Step | Hyp | Ref
| Expression |
1 | | fconst6g 6094 |
. . . . 5
⊢ (𝑘 ∈ ℕ0
→ (1𝑜 × {𝑘}):1𝑜⟶ℕ0) |
2 | | nn0ex 11298 |
. . . . . 6
⊢
ℕ0 ∈ V |
3 | | 1on 7567 |
. . . . . . 7
⊢
1𝑜 ∈ On |
4 | 3 | elexi 3213 |
. . . . . 6
⊢
1𝑜 ∈ V |
5 | 2, 4 | elmap 7886 |
. . . . 5
⊢
((1𝑜 × {𝑘}) ∈ (ℕ0
↑𝑚 1𝑜) ↔ (1𝑜
× {𝑘}):1𝑜⟶ℕ0) |
6 | 1, 5 | sylibr 224 |
. . . 4
⊢ (𝑘 ∈ ℕ0
→ (1𝑜 × {𝑘}) ∈ (ℕ0
↑𝑚 1𝑜)) |
7 | 6 | adantl 482 |
. . 3
⊢ (((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑘 ∈ ℕ0) →
(1𝑜 × {𝑘}) ∈ (ℕ0
↑𝑚 1𝑜)) |
8 | | eqidd 2623 |
. . 3
⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → (𝑘 ∈ ℕ0 ↦
(1𝑜 × {𝑘})) = (𝑘 ∈ ℕ0 ↦
(1𝑜 × {𝑘}))) |
9 | | eqid 2622 |
. . . 4
⊢
(1𝑜 mPwSer 𝑅) = (1𝑜 mPwSer 𝑅) |
10 | | coe1mul2.s |
. . . . 5
⊢ 𝑆 =
(PwSer1‘𝑅) |
11 | | coe1mul2.b |
. . . . 5
⊢ 𝐵 = (Base‘𝑆) |
12 | 10, 11, 9 | psr1bas2 19560 |
. . . 4
⊢ 𝐵 =
(Base‘(1𝑜 mPwSer 𝑅)) |
13 | | coe1mul2.u |
. . . 4
⊢ · =
(.r‘𝑅) |
14 | | coe1mul2.t |
. . . . 5
⊢ ∙ =
(.r‘𝑆) |
15 | 10, 9, 14 | psr1mulr 19594 |
. . . 4
⊢ ∙ =
(.r‘(1𝑜 mPwSer 𝑅)) |
16 | | psr1baslem 19555 |
. . . 4
⊢
(ℕ0 ↑𝑚 1𝑜) =
{𝑎 ∈
(ℕ0 ↑𝑚 1𝑜) ∣
(◡𝑎 “ ℕ) ∈
Fin} |
17 | | simp2 1062 |
. . . 4
⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → 𝐹 ∈ 𝐵) |
18 | | simp3 1063 |
. . . 4
⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → 𝐺 ∈ 𝐵) |
19 | 9, 12, 13, 15, 16, 17, 18 | psrmulfval 19385 |
. . 3
⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → (𝐹 ∙ 𝐺) = (𝑏 ∈ (ℕ0
↑𝑚 1𝑜) ↦ (𝑅 Σg (𝑐 ∈ {𝑑 ∈ (ℕ0
↑𝑚 1𝑜) ∣ 𝑑 ∘𝑟 ≤ 𝑏} ↦ ((𝐹‘𝑐) · (𝐺‘(𝑏 ∘𝑓 − 𝑐))))))) |
20 | | breq2 4657 |
. . . . . 6
⊢ (𝑏 = (1𝑜
× {𝑘}) → (𝑑 ∘𝑟
≤ 𝑏 ↔ 𝑑 ∘𝑟
≤ (1𝑜 × {𝑘}))) |
21 | 20 | rabbidv 3189 |
. . . . 5
⊢ (𝑏 = (1𝑜
× {𝑘}) → {𝑑 ∈ (ℕ0
↑𝑚 1𝑜) ∣ 𝑑 ∘𝑟 ≤ 𝑏} = {𝑑 ∈ (ℕ0
↑𝑚 1𝑜) ∣ 𝑑 ∘𝑟 ≤
(1𝑜 × {𝑘})}) |
22 | | oveq1 6657 |
. . . . . . 7
⊢ (𝑏 = (1𝑜
× {𝑘}) → (𝑏 ∘𝑓
− 𝑐) =
((1𝑜 × {𝑘}) ∘𝑓 − 𝑐)) |
23 | 22 | fveq2d 6195 |
. . . . . 6
⊢ (𝑏 = (1𝑜
× {𝑘}) → (𝐺‘(𝑏 ∘𝑓 − 𝑐)) = (𝐺‘((1𝑜 ×
{𝑘})
∘𝑓 − 𝑐))) |
24 | 23 | oveq2d 6666 |
. . . . 5
⊢ (𝑏 = (1𝑜
× {𝑘}) → ((𝐹‘𝑐) · (𝐺‘(𝑏 ∘𝑓 − 𝑐))) = ((𝐹‘𝑐) · (𝐺‘((1𝑜 ×
{𝑘})
∘𝑓 − 𝑐)))) |
