Step | Hyp | Ref
| Expression |
1 | | mplmon.s |
. . 3
⊢ 𝑃 = (𝐼 mPoly 𝑅) |
2 | | mplmon.b |
. . 3
⊢ 𝐵 = (Base‘𝑃) |
3 | | eqid 2610 |
. . 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 19284 |
. . 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 19284 |
. . 3
⊢ (𝜑 → (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 )) ∈ 𝐵) |
14 | 1, 2, 3, 4, 5, 11,
13 | mplmul 19264 |
. 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 2614 |
. . . . 5
⊢ (𝑦 = 𝑘 → (𝑦 = (𝑋 ∘𝑓 + 𝑌) ↔ 𝑘 = (𝑋 ∘𝑓 + 𝑌))) |
16 | 15 | ifbid 4058 |
. . . 4
⊢ (𝑦 = 𝑘 → if(𝑦 = (𝑋 ∘𝑓 + 𝑌), 1 , 0 ) = if(𝑘 = (𝑋 ∘𝑓 + 𝑌), 1 , 0 )) |
17 | 16 | cbvmptv 4678 |
. . 3
⊢ (𝑦 ∈ 𝐷 ↦ if(𝑦 = (𝑋 ∘𝑓 + 𝑌), 1 , 0 )) = (𝑘 ∈ 𝐷 ↦ if(𝑘 = (𝑋 ∘𝑓 + 𝑌), 1 , 0 )) |
18 | | simpr 476 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑋 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ 𝑘}) → 𝑋 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ 𝑘}) |
19 | 18 | snssd 4281 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑋 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ 𝑘}) → {𝑋} ⊆ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ 𝑘}) |
20 | 19 | resmptd 5371 |
. . . . . . . 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 6565 |
. . . . . . 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 758 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑋 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ 𝑘}) → 𝑅 ∈ Ring) |
23 | | ringmnd 18379 |
. . . . . . . . 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 758 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑋 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ 𝑘}) → 𝑋 ∈ 𝐷) |
26 | | iftrue 4042 |
. . . . . . . . . . . . 13
⊢ (𝑦 = 𝑋 → if(𝑦 = 𝑋, 1 , 0 ) = 1 ) |
27 | | eqid 2610 |
. . . . . . . . . . . . 13
⊢ (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 )) = (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 )) |
28 | | fvex 6113 |
. . . . . . . . . . . . . 14
⊢
(1r‘𝑅) ∈ V |
29 | 7, 28 | eqeltri 2684 |
. . . . . . . . . . . . 13
⊢ 1 ∈
V |
30 | 26, 27, 29 | fvmpt 6191 |
. . . . . . . . . . . 12
⊢ (𝑋 ∈ 𝐷 → ((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))‘𝑋) = 1 ) |
31 | 25, 30 | syl 17 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑋 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ 𝑘}) → ((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))‘𝑋) = 1 ) |
32 | | ssrab2 3650 |
. . . . . . . . . . . . 13
⊢ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ 𝑘} ⊆ 𝐷 |
33 | 8 | ad2antrr 758 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑋 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ 𝑘}) → 𝐼 ∈ 𝑊) |
34 | | simplr 788 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑋 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ 𝑘}) → 𝑘 ∈ 𝐷) |
35 | | eqid 2610 |
. . . . . . . . . . . . . . 15
⊢ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ 𝑘} = {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ 𝑘} |
36 | 5, 35 | psrbagconcl 19194 |
