Step | Hyp | Ref
| Expression |
1 | | psdadd.s |
. . . . 5
⊢ 𝑆 = (𝐼 mPwSer 𝑅) |
2 | | psdadd.b |
. . . . 5
⊢ 𝐵 = (Base‘𝑆) |
3 | | eqid 2735 |
. . . . 5
⊢ {ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin} = {ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin} |
4 | | psdadd.x |
. . . . 5
⊢ (𝜑 → 𝑋 ∈ 𝐼) |
5 | | psdadd.f |
. . . . 5
⊢ (𝜑 → 𝐹 ∈ 𝐵) |
6 | 1, 2, 3, 4, 5 | psdval 22181 |
. . . 4
⊢ (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))))))) |
7 | | psdadd.g |
. . . . 5
⊢ (𝜑 → 𝐺 ∈ 𝐵) |
8 | 1, 2, 3, 4, 7 | psdval 22181 |
. . . 4
⊢ (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))))))) |
9 | 6, 8 | oveq12d 7449 |
. . 3
⊢ (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)))))))) |
10 | | ovex 7464 |
. . . . . 6
⊢ (((𝑏‘𝑋) + 1)(.g‘𝑅)(𝐹‘(𝑏 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))))) ∈ V |
11 | | eqid 2735 |
. . . . . 6
⊢ (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)))))) |
12 | 10, 11 | fnmpti 6712 |
. . . . 5
⊢ (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} |
13 | 12 | a1i 11 |
. . . 4
⊢ (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}) |
14 | | ovex 7464 |
. . . . . 6
⊢ (((𝑏‘𝑋) + 1)(.g‘𝑅)(𝐺‘(𝑏 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))))) ∈ V |
15 | | eqid 2735 |
. . . . . 6
⊢ (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)))))) |
16 | 14, 15 | fnmpti 6712 |
. . . . 5
⊢ (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} |
17 | 16 | a1i 11 |
. . . 4
⊢ (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}) |
18 | | ovex 7464 |
. . . . . 6
⊢
(ℕ0 ↑m 𝐼) ∈ V |
19 | 18 | rabex 5345 |
. . . . 5
⊢ {ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin} ∈
V |
20 | 19 | a1i 11 |
. . . 4
⊢ (𝜑 → {ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin} ∈
V) |
21 | | inidm 4235 |
. . . 4
⊢ ({ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin} ∩ {ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin}) = {ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin} |
22 | | fveq1 6906 |
. . . . . . 7
⊢ (𝑏 = 𝑑 → (𝑏‘𝑋) = (𝑑‘𝑋)) |
23 | 22 | oveq1d 7446 |
. . . . . 6
⊢ (𝑏 = 𝑑 → ((𝑏‘𝑋) + 1) = ((𝑑‘𝑋) + 1)) |
24 | | fvoveq1 7454 |
. . . . . 6
⊢ (𝑏 = 𝑑 → (𝐹‘(𝑏 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))) = (𝐹‘(𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))))) |
25 | 23, 24 | oveq12d 7449 |
. . . . 5
⊢ (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)))))) |
26 | | simpr 484 |
. . . . 5
⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin}) → 𝑑 ∈ {ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈
Fin}) |
27 | | ovexd 7466 |
. . . . 5
⊢ ((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) |
28 | 11, 25, 26, 27 | fvmptd3 7039 |
. . . 4
⊢ ((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)))))) |
29 | | fvoveq1 7454 |
. . . . . 6
⊢ (𝑏 = 𝑑 → (𝐺‘(𝑏 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))) = (𝐺‘(𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))))) |
30 | 23, 29 | oveq12d 7449 |
. . . . 5
⊢ (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)))))) |
31 | | ovexd 7466 |
. . . . 5
⊢ ((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) |
32 | 15, 30, 26, 31 | fvmptd3 7039 |
. . . 4
⊢ ((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)))))) |
33 | 13, 17, 20, 20, 21, 28, 32 | offval 7706 |
. . 3
⊢ (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)))))))) |
34 | | eqid 2735 |
. . . . . . . . . 10
⊢
(+g‘𝑅) = (+g‘𝑅) |
35 | | psdadd.p |
. . . . . . . . . 10
⊢ + =
(+g‘𝑆) |
36 | 1, 2, 34, 35, 5, 7 | psradd 21975 |
. . . . . . . . 9
⊢ (𝜑 → (𝐹 + 𝐺) = (𝐹 ∘f
(+g‘𝑅)𝐺)) |
37 | 36 | adantr 480 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin}) → (𝐹 + 𝐺) = (𝐹 ∘f
(+g‘𝑅)𝐺)) |
38 | 37 | fveq1d 6909 |
. . . . . . 7
⊢ ((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))))) |
39 | | reldmpsr 21952 |
. . . . . . . . . . . . 13
⊢ Rel dom
mPwSer |
40 | 1, 2, 39 | strov2rcl 17253 |
. . . . . . . . . . . 12
⊢ (𝐹 ∈ 𝐵 → 𝐼 ∈ V) |
41 | 5, 40 | syl 17 |
. . . . . . . . . . 11
⊢ (𝜑 → 𝐼 ∈ V) |
42 | 3 | psrbagsn 22105 |
. . . . . . . . . . 11
⊢ (𝐼 ∈ V → (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)) ∈ {ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈
Fin}) |
43 | 41, 42 | syl 17 |
. . . . . . . . . 10
⊢ (𝜑 → (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)) ∈ {ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈
Fin}) |
44 | 43 | adantr 480 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin}) → (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)) ∈ {ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈
Fin}) |
45 | 3 | psrbagaddcl 21962 |
. . . . . . . . 9
⊢ ((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}) |
46 | 26, 44, 45 | syl2anc 584 |
. . . . . . . 8
⊢ ((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}) |
47 | | eqid 2735 |
. . . . . . . . . . 11
⊢
(Base‘𝑅) =
(Base‘𝑅) |
48 | 1, 47, 3, 2, 5 | psrelbas 21972 |
. . . . . . . . . 10
⊢ (𝜑 → 𝐹:{ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈
Fin}⟶(Base‘𝑅)) |
49 | 48 | ffnd 6738 |
. . . . . . . . 9
⊢ (𝜑 → 𝐹 Fn {ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈
Fin}) |
50 | 1, 47, 3, 2, 7 | psrelbas 21972 |
. . . . . . . . . 10
⊢ (𝜑 → 𝐺:{ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈
Fin}⟶(Base‘𝑅)) |
51 | 50 | ffnd 6738 |
. . . . . . . . 9
⊢ (𝜑 → 𝐺 Fn {ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈
Fin}) |
52 | | eqidd 2736 |
. . . . . . . . 9
⊢ ((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))))) |
53 | | eqidd 2736 |
. . . . . . . . 9
⊢ ((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))))) |
54 | 49, 51, 20, 20, 21, 52, 53 | ofval 7708 |
. . . . . . . 8
⊢ ((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)))))) |
55 | 46, 54 | syldan 591 |
. . . . . . 7
⊢ ((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)))))) |
56 | 38, 55 | eqtrd 2775 |
. . . . . 6
⊢ ((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)))))) |
57 | 56 | oveq2d 7447 |
. . . . 5
⊢ ((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))))))) |
58 | | psdadd.r |
. . . . . . 7
⊢ (𝜑 → 𝑅 ∈ CMnd) |
59 | 58 | adantr 480 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin}) → 𝑅 ∈ CMnd) |
60 | 3 | psrbagf 21956 |
. . . . . . . . 9
⊢ (𝑑 ∈ {ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin} → 𝑑:𝐼⟶ℕ0) |
61 | 60 | adantl 481 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin}) → 𝑑:𝐼⟶ℕ0) |
62 | 4 | adantr 480 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin}) → 𝑋 ∈ 𝐼) |
63 | 61, 62 | ffvelcdmd 7105 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin}) → (𝑑‘𝑋) ∈
ℕ0) |
64 | | peano2nn0 12564 |
. . . . . . 7
⊢ ((𝑑‘𝑋) ∈ ℕ0 → ((𝑑‘𝑋) + 1) ∈
ℕ0) |
65 | 63, 64 | syl 17 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin}) → ((𝑑‘𝑋) + 1) ∈
ℕ0) |
66 | 5 | adantr 480 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin}) → 𝐹 ∈ 𝐵) |
67 | 1, 47, 3, 2, 66 | psrelbas 21972 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin}) → 𝐹:{ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈
Fin}⟶(Base‘𝑅)) |
68 | 67, 46 | ffvelcdmd 7105 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin}) → (𝐹‘(𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))) ∈ (Base‘𝑅)) |
69 | 50 | adantr 480 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin}) → 𝐺:{ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈
Fin}⟶(Base‘𝑅)) |
70 | 69, 46 | ffvelcdmd 7105 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin}) → (𝐺‘(𝑑 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))) ∈ (Base‘𝑅)) |
71 | | eqid 2735 |
. . . . . . 7
⊢
(.g‘𝑅) = (.g‘𝑅) |
72 | 47, 71, 34 | mulgnn0di 19858 |
. . . . . 6
⊢ ((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 1371 |
. . . . 5
⊢ ((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 | 57, 73 | eqtr2d 2776 |
. . . 4
⊢ ((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 5248 |
. . 3
⊢ (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 | 9, 33, 75 | 3eqtrd 2779 |
. 2
⊢ (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 | 58 | cmnmndd 19837 |
. . . . 5
⊢ (𝜑 → 𝑅 ∈ Mnd) |
78 | | mndmgm 18767 |
. . . . 5
⊢ (𝑅 ∈ Mnd → 𝑅 ∈ Mgm) |
79 | 77, 78 | syl 17 |
. . . 4
⊢ (𝜑 → 𝑅 ∈ Mgm) |
80 | 1, 2, 79, 4, 5 | psdcl 22183 |
. . 3
⊢ (𝜑 → (((𝐼 mPSDer 𝑅)‘𝑋)‘𝐹) ∈ 𝐵) |
81 | 1, 2, 79, 4, 7 | psdcl 22183 |
. . 3
⊢ (𝜑 → (((𝐼 mPSDer 𝑅)‘𝑋)‘𝐺) ∈ 𝐵) |
82 | 1, 2, 34, 35, 80, 81 | psradd 21975 |
. 2
⊢ (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, 35, 79, 5, 7 | psraddcl 21976 |
. . 3
⊢ (𝜑 → (𝐹 + 𝐺) ∈ 𝐵) |
84 | 1, 2, 3, 4, 83 | psdval 22181 |
. 2
⊢ (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 2786 |
1
⊢ (𝜑 → (((𝐼 mPSDer 𝑅)‘𝑋)‘(𝐹 + 𝐺)) = ((((𝐼 mPSDer 𝑅)‘𝑋)‘𝐹) + (((𝐼 mPSDer 𝑅)‘𝑋)‘𝐺))) |