Step | Hyp | Ref
| Expression |
1 | | mplmon.s |
. . 3
⊢ 𝑃 = (𝐼 mPoly 𝑅) |
2 | | mplmon.b |
. . 3
⊢ 𝐵 = (Base‘𝑃) |
3 | | eqid 2622 |
. . 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 19463 |
. . 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 19463 |
. . 3
⊢ (𝜑 → (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 )) ∈ 𝐵) |
14 | 1, 2, 3, 4, 5, 11,
13 | mplmul 19443 |
. 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 2626 |
. . . . 5
⊢ (𝑦 = 𝑘 → (𝑦 = (𝑋 ∘𝑓 + 𝑌) ↔ 𝑘 = (𝑋 ∘𝑓 + 𝑌))) |
16 | 15 | ifbid 4108 |
. . . 4
⊢ (𝑦 = 𝑘 → if(𝑦 = (𝑋 ∘𝑓 + 𝑌), 1 , 0 ) = if(𝑘 = (𝑋 ∘𝑓 + 𝑌), 1 , 0 )) |
17 | 16 | cbvmptv 4750 |
. . 3
⊢ (𝑦 ∈ 𝐷 ↦ if(𝑦 = (𝑋 ∘𝑓 + 𝑌), 1 , 0 )) = (𝑘 ∈ 𝐷 ↦ if(𝑘 = (𝑋 ∘𝑓 + 𝑌), 1 , 0 )) |
18 | | simpr 477 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑋 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ 𝑘}) → 𝑋 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ 𝑘}) |
19 | 18 | snssd 4340 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑋 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ 𝑘}) → {𝑋} ⊆ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ 𝑘}) |
20 | 19 | resmptd 5452 |
. . . . . . . 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 6666 |
. . . . . . 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 762 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑋 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ 𝑘}) → 𝑅 ∈ Ring) |
23 | | ringmnd 18556 |
. . . . . . . . 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 762 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑋 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ 𝑘}) → 𝑋 ∈ 𝐷) |
26 | | iftrue 4092 |
. . . . . . . . . . . . 13
⊢ (𝑦 = 𝑋 → if(𝑦 = 𝑋, 1 , 0 ) = 1 ) |
27 | | eqid 2622 |
. . . . . . . . . . . . 13
⊢ (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 )) = (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 )) |
28 | | fvex 6201 |
. . . . . . . . . . . . . 14
⊢
(1r‘𝑅) ∈ V |
29 | 7, 28 | eqeltri 2697 |
. . . . . . . . . . . . 13
⊢ 1 ∈
V |
30 | 26, 27, 29 | fvmpt 6282 |
. . . . . . . . . . . 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 3687 |
. . . . . . . . . . . . 13
⊢ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ 𝑘} ⊆ 𝐷 |
33 | 8 | ad2antrr 762 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑋 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ 𝑘}) → 𝐼 ∈ 𝑊) |
34 | | simplr 792 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑋 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ 𝑘}) → 𝑘 ∈ 𝐷) |
35 | | eqid 2622 |
. . . . . . . . . . . . . . 15
⊢ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ 𝑘} = {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ 𝑘} |
36 | 5, 35 | psrbagconcl 19373 |
. . . . . . . . . . . . . 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 1326 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑋 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ 𝑘}) → (𝑘 ∘𝑓 − 𝑋) ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ 𝑘}) |
38 | 32, 37 | sseldi 3601 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑋 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ 𝑘}) → (𝑘 ∘𝑓 − 𝑋) ∈ 𝐷) |
39 | | eqeq1 2626 |
