Step | Hyp | Ref
| Expression |
1 | | zex 11386 |
. . . . . 6
⊢ ℤ
∈ V |
2 | | difexg 4808 |
. . . . . 6
⊢ (ℤ
∈ V → (ℤ ∖ ℕ) ∈ V) |
3 | 1, 2 | ax-mp 5 |
. . . . 5
⊢ (ℤ
∖ ℕ) ∈ V |
4 | | ominf 8172 |
. . . . . 6
⊢ ¬
ω ∈ Fin |
5 | | nnuz 11723 |
. . . . . . . . . 10
⊢ ℕ =
(ℤ≥‘1) |
6 | | 0p1e1 11132 |
. . . . . . . . . . 11
⊢ (0 + 1) =
1 |
7 | 6 | fveq2i 6194 |
. . . . . . . . . 10
⊢
(ℤ≥‘(0 + 1)) =
(ℤ≥‘1) |
8 | 5, 7 | eqtr4i 2647 |
. . . . . . . . 9
⊢ ℕ =
(ℤ≥‘(0 + 1)) |
9 | 8 | difeq2i 3725 |
. . . . . . . 8
⊢ (ℤ
∖ ℕ) = (ℤ ∖ (ℤ≥‘(0 +
1))) |
10 | | 0z 11388 |
. . . . . . . . 9
⊢ 0 ∈
ℤ |
11 | | lzenom 37333 |
. . . . . . . . 9
⊢ (0 ∈
ℤ → (ℤ ∖ (ℤ≥‘(0 + 1))) ≈
ω) |
12 | 10, 11 | ax-mp 5 |
. . . . . . . 8
⊢ (ℤ
∖ (ℤ≥‘(0 + 1))) ≈
ω |
13 | 9, 12 | eqbrtri 4674 |
. . . . . . 7
⊢ (ℤ
∖ ℕ) ≈ ω |
14 | | enfi 8176 |
. . . . . . 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 313 |
. . . . 5
⊢ ¬
(ℤ ∖ ℕ) ∈ Fin |
17 | | incom 3805 |
. . . . . 6
⊢ ((ℤ
∖ ℕ) ∩ ℕ) = (ℕ ∩ (ℤ ∖
ℕ)) |
18 | | disjdif 4040 |
. . . . . 6
⊢ (ℕ
∩ (ℤ ∖ ℕ)) = ∅ |
19 | 17, 18 | eqtri 2644 |
. . . . 5
⊢ ((ℤ
∖ ℕ) ∩ ℕ) = ∅ |
20 | 3, 16, 19 | eldioph4b 37375 |
. . . 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 807 |
. . . . . . . . . . . 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 6678 |
. . . . . . . . . . . . 13
⊢
(1...𝑁) ∈
V |
24 | 23 | mapco2 37278 |
. . . . . . . . . . . 12
⊢ ((𝑎 ∈ (ℕ0
↑𝑚 (1...𝑀)) ∧ 𝐹:(1...𝑁)⟶(1...𝑀)) → (𝑎 ∘ 𝐹) ∈ (ℕ0
↑𝑚 (1...𝑁))) |
25 | 21, 22, 24 | syl2anc 693 |
. . . . . . . . . . 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 3760 |
. . . . . . . . . . . . . . 15
⊢ (𝑐 = (𝑎 ∘ 𝐹) → (𝑐 ∪ 𝑑) = ((𝑎 ∘ 𝐹) ∪ 𝑑)) |
27 | 26 | fveq2d 6195 |
. . . . . . . . . . . . . 14
⊢ (𝑐 = (𝑎 ∘ 𝐹) → (𝑏‘(𝑐 ∪ 𝑑)) = (𝑏‘((𝑎 ∘ 𝐹) ∪ 𝑑))) |
28 | 27 | eqeq1d 2624 |
. . . . . . . . . . . . 13
⊢ (𝑐 = (𝑎 ∘ 𝐹) → ((𝑏‘(𝑐 ∪ 𝑑)) = 0 ↔ (𝑏‘((𝑎 ∘ 𝐹) ∪ 𝑑)) = 0)) |
29 | 28 | rexbidv 3052 |
. . . . . . . . . . . 12
⊢ (𝑐 = (𝑎 ∘ 𝐹) → (∃𝑑 ∈ (ℕ0
↑𝑚 (ℤ ∖ ℕ))(𝑏‘(𝑐 ∪ 𝑑)) = 0 ↔ ∃𝑑 ∈ (ℕ0
↑𝑚 (ℤ ∖ ℕ))(𝑏‘((𝑎 ∘ 𝐹) ∪ 𝑑)) = 0)) |
30 | 29 | elrab3 3364 |
. . . . . . . . . . 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)) |