25 | 21, 24 | mpteq12dv 4733 |
. . . 4
⊢ (𝑏 = (1𝑜
× {𝑘}) → (𝑐 ∈ {𝑑 ∈ (ℕ0
↑𝑚 1𝑜) ∣ 𝑑 ∘𝑟 ≤ 𝑏} ↦ ((𝐹‘𝑐) · (𝐺‘(𝑏 ∘𝑓 − 𝑐)))) = (𝑐 ∈ {𝑑 ∈ (ℕ0
↑𝑚 1𝑜) ∣ 𝑑 ∘𝑟 ≤
(1𝑜 × {𝑘})} ↦ ((𝐹‘𝑐) · (𝐺‘((1𝑜 ×
{𝑘})
∘𝑓 − 𝑐))))) |
26 | 25 | oveq2d 6666 |
. . 3
⊢ (𝑏 = (1𝑜
× {𝑘}) → (𝑅 Σg
(𝑐 ∈ {𝑑 ∈ (ℕ0
↑𝑚 1𝑜) ∣ 𝑑 ∘𝑟 ≤ 𝑏} ↦ ((𝐹‘𝑐) · (𝐺‘(𝑏 ∘𝑓 − 𝑐))))) = (𝑅 Σg (𝑐 ∈ {𝑑 ∈ (ℕ0
↑𝑚 1𝑜) ∣ 𝑑 ∘𝑟 ≤
(1𝑜 × {𝑘})} ↦ ((𝐹‘𝑐) · (𝐺‘((1𝑜 ×
{𝑘})
∘𝑓 − 𝑐)))))) |
27 | 7, 8, 19, 26 | fmptco 6396 |
. 2
⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → ((𝐹 ∙ 𝐺) ∘ (𝑘 ∈ ℕ0 ↦
(1𝑜 × {𝑘}))) = (𝑘 ∈ ℕ0 ↦ (𝑅 Σg
(𝑐 ∈ {𝑑 ∈ (ℕ0
↑𝑚 1𝑜) ∣ 𝑑 ∘𝑟 ≤
(1𝑜 × {𝑘})} ↦ ((𝐹‘𝑐) · (𝐺‘((1𝑜 ×
{𝑘})
∘𝑓 − 𝑐))))))) |
28 | 10 | psr1ring 19617 |
. . . 4
⊢ (𝑅 ∈ Ring → 𝑆 ∈ Ring) |
29 | 11, 14 | ringcl 18561 |
. . . 4
⊢ ((𝑆 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → (𝐹 ∙ 𝐺) ∈ 𝐵) |
30 | 28, 29 | syl3an1 1359 |
. . 3
⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → (𝐹 ∙ 𝐺) ∈ 𝐵) |
31 | | eqid 2622 |
. . . 4
⊢
(coe1‘(𝐹 ∙ 𝐺)) = (coe1‘(𝐹 ∙ 𝐺)) |
32 | | eqid 2622 |
. . . 4
⊢ (𝑘 ∈ ℕ0
↦ (1𝑜 × {𝑘})) = (𝑘 ∈ ℕ0 ↦
(1𝑜 × {𝑘})) |
33 | 31, 11, 10, 32 | coe1fval3 19578 |
. . 3
⊢ ((𝐹 ∙ 𝐺) ∈ 𝐵 → (coe1‘(𝐹 ∙ 𝐺)) = ((𝐹 ∙ 𝐺) ∘ (𝑘 ∈ ℕ0 ↦
(1𝑜 × {𝑘})))) |
34 | 30, 33 | syl 17 |
. 2
⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → (coe1‘(𝐹 ∙ 𝐺)) = ((𝐹 ∙ 𝐺) ∘ (𝑘 ∈ ℕ0 ↦
(1𝑜 × {𝑘})))) |
35 | | eqid 2622 |
. . . . 5
⊢
(Base‘𝑅) =
(Base‘𝑅) |
36 | | eqid 2622 |
. . . . 5
⊢
(0g‘𝑅) = (0g‘𝑅) |
37 | | simpl1 1064 |
. . . . . 6
⊢ (((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑘 ∈ ℕ0) → 𝑅 ∈ Ring) |
38 | | ringcmn 18581 |
. . . . . 6
⊢ (𝑅 ∈ Ring → 𝑅 ∈ CMnd) |
39 | 37, 38 | syl 17 |
. . . . 5
⊢ (((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑘 ∈ ℕ0) → 𝑅 ∈ CMnd) |
40 | | fzfi 12771 |
. . . . . 6
⊢
(0...𝑘) ∈
Fin |
41 | 40 | a1i 11 |
. . . . 5
⊢ (((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑘 ∈ ℕ0) →
(0...𝑘) ∈
Fin) |
42 | | simpll1 1100 |
. . . . . . 7
⊢ ((((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑥 ∈ (0...𝑘)) → 𝑅 ∈ Ring) |
43 | | simpll2 1101 |
