Step | Hyp | Ref
| Expression |
1 | | zex 11800 |
. . . . . 6
⊢ ℤ
∈ V |
2 | | difexg 5083 |
. . . . . 6
⊢ (ℤ
∈ V → (ℤ ∖ ℕ) ∈ V) |
3 | 1, 2 | ax-mp 5 |
. . . . 5
⊢ (ℤ
∖ ℕ) ∈ V |
4 | | ominf 8523 |
. . . . . 6
⊢ ¬
ω ∈ Fin |
5 | | nnuz 12093 |
. . . . . . . . . 10
⊢ ℕ =
(ℤ≥‘1) |
6 | | 0p1e1 11567 |
. . . . . . . . . . 11
⊢ (0 + 1) =
1 |
7 | 6 | fveq2i 6499 |
. . . . . . . . . 10
⊢
(ℤ≥‘(0 + 1)) =
(ℤ≥‘1) |
8 | 5, 7 | eqtr4i 2799 |
. . . . . . . . 9
⊢ ℕ =
(ℤ≥‘(0 + 1)) |
9 | 8 | difeq2i 3980 |
. . . . . . . 8
⊢ (ℤ
∖ ℕ) = (ℤ ∖ (ℤ≥‘(0 +
1))) |
10 | | 0z 11802 |
. . . . . . . . 9
⊢ 0 ∈
ℤ |
11 | | lzenom 38791 |
. . . . . . . . 9
⊢ (0 ∈
ℤ → (ℤ ∖ (ℤ≥‘(0 + 1))) ≈
ω) |
12 | 10, 11 | ax-mp 5 |
. . . . . . . 8
⊢ (ℤ
∖ (ℤ≥‘(0 + 1))) ≈
ω |
13 | 9, 12 | eqbrtri 4946 |
. . . . . . 7
⊢ (ℤ
∖ ℕ) ≈ ω |
14 | | enfi 8527 |
. . . . . . 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 315 |
. . . . 5
⊢ ¬
(ℤ ∖ ℕ) ∈ Fin |
17 | | incom 4060 |
. . . . . 6
⊢ ((ℤ
∖ ℕ) ∩ ℕ) = (ℕ ∩ (ℤ ∖
ℕ)) |
18 | | disjdif 4298 |
. . . . . 6
⊢ (ℕ
∩ (ℤ ∖ ℕ)) = ∅ |
19 | 17, 18 | eqtri 2796 |
. . . . 5
⊢ ((ℤ
∖ ℕ) ∩ ℕ) = ∅ |
20 | 3, 16, 19 | eldioph4b 38833 |
. . . 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})) |
21 | | simpr 477 |
. . . . . . . . . . . 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))) |
22 | | simp-4r 771 |
. . . . . . . . . . . 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)) |
23 | | ovex 7006 |
. . . . . . . . . . . . 13
⊢
(1...𝑁) ∈
V |
24 | 23 | mapco2 38736 |
. . . . . . . . . . . 12
⊢ ((𝑎 ∈ (ℕ0
↑𝑚 (1...𝑀)) ∧ 𝐹:(1...𝑁)⟶(1...𝑀)) → (𝑎 ∘ 𝐹) ∈ (ℕ0
↑𝑚 (1...𝑁))) |
25 | 21, 22, 24 | syl2anc 576 |
. . . . . . . . . . 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))) |
26 | | uneq1 4015 |
. . . . . . . . . . . . . 14
⊢ (𝑐 = (𝑎 ∘ 𝐹) → (𝑐 ∪ 𝑑) = ((𝑎 ∘ 𝐹) ∪ 𝑑)) |
27 | 26 | fveqeq2d 6504 |
. . . . . . . . . . . . 13
⊢ (𝑐 = (𝑎 ∘ 𝐹) → ((𝑏‘(𝑐 ∪ 𝑑)) = 0 ↔ (𝑏‘((𝑎 ∘ 𝐹) ∪ 𝑑)) = 0)) |
28 | 27 | rexbidv 3236 |
. . . . . . . . . . . 12
⊢ (𝑐 = (𝑎 ∘ 𝐹) → (∃𝑑 ∈ (ℕ0
↑𝑚 (ℤ ∖ ℕ))(𝑏‘(𝑐 ∪ 𝑑)) = 0 ↔ ∃𝑑 ∈ (ℕ0
↑𝑚 (ℤ ∖ ℕ))(𝑏‘((𝑎 ∘ 𝐹) ∪ 𝑑)) = 0)) |
29 | 28 | elrab3 3591 |
. . . . . . . . . . 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)) |
