Step |
Hyp |
Ref |
Expression |
1 |
|
coe1sclmul.p |
⊢ 𝑃 = ( Poly1 ‘ 𝑅 ) |
2 |
|
coe1sclmul.b |
⊢ 𝐵 = ( Base ‘ 𝑃 ) |
3 |
|
coe1sclmul.k |
⊢ 𝐾 = ( Base ‘ 𝑅 ) |
4 |
|
coe1sclmul.a |
⊢ 𝐴 = ( algSc ‘ 𝑃 ) |
5 |
|
coe1sclmul.t |
⊢ ∙ = ( .r ‘ 𝑃 ) |
6 |
|
coe1sclmul.u |
⊢ · = ( .r ‘ 𝑅 ) |
7 |
1 2 3 4 5 6
|
coe1sclmul |
⊢ ( ( 𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵 ) → ( coe1 ‘ ( ( 𝐴 ‘ 𝑋 ) ∙ 𝑌 ) ) = ( ( ℕ0 × { 𝑋 } ) ∘f · ( coe1 ‘ 𝑌 ) ) ) |
8 |
7
|
3expb |
⊢ ( ( 𝑅 ∈ Ring ∧ ( 𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵 ) ) → ( coe1 ‘ ( ( 𝐴 ‘ 𝑋 ) ∙ 𝑌 ) ) = ( ( ℕ0 × { 𝑋 } ) ∘f · ( coe1 ‘ 𝑌 ) ) ) |
9 |
8
|
3adant3 |
⊢ ( ( 𝑅 ∈ Ring ∧ ( 𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵 ) ∧ 0 ∈ ℕ0 ) → ( coe1 ‘ ( ( 𝐴 ‘ 𝑋 ) ∙ 𝑌 ) ) = ( ( ℕ0 × { 𝑋 } ) ∘f · ( coe1 ‘ 𝑌 ) ) ) |
10 |
9
|
fveq1d |
⊢ ( ( 𝑅 ∈ Ring ∧ ( 𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵 ) ∧ 0 ∈ ℕ0 ) → ( ( coe1 ‘ ( ( 𝐴 ‘ 𝑋 ) ∙ 𝑌 ) ) ‘ 0 ) = ( ( ( ℕ0 × { 𝑋 } ) ∘f · ( coe1 ‘ 𝑌 ) ) ‘ 0 ) ) |
11 |
|
simp3 |
⊢ ( ( 𝑅 ∈ Ring ∧ ( 𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵 ) ∧ 0 ∈ ℕ0 ) → 0 ∈ ℕ0 ) |
12 |
|
nn0ex |
⊢ ℕ0 ∈ V |
13 |
12
|
a1i |
⊢ ( ( 𝑅 ∈ Ring ∧ ( 𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵 ) ∧ 0 ∈ ℕ0 ) → ℕ0 ∈ V ) |
14 |
|
simp2l |
⊢ ( ( 𝑅 ∈ Ring ∧ ( 𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵 ) ∧ 0 ∈ ℕ0 ) → 𝑋 ∈ 𝐾 ) |
15 |
|
simp2r |
⊢ ( ( 𝑅 ∈ Ring ∧ ( 𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵 ) ∧ 0 ∈ ℕ0 ) → 𝑌 ∈ 𝐵 ) |
16 |
|
eqid |
⊢ ( coe1 ‘ 𝑌 ) = ( coe1 ‘ 𝑌 ) |
17 |
|
eqid |
⊢ ( Base ‘ 𝑅 ) = ( Base ‘ 𝑅 ) |
18 |
16 2 1 17
|
coe1f |
⊢ ( 𝑌 ∈ 𝐵 → ( coe1 ‘ 𝑌 ) : ℕ0 ⟶ ( Base ‘ 𝑅 ) ) |
19 |
|
ffn |
⊢ ( ( coe1 ‘ 𝑌 ) : ℕ0 ⟶ ( Base ‘ 𝑅 ) → ( coe1 ‘ 𝑌 ) Fn ℕ0 ) |
20 |
15 18 19
|
3syl |
⊢ ( ( 𝑅 ∈ Ring ∧ ( 𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵 ) ∧ 0 ∈ ℕ0 ) → ( coe1 ‘ 𝑌 ) Fn ℕ0 ) |
21 |
|
eqidd |
⊢ ( ( ( 𝑅 ∈ Ring ∧ ( 𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵 ) ∧ 0 ∈ ℕ0 ) ∧ 0 ∈ ℕ0 ) → ( ( coe1 ‘ 𝑌 ) ‘ 0 ) = ( ( coe1 ‘ 𝑌 ) ‘ 0 ) ) |
22 |
13 14 20 21
|
ofc1 |
⊢ ( ( ( 𝑅 ∈ Ring ∧ ( 𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵 ) ∧ 0 ∈ ℕ0 ) ∧ 0 ∈ ℕ0 ) → ( ( ( ℕ0 × { 𝑋 } ) ∘f · ( coe1 ‘ 𝑌 ) ) ‘ 0 ) = ( 𝑋 · ( ( coe1 ‘ 𝑌 ) ‘ 0 ) ) ) |
23 |
11 22
|
mpdan |
⊢ ( ( 𝑅 ∈ Ring ∧ ( 𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵 ) ∧ 0 ∈ ℕ0 ) → ( ( ( ℕ0 × { 𝑋 } ) ∘f · ( coe1 ‘ 𝑌 ) ) ‘ 0 ) = ( 𝑋 · ( ( coe1 ‘ 𝑌 ) ‘ 0 ) ) ) |
24 |
10 23
|
eqtrd |
⊢ ( ( 𝑅 ∈ Ring ∧ ( 𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵 ) ∧ 0 ∈ ℕ0 ) → ( ( coe1 ‘ ( ( 𝐴 ‘ 𝑋 ) ∙ 𝑌 ) ) ‘ 0 ) = ( 𝑋 · ( ( coe1 ‘ 𝑌 ) ‘ 0 ) ) ) |