. . . . . . . . . . . . . 14
⊢ ((𝐼 ∈ 𝑊 ∧ 𝑘 ∈ 𝐷 ∧ 𝑋 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ 𝑘}) → (𝑘 ∘𝑓 − 𝑋) ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ 𝑘}) |
37 | 33, 34, 18, 36 | syl3anc 1318 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑋 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ 𝑘}) → (𝑘 ∘𝑓 − 𝑋) ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ 𝑘}) |
38 | 32, 37 | sseldi 3566 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑋 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ 𝑘}) → (𝑘 ∘𝑓 − 𝑋) ∈ 𝐷) |
39 | | eqeq1 2614 |
. . . . . . . . . . . . . 14
⊢ (𝑦 = (𝑘 ∘𝑓 − 𝑋) → (𝑦 = 𝑌 ↔ (𝑘 ∘𝑓 − 𝑋) = 𝑌)) |
40 | 39 | ifbid 4058 |
. . . . . . . . . . . . 13
⊢ (𝑦 = (𝑘 ∘𝑓 − 𝑋) → if(𝑦 = 𝑌, 1 , 0 ) = if((𝑘 ∘𝑓 − 𝑋) = 𝑌, 1 , 0 )) |
41 | | eqid 2610 |
. . . . . . . . . . . . 13
⊢ (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 )) = (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 )) |
42 | | fvex 6113 |
. . . . . . . . . . . . . . 15
⊢
(0g‘𝑅) ∈ V |
43 | 6, 42 | eqeltri 2684 |
. . . . . . . . . . . . . 14
⊢ 0 ∈
V |
44 | 29, 43 | ifex 4106 |
. . . . . . . . . . . . 13
⊢ if((𝑘 ∘𝑓
− 𝑋) = 𝑌, 1 , 0 ) ∈
V |
45 | 40, 41, 44 | fvmpt 6191 |
. . . . . . . . . . . 12
⊢ ((𝑘 ∘𝑓
− 𝑋) ∈ 𝐷 → ((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘 ∘𝑓
− 𝑋)) = if((𝑘 ∘𝑓
− 𝑋) = 𝑌, 1 , 0 )) |
46 | 38, 45 | 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 )) |
47 | 31, 46 | oveq12d 6567 |
. . . . . . . . . 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 ))) |
48 | | eqid 2610 |
. . . . . . . . . . . . . 14
⊢
(Base‘𝑅) =
(Base‘𝑅) |
49 | 48, 7 | ringidcl 18391 |
. . . . . . . . . . . . 13
⊢ (𝑅 ∈ Ring → 1 ∈
(Base‘𝑅)) |
50 | 48, 6 | ring0cl 18392 |
. . . . . . . . . . . . 13
⊢ (𝑅 ∈ Ring → 0 ∈
(Base‘𝑅)) |
51 | 49, 50 | ifcld 4081 |
. . . . . . . . . . . 12
⊢ (𝑅 ∈ Ring → if((𝑘 ∘𝑓
− 𝑋) = 𝑌, 1 , 0 ) ∈ (Base‘𝑅)) |
52 | 22, 51 | syl 17 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑋 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ 𝑘}) → if((𝑘 ∘𝑓 − 𝑋) = 𝑌, 1 , 0 ) ∈ (Base‘𝑅)) |
53 | 48, 3, 7 | ringlidm 18394 |
. . . . . . . . . . 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 )) |
54 | 22, 52, 53 | syl2anc 691 |
. . . . . . . . . 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 )) |
55 | 5 | psrbagf 19186 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝐼 ∈ 𝑊 ∧ 𝑘 ∈ 𝐷) → 𝑘:𝐼⟶ℕ0) |
56 | 33, 34, 55 | syl2anc 691 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑋 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ 𝑘}) → 𝑘:𝐼⟶ℕ0) |
57 | 56 | ffvelrnda 6267 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑋 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ 𝑘}) ∧ 𝑧 ∈ 𝐼) → (𝑘‘𝑧) ∈
ℕ0) |
58 | 8 | adantr 480 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ 𝑘 ∈ 𝐷) → 𝐼 ∈ 𝑊) |
59 | 10 | adantr 480 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ 𝑘 ∈ 𝐷) → 𝑋 ∈ 𝐷) |
60 | 5 | psrbagf 19186 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝐼 ∈ 𝑊 ∧ 𝑋 ∈ 𝐷) → 𝑋:𝐼⟶ℕ0) |
