Step | Hyp | Ref
| Expression |
1 | | zex 12185 |
. . . . . 6
⊢ ℤ
∈ V |
2 | | difexg 5220 |
. . . . . 6
⊢ (ℤ
∈ V → (ℤ ∖ ℕ) ∈ V) |
3 | 1, 2 | ax-mp 5 |
. . . . 5
⊢ (ℤ
∖ ℕ) ∈ V |
4 | | ominf 8890 |
. . . . . 6
⊢ ¬
ω ∈ Fin |
5 | | nnuz 12477 |
. . . . . . . . . 10
⊢ ℕ =
(ℤ≥‘1) |
6 | | 0p1e1 11952 |
. . . . . . . . . . 11
⊢ (0 + 1) =
1 |
7 | 6 | fveq2i 6720 |
. . . . . . . . . 10
⊢
(ℤ≥‘(0 + 1)) =
(ℤ≥‘1) |
8 | 5, 7 | eqtr4i 2768 |
. . . . . . . . 9
⊢ ℕ =
(ℤ≥‘(0 + 1)) |
9 | 8 | difeq2i 4034 |
. . . . . . . 8
⊢ (ℤ
∖ ℕ) = (ℤ ∖ (ℤ≥‘(0 +
1))) |
10 | | 0z 12187 |
. . . . . . . . 9
⊢ 0 ∈
ℤ |
11 | | lzenom 40295 |
. . . . . . . . 9
⊢ (0 ∈
ℤ → (ℤ ∖ (ℤ≥‘(0 + 1))) ≈
ω) |
12 | 10, 11 | ax-mp 5 |
. . . . . . . 8
⊢ (ℤ
∖ (ℤ≥‘(0 + 1))) ≈
ω |
13 | 9, 12 | eqbrtri 5074 |
. . . . . . 7
⊢ (ℤ
∖ ℕ) ≈ ω |
14 | | enfi 8865 |
. . . . . . 7
⊢ ((ℤ
∖ ℕ) ≈ ω → ((ℤ ∖ ℕ) ∈ Fin
↔ ω ∈ Fin)) |
15 | 13, 14 | ax-mp 5 |
. . . . . 6
⊢ ((ℤ
∖ ℕ) ∈ Fin ↔ ω ∈ Fin) |
16 | 4, 15 | mtbir 326 |
. . . . 5
⊢ ¬
(ℤ ∖ ℕ) ∈ Fin |
17 | | disjdifr 4387 |
. . . . 5
⊢ ((ℤ
∖ ℕ) ∩ ℕ) = ∅ |
18 | 3, 16, 17 | eldioph4b 40336 |
. . . 4
⊢ (𝑆 ∈ (Dioph‘𝑁) ↔ (𝑁 ∈ ℕ0 ∧
∃𝑏 ∈
(mzPoly‘((ℤ ∖ ℕ) ∪ (1...𝑁)))𝑆 = {𝑐 ∈ (ℕ0
↑m (1...𝑁))
∣ ∃𝑑 ∈
(ℕ0 ↑m (ℤ ∖ ℕ))(𝑏‘(𝑐 ∪ 𝑑)) = 0})) |
19 | | simpr 488 |
. . . . . . . . . . . 12
⊢
(((((𝑀 ∈
ℕ0 ∧ 𝐹:(1...𝑁)⟶(1...𝑀)) ∧ 𝑁 ∈ ℕ0) ∧ 𝑏 ∈ (mzPoly‘((ℤ
∖ ℕ) ∪ (1...𝑁)))) ∧ 𝑎 ∈ (ℕ0
↑m (1...𝑀))) → 𝑎 ∈ (ℕ0
↑m (1...𝑀))) |
20 | | simp-4r 784 |
. . . . . . . . . . . 12
⊢
(((((𝑀 ∈
ℕ0 ∧ 𝐹:(1...𝑁)⟶(1...𝑀)) ∧ 𝑁 ∈ ℕ0) ∧ 𝑏 ∈ (mzPoly‘((ℤ
∖ ℕ) ∪ (1...𝑁)))) ∧ 𝑎 ∈ (ℕ0
↑m (1...𝑀))) → 𝐹:(1...𝑁)⟶(1...𝑀)) |
21 | | ovex 7246 |
. . . . . . . . . . . . 13
⊢
(1...𝑁) ∈
V |
22 | 21 | mapco2 40240 |
. . . . . . . . . . . 12
⊢ ((𝑎 ∈ (ℕ0
↑m (1...𝑀))
∧ 𝐹:(1...𝑁)⟶(1...𝑀)) → (𝑎 ∘ 𝐹) ∈ (ℕ0
↑m (1...𝑁))) |
23 | 19, 20, 22 | syl2anc 587 |
. . . . . . . . . . 11
⊢
(((((𝑀 ∈
ℕ0 ∧ 𝐹:(1...𝑁)⟶(1...𝑀)) ∧ 𝑁 ∈ ℕ0) ∧ 𝑏 ∈ (mzPoly‘((ℤ
∖ ℕ) ∪ (1...𝑁)))) ∧ 𝑎 ∈ (ℕ0
↑m (1...𝑀))) → (𝑎 ∘ 𝐹) ∈ (ℕ0
↑m (1...𝑁))) |
24 | | uneq1 4070 |
. . . . . . . . . . . . . 14
⊢ (𝑐 = (𝑎 ∘ 𝐹) → (𝑐 ∪ 𝑑) = ((𝑎 ∘ 𝐹) ∪ 𝑑)) |
25 | 24 | fveqeq2d 6725 |
. . . . . . . . . . . . 13
⊢ (𝑐 = (𝑎 ∘ 𝐹) → ((𝑏‘(𝑐 ∪ 𝑑)) = 0 ↔ (𝑏‘((𝑎 ∘ 𝐹) ∪ 𝑑)) = 0)) |
26 | 25 | rexbidv 3216 |
. . . . . . . . . . . 12
⊢ (𝑐 = (𝑎 ∘ 𝐹) → (∃𝑑 ∈ (ℕ0
↑m (ℤ ∖ ℕ))(𝑏‘(𝑐 ∪ 𝑑)) = 0 ↔ ∃𝑑 ∈ (ℕ0
↑m (ℤ ∖ ℕ))(𝑏‘((𝑎 ∘ 𝐹) ∪ 𝑑)) = 0)) |
27 | 26 | elrab3 3603 |
. . . . . . . . . . 11
⊢ ((𝑎 ∘ 𝐹) ∈ (ℕ0
↑m (1...𝑁))
→ ((𝑎 ∘ 𝐹) ∈ {𝑐 ∈ (ℕ0
↑m (1...𝑁))
∣ ∃𝑑 ∈
(ℕ0 ↑m (ℤ ∖ ℕ))(𝑏‘(𝑐 ∪ 𝑑)) = 0} ↔ ∃𝑑 ∈ (ℕ0
↑m (ℤ ∖ ℕ))(𝑏‘((𝑎 ∘ 𝐹) ∪ 𝑑)) = 0)) |
