Step |
Hyp |
Ref |
Expression |
1 |
|
psdadd.s |
⊢ 𝑆 = ( 𝐼 mPwSer 𝑅 ) |
2 |
|
psdadd.b |
⊢ 𝐵 = ( Base ‘ 𝑆 ) |
3 |
|
psdadd.p |
⊢ + = ( +g ‘ 𝑆 ) |
4 |
|
psdadd.r |
⊢ ( 𝜑 → 𝑅 ∈ CMnd ) |
5 |
|
psdadd.x |
⊢ ( 𝜑 → 𝑋 ∈ 𝐼 ) |
6 |
|
psdadd.f |
⊢ ( 𝜑 → 𝐹 ∈ 𝐵 ) |
7 |
|
psdadd.g |
⊢ ( 𝜑 → 𝐺 ∈ 𝐵 ) |
8 |
|
eqid |
⊢ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } = { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } |
9 |
1 2 8 5 6
|
psdval |
⊢ ( 𝜑 → ( ( ( 𝐼 mPSDer 𝑅 ) ‘ 𝑋 ) ‘ 𝐹 ) = ( 𝑏 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ↦ ( ( ( 𝑏 ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( 𝐹 ‘ ( 𝑏 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ) ) ) ) |
10 |
1 2 8 5 7
|
psdval |
⊢ ( 𝜑 → ( ( ( 𝐼 mPSDer 𝑅 ) ‘ 𝑋 ) ‘ 𝐺 ) = ( 𝑏 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ↦ ( ( ( 𝑏 ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( 𝐺 ‘ ( 𝑏 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ) ) ) ) |
11 |
9 10
|
oveq12d |
⊢ ( 𝜑 → ( ( ( ( 𝐼 mPSDer 𝑅 ) ‘ 𝑋 ) ‘ 𝐹 ) ∘f ( +g ‘ 𝑅 ) ( ( ( 𝐼 mPSDer 𝑅 ) ‘ 𝑋 ) ‘ 𝐺 ) ) = ( ( 𝑏 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ↦ ( ( ( 𝑏 ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( 𝐹 ‘ ( 𝑏 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ) ) ) ∘f ( +g ‘ 𝑅 ) ( 𝑏 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ↦ ( ( ( 𝑏 ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( 𝐺 ‘ ( 𝑏 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ) ) ) ) ) |
12 |
|
ovex |
⊢ ( ( ( 𝑏 ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( 𝐹 ‘ ( 𝑏 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ) ) ∈ V |
13 |
|
eqid |
⊢ ( 𝑏 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ↦ ( ( ( 𝑏 ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( 𝐹 ‘ ( 𝑏 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ) ) ) = ( 𝑏 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ↦ ( ( ( 𝑏 ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( 𝐹 ‘ ( 𝑏 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ) ) ) |
14 |
12 13
|
fnmpti |
⊢ ( 𝑏 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ↦ ( ( ( 𝑏 ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( 𝐹 ‘ ( 𝑏 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ) ) ) Fn { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } |
15 |
14
|
a1i |
⊢ ( 𝜑 → ( 𝑏 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ↦ ( ( ( 𝑏 ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( 𝐹 ‘ ( 𝑏 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ) ) ) Fn { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) |
16 |
|
ovex |
⊢ ( ( ( 𝑏 ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( 𝐺 ‘ ( 𝑏 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ) ) ∈ V |
17 |
|
eqid |
⊢ ( 𝑏 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ↦ ( ( ( 𝑏 ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( 𝐺 ‘ ( 𝑏 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ) ) ) = ( 𝑏 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ↦ ( ( ( 𝑏 ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( 𝐺 ‘ ( 𝑏 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ) ) ) |
18 |
16 17
|
fnmpti |
⊢ ( 𝑏 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ↦ ( ( ( 𝑏 ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( 𝐺 ‘ ( 𝑏 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ) ) ) Fn { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } |
19 |
18
|
a1i |
⊢ ( 𝜑 → ( 𝑏 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ↦ ( ( ( 𝑏 ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( 𝐺 ‘ ( 𝑏 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ) ) ) Fn { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) |
20 |
|
ovex |
⊢ ( ℕ0 ↑m 𝐼 ) ∈ V |
21 |
20
|
rabex |