30 | 25, 29 | 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)) |
31 | | simp-5r 773 |
. . . . . . . . . . . . . . 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)) |
32 | | simplr 756 |
. . . . . . . . . . . . . . 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))) |
33 | | simpr 477 |
. . . . . . . . . . . . . . 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))) |
34 | | coundi 5936 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑎 ∪ 𝑑) ∘ (𝐹 ∪ ( I ↾ (ℤ ∖
ℕ)))) = (((𝑎 ∪
𝑑) ∘ 𝐹) ∪ ((𝑎 ∪ 𝑑) ∘ ( I ↾ (ℤ ∖
ℕ)))) |
35 | | coundir 5937 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑎 ∪ 𝑑) ∘ 𝐹) = ((𝑎 ∘ 𝐹) ∪ (𝑑 ∘ 𝐹)) |
36 | | elmapi 8226 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑑 ∈ (ℕ0
↑𝑚 (ℤ ∖ ℕ)) → 𝑑:(ℤ ∖
ℕ)⟶ℕ0) |
37 | 36 | 3ad2ant3 1115 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝐹:(1...𝑁)⟶(1...𝑀) ∧ 𝑎 ∈ (ℕ0
↑𝑚 (1...𝑀)) ∧ 𝑑 ∈ (ℕ0
↑𝑚 (ℤ ∖ ℕ))) → 𝑑:(ℤ ∖
ℕ)⟶ℕ0) |
38 | | simp1 1116 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝐹:(1...𝑁)⟶(1...𝑀) ∧ 𝑎 ∈ (ℕ0
↑𝑚 (1...𝑀)) ∧ 𝑑 ∈ (ℕ0
↑𝑚 (ℤ ∖ ℕ))) → 𝐹:(1...𝑁)⟶(1...𝑀)) |
39 | | incom 4060 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((ℤ
∖ ℕ) ∩ (1...𝑀)) = ((1...𝑀) ∩ (ℤ ∖
ℕ)) |
40 | | fz1ssnn 12752 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢
(1...𝑀) ⊆
ℕ |
41 | | ssdisj 4286 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢
(((1...𝑀) ⊆
ℕ ∧ (ℕ ∩ (ℤ ∖ ℕ)) = ∅) →
((1...𝑀) ∩ (ℤ
∖ ℕ)) = ∅) |
42 | 40, 18, 41 | mp2an 679 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢
((1...𝑀) ∩
(ℤ ∖ ℕ)) = ∅ |
43 | 39, 42 | eqtri 2796 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((ℤ
∖ ℕ) ∩ (1...𝑀)) = ∅ |
44 | 43 | a1i 11 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝐹:(1...𝑁)⟶(1...𝑀) ∧ 𝑎 ∈ (ℕ0
↑𝑚 (1...𝑀)) ∧ 𝑑 ∈ (ℕ0
↑𝑚 (ℤ ∖ ℕ))) → ((ℤ ∖
ℕ) ∩ (1...𝑀)) =
∅) |
45 | | coeq0i 38774 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝑑:(ℤ ∖
ℕ)⟶ℕ0 ∧ 𝐹:(1...𝑁)⟶(1...𝑀) ∧ ((ℤ ∖ ℕ) ∩
(1...𝑀)) = ∅) →
(𝑑 ∘ 𝐹) = ∅) |
46 | 37, 38, 44, 45 | syl3anc 1351 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝐹:(1...𝑁)⟶(1...𝑀) ∧ 𝑎 ∈ (ℕ0
↑𝑚 (1...𝑀)) ∧ 𝑑 ∈ (ℕ0
