Step | Hyp | Ref
| Expression |
1 | | mplmon.s |
. . 3
⊢ 𝑃 = (𝐼 mPoly 𝑅) |
2 | | mplmon.b |
. . 3
⊢ 𝐵 = (Base‘𝑃) |
3 | | eqid 2821 |
. . 3
⊢
(.r‘𝑅) = (.r‘𝑅) |
4 | | mplmonmul.t |
. . 3
⊢ · =
(.r‘𝑃) |
5 | | mplmon.d |
. . 3
⊢ 𝐷 = {𝑓 ∈ (ℕ0
↑m 𝐼)
∣ (◡𝑓 “ ℕ) ∈
Fin} |
6 | | mplmon.z |
. . . 4
⊢ 0 =
(0g‘𝑅) |
7 | | mplmon.o |
. . . 4
⊢ 1 =
(1r‘𝑅) |
8 | | mplmon.i |
. . . 4
⊢ (𝜑 → 𝐼 ∈ 𝑊) |
9 | | mplmon.r |
. . . 4
⊢ (𝜑 → 𝑅 ∈ Ring) |
10 | | mplmon.x |
. . . 4
⊢ (𝜑 → 𝑋 ∈ 𝐷) |
11 | 1, 2, 6, 7, 5, 8, 9, 10 | mplmon 20238 |
. . 3
⊢ (𝜑 → (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 )) ∈ 𝐵) |
12 | | mplmonmul.x |
. . . 4
⊢ (𝜑 → 𝑌 ∈ 𝐷) |
13 | 1, 2, 6, 7, 5, 8, 9, 12 | mplmon 20238 |
. . 3
⊢ (𝜑 → (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 )) ∈ 𝐵) |
14 | 1, 2, 3, 4, 5, 11,
13 | mplmul 20217 |
. 2
⊢ (𝜑 → ((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 )) · (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))) = (𝑘 ∈ 𝐷 ↦ (𝑅 Σg (𝑗 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘} ↦ (((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))‘𝑗)(.r‘𝑅)((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘 ∘f −
𝑗))))))) |
15 | | eqeq1 2825 |
. . . . 5
⊢ (𝑦 = 𝑘 → (𝑦 = (𝑋 ∘f + 𝑌) ↔ 𝑘 = (𝑋 ∘f + 𝑌))) |
16 | 15 | ifbid 4488 |
. . . 4
⊢ (𝑦 = 𝑘 → if(𝑦 = (𝑋 ∘f + 𝑌), 1 , 0 ) = if(𝑘 = (𝑋 ∘f + 𝑌), 1 , 0 )) |
17 | 16 | cbvmptv 5161 |
. . 3
⊢ (𝑦 ∈ 𝐷 ↦ if(𝑦 = (𝑋 ∘f + 𝑌), 1 , 0 )) = (𝑘 ∈ 𝐷 ↦ if(𝑘 = (𝑋 ∘f + 𝑌), 1 , 0 )) |
18 | | simpr 487 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑋 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘}) → 𝑋 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘}) |
19 | 18 | snssd 4735 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑋 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘}) → {𝑋} ⊆ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘}) |
20 | 19 | resmptd 5902 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑋 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘}) → ((𝑗 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘} ↦ (((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))‘𝑗)(.r‘𝑅)((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘 ∘f −
𝑗)))) ↾ {𝑋}) = (𝑗 ∈ {𝑋} ↦ (((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))‘𝑗)(.r‘𝑅)((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘 ∘f −
𝑗))))) |
21 | 20 | oveq2d 7166 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑋 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘}) → (𝑅 Σg ((𝑗 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘} ↦ (((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))‘𝑗)(.r‘𝑅)((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘 ∘f −
𝑗)))) ↾ {𝑋})) = (𝑅 Σg (𝑗 ∈ {𝑋} ↦ (((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))‘𝑗)(.r‘𝑅)((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘 ∘f −
𝑗)))))) |
22 | 9 | ad2antrr 724 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑋 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘}) → 𝑅 ∈ Ring) |
23 | | ringmnd 19300 |
. . . . . . . . 9
⊢ (𝑅 ∈ Ring → 𝑅 ∈ Mnd) |
24 | 22, 23 | syl 17 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑋 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘}) → 𝑅 ∈ Mnd) |
25 | 10 | ad2antrr 724 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑋 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘}) → 𝑋 ∈ 𝐷) |
26 | | iftrue 4472 |
. . . . . . . . . . . . 13
⊢ (𝑦 = 𝑋 → if(𝑦 = 𝑋, 1 , 0 ) = 1 ) |
27 | | eqid 2821 |
. . . . . . . . . . . . 13
⊢ (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 )) = (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 )) |
28 | 7 | fvexi 6678 |
. . . . . . . . . . . . 13
⊢ 1 ∈
V |
29 | 26, 27, 28 | fvmpt 6762 |
. . . . . . . . . . . 12
⊢ (𝑋 ∈ 𝐷 → ((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))‘𝑋) = 1 ) |
30 | 25, 29 | syl 17 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑋 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘}) → ((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))‘𝑋) = 1 ) |
31 | | ssrab2 4055 |
. . . . . . . . . . . . 13
⊢ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘} ⊆ 𝐷 |
32 | 8 | ad2antrr 724 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑋 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘}) → 𝐼 ∈ 𝑊) |