. . . . . . . . 9
⊢ ((((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑥 ∈ (0...𝑘)) → 𝐹 ∈ 𝐵) |
44 | | eqid 2622 |
. . . . . . . . . 10
⊢
(coe1‘𝐹) = (coe1‘𝐹) |
45 | 44, 11, 10, 35 | coe1f2 19579 |
. . . . . . . . 9
⊢ (𝐹 ∈ 𝐵 → (coe1‘𝐹):ℕ0⟶(Base‘𝑅)) |
46 | 43, 45 | syl 17 |
. . . . . . . 8
⊢ ((((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑥 ∈ (0...𝑘)) → (coe1‘𝐹):ℕ0⟶(Base‘𝑅)) |
47 | | elfznn0 12433 |
. . . . . . . . 9
⊢ (𝑥 ∈ (0...𝑘) → 𝑥 ∈ ℕ0) |
48 | 47 | adantl 482 |
. . . . . . . 8
⊢ ((((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑥 ∈ (0...𝑘)) → 𝑥 ∈ ℕ0) |
49 | 46, 48 | ffvelrnd 6360 |
. . . . . . 7
⊢ ((((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑥 ∈ (0...𝑘)) → ((coe1‘𝐹)‘𝑥) ∈ (Base‘𝑅)) |
50 | | simpll3 1102 |
. . . . . . . . 9
⊢ ((((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑥 ∈ (0...𝑘)) → 𝐺 ∈ 𝐵) |
51 | | eqid 2622 |
. . . . . . . . . 10
⊢
(coe1‘𝐺) = (coe1‘𝐺) |
52 | 51, 11, 10, 35 | coe1f2 19579 |
. . . . . . . . 9
⊢ (𝐺 ∈ 𝐵 → (coe1‘𝐺):ℕ0⟶(Base‘𝑅)) |
53 | 50, 52 | syl 17 |
. . . . . . . 8
⊢ ((((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑥 ∈ (0...𝑘)) → (coe1‘𝐺):ℕ0⟶(Base‘𝑅)) |
54 | | fznn0sub 12373 |
. . . . . . . . 9
⊢ (𝑥 ∈ (0...𝑘) → (𝑘 − 𝑥) ∈
ℕ0) |
55 | 54 | adantl 482 |
. . . . . . . 8
⊢ ((((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑥 ∈ (0...𝑘)) → (𝑘 − 𝑥) ∈
ℕ0) |
56 | 53, 55 | ffvelrnd 6360 |
. . . . . . 7
⊢ ((((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑥 ∈ (0...𝑘)) → ((coe1‘𝐺)‘(𝑘 − 𝑥)) ∈ (Base‘𝑅)) |
57 | 35, 13 | ringcl 18561 |
. . . . . . 7
⊢ ((𝑅 ∈ Ring ∧
((coe1‘𝐹)‘𝑥) ∈ (Base‘𝑅) ∧ ((coe1‘𝐺)‘(𝑘 − 𝑥)) ∈ (Base‘𝑅)) → (((coe1‘𝐹)‘𝑥) ·
((coe1‘𝐺)‘(𝑘 − 𝑥))) ∈ (Base‘𝑅)) |
58 | 42, 49, 56, 57 | syl3anc 1326 |
. . . . . 6
⊢ ((((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑥 ∈ (0...𝑘)) → (((coe1‘𝐹)‘𝑥) ·
((coe1‘𝐺)‘(𝑘 − 𝑥))) ∈ (Base‘𝑅)) |
59 | | eqid 2622 |
. . . . . 6
⊢ (𝑥 ∈ (0...𝑘) ↦ (((coe1‘𝐹)‘𝑥) ·
((coe1‘𝐺)‘(𝑘 − 𝑥)))) = (𝑥 ∈ (0...𝑘) ↦ (((coe1‘𝐹)‘𝑥) ·
((coe1‘𝐺)‘(𝑘 − 𝑥)))) |
60 | 58, 59 | fmptd 6385 |
. . . . 5