28 | 23, 27 | syl 17 |
. . . . . . . . . 10
⊢
(((((𝑀 ∈
ℕ0 ∧ 𝐹:(1...𝑁)⟶(1...𝑀)) ∧ 𝑁 ∈ ℕ0) ∧ 𝑏 ∈ (mzPoly‘((ℤ
∖ ℕ) ∪ (1...𝑁)))) ∧ 𝑎 ∈ (ℕ0
↑m (1...𝑀))) → ((𝑎 ∘ 𝐹) ∈ {𝑐 ∈ (ℕ0
↑m (1...𝑁))
∣ ∃𝑑 ∈
(ℕ0 ↑m (ℤ ∖ ℕ))(𝑏‘(𝑐 ∪ 𝑑)) = 0} ↔ ∃𝑑 ∈ (ℕ0
↑m (ℤ ∖ ℕ))(𝑏‘((𝑎 ∘ 𝐹) ∪ 𝑑)) = 0)) |
29 | | simp-5r 786 |
. . . . . . . . . . . . . . 15
⊢
((((((𝑀 ∈
ℕ0 ∧ 𝐹:(1...𝑁)⟶(1...𝑀)) ∧ 𝑁 ∈ ℕ0) ∧ 𝑏 ∈ (mzPoly‘((ℤ
∖ ℕ) ∪ (1...𝑁)))) ∧ 𝑎 ∈ (ℕ0
↑m (1...𝑀))) ∧ 𝑑 ∈ (ℕ0
↑m (ℤ ∖ ℕ))) → 𝐹:(1...𝑁)⟶(1...𝑀)) |
30 | | simplr 769 |
. . . . . . . . . . . . . . 15
⊢
((((((𝑀 ∈
ℕ0 ∧ 𝐹:(1...𝑁)⟶(1...𝑀)) ∧ 𝑁 ∈ ℕ0) ∧ 𝑏 ∈ (mzPoly‘((ℤ
∖ ℕ) ∪ (1...𝑁)))) ∧ 𝑎 ∈ (ℕ0
↑m (1...𝑀))) ∧ 𝑑 ∈ (ℕ0
↑m (ℤ ∖ ℕ))) → 𝑎 ∈ (ℕ0
↑m (1...𝑀))) |
31 | | simpr 488 |
. . . . . . . . . . . . . . 15
⊢
((((((𝑀 ∈
ℕ0 ∧ 𝐹:(1...𝑁)⟶(1...𝑀)) ∧ 𝑁 ∈ ℕ0) ∧ 𝑏 ∈ (mzPoly‘((ℤ
∖ ℕ) ∪ (1...𝑁)))) ∧ 𝑎 ∈ (ℕ0
↑m (1...𝑀))) ∧ 𝑑 ∈ (ℕ0
↑m (ℤ ∖ ℕ))) → 𝑑 ∈ (ℕ0
↑m (ℤ ∖ ℕ))) |
32 | | coundi 6111 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑎 ∪ 𝑑) ∘ (𝐹 ∪ ( I ↾ (ℤ ∖
ℕ)))) = (((𝑎 ∪
𝑑) ∘ 𝐹) ∪ ((𝑎 ∪ 𝑑) ∘ ( I ↾ (ℤ ∖
ℕ)))) |
33 | | coundir 6112 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑎 ∪ 𝑑) ∘ 𝐹) = ((𝑎 ∘ 𝐹) ∪ (𝑑 ∘ 𝐹)) |
34 | | elmapi 8530 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑑 ∈ (ℕ0
↑m (ℤ ∖ ℕ)) → 𝑑:(ℤ ∖
ℕ)⟶ℕ0) |
35 | 34 | 3ad2ant3 1137 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝐹:(1...𝑁)⟶(1...𝑀) ∧ 𝑎 ∈ (ℕ0
↑m (1...𝑀))
∧ 𝑑 ∈
(ℕ0 ↑m (ℤ ∖ ℕ))) →
𝑑:(ℤ ∖
ℕ)⟶ℕ0) |
36 | | simp1 1138 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝐹:(1...𝑁)⟶(1...𝑀) ∧ 𝑎 ∈ (ℕ0
↑m (1...𝑀))
∧ 𝑑 ∈
(ℕ0 ↑m (ℤ ∖ ℕ))) →
𝐹:(1...𝑁)⟶(1...𝑀)) |
37 | | incom 4115 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((ℤ
∖ ℕ) ∩ (1...𝑀)) = ((1...𝑀) ∩ (ℤ ∖
ℕ)) |
38 | | fz1ssnn 13143 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢
(1...𝑀) ⊆
ℕ |
39 | | disjdif 4386 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (ℕ
∩ (ℤ ∖ ℕ)) = ∅ |
40 | | ssdisj 4374 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢
(((1...𝑀) ⊆
ℕ ∧ (ℕ ∩ (ℤ ∖ ℕ)) = ∅) →
((1...𝑀) ∩ (ℤ
∖ ℕ)) = ∅) |
41 | 38, 39, 40 | mp2an 692 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢
((1...𝑀) ∩
(ℤ ∖ ℕ)) = ∅ |
42 | 37, 41 | eqtri 2765 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((ℤ
∖ ℕ) ∩ (1...𝑀)) = ∅ |
43 | 42 | a1i 11 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝐹:(1...𝑁)⟶(1...𝑀) ∧ 𝑎 ∈ (ℕ0
↑m (1...𝑀))
∧ 𝑑 ∈
(ℕ0 ↑m (ℤ ∖ ℕ))) →
((ℤ ∖ ℕ) ∩ (1...𝑀)) = ∅) |
44 | | coeq0i 40278 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝑑:(ℤ ∖