↑𝑚 (ℤ ∖ ℕ))) → (𝑑 ∘ 𝐹) = ∅) |
47 | 46 | uneq2d 4022 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝐹:(1...𝑁)⟶(1...𝑀) ∧ 𝑎 ∈ (ℕ0
↑𝑚 (1...𝑀)) ∧ 𝑑 ∈ (ℕ0
↑𝑚 (ℤ ∖ ℕ))) → ((𝑎 ∘ 𝐹) ∪ (𝑑 ∘ 𝐹)) = ((𝑎 ∘ 𝐹) ∪ ∅)) |
48 | 35, 47 | syl5eq 2820 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝐹:(1...𝑁)⟶(1...𝑀) ∧ 𝑎 ∈ (ℕ0
↑𝑚 (1...𝑀)) ∧ 𝑑 ∈ (ℕ0
↑𝑚 (ℤ ∖ ℕ))) → ((𝑎 ∪ 𝑑) ∘ 𝐹) = ((𝑎 ∘ 𝐹) ∪ ∅)) |
49 | | un0 4224 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑎 ∘ 𝐹) ∪ ∅) = (𝑎 ∘ 𝐹) |
50 | 48, 49 | syl6eq 2824 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝐹:(1...𝑁)⟶(1...𝑀) ∧ 𝑎 ∈ (ℕ0
↑𝑚 (1...𝑀)) ∧ 𝑑 ∈ (ℕ0
↑𝑚 (ℤ ∖ ℕ))) → ((𝑎 ∪ 𝑑) ∘ 𝐹) = (𝑎 ∘ 𝐹)) |
51 | | coundir 5937 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑎 ∪ 𝑑) ∘ ( I ↾ (ℤ ∖
ℕ))) = ((𝑎 ∘ (
I ↾ (ℤ ∖ ℕ))) ∪ (𝑑 ∘ ( I ↾ (ℤ ∖
ℕ)))) |
52 | | elmapi 8226 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑎 ∈ (ℕ0
↑𝑚 (1...𝑀)) → 𝑎:(1...𝑀)⟶ℕ0) |
53 | 52 | 3ad2ant2 1114 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝐹:(1...𝑁)⟶(1...𝑀) ∧ 𝑎 ∈ (ℕ0
↑𝑚 (1...𝑀)) ∧ 𝑑 ∈ (ℕ0
↑𝑚 (ℤ ∖ ℕ))) → 𝑎:(1...𝑀)⟶ℕ0) |
54 | | f1oi 6478 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ( I
↾ (ℤ ∖ ℕ)):(ℤ ∖ ℕ)–1-1-onto→(ℤ ∖ ℕ) |
55 | | f1of 6441 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (( I
↾ (ℤ ∖ ℕ)):(ℤ ∖ ℕ)–1-1-onto→(ℤ ∖ ℕ) → ( I ↾
(ℤ ∖ ℕ)):(ℤ ∖ ℕ)⟶(ℤ ∖
ℕ)) |
56 | 54, 55 | ax-mp 5 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ( I
↾ (ℤ ∖ ℕ)):(ℤ ∖ ℕ)⟶(ℤ
∖ ℕ) |
57 | | coeq0i 38774 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝑎:(1...𝑀)⟶ℕ0 ∧ ( I
↾ (ℤ ∖ ℕ)):(ℤ ∖ ℕ)⟶(ℤ
∖ ℕ) ∧ ((1...𝑀) ∩ (ℤ ∖ ℕ)) =
∅) → (𝑎 ∘
( I ↾ (ℤ ∖ ℕ))) = ∅) |
58 | 56, 42, 57 | mp3an23 1432 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑎:(1...𝑀)⟶ℕ0 → (𝑎 ∘ ( I ↾ (ℤ
∖ ℕ))) = ∅) |
59 | 53, 58 | syl 17 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝐹:(1...𝑁)⟶(1...𝑀) ∧ 𝑎 ∈ (ℕ0
↑𝑚 (1...𝑀)) ∧ 𝑑 ∈ (ℕ0
↑𝑚 (ℤ ∖ ℕ))) → (𝑎 ∘ ( I ↾ (ℤ
∖ ℕ))) = ∅) |
60 | | coires1 5953 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑑 ∘ ( I ↾ (ℤ
∖ ℕ))) = (𝑑
↾ (ℤ ∖ ℕ)) |
61 | | ffn 6341 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑑:(ℤ ∖
