Step | Hyp | Ref
| Expression |
1 | | ssrab2 3650 |
. . . . . 6
⊢ {𝑣 ∈ ((Base‘𝑅) ↑𝑚
ℕ0) ∣ 𝑣 finSupp 0 } ⊆
((Base‘𝑅)
↑𝑚 ℕ0) |
2 | 1 | a1i 11 |
. . . . 5
⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → {𝑣 ∈ ((Base‘𝑅) ↑𝑚
ℕ0) ∣ 𝑣 finSupp 0 } ⊆
((Base‘𝑅)
↑𝑚 ℕ0)) |
3 | 2 | olcd 407 |
. . . 4
⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → (∪ 𝑢 ∈ (𝑁 × 𝑁){(coe1‘(𝑀‘𝑢))} ⊆ ((Base‘𝑅) ↑𝑚
ℕ0) ∨ {𝑣 ∈ ((Base‘𝑅) ↑𝑚
ℕ0) ∣ 𝑣 finSupp 0 } ⊆
((Base‘𝑅)
↑𝑚 ℕ0))) |
4 | | inss 3804 |
. . . 4
⊢
((∪ 𝑢 ∈ (𝑁 × 𝑁){(coe1‘(𝑀‘𝑢))} ⊆ ((Base‘𝑅) ↑𝑚
ℕ0) ∨ {𝑣 ∈ ((Base‘𝑅) ↑𝑚
ℕ0) ∣ 𝑣 finSupp 0 } ⊆
((Base‘𝑅)
↑𝑚 ℕ0)) → (∪ 𝑢 ∈ (𝑁 × 𝑁){(coe1‘(𝑀‘𝑢))} ∩ {𝑣 ∈ ((Base‘𝑅) ↑𝑚
ℕ0) ∣ 𝑣 finSupp 0 }) ⊆
((Base‘𝑅)
↑𝑚 ℕ0)) |
5 | 3, 4 | syl 17 |
. . 3
⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → (∪ 𝑢 ∈ (𝑁 × 𝑁){(coe1‘(𝑀‘𝑢))} ∩ {𝑣 ∈ ((Base‘𝑅) ↑𝑚
ℕ0) ∣ 𝑣 finSupp 0 }) ⊆
((Base‘𝑅)
↑𝑚 ℕ0)) |
6 | | xpfi 8116 |
. . . . . . 7
⊢ ((𝑁 ∈ Fin ∧ 𝑁 ∈ Fin) → (𝑁 × 𝑁) ∈ Fin) |
7 | 6 | anidms 675 |
. . . . . 6
⊢ (𝑁 ∈ Fin → (𝑁 × 𝑁) ∈ Fin) |
8 | | snfi 7923 |
. . . . . . . 8
⊢
{(coe1‘(𝑀‘𝑢))} ∈ Fin |
9 | 8 | a1i 11 |
. . . . . . 7
⊢ ((𝑁 ∈ Fin ∧ 𝑢 ∈ (𝑁 × 𝑁)) → {(coe1‘(𝑀‘𝑢))} ∈ Fin) |
10 | 9 | ralrimiva 2949 |
. . . . . 6
⊢ (𝑁 ∈ Fin → ∀𝑢 ∈ (𝑁 × 𝑁){(coe1‘(𝑀‘𝑢))} ∈ Fin) |
11 | 7, 10 | jca 553 |
. . . . 5
⊢ (𝑁 ∈ Fin → ((𝑁 × 𝑁) ∈ Fin ∧ ∀𝑢 ∈ (𝑁 × 𝑁){(coe1‘(𝑀‘𝑢))} ∈ Fin)) |
12 | 11 | 3ad2ant1 1075 |
. . . 4
⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → ((𝑁 × 𝑁) ∈ Fin ∧ ∀𝑢 ∈ (𝑁 × 𝑁){(coe1‘(𝑀‘𝑢))} ∈ Fin)) |
13 | | iunfi 8137 |
. . . 4
⊢ (((𝑁 × 𝑁) ∈ Fin ∧ ∀𝑢 ∈ (𝑁 × 𝑁){(coe1‘(𝑀‘𝑢))} ∈ Fin) → ∪ 𝑢 ∈ (𝑁 × 𝑁){(coe1‘(𝑀‘𝑢))} ∈ Fin) |
14 | | infi 8069 |
. . . 4
⊢ (∪ 𝑢 ∈ (𝑁 × 𝑁){(coe1‘(𝑀‘𝑢))} ∈ Fin → (∪ 𝑢 ∈ (𝑁 × 𝑁){(coe1‘(𝑀‘𝑢))} ∩ {𝑣 ∈ ((Base‘𝑅) ↑𝑚
ℕ0) ∣ 𝑣 finSupp 0 }) ∈
Fin) |
15 | 12, 13, 14 | 3syl 18 |
. . 3
⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → (∪ 𝑢 ∈ (𝑁 × 𝑁){(coe1‘(𝑀‘𝑢))} ∩ {𝑣 ∈ ((Base‘𝑅) ↑𝑚
ℕ0) ∣ 𝑣 finSupp 0 }) ∈
Fin) |
16 | | pmatcoe1fsupp.0 |
. . . . 5
⊢ 0 =
(0g‘𝑅) |
17 | | fvex 6113 |
. . . . 5
⊢
(0g‘𝑅) ∈ V |
18 | 16, 17 | eqeltri 2684 |
. . . 4
⊢ 0 ∈
V |
19 | 18 | a1i 11 |
. . 3
⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → 0 ∈ V) |
20 | | elin 3758 |
. . . . . 6
⊢ (𝑤 ∈ (∪ 𝑢 ∈ (𝑁 × 𝑁){(coe1‘(𝑀‘𝑢))} ∩ {𝑣 ∈ ((Base‘𝑅) ↑𝑚
ℕ0) ∣ 𝑣 finSupp 0 }) ↔ (𝑤 ∈ ∪ 𝑢 ∈ (𝑁 × 𝑁){(coe1‘(𝑀‘𝑢))} ∧ 𝑤 ∈ {𝑣 ∈ ((Base‘𝑅) ↑𝑚
ℕ0) ∣ 𝑣 finSupp 0 })) |
21 | | breq1 4586 |
. . . . . . . 8
⊢ (𝑣 = 𝑤 → (𝑣 finSupp 0 ↔ 𝑤 finSupp 0 )) |
22 | 21 | elrab 3331 |
. . . . . . 7
⊢ (𝑤 ∈ {𝑣 ∈ ((Base‘𝑅) ↑𝑚
ℕ0) ∣ 𝑣 finSupp 0 } ↔ (𝑤 ∈ ((Base‘𝑅) ↑𝑚
ℕ0) ∧ 𝑤 finSupp 0 )) |
23 | 22 | simprbi 479 |
. . . . . 6
⊢ (𝑤 ∈ {𝑣 ∈ ((Base‘𝑅) ↑𝑚
ℕ0) ∣ 𝑣 finSupp 0 } → 𝑤 finSupp 0 ) |
24 | 20, 23 | simplbiim 657 |
. . . . 5
⊢ (𝑤 ∈ (∪ 𝑢 ∈ (𝑁 × 𝑁){(coe1‘(𝑀‘𝑢))} ∩ {𝑣 ∈ ((Base‘𝑅) ↑𝑚
ℕ0) ∣ 𝑣 finSupp 0 }) → 𝑤 finSupp 0 ) |
25 | 24 | rgen 2906 |
. . . 4
⊢
∀𝑤 ∈
(∪ 𝑢 ∈ (𝑁 × 𝑁){(coe1‘(𝑀‘𝑢))} ∩ {𝑣 ∈ ((Base‘𝑅) ↑𝑚
ℕ0) ∣ 𝑣 finSupp 0 })𝑤 finSupp 0 |
26 | 25 | a1i 11 |
. . 3
⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → ∀𝑤 ∈ (∪
𝑢 ∈ (𝑁 × 𝑁){(coe1‘(𝑀‘𝑢))} ∩ {𝑣 ∈ ((Base‘𝑅) ↑𝑚
ℕ0) ∣ 𝑣 finSupp 0 })𝑤 finSupp 0 ) |
27 | | fsuppmapnn0fiub0 12655 |
. . . 4
⊢
(((∪ 𝑢 ∈ (𝑁 × 𝑁){(coe1‘(𝑀‘𝑢))} ∩ {𝑣 ∈ ((Base‘𝑅) ↑𝑚
ℕ0) ∣ 𝑣 finSupp 0 }) ⊆
((Base‘𝑅)
↑𝑚 ℕ0) ∧ (∪ 𝑢 ∈ (𝑁 × 𝑁){(coe1‘(𝑀‘𝑢))} ∩ {𝑣 ∈ ((Base‘𝑅) ↑𝑚
ℕ0) ∣ 𝑣 finSupp 0 }) ∈ Fin ∧ 0 ∈ V)
→ (∀𝑤 ∈
(∪ 𝑢 ∈ (𝑁 × 𝑁){(coe1‘(𝑀‘𝑢))} ∩ {𝑣 ∈ ((Base‘𝑅) ↑𝑚
ℕ0) ∣ 𝑣 finSupp 0 })𝑤 finSupp 0 → ∃𝑠 ∈ ℕ0
∀𝑤 ∈ (∪ 𝑢 ∈ (𝑁 × 𝑁){(coe1‘(𝑀‘𝑢))} ∩ {𝑣 ∈ ((Base‘𝑅) ↑𝑚
ℕ0) ∣ 𝑣 finSupp 0 })∀𝑧 ∈ ℕ0
(𝑠 < 𝑧 → (𝑤‘𝑧) = 0 ))) |
28 | 27 | imp 444 |
. . 3
⊢
((((∪ 𝑢 ∈ (𝑁 × 𝑁){(coe1‘(𝑀‘𝑢))} ∩ {𝑣 ∈ ((Base‘𝑅) ↑𝑚