ℕ)⟶ℕ0 ∧ 𝐹:(1...𝑁)⟶(1...𝑀) ∧ ((ℤ ∖ ℕ) ∩
(1...𝑀)) = ∅) →
(𝑑 ∘ 𝐹) = ∅) |
45 | 35, 36, 43, 44 | syl3anc 1373 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝐹:(1...𝑁)⟶(1...𝑀) ∧ 𝑎 ∈ (ℕ0
↑m (1...𝑀))
∧ 𝑑 ∈
(ℕ0 ↑m (ℤ ∖ ℕ))) →
(𝑑 ∘ 𝐹) = ∅) |
46 | 45 | uneq2d 4077 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝐹:(1...𝑁)⟶(1...𝑀) ∧ 𝑎 ∈ (ℕ0
↑m (1...𝑀))
∧ 𝑑 ∈
(ℕ0 ↑m (ℤ ∖ ℕ))) →
((𝑎 ∘ 𝐹) ∪ (𝑑 ∘ 𝐹)) = ((𝑎 ∘ 𝐹) ∪ ∅)) |
47 | 33, 46 | syl5eq 2790 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝐹:(1...𝑁)⟶(1...𝑀) ∧ 𝑎 ∈ (ℕ0
↑m (1...𝑀))
∧ 𝑑 ∈
(ℕ0 ↑m (ℤ ∖ ℕ))) →
((𝑎 ∪ 𝑑) ∘ 𝐹) = ((𝑎 ∘ 𝐹) ∪ ∅)) |
48 | | un0 4305 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑎 ∘ 𝐹) ∪ ∅) = (𝑎 ∘ 𝐹) |
49 | 47, 48 | eqtrdi 2794 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝐹:(1...𝑁)⟶(1...𝑀) ∧ 𝑎 ∈ (ℕ0
↑m (1...𝑀))
∧ 𝑑 ∈
(ℕ0 ↑m (ℤ ∖ ℕ))) →
((𝑎 ∪ 𝑑) ∘ 𝐹) = (𝑎 ∘ 𝐹)) |
50 | | coundir 6112 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑎 ∪ 𝑑) ∘ ( I ↾ (ℤ ∖
ℕ))) = ((𝑎 ∘ (
I ↾ (ℤ ∖ ℕ))) ∪ (𝑑 ∘ ( I ↾ (ℤ ∖
ℕ)))) |
51 | | elmapi 8530 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑎 ∈ (ℕ0
↑m (1...𝑀))
→ 𝑎:(1...𝑀)⟶ℕ0) |
52 | 51 | 3ad2ant2 1136 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝐹:(1...𝑁)⟶(1...𝑀) ∧ 𝑎 ∈ (ℕ0
↑m (1...𝑀))
∧ 𝑑 ∈
(ℕ0 ↑m (ℤ ∖ ℕ))) →
𝑎:(1...𝑀)⟶ℕ0) |
53 | | f1oi 6698 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ( I
↾ (ℤ ∖ ℕ)):(ℤ ∖ ℕ)–1-1-onto→(ℤ ∖ ℕ) |
54 | | f1of 6661 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (( I
↾ (ℤ ∖ ℕ)):(ℤ ∖ ℕ)–1-1-onto→(ℤ ∖ ℕ) → ( I ↾
(ℤ ∖ ℕ)):(ℤ ∖ ℕ)⟶(ℤ ∖
ℕ)) |
55 | 53, 54 | ax-mp 5 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ( I
↾ (ℤ ∖ ℕ)):(ℤ ∖ ℕ)⟶(ℤ
∖ ℕ) |
56 | | coeq0i 40278 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝑎:(1...𝑀)⟶ℕ0 ∧ ( I
↾ (ℤ ∖ ℕ)):(ℤ ∖ ℕ)⟶(ℤ
∖ ℕ) ∧ ((1...𝑀) ∩ (ℤ ∖ ℕ)) =
∅) → (𝑎 ∘
( I ↾ (ℤ ∖ ℕ))) = ∅) |
57 | 55, 41, 56 | mp3an23 1455 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑎:(1...𝑀)⟶ℕ0 → (𝑎 ∘ ( I ↾ (ℤ
∖ ℕ))) = ∅) |
58 | 52, 57 | syl 17 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝐹:(1...𝑁)⟶(1...𝑀) ∧ 𝑎 ∈ (ℕ0
↑m (1...𝑀))
∧ 𝑑 ∈
(ℕ0 ↑m (ℤ ∖ ℕ))) →
(𝑎 ∘ ( I ↾
(ℤ ∖ ℕ))) = ∅) |
59 | | coires1 6128 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑑 ∘ ( I ↾ (ℤ
∖ ℕ))) = (𝑑
↾ (ℤ ∖ ℕ)) |
60 | | ffn 6545 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑑:(ℤ ∖
ℕ)⟶ℕ0 → 𝑑 Fn (ℤ ∖
ℕ)) |
61 | | fnresdm 6496 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑑 Fn (ℤ ∖ ℕ)
→ (𝑑 ↾ (ℤ
∖ ℕ)) = 𝑑) |
62 | 34, 60, 61 | 3syl 18 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑑 ∈ (ℕ0