. . . . . . . . . . . . . 14
⊢ (𝑦 = (𝑘 ∘𝑓 − 𝑋) → (𝑦 = 𝑌 ↔ (𝑘 ∘𝑓 − 𝑋) = 𝑌)) |
40 | 39 | ifbid 4108 |
. . . . . . . . . . . . 13
⊢ (𝑦 = (𝑘 ∘𝑓 − 𝑋) → if(𝑦 = 𝑌, 1 , 0 ) = if((𝑘 ∘𝑓 − 𝑋) = 𝑌, 1 , 0 )) |
41 | | eqid 2622 |
. . . . . . . . . . . . 13
⊢ (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 )) = (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 )) |
42 | | fvex 6201 |
. . . . . . . . . . . . . . 15
⊢
(0g‘𝑅) ∈ V |
43 | 6, 42 | eqeltri 2697 |
. . . . . . . . . . . . . 14
⊢ 0 ∈
V |
44 | 29, 43 | ifex 4156 |
. . . . . . . . . . . . 13
⊢ if((𝑘 ∘𝑓
− 𝑋) = 𝑌, 1 , 0 ) ∈
V |
45 | 40, 41, 44 | fvmpt 6282 |
. . . . . . . . . . . 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 6668 |
. . . . . . . . . 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 2622 |
. . . . . . . . . . . . . 14
⊢
(Base‘𝑅) =
(Base‘𝑅) |
49 | 48, 7 | ringidcl 18568 |
. . . . . . . . . . . . 13
⊢ (𝑅 ∈ Ring → 1 ∈
(Base‘𝑅)) |
50 | 48, 6 | ring0cl 18569 |
. . . . . . . . . . . . 13
⊢ (𝑅 ∈ Ring → 0 ∈
(Base‘𝑅)) |
51 | 49, 50 | ifcld 4131 |
. . . . . . . . . . . 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 18571 |
. . . . . . . . . . 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 693 |
. . . . . . . . . 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 19365 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝐼 ∈ 𝑊 ∧ 𝑘 ∈ 𝐷) → 𝑘:𝐼⟶ℕ0) |
56 | 33, 34, 55 | syl2anc 693 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑋 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ 𝑘}) → 𝑘:𝐼⟶ℕ0) |
57 | 56 | ffvelrnda 6359 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑋 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ 𝑘}) ∧ 𝑧 ∈ 𝐼) → (𝑘‘𝑧) ∈
ℕ0) |
58 | 8 | adantr 481 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ 𝑘 ∈ 𝐷) → 𝐼 ∈ 𝑊) |
59 | 10 | adantr 481 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ 𝑘 ∈ 𝐷) → 𝑋 ∈ 𝐷) |
60 | 5 | psrbagf 19365 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝐼 ∈ 𝑊 ∧ 𝑋 ∈ 𝐷) → 𝑋:𝐼⟶ℕ0) |
61 | 58, 59, 60 | syl2anc 693 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑘 ∈ 𝐷) → 𝑋:𝐼⟶ℕ0) |
62 | 61 | ffvelrnda 6359 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑧 ∈ 𝐼) → (𝑋‘𝑧) ∈
ℕ0) |
63 | 62 | adantlr 751 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑋 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ 𝑘}) ∧ 𝑧 ∈ 𝐼) → (𝑋‘𝑧) ∈
ℕ0) |
64 | 5 | psrbagf 19365 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝐼 ∈ 𝑊 ∧ 𝑌 ∈ 𝐷) → 𝑌:𝐼⟶ℕ0) |
65 | 8, 12, 64 | syl2anc 693 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 → 𝑌:𝐼⟶ℕ0) |
66 | 65 | adantr 481 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑘 ∈ 𝐷) → 𝑌:𝐼⟶ℕ0) |
67 | 66 | ffvelrnda 6359 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑧 ∈ 𝐼) → (𝑌‘𝑧) ∈
ℕ0) |
68 | 67 | adantlr 751 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑋 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ 𝑘}) ∧ 𝑧 ∈ 𝐼) → (𝑌‘𝑧) ∈
ℕ0) |
69 | | nn0cn 11302 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑘‘𝑧) ∈ ℕ0 → (𝑘‘𝑧) ∈ ℂ) |
70 | | nn0cn 11302 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑋‘𝑧) ∈ ℕ0 → (𝑋‘𝑧) ∈ ℂ) |
71 | | nn0cn 11302 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑌‘𝑧) ∈ ℕ0 → (𝑌‘𝑧) ∈ ℂ) |