ℕ0) ∣ 𝑣 finSupp 0 }) ⊆
((Base‘𝑅)
↑𝑚 ℕ0) ∧ (∪ 𝑢 ∈ (𝑁 × 𝑁){(coe1‘(𝑀‘𝑢))} ∩ {𝑣 ∈ ((Base‘𝑅) ↑𝑚
ℕ0) ∣ 𝑣 finSupp 0 }) ∈ Fin ∧ 0 ∈ V)
∧ ∀𝑤 ∈
(∪ 𝑢 ∈ (𝑁 × 𝑁){(coe1‘(𝑀‘𝑢))} ∩ {𝑣 ∈ ((Base‘𝑅) ↑𝑚
ℕ0) ∣ 𝑣 finSupp 0 })𝑤 finSupp 0 ) → ∃𝑠 ∈ ℕ0
∀𝑤 ∈ (∪ 𝑢 ∈ (𝑁 × 𝑁){(coe1‘(𝑀‘𝑢))} ∩ {𝑣 ∈ ((Base‘𝑅) ↑𝑚
ℕ0) ∣ 𝑣 finSupp 0 })∀𝑧 ∈ ℕ0
(𝑠 < 𝑧 → (𝑤‘𝑧) = 0 )) |
29 | 5, 15, 19, 26, 28 | syl31anc 1321 |
. 2
⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → ∃𝑠 ∈ ℕ0 ∀𝑤 ∈ (∪ 𝑢 ∈ (𝑁 × 𝑁){(coe1‘(𝑀‘𝑢))} ∩ {𝑣 ∈ ((Base‘𝑅) ↑𝑚
ℕ0) ∣ 𝑣 finSupp 0 })∀𝑧 ∈ ℕ0
(𝑠 < 𝑧 → (𝑤‘𝑧) = 0 )) |
30 | | opelxpi 5072 |
. . . . . . . . . . . . . . 15
⊢ ((𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → 〈𝑖, 𝑗〉 ∈ (𝑁 × 𝑁)) |
31 | | df-ov 6552 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑖𝑀𝑗) = (𝑀‘〈𝑖, 𝑗〉) |
32 | 31 | fveq2i 6106 |
. . . . . . . . . . . . . . . . 17
⊢
(coe1‘(𝑖𝑀𝑗)) = (coe1‘(𝑀‘〈𝑖, 𝑗〉)) |
33 | | fvex 6113 |
. . . . . . . . . . . . . . . . . 18
⊢
(coe1‘(𝑀‘〈𝑖, 𝑗〉)) ∈ V |
34 | 33 | snid 4155 |
. . . . . . . . . . . . . . . . 17
⊢
(coe1‘(𝑀‘〈𝑖, 𝑗〉)) ∈
{(coe1‘(𝑀‘〈𝑖, 𝑗〉))} |
35 | 32, 34 | eqeltri 2684 |
. . . . . . . . . . . . . . . 16
⊢
(coe1‘(𝑖𝑀𝑗)) ∈ {(coe1‘(𝑀‘〈𝑖, 𝑗〉))} |
36 | 35 | a1i 11 |
. . . . . . . . . . . . . . 15
⊢ ((𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → (coe1‘(𝑖𝑀𝑗)) ∈ {(coe1‘(𝑀‘〈𝑖, 𝑗〉))}) |
37 | | fveq2 6103 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑢 = 〈𝑖, 𝑗〉 → (𝑀‘𝑢) = (𝑀‘〈𝑖, 𝑗〉)) |
38 | 37 | fveq2d 6107 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑢 = 〈𝑖, 𝑗〉 → (coe1‘(𝑀‘𝑢)) = (coe1‘(𝑀‘〈𝑖, 𝑗〉))) |
39 | 38 | sneqd 4137 |
. . . . . . . . . . . . . . . 16
⊢ (𝑢 = 〈𝑖, 𝑗〉 → {(coe1‘(𝑀‘𝑢))} = {(coe1‘(𝑀‘〈𝑖, 𝑗〉))}) |
40 | 39 | eliuni 4462 |
. . . . . . . . . . . . . . 15
⊢
((〈𝑖, 𝑗〉 ∈ (𝑁 × 𝑁) ∧ (coe1‘(𝑖𝑀𝑗)) ∈ {(coe1‘(𝑀‘〈𝑖, 𝑗〉))}) →
(coe1‘(𝑖𝑀𝑗)) ∈ ∪