31 | 25, 30 | 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)) |
32 | | simp-5r 809 |
. . . . . . . . . . . . . . 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)) |
33 | | simplr 792 |
. . . . . . . . . . . . . . 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))) |
34 | | 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))) |
35 | | coundi 5636 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑎 ∪ 𝑑) ∘ (𝐹 ∪ ( I ↾ (ℤ ∖
ℕ)))) = (((𝑎 ∪
𝑑) ∘ 𝐹) ∪ ((𝑎 ∪ 𝑑) ∘ ( I ↾ (ℤ ∖
ℕ)))) |
36 | | coundir 5637 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑎 ∪ 𝑑) ∘ 𝐹) = ((𝑎 ∘ 𝐹) ∪ (𝑑 ∘ 𝐹)) |
37 | | elmapi 7879 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑑 ∈ (ℕ0
↑𝑚 (ℤ ∖ ℕ)) → 𝑑:(ℤ ∖
ℕ)⟶ℕ0) |
38 | 37 | 3ad2ant3 1084 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝐹:(1...𝑁)⟶(1...𝑀) ∧ 𝑎 ∈ (ℕ0
↑𝑚 (1...𝑀)) ∧ 𝑑 ∈ (ℕ0
↑𝑚 (ℤ ∖ ℕ))) → 𝑑:(ℤ ∖
ℕ)⟶ℕ0) |
39 | | simp1 1061 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝐹:(1...𝑁)⟶(1...𝑀) ∧ 𝑎 ∈ (ℕ0
↑𝑚 (1...𝑀)) ∧ 𝑑 ∈ (ℕ0
↑𝑚 (ℤ ∖ ℕ))) → 𝐹:(1...𝑁)⟶(1...𝑀)) |
40 | | incom 3805 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((ℤ
∖ ℕ) ∩ (1...𝑀)) = ((1...𝑀) ∩ (ℤ ∖
ℕ)) |
41 | | fz1ssnn 12372 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢
(1...𝑀) ⊆
ℕ |
42 | | ssdisj 4026 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢
(((1...𝑀) ⊆
ℕ ∧ (ℕ ∩ (ℤ ∖ ℕ)) = ∅) →
((1...𝑀) ∩ (ℤ
∖ ℕ)) = ∅) |
43 | 41, 18, 42 | mp2an 708 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢
((1...𝑀) ∩
(ℤ ∖ ℕ)) = ∅ |
44 | 40, 43 | eqtri 2644 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((ℤ
∖ ℕ) ∩ (1...𝑀)) = ∅ |
45 | 44 | a1i 11 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝐹:(1...𝑁)⟶(1...𝑀) ∧ 𝑎 ∈ (ℕ0
↑𝑚 (1...𝑀)) ∧ 𝑑 ∈ (ℕ0
↑𝑚 (ℤ ∖ ℕ))) → ((ℤ ∖
ℕ) ∩ (1...𝑀)) =
∅) |
46 | | coeq0i 37316 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝑑:(ℤ ∖
ℕ)⟶ℕ0 ∧ 𝐹:(1...𝑁)⟶(1...𝑀) ∧ ((ℤ ∖ ℕ) ∩
(1...𝑀)) = ∅) →
(𝑑 ∘ 𝐹) = ∅) |
47 | 38, 39, 45, 46 | syl3anc 1326 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝐹:(1...𝑁)⟶(1...𝑀) ∧ 𝑎 ∈ (ℕ0
↑𝑚 (1...𝑀)) ∧ 𝑑 ∈ (ℕ0
↑𝑚 (ℤ ∖ ℕ))) → (𝑑 ∘ 𝐹) = ∅) |