↑m (ℤ ∖ ℕ)) → (𝑑 ↾ (ℤ ∖ ℕ)) = 𝑑) |
63 | 59, 62 | syl5eq 2790 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑑 ∈ (ℕ0
↑m (ℤ ∖ ℕ)) → (𝑑 ∘ ( I ↾ (ℤ ∖
ℕ))) = 𝑑) |
64 | 63 | 3ad2ant3 1137 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝐹:(1...𝑁)⟶(1...𝑀) ∧ 𝑎 ∈ (ℕ0
↑m (1...𝑀))
∧ 𝑑 ∈
(ℕ0 ↑m (ℤ ∖ ℕ))) →
(𝑑 ∘ ( I ↾
(ℤ ∖ ℕ))) = 𝑑) |
65 | 58, 64 | uneq12d 4078 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝐹:(1...𝑁)⟶(1...𝑀) ∧ 𝑎 ∈ (ℕ0
↑m (1...𝑀))
∧ 𝑑 ∈
(ℕ0 ↑m (ℤ ∖ ℕ))) →
((𝑎 ∘ ( I ↾
(ℤ ∖ ℕ))) ∪ (𝑑 ∘ ( I ↾ (ℤ ∖
ℕ)))) = (∅ ∪ 𝑑)) |
66 | 50, 65 | syl5eq 2790 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝐹:(1...𝑁)⟶(1...𝑀) ∧ 𝑎 ∈ (ℕ0
↑m (1...𝑀))
∧ 𝑑 ∈
(ℕ0 ↑m (ℤ ∖ ℕ))) →
((𝑎 ∪ 𝑑) ∘ ( I ↾ (ℤ
∖ ℕ))) = (∅ ∪ 𝑑)) |
67 | | uncom 4067 |
. . . . . . . . . . . . . . . . . . 19
⊢ (∅
∪ 𝑑) = (𝑑 ∪ ∅) |
68 | | un0 4305 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑑 ∪ ∅) = 𝑑 |
69 | 67, 68 | eqtri 2765 |
. . . . . . . . . . . . . . . . . 18
⊢ (∅
∪ 𝑑) = 𝑑 |
70 | 66, 69 | eqtrdi 2794 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝐹:(1...𝑁)⟶(1...𝑀) ∧ 𝑎 ∈ (ℕ0
↑m (1...𝑀))
∧ 𝑑 ∈
(ℕ0 ↑m (ℤ ∖ ℕ))) →
((𝑎 ∪ 𝑑) ∘ ( I ↾ (ℤ
∖ ℕ))) = 𝑑) |
71 | 49, 70 | uneq12d 4078 |
. . . . . . . . . . . . . . . 16
⊢ ((𝐹:(1...𝑁)⟶(1...𝑀) ∧ 𝑎 ∈ (ℕ0
↑m (1...𝑀))
∧ 𝑑 ∈
(ℕ0 ↑m (ℤ ∖ ℕ))) →
(((𝑎 ∪ 𝑑) ∘ 𝐹) ∪ ((𝑎 ∪ 𝑑) ∘ ( I ↾ (ℤ ∖
ℕ)))) = ((𝑎 ∘
𝐹) ∪ 𝑑)) |
72 | 32, 71 | eqtr2id 2791 |
. . . . . . . . . . . . . . 15
⊢ ((𝐹:(1...𝑁)⟶(1...𝑀) ∧ 𝑎 ∈ (ℕ0
↑m (1...𝑀))
∧ 𝑑 ∈
(ℕ0 ↑m (ℤ ∖ ℕ))) →
((𝑎 ∘ 𝐹) ∪ 𝑑) = ((𝑎 ∪ 𝑑) ∘ (𝐹 ∪ ( I ↾ (ℤ ∖
ℕ))))) |
73 | 29, 30, 31, 72 | syl3anc 1373 |
. . . . . . . . . . . . . 14
⊢
((((((𝑀 ∈
ℕ0 ∧ 𝐹:(1...𝑁)⟶(1...𝑀)) ∧ 𝑁 ∈ ℕ0) ∧ 𝑏 ∈ (mzPoly‘((ℤ
∖ ℕ) ∪ (1...𝑁)))) ∧ 𝑎 ∈ (ℕ0
↑m (1...𝑀))) ∧ 𝑑 ∈ (ℕ0
↑m (ℤ ∖ ℕ))) → ((𝑎 ∘ 𝐹) ∪ 𝑑) = ((𝑎 ∪ 𝑑) ∘ (𝐹 ∪ ( I ↾ (ℤ ∖
ℕ))))) |
74 | 73 | fveq2d 6721 |
. . . . . . . . . . . . 13
⊢
((((((𝑀 ∈
ℕ0 ∧ 𝐹:(1...𝑁)⟶(1...𝑀)) ∧ 𝑁 ∈ ℕ0) ∧ 𝑏 ∈ (mzPoly‘((ℤ
∖ ℕ) ∪ (1...𝑁)))) ∧ 𝑎 ∈ (ℕ0
↑m (1...𝑀))) ∧ 𝑑 ∈ (ℕ0
↑m (ℤ ∖ ℕ))) → (𝑏‘((𝑎 ∘ 𝐹) ∪ 𝑑)) = (𝑏‘((𝑎 ∪ 𝑑) ∘ (𝐹 ∪ ( I ↾ (ℤ ∖
ℕ)))))) |
75 | | nn0ssz 12198 |
. . . . . . . . . . . . . . . . 17
⊢
ℕ0 ⊆ ℤ |
76 | | mapss 8570 |
. . . . . . . . . . . . . . . . 17
⊢ ((ℤ
∈ V ∧ ℕ0 ⊆ ℤ) → (ℕ0
↑m ((ℤ ∖ ℕ) ∪ (1...𝑀))) ⊆ (ℤ ↑m
((ℤ ∖ ℕ) ∪ (1...𝑀)))) |
77 | 1, 75, 76 | mp2an 692 |
. . . . . . . . . . . . . . . 16
⊢
(ℕ0 ↑m ((ℤ ∖ ℕ) ∪
(1...𝑀))) ⊆ (ℤ
↑m ((ℤ ∖ ℕ) ∪ (1...𝑀))) |
78 | 41 | reseq2i 5848 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑎 ↾ ((1...𝑀) ∩ (ℤ ∖ ℕ))) = (𝑎 ↾