⊢ (((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑘 ∈ ℕ0) → (𝑥 ∈ (0...𝑘) ↦ (((coe1‘𝐹)‘𝑥) ·
((coe1‘𝐺)‘(𝑘 − 𝑥)))):(0...𝑘)⟶(Base‘𝑅)) |
61 | 40 | elexi 3213 |
. . . . . . . . 9
⊢
(0...𝑘) ∈
V |
62 | 61 | mptex 6486 |
. . . . . . . 8
⊢ (𝑥 ∈ (0...𝑘) ↦ (((coe1‘𝐹)‘𝑥) ·
((coe1‘𝐺)‘(𝑘 − 𝑥)))) ∈ V |
63 | | funmpt 5926 |
. . . . . . . 8
⊢ Fun
(𝑥 ∈ (0...𝑘) ↦
(((coe1‘𝐹)‘𝑥) ·
((coe1‘𝐺)‘(𝑘 − 𝑥)))) |
64 | | fvex 6201 |
. . . . . . . 8
⊢
(0g‘𝑅) ∈ V |
65 | 62, 63, 64 | 3pm3.2i 1239 |
. . . . . . 7
⊢ ((𝑥 ∈ (0...𝑘) ↦ (((coe1‘𝐹)‘𝑥) ·
((coe1‘𝐺)‘(𝑘 − 𝑥)))) ∈ V ∧ Fun (𝑥 ∈ (0...𝑘) ↦ (((coe1‘𝐹)‘𝑥) ·
((coe1‘𝐺)‘(𝑘 − 𝑥)))) ∧ (0g‘𝑅) ∈ V) |
66 | | suppssdm 7308 |
. . . . . . . . 9
⊢ ((𝑥 ∈ (0...𝑘) ↦ (((coe1‘𝐹)‘𝑥) ·
((coe1‘𝐺)‘(𝑘 − 𝑥)))) supp (0g‘𝑅)) ⊆ dom (𝑥 ∈ (0...𝑘) ↦ (((coe1‘𝐹)‘𝑥) ·
((coe1‘𝐺)‘(𝑘 − 𝑥)))) |
67 | 59 | dmmptss 5631 |
. . . . . . . . 9
⊢ dom
(𝑥 ∈ (0...𝑘) ↦
(((coe1‘𝐹)‘𝑥) ·
((coe1‘𝐺)‘(𝑘 − 𝑥)))) ⊆ (0...𝑘) |
68 | 66, 67 | sstri 3612 |
. . . . . . . 8
⊢ ((𝑥 ∈ (0...𝑘) ↦ (((coe1‘𝐹)‘𝑥) ·
((coe1‘𝐺)‘(𝑘 − 𝑥)))) supp (0g‘𝑅)) ⊆ (0...𝑘) |
69 | 40, 68 | pm3.2i 471 |
. . . . . . 7
⊢
((0...𝑘) ∈ Fin
∧ ((𝑥 ∈ (0...𝑘) ↦
(((coe1‘𝐹)‘𝑥) ·
((coe1‘𝐺)‘(𝑘 − 𝑥)))) supp (0g‘𝑅)) ⊆ (0...𝑘)) |
70 | | suppssfifsupp 8290 |
. . . . . . 7
⊢ ((((𝑥 ∈ (0...𝑘) ↦ (((coe1‘𝐹)‘𝑥) ·
((coe1‘𝐺)‘(𝑘 − 𝑥)))) ∈ V ∧ Fun (𝑥 ∈ (0...𝑘) ↦ (((coe1‘𝐹)‘𝑥) ·
((coe1‘𝐺)‘(𝑘 − 𝑥)))) ∧ (0g‘𝑅) ∈ V) ∧ ((0...𝑘) ∈ Fin ∧ ((𝑥 ∈ (0...𝑘) ↦ (((coe1‘𝐹)‘𝑥) ·
((coe1‘𝐺)‘(𝑘 − 𝑥)))) supp (0g‘𝑅)) ⊆ (0...𝑘))) → (𝑥 ∈ (0...𝑘) ↦ (((coe1‘𝐹)‘𝑥) ·
((coe1‘𝐺)‘(𝑘 − 𝑥)))) finSupp (0g‘𝑅)) |
71 | 65, 69, 70 | mp2an 708 |
. . . . . 6
⊢ (𝑥 ∈ (0...𝑘) ↦ (((coe1‘𝐹)‘𝑥) ·
((coe1‘𝐺)‘(𝑘 − 𝑥)))) finSupp (0g‘𝑅) |
72 | 71 | a1i 11 |
. . . . 5
⊢ (((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑘 ∈ ℕ0) → (𝑥 ∈ (0...𝑘) ↦ (((coe1‘𝐹)‘𝑥) ·
((coe1‘𝐺)‘(𝑘 − 𝑥)))) finSupp (0g‘𝑅)) |
73 | | eqid 2622 |
. . . . . . 7
⊢ {𝑑 ∈ (ℕ0
↑𝑚 1𝑜) ∣ 𝑑 ∘𝑟 ≤
(1𝑜 × {𝑘})} = {𝑑 ∈ (ℕ0
↑𝑚 1𝑜) ∣ 𝑑 ∘𝑟 ≤
(1𝑜 × {𝑘})} |
74 | 73 | coe1mul2lem2 19638 |
. . . . . 6
⊢ (𝑘 ∈ ℕ0
→ (𝑐 ∈ {𝑑 ∈ (ℕ0
↑𝑚 1𝑜) ∣ 𝑑 ∘𝑟 ≤
(1𝑜 × {𝑘})} ↦ (𝑐‘∅)):{𝑑 ∈ (ℕ0