48 | 47 | uneq2d 3767 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝐹:(1...𝑁)⟶(1...𝑀) ∧ 𝑎 ∈ (ℕ0
↑𝑚 (1...𝑀)) ∧ 𝑑 ∈ (ℕ0
↑𝑚 (ℤ ∖ ℕ))) → ((𝑎 ∘ 𝐹) ∪ (𝑑 ∘ 𝐹)) = ((𝑎 ∘ 𝐹) ∪ ∅)) |
49 | 36, 48 | syl5eq 2668 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝐹:(1...𝑁)⟶(1...𝑀) ∧ 𝑎 ∈ (ℕ0
↑𝑚 (1...𝑀)) ∧ 𝑑 ∈ (ℕ0
↑𝑚 (ℤ ∖ ℕ))) → ((𝑎 ∪ 𝑑) ∘ 𝐹) = ((𝑎 ∘ 𝐹) ∪ ∅)) |
50 | | un0 3967 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑎 ∘ 𝐹) ∪ ∅) = (𝑎 ∘ 𝐹) |
51 | 49, 50 | syl6eq 2672 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝐹:(1...𝑁)⟶(1...𝑀) ∧ 𝑎 ∈ (ℕ0
↑𝑚 (1...𝑀)) ∧ 𝑑 ∈ (ℕ0
↑𝑚 (ℤ ∖ ℕ))) → ((𝑎 ∪ 𝑑) ∘ 𝐹) = (𝑎 ∘ 𝐹)) |
52 | | coundir 5637 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑎 ∪ 𝑑) ∘ ( I ↾ (ℤ ∖
ℕ))) = ((𝑎 ∘ (
I ↾ (ℤ ∖ ℕ))) ∪ (𝑑 ∘ ( I ↾ (ℤ ∖
ℕ)))) |
53 | | elmapi 7879 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑎 ∈ (ℕ0
↑𝑚 (1...𝑀)) → 𝑎:(1...𝑀)⟶ℕ0) |
54 | 53 | 3ad2ant2 1083 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝐹:(1...𝑁)⟶(1...𝑀) ∧ 𝑎 ∈ (ℕ0
↑𝑚 (1...𝑀)) ∧ 𝑑 ∈ (ℕ0
↑𝑚 (ℤ ∖ ℕ))) → 𝑎:(1...𝑀)⟶ℕ0) |
55 | | f1oi 6174 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ( I
↾ (ℤ ∖ ℕ)):(ℤ ∖ ℕ)–1-1-onto→(ℤ ∖ ℕ) |
56 | | f1of 6137 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (( I
↾ (ℤ ∖ ℕ)):(ℤ ∖ ℕ)–1-1-onto→(ℤ ∖ ℕ) → ( I ↾
(ℤ ∖ ℕ)):(ℤ ∖ ℕ)⟶(ℤ ∖
ℕ)) |
57 | 55, 56 | ax-mp 5 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ( I
↾ (ℤ ∖ ℕ)):(ℤ ∖ ℕ)⟶(ℤ
∖ ℕ) |
58 | | coeq0i 37316 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝑎:(1...𝑀)⟶ℕ0 ∧ ( I
↾ (ℤ ∖ ℕ)):(ℤ ∖ ℕ)⟶(ℤ
∖ ℕ) ∧ ((1...𝑀) ∩ (ℤ ∖ ℕ)) =
∅) → (𝑎 ∘
( I ↾ (ℤ ∖ ℕ))) = ∅) |
59 | 57, 43, 58 | mp3an23 1416 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑎:(1...𝑀)⟶ℕ0 → (𝑎 ∘ ( I ↾ (ℤ
∖ ℕ))) = ∅) |
60 | 54, 59 | syl 17 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝐹:(1...𝑁)⟶(1...𝑀) ∧ 𝑎 ∈ (ℕ0
↑𝑚 (1...𝑀)) ∧ 𝑑 ∈ (ℕ0
↑𝑚 (ℤ ∖ ℕ))) → (𝑎 ∘ ( I ↾ (ℤ
∖ ℕ))) = ∅) |
61 | | coires1 5653 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑑 ∘ ( I ↾ (ℤ
∖ ℕ))) = (𝑑
↾ (ℤ ∖ ℕ)) |
62 | | ffn 6045 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑑:(ℤ ∖
ℕ)⟶ℕ0 → 𝑑 Fn (ℤ ∖
ℕ)) |
