Step |
Hyp |
Ref |
Expression |
1 |
|
mplmon2.p |
⊢ 𝑃 = ( 𝐼 mPoly 𝑅 ) |
2 |
|
mplmon2.v |
⊢ · = ( ·𝑠 ‘ 𝑃 ) |
3 |
|
mplmon2.d |
⊢ 𝐷 = { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } |
4 |
|
mplmon2.o |
⊢ 1 = ( 1r ‘ 𝑅 ) |
5 |
|
mplmon2.z |
⊢ 0 = ( 0g ‘ 𝑅 ) |
6 |
|
mplmon2.b |
⊢ 𝐵 = ( Base ‘ 𝑅 ) |
7 |
|
mplmon2.i |
⊢ ( 𝜑 → 𝐼 ∈ 𝑊 ) |
8 |
|
mplmon2.r |
⊢ ( 𝜑 → 𝑅 ∈ Ring ) |
9 |
|
mplmon2.k |
⊢ ( 𝜑 → 𝐾 ∈ 𝐷 ) |
10 |
|
mplmon2.x |
⊢ ( 𝜑 → 𝑋 ∈ 𝐵 ) |
11 |
|
eqid |
⊢ ( Base ‘ 𝑃 ) = ( Base ‘ 𝑃 ) |
12 |
|
eqid |
⊢ ( .r ‘ 𝑅 ) = ( .r ‘ 𝑅 ) |
13 |
1 11 5 4 3 7 8 9
|
mplmon |
⊢ ( 𝜑 → ( 𝑦 ∈ 𝐷 ↦ if ( 𝑦 = 𝐾 , 1 , 0 ) ) ∈ ( Base ‘ 𝑃 ) ) |
14 |
1 2 6 11 12 3 10 13
|
mplvsca |
⊢ ( 𝜑 → ( 𝑋 · ( 𝑦 ∈ 𝐷 ↦ if ( 𝑦 = 𝐾 , 1 , 0 ) ) ) = ( ( 𝐷 × { 𝑋 } ) ∘f ( .r ‘ 𝑅 ) ( 𝑦 ∈ 𝐷 ↦ if ( 𝑦 = 𝐾 , 1 , 0 ) ) ) ) |
15 |
|
ovex |
⊢ ( ℕ0 ↑m 𝐼 ) ∈ V |
16 |
3 15
|
rabex2 |
⊢ 𝐷 ∈ V |
17 |
16
|
a1i |
⊢ ( 𝜑 → 𝐷 ∈ V ) |
18 |
10
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐷 ) → 𝑋 ∈ 𝐵 ) |
19 |
4
|
fvexi |
⊢ 1 ∈ V |
20 |
5
|
fvexi |
⊢ 0 ∈ V |
21 |
19 20
|
ifex |
⊢ if ( 𝑦 = 𝐾 , 1 , 0 ) ∈ V |
22 |
21
|
a1i |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐷 ) → if ( 𝑦 = 𝐾 , 1 , 0 ) ∈ V ) |
23 |
|
fconstmpt |
⊢ ( 𝐷 × { 𝑋 } ) = ( 𝑦 ∈ 𝐷 ↦ 𝑋 ) |
24 |
23
|
a1i |
⊢ ( 𝜑 → ( 𝐷 × { 𝑋 } ) = ( 𝑦 ∈ 𝐷 ↦ 𝑋 ) ) |
25 |
|
eqidd |
⊢ ( 𝜑 → ( 𝑦 ∈ 𝐷 ↦ if ( 𝑦 = 𝐾 , 1 , 0 ) ) = ( 𝑦 ∈ 𝐷 ↦ if ( 𝑦 = 𝐾 , 1 , 0 ) ) ) |
26 |
17 18 22 24 25
|
offval2 |
⊢ ( 𝜑 → ( ( 𝐷 × { 𝑋 } ) ∘f ( .r ‘ 𝑅 ) ( 𝑦 ∈ 𝐷 ↦ if ( 𝑦 = 𝐾 , 1 , 0 ) ) ) = ( 𝑦 ∈ 𝐷 ↦ ( 𝑋 ( .r ‘ 𝑅 ) if ( 𝑦 = 𝐾 , 1 , 0 ) ) ) ) |
27 |
|
oveq2 |
