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 |
|
eqid |
⊢ ( 0g ‘ 𝑅 ) = ( 0g ‘ 𝑅 ) |
8 |
|
eqid |
⊢ ( var1 ‘ 𝑅 ) = ( var1 ‘ 𝑅 ) |
9 |
|
eqid |
⊢ ( ·𝑠 ‘ 𝑃 ) = ( ·𝑠 ‘ 𝑃 ) |
10 |
|
eqid |
⊢ ( mulGrp ‘ 𝑃 ) = ( mulGrp ‘ 𝑃 ) |
11 |
|
eqid |
⊢ ( .g ‘ ( mulGrp ‘ 𝑃 ) ) = ( .g ‘ ( mulGrp ‘ 𝑃 ) ) |
12 |
|
simp3 |
⊢ ( ( 𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵 ) → 𝑌 ∈ 𝐵 ) |
13 |
|
simp1 |
⊢ ( ( 𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵 ) → 𝑅 ∈ Ring ) |
14 |
|
simp2 |
⊢ ( ( 𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵 ) → 𝑋 ∈ 𝐾 ) |
15 |
|
0nn0 |
⊢ 0 ∈ ℕ0 |
16 |
15
|
a1i |
⊢ ( ( 𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵 ) → 0 ∈ ℕ0 ) |
17 |
7 3 1 8 9 10 11 2 5 6 12 13 14 16
|
coe1tmmul |
⊢ ( ( 𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵 ) → ( coe1 ‘ ( ( 𝑋 ( ·𝑠 ‘ 𝑃 ) ( 0 ( .g ‘ ( mulGrp ‘ 𝑃 ) ) ( var1 ‘ 𝑅 ) ) ) ∙ 𝑌 ) ) = ( 𝑥 ∈ ℕ0 ↦ if ( 0 ≤ 𝑥 , ( 𝑋 · ( ( coe1 ‘ 𝑌 ) ‘ ( 𝑥 − 0 ) ) ) , ( 0g ‘ 𝑅 ) ) ) ) |
18 |
3 1 8 9 10 11 4
|
ply1scltm |
⊢ ( ( 𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐾 ) → ( 𝐴 ‘ 𝑋 ) = ( 𝑋 ( ·𝑠 ‘ 𝑃 ) ( 0 ( .g ‘ ( mulGrp ‘ 𝑃 ) ) ( var1 ‘ 𝑅 ) ) ) ) |
19 |
18
|
3adant3 |
⊢ ( ( 𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵 ) → ( 𝐴 ‘ 𝑋 ) = ( 𝑋 ( ·𝑠 ‘ 𝑃 ) ( 0 ( .g ‘ ( mulGrp ‘ 𝑃 ) ) ( var1 ‘ 𝑅 ) ) ) ) |
20 |
19
|
fvoveq1d |
⊢ ( ( 𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵 ) → ( coe1 ‘ ( ( 𝐴 ‘ 𝑋 ) ∙ 𝑌 ) ) = ( coe1 ‘ ( ( 𝑋 ( ·𝑠 ‘ 𝑃 ) ( 0 ( .g ‘ ( mulGrp ‘ 𝑃 ) ) ( var1 ‘ 𝑅 ) ) ) ∙ 𝑌 ) ) ) |
21 |
|
nn0ex |
⊢ ℕ0 ∈ V |
22 |
21
|
a1i |
⊢ ( ( 𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵 ) → ℕ0 ∈ V ) |
23 |
|
simpl2 |
⊢ ( ( ( 𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝑥 ∈ ℕ0 ) → 𝑋 ∈ 𝐾 ) |
24 |
|
fvexd |
⊢ ( ( ( 𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝑥 ∈ ℕ0 ) → ( ( coe1 ‘ 𝑌 ) ‘ 𝑥 ) ∈ V ) |
