐷 ) = ( .r ‘ ( Scalar ‘ 𝑊 ) ) )
24 7 23 eqtrd ( 𝜑× = ( .r ‘ ( Scalar ‘ 𝑊 ) ) )
25 24 oveqd ( 𝜑 → ( 𝑟 × ( 𝐺𝑥 ) ) = ( 𝑟 ( .r ‘ ( Scalar ‘ 𝑊 ) ) ( 𝐺𝑥 ) ) )
26 eqidd ( 𝜑 → ( 𝐺𝑦 ) = ( 𝐺𝑦 ) )
27 22 25 26 oveq123d ( 𝜑 → ( ( 𝑟 × ( 𝐺𝑥 ) ) ( 𝐺𝑦 ) ) = ( ( 𝑟 ( .r ‘ ( Scalar ‘ 𝑊 ) ) ( 𝐺𝑥 ) ) ( +g ‘ ( Scalar ‘ 𝑊 ) ) ( 𝐺𝑦 ) ) )
28 20 27 eqeq12d ( 𝜑 → ( ( 𝐺 ‘ ( ( 𝑟 · 𝑥 ) + 𝑦 ) ) = ( ( 𝑟 × ( 𝐺𝑥 ) ) ( 𝐺𝑦 ) ) ↔ ( 𝐺 ‘ ( ( 𝑟 ( ·𝑠𝑊 ) 𝑥 ) ( +g𝑊 ) 𝑦 ) ) = ( ( 𝑟 ( .r ‘ ( Scalar ‘ 𝑊 ) ) ( 𝐺𝑥 ) ) ( +g ‘ ( Scalar ‘ 𝑊 ) ) ( 𝐺𝑦 ) ) ) )
29 1 28 raleqbidv ( 𝜑 → ( ∀ 𝑦𝑉 ( 𝐺 ‘ ( ( 𝑟 · 𝑥 ) + 𝑦 ) ) = ( ( 𝑟 × ( 𝐺𝑥 ) ) ( 𝐺𝑦 ) ) ↔ ∀ 𝑦 ∈ ( Base ‘ 𝑊 ) ( 𝐺 ‘ ( ( 𝑟 ( ·𝑠𝑊 ) 𝑥 ) ( +g𝑊 ) 𝑦 ) ) = ( ( 𝑟 ( .r ‘ ( Scalar ‘ 𝑊 ) ) ( 𝐺𝑥 ) ) ( +g ‘ ( Scalar ‘ 𝑊 ) ) ( 𝐺𝑦 ) ) ) )
30 1 29 raleqbidv ( 𝜑 → ( ∀ 𝑥𝑉𝑦𝑉 ( 𝐺 ‘ ( ( 𝑟 · 𝑥 ) + 𝑦 ) ) = ( ( 𝑟 × ( 𝐺𝑥 ) ) ( 𝐺𝑦 ) ) ↔ ∀ 𝑥 ∈ ( Base ‘ 𝑊 ) ∀ 𝑦 ∈ ( Base ‘ 𝑊 ) ( 𝐺 ‘ ( ( 𝑟 ( ·𝑠𝑊 ) 𝑥 ) ( +g𝑊 ) 𝑦 ) ) = ( ( 𝑟 ( .r ‘ ( Scalar ‘ 𝑊 ) ) ( 𝐺𝑥 ) ) ( +g ‘ ( Scalar ‘ 𝑊 ) ) ( 𝐺𝑦 ) ) ) )
31 13 30 raleqbidv ( 𝜑 → ( ∀ 𝑟𝐾𝑥𝑉𝑦𝑉 ( 𝐺 ‘ ( ( 𝑟 · 𝑥 ) + 𝑦 ) ) = ( ( 𝑟 × ( 𝐺𝑥 ) ) ( 𝐺𝑦 ) ) ↔ ∀ 𝑟 ∈ ( Base ‘ ( Scalar ‘ 𝑊 ) ) ∀ 𝑥 ∈ ( Base ‘ 𝑊 ) ∀ 𝑦 ∈ ( Base ‘ 𝑊 ) ( 𝐺 ‘ ( ( 𝑟 ( ·𝑠𝑊 ) 𝑥 ) ( +g𝑊 ) 𝑦 ) ) = ( ( 𝑟 ( .r ‘ ( Scalar ‘ 𝑊 ) ) ( 𝐺𝑥 ) ) ( +g ‘ ( Scalar ‘ 𝑊 ) ) ( 𝐺𝑦 ) ) ) )