72 | | subadd 10284 |
. . . . . . . . . . . . . . . . 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 1368 |
. . . . . . . . . . . . . . . 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 1326 |
. . . . . . . . . . . . . . 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 2629 |
. . . . . . . . . . . . . . 15
⊢ (((𝑋‘𝑧) + (𝑌‘𝑧)) = (𝑘‘𝑧) ↔ (𝑘‘𝑧) = ((𝑋‘𝑧) + (𝑌‘𝑧))) |
76 | 74, 75 | syl6bb 276 |
. . . . . . . . . . . . . 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 2985 |
. . . . . . . . . . . . 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 6299 |
. . . . . . . . . . . . . 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 | | ovexd 6680 |
. . . . . . . . . . . . . 14
⊢ (𝑧 ∈ 𝐼 → ((𝑘‘𝑧) − (𝑋‘𝑧)) ∈ V) |
80 | 78, 79 | mprg 2926 |
. . . . . . . . . . . . 13
⊢ ((𝑧 ∈ 𝐼 ↦ ((𝑘‘𝑧) − (𝑋‘𝑧))) = (𝑧 ∈ 𝐼 ↦ (𝑌‘𝑧)) ↔ ∀𝑧 ∈ 𝐼 ((𝑘‘𝑧) − (𝑋‘𝑧)) = (𝑌‘𝑧)) |
81 | | mpteqb 6299 |
. . . . . . . . . . . . . 14
⊢
(∀𝑧 ∈
𝐼 (𝑘‘𝑧) ∈ V → ((𝑧 ∈ 𝐼 ↦ (𝑘‘𝑧)) = (𝑧 ∈ 𝐼 ↦ ((𝑋‘𝑧) + (𝑌‘𝑧))) ↔ ∀𝑧 ∈ 𝐼 (𝑘‘𝑧) = ((𝑋‘𝑧) + (𝑌‘𝑧)))) |
82 | | fvex 6201 |
. . . . . . . . . . . . . . 15
⊢ (𝑘‘𝑧) ∈ V |
83 | 82 | a1i 11 |
. . . . . . . . . . . . . 14
⊢ (𝑧 ∈ 𝐼 → (𝑘‘𝑧) ∈ V) |
84 | 81, 83 | mprg 2926 |
. . . . . . . . . . . . 13
⊢ ((𝑧 ∈ 𝐼 ↦ (𝑘‘𝑧)) = (𝑧 ∈ 𝐼 ↦ ((𝑋‘𝑧) + (𝑌‘𝑧))) ↔ ∀𝑧 ∈ 𝐼 (𝑘‘𝑧) = ((𝑋‘𝑧) + (𝑌‘𝑧))) |
85 | 77, 80, 84 | 3bitr4g 303 |
. . . . . . . . . . . 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))))) |
86 | 56 | feqmptd 6249 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑋 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ 𝑘}) → 𝑘 = (𝑧 ∈ 𝐼 ↦ (𝑘‘𝑧))) |
87 | 61 | feqmptd 6249 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑘 ∈ 𝐷) → 𝑋 = (𝑧 ∈ 𝐼 ↦ (𝑋‘𝑧))) |
88 | 87 | adantr 481 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑋 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ 𝑘}) → 𝑋 = (𝑧 ∈ 𝐼 ↦ (𝑋‘𝑧))) |
89 | 33, 57, 63, 86, 88 | offval2 6914 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑋 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ 𝑘}) → (𝑘 ∘𝑓 − 𝑋) = (𝑧 ∈ 𝐼 ↦ ((𝑘‘𝑧) − (𝑋‘𝑧)))) |
90 | 66 | feqmptd 6249 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑘 ∈ 𝐷) → 𝑌 = (𝑧 ∈ 𝐼 ↦ (𝑌‘𝑧))) |
91 | 90 | adantr 481 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑋 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ 𝑘}) → 𝑌 = (𝑧 ∈ 𝐼 ↦ (𝑌‘𝑧))) |
92 | 89, 91 | eqeq12d 2637 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑋 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ 𝑘}) → ((𝑘 ∘𝑓 − 𝑋) = 𝑌 ↔ (𝑧 ∈ 𝐼 ↦ ((𝑘‘𝑧) − (𝑋‘𝑧))) = (𝑧 ∈ 𝐼 ↦ (𝑌‘𝑧)))) |
93 | 58, 62, 67, 87, 90 | offval2 6914 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑘 ∈ 𝐷) → (𝑋 ∘𝑓 + 𝑌) = (𝑧 ∈ 𝐼 ↦ ((𝑋‘𝑧) + (𝑌‘𝑧)))) |
94 | 93 | adantr 481 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑋 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ 𝑘}) → (𝑋 ∘𝑓 + 𝑌) = (𝑧 ∈ 𝐼 ↦ ((𝑋‘𝑧) + (𝑌‘𝑧)))) |
95 | 86, 94 | eqeq12d 2637 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑋 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ 𝑘}) → (𝑘 = (𝑋 ∘𝑓 + 𝑌) ↔ (𝑧 ∈ 𝐼 ↦ (𝑘‘𝑧)) = (𝑧 ∈ 𝐼 ↦ ((𝑋‘𝑧) + (𝑌‘𝑧))))) |
96 | 85, 92, 95 | 3bitr4d 300 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑋 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ 𝑘}) → ((𝑘 ∘𝑓 − 𝑋) = 𝑌 ↔ 𝑘 = (𝑋 ∘𝑓 + 𝑌))) |