↑𝑚 1𝑜) ∣ 𝑑 ∘𝑟 ≤
(1𝑜 × {𝑘})}–1-1-onto→(0...𝑘)) |
75 | 74 | adantl 482 |
. . . . 5
⊢ (((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑘 ∈ ℕ0) → (𝑐 ∈ {𝑑 ∈ (ℕ0
↑𝑚 1𝑜) ∣ 𝑑 ∘𝑟 ≤
(1𝑜 × {𝑘})} ↦ (𝑐‘∅)):{𝑑 ∈ (ℕ0
↑𝑚 1𝑜) ∣ 𝑑 ∘𝑟 ≤
(1𝑜 × {𝑘})}–1-1-onto→(0...𝑘)) |
76 | 35, 36, 39, 41, 60, 72, 75 | gsumf1o 18317 |
. . . 4
⊢ (((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑘 ∈ ℕ0) → (𝑅 Σg
(𝑥 ∈ (0...𝑘) ↦
(((coe1‘𝐹)‘𝑥) ·
((coe1‘𝐺)‘(𝑘 − 𝑥))))) = (𝑅 Σg ((𝑥 ∈ (0...𝑘) ↦ (((coe1‘𝐹)‘𝑥) ·
((coe1‘𝐺)‘(𝑘 − 𝑥)))) ∘ (𝑐 ∈ {𝑑 ∈ (ℕ0
↑𝑚 1𝑜) ∣ 𝑑 ∘𝑟 ≤
(1𝑜 × {𝑘})} ↦ (𝑐‘∅))))) |
77 | | breq1 4656 |
. . . . . . . . . . 11
⊢ (𝑑 = 𝑐 → (𝑑 ∘𝑟 ≤
(1𝑜 × {𝑘}) ↔ 𝑐 ∘𝑟 ≤
(1𝑜 × {𝑘}))) |
78 | 77 | elrab 3363 |
. . . . . . . . . 10
⊢ (𝑐 ∈ {𝑑 ∈ (ℕ0
↑𝑚 1𝑜) ∣ 𝑑 ∘𝑟 ≤
(1𝑜 × {𝑘})} ↔ (𝑐 ∈ (ℕ0
↑𝑚 1𝑜) ∧ 𝑐 ∘𝑟 ≤
(1𝑜 × {𝑘}))) |
79 | 78 | simprbi 480 |
. . . . . . . . 9
⊢ (𝑐 ∈ {𝑑 ∈ (ℕ0
↑𝑚 1𝑜) ∣ 𝑑 ∘𝑟 ≤
(1𝑜 × {𝑘})} → 𝑐 ∘𝑟 ≤
(1𝑜 × {𝑘})) |
80 | 79 | adantl 482 |
. . . . . . . 8
⊢ ((((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑐 ∈ {𝑑 ∈ (ℕ0
↑𝑚 1𝑜) ∣ 𝑑 ∘𝑟 ≤
(1𝑜 × {𝑘})}) → 𝑐 ∘𝑟 ≤
(1𝑜 × {𝑘})) |
81 | | simplr 792 |
. . . . . . . . 9
⊢ ((((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑐 ∈ {𝑑 ∈ (ℕ0
↑𝑚 1𝑜) ∣ 𝑑 ∘𝑟 ≤
(1𝑜 × {𝑘})}) → 𝑘 ∈ ℕ0) |
82 | | elrabi 3359 |
. . . . . . . . . 10
⊢ (𝑐 ∈ {𝑑 ∈ (ℕ0
↑𝑚 1𝑜) ∣ 𝑑 ∘𝑟 ≤
(1𝑜 × {𝑘})} → 𝑐 ∈ (ℕ0
↑𝑚 1𝑜)) |
83 | 82 | adantl 482 |
. . . . . . . . 9
⊢ ((((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑐 ∈ {𝑑 ∈ (ℕ0
↑𝑚 1𝑜) ∣ 𝑑 ∘𝑟 ≤
(1𝑜 × {𝑘})}) → 𝑐 ∈ (ℕ0
↑𝑚 1𝑜)) |
84 | | coe1mul2lem1 19637 |
. . . . . . . . 9
⊢ ((𝑘 ∈ ℕ0
∧ 𝑐 ∈
(ℕ0 ↑𝑚 1𝑜)) →
(𝑐
∘𝑟 ≤ (1𝑜 × {𝑘}) ↔ (𝑐‘∅) ∈ (0...𝑘))) |
85 | 81, 83, 84 | syl2anc 693 |
. . . . . . . 8
⊢ ((((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑐 ∈ {𝑑 ∈ (ℕ0
↑𝑚 1𝑜) ∣ 𝑑 ∘𝑟 ≤
(1𝑜 × {𝑘})}) → (𝑐 ∘𝑟 ≤
(1𝑜 × {𝑘}) ↔ (𝑐‘∅) ∈ (0...𝑘))) |
86 | 80, 85 | mpbid 222 |
. . . . . . 7
⊢ ((((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑐 ∈ {𝑑 ∈ (ℕ0
↑𝑚 1𝑜) ∣ 𝑑 ∘𝑟 ≤
(1𝑜 × {𝑘})}) → (𝑐‘∅) ∈ (0...𝑘)) |
87 | | eqidd 2623 |
. . . . . . 7