33 | | simplr 767 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑋 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘}) → 𝑘 ∈ 𝐷) |
34 | | eqid 2821 |
. . . . . . . . . . . . . . 15
⊢ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘} = {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘} |
35 | 5, 34 | psrbagconcl 20147 |
. . . . . . . . . . . . . 14
⊢ ((𝐼 ∈ 𝑊 ∧ 𝑘 ∈ 𝐷 ∧ 𝑋 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘}) → (𝑘 ∘f − 𝑋) ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘}) |
36 | 32, 33, 18, 35 | syl3anc 1367 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑋 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘}) → (𝑘 ∘f − 𝑋) ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘}) |
37 | 31, 36 | sseldi 3964 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑋 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘}) → (𝑘 ∘f − 𝑋) ∈ 𝐷) |
38 | | eqeq1 2825 |
. . . . . . . . . . . . . 14
⊢ (𝑦 = (𝑘 ∘f − 𝑋) → (𝑦 = 𝑌 ↔ (𝑘 ∘f − 𝑋) = 𝑌)) |
39 | 38 | ifbid 4488 |
. . . . . . . . . . . . 13
⊢ (𝑦 = (𝑘 ∘f − 𝑋) → if(𝑦 = 𝑌, 1 , 0 ) = if((𝑘 ∘f − 𝑋) = 𝑌, 1 , 0 )) |
40 | | eqid 2821 |
. . . . . . . . . . . . 13
⊢ (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 )) = (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 )) |
41 | 6 | fvexi 6678 |
. . . . . . . . . . . . . 14
⊢ 0 ∈
V |
42 | 28, 41 | ifex 4514 |
. . . . . . . . . . . . 13
⊢ if((𝑘 ∘f −
𝑋) = 𝑌, 1 , 0 ) ∈
V |
43 | 39, 40, 42 | fvmpt 6762 |
. . . . . . . . . . . 12
⊢ ((𝑘 ∘f −
𝑋) ∈ 𝐷 → ((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘 ∘f −
𝑋)) = if((𝑘 ∘f −
𝑋) = 𝑌, 1 , 0 )) |
44 | 37, 43 | syl 17 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑋 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘}) → ((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘 ∘f −
𝑋)) = if((𝑘 ∘f −
𝑋) = 𝑌, 1 , 0 )) |
45 | 30, 44 | oveq12d 7168 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑋 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘}) → (((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))‘𝑋)(.r‘𝑅)((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘 ∘f −
𝑋))) = ( 1 (.r‘𝑅)if((𝑘 ∘f − 𝑋) = 𝑌, 1 , 0 ))) |
46 | | eqid 2821 |
. . . . . . . . . . . . . 14
⊢
(Base‘𝑅) =
(Base‘𝑅) |
47 | 46, 7 | ringidcl 19312 |
. . . . . . . . . . . . 13
⊢ (𝑅 ∈ Ring → 1 ∈
(Base‘𝑅)) |
48 | 46, 6 | ring0cl 19313 |
. . . . . . . . . . . . 13
⊢ (𝑅 ∈ Ring → 0 ∈
(Base‘𝑅)) |
49 | 47, 48 | ifcld 4511 |
. . . . . . . . . . . 12
⊢ (𝑅 ∈ Ring → if((𝑘 ∘f −
𝑋) = 𝑌, 1 , 0 ) ∈ (Base‘𝑅)) |
50 | 22, 49 | syl 17 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑋 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘}) → if((𝑘 ∘f − 𝑋) = 𝑌, 1 , 0 ) ∈ (Base‘𝑅)) |
51 | 46, 3, 7 | ringlidm 19315 |
. . . . . . . . . . 11
⊢ ((𝑅 ∈ Ring ∧ if((𝑘 ∘f −
𝑋) = 𝑌, 1 , 0 ) ∈ (Base‘𝑅)) → ( 1 (.r‘𝑅)if((𝑘 ∘f − 𝑋) = 𝑌, 1 , 0 )) = if((𝑘 ∘f −
𝑋) = 𝑌, 1 , 0 )) |
52 | 22, 50, 51 | syl2anc 586 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑋 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘}) → ( 1 (.r‘𝑅)if((𝑘 ∘f − 𝑋) = 𝑌, 1 , 0 )) = if((𝑘 ∘f −
𝑋) = 𝑌, 1 , 0 )) |
53 | 5 | psrbagf 20139 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝐼 ∈ 𝑊 ∧ 𝑘 ∈ 𝐷) → 𝑘:𝐼⟶ℕ0) |
54 | 32, 33, 53 | syl2anc 586 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑋 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘}) → 𝑘:𝐼⟶ℕ0) |
55 | 54 | ffvelrnda 6845 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑋 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘}) ∧ 𝑧 ∈ 𝐼) → (𝑘‘𝑧) ∈
ℕ0) |
56 | 8 | adantr 483 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ 𝑘 ∈ 𝐷) → 𝐼 ∈ 𝑊) |
57 | 10 | adantr 483 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ 𝑘 ∈ 𝐷) → 𝑋 ∈ 𝐷) |
58 | 5 | psrbagf 20139 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝐼 ∈ 𝑊 ∧ 𝑋 ∈ 𝐷) → 𝑋:𝐼⟶ℕ0) |
59 | 56, 57, 58 | syl2anc 586 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑘 ∈ 𝐷) → 𝑋:𝐼⟶ℕ0) |
60 | 59 | ffvelrnda 6845 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑧 ∈ 𝐼) → (𝑋‘𝑧) ∈
ℕ0) |
61 | 60 | adantlr 713 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑋 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘}) ∧ 𝑧 ∈ 𝐼) → (𝑋‘𝑧) ∈
ℕ0) |
62 | 5 | psrbagf 20139 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝐼 ∈ 𝑊 ∧ 𝑌 ∈ 𝐷) → 𝑌:𝐼⟶ℕ0) |
63 | 8, 12, 62 | syl2anc 586 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 → 𝑌:𝐼⟶ℕ0) |