97 | 96 | ifbid 4108 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑋 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ 𝑘}) → if((𝑘 ∘𝑓 − 𝑋) = 𝑌, 1 , 0 ) = if(𝑘 = (𝑋 ∘𝑓 + 𝑌), 1 , 0 )) |
98 | 47, 54, 97 | 3eqtrd 2660 |
. . . . . . . . 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 )) |
99 | 97, 52 | eqeltrrd 2702 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑋 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ 𝑘}) → if(𝑘 = (𝑋 ∘𝑓 + 𝑌), 1 , 0 ) ∈ (Base‘𝑅)) |
100 | 98, 99 | eqeltrd 2701 |
. . . . . . . 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)) |
101 | | fveq2 6191 |
. . . . . . . . . 10
⊢ (𝑗 = 𝑋 → ((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))‘𝑗) = ((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))‘𝑋)) |
102 | | oveq2 6658 |
. . . . . . . . . . 11
⊢ (𝑗 = 𝑋 → (𝑘 ∘𝑓 − 𝑗) = (𝑘 ∘𝑓 − 𝑋)) |
103 | 102 | fveq2d 6195 |
. . . . . . . . . 10
⊢ (𝑗 = 𝑋 → ((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘 ∘𝑓
− 𝑗)) = ((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘 ∘𝑓
− 𝑋))) |
104 | 101, 103 | oveq12d 6668 |
. . . . . . . . 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)))) |
105 | 48, 104 | gsumsn 18354 |
. . . . . . . 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)))) |
106 | 24, 25, 100, 105 | syl3anc 1326 |
. . . . . . 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)))) |
107 | 21, 106, 98 | 3eqtrd 2660 |
. . . . . 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 )) |
108 | 6 | gsum0 17278 |
. . . . . . 7
⊢ (𝑅 Σg
∅) = 0 |
109 | | disjsn 4246 |
. . . . . . . . 9
⊢ (({𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ 𝑘} ∩ {𝑋}) = ∅ ↔ ¬ 𝑋 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ 𝑘}) |
110 | 9 | ad2antrr 762 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑗 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ 𝑘}) → 𝑅 ∈ Ring) |
111 | 1, 48, 2, 5, 11 | mplelf 19433 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 )):𝐷⟶(Base‘𝑅)) |
112 | 111 | ad2antrr 762 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑗 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ 𝑘}) → (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 )):𝐷⟶(Base‘𝑅)) |
113 | | simpr 477 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑗 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ 𝑘}) → 𝑗 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ 𝑘}) |
114 | 32, 113 | sseldi 3601 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑗 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ 𝑘}) → 𝑗 ∈ 𝐷) |
115 | 112, 114 | ffvelrnd 6360 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑗 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ 𝑘}) → ((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))‘𝑗) ∈ (Base‘𝑅)) |
116 | 1, 48, 2, 5, 13 | mplelf 19433 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 )):𝐷⟶(Base‘𝑅)) |
117 | 116 | ad2antrr 762 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑗 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ 𝑘}) → (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 )):𝐷⟶(Base‘𝑅)) |