63 | | fnresdm 6000 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑑 Fn (ℤ ∖ ℕ)
→ (𝑑 ↾ (ℤ
∖ ℕ)) = 𝑑) |
64 | 37, 62, 63 | 3syl 18 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑑 ∈ (ℕ0
↑𝑚 (ℤ ∖ ℕ)) → (𝑑 ↾ (ℤ ∖
ℕ)) = 𝑑) |
65 | 61, 64 | syl5eq 2668 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑑 ∈ (ℕ0
↑𝑚 (ℤ ∖ ℕ)) → (𝑑 ∘ ( I ↾ (ℤ
∖ ℕ))) = 𝑑) |
66 | 65 | 3ad2ant3 1084 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝐹:(1...𝑁)⟶(1...𝑀) ∧ 𝑎 ∈ (ℕ0
↑𝑚 (1...𝑀)) ∧ 𝑑 ∈ (ℕ0
↑𝑚 (ℤ ∖ ℕ))) → (𝑑 ∘ ( I ↾ (ℤ
∖ ℕ))) = 𝑑) |
67 | 60, 66 | uneq12d 3768 |
. . . . . . . . . . . . . . . . . . 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)) |
68 | 52, 67 | syl5eq 2668 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝐹:(1...𝑁)⟶(1...𝑀) ∧ 𝑎 ∈ (ℕ0
↑𝑚 (1...𝑀)) ∧ 𝑑 ∈ (ℕ0
↑𝑚 (ℤ ∖ ℕ))) → ((𝑎 ∪ 𝑑) ∘ ( I ↾ (ℤ ∖
ℕ))) = (∅ ∪ 𝑑)) |
69 | | uncom 3757 |
. . . . . . . . . . . . . . . . . . 19
⊢ (∅
∪ 𝑑) = (𝑑 ∪ ∅) |
70 | | un0 3967 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑑 ∪ ∅) = 𝑑 |
71 | 69, 70 | eqtri 2644 |
. . . . . . . . . . . . . . . . . 18
⊢ (∅
∪ 𝑑) = 𝑑 |
72 | 68, 71 | syl6eq 2672 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝐹:(1...𝑁)⟶(1...𝑀) ∧ 𝑎 ∈ (ℕ0
↑𝑚 (1...𝑀)) ∧ 𝑑 ∈ (ℕ0
↑𝑚 (ℤ ∖ ℕ))) → ((𝑎 ∪ 𝑑) ∘ ( I ↾ (ℤ ∖
ℕ))) = 𝑑) |
73 | 51, 72 | uneq12d 3768 |
. . . . . . . . . . . . . . . 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)) |
74 | 35, 73 | syl5req 2669 |
. . . . . . . . . . . . . . 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))))) |
75 | 32, 33, 34, 74 | syl3anc 1326 |
. . . . . . . . . . . . . 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))))) |
76 | 75 | fveq2d 6195 |
. . . . . . . . . . . . 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)))))) |
77 | | nn0ssz 11398 |
. . . . . . . . . . . . . . . . 17
⊢
ℕ0 ⊆ ℤ |
78 | | mapss 7900 |
. . . . . . . . . . . . . . . . 17
⊢ ((ℤ
∈ V ∧ ℕ0 ⊆ ℤ) → (ℕ0
↑𝑚 ((ℤ ∖ ℕ) ∪ (1...𝑀))) ⊆ (ℤ
↑𝑚 ((ℤ ∖ ℕ) ∪ (1...𝑀)))) |
79 | 1, 77, 78 | mp2an 708 |
. . . . . . . . . . . . . . . 16
⊢
(ℕ0 ↑𝑚 ((ℤ ∖
ℕ) ∪ (1...𝑀)))
⊆ (ℤ ↑𝑚 ((ℤ ∖ ℕ) ∪
(1...𝑀))) |
80 | 43 | reseq2i 5393 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑎 ↾ ((1...𝑀) ∩ (ℤ ∖ ℕ))) = (𝑎 ↾
