Step |
Hyp |
Ref |
Expression |
1 |
|
islfld.v |
⊢ ( 𝜑 → 𝑉 = ( Base ‘ 𝑊 ) ) |
2 |
|
islfld.a |
⊢ ( 𝜑 → + = ( +g ‘ 𝑊 ) ) |
3 |
|
islfld.d |
⊢ ( 𝜑 → 𝐷 = ( Scalar ‘ 𝑊 ) ) |
4 |
|
islfld.s |
⊢ ( 𝜑 → · = ( ·𝑠 ‘ 𝑊 ) ) |
5 |
|
islfld.k |
⊢ ( 𝜑 → 𝐾 = ( Base ‘ 𝐷 ) ) |
6 |
|
islfld.p |
⊢ ( 𝜑 → ⨣ = ( +g ‘ 𝐷 ) ) |
7 |
|
islfld.t |
⊢ ( 𝜑 → × = ( .r ‘ 𝐷 ) ) |
8 |
|
islfld.f |
⊢ ( 𝜑 → 𝐹 = ( LFnl ‘ 𝑊 ) ) |
9 |
|
islfld.u |
⊢ ( 𝜑 → 𝐺 : 𝑉 ⟶ 𝐾 ) |
10 |
|
islfld.l |
⊢ ( ( 𝜑 ∧ ( 𝑟 ∈ 𝐾 ∧ 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ) ) → ( 𝐺 ‘ ( ( 𝑟 · 𝑥 ) + 𝑦 ) ) = ( ( 𝑟 × ( 𝐺 ‘ 𝑥 ) ) ⨣ ( 𝐺 ‘ 𝑦 ) ) ) |
11 |
|
islfld.w |
⊢ ( 𝜑 → 𝑊 ∈ 𝑋 ) |
12 |
3
|
fveq2d |
⊢ ( 𝜑 → ( Base ‘ 𝐷 ) = ( Base ‘ ( Scalar ‘ 𝑊 ) ) ) |
13 |
5 12
|
eqtrd |
⊢ ( 𝜑 → 𝐾 = ( Base ‘ ( Scalar ‘ 𝑊 ) ) ) |
14 |
1 13
|
feq23d |
⊢ ( 𝜑 → ( 𝐺 : 𝑉 ⟶ 𝐾 ↔ 𝐺 : ( Base ‘ 𝑊 ) ⟶ ( Base ‘ ( Scalar ‘ 𝑊 ) ) ) ) |
15 |
9 14
|
mpbid |
⊢ ( 𝜑 → 𝐺 : ( Base ‘ 𝑊 ) ⟶ ( Base ‘ ( Scalar ‘ 𝑊 ) ) ) |
16 |
10
|
ralrimivvva |
⊢ ( 𝜑 → ∀ 𝑟 ∈ 𝐾 ∀ 𝑥 ∈ 𝑉 ∀ 𝑦 ∈ 𝑉 ( 𝐺 ‘ ( ( 𝑟 · 𝑥 ) + 𝑦 ) ) = ( ( 𝑟 × ( 𝐺 ‘ 𝑥 ) ) ⨣ ( 𝐺 ‘ 𝑦 ) ) ) |
17 |
4
|
oveqd |
⊢ ( 𝜑 → ( 𝑟 · 𝑥 ) = ( 𝑟 ( ·𝑠 ‘ 𝑊 ) 𝑥 ) ) |
18 |
|
eqidd |
⊢ ( 𝜑 → 𝑦 = 𝑦 ) |
19 |
2 17 18
|
oveq123d |
⊢ ( 𝜑 → ( ( 𝑟 · 𝑥 ) + 𝑦 ) = ( ( 𝑟 ( ·𝑠 ‘ 𝑊 ) 𝑥 ) ( +g ‘ 𝑊 ) 𝑦 ) ) |
20 |
19
|
fveq2d |
⊢ ( 𝜑 → ( 𝐺 ‘ ( ( 𝑟 · 𝑥 ) + 𝑦 ) ) = ( 𝐺 ‘ ( ( 𝑟 ( ·𝑠 ‘ 𝑊 ) 𝑥 ) ( +g ‘ 𝑊 ) 𝑦 ) ) ) |
21 |
3
|
fveq2d |