61 | 58, 59, 60 | syl2anc 691 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑘 ∈ 𝐷) → 𝑋:𝐼⟶ℕ0) |
62 | 61 | ffvelrnda 6267 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑧 ∈ 𝐼) → (𝑋‘𝑧) ∈
ℕ0) |
63 | 62 | adantlr 747 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑋 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ 𝑘}) ∧ 𝑧 ∈ 𝐼) → (𝑋‘𝑧) ∈
ℕ0) |
64 | 5 | psrbagf 19186 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝐼 ∈ 𝑊 ∧ 𝑌 ∈ 𝐷) → 𝑌:𝐼⟶ℕ0) |
65 | 8, 12, 64 | syl2anc 691 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 → 𝑌:𝐼⟶ℕ0) |
66 | 65 | adantr 480 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑘 ∈ 𝐷) → 𝑌:𝐼⟶ℕ0) |
67 | 66 | ffvelrnda 6267 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑧 ∈ 𝐼) → (𝑌‘𝑧) ∈
ℕ0) |
68 | 67 | adantlr 747 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑋 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ 𝑘}) ∧ 𝑧 ∈ 𝐼) → (𝑌‘𝑧) ∈
ℕ0) |
69 | | nn0cn 11179 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑘‘𝑧) ∈ ℕ0 → (𝑘‘𝑧) ∈ ℂ) |
70 | | nn0cn 11179 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑋‘𝑧) ∈ ℕ0 → (𝑋‘𝑧) ∈ ℂ) |
71 | | nn0cn 11179 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑌‘𝑧) ∈ ℕ0 → (𝑌‘𝑧) ∈ ℂ) |
72 | | subadd 10163 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝑘‘𝑧) ∈ ℂ ∧ (𝑋‘𝑧) ∈ ℂ ∧ (𝑌‘𝑧) ∈ ℂ) → (((𝑘‘𝑧) − (𝑋‘𝑧)) = (𝑌‘𝑧) ↔ ((𝑋‘𝑧) + (𝑌‘𝑧)) = (𝑘‘𝑧))) |
73 | 69, 70, 71, 72 | syl3an 1360 |
. . . . . . . . . . . . . . . 16
⊢ (((𝑘‘𝑧) ∈ ℕ0 ∧ (𝑋‘𝑧) ∈ ℕ0 ∧ (𝑌‘𝑧) ∈ ℕ0) → (((𝑘‘𝑧) − (𝑋‘𝑧)) = (𝑌‘𝑧) ↔ ((𝑋‘𝑧) + (𝑌‘𝑧)) = (𝑘‘𝑧))) |
74 | 57, 63, 68, 73 | syl3anc 1318 |
. . . . . . . . . . . . . . 15
⊢ ((((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑋 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ 𝑘}) ∧ 𝑧 ∈ 𝐼) → (((𝑘‘𝑧) − (𝑋‘𝑧)) = (𝑌‘𝑧) ↔ ((𝑋‘𝑧) + (𝑌‘𝑧)) = (𝑘‘𝑧))) |
75 | | eqcom 2617 |
. . . . . . . . . . . . . . 15
⊢ (((𝑋‘𝑧) + (𝑌‘𝑧)) = (𝑘‘𝑧) ↔ (𝑘‘𝑧) = ((𝑋‘𝑧) + (𝑌‘𝑧))) |
76 | 74, 75 | syl6bb 275 |
. . . . . . . . . . . . . 14
⊢ ((((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑋 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ 𝑘}) ∧ 𝑧 ∈ 𝐼) → (((𝑘‘𝑧) − (𝑋‘𝑧)) = (𝑌‘𝑧) ↔ (𝑘‘𝑧) = ((𝑋‘𝑧) + (𝑌‘𝑧)))) |
77 | 76 | ralbidva 2968 |
. . . . . . . . . . . . 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)))) |
78 | | mpteqb 6207 |
. . . . . . . . . . . . . 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))) |
79 | | ovex 6577 |
. . . . . . . . . . . . . . 15
⊢ ((𝑘‘𝑧) − (𝑋‘𝑧)) ∈ V |
80 | 79 | a1i 11 |
. . . . . . . . . . . . . 14
⊢ (𝑧 ∈ 𝐼 → ((𝑘‘𝑧) − (𝑋‘𝑧)) ∈ V) |
81 | 78, 80 | mprg 2910 |
. . . . . . . . . . . . 13
⊢ ((𝑧 ∈ 𝐼 ↦ ((𝑘‘𝑧) − (𝑋‘𝑧))) = (𝑧 ∈ 𝐼 ↦ (𝑌‘𝑧)) ↔ ∀𝑧 ∈ 𝐼 ((𝑘‘𝑧) − (𝑋‘𝑧)) = (𝑌‘𝑧)) |
82 | | mpteqb 6207 |
. . . . . . . . . . . . . 14
⊢
(∀𝑧 ∈
𝐼 (𝑘‘𝑧) ∈ V → ((𝑧 ∈ 𝐼 ↦ (𝑘‘𝑧)) = (𝑧 ∈ 𝐼 ↦ ((𝑋‘𝑧) + (𝑌‘𝑧))) ↔ ∀𝑧 ∈ 𝐼 (𝑘‘𝑧) = ((𝑋‘𝑧) + (𝑌‘𝑧)))) |
83 | | fvex 6113 |
. . . . . . . . . . . . . . 15
⊢ (𝑘‘𝑧) ∈ V |
84 | 83 | a1i 11 |
. . . . . . . . . . . . . 14
⊢ (𝑧 ∈ 𝐼 → (𝑘‘𝑧) ∈ V) |