118 | 8 | ad2antrr 762 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑗 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ 𝑘}) → 𝐼 ∈ 𝑊) |
119 | | simplr 792 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑗 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ 𝑘}) → 𝑘 ∈ 𝐷) |
120 | 5, 35 | psrbagconcl 19373 |
. . . . . . . . . . . . . . . 16
⊢ ((𝐼 ∈ 𝑊 ∧ 𝑘 ∈ 𝐷 ∧ 𝑗 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ 𝑘}) → (𝑘 ∘𝑓 − 𝑗) ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ 𝑘}) |
121 | 118, 119,
113, 120 | syl3anc 1326 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑗 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ 𝑘}) → (𝑘 ∘𝑓 − 𝑗) ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ 𝑘}) |
122 | 32, 121 | sseldi 3601 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑗 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ 𝑘}) → (𝑘 ∘𝑓 − 𝑗) ∈ 𝐷) |
123 | 117, 122 | ffvelrnd 6360 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑗 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ 𝑘}) → ((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘 ∘𝑓
− 𝑗)) ∈
(Base‘𝑅)) |
124 | 48, 3 | ringcl 18561 |
. . . . . . . . . . . . 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)) |
125 | 110, 115,
123, 124 | syl3anc 1326 |
. . . . . . . . . . . 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)) |
126 | | eqid 2622 |
. . . . . . . . . . . 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)))) |
127 | 125, 126 | fmptd 6385 |
. . . . . . . . . . 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)) |
128 | | ffn 6045 |
. . . . . . . . . . 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}) |
129 | | fnresdisj 6001 |
. . . . . . . . . . 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}) =
∅)) |
130 | 127, 128,
129 | 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}) =
∅)) |
131 | 130 | biimpa 501 |
. . . . . . . . 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}) =
∅) |
132 | 109, 131 | sylan2br 493 |
. . . . . . . 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}) =
∅) |
133 | 132 | oveq2d 6666 |
. . . . . . 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
∅)) |
134 | 62 | nn0red 11352 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑧 ∈ 𝐼) → (𝑋‘𝑧) ∈ ℝ) |
135 | | nn0addge1 11339 |
. . . . . . . . . . . . . 14
⊢ (((𝑋‘𝑧) ∈ ℝ ∧ (𝑌‘𝑧) ∈ ℕ0) → (𝑋‘𝑧) ≤ ((𝑋‘𝑧) + (𝑌‘𝑧))) |
136 | 134, 67, 135 | syl2anc 693 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑧 ∈ 𝐼) → (𝑋‘𝑧) ≤ ((𝑋‘𝑧) + (𝑌‘𝑧))) |
137 | 136 | ralrimiva 2966 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑘 ∈ 𝐷) → ∀𝑧 ∈ 𝐼 (𝑋‘𝑧) ≤ ((𝑋‘𝑧) + (𝑌‘𝑧))) |
138 | | ovexd 6680 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑧 ∈ 𝐼) → ((𝑋‘𝑧) + (𝑌‘𝑧)) ∈ V) |
139 | 58, 62, 138, 87, 93 | ofrfval2 6915 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑘 ∈ 𝐷) → (𝑋 ∘𝑟 ≤ (𝑋 ∘𝑓 +
𝑌) ↔ ∀𝑧 ∈ 𝐼 (𝑋‘𝑧) ≤ ((𝑋‘𝑧) + (𝑌‘𝑧)))) |
140 | 137, 139 | mpbird 247 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑘 ∈ 𝐷) → 𝑋 ∘𝑟 ≤ (𝑋 ∘𝑓 +
𝑌)) |
141 | | breq1 4656 |
. . . . . . . . . . . 12
⊢ (𝑥 = 𝑋 → (𝑥 ∘𝑟 ≤ (𝑋 ∘𝑓 +
𝑌) ↔ 𝑋 ∘𝑟 ≤ (𝑋 ∘𝑓 +
𝑌))) |
142 | 141 | elrab 3363 |
. . . . . . . . . . 11
⊢ (𝑋 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ (𝑋 ∘𝑓 +