⊢ ( 𝜑 → ( +g ‘ 𝐷 ) = ( +g ‘ ( Scalar ‘ 𝑊 ) ) ) |
22 |
6 21
|
eqtrd |
⊢ ( 𝜑 → ⨣ = ( +g ‘ ( Scalar ‘ 𝑊 ) ) ) |
23 |
3
|
fveq2d |
⊢ ( 𝜑 → ( .r ‘ 𝐷 ) = ( .r ‘ ( Scalar ‘ 𝑊 ) ) ) |
24 |
7 23
|
eqtrd |
⊢ ( 𝜑 → × = ( .r ‘ ( Scalar ‘ 𝑊 ) ) ) |
25 |
24
|
oveqd |
⊢ ( 𝜑 → ( 𝑟 × ( 𝐺 ‘ 𝑥 ) ) = ( 𝑟 ( .r ‘ ( Scalar ‘ 𝑊 ) ) ( 𝐺 ‘ 𝑥 ) ) ) |
26 |
|
eqidd |
⊢ ( 𝜑 → ( 𝐺 ‘ 𝑦 ) = ( 𝐺 ‘ 𝑦 ) ) |
27 |
22 25 26
|
oveq123d |
⊢ ( 𝜑 → ( ( 𝑟 × ( 𝐺 ‘ 𝑥 ) ) ⨣ ( 𝐺 ‘ 𝑦 ) ) = ( ( 𝑟 ( .r ‘ ( Scalar ‘ 𝑊 ) ) ( 𝐺 ‘ 𝑥 ) ) ( +g ‘ ( Scalar ‘ 𝑊 ) ) ( 𝐺 ‘ 𝑦 ) ) ) |
28 |
20 27
|
eqeq12d |
⊢ ( 𝜑 → ( ( 𝐺 ‘ ( ( 𝑟 · 𝑥 ) + 𝑦 ) ) = ( ( 𝑟 × ( 𝐺 ‘ 𝑥 ) ) ⨣ ( 𝐺 ‘ 𝑦 ) ) ↔ ( 𝐺 ‘ ( ( 𝑟 ( ·𝑠 ‘ 𝑊 ) 𝑥 ) ( +g ‘ 𝑊 ) 𝑦 ) ) = ( ( 𝑟 ( .r ‘ ( Scalar ‘ 𝑊 ) ) ( 𝐺 ‘ 𝑥 ) ) ( +g ‘ ( Scalar ‘ 𝑊 ) ) ( 𝐺 ‘ 𝑦 ) ) ) ) |
29 |
1 28
|
raleqbidv |
⊢ ( 𝜑 → ( ∀ 𝑦 ∈ 𝑉 ( 𝐺 ‘ ( ( 𝑟 · 𝑥 ) + 𝑦 ) ) = ( ( 𝑟 × ( 𝐺 ‘ 𝑥 ) ) ⨣ ( 𝐺 ‘ 𝑦 ) ) ↔ ∀ 𝑦 ∈ ( Base ‘ 𝑊 ) ( 𝐺 ‘ ( ( 𝑟 ( ·𝑠 ‘ 𝑊 ) 𝑥 ) ( +g ‘ 𝑊 ) 𝑦 ) ) = ( ( 𝑟 ( .r ‘ ( Scalar ‘ 𝑊 ) ) ( 𝐺 ‘ 𝑥 ) ) ( +g ‘ ( Scalar ‘ 𝑊 ) ) ( 𝐺 ‘ 𝑦 ) ) ) ) |
30 |
1 29
|
raleqbidv |
⊢ ( 𝜑 → ( ∀ 𝑥 ∈ 𝑉 ∀ 𝑦 ∈ 𝑉 ( 𝐺 ‘ ( ( 𝑟 · 𝑥 ) + 𝑦 ) ) = ( ( 𝑟 × ( 𝐺 ‘ 𝑥 ) ) ⨣ ( 𝐺 ‘ 𝑦 ) ) ↔ ∀ 𝑥 ∈ ( Base ‘ 𝑊 ) ∀ 𝑦 ∈ ( Base ‘ 𝑊 ) ( 𝐺 ‘ ( ( 𝑟 ( ·𝑠 ‘ 𝑊 ) 𝑥 ) ( +g ‘ 𝑊 ) 𝑦 ) ) = ( ( 𝑟 ( .r ‘ ( Scalar ‘ 𝑊 ) ) ( 𝐺 ‘ 𝑥 ) ) ( +g ‘ ( Scalar ‘ 𝑊 ) ) ( 𝐺 ‘ 𝑦 ) ) ) ) |
31 |
13 30
|
raleqbidv |