∅) |
81 | | res0 5400 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑎 ↾ ∅) =
∅ |
82 | 80, 81 | eqtri 2644 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑎 ↾ ((1...𝑀) ∩ (ℤ ∖ ℕ))) =
∅ |
83 | 43 | reseq2i 5393 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑑 ↾ ((1...𝑀) ∩ (ℤ ∖ ℕ))) = (𝑑 ↾
∅) |
84 | | res0 5400 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑑 ↾ ∅) =
∅ |
85 | 83, 84 | eqtri 2644 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑑 ↾ ((1...𝑀) ∩ (ℤ ∖ ℕ))) =
∅ |
86 | 82, 85 | eqtr4i 2647 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑎 ↾ ((1...𝑀) ∩ (ℤ ∖ ℕ))) = (𝑑 ↾ ((1...𝑀) ∩ (ℤ ∖
ℕ))) |
87 | | elmapresaun 37334 |
. . . . . . . . . . . . . . . . . 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)))) |
88 | | uncom 3757 |
. . . . . . . . . . . . . . . . . . 19
⊢
((1...𝑀) ∪
(ℤ ∖ ℕ)) = ((ℤ ∖ ℕ) ∪ (1...𝑀)) |
89 | 88 | oveq2i 6661 |
. . . . . . . . . . . . . . . . . 18
⊢
(ℕ0 ↑𝑚 ((1...𝑀) ∪ (ℤ ∖ ℕ))) =
(ℕ0 ↑𝑚 ((ℤ ∖ ℕ)
∪ (1...𝑀))) |
90 | 87, 89 | syl6eleq 2711 |
. . . . . . . . . . . . . . . . 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)))) |
91 | 86, 90 | mp3an3 1413 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑎 ∈ (ℕ0
↑𝑚 (1...𝑀)) ∧ 𝑑 ∈ (ℕ0
↑𝑚 (ℤ ∖ ℕ))) → (𝑎 ∪ 𝑑) ∈ (ℕ0
↑𝑚 ((ℤ ∖ ℕ) ∪ (1...𝑀)))) |
92 | 79, 91 | sseldi 3601 |
. . . . . . . . . . . . . . 15
⊢ ((𝑎 ∈ (ℕ0
↑𝑚 (1...𝑀)) ∧ 𝑑 ∈ (ℕ0
↑𝑚 (ℤ ∖ ℕ))) → (𝑎 ∪ 𝑑) ∈ (ℤ ↑𝑚
((ℤ ∖ ℕ) ∪ (1...𝑀)))) |
93 | 92 | adantll 750 |
. . . . . . . . . . . . . 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)))) |
94 | | coeq1 5279 |
. . . . . . . . . . . . . . . 16
⊢ (𝑒 = (𝑎 ∪ 𝑑) → (𝑒 ∘ (𝐹 ∪ ( I ↾ (ℤ ∖
ℕ)))) = ((𝑎 ∪
𝑑) ∘ (𝐹 ∪ ( I ↾ (ℤ
∖ ℕ))))) |
95 | 94 | fveq2d 6195 |
. . . . . . . . . . . . . . 15
⊢ (𝑒 = (𝑎 ∪ 𝑑) → (𝑏‘(𝑒 ∘ (𝐹 ∪ ( I ↾ (ℤ ∖
ℕ))))) = (𝑏‘((𝑎 ∪ 𝑑) ∘ (𝐹 ∪ ( I ↾ (ℤ ∖
ℕ)))))) |
96 | | eqid 2622 |
. . . . . . . . . . . . . . 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)))))) |
97 | | fvex 6201 |
. . . . . . . . . . . . . . 15
⊢ (𝑏‘((𝑎 ∪ 𝑑) ∘ (𝐹 ∪ ( I ↾ (ℤ ∖
ℕ))))) ∈ V |
98 | 95, 96, 97 | fvmpt 6282 |