ℕ)⟶ℕ0 → 𝑑 Fn (ℤ ∖
ℕ)) |
62 | | fnresdm 6296 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑑 Fn (ℤ ∖ ℕ)
→ (𝑑 ↾ (ℤ
∖ ℕ)) = 𝑑) |
63 | 36, 61, 62 | 3syl 18 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑑 ∈ (ℕ0
↑𝑚 (ℤ ∖ ℕ)) → (𝑑 ↾ (ℤ ∖
ℕ)) = 𝑑) |
64 | 60, 63 | syl5eq 2820 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑑 ∈ (ℕ0
↑𝑚 (ℤ ∖ ℕ)) → (𝑑 ∘ ( I ↾ (ℤ
∖ ℕ))) = 𝑑) |
65 | 64 | 3ad2ant3 1115 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝐹:(1...𝑁)⟶(1...𝑀) ∧ 𝑎 ∈ (ℕ0
↑𝑚 (1...𝑀)) ∧ 𝑑 ∈ (ℕ0
↑𝑚 (ℤ ∖ ℕ))) → (𝑑 ∘ ( I ↾ (ℤ
∖ ℕ))) = 𝑑) |
66 | 59, 65 | uneq12d 4023 |
. . . . . . . . . . . . . . . . . . 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)) |
67 | 51, 66 | syl5eq 2820 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝐹:(1...𝑁)⟶(1...𝑀) ∧ 𝑎 ∈ (ℕ0
↑𝑚 (1...𝑀)) ∧ 𝑑 ∈ (ℕ0
↑𝑚 (ℤ ∖ ℕ))) → ((𝑎 ∪ 𝑑) ∘ ( I ↾ (ℤ ∖
ℕ))) = (∅ ∪ 𝑑)) |
68 | | uncom 4012 |
. . . . . . . . . . . . . . . . . . 19
⊢ (∅
∪ 𝑑) = (𝑑 ∪ ∅) |
69 | | un0 4224 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑑 ∪ ∅) = 𝑑 |
70 | 68, 69 | eqtri 2796 |
. . . . . . . . . . . . . . . . . 18
⊢ (∅
∪ 𝑑) = 𝑑 |
71 | 67, 70 | syl6eq 2824 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝐹:(1...𝑁)⟶(1...𝑀) ∧ 𝑎 ∈ (ℕ0
↑𝑚 (1...𝑀)) ∧ 𝑑 ∈ (ℕ0
↑𝑚 (ℤ ∖ ℕ))) → ((𝑎 ∪ 𝑑) ∘ ( I ↾ (ℤ ∖
ℕ))) = 𝑑) |
72 | 50, 71 | uneq12d 4023 |
. . . . . . . . . . . . . . . 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)) |
73 | 34, 72 | syl5req 2821 |
. . . . . . . . . . . . . . 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))))) |
74 | 31, 32, 33, 73 | syl3anc 1351 |
. . . . . . . . . . . . . 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))))) |
75 | 74 | fveq2d 6500 |
. . . . . . . . . . . . 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)))))) |
76 | | nn0ssz 11814 |
. . . . . . . . . . . . . . . . 17
⊢
ℕ0 ⊆ ℤ |
77 | | mapss 8249 |
. . . . . . . . . . . . . . . . 17
⊢ ((ℤ
∈ V ∧ ℕ0 ⊆ ℤ) → (ℕ0
↑𝑚 ((ℤ ∖ ℕ) ∪ (1...𝑀))) ⊆ (ℤ
↑𝑚 ((ℤ ∖ ℕ) ∪ (1...𝑀)))) |
78 | 1, 76, 77 | mp2an 679 |
. . . . . . . . . . . . . . . 16
⊢
(ℕ0 ↑𝑚 ((ℤ ∖
ℕ) ∪ (1...𝑀)))
⊆ (ℤ ↑𝑚 ((ℤ ∖ ℕ) ∪
(1...𝑀))) |