64 | 63 | adantr 483 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑘 ∈ 𝐷) → 𝑌:𝐼⟶ℕ0) |
65 | 64 | ffvelrnda 6845 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑧 ∈ 𝐼) → (𝑌‘𝑧) ∈
ℕ0) |
66 | 65 | adantlr 713 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑋 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘}) ∧ 𝑧 ∈ 𝐼) → (𝑌‘𝑧) ∈
ℕ0) |
67 | | nn0cn 11901 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑘‘𝑧) ∈ ℕ0 → (𝑘‘𝑧) ∈ ℂ) |
68 | | nn0cn 11901 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑋‘𝑧) ∈ ℕ0 → (𝑋‘𝑧) ∈ ℂ) |
69 | | nn0cn 11901 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑌‘𝑧) ∈ ℕ0 → (𝑌‘𝑧) ∈ ℂ) |
70 | | subadd 10883 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝑘‘𝑧) ∈ ℂ ∧ (𝑋‘𝑧) ∈ ℂ ∧ (𝑌‘𝑧) ∈ ℂ) → (((𝑘‘𝑧) − (𝑋‘𝑧)) = (𝑌‘𝑧) ↔ ((𝑋‘𝑧) + (𝑌‘𝑧)) = (𝑘‘𝑧))) |
71 | 67, 68, 69, 70 | syl3an 1156 |
. . . . . . . . . . . . . . . 16
⊢ (((𝑘‘𝑧) ∈ ℕ0 ∧ (𝑋‘𝑧) ∈ ℕ0 ∧ (𝑌‘𝑧) ∈ ℕ0) → (((𝑘‘𝑧) − (𝑋‘𝑧)) = (𝑌‘𝑧) ↔ ((𝑋‘𝑧) + (𝑌‘𝑧)) = (𝑘‘𝑧))) |
72 | 55, 61, 66, 71 | syl3anc 1367 |
. . . . . . . . . . . . . . 15
⊢ ((((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑋 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘}) ∧ 𝑧 ∈ 𝐼) → (((𝑘‘𝑧) − (𝑋‘𝑧)) = (𝑌‘𝑧) ↔ ((𝑋‘𝑧) + (𝑌‘𝑧)) = (𝑘‘𝑧))) |
73 | | eqcom 2828 |
. . . . . . . . . . . . . . 15
⊢ (((𝑋‘𝑧) + (𝑌‘𝑧)) = (𝑘‘𝑧) ↔ (𝑘‘𝑧) = ((𝑋‘𝑧) + (𝑌‘𝑧))) |
74 | 72, 73 | syl6bb 289 |
. . . . . . . . . . . . . 14
⊢ ((((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑋 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘}) ∧ 𝑧 ∈ 𝐼) → (((𝑘‘𝑧) − (𝑋‘𝑧)) = (𝑌‘𝑧) ↔ (𝑘‘𝑧) = ((𝑋‘𝑧) + (𝑌‘𝑧)))) |
75 | 74 | ralbidva 3196 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑋 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘}) → (∀𝑧 ∈ 𝐼 ((𝑘‘𝑧) − (𝑋‘𝑧)) = (𝑌‘𝑧) ↔ ∀𝑧 ∈ 𝐼 (𝑘‘𝑧) = ((𝑋‘𝑧) + (𝑌‘𝑧)))) |
76 | | mpteqb 6781 |
. . . . . . . . . . . . . 14
⊢
(∀𝑧 ∈
𝐼 ((𝑘‘𝑧) − (𝑋‘𝑧)) ∈ V → ((𝑧 ∈ 𝐼 ↦ ((𝑘‘𝑧) − (𝑋‘𝑧))) = (𝑧 ∈ 𝐼 ↦ (𝑌‘𝑧)) ↔ ∀𝑧 ∈ 𝐼 ((𝑘‘𝑧) − (𝑋‘𝑧)) = (𝑌‘𝑧))) |
77 | | ovexd 7185 |
. . . . . . . . . . . . . 14
⊢ (𝑧 ∈ 𝐼 → ((𝑘‘𝑧) − (𝑋‘𝑧)) ∈ V) |
78 | 76, 77 | mprg 3152 |
. . . . . . . . . . . . 13
⊢ ((𝑧 ∈ 𝐼 ↦ ((𝑘‘𝑧) − (𝑋‘𝑧))) = (𝑧 ∈ 𝐼 ↦ (𝑌‘𝑧)) ↔ ∀𝑧 ∈ 𝐼 ((𝑘‘𝑧) − (𝑋‘𝑧)) = (𝑌‘𝑧)) |
79 | | mpteqb 6781 |
. . . . . . . . . . . . . 14
⊢
(∀𝑧 ∈
𝐼 (𝑘‘𝑧) ∈ V → ((𝑧 ∈ 𝐼 ↦ (𝑘‘𝑧)) = (𝑧 ∈ 𝐼 ↦ ((𝑋‘𝑧) + (𝑌‘𝑧))) ↔ ∀𝑧 ∈ 𝐼 (𝑘‘𝑧) = ((𝑋‘𝑧) + (𝑌‘𝑧)))) |
80 | | fvexd 6679 |
. . . . . . . . . . . . . 14
⊢ (𝑧 ∈ 𝐼 → (𝑘‘𝑧) ∈ V) |
81 | 79, 80 | mprg 3152 |
. . . . . . . . . . . . 13
⊢ ((𝑧 ∈ 𝐼 ↦ (𝑘‘𝑧)) = (𝑧 ∈ 𝐼 ↦ ((𝑋‘𝑧) + (𝑌‘𝑧))) ↔ ∀𝑧 ∈ 𝐼 (𝑘‘𝑧) = ((𝑋‘𝑧) + (𝑌‘𝑧))) |
82 | 75, 78, 81 | 3bitr4g 316 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑋 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘}) → ((𝑧 ∈ 𝐼 ↦ ((𝑘‘𝑧) − (𝑋‘𝑧))) = (𝑧 ∈ 𝐼 ↦ (𝑌‘𝑧)) ↔ (𝑧 ∈ 𝐼 ↦ (𝑘‘𝑧)) = (𝑧 ∈ 𝐼 ↦ ((𝑋‘𝑧) + (𝑌‘𝑧))))) |
83 | 54 | feqmptd 6727 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑋 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘}) → 𝑘 = (𝑧 ∈ 𝐼 ↦ (𝑘‘𝑧))) |
84 | 59 | feqmptd 6727 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑘 ∈ 𝐷) → 𝑋 = (𝑧 ∈ 𝐼 ↦ (𝑋‘𝑧))) |
85 | 84 | adantr 483 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑋 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘}) → 𝑋 = (𝑧 ∈ 𝐼 ↦ (𝑋‘𝑧))) |
86 | 32, 55, 61, 83, 85 | offval2 7420 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑋 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘}) → (𝑘 ∘f − 𝑋) = (𝑧 ∈ 𝐼 ↦ ((𝑘‘𝑧) − (𝑋‘𝑧)))) |
87 | 64 | feqmptd 6727 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑘 ∈ 𝐷) → 𝑌 = (𝑧 ∈ 𝐼 ↦ (𝑌‘𝑧))) |
88 | 87 | adantr 483 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑋 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘}) → 𝑌 = (𝑧 ∈ 𝐼 ↦ (𝑌‘𝑧))) |
89 | 86, 88 | eqeq12d 2837 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑋 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘}) → ((𝑘 ∘f − 𝑋) = 𝑌 ↔ (𝑧 ∈ 𝐼 ↦ ((𝑘‘𝑧) − (𝑋‘𝑧))) = (𝑧 ∈ 𝐼 ↦ (𝑌‘𝑧)))) |
90 | 56, 60, 65, 84, 87 | offval2 7420 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑘 ∈ 𝐷) → (𝑋 ∘f + 𝑌) = (𝑧 ∈ 𝐼 ↦ ((𝑋‘𝑧) + (𝑌‘𝑧)))) |
91 | 90 | adantr 483 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑋 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘}) → (𝑋 ∘f + 𝑌) = (𝑧 ∈ 𝐼 ↦ ((𝑋‘𝑧) + (𝑌‘𝑧)))) |