85 | 82, 84 | mprg 2910 |
. . . . . . . . . . . . 13
⊢ ((𝑧 ∈ 𝐼 ↦ (𝑘‘𝑧)) = (𝑧 ∈ 𝐼 ↦ ((𝑋‘𝑧) + (𝑌‘𝑧))) ↔ ∀𝑧 ∈ 𝐼 (𝑘‘𝑧) = ((𝑋‘𝑧) + (𝑌‘𝑧))) |
86 | 77, 81, 85 | 3bitr4g 302 |
. . . . . . . . . . . 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))))) |
87 | 56 | feqmptd 6159 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑋 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ 𝑘}) → 𝑘 = (𝑧 ∈ 𝐼 ↦ (𝑘‘𝑧))) |
88 | 61 | feqmptd 6159 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑘 ∈ 𝐷) → 𝑋 = (𝑧 ∈ 𝐼 ↦ (𝑋‘𝑧))) |
89 | 88 | adantr 480 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑋 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ 𝑘}) → 𝑋 = (𝑧 ∈ 𝐼 ↦ (𝑋‘𝑧))) |
90 | 33, 57, 63, 87, 89 | offval2 6812 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑋 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ 𝑘}) → (𝑘 ∘𝑓 − 𝑋) = (𝑧 ∈ 𝐼 ↦ ((𝑘‘𝑧) − (𝑋‘𝑧)))) |
91 | 66 | feqmptd 6159 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑘 ∈ 𝐷) → 𝑌 = (𝑧 ∈ 𝐼 ↦ (𝑌‘𝑧))) |
92 | 91 | adantr 480 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑋 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ 𝑘}) → 𝑌 = (𝑧 ∈ 𝐼 ↦ (𝑌‘𝑧))) |
93 | 90, 92 | eqeq12d 2625 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑋 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ 𝑘}) → ((𝑘 ∘𝑓 − 𝑋) = 𝑌 ↔ (𝑧 ∈ 𝐼 ↦ ((𝑘‘𝑧) − (𝑋‘𝑧))) = (𝑧 ∈ 𝐼 ↦ (𝑌‘𝑧)))) |
94 | 58, 62, 67, 88, 91 | offval2 6812 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑘 ∈ 𝐷) → (𝑋 ∘𝑓 + 𝑌) = (𝑧 ∈ 𝐼 ↦ ((𝑋‘𝑧) + (𝑌‘𝑧)))) |
95 | 94 | adantr 480 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑋 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ 𝑘}) → (𝑋 ∘𝑓 + 𝑌) = (𝑧 ∈ 𝐼 ↦ ((𝑋‘𝑧) + (𝑌‘𝑧)))) |
96 | 87, 95 | eqeq12d 2625 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑋 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ 𝑘}) → (𝑘 = (𝑋 ∘𝑓 + 𝑌) ↔ (𝑧 ∈ 𝐼 ↦ (𝑘‘𝑧)) = (𝑧 ∈ 𝐼 ↦ ((𝑋‘𝑧) + (𝑌‘𝑧))))) |
97 | 86, 93, 96 | 3bitr4d 299 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑋 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ 𝑘}) → ((𝑘 ∘𝑓 − 𝑋) = 𝑌 ↔ 𝑘 = (𝑋 ∘𝑓 + 𝑌))) |
98 | 97 | ifbid 4058 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑋 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ 𝑘}) → if((𝑘 ∘𝑓 − 𝑋) = 𝑌, 1 , 0 ) = if(𝑘 = (𝑋 ∘𝑓 + 𝑌), 1 , 0 )) |
99 | 47, 54, 98 | 3eqtrd 2648 |
. . . . . . . . 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 )) |
100 | 98, 52 | eqeltrrd 2689 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑋 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ 𝑘}) → if(𝑘 = (𝑋 ∘𝑓 + 𝑌), 1 , 0 ) ∈ (Base‘𝑅)) |
101 | 99, 100 | eqeltrd 2688 |
. . . . . . . 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)) |
102 | | fveq2 6103 |
. . . . . . . . . 10
⊢ (𝑗 = 𝑋 → ((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))‘𝑗) = ((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))‘𝑋)) |
103 | | oveq2 6557 |
. . . . . . . . . . 11
⊢ (𝑗 = 𝑋 → (𝑘 ∘𝑓 − 𝑗) = (𝑘 ∘𝑓 − 𝑋)) |
104 | 103 | fveq2d 6107 |