⊢ ( 1 = if ( 𝑦 = 𝐾 , 1 , 0 ) → ( 𝑋 ( .r ‘ 𝑅 ) 1 ) = ( 𝑋 ( .r ‘ 𝑅 ) if ( 𝑦 = 𝐾 , 1 , 0 ) ) ) |
28 |
27
|
eqeq1d |
⊢ ( 1 = if ( 𝑦 = 𝐾 , 1 , 0 ) → ( ( 𝑋 ( .r ‘ 𝑅 ) 1 ) = if ( 𝑦 = 𝐾 , 𝑋 , 0 ) ↔ ( 𝑋 ( .r ‘ 𝑅 ) if ( 𝑦 = 𝐾 , 1 , 0 ) ) = if ( 𝑦 = 𝐾 , 𝑋 , 0 ) ) ) |
29 |
|
oveq2 |
⊢ ( 0 = if ( 𝑦 = 𝐾 , 1 , 0 ) → ( 𝑋 ( .r ‘ 𝑅 ) 0 ) = ( 𝑋 ( .r ‘ 𝑅 ) if ( 𝑦 = 𝐾 , 1 , 0 ) ) ) |
30 |
29
|
eqeq1d |
⊢ ( 0 = if ( 𝑦 = 𝐾 , 1 , 0 ) → ( ( 𝑋 ( .r ‘ 𝑅 ) 0 ) = if ( 𝑦 = 𝐾 , 𝑋 , 0 ) ↔ ( 𝑋 ( .r ‘ 𝑅 ) if ( 𝑦 = 𝐾 , 1 , 0 ) ) = if ( 𝑦 = 𝐾 , 𝑋 , 0 ) ) ) |
31 |
6 12 4
|
ringridm |
⊢ ( ( 𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵 ) → ( 𝑋 ( .r ‘ 𝑅 ) 1 ) = 𝑋 ) |
32 |
8 10 31
|
syl2anc |
⊢ ( 𝜑 → ( 𝑋 ( .r ‘ 𝑅 ) 1 ) = 𝑋 ) |
33 |
|
iftrue |
⊢ ( 𝑦 = 𝐾 → if ( 𝑦 = 𝐾 , 𝑋 , 0 ) = 𝑋 ) |
34 |
33
|
eqcomd |
⊢ ( 𝑦 = 𝐾 → 𝑋 = if ( 𝑦 = 𝐾 , 𝑋 , 0 ) ) |
35 |
32 34
|
sylan9eq |
⊢ ( ( 𝜑 ∧ 𝑦 = 𝐾 ) → ( 𝑋 ( .r ‘ 𝑅 ) 1 ) = if ( 𝑦 = 𝐾 , 𝑋 , 0 ) ) |
36 |
6 12 5
|
ringrz |
⊢ ( ( 𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵 ) → ( 𝑋 ( .r ‘ 𝑅 ) 0 ) = 0 ) |
37 |
8 10 36
|
syl2anc |
⊢ ( 𝜑 → ( 𝑋 ( .r ‘ 𝑅 ) 0 ) = 0 ) |
38 |
|
iffalse |
⊢ ( ¬ 𝑦 = 𝐾 → if ( 𝑦 = 𝐾 , 𝑋 , 0 ) = 0 ) |
39 |
38
|
eqcomd |
⊢ ( ¬ 𝑦 = 𝐾 → 0 = if ( 𝑦 = 𝐾 , 𝑋 , 0 ) ) |
40 |
37 39
|
sylan9eq |
⊢ ( ( 𝜑 ∧ ¬ 𝑦 = 𝐾 ) → ( 𝑋 ( .r ‘ 𝑅 ) 0 ) = if ( 𝑦 = 𝐾 , 𝑋 , 0 ) ) |
41 |
28 30 35 40
|
ifbothda |
⊢ ( 𝜑 → ( 𝑋 ( .r ‘ 𝑅 ) if ( 𝑦 = 𝐾 , 1 , 0 ) ) = if ( 𝑦 = 𝐾 , 𝑋 , 0 ) ) |
42 |
41
|
mpteq2dv |
⊢ ( 𝜑 → ( 𝑦 ∈ 𝐷 ↦ ( 𝑋 ( .r ‘ 𝑅 ) if ( 𝑦 = 𝐾 , 1 , 0 ) ) ) = ( 𝑦 ∈ 𝐷 ↦ if ( 𝑦 = 𝐾 , 𝑋 , 0 ) ) ) |
43 |
14 26 42
|
3eqtrd |
⊢ ( 𝜑 → ( 𝑋 · ( 𝑦 ∈ 𝐷 ↦ if ( 𝑦 = 𝐾 , 1 , 0 ) ) ) = ( 𝑦 ∈ 𝐷 ↦ if ( 𝑦 = 𝐾 , 𝑋 , 0 ) ) ) |