⊢ ( 𝜑 → ( ∀ 𝑟 ∈ 𝐾 ∀ 𝑥 ∈ 𝑉 ∀ 𝑦 ∈ 𝑉 ( 𝐺 ‘ ( ( 𝑟 · 𝑥 ) + 𝑦 ) ) = ( ( 𝑟 × ( 𝐺 ‘ 𝑥 ) ) ⨣ ( 𝐺 ‘ 𝑦 ) ) ↔ ∀ 𝑟 ∈ ( Base ‘ ( Scalar ‘ 𝑊 ) ) ∀ 𝑥 ∈ ( Base ‘ 𝑊 ) ∀ 𝑦 ∈ ( Base ‘ 𝑊 ) ( 𝐺 ‘ ( ( 𝑟 ( ·𝑠 ‘ 𝑊 ) 𝑥 ) ( +g ‘ 𝑊 ) 𝑦 ) ) = ( ( 𝑟 ( .r ‘ ( Scalar ‘ 𝑊 ) ) ( 𝐺 ‘ 𝑥 ) ) ( +g ‘ ( Scalar ‘ 𝑊 ) ) ( 𝐺 ‘ 𝑦 ) ) ) ) |
32 |
16 31
|
mpbid |
⊢ ( 𝜑 → ∀ 𝑟 ∈ ( Base ‘ ( Scalar ‘ 𝑊 ) ) ∀ 𝑥 ∈ ( Base ‘ 𝑊 ) ∀ 𝑦 ∈ ( Base ‘ 𝑊 ) ( 𝐺 ‘ ( ( 𝑟 ( ·𝑠 ‘ 𝑊 ) 𝑥 ) ( +g ‘ 𝑊 ) 𝑦 ) ) = ( ( 𝑟 ( .r ‘ ( Scalar ‘ 𝑊 ) ) ( 𝐺 ‘ 𝑥 ) ) ( +g ‘ ( Scalar ‘ 𝑊 ) ) ( 𝐺 ‘ 𝑦 ) ) ) |
33 |
|
eqid |
⊢ ( Base ‘ 𝑊 ) = ( Base ‘ 𝑊 ) |
34 |
|
eqid |
⊢ ( +g ‘ 𝑊 ) = ( +g ‘ 𝑊 ) |
35 |
|
eqid |
⊢ ( Scalar ‘ 𝑊 ) = ( Scalar ‘ 𝑊 ) |
36 |
|
eqid |
⊢ ( ·𝑠 ‘ 𝑊 ) = ( ·𝑠 ‘ 𝑊 ) |
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 |
⊢ ( 𝑊 ∈ 𝑋 → ( 𝐺 ∈ ( LFnl ‘ 𝑊 ) ↔ ( 𝐺 : ( Base ‘ 𝑊 ) ⟶ ( Base ‘ ( Scalar ‘ 𝑊 ) ) ∧ ∀ 𝑟 ∈ ( Base ‘ ( Scalar ‘ 𝑊 ) ) ∀ 𝑥 ∈ ( Base ‘ 𝑊 ) ∀ 𝑦 ∈ ( Base ‘ 𝑊 ) ( 𝐺 ‘ ( ( 𝑟 ( ·𝑠 ‘ 𝑊 ) 𝑥 ) ( +g ‘ 𝑊 ) 𝑦 ) ) = ( ( 𝑟 ( .r ‘ ( Scalar ‘ 𝑊 ) ) ( 𝐺 ‘ 𝑥 ) ) ( +g ‘ ( Scalar ‘ 𝑊 ) ) ( 𝐺 ‘ 𝑦 ) ) ) ) ) |
42 |
41
|
biimpar |
⊢ ( ( 𝑊 ∈ 𝑋 ∧ ( 𝐺 : ( Base ‘ 𝑊 ) ⟶ ( Base ‘ ( Scalar ‘ 𝑊 ) ) ∧ ∀ 𝑟 ∈ ( Base ‘ ( Scalar ‘ 𝑊 ) ) ∀ 𝑥 ∈ ( Base ‘ 𝑊 ) ∀ 𝑦 ∈ ( Base ‘ 𝑊 ) ( 𝐺 ‘ ( ( 𝑟 ( ·𝑠 ‘ 𝑊 ) 𝑥 ) ( +g ‘ 𝑊 ) 𝑦 ) ) = ( ( 𝑟 ( .r ‘ ( Scalar ‘ 𝑊 ) ) ( 𝐺 ‘ 𝑥 ) ) ( +g ‘ ( Scalar ‘ 𝑊 ) ) ( 𝐺 ‘ 𝑦 ) ) ) ) → 𝐺 ∈ ( LFnl ‘ 𝑊 ) ) |
43 |
11 15 32 42
|
syl12anc |
⊢ ( 𝜑 → 𝐺 ∈ ( LFnl ‘ 𝑊 ) ) |
44 |
43 8
|
eleqtrrd |
⊢ ( 𝜑 → 𝐺 ∈ 𝐹 ) |