. . . . . . . . . . . . . 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)))))) |
99 | 93, 98 | 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)))))) |
100 | 76, 99 | eqtr4d 2659 |
. . . . . . . . . . . 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))) |
101 | 100 | eqeq1d 2624 |
. . . . . . . . . . 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)) |
102 | 101 | rexbidva 3049 |
. . . . . . . . . 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)) |
103 | 31, 102 | bitrd 268 |
. . . . . . . . 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)) |
104 | 103 | rabbidva 3188 |
. . . . . . . 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}) |
105 | | simplll 798 |
. . . . . . . . 9
⊢ ((((𝑀 ∈ ℕ0
∧ 𝐹:(1...𝑁)⟶(1...𝑀)) ∧ 𝑁 ∈ ℕ0) ∧ 𝑏 ∈ (mzPoly‘((ℤ
∖ ℕ) ∪ (1...𝑁)))) → 𝑀 ∈
ℕ0) |
106 | | ovex 6678 |
. . . . . . . . . . . 12
⊢
(1...𝑀) ∈
V |
107 | 3, 106 | unex 6956 |
. . . . . . . . . . 11
⊢ ((ℤ
∖ ℕ) ∪ (1...𝑀)) ∈ V |
108 | 107 | a1i 11 |
. . . . . . . . . 10
⊢ ((((𝑀 ∈ ℕ0
∧ 𝐹:(1...𝑁)⟶(1...𝑀)) ∧ 𝑁 ∈ ℕ0) ∧ 𝑏 ∈ (mzPoly‘((ℤ
∖ ℕ) ∪ (1...𝑁)))) → ((ℤ ∖ ℕ) ∪
(1...𝑀)) ∈
V) |
109 | | simpr 477 |
. . . . . . . . . 10
⊢ ((((𝑀 ∈ ℕ0
∧ 𝐹:(1...𝑁)⟶(1...𝑀)) ∧ 𝑁 ∈ ℕ0) ∧ 𝑏 ∈ (mzPoly‘((ℤ
∖ ℕ) ∪ (1...𝑁)))) → 𝑏 ∈ (mzPoly‘((ℤ ∖
ℕ) ∪ (1...𝑁)))) |
110 | 57 | a1i 11 |
. . . . . . . . . . . . 13
⊢ (𝐹:(1...𝑁)⟶(1...𝑀) → ( I ↾ (ℤ ∖
ℕ)):(ℤ ∖ ℕ)⟶(ℤ ∖
ℕ)) |
111 | | id 22 |
. . . . . . . . . . . . 13
⊢ (𝐹:(1...𝑁)⟶(1...𝑀) → 𝐹:(1...𝑁)⟶(1...𝑀)) |
112 | | incom 3805 |
. . . . . . . . . . . . . . 15
⊢ ((ℤ
∖ ℕ) ∩ (1...𝑁)) = ((1...𝑁) ∩ (ℤ ∖
ℕ)) |
113 | | fz1ssnn 12372 |
. . . . . . . . . . . . . . . 16
⊢
(1...𝑁) ⊆
ℕ |
114 | | ssdisj 4026 |
. . . . . . . . . . . . . . . 16
⊢
(((1...𝑁) ⊆
ℕ ∧ (ℕ ∩ (ℤ ∖ ℕ)) = ∅) →
((1...𝑁) ∩ (ℤ
∖ ℕ)) = ∅) |
115 | 113, 18, 114 | mp2an 708 |
. . . . . . . . . . . . . . 15
⊢
((1...𝑁) ∩
(ℤ ∖ ℕ)) = ∅ |
116 | 112, 115 | eqtri 2644 |
. . . . . . . . . . . . . 14
⊢ ((ℤ
∖ ℕ) ∩ (1...𝑁)) = ∅ |
117 | 116 | a1i 11 |
. . . . . . . . . . . . 13
⊢ (𝐹:(1...𝑁)⟶(1...𝑀) → ((ℤ ∖ ℕ) ∩
(1...𝑁)) =
∅) |
118 | | fun 6066 |
. . . . . . . . . . . . 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))) |
119 | 110, 111,