25 |
|
fconstmpt |
⊢ ( ℕ0 × { 𝑋 } ) = ( 𝑥 ∈ ℕ0 ↦ 𝑋 ) |
26 |
25
|
a1i |
⊢ ( ( 𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵 ) → ( ℕ0 × { 𝑋 } ) = ( 𝑥 ∈ ℕ0 ↦ 𝑋 ) ) |
27 |
|
eqid |
⊢ ( coe1 ‘ 𝑌 ) = ( coe1 ‘ 𝑌 ) |
28 |
27 2 1 3
|
coe1f |
⊢ ( 𝑌 ∈ 𝐵 → ( coe1 ‘ 𝑌 ) : ℕ0 ⟶ 𝐾 ) |
29 |
28
|
3ad2ant3 |
⊢ ( ( 𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵 ) → ( coe1 ‘ 𝑌 ) : ℕ0 ⟶ 𝐾 ) |
30 |
29
|
feqmptd |
⊢ ( ( 𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵 ) → ( coe1 ‘ 𝑌 ) = ( 𝑥 ∈ ℕ0 ↦ ( ( coe1 ‘ 𝑌 ) ‘ 𝑥 ) ) ) |
31 |
22 23 24 26 30
|
offval2 |
⊢ ( ( 𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵 ) → ( ( ℕ0 × { 𝑋 } ) ∘f · ( coe1 ‘ 𝑌 ) ) = ( 𝑥 ∈ ℕ0 ↦ ( 𝑋 · ( ( coe1 ‘ 𝑌 ) ‘ 𝑥 ) ) ) ) |
32 |
|
nn0ge0 |
⊢ ( 𝑥 ∈ ℕ0 → 0 ≤ 𝑥 ) |
33 |
32
|
iftrued |
⊢ ( 𝑥 ∈ ℕ0 → if ( 0 ≤ 𝑥 , ( 𝑋 · ( ( coe1 ‘ 𝑌 ) ‘ ( 𝑥 − 0 ) ) ) , ( 0g ‘ 𝑅 ) ) = ( 𝑋 · ( ( coe1 ‘ 𝑌 ) ‘ ( 𝑥 − 0 ) ) ) ) |
34 |
|
nn0cn |
⊢ ( 𝑥 ∈ ℕ0 → 𝑥 ∈ ℂ ) |
35 |
34
|
subid1d |
⊢ ( 𝑥 ∈ ℕ0 → ( 𝑥 − 0 ) = 𝑥 ) |
36 |
35
|
fveq2d |
⊢ ( 𝑥 ∈ ℕ0 → ( ( coe1 ‘ 𝑌 ) ‘ ( 𝑥 − 0 ) ) = ( ( coe1 ‘ 𝑌 ) ‘ 𝑥 ) ) |
37 |
36
|
oveq2d |
⊢ ( 𝑥 ∈ ℕ0 → ( 𝑋 · ( ( coe1 ‘ 𝑌 ) ‘ ( 𝑥 − 0 ) ) ) = ( 𝑋 · ( ( coe1 ‘ 𝑌 ) ‘ 𝑥 ) ) ) |
38 |
33 37
|
eqtrd |
⊢ ( 𝑥 ∈ ℕ0 → if ( 0 ≤ 𝑥 , ( 𝑋 · ( ( coe1 ‘ 𝑌 ) ‘ ( 𝑥 − 0 ) ) ) , ( 0g ‘ 𝑅 ) ) = ( 𝑋 · ( ( coe1 ‘ 𝑌 ) ‘ 𝑥 ) ) ) |
39 |
38
|
mpteq2ia |
⊢ ( 𝑥 ∈ ℕ0 ↦ if ( 0 ≤ 𝑥 , ( 𝑋 · ( ( coe1 ‘ 𝑌 ) ‘ ( 𝑥 − 0 ) ) ) , ( 0g ‘ 𝑅 ) ) ) = ( 𝑥 ∈ ℕ0 ↦ ( 𝑋 · ( ( coe1 ‘ 𝑌 ) ‘ 𝑥 ) ) ) |
40 |
31 39
|
eqtr4di |
⊢ ( ( 𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵 ) → ( ( ℕ0 × { 𝑋 } ) ∘f · ( coe1 ‘ 𝑌 ) ) = ( 𝑥 ∈ ℕ0 ↦ if ( 0 ≤ 𝑥 , ( 𝑋 · ( ( coe1 ‘ 𝑌 ) ‘ ( 𝑥 − 0 ) ) ) , ( 0g ‘ 𝑅 ) ) ) ) |
41 |
17 20 40
|
3eqtr4d |
⊢ ( ( 𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵 ) → ( coe1 ‘ ( ( 𝐴 ‘ 𝑋 ) ∙ 𝑌 ) ) = ( ( ℕ0 × { 𝑋 } ) ∘f · ( coe1 ‘ 𝑌 ) ) ) |