∅) |
79 | | res0 5855 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑎 ↾ ∅) =
∅ |
80 | 78, 79 | eqtri 2765 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑎 ↾ ((1...𝑀) ∩ (ℤ ∖ ℕ))) =
∅ |
81 | 41 | reseq2i 5848 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑑 ↾ ((1...𝑀) ∩ (ℤ ∖ ℕ))) = (𝑑 ↾
∅) |
82 | | res0 5855 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑑 ↾ ∅) =
∅ |
83 | 81, 82 | eqtri 2765 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑑 ↾ ((1...𝑀) ∩ (ℤ ∖ ℕ))) =
∅ |
84 | 80, 83 | eqtr4i 2768 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑎 ↾ ((1...𝑀) ∩ (ℤ ∖ ℕ))) = (𝑑 ↾ ((1...𝑀) ∩ (ℤ ∖
ℕ))) |
85 | | elmapresaun 8561 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑎 ∈ (ℕ0
↑m (1...𝑀))
∧ 𝑑 ∈
(ℕ0 ↑m (ℤ ∖ ℕ)) ∧ (𝑎 ↾ ((1...𝑀) ∩ (ℤ ∖ ℕ))) = (𝑑 ↾ ((1...𝑀) ∩ (ℤ ∖ ℕ)))) →
(𝑎 ∪ 𝑑) ∈ (ℕ0
↑m ((1...𝑀)
∪ (ℤ ∖ ℕ)))) |
86 | | uncom 4067 |
. . . . . . . . . . . . . . . . . . 19
⊢
((1...𝑀) ∪
(ℤ ∖ ℕ)) = ((ℤ ∖ ℕ) ∪ (1...𝑀)) |
87 | 86 | oveq2i 7224 |
. . . . . . . . . . . . . . . . . 18
⊢
(ℕ0 ↑m ((1...𝑀) ∪ (ℤ ∖ ℕ))) =
(ℕ0 ↑m ((ℤ ∖ ℕ) ∪
(1...𝑀))) |
88 | 85, 87 | eleqtrdi 2848 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑎 ∈ (ℕ0
↑m (1...𝑀))
∧ 𝑑 ∈
(ℕ0 ↑m (ℤ ∖ ℕ)) ∧ (𝑎 ↾ ((1...𝑀) ∩ (ℤ ∖ ℕ))) = (𝑑 ↾ ((1...𝑀) ∩ (ℤ ∖ ℕ)))) →
(𝑎 ∪ 𝑑) ∈ (ℕ0
↑m ((ℤ ∖ ℕ) ∪ (1...𝑀)))) |
89 | 84, 88 | mp3an3 1452 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑎 ∈ (ℕ0
↑m (1...𝑀))
∧ 𝑑 ∈
(ℕ0 ↑m (ℤ ∖ ℕ))) →
(𝑎 ∪ 𝑑) ∈ (ℕ0
↑m ((ℤ ∖ ℕ) ∪ (1...𝑀)))) |
90 | 77, 89 | sseldi 3899 |
. . . . . . . . . . . . . . 15
⊢ ((𝑎 ∈ (ℕ0
↑m (1...𝑀))
∧ 𝑑 ∈
(ℕ0 ↑m (ℤ ∖ ℕ))) →
(𝑎 ∪ 𝑑) ∈ (ℤ ↑m
((ℤ ∖ ℕ) ∪ (1...𝑀)))) |
91 | 90 | adantll 714 |
. . . . . . . . . . . . . 14
⊢
((((((𝑀 ∈
ℕ0 ∧ 𝐹:(1...𝑁)⟶(1...𝑀)) ∧ 𝑁 ∈ ℕ0) ∧ 𝑏 ∈ (mzPoly‘((ℤ
∖ ℕ) ∪ (1...𝑁)))) ∧ 𝑎 ∈ (ℕ0
↑m (1...𝑀))) ∧ 𝑑 ∈ (ℕ0
↑m (ℤ ∖ ℕ))) → (𝑎 ∪ 𝑑) ∈ (ℤ ↑m
((ℤ ∖ ℕ) ∪ (1...𝑀)))) |
92 | | coeq1 5726 |
. . . . . . . . . . . . . . . 16
⊢ (𝑒 = (𝑎 ∪ 𝑑) → (𝑒 ∘ (𝐹 ∪ ( I ↾ (ℤ ∖
ℕ)))) = ((𝑎 ∪
𝑑) ∘ (𝐹 ∪ ( I ↾ (ℤ
∖ ℕ))))) |
93 | 92 | fveq2d 6721 |
. . . . . . . . . . . . . . 15
⊢ (𝑒 = (𝑎 ∪ 𝑑) → (𝑏‘(𝑒 ∘ (𝐹 ∪ ( I ↾ (ℤ ∖
ℕ))))) = (𝑏‘((𝑎 ∪ 𝑑) ∘ (𝐹 ∪ ( I ↾ (ℤ ∖
ℕ)))))) |
94 | | eqid 2737 |
. . . . . . . . . . . . . . 15
⊢ (𝑒 ∈ (ℤ
↑m ((ℤ ∖ ℕ) ∪ (1...𝑀))) ↦ (𝑏‘(𝑒 ∘ (𝐹 ∪ ( I ↾ (ℤ ∖
ℕ)))))) = (𝑒 ∈
(ℤ ↑m ((ℤ ∖ ℕ) ∪ (1...𝑀))) ↦ (𝑏‘(𝑒 ∘ (𝐹 ∪ ( I ↾ (ℤ ∖
ℕ)))))) |
95 | | fvex 6730 |
. . . . . . . . . . . . . . 15
⊢ (𝑏‘((𝑎 ∪ 𝑑) ∘ (𝐹 ∪ ( I ↾ (ℤ ∖
ℕ))))) ∈ V |
96 | 93, 94, 95 | fvmpt 6818 |
. . . . . . . . . . . . . 14