117, 118 | syl21anc 1325 |
. . . . . . . . . . . 12
⊢ (𝐹:(1...𝑁)⟶(1...𝑀) → (( I ↾ (ℤ ∖
ℕ)) ∪ 𝐹):((ℤ ∖ ℕ) ∪
(1...𝑁))⟶((ℤ
∖ ℕ) ∪ (1...𝑀))) |
120 | | uncom 3757 |
. . . . . . . . . . . . 13
⊢ (( I
↾ (ℤ ∖ ℕ)) ∪ 𝐹) = (𝐹 ∪ ( I ↾ (ℤ ∖
ℕ))) |
121 | 120 | feq1i 6036 |
. . . . . . . . . . . 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))) |
122 | 119, 121 | sylib 208 |
. . . . . . . . . . 11
⊢ (𝐹:(1...𝑁)⟶(1...𝑀) → (𝐹 ∪ ( I ↾ (ℤ ∖
ℕ))):((ℤ ∖ ℕ) ∪ (1...𝑁))⟶((ℤ ∖ ℕ) ∪
(1...𝑀))) |
123 | 122 | ad3antlr 767 |
. . . . . . . . . 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))) |
124 | | mzprename 37312 |
. . . . . . . . . 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)))) |
125 | 108, 109,
123, 124 | syl3anc 1326 |
. . . . . . . . 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)))) |
126 | 3, 16, 19 | eldioph4i 37376 |
. . . . . . . . 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)) |
127 | 105, 125,
126 | syl2anc 693 |
. . . . . . . 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)) |
128 | 104, 127 | eqeltrd 2701 |
. . . . . . 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)) |
129 | | eleq2 2690 |
. . . . . . . . 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})) |
130 | 129 | rabbidv 3189 |
. . . . . . . 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}}) |
131 | 130 | eleq1d 2686 |
. . . . . . 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))) |
132 | 128, 131 | syl5ibrcom 237 |
. . . . . 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))) |
133 | 132 | rexlimdva 3031 |
. . . . 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))) |
134 | 133 | expimpd 629 |
. . . 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))) |
135 | 20, 134 | syl5bi 232 |
. . 3
⊢ ((𝑀 ∈ ℕ0
∧ 𝐹:(1...𝑁)⟶(1...𝑀)) → (𝑆 ∈ (Dioph‘𝑁) → {𝑎 ∈ (ℕ0
↑𝑚 (1...𝑀)) ∣ (𝑎 ∘ 𝐹) ∈ 𝑆} ∈ (Dioph‘𝑀))) |
136 | 135 | impcom 446 |
. 2
⊢ ((𝑆 ∈ (Dioph‘𝑁) ∧ (𝑀 ∈ ℕ0 ∧ 𝐹:(1...𝑁)⟶(1...𝑀))) → {𝑎 ∈ (ℕ0
↑𝑚 (1...𝑀)) ∣ (𝑎 ∘ 𝐹) ∈ 𝑆} ∈ (Dioph‘𝑀)) |
137 | 136 | 3impb 1260 |
1
⊢ ((𝑆 ∈ (Dioph‘𝑁) ∧ 𝑀 ∈ ℕ0 ∧ 𝐹:(1...𝑁)⟶(1...𝑀)) → {𝑎 ∈ (ℕ0
↑𝑚 (1...𝑀)) ∣ (𝑎 ∘ 𝐹) ∈ 𝑆} ∈ (Dioph‘𝑀)) |