𝑌)} ↔ (𝑋 ∈ 𝐷 ∧ 𝑋 ∘𝑟 ≤ (𝑋 ∘𝑓 +
𝑌))) |
143 | 59, 140, 142 | sylanbrc 698 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑘 ∈ 𝐷) → 𝑋 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ (𝑋 ∘𝑓 +
𝑌)}) |
144 | | breq2 4657 |
. . . . . . . . . . . 12
⊢ (𝑘 = (𝑋 ∘𝑓 + 𝑌) → (𝑥 ∘𝑟 ≤ 𝑘 ↔ 𝑥 ∘𝑟 ≤ (𝑋 ∘𝑓 +
𝑌))) |
145 | 144 | rabbidv 3189 |
. . . . . . . . . . 11
⊢ (𝑘 = (𝑋 ∘𝑓 + 𝑌) → {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ 𝑘} = {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ (𝑋 ∘𝑓 +
𝑌)}) |
146 | 145 | eleq2d 2687 |
. . . . . . . . . 10
⊢ (𝑘 = (𝑋 ∘𝑓 + 𝑌) → (𝑋 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ 𝑘} ↔ 𝑋 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ (𝑋 ∘𝑓 +
𝑌)})) |
147 | 143, 146 | syl5ibrcom 237 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑘 ∈ 𝐷) → (𝑘 = (𝑋 ∘𝑓 + 𝑌) → 𝑋 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ 𝑘})) |
148 | 147 | con3dimp 457 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ ¬ 𝑋 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ 𝑘}) → ¬ 𝑘 = (𝑋 ∘𝑓 + 𝑌)) |
149 | 148 | iffalsed 4097 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ ¬ 𝑋 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ 𝑘}) → if(𝑘 = (𝑋 ∘𝑓 + 𝑌), 1 , 0 ) = 0 ) |
150 | 108, 133,
149 | 3eqtr4a 2682 |
. . . . . 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 )) |
151 | 107, 150 | pm2.61dan 832 |
. . . . 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 )) |
152 | 9 | adantr 481 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑘 ∈ 𝐷) → 𝑅 ∈ Ring) |
153 | | ringcmn 18581 |
. . . . . . 7
⊢ (𝑅 ∈ Ring → 𝑅 ∈ CMnd) |
154 | 152, 153 | syl 17 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑘 ∈ 𝐷) → 𝑅 ∈ CMnd) |
155 | 5 | psrbaglefi 19372 |
. . . . . . 7
⊢ ((𝐼 ∈ 𝑊 ∧ 𝑘 ∈ 𝐷) → {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ 𝑘} ∈ Fin) |
156 | 8, 155 | sylan 488 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑘 ∈ 𝐷) → {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ 𝑘} ∈ Fin) |
157 | | ssdif 3745 |
. . . . . . . . . . . 12
⊢ ({𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ 𝑘} ⊆ 𝐷 → ({𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ 𝑘} ∖ {𝑋}) ⊆ (𝐷 ∖ {𝑋})) |
158 | 32, 157 | ax-mp 5 |
. . . . . . . . . . 11
⊢ ({𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ 𝑘} ∖ {𝑋}) ⊆ (𝐷 ∖ {𝑋}) |
159 | 158 | sseli 3599 |
. . . . . . . . . 10
⊢ (𝑗 ∈ ({𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ 𝑘} ∖ {𝑋}) → 𝑗 ∈ (𝐷 ∖ {𝑋})) |
160 | 111 | adantr 481 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑘 ∈ 𝐷) → (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 )):𝐷⟶(Base‘𝑅)) |
161 | | eldifsni 4320 |
. . . . . . . . . . . . . . 15
⊢ (𝑦 ∈ (𝐷 ∖ {𝑋}) → 𝑦 ≠ 𝑋) |
162 | 161 | adantl 482 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑦 ∈ (𝐷 ∖ {𝑋})) → 𝑦 ≠ 𝑋) |
163 | 162 | neneqd 2799 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑦 ∈ (𝐷 ∖ {𝑋})) → ¬ 𝑦 = 𝑋) |
164 | 163 | iffalsed 4097 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑦 ∈ (𝐷 ∖ {𝑋})) → if(𝑦 = 𝑋, 1 , 0 ) = 0 ) |
165 | | ovex 6678 |
. . . . . . . . . . . . . 14
⊢
(ℕ0 ↑𝑚 𝐼) ∈ V |
166 | 5, 165 | rabex2 4815 |
. . . . . . . . . . . . 13