92 | 83, 91 | eqeq12d 2837 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑋 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘}) → (𝑘 = (𝑋 ∘f + 𝑌) ↔ (𝑧 ∈ 𝐼 ↦ (𝑘‘𝑧)) = (𝑧 ∈ 𝐼 ↦ ((𝑋‘𝑧) + (𝑌‘𝑧))))) |
93 | 82, 89, 92 | 3bitr4d 313 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑋 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘}) → ((𝑘 ∘f − 𝑋) = 𝑌 ↔ 𝑘 = (𝑋 ∘f + 𝑌))) |
94 | 93 | ifbid 4488 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑋 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘}) → if((𝑘 ∘f − 𝑋) = 𝑌, 1 , 0 ) = if(𝑘 = (𝑋 ∘f + 𝑌), 1 , 0 )) |
95 | 45, 52, 94 | 3eqtrd 2860 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑋 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘}) → (((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))‘𝑋)(.r‘𝑅)((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘 ∘f −
𝑋))) = if(𝑘 = (𝑋 ∘f + 𝑌), 1 , 0 )) |
96 | 94, 50 | eqeltrrd 2914 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑋 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘}) → if(𝑘 = (𝑋 ∘f + 𝑌), 1 , 0 ) ∈ (Base‘𝑅)) |
97 | 95, 96 | eqeltrd 2913 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑋 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘}) → (((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))‘𝑋)(.r‘𝑅)((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘 ∘f −
𝑋))) ∈
(Base‘𝑅)) |
98 | | fveq2 6664 |
. . . . . . . . . 10
⊢ (𝑗 = 𝑋 → ((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))‘𝑗) = ((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))‘𝑋)) |
99 | | oveq2 7158 |
. . . . . . . . . . 11
⊢ (𝑗 = 𝑋 → (𝑘 ∘f − 𝑗) = (𝑘 ∘f − 𝑋)) |
100 | 99 | fveq2d 6668 |
. . . . . . . . . 10
⊢ (𝑗 = 𝑋 → ((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘 ∘f −
𝑗)) = ((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘 ∘f −
𝑋))) |
101 | 98, 100 | oveq12d 7168 |
. . . . . . . . 9
⊢ (𝑗 = 𝑋 → (((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))‘𝑗)(.r‘𝑅)((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘 ∘f −
𝑗))) = (((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))‘𝑋)(.r‘𝑅)((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘 ∘f −
𝑋)))) |
102 | 46, 101 | gsumsn 19068 |
. . . . . . . 8
⊢ ((𝑅 ∈ Mnd ∧ 𝑋 ∈ 𝐷 ∧ (((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))‘𝑋)(.r‘𝑅)((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘 ∘f −
𝑋))) ∈
(Base‘𝑅)) →
(𝑅
Σg (𝑗 ∈ {𝑋} ↦ (((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))‘𝑗)(.r‘𝑅)((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘 ∘f −
𝑗))))) = (((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))‘𝑋)(.r‘𝑅)((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘 ∘f −
𝑋)))) |
103 | 24, 25, 97, 102 | syl3anc 1367 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑋 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘}) → (𝑅 Σg (𝑗 ∈ {𝑋} ↦ (((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))‘𝑗)(.r‘𝑅)((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘 ∘f −
𝑗))))) = (((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))‘𝑋)(.r‘𝑅)((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘 ∘f −
𝑋)))) |
104 | 21, 103, 95 | 3eqtrd 2860 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑋 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘}) → (𝑅 Σg ((𝑗 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘} ↦ (((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))‘𝑗)(.r‘𝑅)((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘 ∘f −
𝑗)))) ↾ {𝑋})) = if(𝑘 = (𝑋 ∘f + 𝑌), 1 , 0 )) |
105 | 6 | gsum0 17888 |
. . . . . . 7
⊢ (𝑅 Σg
∅) = 0 |
106 | | disjsn 4640 |
. . . . . . . . 9
⊢ (({𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘} ∩ {𝑋}) = ∅ ↔ ¬ 𝑋 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘}) |
107 | 9 | ad2antrr 724 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑗 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘}) → 𝑅 ∈ Ring) |
108 | 1, 46, 2, 5, 11 | mplelf 20207 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 )):𝐷⟶(Base‘𝑅)) |
109 | 108 | ad2antrr 724 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑗 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘}) → (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 )):𝐷⟶(Base‘𝑅)) |
110 | | simpr 487 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑗 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘}) → 𝑗 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘}) |