. . . . . . . . . 10
⊢ (𝑗 = 𝑋 → ((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘 ∘𝑓
− 𝑗)) = ((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘 ∘𝑓
− 𝑋))) |
105 | 102, 104 | oveq12d 6567 |
. . . . . . . . 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)))) |
106 | 48, 105 | gsumsn 18177 |
. . . . . . . 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)))) |
107 | 24, 25, 101, 106 | syl3anc 1318 |
. . . . . . 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)))) |
108 | 21, 107, 99 | 3eqtrd 2648 |
. . . . . 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 )) |
109 | 6 | gsum0 17101 |
. . . . . . 7
⊢ (𝑅 Σg
∅) = 0 |
110 | | disjsn 4192 |
. . . . . . . . 9
⊢ (({𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ 𝑘} ∩ {𝑋}) = ∅ ↔ ¬ 𝑋 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ 𝑘}) |
111 | 9 | ad2antrr 758 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑗 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ 𝑘}) → 𝑅 ∈ Ring) |
112 | 1, 48, 2, 5, 11 | mplelf 19254 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 )):𝐷⟶(Base‘𝑅)) |
113 | 112 | ad2antrr 758 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑗 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ 𝑘}) → (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 )):𝐷⟶(Base‘𝑅)) |
114 | | simpr 476 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑗 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ 𝑘}) → 𝑗 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ 𝑘}) |
115 | 32, 114 | sseldi 3566 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑗 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ 𝑘}) → 𝑗 ∈ 𝐷) |
116 | 113, 115 | ffvelrnd 6268 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑗 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ 𝑘}) → ((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))‘𝑗) ∈ (Base‘𝑅)) |
117 | 1, 48, 2, 5, 13 | mplelf 19254 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 )):𝐷⟶(Base‘𝑅)) |
118 | 117 | ad2antrr 758 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑗 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ 𝑘}) → (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 )):𝐷⟶(Base‘𝑅)) |
119 | 8 | ad2antrr 758 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑗 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ 𝑘}) → 𝐼 ∈ 𝑊) |
120 | | simplr 788 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑗 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ 𝑘}) → 𝑘 ∈ 𝐷) |
121 | 5, 35 | psrbagconcl 19194 |
. . . . . . . . . . . . . . . 16
⊢ ((𝐼 ∈ 𝑊 ∧ 𝑘 ∈ 𝐷 ∧ 𝑗 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ 𝑘}) → (𝑘 ∘𝑓 − 𝑗) ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ 𝑘}) |
122 | 119, 120,
114, 121 | syl3anc 1318 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑗 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ 𝑘}) → (𝑘 ∘𝑓 − 𝑗) ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ 𝑘}) |