𝑢 ∈ (𝑁 × 𝑁){(coe1‘(𝑀‘𝑢))}) |
41 | 30, 36, 40 | syl2anc 691 |
. . . . . . . . . . . . . 14
⊢ ((𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → (coe1‘(𝑖𝑀𝑗)) ∈ ∪
𝑢 ∈ (𝑁 × 𝑁){(coe1‘(𝑀‘𝑢))}) |
42 | 41 | adantl 481 |
. . . . . . . . . . . . 13
⊢
(((((𝑁 ∈ Fin
∧ 𝑅 ∈ Ring ∧
𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ0) ∧ 𝑥 ∈ ℕ0)
∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → (coe1‘(𝑖𝑀𝑗)) ∈ ∪
𝑢 ∈ (𝑁 × 𝑁){(coe1‘(𝑀‘𝑢))}) |
43 | | pmatcoe1fsupp.c |
. . . . . . . . . . . . . . . 16
⊢ 𝐶 = (𝑁 Mat 𝑃) |
44 | | eqid 2610 |
. . . . . . . . . . . . . . . 16
⊢
(Base‘𝑃) =
(Base‘𝑃) |
45 | | pmatcoe1fsupp.b |
. . . . . . . . . . . . . . . 16
⊢ 𝐵 = (Base‘𝐶) |
46 | | simprl 790 |
. . . . . . . . . . . . . . . 16
⊢
(((((𝑁 ∈ Fin
∧ 𝑅 ∈ Ring ∧
𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ0) ∧ 𝑥 ∈ ℕ0)
∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → 𝑖 ∈ 𝑁) |
47 | | simprr 792 |
. . . . . . . . . . . . . . . 16
⊢
(((((𝑁 ∈ Fin
∧ 𝑅 ∈ Ring ∧
𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ0) ∧ 𝑥 ∈ ℕ0)
∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → 𝑗 ∈ 𝑁) |
48 | 45 | eleq2i 2680 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑀 ∈ 𝐵 ↔ 𝑀 ∈ (Base‘𝐶)) |
49 | 48 | biimpi 205 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑀 ∈ 𝐵 → 𝑀 ∈ (Base‘𝐶)) |
50 | 49 | 3ad2ant3 1077 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → 𝑀 ∈ (Base‘𝐶)) |
51 | 50 | ad3antrrr 762 |
. . . . . . . . . . . . . . . . 17
⊢
(((((𝑁 ∈ Fin
∧ 𝑅 ∈ Ring ∧
𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ0) ∧ 𝑥 ∈ ℕ0)
∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → 𝑀 ∈ (Base‘𝐶)) |
52 | 51, 45 | syl6eleqr 2699 |
. . . . . . . . . . . . . . . 16
⊢
(((((𝑁 ∈ Fin
∧ 𝑅 ∈ Ring ∧
𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ0) ∧ 𝑥 ∈ ℕ0)
∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → 𝑀 ∈ 𝐵) |
53 | 43, 44, 45, 46, 47, 52 | matecld 20051 |
. . . . . . . . . . . . . . 15
⊢
(((((𝑁 ∈ Fin
∧ 𝑅 ∈ Ring ∧
𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ0) ∧ 𝑥 ∈ ℕ0)
∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → (𝑖𝑀𝑗) ∈ (Base‘𝑃)) |
54 | | eqid 2610 |
. . . . . . . . . . . . . . . 16
⊢
(coe1‘(𝑖𝑀𝑗)) = (coe1‘(𝑖𝑀𝑗)) |
55 | | pmatcoe1fsupp.p |
. . . . . . . . . . . . . . . 16
⊢ 𝑃 = (Poly1‘𝑅) |
56 | | eqid 2610 |
. . . . . . . . . . . . . . . 16
⊢
(0g‘𝑅) = (0g‘𝑅) |
57 | | eqid 2610 |
. . . . . . . . . . . . . . . 16
⊢
(Base‘𝑅) =
(Base‘𝑅) |
58 | 54, 44, 55, 56, 57 | coe1fsupp 19405 |
. . . . . . . . . . . . . . 15
⊢ ((𝑖𝑀𝑗) ∈ (Base‘𝑃) → (coe1‘(𝑖𝑀𝑗)) ∈ {𝑣 ∈ ((Base‘𝑅) ↑𝑚
ℕ0) ∣ 𝑣 finSupp (0g‘𝑅)}) |
59 | 53, 58 | syl 17 |
. . . . . . . . . . . . . 14
⊢
(((((𝑁 ∈ Fin
∧ 𝑅 ∈ Ring ∧
𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ0) ∧ 𝑥 ∈ ℕ0)
∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → (coe1‘(𝑖𝑀𝑗)) ∈ {𝑣 ∈ ((Base‘𝑅) ↑𝑚
ℕ0) ∣ 𝑣 finSupp (0g‘𝑅)}) |
60 | 16 | a1i 11 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → 0 =
(0g‘𝑅)) |
61 | 60 | breq2d 4595 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → (𝑣 finSupp 0 ↔ 𝑣 finSupp (0g‘𝑅))) |
62 | 61 | rabbidv 3164 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → {𝑣 ∈ ((Base‘𝑅) ↑𝑚
ℕ0) ∣ 𝑣 finSupp 0 } = {𝑣 ∈ ((Base‘𝑅) ↑𝑚
ℕ0) ∣ 𝑣 finSupp (0g‘𝑅)}) |
63 | 62 | eleq2d 2673 |
. . . . . . . . . . . . . . 15
⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → ((coe1‘(𝑖𝑀𝑗)) ∈ {𝑣 ∈ ((Base‘𝑅) ↑𝑚
ℕ0) ∣ 𝑣 finSupp 0 } ↔
(coe1‘(𝑖𝑀𝑗)) ∈ {𝑣 ∈ ((Base‘𝑅) ↑𝑚
ℕ0) ∣ 𝑣 finSupp (0g‘𝑅)})) |
64 | 63 | ad3antrrr 762 |
. . . . . . . . . . . . . 14
⊢
(((((𝑁 ∈ Fin
∧ 𝑅 ∈ Ring ∧
𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ0) ∧ 𝑥 ∈ ℕ0)
∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → ((coe1‘(𝑖𝑀𝑗)) ∈ {𝑣 ∈ ((Base‘𝑅) ↑𝑚
ℕ0) ∣ 𝑣 finSupp 0 } ↔
(coe1‘(𝑖𝑀𝑗)) ∈ {𝑣 ∈ ((Base‘𝑅) ↑𝑚
ℕ0) ∣ 𝑣 finSupp (0g‘𝑅)})) |
65 | 59, 64 | mpbird 246 |
. . . . . . . . . . . . 13
⊢
(((((𝑁 ∈ Fin
∧ 𝑅 ∈ Ring ∧
𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ0) ∧ 𝑥 ∈ ℕ0)
∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → (coe1‘(𝑖𝑀𝑗)) ∈ {𝑣 ∈ ((Base‘𝑅) ↑𝑚
ℕ0) ∣ 𝑣 finSupp 0 }) |
66 | 42, 65 | elind 3760 |
. . . . . . . . . . . 12
⊢
(((((𝑁 ∈ Fin
∧ 𝑅 ∈ Ring ∧
𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ0) ∧ 𝑥 ∈ ℕ0)
∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → (coe1‘(𝑖𝑀𝑗)) ∈ (∪ 𝑢 ∈ (𝑁 × 𝑁){(coe1‘(𝑀‘𝑢))} ∩ {𝑣 ∈ ((Base‘𝑅) ↑𝑚
ℕ0) ∣ 𝑣 finSupp 0 })) |
67 | | simplr 788 |
. . . . . . . . . . . 12
⊢
(((((𝑁 ∈ Fin
∧ 𝑅 ∈ Ring ∧
𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ0) ∧ 𝑥 ∈ ℕ0)
∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → 𝑥 ∈ ℕ0) |
68 | | fveq1 6102 |
. . . . . . . . . . . . . . 15
⊢ (𝑤 = (coe1‘(𝑖𝑀𝑗)) → (𝑤‘𝑧) = ((coe1‘(𝑖𝑀𝑗))‘𝑧)) |
69 | 68 | eqeq1d 2612 |
. . . . . . . . . . . . . 14
⊢ (𝑤 = (coe1‘(𝑖𝑀𝑗)) → ((𝑤‘𝑧) = 0 ↔
((coe1‘(𝑖𝑀𝑗))‘𝑧) = 0 )) |
70 | 69 | imbi2d 329 |
. . . . . . . . . . . . 13
⊢ (𝑤 = (coe1‘(𝑖𝑀𝑗)) → ((𝑠 < 𝑧 → (𝑤‘𝑧) = 0 ) ↔ (𝑠 < 𝑧 → ((coe1‘(𝑖𝑀𝑗))‘𝑧) = 0 ))) |
71 | | breq2 4587 |
. . . . . . . . . . . . . 14
⊢ (𝑧 = 𝑥 → (𝑠 < 𝑧 ↔ 𝑠 < 𝑥)) |
72 | | fveq2 6103 |
. . . . . . . . . . . . . . 15
⊢ (𝑧 = 𝑥 → ((coe1‘(𝑖𝑀𝑗))‘𝑧) = ((coe1‘(𝑖𝑀𝑗))‘𝑥)) |
73 | 72 | eqeq1d 2612 |
. . . . . . . . . . . . . 14
⊢ (𝑧 = 𝑥 → (((coe1‘(𝑖𝑀𝑗))‘𝑧) = 0 ↔
((coe1‘(𝑖𝑀𝑗))‘𝑥) = 0 )) |
74 | 71, 73 | imbi12d 333 |
. . . . . . . . . . . . 13
⊢ (𝑧 = 𝑥 → ((𝑠 < 𝑧 → ((coe1‘(𝑖𝑀𝑗))‘𝑧) = 0 ) ↔ (𝑠 < 𝑥 → ((coe1‘(𝑖𝑀𝑗))‘𝑥) = 0 ))) |
75 | 70, 74 | rspc2v 3293 |
. . . . . . . . . . . 12
⊢