79 | 42 | reseq2i 5689 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑎 ↾ ((1...𝑀) ∩ (ℤ ∖ ℕ))) = (𝑎 ↾
∅) |
80 | | res0 5696 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑎 ↾ ∅) =
∅ |
81 | 79, 80 | eqtri 2796 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑎 ↾ ((1...𝑀) ∩ (ℤ ∖ ℕ))) =
∅ |
82 | 42 | reseq2i 5689 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑑 ↾ ((1...𝑀) ∩ (ℤ ∖ ℕ))) = (𝑑 ↾
∅) |
83 | | res0 5696 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑑 ↾ ∅) =
∅ |
84 | 82, 83 | eqtri 2796 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑑 ↾ ((1...𝑀) ∩ (ℤ ∖ ℕ))) =
∅ |
85 | 81, 84 | eqtr4i 2799 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑎 ↾ ((1...𝑀) ∩ (ℤ ∖ ℕ))) = (𝑑 ↾ ((1...𝑀) ∩ (ℤ ∖
ℕ))) |
86 | | elmapresaun 38792 |
. . . . . . . . . . . . . . . . . 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)))) |
87 | | uncom 4012 |
. . . . . . . . . . . . . . . . . . 19
⊢
((1...𝑀) ∪
(ℤ ∖ ℕ)) = ((ℤ ∖ ℕ) ∪ (1...𝑀)) |
88 | 87 | oveq2i 6985 |
. . . . . . . . . . . . . . . . . 18
⊢
(ℕ0 ↑𝑚 ((1...𝑀) ∪ (ℤ ∖ ℕ))) =
(ℕ0 ↑𝑚 ((ℤ ∖ ℕ)
∪ (1...𝑀))) |
89 | 86, 88 | syl6eleq 2870 |
. . . . . . . . . . . . . . . . 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)))) |
90 | 85, 89 | mp3an3 1429 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑎 ∈ (ℕ0
↑𝑚 (1...𝑀)) ∧ 𝑑 ∈ (ℕ0
↑𝑚 (ℤ ∖ ℕ))) → (𝑎 ∪ 𝑑) ∈ (ℕ0
↑𝑚 ((ℤ ∖ ℕ) ∪ (1...𝑀)))) |
91 | 78, 90 | sseldi 3850 |
. . . . . . . . . . . . . . 15
⊢ ((𝑎 ∈ (ℕ0
↑𝑚 (1...𝑀)) ∧ 𝑑 ∈ (ℕ0
↑𝑚 (ℤ ∖ ℕ))) → (𝑎 ∪ 𝑑) ∈ (ℤ ↑𝑚
((ℤ ∖ ℕ) ∪ (1...𝑀)))) |
92 | 91 | adantll 701 |
. . . . . . . . . . . . . 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)))) |
93 | | coeq1 5574 |
. . . . . . . . . . . . . . . 16
⊢ (𝑒 = (𝑎 ∪ 𝑑) → (𝑒 ∘ (𝐹 ∪ ( I ↾ (ℤ ∖
ℕ)))) = ((𝑎 ∪
𝑑) ∘ (𝐹 ∪ ( I ↾ (ℤ
∖ ℕ))))) |
94 | 93 | fveq2d 6500 |
. . . . . . . . . . . . . . 15
⊢ (𝑒 = (𝑎 ∪ 𝑑) → (𝑏‘(𝑒 ∘ (𝐹 ∪ ( I ↾ (ℤ ∖
ℕ))))) = (𝑏‘((𝑎 ∪ 𝑑) ∘ (𝐹 ∪ ( I ↾ (ℤ ∖
ℕ)))))) |
95 | | eqid 2772 |
. . . . . . . . . . . . . . 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)))))) |
96 | | fvex 6509 |
. . . . . . . . . . . . . . 15
⊢ (𝑏‘((𝑎 ∪ 𝑑) ∘ (𝐹 ∪ ( I ↾ (ℤ ∖
ℕ))))) ∈ V |
97 | 94, 95, 96 | fvmpt 6593 |