⊢ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∈ V |
22 |
21
|
a1i |
⊢ ( 𝜑 → { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∈ V ) |
23 |
|
inidm |
⊢ ( { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∩ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) = { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } |
24 |
|
fveq1 |
⊢ ( 𝑏 = 𝑑 → ( 𝑏 ‘ 𝑋 ) = ( 𝑑 ‘ 𝑋 ) ) |
25 |
24
|
oveq1d |
⊢ ( 𝑏 = 𝑑 → ( ( 𝑏 ‘ 𝑋 ) + 1 ) = ( ( 𝑑 ‘ 𝑋 ) + 1 ) ) |
26 |
|
fvoveq1 |
⊢ ( 𝑏 = 𝑑 → ( 𝐹 ‘ ( 𝑏 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ) = ( 𝐹 ‘ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ) ) |
27 |
25 26
|
oveq12d |
⊢ ( 𝑏 = 𝑑 → ( ( ( 𝑏 ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( 𝐹 ‘ ( 𝑏 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ) ) = ( ( ( 𝑑 ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( 𝐹 ‘ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ) ) ) |
28 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) |
29 |
|
ovexd |
⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( ( ( 𝑑 ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( 𝐹 ‘ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ) ) ∈ V ) |
30 |
13 27 28 29
|
fvmptd3 |
⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( ( 𝑏 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ↦ ( ( ( 𝑏 ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( 𝐹 ‘ ( 𝑏 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ) ) ) ‘ 𝑑 ) = ( ( ( 𝑑 ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( 𝐹 ‘ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ) ) ) |
31 |
|
fvoveq1 |
⊢ ( 𝑏 = 𝑑 → ( 𝐺 ‘ ( 𝑏 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ) = ( 𝐺 ‘ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ) ) |
32 |
25 31
|
oveq12d |
⊢ ( 𝑏 = 𝑑 → ( ( ( 𝑏 ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( 𝐺 ‘ ( 𝑏 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ) ) = ( ( ( 𝑑 ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( 𝐺 ‘ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ) ) ) |
33 |
|
ovexd |
⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( ( ( 𝑑 ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( 𝐺 ‘ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ) ) ∈ V ) |
34 |
17 32 28 33
|
fvmptd3 |
⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( ( 𝑏 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ↦ ( ( ( 𝑏 ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( 𝐺 ‘ ( 𝑏 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ) ) ) ‘ 𝑑 ) = ( ( ( 𝑑 ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( 𝐺 ‘ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ) ) ) |
35 |
15 19 22 22 23 30 34
|
offval |
⊢ ( 𝜑 → ( ( 𝑏 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ↦ ( ( ( 𝑏 ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( 𝐹 ‘ ( 𝑏 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ) ) ) ∘f ( +g ‘ 𝑅 ) ( 𝑏 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ↦ ( ( ( 𝑏 ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( 𝐺 ‘ ( 𝑏 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ) ) ) ) = ( 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ↦ ( ( ( ( 𝑑 ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( 𝐹 ‘ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ) ) ( +g ‘ 𝑅 ) ( ( ( 𝑑 ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( 𝐺 ‘ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ) ) ) ) ) |
36 |
|
eqid |
⊢ ( +g ‘ 𝑅 ) = ( +g ‘ 𝑅 ) |
37 |
1 2 36 3 6 7
|
psradd |
⊢ ( 𝜑 → ( 𝐹 + 𝐺 ) = ( 𝐹 ∘f ( +g ‘ 𝑅 ) 𝐺 ) ) |
38 |
37
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( 𝐹 + 𝐺 ) = ( 𝐹 ∘f ( +g ‘ 𝑅 ) 𝐺 ) ) |
39 |
38
|
fveq1d |
⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( ( 𝐹 + 𝐺 ) ‘ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ) = ( ( 𝐹 ∘f ( +g ‘ 𝑅 ) 𝐺 ) ‘ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ) ) |
40 |
|
reldmpsr |
⊢ Rel dom mPwSer |
41 |
1 2 40
|
strov2rcl |
⊢ ( 𝐹 ∈ 𝐵 → 𝐼 ∈ V ) |
42 |
6 41
|
syl |
⊢ ( 𝜑 → 𝐼 ∈ V ) |
43 |
8
|
psrbagsn |
⊢ ( 𝐼 ∈ V → ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) |
44 |
42 43
|
syl |
⊢ ( 𝜑 → ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) |
45 |
44
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) |
46 |
8
|
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 } ) |
47 |
28 45 46
|
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 } ) |
48 |
|
eqid |
⊢ ( Base ‘ 𝑅 ) = ( Base ‘ 𝑅 ) |
49 |
1 48 8 2 6
|
psrelbas |
⊢ ( 𝜑 → 𝐹 : { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ⟶ ( Base ‘ 𝑅 ) ) |
50 |
49
|
ffnd |
⊢ ( 𝜑 → 𝐹 Fn { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) |
51 |
1 48 8 2 7
|
psrelbas |
⊢ ( 𝜑 → 𝐺 : { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ⟶ ( Base ‘ 𝑅 ) ) |
52 |
51
|
ffnd |
⊢ ( 𝜑 → 𝐺 Fn { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) |
53 |
|
eqidd |
⊢ ( ( 𝜑 ∧ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( 𝐹 ‘ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ) = ( 𝐹 ‘ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ) ) |
54 |
|
eqidd |
⊢ ( ( 𝜑 ∧ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( 𝐺 ‘ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ) = ( 𝐺 ‘ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ) ) |
55 |
50 52 22 22 23 53 54
|
ofval |
⊢ ( ( 𝜑 ∧ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( ( 𝐹 ∘f ( +g ‘ 𝑅 ) 𝐺 ) ‘ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ) = ( ( 𝐹 ‘ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ) ( +g ‘ 𝑅 ) ( 𝐺 ‘ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ) ) ) |
56 |
47 55
|
syldan |
⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( ( 𝐹 ∘f ( +g ‘ 𝑅 ) 𝐺 ) ‘ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ) = ( ( 𝐹 ‘ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ) ( +g ‘ 𝑅 ) ( 𝐺 ‘ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ) ) ) |
57 |
39 56
|
eqtrd |
⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( ( 𝐹 + 𝐺 ) ‘ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ) = ( ( 𝐹 ‘ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ) ( +g ‘ 𝑅 ) ( 𝐺 ‘ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ) ) ) |
58 |
57
|
oveq2d |
⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( ( ( 𝑑 ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( ( 𝐹 + 𝐺 ) ‘ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ) ) = ( ( ( 𝑑 ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ) ( +g ‘ 𝑅 ) ( 𝐺 ‘ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ) ) ) ) |
59 |
4
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → 𝑅 ∈ CMnd ) |
60 |
8
|
psrbagf |
⊢ ( 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } → 𝑑 : 𝐼 ⟶ ℕ0 ) |
61 |
60
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → 𝑑 : 𝐼 ⟶ ℕ0 ) |
62 |
5
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → 𝑋 ∈ 𝐼 ) |
63 |
61 62
|
ffvelcdmd |
⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( 𝑑 ‘ 𝑋 ) ∈ ℕ0 ) |
64 |
|
peano2nn0 |
⊢ ( ( 𝑑 ‘ 𝑋 ) ∈ ℕ0 → ( ( 𝑑 ‘ 𝑋 ) + 1 ) ∈ ℕ0 ) |
65 |
63 64
|
syl |
⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( ( 𝑑 ‘ 𝑋 ) + 1 ) ∈ ℕ0 ) |
66 |
6
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → 𝐹 ∈ 𝐵 ) |
67 |
1 48 8 2 66
|
psrelbas |
⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → 𝐹 : { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ⟶ ( Base ‘ 𝑅 ) ) |
68 |
67 47
|
ffvelcdmd |
⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( 𝐹 ‘ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ) ∈ ( Base ‘ 𝑅 ) ) |
69 |
51