⊢ ((𝑎 ∪ 𝑑) ∈ (ℤ ↑m
((ℤ ∖ ℕ) ∪ (1...𝑀))) → ((𝑒 ∈ (ℤ ↑m ((ℤ
∖ ℕ) ∪ (1...𝑀))) ↦ (𝑏‘(𝑒 ∘ (𝐹 ∪ ( I ↾ (ℤ ∖
ℕ))))))‘(𝑎
∪ 𝑑)) = (𝑏‘((𝑎 ∪ 𝑑) ∘ (𝐹 ∪ ( I ↾ (ℤ ∖
ℕ)))))) |
97 | 91, 96 | syl 17 |
. . . . . . . . . . . . 13
⊢
((((((𝑀 ∈
ℕ0 ∧ 𝐹:(1...𝑁)⟶(1...𝑀)) ∧ 𝑁 ∈ ℕ0) ∧ 𝑏 ∈ (mzPoly‘((ℤ
∖ ℕ) ∪ (1...𝑁)))) ∧ 𝑎 ∈ (ℕ0
↑m (1...𝑀))) ∧ 𝑑 ∈ (ℕ0
↑m (ℤ ∖ ℕ))) → ((𝑒 ∈ (ℤ ↑m ((ℤ
∖ ℕ) ∪ (1...𝑀))) ↦ (𝑏‘(𝑒 ∘ (𝐹 ∪ ( I ↾ (ℤ ∖
ℕ))))))‘(𝑎
∪ 𝑑)) = (𝑏‘((𝑎 ∪ 𝑑) ∘ (𝐹 ∪ ( I ↾ (ℤ ∖
ℕ)))))) |
98 | 74, 97 | eqtr4d 2780 |
. . . . . . . . . . . 12
⊢
((((((𝑀 ∈
ℕ0 ∧ 𝐹:(1...𝑁)⟶(1...𝑀)) ∧ 𝑁 ∈ ℕ0) ∧ 𝑏 ∈ (mzPoly‘((ℤ
∖ ℕ) ∪ (1...𝑁)))) ∧ 𝑎 ∈ (ℕ0
↑m (1...𝑀))) ∧ 𝑑 ∈ (ℕ0
↑m (ℤ ∖ ℕ))) → (𝑏‘((𝑎 ∘ 𝐹) ∪ 𝑑)) = ((𝑒 ∈ (ℤ ↑m ((ℤ
∖ ℕ) ∪ (1...𝑀))) ↦ (𝑏‘(𝑒 ∘ (𝐹 ∪ ( I ↾ (ℤ ∖
ℕ))))))‘(𝑎
∪ 𝑑))) |
99 | 98 | eqeq1d 2739 |
. . . . . . . . . . 11
⊢
((((((𝑀 ∈
ℕ0 ∧ 𝐹:(1...𝑁)⟶(1...𝑀)) ∧ 𝑁 ∈ ℕ0) ∧ 𝑏 ∈ (mzPoly‘((ℤ
∖ ℕ) ∪ (1...𝑁)))) ∧ 𝑎 ∈ (ℕ0
↑m (1...𝑀))) ∧ 𝑑 ∈ (ℕ0
↑m (ℤ ∖ ℕ))) → ((𝑏‘((𝑎 ∘ 𝐹) ∪ 𝑑)) = 0 ↔ ((𝑒 ∈ (ℤ ↑m ((ℤ
∖ ℕ) ∪ (1...𝑀))) ↦ (𝑏‘(𝑒 ∘ (𝐹 ∪ ( I ↾ (ℤ ∖
ℕ))))))‘(𝑎
∪ 𝑑)) =
0)) |
100 | 99 | rexbidva 3215 |
. . . . . . . . . 10
⊢
(((((𝑀 ∈
ℕ0 ∧ 𝐹:(1...𝑁)⟶(1...𝑀)) ∧ 𝑁 ∈ ℕ0) ∧ 𝑏 ∈ (mzPoly‘((ℤ
∖ ℕ) ∪ (1...𝑁)))) ∧ 𝑎 ∈ (ℕ0
↑m (1...𝑀))) → (∃𝑑 ∈ (ℕ0
↑m (ℤ ∖ ℕ))(𝑏‘((𝑎 ∘ 𝐹) ∪ 𝑑)) = 0 ↔ ∃𝑑 ∈ (ℕ0
↑m (ℤ ∖ ℕ))((𝑒 ∈ (ℤ ↑m ((ℤ
∖ ℕ) ∪ (1...𝑀))) ↦ (𝑏‘(𝑒 ∘ (𝐹 ∪ ( I ↾ (ℤ ∖
ℕ))))))‘(𝑎
∪ 𝑑)) =
0)) |
101 | 28, 100 | bitrd 282 |
. . . . . . . . 9
⊢
(((((𝑀 ∈
ℕ0 ∧ 𝐹:(1...𝑁)⟶(1...𝑀)) ∧ 𝑁 ∈ ℕ0) ∧ 𝑏 ∈ (mzPoly‘((ℤ
∖ ℕ) ∪ (1...𝑁)))) ∧ 𝑎 ∈ (ℕ0
↑m (1...𝑀))) → ((𝑎 ∘ 𝐹) ∈ {𝑐 ∈ (ℕ0
↑m (1...𝑁))
∣ ∃𝑑 ∈
(ℕ0 ↑m (ℤ ∖ ℕ))(𝑏‘(𝑐 ∪ 𝑑)) = 0} ↔ ∃𝑑 ∈ (ℕ0
↑m (ℤ ∖ ℕ))((𝑒 ∈ (ℤ ↑m ((ℤ
∖ ℕ) ∪ (1...𝑀))) ↦ (𝑏‘(𝑒 ∘ (𝐹 ∪ ( I ↾ (ℤ ∖
ℕ))))))‘(𝑎
∪ 𝑑)) =
0)) |
102 | 101 | rabbidva 3388 |
. . . . . . . 8
⊢ ((((𝑀 ∈ ℕ0
∧ 𝐹:(1...𝑁)⟶(1...𝑀)) ∧ 𝑁 ∈ ℕ0) ∧ 𝑏 ∈ (mzPoly‘((ℤ
∖ ℕ) ∪ (1...𝑁)))) → {𝑎 ∈ (ℕ0
↑m (1...𝑀))
∣ (𝑎 ∘ 𝐹) ∈ {𝑐 ∈ (ℕ0
↑m (1...𝑁))
∣ ∃𝑑 ∈
(ℕ0 ↑m (ℤ ∖ ℕ))(𝑏‘(𝑐 ∪ 𝑑)) = 0}} = {𝑎 ∈ (ℕ0
↑m (1...𝑀))
∣ ∃𝑑 ∈
(ℕ0 ↑m (ℤ ∖ ℕ))((𝑒 ∈ (ℤ
↑m ((ℤ ∖ ℕ) ∪ (1...𝑀))) ↦ (𝑏‘(𝑒 ∘ (𝐹 ∪ ( I ↾ (ℤ ∖
ℕ))))))‘(𝑎
∪ 𝑑)) =
0}) |
103 | | simplll 775 |
. . . . . . . . 9
⊢ ((((𝑀 ∈ ℕ0
∧ 𝐹:(1...𝑁)⟶(1...𝑀)) ∧ 𝑁 ∈ ℕ0) ∧ 𝑏 ∈ (mzPoly‘((ℤ
∖ ℕ) ∪ (1...𝑁)))) → 𝑀 ∈
ℕ0) |
104 | | ovex 7246 |
. . . . . . . . . . . 12
⊢
(1...𝑀) ∈
V |
105 | 3, 104 | unex 7531 |
. . . . . . . . . . 11
⊢ ((ℤ
∖ ℕ) ∪ (1...𝑀)) ∈ V |
106 | 105 | a1i 11 |
. . . . . . . . . 10
⊢ ((((𝑀 ∈ ℕ0
∧ 𝐹:(1...𝑁)⟶(1...𝑀)) ∧ 𝑁 ∈ ℕ0) ∧ 𝑏 ∈ (mzPoly‘((ℤ
∖ ℕ) ∪ (1...𝑁)))) → ((ℤ ∖ ℕ) ∪
(1...𝑀)) ∈
V) |
107 | | simpr 488 |
. . . . . . . . . 10
⊢ ((((𝑀 ∈ ℕ0