. . . . . . . . . . . . . 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)))))) |
98 | 92, 97 | 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)))))) |
99 | 75, 98 | eqtr4d 2811 |
. . . . . . . . . . . 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))) |
100 | 99 | eqeq1d 2774 |
. . . . . . . . . . 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)) |
101 | 100 | rexbidva 3235 |
. . . . . . . . . 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)) |
102 | 30, 101 | bitrd 271 |
. . . . . . . . 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)) |
103 | 102 | rabbidva 3396 |
. . . . . . . 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}) |
104 | | simplll 762 |
. . . . . . . . 9
⊢ ((((𝑀 ∈ ℕ0
∧ 𝐹:(1...𝑁)⟶(1...𝑀)) ∧ 𝑁 ∈ ℕ0) ∧ 𝑏 ∈ (mzPoly‘((ℤ
∖ ℕ) ∪ (1...𝑁)))) → 𝑀 ∈
ℕ0) |
105 | | ovex 7006 |
. . . . . . . . . . . 12
⊢
(1...𝑀) ∈
V |
106 | 3, 105 | unex 7284 |
. . . . . . . . . . 11
⊢ ((ℤ
∖ ℕ) ∪ (1...𝑀)) ∈ V |
107 | 106 | a1i 11 |
. . . . . . . . . 10
⊢ ((((𝑀 ∈ ℕ0
∧ 𝐹:(1...𝑁)⟶(1...𝑀)) ∧ 𝑁 ∈ ℕ0) ∧ 𝑏 ∈ (mzPoly‘((ℤ
∖ ℕ) ∪ (1...𝑁)))) → ((ℤ ∖ ℕ) ∪
(1...𝑀)) ∈
V) |
108 | | simpr 477 |
. . . . . . . . . 10
⊢ ((((𝑀 ∈ ℕ0
∧ 𝐹:(1...𝑁)⟶(1...𝑀)) ∧ 𝑁 ∈ ℕ0) ∧ 𝑏 ∈ (mzPoly‘((ℤ
∖ ℕ) ∪ (1...𝑁)))) → 𝑏 ∈ (mzPoly‘((ℤ ∖
ℕ) ∪ (1...𝑁)))) |
109 | 56 | a1i 11 |
. . . . . . . . . . . . 13
⊢ (𝐹:(1...𝑁)⟶(1...𝑀) → ( I ↾ (ℤ ∖
ℕ)):(ℤ ∖ ℕ)⟶(ℤ ∖
ℕ)) |
110 | | id 22 |
. . . . . . . . . . . . 13
⊢ (𝐹:(1...𝑁)⟶(1...𝑀) → 𝐹:(1...𝑁)⟶(1...𝑀)) |
111 | | incom 4060 |
. . . . . . . . . . . . . . 15
⊢ ((ℤ
∖ ℕ) ∩ (1...𝑁)) = ((1...𝑁) ∩ (ℤ ∖
ℕ)) |
112 | | fz1ssnn 12752 |
. . . . . . . . . . . . . . . 16
⊢
(1...𝑁) ⊆
ℕ |
113 | | ssdisj 4286 |
. . . . . . . . . . . . . . . 16
⊢
(((1...𝑁) ⊆
ℕ ∧ (ℕ ∩ (ℤ ∖ ℕ)) = ∅) →
((1...𝑁) ∩ (ℤ
∖ ℕ)) = ∅) |
114 | 112, 18, 113 | mp2an 679 |
. . . . . . . . . . . . . . 15
⊢
((1...𝑁) ∩
(ℤ ∖ ℕ)) = ∅ |
115 | 111, 114 | eqtri 2796 |
. . . . . . . . . . . . . 14
⊢ ((ℤ
∖ ℕ) ∩ (1...𝑁)) = ∅ |
116 | 115 | a1i 11 |
. . . . . . . . . . . . 13
⊢ (𝐹:(1...𝑁)⟶(1...𝑀) → ((ℤ ∖ ℕ) ∩
(1...𝑁)) =
∅) |
117 | | fun 6366 |
. . . . . . . . . . . . 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))) |