(((coe1‘(𝑖𝑀𝑗)) ∈ (∪ 𝑢 ∈ (𝑁 × 𝑁){(coe1‘(𝑀‘𝑢))} ∩ {𝑣 ∈ ((Base‘𝑅) ↑𝑚
ℕ0) ∣ 𝑣 finSupp 0 }) ∧ 𝑥 ∈ ℕ0)
→ (∀𝑤 ∈
(∪ 𝑢 ∈ (𝑁 × 𝑁){(coe1‘(𝑀‘𝑢))} ∩ {𝑣 ∈ ((Base‘𝑅) ↑𝑚
ℕ0) ∣ 𝑣 finSupp 0 })∀𝑧 ∈ ℕ0
(𝑠 < 𝑧 → (𝑤‘𝑧) = 0 ) → (𝑠 < 𝑥 → ((coe1‘(𝑖𝑀𝑗))‘𝑥) = 0 ))) |
76 | 66, 67, 75 | syl2anc 691 |
. . . . . . . . . . 11
⊢
(((((𝑁 ∈ Fin
∧ 𝑅 ∈ Ring ∧
𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ0) ∧ 𝑥 ∈ ℕ0)
∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → (∀𝑤 ∈ (∪
𝑢 ∈ (𝑁 × 𝑁){(coe1‘(𝑀‘𝑢))} ∩ {𝑣 ∈ ((Base‘𝑅) ↑𝑚
ℕ0) ∣ 𝑣 finSupp 0 })∀𝑧 ∈ ℕ0
(𝑠 < 𝑧 → (𝑤‘𝑧) = 0 ) → (𝑠 < 𝑥 → ((coe1‘(𝑖𝑀𝑗))‘𝑥) = 0 ))) |
77 | 76 | ex 449 |
. . . . . . . . . 10
⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ0) ∧ 𝑥 ∈ ℕ0)
→ ((𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → (∀𝑤 ∈ (∪
𝑢 ∈ (𝑁 × 𝑁){(coe1‘(𝑀‘𝑢))} ∩ {𝑣 ∈ ((Base‘𝑅) ↑𝑚
ℕ0) ∣ 𝑣 finSupp 0 })∀𝑧 ∈ ℕ0
(𝑠 < 𝑧 → (𝑤‘𝑧) = 0 ) → (𝑠 < 𝑥 → ((coe1‘(𝑖𝑀𝑗))‘𝑥) = 0 )))) |
78 | 77 | com23 84 |
. . . . . . . . 9
⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ0) ∧ 𝑥 ∈ ℕ0)
→ (∀𝑤 ∈
(∪ 𝑢 ∈ (𝑁 × 𝑁){(coe1‘(𝑀‘𝑢))} ∩ {𝑣 ∈ ((Base‘𝑅) ↑𝑚
ℕ0) ∣ 𝑣 finSupp 0 })∀𝑧 ∈ ℕ0
(𝑠 < 𝑧 → (𝑤‘𝑧) = 0 ) → ((𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → (𝑠 < 𝑥 → ((coe1‘(𝑖𝑀𝑗))‘𝑥) = 0 )))) |
79 | 78 | impancom 455 |
. . . . . . . 8
⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ0) ∧
∀𝑤 ∈ (∪ 𝑢 ∈ (𝑁 × 𝑁){(coe1‘(𝑀‘𝑢))} ∩ {𝑣 ∈ ((Base‘𝑅) ↑𝑚
ℕ0) ∣ 𝑣 finSupp 0 })∀𝑧 ∈ ℕ0
(𝑠 < 𝑧 → (𝑤‘𝑧) = 0 )) → (𝑥 ∈ ℕ0
→ ((𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → (𝑠 < 𝑥 → ((coe1‘(𝑖𝑀𝑗))‘𝑥) = 0 )))) |
80 | 79 | imp 444 |
. . . . . . 7
⊢
(((((𝑁 ∈ Fin
∧ 𝑅 ∈ Ring ∧
𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ0) ∧
∀𝑤 ∈ (∪ 𝑢 ∈ (𝑁 × 𝑁){(coe1‘(𝑀‘𝑢))} ∩ {𝑣 ∈ ((Base‘𝑅) ↑𝑚
ℕ0) ∣ 𝑣 finSupp 0 })∀𝑧 ∈ ℕ0
(𝑠 < 𝑧 → (𝑤‘𝑧) = 0 )) ∧ 𝑥 ∈ ℕ0)
→ ((𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → (𝑠 < 𝑥 → ((coe1‘(𝑖𝑀𝑗))‘𝑥) = 0 ))) |