111 | 31, 110 | sseldi 3964 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑗 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘}) → 𝑗 ∈ 𝐷) |
112 | 109, 111 | ffvelrnd 6846 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑗 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘}) → ((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))‘𝑗) ∈ (Base‘𝑅)) |
113 | 1, 46, 2, 5, 13 | mplelf 20207 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 )):𝐷⟶(Base‘𝑅)) |
114 | 113 | ad2antrr 724 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑗 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘}) → (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 )):𝐷⟶(Base‘𝑅)) |
115 | 8 | ad2antrr 724 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑗 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘}) → 𝐼 ∈ 𝑊) |
116 | | simplr 767 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑗 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘}) → 𝑘 ∈ 𝐷) |
117 | 5, 34 | psrbagconcl 20147 |
. . . . . . . . . . . . . . . 16
⊢ ((𝐼 ∈ 𝑊 ∧ 𝑘 ∈ 𝐷 ∧ 𝑗 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘}) → (𝑘 ∘f − 𝑗) ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘}) |
118 | 115, 116,
110, 117 | syl3anc 1367 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑗 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘}) → (𝑘 ∘f − 𝑗) ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘}) |
119 | 31, 118 | sseldi 3964 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑗 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘}) → (𝑘 ∘f − 𝑗) ∈ 𝐷) |
120 | 114, 119 | ffvelrnd 6846 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑗 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘}) → ((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘 ∘f −
𝑗)) ∈
(Base‘𝑅)) |
121 | 46, 3 | ringcl 19305 |
. . . . . . . . . . . . 13
⊢ ((𝑅 ∈ Ring ∧ ((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))‘𝑗) ∈ (Base‘𝑅) ∧ ((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘 ∘f −
𝑗)) ∈
(Base‘𝑅)) →
(((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))‘𝑗)(.r‘𝑅)((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘 ∘f −
𝑗))) ∈
(Base‘𝑅)) |
122 | 107, 112,
120, 121 | syl3anc 1367 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑗 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘}) → (((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))‘𝑗)(.r‘𝑅)((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘 ∘f −
𝑗))) ∈
(Base‘𝑅)) |
123 | 122 | fmpttd 6873 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑘 ∈ 𝐷) → (𝑗 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘} ↦ (((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))‘𝑗)(.r‘𝑅)((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘 ∘f −
𝑗)))):{𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘}⟶(Base‘𝑅)) |
124 | | ffn 6508 |
. . . . . . . . . . 11
⊢ ((𝑗 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘} ↦ (((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))‘𝑗)(.r‘𝑅)((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘 ∘f −
𝑗)))):{𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘}⟶(Base‘𝑅) → (𝑗 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘} ↦ (((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))‘𝑗)(.r‘𝑅)((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘 ∘f −
𝑗)))) Fn {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘}) |
125 | | fnresdisj 6461 |
. . . . . . . . . . 11
⊢ ((𝑗 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘} ↦ (((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))‘𝑗)(.r‘𝑅)((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘 ∘f −
𝑗)))) Fn {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘} → (({𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘} ∩ {𝑋}) = ∅ ↔ ((𝑗 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘} ↦ (((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))‘𝑗)(.r‘𝑅)((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘 ∘f −
𝑗)))) ↾ {𝑋}) = ∅)) |
126 | 123, 124,
125 | 3syl 18 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑘 ∈ 𝐷) → (({𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘} ∩ {𝑋}) = ∅ ↔ ((𝑗 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘} ↦ (((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))‘𝑗)(.r‘𝑅)((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘 ∘f −
𝑗)))) ↾ {𝑋}) = ∅)) |
127 | 126 | biimpa 479 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ ({𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘} ∩ {𝑋}) = ∅) → ((𝑗 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘} ↦ (((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))‘𝑗)(.r‘𝑅)((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘 ∘f −
𝑗)))) ↾ {𝑋}) = ∅) |
128 | 106, 127 | sylan2br 596 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ ¬ 𝑋 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘}) → ((𝑗 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘} ↦ (((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))‘𝑗)(.r‘𝑅)((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘 ∘f −
𝑗)))) ↾ {𝑋}) = ∅) |
129 | 128 | oveq2d 7166 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ ¬ 𝑋 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘}) → (𝑅 Σg ((𝑗 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘} ↦ (((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))‘𝑗)(.r‘𝑅)((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘 ∘f −
𝑗)))) ↾ {𝑋})) = (𝑅 Σg
∅)) |
130 | | breq1 5061 |
. . . . . . . . . . 11
⊢ (𝑥 = 𝑋 → (𝑥 ∘r ≤ (𝑋 ∘f + 𝑌) ↔ 𝑋 ∘r ≤ (𝑋 ∘f + 𝑌))) |
131 | 60 | nn0red 11950 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑧 ∈ 𝐼) → (𝑋‘𝑧) ∈ ℝ) |
132 | | nn0addge1 11937 |
. . . . . . . . . . . . . 14
⊢ (((𝑋‘𝑧) ∈ ℝ ∧ (𝑌‘𝑧) ∈ ℕ0) → (𝑋‘𝑧) ≤ ((𝑋‘𝑧) + (𝑌‘𝑧))) |
133 | 131, 65, 132 | syl2anc 586 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑧 ∈ 𝐼) → (𝑋‘𝑧) ≤ ((𝑋‘𝑧) + (𝑌‘𝑧))) |
134 | 133 | ralrimiva 3182 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑘 ∈ 𝐷) → ∀𝑧 ∈ 𝐼 (𝑋‘𝑧) ≤ ((𝑋‘𝑧) + (𝑌‘𝑧))) |
135 | | ovexd 7185 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑧 ∈ 𝐼) → ((𝑋‘𝑧) + (𝑌‘𝑧)) ∈ V) |
136 | 56, 60, 135, 84, 90 | ofrfval2 7421 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑘 ∈ 𝐷) → (𝑋 ∘r ≤ (𝑋 ∘f + 𝑌) ↔ ∀𝑧 ∈ 𝐼 (𝑋‘𝑧) ≤ ((𝑋‘𝑧) + (𝑌‘𝑧)))) |
137 | 134, 136 | mpbird 259 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑘 ∈ 𝐷) → 𝑋 ∘r ≤ (𝑋 ∘f + 𝑌)) |
138 | 130, 57, 137 | elrabd 3681 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑘 ∈ 𝐷) → 𝑋 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ (𝑋 ∘f + 𝑌)}) |
139 | | breq2 5062 |
. . . . . . . . . . . 12
⊢ (𝑘 = (𝑋 ∘f + 𝑌) → (𝑥 ∘r ≤ 𝑘 ↔ 𝑥 ∘r ≤ (𝑋 ∘f + 𝑌))) |
140 | 139 | rabbidv 3480 |
. . . . . . . . . . 11
⊢ (𝑘 = (𝑋 ∘f + 𝑌) → {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘} = {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ (𝑋 ∘f + 𝑌)}) |
141 | 140 | eleq2d 2898 |
. . . . . . . . . 10
⊢ (𝑘 = (𝑋 ∘f + 𝑌) → (𝑋 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘} ↔ 𝑋 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ (𝑋 ∘f + 𝑌)})) |
142 | 138, 141 | syl5ibrcom 249 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑘 ∈ 𝐷) → (𝑘 = (𝑋 ∘f + 𝑌) → 𝑋 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘})) |
143 | 142 | con3dimp 411 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ ¬ 𝑋 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘}) → ¬ 𝑘 = (𝑋 ∘f + 𝑌)) |
144 | 143 | iffalsed 4477 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ ¬ 𝑋 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘}) → if(𝑘 = (𝑋 ∘f + 𝑌), 1 , 0 ) = 0 ) |
145 | 105, 129,
144 | 3eqtr4a 2882 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ ¬ 𝑋 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘}) → (𝑅 Σg ((𝑗 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘} ↦ (((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))‘𝑗)(.r‘𝑅)((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘 ∘f −
𝑗)))) ↾ {𝑋})) = if(𝑘 = (𝑋 ∘f + 𝑌), 1 , 0 )) |
146 | 104, 145 | pm2.61dan 811 |
. . . . 5
⊢ ((𝜑 ∧ 𝑘 ∈ 𝐷) → (𝑅 Σg ((𝑗 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘} ↦ (((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))‘𝑗)(.r‘𝑅)((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘 ∘f −
𝑗)))) ↾ {𝑋})) = if(𝑘 = (𝑋 ∘f + 𝑌), 1 , 0 )) |
147 | 9 | adantr 483 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑘 ∈ 𝐷) → 𝑅 ∈ Ring) |
148 | | ringcmn 19325 |
. . . . . . 7
⊢ (𝑅 ∈ Ring → 𝑅 ∈ CMnd) |
149 | 147, 148 | syl 17 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑘 ∈ 𝐷) → 𝑅 ∈ CMnd) |
150 | 5 | psrbaglefi 20146 |
. . . . . . 7
⊢ ((𝐼 ∈ 𝑊 ∧ 𝑘 ∈ 𝐷) → {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘} ∈ Fin) |