118 | 109, 110,
116, 117 | syl21anc 825 |
. . . . . . . . . . . 12
⊢ (𝐹:(1...𝑁)⟶(1...𝑀) → (( I ↾ (ℤ ∖
ℕ)) ∪ 𝐹):((ℤ ∖ ℕ) ∪
(1...𝑁))⟶((ℤ
∖ ℕ) ∪ (1...𝑀))) |
119 | | uncom 4012 |
. . . . . . . . . . . . 13
⊢ (( I
↾ (ℤ ∖ ℕ)) ∪ 𝐹) = (𝐹 ∪ ( I ↾ (ℤ ∖
ℕ))) |
120 | 119 | feq1i 6332 |
. . . . . . . . . . . 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))) |
121 | 118, 120 | sylib 210 |
. . . . . . . . . . 11
⊢ (𝐹:(1...𝑁)⟶(1...𝑀) → (𝐹 ∪ ( I ↾ (ℤ ∖
ℕ))):((ℤ ∖ ℕ) ∪ (1...𝑁))⟶((ℤ ∖ ℕ) ∪
(1...𝑀))) |
122 | 121 | ad3antlr 718 |
. . . . . . . . . 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))) |
123 | | mzprename 38770 |
. . . . . . . . . 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)))) |
124 | 107, 108,
122, 123 | syl3anc 1351 |
. . . . . . . . 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)))) |
125 | 3, 16, 19 | eldioph4i 38834 |
. . . . . . . . 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)) |
126 | 104, 124,
125 | syl2anc 576 |
. . . . . . . 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)) |
127 | 103, 126 | eqeltrd 2860 |
. . . . . . 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)) |
128 | | eleq2 2848 |
. . . . . . . . 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})) |
129 | 128 | rabbidv 3397 |
. . . . . . . 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}}) |
130 | 129 | eleq1d 2844 |
. . . . . . 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))) |
131 | 127, 130 | syl5ibrcom 239 |
. . . . . 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))) |
132 | 131 | rexlimdva 3223 |
. . . . 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))) |
133 | 132 | expimpd 446 |
. . . 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))) |
134 | 20, 133 | syl5bi 234 |
. . 3
⊢ ((𝑀 ∈ ℕ0
∧ 𝐹:(1...𝑁)⟶(1...𝑀)) → (𝑆 ∈ (Dioph‘𝑁) → {𝑎 ∈ (ℕ0
↑𝑚 (1...𝑀)) ∣ (𝑎 ∘ 𝐹) ∈ 𝑆} ∈ (Dioph‘𝑀))) |
135 | 134 | impcom 399 |
. 2
⊢ ((𝑆 ∈ (Dioph‘𝑁) ∧ (𝑀 ∈ ℕ0 ∧ 𝐹:(1...𝑁)⟶(1...𝑀))) → {𝑎 ∈ (ℕ0
↑𝑚 (1...𝑀)) ∣ (𝑎 ∘ 𝐹) ∈ 𝑆} ∈ (Dioph‘𝑀)) |
136 | 135 | 3impb 1095 |
1
⊢ ((𝑆 ∈ (Dioph‘𝑁) ∧ 𝑀 ∈ ℕ0 ∧ 𝐹:(1...𝑁)⟶(1...𝑀)) → {𝑎 ∈ (ℕ0
↑𝑚 (1...𝑀)) ∣ (𝑎 ∘ 𝐹) ∈ 𝑆} ∈ (Dioph‘𝑀)) |