81 | 80 | com23 84 |
. . . . . 6
⊢
(((((𝑁 ∈ Fin
∧ 𝑅 ∈ Ring ∧
𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ0) ∧
∀𝑤 ∈ (∪ 𝑢 ∈ (𝑁 × 𝑁){(coe1‘(𝑀‘𝑢))} ∩ {𝑣 ∈ ((Base‘𝑅) ↑𝑚
ℕ0) ∣ 𝑣 finSupp 0 })∀𝑧 ∈ ℕ0
(𝑠 < 𝑧 → (𝑤‘𝑧) = 0 )) ∧ 𝑥 ∈ ℕ0)
→ (𝑠 < 𝑥 → ((𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → ((coe1‘(𝑖𝑀𝑗))‘𝑥) = 0 ))) |
82 | 81 | ralrimdvv 2956 |
. . . . 5
⊢
(((((𝑁 ∈ Fin
∧ 𝑅 ∈ Ring ∧
𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ0) ∧
∀𝑤 ∈ (∪ 𝑢 ∈ (𝑁 × 𝑁){(coe1‘(𝑀‘𝑢))} ∩ {𝑣 ∈ ((Base‘𝑅) ↑𝑚
ℕ0) ∣ 𝑣 finSupp 0 })∀𝑧 ∈ ℕ0
(𝑠 < 𝑧 → (𝑤‘𝑧) = 0 )) ∧ 𝑥 ∈ ℕ0)
→ (𝑠 < 𝑥 → ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 ((coe1‘(𝑖𝑀𝑗))‘𝑥) = 0 )) |
83 | 82 | ralrimiva 2949 |
. . . 4
⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ0) ∧
∀𝑤 ∈ (∪ 𝑢 ∈ (𝑁 × 𝑁){(coe1‘(𝑀‘𝑢))} ∩ {𝑣 ∈ ((Base‘𝑅) ↑𝑚
ℕ0) ∣ 𝑣 finSupp 0 })∀𝑧 ∈ ℕ0
(𝑠 < 𝑧 → (𝑤‘𝑧) = 0 )) → ∀𝑥 ∈ ℕ0
(𝑠 < 𝑥 → ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 ((coe1‘(𝑖𝑀𝑗))‘𝑥) = 0 )) |
84 | 83 | ex 449 |
. . 3
⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ0) →
(∀𝑤 ∈ (∪ 𝑢 ∈ (𝑁 × 𝑁){(coe1‘(𝑀‘𝑢))} ∩ {𝑣 ∈ ((Base‘𝑅) ↑𝑚
ℕ0) ∣ 𝑣 finSupp 0 })∀𝑧 ∈ ℕ0
(𝑠 < 𝑧 → (𝑤‘𝑧) = 0 ) → ∀𝑥 ∈ ℕ0
(𝑠 < 𝑥 → ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 ((coe1‘(𝑖𝑀𝑗))‘𝑥) = 0 ))) |
85 | 84 | reximdva 3000 |
. 2
⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → (∃𝑠 ∈ ℕ0 ∀𝑤 ∈ (∪ 𝑢 ∈ (𝑁 × 𝑁){(coe1‘(𝑀‘𝑢))} ∩ {𝑣 ∈ ((Base‘𝑅) ↑𝑚
ℕ0) ∣ 𝑣 finSupp 0 })∀𝑧 ∈ ℕ0
(𝑠 < 𝑧 → (𝑤‘𝑧) = 0 ) → ∃𝑠 ∈ ℕ0
∀𝑥 ∈
ℕ0 (𝑠 <
𝑥 → ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 ((coe1‘(𝑖𝑀𝑗))‘𝑥) = 0 ))) |
86 | 29, 85 | mpd 15 |
1
⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → ∃𝑠 ∈ ℕ0 ∀𝑥 ∈ ℕ0
(𝑠 < 𝑥 → ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 ((coe1‘(𝑖𝑀𝑗))‘𝑥) = 0 )) |