151 | 8, 150 | sylan 582 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑘 ∈ 𝐷) → {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘} ∈ Fin) |
152 | | ssdif 4115 |
. . . . . . . . . . . 12
⊢ ({𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘} ⊆ 𝐷 → ({𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘} ∖ {𝑋}) ⊆ (𝐷 ∖ {𝑋})) |
153 | 31, 152 | ax-mp 5 |
. . . . . . . . . . 11
⊢ ({𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘} ∖ {𝑋}) ⊆ (𝐷 ∖ {𝑋}) |
154 | 153 | sseli 3962 |
. . . . . . . . . 10
⊢ (𝑗 ∈ ({𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘} ∖ {𝑋}) → 𝑗 ∈ (𝐷 ∖ {𝑋})) |
155 | 108 | adantr 483 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑘 ∈ 𝐷) → (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 )):𝐷⟶(Base‘𝑅)) |
156 | | eldifsni 4715 |
. . . . . . . . . . . . . . 15
⊢ (𝑦 ∈ (𝐷 ∖ {𝑋}) → 𝑦 ≠ 𝑋) |
157 | 156 | adantl 484 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑦 ∈ (𝐷 ∖ {𝑋})) → 𝑦 ≠ 𝑋) |
158 | 157 | neneqd 3021 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑦 ∈ (𝐷 ∖ {𝑋})) → ¬ 𝑦 = 𝑋) |
159 | 158 | iffalsed 4477 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑦 ∈ (𝐷 ∖ {𝑋})) → if(𝑦 = 𝑋, 1 , 0 ) = 0 ) |
160 | | ovex 7183 |
. . . . . . . . . . . . . 14
⊢
(ℕ0 ↑m 𝐼) ∈ V |
161 | 5, 160 | rabex2 5229 |
. . . . . . . . . . . . 13
⊢ 𝐷 ∈ V |
162 | 161 | a1i 11 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑘 ∈ 𝐷) → 𝐷 ∈ V) |
163 | 159, 162 | suppss2 7858 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑘 ∈ 𝐷) → ((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 )) supp 0 ) ⊆ {𝑋}) |
164 | 41 | a1i 11 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑘 ∈ 𝐷) → 0 ∈ V) |
165 | 155, 163,
162, 164 | suppssr 7855 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑗 ∈ (𝐷 ∖ {𝑋})) → ((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))‘𝑗) = 0 ) |
166 | 154, 165 | sylan2 594 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑗 ∈ ({𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘} ∖ {𝑋})) → ((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))‘𝑗) = 0 ) |
167 | 166 | oveq1d 7165 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑗 ∈ ({𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘} ∖ {𝑋})) → (((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))‘𝑗)(.r‘𝑅)((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘 ∘f −
𝑗))) = ( 0 (.r‘𝑅)((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘 ∘f −
𝑗)))) |
168 | | eldifi 4102 |
. . . . . . . . 9
⊢ (𝑗 ∈ ({𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘} ∖ {𝑋}) → 𝑗 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘}) |
169 | 46, 3, 6 | ringlz 19331 |
. . . . . . . . . 10
⊢ ((𝑅 ∈ Ring ∧ ((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘 ∘f −
𝑗)) ∈
(Base‘𝑅)) → (
0
(.r‘𝑅)((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘 ∘f −
𝑗))) = 0 ) |
170 | 107, 120,
169 | syl2anc 586 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑗 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘}) → ( 0 (.r‘𝑅)((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘 ∘f −
𝑗))) = 0 ) |
171 | 168, 170 | sylan2 594 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑗 ∈ ({𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘} ∖ {𝑋})) → ( 0 (.r‘𝑅)((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘 ∘f −
𝑗))) = 0 ) |
172 | 167, 171 | eqtrd 2856 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑗 ∈ ({𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘} ∖ {𝑋})) → (((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))‘𝑗)(.r‘𝑅)((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘 ∘f −
𝑗))) = 0 ) |
173 | 161 | rabex 5227 |
. . . . . . . 8
⊢ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘} ∈ V |
174 | 173 | a1i 11 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑘 ∈ 𝐷) → {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘} ∈ V) |
175 | 172, 174 | suppss2 7858 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑘 ∈ 𝐷) → ((𝑗 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘} ↦ (((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))‘𝑗)(.r‘𝑅)((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘 ∘f −
𝑗)))) supp 0 ) ⊆
{𝑋}) |
176 | 161 | mptrabex 6982 |
. . . . . . . . 9