⊢ 𝐷 ∈ V |
167 | 166 | a1i 11 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑘 ∈ 𝐷) → 𝐷 ∈ V) |
168 | 164, 167 | suppss2 7329 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑘 ∈ 𝐷) → ((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 )) supp 0 ) ⊆ {𝑋}) |
169 | 43 | a1i 11 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑘 ∈ 𝐷) → 0 ∈ V) |
170 | 160, 168,
167, 169 | suppssr 7326 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑗 ∈ (𝐷 ∖ {𝑋})) → ((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))‘𝑗) = 0 ) |
171 | 159, 170 | sylan2 491 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑗 ∈ ({𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ 𝑘} ∖ {𝑋})) → ((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))‘𝑗) = 0 ) |
172 | 171 | oveq1d 6665 |
. . . . . . . 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)))) |
173 | | eldifi 3732 |
. . . . . . . . 9
⊢ (𝑗 ∈ ({𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ 𝑘} ∖ {𝑋}) → 𝑗 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ 𝑘}) |
174 | 48, 3, 6 | ringlz 18587 |
. . . . . . . . . 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
) |
175 | 110, 123,
174 | syl2anc 693 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑗 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ 𝑘}) → ( 0 (.r‘𝑅)((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘 ∘𝑓
− 𝑗))) = 0
) |
176 | 173, 175 | sylan2 491 |
. . . . . . . 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
) |
177 | 172, 176 | eqtrd 2656 |
. . . . . . 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
) |
178 | 166 | rabex 4813 |
. . . . . . . 8
⊢ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ 𝑘} ∈ V |
179 | 178 | a1i 11 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑘 ∈ 𝐷) → {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ 𝑘} ∈ V) |
180 | 177, 179 | suppss2 7329 |
. . . . . 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}) |
181 | 166 | mptrabex 6488 |
. . . . . . . . 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 |
182 | | funmpt 5926 |
. . . . . . . . 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)))) |
183 | 181, 182,
43 | 3pm3.2i 1239 |
. . . . . . . 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) |
184 | 183 | 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)) |
185 | | snfi 8038 |
. . . . . . . 8
⊢ {𝑋} ∈ Fin |
186 | 185 | a1i 11 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑘 ∈ 𝐷) → {𝑋} ∈ Fin) |
187 | | suppssfifsupp 8290 |
. . . . . . 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
) |
188 | 184, 186,
180, 187 | syl12anc 1324 |
. . . . . 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
) |
189 | 48, 6, 154, 156, 127, 180, 188 | gsumres 18314 |
. . . . 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)))))) |
190 | 151, 189 | eqtr3d 2658 |
. . . 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)))))) |
191 | 190 | mpteq2dva 4744 |
. . 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))))))) |
192 | 17, 191 | syl5eq 2668 |
. 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))))))) |
193 | 14, 192 | eqtr4d 2659 |
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 ))) |