⊢ (((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑘 ∈ ℕ0) → (𝑐 ∈ {𝑑 ∈ (ℕ0
↑𝑚 1𝑜) ∣ 𝑑 ∘𝑟 ≤
(1𝑜 × {𝑘})} ↦ (𝑐‘∅)) = (𝑐 ∈ {𝑑 ∈ (ℕ0
↑𝑚 1𝑜) ∣ 𝑑 ∘𝑟 ≤
(1𝑜 × {𝑘})} ↦ (𝑐‘∅))) |
88 | | eqidd 2623 |
. . . . . . 7
⊢ (((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑘 ∈ ℕ0) → (𝑥 ∈ (0...𝑘) ↦ (((coe1‘𝐹)‘𝑥) ·
((coe1‘𝐺)‘(𝑘 − 𝑥)))) = (𝑥 ∈ (0...𝑘) ↦ (((coe1‘𝐹)‘𝑥) ·
((coe1‘𝐺)‘(𝑘 − 𝑥))))) |
89 | | fveq2 6191 |
. . . . . . . 8
⊢ (𝑥 = (𝑐‘∅) →
((coe1‘𝐹)‘𝑥) = ((coe1‘𝐹)‘(𝑐‘∅))) |
90 | | oveq2 6658 |
. . . . . . . . 9
⊢ (𝑥 = (𝑐‘∅) → (𝑘 − 𝑥) = (𝑘 − (𝑐‘∅))) |
91 | 90 | fveq2d 6195 |
. . . . . . . 8
⊢ (𝑥 = (𝑐‘∅) →
((coe1‘𝐺)‘(𝑘 − 𝑥)) = ((coe1‘𝐺)‘(𝑘 − (𝑐‘∅)))) |
92 | 89, 91 | oveq12d 6668 |
. . . . . . 7
⊢ (𝑥 = (𝑐‘∅) →
(((coe1‘𝐹)‘𝑥) ·
((coe1‘𝐺)‘(𝑘 − 𝑥))) = (((coe1‘𝐹)‘(𝑐‘∅)) ·
((coe1‘𝐺)‘(𝑘 − (𝑐‘∅))))) |
93 | 86, 87, 88, 92 | fmptco 6396 |
. . . . . 6
⊢ (((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑘 ∈ ℕ0) → ((𝑥 ∈ (0...𝑘) ↦ (((coe1‘𝐹)‘𝑥) ·
((coe1‘𝐺)‘(𝑘 − 𝑥)))) ∘ (𝑐 ∈ {𝑑 ∈ (ℕ0
↑𝑚 1𝑜) ∣ 𝑑 ∘𝑟 ≤
(1𝑜 × {𝑘})} ↦ (𝑐‘∅))) = (𝑐 ∈ {𝑑 ∈ (ℕ0
↑𝑚 1𝑜) ∣ 𝑑 ∘𝑟 ≤
(1𝑜 × {𝑘})} ↦ (((coe1‘𝐹)‘(𝑐‘∅)) ·
((coe1‘𝐺)‘(𝑘 − (𝑐‘∅)))))) |
94 | | simpll2 1101 |
. . . . . . . . 9
⊢ ((((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑐 ∈ {𝑑 ∈ (ℕ0
↑𝑚 1𝑜) ∣ 𝑑 ∘𝑟 ≤
(1𝑜 × {𝑘})}) → 𝐹 ∈ 𝐵) |
95 | 44 | fvcoe1 19577 |
. . . . . . . . 9
⊢ ((𝐹 ∈ 𝐵 ∧ 𝑐 ∈ (ℕ0
↑𝑚 1𝑜)) → (𝐹‘𝑐) = ((coe1‘𝐹)‘(𝑐‘∅))) |
96 | 94, 83, 95 | syl2anc 693 |
. . . . . . . 8
⊢ ((((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑐 ∈ {𝑑 ∈ (ℕ0
↑𝑚 1𝑜) ∣ 𝑑 ∘𝑟 ≤
(1𝑜 × {𝑘})}) → (𝐹‘𝑐) = ((coe1‘𝐹)‘(𝑐‘∅))) |
97 | | df1o2 7572 |
. . . . . . . . . . . . . 14
⊢
1𝑜 = {∅} |
98 | | 0ex 4790 |
. . . . . . . . . . . . . 14
⊢ ∅
∈ V |
99 | 97, 2, 98 | mapsnconst 7903 |
. . . . . . . . . . . . 13
⊢ (𝑐 ∈ (ℕ0
↑𝑚 1𝑜) → 𝑐 = (1𝑜 × {(𝑐‘∅)})) |
100 | 83, 99 | syl 17 |
. . . . . . . . . . . 12
⊢ ((((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑐 ∈ {𝑑 ∈ (ℕ0
↑𝑚 1𝑜) ∣ 𝑑 ∘𝑟 ≤
(1𝑜 × {𝑘})}) → 𝑐 = (1𝑜 × {(𝑐‘∅)})) |
101 | 100 | oveq2d 6666 |
. . . . . . . . . . 11
⊢ ((((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑐 ∈ {𝑑 ∈ (ℕ0