123 | 32, 122 | sseldi 3566 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑗 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ 𝑘}) → (𝑘 ∘𝑓 − 𝑗) ∈ 𝐷) |
124 | 118, 123 | ffvelrnd 6268 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑗 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ 𝑘}) → ((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘 ∘𝑓
− 𝑗)) ∈
(Base‘𝑅)) |
125 | 48, 3 | ringcl 18384 |
. . . . . . . . . . . . 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)) |
126 | 111, 116,
124, 125 | syl3anc 1318 |
. . . . . . . . . . . 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)) |
127 | | eqid 2610 |
. . . . . . . . . . . 12
⊢ (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)))) = (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)))) |
128 | 126, 127 | fmptd 6292 |
. . . . . . . . . . 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)) |
129 | | ffn 5958 |
. . . . . . . . . . 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}) |
130 | | fnresdisj 5915 |
. . . . . . . . . . 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}) =
∅)) |
131 | 128, 129,
130 | 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}) =
∅)) |
132 | 131 | biimpa 500 |
. . . . . . . . 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}) =
∅) |
133 | 110, 132 | sylan2br 492 |
. . . . . . . 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}) =
∅) |
134 | 133 | oveq2d 6565 |
. . . . . . 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
∅)) |
135 | 62 | nn0red 11229 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑧 ∈ 𝐼) → (𝑋‘𝑧) ∈ ℝ) |
136 | | nn0addge1 11216 |
. . . . . . . . . . . . . 14
⊢ (((𝑋‘𝑧) ∈ ℝ ∧ (𝑌‘𝑧) ∈ ℕ0) → (𝑋‘𝑧) ≤ ((𝑋‘𝑧) + (𝑌‘𝑧))) |
137 | 135, 67, 136 | syl2anc 691 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑧 ∈ 𝐼) → (𝑋‘𝑧) ≤ ((𝑋‘𝑧) + (𝑌‘𝑧))) |
138 | 137 | ralrimiva 2949 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑘 ∈ 𝐷) → ∀𝑧 ∈ 𝐼 (𝑋‘𝑧) ≤ ((𝑋‘𝑧) + (𝑌‘𝑧))) |
139 | | ovex 6577 |
. . . . . . . . . . . . . 14
⊢ ((𝑋‘𝑧) + (𝑌‘𝑧)) ∈ V |
140 | 139 | a1i 11 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑧 ∈ 𝐼) → ((𝑋‘𝑧) + (𝑌‘𝑧)) ∈ V) |
141 | 58, 62, 140, 88, 94 | ofrfval2 6813 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑘 ∈ 𝐷) → (𝑋 ∘𝑟 ≤ (𝑋 ∘𝑓 +
𝑌) ↔ ∀𝑧 ∈ 𝐼 (𝑋‘𝑧) ≤ ((𝑋‘𝑧) + (𝑌‘𝑧)))) |
142 | 138, 141 | mpbird 246 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑘 ∈ 𝐷) → 𝑋 ∘𝑟 ≤ (𝑋 ∘𝑓 +
𝑌)) |
143 | | breq1 4586 |
. . . . . . . . . . . 12
⊢ (𝑥 = 𝑋 → (𝑥 ∘𝑟 ≤ (𝑋 ∘𝑓 +
𝑌) ↔ 𝑋 ∘𝑟 ≤ (𝑋 ∘𝑓 +
𝑌))) |
144 | 143 | elrab 3331 |
. . . . . . . . . . 11
⊢ (𝑋 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ (𝑋 ∘𝑓 +
𝑌)} ↔ (𝑋 ∈ 𝐷 ∧ 𝑋 ∘𝑟 ≤ (𝑋 ∘𝑓 +
𝑌))) |
145 | 59, 142, 144 | sylanbrc 695 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑘 ∈ 𝐷) → 𝑋 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ (𝑋 ∘𝑓 +
𝑌)}) |
146 | | breq2 4587 |
. . . . . . . . . . . 12
⊢ (𝑘 = (𝑋 ∘𝑓 + 𝑌) → (𝑥 ∘𝑟 ≤ 𝑘 ↔ 𝑥 ∘𝑟 ≤ (𝑋 ∘𝑓 +
𝑌))) |
147 | 146 | rabbidv 3164 |
. . . . . . . . . . 11