∧ 𝐹:(1...𝑁)⟶(1...𝑀)) ∧ 𝑁 ∈ ℕ0) ∧ 𝑏 ∈ (mzPoly‘((ℤ
∖ ℕ) ∪ (1...𝑁)))) → 𝑏 ∈ (mzPoly‘((ℤ ∖
ℕ) ∪ (1...𝑁)))) |
108 | 55 | a1i 11 |
. . . . . . . . . . . . 13
⊢ (𝐹:(1...𝑁)⟶(1...𝑀) → ( I ↾ (ℤ ∖
ℕ)):(ℤ ∖ ℕ)⟶(ℤ ∖
ℕ)) |
109 | | id 22 |
. . . . . . . . . . . . 13
⊢ (𝐹:(1...𝑁)⟶(1...𝑀) → 𝐹:(1...𝑁)⟶(1...𝑀)) |
110 | | incom 4115 |
. . . . . . . . . . . . . . 15
⊢ ((ℤ
∖ ℕ) ∩ (1...𝑁)) = ((1...𝑁) ∩ (ℤ ∖
ℕ)) |
111 | | fz1ssnn 13143 |
. . . . . . . . . . . . . . . 16
⊢
(1...𝑁) ⊆
ℕ |
112 | | ssdisj 4374 |
. . . . . . . . . . . . . . . 16
⊢
(((1...𝑁) ⊆
ℕ ∧ (ℕ ∩ (ℤ ∖ ℕ)) = ∅) →
((1...𝑁) ∩ (ℤ
∖ ℕ)) = ∅) |
113 | 111, 39, 112 | mp2an 692 |
. . . . . . . . . . . . . . 15
⊢
((1...𝑁) ∩
(ℤ ∖ ℕ)) = ∅ |
114 | 110, 113 | eqtri 2765 |
. . . . . . . . . . . . . 14
⊢ ((ℤ
∖ ℕ) ∩ (1...𝑁)) = ∅ |
115 | 114 | a1i 11 |
. . . . . . . . . . . . 13
⊢ (𝐹:(1...𝑁)⟶(1...𝑀) → ((ℤ ∖ ℕ) ∩
(1...𝑁)) =
∅) |
116 | | fun 6581 |
. . . . . . . . . . . . 13
⊢ (((( I
↾ (ℤ ∖ ℕ)):(ℤ ∖ ℕ)⟶(ℤ
∖ ℕ) ∧ 𝐹:(1...𝑁)⟶(1...𝑀)) ∧ ((ℤ ∖ ℕ) ∩
(1...𝑁)) = ∅) →
(( I ↾ (ℤ ∖ ℕ)) ∪ 𝐹):((ℤ ∖ ℕ) ∪
(1...𝑁))⟶((ℤ
∖ ℕ) ∪ (1...𝑀))) |
117 | 108, 109,
115, 116 | syl21anc 838 |
. . . . . . . . . . . 12
⊢ (𝐹:(1...𝑁)⟶(1...𝑀) → (( I ↾ (ℤ ∖
ℕ)) ∪ 𝐹):((ℤ ∖ ℕ) ∪
(1...𝑁))⟶((ℤ
∖ ℕ) ∪ (1...𝑀))) |
118 | | uncom 4067 |
. . . . . . . . . . . . 13
⊢ (( I
↾ (ℤ ∖ ℕ)) ∪ 𝐹) = (𝐹 ∪ ( I ↾ (ℤ ∖
ℕ))) |
119 | 118 | feq1i 6536 |
. . . . . . . . . . . 12
⊢ ((( I
↾ (ℤ ∖ ℕ)) ∪ 𝐹):((ℤ ∖ ℕ) ∪
(1...𝑁))⟶((ℤ
∖ ℕ) ∪ (1...𝑀)) ↔ (𝐹 ∪ ( I ↾ (ℤ ∖
ℕ))):((ℤ ∖ ℕ) ∪ (1...𝑁))⟶((ℤ ∖ ℕ) ∪
(1...𝑀))) |
120 | 117, 119 | sylib 221 |
. . . . . . . . . . 11
⊢ (𝐹:(1...𝑁)⟶(1...𝑀) → (𝐹 ∪ ( I ↾ (ℤ ∖
ℕ))):((ℤ ∖ ℕ) ∪ (1...𝑁))⟶((ℤ ∖ ℕ) ∪
(1...𝑀))) |
121 | 120 | ad3antlr 731 |
. . . . . . . . . 10
⊢ ((((𝑀 ∈ ℕ0
∧ 𝐹:(1...𝑁)⟶(1...𝑀)) ∧ 𝑁 ∈ ℕ0) ∧ 𝑏 ∈ (mzPoly‘((ℤ
∖ ℕ) ∪ (1...𝑁)))) → (𝐹 ∪ ( I ↾ (ℤ ∖
ℕ))):((ℤ ∖ ℕ) ∪ (1...𝑁))⟶((ℤ ∖ ℕ) ∪
(1...𝑀))) |
122 | | mzprename 40274 |
. . . . . . . . . 10
⊢
((((ℤ ∖ ℕ) ∪ (1...𝑀)) ∈ V ∧ 𝑏 ∈ (mzPoly‘((ℤ ∖
ℕ) ∪ (1...𝑁)))
∧ (𝐹 ∪ ( I ↾
(ℤ ∖ ℕ))):((ℤ ∖ ℕ) ∪ (1...𝑁))⟶((ℤ ∖
ℕ) ∪ (1...𝑀)))
→ (𝑒 ∈ (ℤ
↑m ((ℤ ∖ ℕ) ∪ (1...𝑀))) ↦ (𝑏‘(𝑒 ∘ (𝐹 ∪ ( I ↾ (ℤ ∖
ℕ)))))) ∈ (mzPoly‘((ℤ ∖ ℕ) ∪ (1...𝑀)))) |
123 | 106, 107,
121, 122 | syl3anc 1373 |
. . . . . . . . 9
⊢ ((((𝑀 ∈ ℕ0
∧ 𝐹:(1...𝑁)⟶(1...𝑀)) ∧ 𝑁 ∈ ℕ0) ∧ 𝑏 ∈ (mzPoly‘((ℤ
∖ ℕ) ∪ (1...𝑁)))) → (𝑒 ∈ (ℤ ↑m ((ℤ
∖ ℕ) ∪ (1...𝑀))) ↦ (𝑏‘(𝑒 ∘ (𝐹 ∪ ( I ↾ (ℤ ∖
ℕ)))))) ∈ (mzPoly‘((ℤ ∖ ℕ) ∪ (1...𝑀)))) |
124 | 3, 16, 17 | eldioph4i 40337 |
. . . . . . . . 9
⊢ ((𝑀 ∈ ℕ0
∧ (𝑒 ∈ (ℤ
↑m ((ℤ ∖ ℕ) ∪ (1...𝑀))) ↦ (𝑏‘(𝑒 ∘ (𝐹 ∪ ( I ↾ (ℤ ∖
ℕ)))))) ∈ (mzPoly‘((ℤ ∖ ℕ) ∪ (1...𝑀)))) → {𝑎 ∈ (ℕ0