↑𝑚 1𝑜) ∣ 𝑑 ∘𝑟 ≤
(1𝑜 × {𝑘})}) → ((1𝑜 ×
{𝑘})
∘𝑓 − 𝑐) = ((1𝑜 × {𝑘}) ∘𝑓
− (1𝑜 × {(𝑐‘∅)}))) |
102 | 3 | a1i 11 |
. . . . . . . . . . . 12
⊢ ((((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑐 ∈ {𝑑 ∈ (ℕ0
↑𝑚 1𝑜) ∣ 𝑑 ∘𝑟 ≤
(1𝑜 × {𝑘})}) → 1𝑜 ∈
On) |
103 | | vex 3203 |
. . . . . . . . . . . . 13
⊢ 𝑘 ∈ V |
104 | 103 | a1i 11 |
. . . . . . . . . . . 12
⊢ ((((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑐 ∈ {𝑑 ∈ (ℕ0
↑𝑚 1𝑜) ∣ 𝑑 ∘𝑟 ≤
(1𝑜 × {𝑘})}) → 𝑘 ∈ V) |
105 | | fvexd 6203 |
. . . . . . . . . . . 12
⊢ ((((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑐 ∈ {𝑑 ∈ (ℕ0
↑𝑚 1𝑜) ∣ 𝑑 ∘𝑟 ≤
(1𝑜 × {𝑘})}) → (𝑐‘∅) ∈ V) |
106 | 102, 104,
105 | ofc12 6922 |
. . . . . . . . . . 11
⊢ ((((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑐 ∈ {𝑑 ∈ (ℕ0
↑𝑚 1𝑜) ∣ 𝑑 ∘𝑟 ≤
(1𝑜 × {𝑘})}) → ((1𝑜 ×
{𝑘})
∘𝑓 − (1𝑜 × {(𝑐‘∅)})) =
(1𝑜 × {(𝑘 − (𝑐‘∅))})) |
107 | 101, 106 | eqtrd 2656 |
. . . . . . . . . 10
⊢ ((((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑐 ∈ {𝑑 ∈ (ℕ0
↑𝑚 1𝑜) ∣ 𝑑 ∘𝑟 ≤
(1𝑜 × {𝑘})}) → ((1𝑜 ×
{𝑘})
∘𝑓 − 𝑐) = (1𝑜 × {(𝑘 − (𝑐‘∅))})) |
108 | 107 | fveq2d 6195 |
. . . . . . . . 9
⊢ ((((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑐 ∈ {𝑑 ∈ (ℕ0
↑𝑚 1𝑜) ∣ 𝑑 ∘𝑟 ≤
(1𝑜 × {𝑘})}) → (𝐺‘((1𝑜 ×
{𝑘})
∘𝑓 − 𝑐)) = (𝐺‘(1𝑜 ×
{(𝑘 − (𝑐‘∅))}))) |
109 | | simpll3 1102 |
. . . . . . . . . 10
⊢ ((((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑐 ∈ {𝑑 ∈ (ℕ0
↑𝑚 1𝑜) ∣ 𝑑 ∘𝑟 ≤
(1𝑜 × {𝑘})}) → 𝐺 ∈ 𝐵) |
110 | | fznn0sub 12373 |
. . . . . . . . . . 11
⊢ ((𝑐‘∅) ∈
(0...𝑘) → (𝑘 − (𝑐‘∅)) ∈
ℕ0) |
111 | 86, 110 | syl 17 |
. . . . . . . . . 10
⊢ ((((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑐 ∈ {𝑑 ∈ (ℕ0
↑𝑚 1𝑜) ∣ 𝑑 ∘𝑟 ≤
(1𝑜 × {𝑘})}) → (𝑘 − (𝑐‘∅)) ∈
ℕ0) |
112 | 51 | coe1fv 19576 |
. . . . . . . . . 10
⊢ ((𝐺 ∈ 𝐵 ∧ (𝑘 − (𝑐‘∅)) ∈ ℕ0)
→ ((coe1‘𝐺)‘(𝑘 − (𝑐‘∅))) = (𝐺‘(1𝑜 ×
{(𝑘 − (𝑐‘∅))}))) |
113 | 109, 111,
112 | syl2anc 693 |
. . . . . . . . 9
⊢ ((((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑐 ∈ {𝑑 ∈ (ℕ0
↑𝑚 1𝑜) ∣ 𝑑 ∘𝑟 ≤
(1𝑜 × {𝑘})}) → ((coe1‘𝐺)‘(𝑘 − (𝑐‘∅))) = (𝐺‘(1𝑜 ×
{(𝑘 − (𝑐‘∅))}))) |
