Step |
Hyp |
Ref |
Expression |
1 |
|
psdvsca.s |
⊢ 𝑆 = ( 𝐼 mPwSer 𝑅 ) |
2 |
|
psdvsca.b |
⊢ 𝐵 = ( Base ‘ 𝑆 ) |
3 |
|
psdvsca.m |
⊢ · = ( ·𝑠 ‘ 𝑆 ) |
4 |
|
psdvsca.k |
⊢ 𝐾 = ( Base ‘ 𝑅 ) |
5 |
|
psdvsca.r |
⊢ ( 𝜑 → 𝑅 ∈ CRing ) |
6 |
|
psdvsca.x |
⊢ ( 𝜑 → 𝑋 ∈ 𝐼 ) |
7 |
|
psdvsca.f |
⊢ ( 𝜑 → 𝐹 ∈ 𝐵 ) |
8 |
|
psdvsca.c |
⊢ ( 𝜑 → 𝐶 ∈ 𝐾 ) |
9 |
|
eqid |
⊢ ( Base ‘ 𝑅 ) = ( Base ‘ 𝑅 ) |
10 |
|
eqid |
⊢ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } = { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } |
11 |
5
|
crngringd |
⊢ ( 𝜑 → 𝑅 ∈ Ring ) |
12 |
|
ringmgm |
⊢ ( 𝑅 ∈ Ring → 𝑅 ∈ Mgm ) |
13 |
11 12
|
syl |
⊢ ( 𝜑 → 𝑅 ∈ Mgm ) |
14 |
1 3 4 2 11 8 7
|
psrvscacl |
⊢ ( 𝜑 → ( 𝐶 · 𝐹 ) ∈ 𝐵 ) |
15 |
1 2 13 6 14
|
psdcl |
⊢ ( 𝜑 → ( ( ( 𝐼 mPSDer 𝑅 ) ‘ 𝑋 ) ‘ ( 𝐶 · 𝐹 ) ) ∈ 𝐵 ) |
16 |
1 9 10 2 15
|
psrelbas |
⊢ ( 𝜑 → ( ( ( 𝐼 mPSDer 𝑅 ) ‘ 𝑋 ) ‘ ( 𝐶 · 𝐹 ) ) : { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ⟶ ( Base ‘ 𝑅 ) ) |
17 |
16
|
ffnd |
⊢ ( 𝜑 → ( ( ( 𝐼 mPSDer 𝑅 ) ‘ 𝑋 ) ‘ ( 𝐶 · 𝐹 ) ) Fn { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) |
18 |
1 2 13 6 7
|
psdcl |
⊢ ( 𝜑 → ( ( ( 𝐼 mPSDer 𝑅 ) ‘ 𝑋 ) ‘ 𝐹 ) ∈ 𝐵 ) |
19 |
1 3 4 2 11 8 18
|
psrvscacl |
⊢ ( 𝜑 → ( 𝐶 · ( ( ( 𝐼 mPSDer 𝑅 ) ‘ 𝑋 ) ‘ 𝐹 ) ) ∈ 𝐵 ) |
20 |
1 9 10 2 19
|
psrelbas |
⊢ ( 𝜑 → ( 𝐶 · ( ( ( 𝐼 mPSDer 𝑅 ) ‘ 𝑋 ) ‘ 𝐹 ) ) : { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ⟶ ( Base ‘ 𝑅 ) ) |
21 |
20
|
ffnd |
⊢ ( 𝜑 → ( 𝐶 · ( ( ( 𝐼 mPSDer 𝑅 ) ‘ 𝑋 ) ‘ 𝐹 ) ) Fn { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) |
22 |
11
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → 𝑅 ∈ Ring ) |
23 |
10
|
psrbagf |
⊢ ( 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } → 𝑑 : 𝐼 ⟶ ℕ0 ) |
24 |
23
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → 𝑑 : 𝐼 ⟶ ℕ0 ) |
25 |
6
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → 𝑋 ∈ 𝐼 ) |
26 |
24 25
|
ffvelcdmd |
⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( 𝑑 ‘ 𝑋 ) ∈ ℕ0 ) |
27 |
|
peano2nn0 |
⊢ ( ( 𝑑 ‘ 𝑋 ) ∈ ℕ0 → ( ( 𝑑 ‘ 𝑋 ) + 1 ) ∈ ℕ0 ) |
28 |
27
|
nn0zd |
⊢ ( ( 𝑑 ‘ 𝑋 ) ∈ ℕ0 → ( ( 𝑑 ‘ 𝑋 ) + 1 ) ∈ ℤ ) |
29 |
26 28
|
syl |
⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( ( 𝑑 ‘ 𝑋 ) + 1 ) ∈ ℤ ) |
30 |
8
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → 𝐶 ∈ 𝐾 ) |
31 |
1 4 10 2 7
|
psrelbas |
⊢ ( 𝜑 → 𝐹 : { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ⟶ 𝐾 ) |
32 |
31
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → 𝐹 : { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ⟶ 𝐾 ) |
33 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) |
34 |
|
reldmpsr |
⊢ Rel dom mPwSer |
35 |
1 2 34
|
strov2rcl |
⊢ ( 𝐹 ∈ 𝐵 → 𝐼 ∈ V ) |
36 |
7 35
|
syl |
⊢ ( 𝜑 → 𝐼 ∈ V ) |
37 |
10
|
psrbagsn |
⊢ ( 𝐼 ∈ V → ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) |
38 |
36 37
|
syl |
⊢ ( 𝜑 → ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) |
39 |
38
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) |
40 |
10
|
psrbagaddcl |
⊢ ( ( 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∧ ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) |
41 |
33 39 40
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) |
42 |
32 41
|
ffvelcdmd |
⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( 𝐹 ‘ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ) ∈ 𝐾 ) |
43 |
|
eqid |
⊢ ( .g ‘ 𝑅 ) = ( .g ‘ 𝑅 ) |