⊢ (𝑗 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘} ↦ (((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))‘𝑗)(.r‘𝑅)((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘 ∘f −
𝑗)))) ∈
V |
177 | | funmpt 6387 |
. . . . . . . . 9
⊢ Fun
(𝑗 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘} ↦ (((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))‘𝑗)(.r‘𝑅)((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘 ∘f −
𝑗)))) |
178 | 176, 177,
41 | 3pm3.2i 1335 |
. . . . . . . 8
⊢ ((𝑗 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘} ↦ (((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))‘𝑗)(.r‘𝑅)((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘 ∘f −
𝑗)))) ∈ V ∧ Fun
(𝑗 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘} ↦ (((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))‘𝑗)(.r‘𝑅)((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘 ∘f −
𝑗)))) ∧ 0 ∈
V) |
179 | 178 | a1i 11 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑘 ∈ 𝐷) → ((𝑗 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘} ↦ (((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))‘𝑗)(.r‘𝑅)((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘 ∘f −
𝑗)))) ∈ V ∧ Fun
(𝑗 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘} ↦ (((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))‘𝑗)(.r‘𝑅)((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘 ∘f −
𝑗)))) ∧ 0 ∈
V)) |
180 | | snfi 8588 |
. . . . . . . 8
⊢ {𝑋} ∈ Fin |
181 | 180 | a1i 11 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑘 ∈ 𝐷) → {𝑋} ∈ Fin) |
182 | | suppssfifsupp 8842 |
. . . . . . 7
⊢ ((((𝑗 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘} ↦ (((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))‘𝑗)(.r‘𝑅)((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘 ∘f −
𝑗)))) ∈ V ∧ Fun
(𝑗 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘} ↦ (((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))‘𝑗)(.r‘𝑅)((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘 ∘f −
𝑗)))) ∧ 0 ∈ V)
∧ ({𝑋} ∈ Fin ∧
((𝑗 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘} ↦ (((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))‘𝑗)(.r‘𝑅)((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘 ∘f −
𝑗)))) supp 0 ) ⊆
{𝑋})) → (𝑗 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘} ↦ (((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))‘𝑗)(.r‘𝑅)((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘 ∘f −
𝑗)))) finSupp 0
) |
183 | 179, 181,
175, 182 | syl12anc 834 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑘 ∈ 𝐷) → (𝑗 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘} ↦ (((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))‘𝑗)(.r‘𝑅)((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘 ∘f −
𝑗)))) finSupp 0
) |
184 | 46, 6, 149, 151, 123, 175, 183 | gsumres 19027 |
. . . . 5
⊢ ((𝜑 ∧ 𝑘 ∈ 𝐷) → (𝑅 Σg ((𝑗 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘} ↦ (((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))‘𝑗)(.r‘𝑅)((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘 ∘f −
𝑗)))) ↾ {𝑋})) = (𝑅 Σg (𝑗 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘} ↦ (((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))‘𝑗)(.r‘𝑅)((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘 ∘f −
𝑗)))))) |
185 | 146, 184 | eqtr3d 2858 |
. . . 4
⊢ ((𝜑 ∧ 𝑘 ∈ 𝐷) → if(𝑘 = (𝑋 ∘f + 𝑌), 1 , 0 ) = (𝑅 Σg (𝑗 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘} ↦ (((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))‘𝑗)(.r‘𝑅)((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘 ∘f −
𝑗)))))) |
186 | 185 | mpteq2dva 5153 |
. . 3
⊢ (𝜑 → (𝑘 ∈ 𝐷 ↦ if(𝑘 = (𝑋 ∘f + 𝑌), 1 , 0 )) = (𝑘 ∈ 𝐷 ↦ (𝑅 Σg (𝑗 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘} ↦ (((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))‘𝑗)(.r‘𝑅)((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘 ∘f −
𝑗))))))) |
187 | 17, 186 | syl5eq 2868 |
. 2
⊢ (𝜑 → (𝑦 ∈ 𝐷 ↦ if(𝑦 = (𝑋 ∘f + 𝑌), 1 , 0 )) = (𝑘 ∈ 𝐷 ↦ (𝑅 Σg (𝑗 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘} ↦ (((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))‘𝑗)(.r‘𝑅)((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘 ∘f −
𝑗))))))) |
188 | 14, 187 | eqtr4d 2859 |
1
⊢ (𝜑 → ((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 )) · (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))) = (𝑦 ∈ 𝐷 ↦ if(𝑦 = (𝑋 ∘f + 𝑌), 1 , 0 ))) |