114 | 108, 113 | eqtr4d 2659 |
. . . . . . . 8
⊢ ((((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑐 ∈ {𝑑 ∈ (ℕ0
↑𝑚 1𝑜) ∣ 𝑑 ∘𝑟 ≤
(1𝑜 × {𝑘})}) → (𝐺‘((1𝑜 ×
{𝑘})
∘𝑓 − 𝑐)) = ((coe1‘𝐺)‘(𝑘 − (𝑐‘∅)))) |
115 | 96, 114 | oveq12d 6668 |
. . . . . . 7
⊢ ((((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑐 ∈ {𝑑 ∈ (ℕ0
↑𝑚 1𝑜) ∣ 𝑑 ∘𝑟 ≤
(1𝑜 × {𝑘})}) → ((𝐹‘𝑐) · (𝐺‘((1𝑜 ×
{𝑘})
∘𝑓 − 𝑐))) = (((coe1‘𝐹)‘(𝑐‘∅)) ·
((coe1‘𝐺)‘(𝑘 − (𝑐‘∅))))) |
116 | 115 | mpteq2dva 4744 |
. . . . . 6
⊢ (((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑘 ∈ ℕ0) → (𝑐 ∈ {𝑑 ∈ (ℕ0
↑𝑚 1𝑜) ∣ 𝑑 ∘𝑟 ≤
(1𝑜 × {𝑘})} ↦ ((𝐹‘𝑐) · (𝐺‘((1𝑜 ×
{𝑘})
∘𝑓 − 𝑐)))) = (𝑐 ∈ {𝑑 ∈ (ℕ0
↑𝑚 1𝑜) ∣ 𝑑 ∘𝑟 ≤
(1𝑜 × {𝑘})} ↦ (((coe1‘𝐹)‘(𝑐‘∅)) ·
((coe1‘𝐺)‘(𝑘 − (𝑐‘∅)))))) |
117 | 93, 116 | eqtr4d 2659 |
. . . . 5
⊢ (((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑘 ∈ ℕ0) → ((𝑥 ∈ (0...𝑘) ↦ (((coe1‘𝐹)‘𝑥) ·
((coe1‘𝐺)‘(𝑘 − 𝑥)))) ∘ (𝑐 ∈ {𝑑 ∈ (ℕ0
↑𝑚 1𝑜) ∣ 𝑑 ∘𝑟 ≤
(1𝑜 × {𝑘})} ↦ (𝑐‘∅))) = (𝑐 ∈ {𝑑 ∈ (ℕ0
↑𝑚 1𝑜) ∣ 𝑑 ∘𝑟 ≤
(1𝑜 × {𝑘})} ↦ ((𝐹‘𝑐) · (𝐺‘((1𝑜 ×
{𝑘})
∘𝑓 − 𝑐))))) |
118 | 117 | oveq2d 6666 |
. . . 4
⊢ (((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑘 ∈ ℕ0) → (𝑅 Σg
((𝑥 ∈ (0...𝑘) ↦
(((coe1‘𝐹)‘𝑥) ·
((coe1‘𝐺)‘(𝑘 − 𝑥)))) ∘ (𝑐 ∈ {𝑑 ∈ (ℕ0
↑𝑚 1𝑜) ∣ 𝑑 ∘𝑟 ≤
(1𝑜 × {𝑘})} ↦ (𝑐‘∅)))) = (𝑅 Σg (𝑐 ∈ {𝑑 ∈ (ℕ0
↑𝑚 1𝑜) ∣ 𝑑 ∘𝑟 ≤
(1𝑜 × {𝑘})} ↦ ((𝐹‘𝑐) · (𝐺‘((1𝑜 ×
{𝑘})
∘𝑓 − 𝑐)))))) |
119 | 76, 118 | eqtrd 2656 |
. . 3
⊢ (((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑘 ∈ ℕ0) → (𝑅 Σg
(𝑥 ∈ (0...𝑘) ↦
(((coe1‘𝐹)‘𝑥) ·
((coe1‘𝐺)‘(𝑘 − 𝑥))))) = (𝑅 Σg (𝑐 ∈ {𝑑 ∈ (ℕ0
↑𝑚 1𝑜) ∣ 𝑑 ∘𝑟 ≤
(1𝑜 × {𝑘})} ↦ ((𝐹‘𝑐) · (𝐺‘((1𝑜 ×
{𝑘})
∘𝑓 − 𝑐)))))) |
120 | 119 | mpteq2dva 4744 |
. 2
⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → (𝑘 ∈ ℕ0 ↦ (𝑅 Σg
(𝑥 ∈ (0...𝑘) ↦
(((coe1‘𝐹)‘𝑥) ·
((coe1‘𝐺)‘(𝑘 − 𝑥)))))) = (𝑘 ∈ ℕ0 ↦ (𝑅 Σg
(𝑐 ∈ {𝑑 ∈ (ℕ0
↑𝑚 1𝑜) ∣ 𝑑 ∘𝑟 ≤
(1𝑜 × {𝑘})} ↦ ((𝐹‘𝑐) · (𝐺‘((1𝑜 ×
{𝑘})
∘𝑓 − 𝑐))))))) |
121 | 27, 34, 120 | 3eqtr4d 2666 |
1
⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → (coe1‘(𝐹 ∙ 𝐺)) = (𝑘 ∈ ℕ0 ↦ (𝑅 Σg
(𝑥 ∈ (0...𝑘) ↦
(((coe1‘𝐹)‘𝑥) ·
((coe1‘𝐺)‘(𝑘 − 𝑥))))))) |