⊢ (𝑘 = (𝑋 ∘𝑓 + 𝑌) → {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ 𝑘} = {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ (𝑋 ∘𝑓 +
𝑌)}) |
148 | 147 | eleq2d 2673 |
. . . . . . . . . 10
⊢ (𝑘 = (𝑋 ∘𝑓 + 𝑌) → (𝑋 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ 𝑘} ↔ 𝑋 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ (𝑋 ∘𝑓 +
𝑌)})) |
149 | 145, 148 | syl5ibrcom 236 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑘 ∈ 𝐷) → (𝑘 = (𝑋 ∘𝑓 + 𝑌) → 𝑋 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ 𝑘})) |
150 | 149 | con3dimp 456 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ ¬ 𝑋 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ 𝑘}) → ¬ 𝑘 = (𝑋 ∘𝑓 + 𝑌)) |
151 | 150 | iffalsed 4047 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ ¬ 𝑋 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ 𝑘}) → if(𝑘 = (𝑋 ∘𝑓 + 𝑌), 1 , 0 ) = 0 ) |
152 | 109, 134,
151 | 3eqtr4a 2670 |
. . . . . 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 )) |
153 | 108, 152 | pm2.61dan 828 |
. . . . 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 )) |
154 | 9 | adantr 480 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑘 ∈ 𝐷) → 𝑅 ∈ Ring) |
155 | | ringcmn 18404 |
. . . . . . 7
⊢ (𝑅 ∈ Ring → 𝑅 ∈ CMnd) |
156 | 154, 155 | syl 17 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑘 ∈ 𝐷) → 𝑅 ∈ CMnd) |
157 | 5 | psrbaglefi 19193 |
. . . . . . 7
⊢ ((𝐼 ∈ 𝑊 ∧ 𝑘 ∈ 𝐷) → {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ 𝑘} ∈ Fin) |
158 | 8, 157 | sylan 487 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑘 ∈ 𝐷) → {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ 𝑘} ∈ Fin) |
159 | | ssdif 3707 |
. . . . . . . . . . . 12
⊢ ({𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ 𝑘} ⊆ 𝐷 → ({𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ 𝑘} ∖ {𝑋}) ⊆ (𝐷 ∖ {𝑋})) |
160 | 32, 159 | ax-mp 5 |
. . . . . . . . . . 11
⊢ ({𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ 𝑘} ∖ {𝑋}) ⊆ (𝐷 ∖ {𝑋}) |
161 | 160 | sseli 3564 |
. . . . . . . . . 10
⊢ (𝑗 ∈ ({𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ 𝑘} ∖ {𝑋}) → 𝑗 ∈ (𝐷 ∖ {𝑋})) |
162 | 112 | adantr 480 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑘 ∈ 𝐷) → (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 )):𝐷⟶(Base‘𝑅)) |
163 | | eldifsni 4261 |
. . . . . . . . . . . . . . 15
⊢ (𝑦 ∈ (𝐷 ∖ {𝑋}) → 𝑦 ≠ 𝑋) |
164 | 163 | adantl 481 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑦 ∈ (𝐷 ∖ {𝑋})) → 𝑦 ≠ 𝑋) |
165 | 164 | neneqd 2787 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑦 ∈ (𝐷 ∖ {𝑋})) → ¬ 𝑦 = 𝑋) |
166 | 165 | iffalsed 4047 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑦 ∈ (𝐷 ∖ {𝑋})) → if(𝑦 = 𝑋, 1 , 0 ) = 0 ) |
167 | | ovex 6577 |
. . . . . . . . . . . . . 14
⊢
(ℕ0 ↑𝑚 𝐼) ∈ V |
168 | 5, 167 | rabex2 4742 |
. . . . . . . . . . . . 13
⊢ 𝐷 ∈ V |
169 | 168 | a1i 11 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑘 ∈ 𝐷) → 𝐷 ∈ V) |
170 | 166, 169 | suppss2 7216 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑘 ∈ 𝐷) → ((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 )) supp 0 ) ⊆ {𝑋}) |