32 16 31 mpbid ( 𝜑 → ∀ 𝑟 ∈ ( Base ‘ ( Scalar ‘ 𝑊 ) ) ∀ 𝑥 ∈ ( Base ‘ 𝑊 ) ∀ 𝑦 ∈ ( Base ‘ 𝑊 ) ( 𝐺 ‘ ( ( 𝑟 ( ·𝑠𝑊 ) 𝑥 ) ( +g𝑊 ) 𝑦 ) ) = ( ( 𝑟 ( .r ‘ ( Scalar ‘ 𝑊 ) ) ( 𝐺𝑥 ) ) ( +g ‘ ( Scalar ‘ 𝑊 ) ) ( 𝐺𝑦 ) ) )
33 eqid ( Base ‘ 𝑊 ) = ( Base ‘ 𝑊 )
34 eqid ( +g𝑊 ) = ( +g𝑊 )
35 eqid ( Scalar ‘ 𝑊 ) = ( Scalar ‘ 𝑊 )
36 eqid ( ·𝑠𝑊 ) = ( ·𝑠𝑊 )
37 eqid ( Base ‘ ( Scalar ‘ 𝑊 ) ) = ( Base ‘ ( Scalar ‘ 𝑊 ) )
38 eqid ( +g ‘ ( Scalar ‘ 𝑊 ) ) = ( +g ‘ ( Scalar ‘ 𝑊 ) )
39 eqid ( .r ‘ ( Scalar ‘ 𝑊 ) ) = ( .r ‘ ( Scalar ‘ 𝑊 ) )
40 eqid ( LFnl ‘ 𝑊 ) = ( LFnl ‘ 𝑊 )
41 33 34 35 36 37 38 39 40 islfl ( 𝑊𝑋 → ( 𝐺 ∈ ( LFnl ‘ 𝑊 ) ↔ ( 𝐺 : ( Base ‘ 𝑊 ) ⟶ ( Base ‘ ( Scalar ‘ 𝑊 ) ) ∧ ∀ 𝑟 ∈ ( Base ‘ ( Scalar ‘ 𝑊 ) ) ∀ 𝑥 ∈ ( Base ‘ 𝑊 ) ∀ 𝑦 ∈ ( Base ‘ 𝑊 ) ( 𝐺 ‘ ( ( 𝑟 ( ·𝑠𝑊 ) 𝑥 ) ( +g𝑊 ) 𝑦 ) ) = ( ( 𝑟 ( .r ‘ ( Scalar ‘ 𝑊 ) ) ( 𝐺𝑥 ) ) ( +g ‘ ( Scalar ‘ 𝑊 ) ) ( 𝐺𝑦 ) ) ) ) )
42 41 biimpar ( ( 𝑊𝑋 ∧ ( 𝐺 : ( Base ‘ 𝑊 ) ⟶ ( Base ‘ ( Scalar ‘ 𝑊 ) ) ∧ ∀ 𝑟 ∈ ( Base ‘ ( Scalar ‘ 𝑊 ) ) ∀ 𝑥 ∈ ( Base ‘ 𝑊 ) ∀ 𝑦 ∈ ( Base ‘ 𝑊 ) ( 𝐺 ‘ ( ( 𝑟 ( ·𝑠𝑊 ) 𝑥 ) ( +g𝑊 ) 𝑦 ) ) = ( ( 𝑟 ( .r ‘ ( Scalar ‘ 𝑊 ) ) ( 𝐺𝑥 ) ) ( +g ‘ ( Scalar ‘ 𝑊 ) ) ( 𝐺𝑦 ) ) ) ) → 𝐺 ∈ ( LFnl ‘ 𝑊 ) )
43 11 15 32 42 syl12anc ( 𝜑𝐺 ∈ ( LFnl ‘ 𝑊 ) )
44 43 8 eleqtrrd ( 𝜑𝐺𝐹 )