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → 𝐺 : { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ⟶ ( Base ‘ 𝑅 ) ) |
70 |
69 47
|
ffvelcdmd |
⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( 𝐺 ‘ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ) ∈ ( Base ‘ 𝑅 ) ) |
71 |
|
eqid |
⊢ ( .g ‘ 𝑅 ) = ( .g ‘ 𝑅 ) |
72 |
48 71 36
|
mulgnn0di |
⊢ ( ( 𝑅 ∈ CMnd ∧ ( ( ( 𝑑 ‘ 𝑋 ) + 1 ) ∈ ℕ0 ∧ ( 𝐹 ‘ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ) ∈ ( Base ‘ 𝑅 ) ∧ ( 𝐺 ‘ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ) ∈ ( Base ‘ 𝑅 ) ) ) → ( ( ( 𝑑 ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ) ( +g ‘ 𝑅 ) ( 𝐺 ‘ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ) ) ) = ( ( ( ( 𝑑 ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( 𝐹 ‘ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ) ) ( +g ‘ 𝑅 ) ( ( ( 𝑑 ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( 𝐺 ‘ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ) ) ) ) |
73 |
59 65 68 70 72
|
syl13anc |
⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( ( ( 𝑑 ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ) ( +g ‘ 𝑅 ) ( 𝐺 ‘ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ) ) ) = ( ( ( ( 𝑑 ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( 𝐹 ‘ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ) ) ( +g ‘ 𝑅 ) ( ( ( 𝑑 ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( 𝐺 ‘ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ) ) ) ) |
74 |
58 73
|
eqtr2d |
⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( ( ( ( 𝑑 ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( 𝐹 ‘ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ) ) ( +g ‘ 𝑅 ) ( ( ( 𝑑 ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( 𝐺 ‘ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ) ) ) = ( ( ( 𝑑 ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( ( 𝐹 + 𝐺 ) ‘ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ) ) ) |
75 |
74
|
mpteq2dva |
⊢ ( 𝜑 → ( 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ↦ ( ( ( ( 𝑑 ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( 𝐹 ‘ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ) ) ( +g ‘ 𝑅 ) ( ( ( 𝑑 ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( 𝐺 ‘ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ) ) ) ) = ( 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ↦ ( ( ( 𝑑 ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( ( 𝐹 + 𝐺 ) ‘ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ) ) ) ) |
76 |
11 35 75
|
3eqtrd |
⊢ ( 𝜑 → ( ( ( ( 𝐼 mPSDer 𝑅 ) ‘ 𝑋 ) ‘ 𝐹 ) ∘f ( +g ‘ 𝑅 ) ( ( ( 𝐼 mPSDer 𝑅 ) ‘ 𝑋 ) ‘ 𝐺 ) ) = ( 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ↦ ( ( ( 𝑑 ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( ( 𝐹 + 𝐺 ) ‘ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ) ) ) ) |
77 |
4
|
cmnmndd |
⊢ ( 𝜑 → 𝑅 ∈ Mnd ) |
78 |
|
mndmgm |
⊢ ( 𝑅 ∈ Mnd → 𝑅 ∈ Mgm ) |
79 |
77 78
|
syl |
⊢ ( 𝜑 → 𝑅 ∈ Mgm ) |
80 |
1 2 79 5 6
|
psdcl |
⊢ ( 𝜑 → ( ( ( 𝐼 mPSDer 𝑅 ) ‘ 𝑋 ) ‘ 𝐹 ) ∈ 𝐵 ) |
81 |
1 2 79 5 7
|
psdcl |
⊢ ( 𝜑 → ( ( ( 𝐼 mPSDer 𝑅 ) ‘ 𝑋 ) ‘ 𝐺 ) ∈ 𝐵 ) |
82 |
1 2 36 3 80 81
|
psradd |
⊢ ( 𝜑 → ( ( ( ( 𝐼 mPSDer 𝑅 ) ‘ 𝑋 ) ‘ 𝐹 ) + ( ( ( 𝐼 mPSDer 𝑅 ) ‘ 𝑋 ) ‘ 𝐺 ) ) = ( ( ( ( 𝐼 mPSDer 𝑅 ) ‘ 𝑋 ) ‘ 𝐹 ) ∘f ( +g ‘ 𝑅 ) ( ( ( 𝐼 mPSDer 𝑅 ) ‘ 𝑋 ) ‘ 𝐺 ) ) ) |
83 |
1 2 3 79 6 7
|
psraddcl |
⊢ ( 𝜑 → ( 𝐹 + 𝐺 ) ∈ 𝐵 ) |
84 |
1 2 8 5 83
|
psdval |
⊢ ( 𝜑 → ( ( ( 𝐼 mPSDer 𝑅 ) ‘ 𝑋 ) ‘ ( 𝐹 + 𝐺 ) ) = ( 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ↦ ( ( ( 𝑑 ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( ( 𝐹 + 𝐺 ) ‘ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ) ) ) ) |
85 |
76 82 84
|
3eqtr4rd |
⊢ ( 𝜑 → ( ( ( 𝐼 mPSDer 𝑅 ) ‘ 𝑋 ) ‘ ( 𝐹 + 𝐺 ) ) = ( ( ( ( 𝐼 mPSDer 𝑅 ) ‘ 𝑋 ) ‘ 𝐹 ) + ( ( ( 𝐼 mPSDer 𝑅 ) ‘ 𝑋 ) ‘ 𝐺 ) ) ) |