↑m (1...𝑀))
∣ ∃𝑑 ∈
(ℕ0 ↑m (ℤ ∖ ℕ))((𝑒 ∈ (ℤ
↑m ((ℤ ∖ ℕ) ∪ (1...𝑀))) ↦ (𝑏‘(𝑒 ∘ (𝐹 ∪ ( I ↾ (ℤ ∖
ℕ))))))‘(𝑎
∪ 𝑑)) = 0} ∈
(Dioph‘𝑀)) |
125 | 103, 123,
124 | syl2anc 587 |
. . . . . . . 8
⊢ ((((𝑀 ∈ ℕ0
∧ 𝐹:(1...𝑁)⟶(1...𝑀)) ∧ 𝑁 ∈ ℕ0) ∧ 𝑏 ∈ (mzPoly‘((ℤ
∖ ℕ) ∪ (1...𝑁)))) → {𝑎 ∈ (ℕ0
↑m (1...𝑀))
∣ ∃𝑑 ∈
(ℕ0 ↑m (ℤ ∖ ℕ))((𝑒 ∈ (ℤ
↑m ((ℤ ∖ ℕ) ∪ (1...𝑀))) ↦ (𝑏‘(𝑒 ∘ (𝐹 ∪ ( I ↾ (ℤ ∖
ℕ))))))‘(𝑎
∪ 𝑑)) = 0} ∈
(Dioph‘𝑀)) |
126 | 102, 125 | eqeltrd 2838 |
. . . . . . 7
⊢ ((((𝑀 ∈ ℕ0
∧ 𝐹:(1...𝑁)⟶(1...𝑀)) ∧ 𝑁 ∈ ℕ0) ∧ 𝑏 ∈ (mzPoly‘((ℤ
∖ ℕ) ∪ (1...𝑁)))) → {𝑎 ∈ (ℕ0
↑m (1...𝑀))
∣ (𝑎 ∘ 𝐹) ∈ {𝑐 ∈ (ℕ0
↑m (1...𝑁))
∣ ∃𝑑 ∈
(ℕ0 ↑m (ℤ ∖ ℕ))(𝑏‘(𝑐 ∪ 𝑑)) = 0}} ∈ (Dioph‘𝑀)) |
127 | | eleq2 2826 |
. . . . . . . . 9
⊢ (𝑆 = {𝑐 ∈ (ℕ0
↑m (1...𝑁))
∣ ∃𝑑 ∈
(ℕ0 ↑m (ℤ ∖ ℕ))(𝑏‘(𝑐 ∪ 𝑑)) = 0} → ((𝑎 ∘ 𝐹) ∈ 𝑆 ↔ (𝑎 ∘ 𝐹) ∈ {𝑐 ∈ (ℕ0
↑m (1...𝑁))
∣ ∃𝑑 ∈
(ℕ0 ↑m (ℤ ∖ ℕ))(𝑏‘(𝑐 ∪ 𝑑)) = 0})) |
128 | 127 | rabbidv 3390 |
. . . . . . . 8
⊢ (𝑆 = {𝑐 ∈ (ℕ0
↑m (1...𝑁))
∣ ∃𝑑 ∈
(ℕ0 ↑m (ℤ ∖ ℕ))(𝑏‘(𝑐 ∪ 𝑑)) = 0} → {𝑎 ∈ (ℕ0
↑m (1...𝑀))
∣ (𝑎 ∘ 𝐹) ∈ 𝑆} = {𝑎 ∈ (ℕ0
↑m (1...𝑀))
∣ (𝑎 ∘ 𝐹) ∈ {𝑐 ∈ (ℕ0
↑m (1...𝑁))
∣ ∃𝑑 ∈
(ℕ0 ↑m (ℤ ∖ ℕ))(𝑏‘(𝑐 ∪ 𝑑)) = 0}}) |
129 | 128 | eleq1d 2822 |
. . . . . . 7
⊢ (𝑆 = {𝑐 ∈ (ℕ0
↑m (1...𝑁))
∣ ∃𝑑 ∈
(ℕ0 ↑m (ℤ ∖ ℕ))(𝑏‘(𝑐 ∪ 𝑑)) = 0} → ({𝑎 ∈ (ℕ0
↑m (1...𝑀))
∣ (𝑎 ∘ 𝐹) ∈ 𝑆} ∈ (Dioph‘𝑀) ↔ {𝑎 ∈ (ℕ0
↑m (1...𝑀))
∣ (𝑎 ∘ 𝐹) ∈ {𝑐 ∈ (ℕ0
↑m (1...𝑁))
∣ ∃𝑑 ∈
(ℕ0 ↑m (ℤ ∖ ℕ))(𝑏‘(𝑐 ∪ 𝑑)) = 0}} ∈ (Dioph‘𝑀))) |
130 | 126, 129 | syl5ibrcom 250 |
. . . . . 6
⊢ ((((𝑀 ∈ ℕ0
∧ 𝐹:(1...𝑁)⟶(1...𝑀)) ∧ 𝑁 ∈ ℕ0) ∧ 𝑏 ∈ (mzPoly‘((ℤ
∖ ℕ) ∪ (1...𝑁)))) → (𝑆 = {𝑐 ∈ (ℕ0
↑m (1...𝑁))
∣ ∃𝑑 ∈
(ℕ0 ↑m (ℤ ∖ ℕ))(𝑏‘(𝑐 ∪ 𝑑)) = 0} → {𝑎 ∈ (ℕ0
↑m (1...𝑀))
∣ (𝑎 ∘ 𝐹) ∈ 𝑆} ∈ (Dioph‘𝑀))) |
131 | 130 | rexlimdva 3203 |
. . . . 5
⊢ (((𝑀 ∈ ℕ0
∧ 𝐹:(1...𝑁)⟶(1...𝑀)) ∧ 𝑁 ∈ ℕ0) →
(∃𝑏 ∈
(mzPoly‘((ℤ ∖ ℕ) ∪ (1...𝑁)))𝑆 = {𝑐 ∈ (ℕ0
↑m (1...𝑁))
∣ ∃𝑑 ∈
(ℕ0 ↑m (ℤ ∖ ℕ))(𝑏‘(𝑐 ∪ 𝑑)) = 0} → {𝑎 ∈ (ℕ0
↑m (1...𝑀))
∣ (𝑎 ∘ 𝐹) ∈ 𝑆} ∈ (Dioph‘𝑀))) |
132 | 131 | expimpd 457 |
. . . 4
⊢ ((𝑀 ∈ ℕ0
∧ 𝐹:(1...𝑁)⟶(1...𝑀)) → ((𝑁 ∈ ℕ0 ∧
∃𝑏 ∈
(mzPoly‘((ℤ ∖ ℕ) ∪ (1...𝑁)))𝑆 = {𝑐 ∈ (ℕ0
↑m (1...𝑁))
∣ ∃𝑑 ∈
(ℕ0 ↑m (ℤ ∖ ℕ))(𝑏‘(𝑐 ∪ 𝑑)) = 0}) → {𝑎 ∈ (ℕ0
↑m (1...𝑀))
∣ (𝑎 ∘ 𝐹) ∈ 𝑆} ∈ (Dioph‘𝑀))) |
133 | 18, 132 | syl5bi 245 |
. . 3
⊢ ((𝑀 ∈ ℕ0
∧ 𝐹:(1...𝑁)⟶(1...𝑀)) → (𝑆 ∈ (Dioph‘𝑁) → {𝑎 ∈ (ℕ0
↑m (1...𝑀))
∣ (𝑎 ∘ 𝐹) ∈ 𝑆} ∈ (Dioph‘𝑀))) |
134 | 133 | impcom 411 |
. 2
⊢ ((𝑆 ∈ (Dioph‘𝑁) ∧ (𝑀 ∈ ℕ0 ∧ 𝐹:(1...𝑁)⟶(1...𝑀))) → {𝑎 ∈ (ℕ0
↑m (1...𝑀))
∣ (𝑎 ∘ 𝐹) ∈ 𝑆} ∈ (Dioph‘𝑀)) |
135 | 134 | 3impb 1117 |
1
⊢ ((𝑆 ∈ (Dioph‘𝑁) ∧ 𝑀 ∈ ℕ0 ∧ 𝐹:(1...𝑁)⟶(1...𝑀)) → {𝑎 ∈ (ℕ0
↑m (1...𝑀))
∣ (𝑎 ∘ 𝐹) ∈ 𝑆} ∈ (Dioph‘𝑀)) |