44 |
|
eqid |
⊢ ( .r ‘ 𝑅 ) = ( .r ‘ 𝑅 ) |
45 |
4 43 44
|
mulgass3 |
⊢ ( ( 𝑅 ∈ Ring ∧ ( ( ( 𝑑 ‘ 𝑋 ) + 1 ) ∈ ℤ ∧ 𝐶 ∈ 𝐾 ∧ ( 𝐹 ‘ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ) ∈ 𝐾 ) ) → ( 𝐶 ( .r ‘ 𝑅 ) ( ( ( 𝑑 ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( 𝐹 ‘ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ) ) ) = ( ( ( 𝑑 ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( 𝐶 ( .r ‘ 𝑅 ) ( 𝐹 ‘ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ) ) ) ) |
46 |
22 29 30 42 45
|
syl13anc |
⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( 𝐶 ( .r ‘ 𝑅 ) ( ( ( 𝑑 ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( 𝐹 ‘ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ) ) ) = ( ( ( 𝑑 ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( 𝐶 ( .r ‘ 𝑅 ) ( 𝐹 ‘ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ) ) ) ) |
47 |
7
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → 𝐹 ∈ 𝐵 ) |
48 |
1 2 10 25 47 33
|
psdcoef |
⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( ( ( ( 𝐼 mPSDer 𝑅 ) ‘ 𝑋 ) ‘ 𝐹 ) ‘ 𝑑 ) = ( ( ( 𝑑 ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( 𝐹 ‘ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ) ) ) |
49 |
48
|
oveq2d |
⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( 𝐶 ( .r ‘ 𝑅 ) ( ( ( ( 𝐼 mPSDer 𝑅 ) ‘ 𝑋 ) ‘ 𝐹 ) ‘ 𝑑 ) ) = ( 𝐶 ( .r ‘ 𝑅 ) ( ( ( 𝑑 ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( 𝐹 ‘ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ) ) ) ) |
50 |
1 3 4 2 44 10 30 47 41
|
psrvscaval |
⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( ( 𝐶 · 𝐹 ) ‘ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ) = ( 𝐶 ( .r ‘ 𝑅 ) ( 𝐹 ‘ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ) ) ) |
51 |
50
|
oveq2d |
⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( ( ( 𝑑 ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( ( 𝐶 · 𝐹 ) ‘ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ) ) = ( ( ( 𝑑 ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( 𝐶 ( .r ‘ 𝑅 ) ( 𝐹 ‘ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ) ) ) ) |
52 |
46 49 51
|
3eqtr4rd |
⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( ( ( 𝑑 ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( ( 𝐶 · 𝐹 ) ‘ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ) ) = ( 𝐶 ( .r ‘ 𝑅 ) ( ( ( ( 𝐼 mPSDer 𝑅 ) ‘ 𝑋 ) ‘ 𝐹 ) ‘ 𝑑 ) ) ) |
53 |
14
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( 𝐶 · 𝐹 ) ∈ 𝐵 ) |
54 |
1 2 10 25 53 33
|
psdcoef |
⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( ( ( ( 𝐼 mPSDer 𝑅 ) ‘ 𝑋 ) ‘ ( 𝐶 · 𝐹 ) ) ‘ 𝑑 ) = ( ( ( 𝑑 ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( ( 𝐶 · 𝐹 ) ‘ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ) ) ) |
55 |
18
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( ( ( 𝐼 mPSDer 𝑅 ) ‘ 𝑋 ) ‘ 𝐹 ) ∈ 𝐵 ) |
56 |
1 3 4 2 44 10 30 55 33
|
psrvscaval |
⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( ( 𝐶 · ( ( ( 𝐼 mPSDer 𝑅 ) ‘ 𝑋 ) ‘ 𝐹 ) ) ‘ 𝑑 ) = ( 𝐶 ( .r ‘ 𝑅 ) ( ( ( ( 𝐼 mPSDer 𝑅 ) ‘ 𝑋 ) ‘ 𝐹 ) ‘ 𝑑 ) ) ) |
57 |
52 54 56
|
3eqtr4d |
⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( ( ( ( 𝐼 mPSDer 𝑅 ) ‘ 𝑋 ) ‘ ( 𝐶 · 𝐹 ) ) ‘ 𝑑 ) = ( ( 𝐶 · ( ( ( 𝐼 mPSDer 𝑅 ) ‘ 𝑋 ) ‘ 𝐹 ) ) ‘ 𝑑 ) ) |
58 |
17 21 57
|
eqfnfvd |
⊢ ( 𝜑 → ( ( ( 𝐼 mPSDer 𝑅 ) ‘ 𝑋 ) ‘ ( 𝐶 · 𝐹 ) ) = ( 𝐶 · ( ( ( 𝐼 mPSDer 𝑅 ) ‘ 𝑋 ) ‘ 𝐹 ) ) ) |