Step |
Hyp |
Ref |
Expression |
1 |
|
lflass.v |
⊢ 𝑉 = ( Base ‘ 𝑊 ) |
2 |
|
lflass.r |
⊢ 𝑅 = ( Scalar ‘ 𝑊 ) |
3 |
|
lflass.k |
⊢ 𝐾 = ( Base ‘ 𝑅 ) |
4 |
|
lflass.t |
⊢ · = ( .r ‘ 𝑅 ) |
5 |
|
lflass.f |
⊢ 𝐹 = ( LFnl ‘ 𝑊 ) |
6 |
|
lflass.w |
⊢ ( 𝜑 → 𝑊 ∈ LMod ) |
7 |
|
lflass.x |
⊢ ( 𝜑 → 𝑋 ∈ 𝐾 ) |
8 |
|
lflass.y |
⊢ ( 𝜑 → 𝑌 ∈ 𝐾 ) |
9 |
|
lflass.g |
⊢ ( 𝜑 → 𝐺 ∈ 𝐹 ) |
10 |
1
|
fvexi |
⊢ 𝑉 ∈ V |
11 |
10
|
a1i |
⊢ ( 𝜑 → 𝑉 ∈ V ) |
12 |
2 3 1 5
|
lflf |
⊢ ( ( 𝑊 ∈ LMod ∧ 𝐺 ∈ 𝐹 ) → 𝐺 : 𝑉 ⟶ 𝐾 ) |
13 |
6 9 12
|
syl2anc |
⊢ ( 𝜑 → 𝐺 : 𝑉 ⟶ 𝐾 ) |
14 |
|
fconst6g |
⊢ ( 𝑋 ∈ 𝐾 → ( 𝑉 × { 𝑋 } ) : 𝑉 ⟶ 𝐾 ) |
15 |
7 14
|
syl |
⊢ ( 𝜑 → ( 𝑉 × { 𝑋 } ) : 𝑉 ⟶ 𝐾 ) |
16 |
|
fconst6g |
⊢ ( 𝑌 ∈ 𝐾 → ( 𝑉 × { 𝑌 } ) : 𝑉 ⟶ 𝐾 ) |
17 |
8 16
|
syl |
⊢ ( 𝜑 → ( 𝑉 × { 𝑌 } ) : 𝑉 ⟶ 𝐾 ) |
18 |
2
|
lmodring |
⊢ ( 𝑊 ∈ LMod → 𝑅 ∈ Ring ) |
19 |
6 18
|
syl |
⊢ ( 𝜑 → 𝑅 ∈ Ring ) |
20 |
3 4
|
ringass |
⊢ ( ( 𝑅 ∈ Ring ∧ ( 𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝐾 ∧ 𝑧 ∈ 𝐾 ) ) → ( ( 𝑥 · 𝑦 ) · 𝑧 ) = ( 𝑥 · ( 𝑦 · 𝑧 ) ) ) |
21 |
19 20
|
sylan |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝐾 ∧ 𝑧 ∈ 𝐾 ) ) → ( ( 𝑥 · 𝑦 ) · 𝑧 ) = ( 𝑥 · ( 𝑦 · 𝑧 ) ) ) |
22 |
11 13 15 17 21
|
caofass |
⊢ ( 𝜑 → ( ( 𝐺 ∘f · ( 𝑉 × { 𝑋 } ) ) ∘f · ( 𝑉 × { 𝑌 } ) ) = ( 𝐺 ∘f · ( ( 𝑉 × { 𝑋 } ) ∘f · ( 𝑉 × { 𝑌 } ) ) ) ) |
23 |
11 7 8
|
ofc12 |
⊢ ( 𝜑 → ( ( 𝑉 × { 𝑋 } ) ∘f · ( 𝑉 × { 𝑌 } ) ) = ( 𝑉 × { ( 𝑋 · 𝑌 ) } ) ) |
24 |
23
|
oveq2d |
⊢ ( 𝜑 → ( 𝐺 ∘f · ( ( 𝑉 × { 𝑋 } ) ∘f · ( 𝑉 × { 𝑌 } ) ) ) = ( 𝐺 ∘f · ( 𝑉 × { ( 𝑋 · 𝑌 ) } ) ) ) |
25 |
22 24
|
eqtr2d |
⊢ ( 𝜑 → ( 𝐺 ∘f · ( 𝑉 × { ( 𝑋 · 𝑌 ) } ) ) = ( ( 𝐺 ∘f · ( 𝑉 × { 𝑋 } ) ) ∘f · ( 𝑉 × { 𝑌 } ) ) ) |