171 | 43 | a1i 11 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑘 ∈ 𝐷) → 0 ∈ V) |
172 | 162, 170,
169, 171 | suppssr 7213 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑗 ∈ (𝐷 ∖ {𝑋})) → ((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))‘𝑗) = 0 ) |
173 | 161, 172 | sylan2 490 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑗 ∈ ({𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ 𝑘} ∖ {𝑋})) → ((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))‘𝑗) = 0 ) |
174 | 173 | oveq1d 6564 |
. . . . . . . 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)))) |
175 | | eldifi 3694 |
. . . . . . . . 9
⊢ (𝑗 ∈ ({𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ 𝑘} ∖ {𝑋}) → 𝑗 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ 𝑘}) |
176 | 48, 3, 6 | ringlz 18410 |
. . . . . . . . . 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
) |
177 | 111, 124,
176 | syl2anc 691 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑗 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ 𝑘}) → ( 0 (.r‘𝑅)((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘 ∘𝑓
− 𝑗))) = 0
) |
178 | 175, 177 | sylan2 490 |
. . . . . . . 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
) |
179 | 174, 178 | eqtrd 2644 |
. . . . . . 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
) |
180 | 168 | rabex 4740 |
. . . . . . . 8
⊢ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ 𝑘} ∈ V |
181 | 180 | a1i 11 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑘 ∈ 𝐷) → {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ 𝑘} ∈ V) |
182 | 179, 181 | suppss2 7216 |
. . . . . 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}) |
183 | 168 | mptrabex 6392 |
. . . . . . . . 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 |
184 | | funmpt 5840 |
. . . . . . . . 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)))) |
185 | 183, 184,
43 | 3pm3.2i 1232 |
. . . . . . . 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) |
186 | 185 | 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)) |
187 | | snfi 7923 |
. . . . . . . 8
⊢ {𝑋} ∈ Fin |
188 | 187 | a1i 11 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑘 ∈ 𝐷) → {𝑋} ∈ Fin) |
189 | | suppssfifsupp 8173 |
. . . . . . 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
) |
190 | 186, 188,
182, 189 | syl12anc 1316 |
. . . . . 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
) |
191 | 48, 6, 156, 158, 128, 182, 190 | gsumres 18137 |
. . . . 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)))))) |
192 | 153, 191 | eqtr3d 2646 |
. . . 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)))))) |
193 | 192 | mpteq2dva 4672 |
. . 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))))))) |
194 | 17, 193 | syl5eq 2656 |
. 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))))))) |
195 | 14, 194 | eqtr4d 2647 |
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 ))) |