| Step | Hyp | Ref
| Expression |
| 1 | | zex 12622 |
. . . . . 6
⊢ ℤ
∈ V |
| 2 | | difexg 5329 |
. . . . . 6
⊢ (ℤ
∈ V → (ℤ ∖ ℕ) ∈ V) |
| 3 | 1, 2 | ax-mp 5 |
. . . . 5
⊢ (ℤ
∖ ℕ) ∈ V |
| 4 | | ominf 9294 |
. . . . . 6
⊢ ¬
ω ∈ Fin |
| 5 | | nnuz 12921 |
. . . . . . . . . 10
⊢ ℕ =
(ℤ≥‘1) |
| 6 | | 0p1e1 12388 |
. . . . . . . . . . 11
⊢ (0 + 1) =
1 |
| 7 | 6 | fveq2i 6909 |
. . . . . . . . . 10
⊢
(ℤ≥‘(0 + 1)) =
(ℤ≥‘1) |
| 8 | 5, 7 | eqtr4i 2768 |
. . . . . . . . 9
⊢ ℕ =
(ℤ≥‘(0 + 1)) |
| 9 | 8 | difeq2i 4123 |
. . . . . . . 8
⊢ (ℤ
∖ ℕ) = (ℤ ∖ (ℤ≥‘(0 +
1))) |
| 10 | | 0z 12624 |
. . . . . . . . 9
⊢ 0 ∈
ℤ |
| 11 | | lzenom 42781 |
. . . . . . . . 9
⊢ (0 ∈
ℤ → (ℤ ∖ (ℤ≥‘(0 + 1))) ≈
ω) |
| 12 | 10, 11 | ax-mp 5 |
. . . . . . . 8
⊢ (ℤ
∖ (ℤ≥‘(0 + 1))) ≈
ω |
| 13 | 9, 12 | eqbrtri 5164 |
. . . . . . 7
⊢ (ℤ
∖ ℕ) ≈ ω |
| 14 | | enfi 9227 |
. . . . . . 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 323 |
. . . . 5
⊢ ¬
(ℤ ∖ ℕ) ∈ Fin |
| 17 | | disjdifr 4473 |
. . . . 5
⊢ ((ℤ
∖ ℕ) ∩ ℕ) = ∅ |
| 18 | 3, 16, 17 | eldioph4b 42822 |
. . . 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 484 |
. . . . . . . . . . . 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 7464 |
. . . . . . . . . . . . 13
⊢
(1...𝑁) ∈
V |
| 22 | 21 | mapco2 42726 |
. . . . . . . . . . . 12
⊢ ((𝑎 ∈ (ℕ0
↑m (1...𝑀))
∧ 𝐹:(1...𝑁)⟶(1...𝑀)) → (𝑎 ∘ 𝐹) ∈ (ℕ0
↑m (1...𝑁))) |
| 23 | 19, 20, 22 | syl2anc 584 |
. . . . . . . . . . 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 4161 |
. . . . . . . . . . . . . 14
⊢ (𝑐 = (𝑎 ∘ 𝐹) → (𝑐 ∪ 𝑑) = ((𝑎 ∘ 𝐹) ∪ 𝑑)) |
| 25 | 24 | fveqeq2d 6914 |
. . . . . . . . . . . . 13
⊢ (𝑐 = (𝑎 ∘ 𝐹) → ((𝑏‘(𝑐 ∪ 𝑑)) = 0 ↔ (𝑏‘((𝑎 ∘ 𝐹) ∪ 𝑑)) = 0)) |
| 26 | 25 | rexbidv 3179 |
. . . . . . . . . . . 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 3693 |
. . . . . . . . . . 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 484 |
. . . . . . . . . . . . . . 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 6267 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑎 ∪ 𝑑) ∘ (𝐹 ∪ ( I ↾ (ℤ ∖
ℕ)))) = (((𝑎 ∪
𝑑) ∘ 𝐹) ∪ ((𝑎 ∪ 𝑑) ∘ ( I ↾ (ℤ ∖
ℕ)))) |
| 33 | | coundir 6268 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑎 ∪ 𝑑) ∘ 𝐹) = ((𝑎 ∘ 𝐹) ∪ (𝑑 ∘ 𝐹)) |
| 34 | | elmapi 8889 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑑 ∈ (ℕ0
↑m (ℤ ∖ ℕ)) → 𝑑:(ℤ ∖
ℕ)⟶ℕ0) |
| 35 | 34 | 3ad2ant3 1136 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝐹:(1...𝑁)⟶(1...𝑀) ∧ 𝑎 ∈ (ℕ0
↑m (1...𝑀))
∧ 𝑑 ∈
(ℕ0 ↑m (ℤ ∖ ℕ))) →
𝑑:(ℤ ∖
ℕ)⟶ℕ0) |
| 36 | | simp1 1137 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝐹:(1...𝑁)⟶(1...𝑀) ∧ 𝑎 ∈ (ℕ0
↑m (1...𝑀))
∧ 𝑑 ∈
(ℕ0 ↑m (ℤ ∖ ℕ))) →
𝐹:(1...𝑁)⟶(1...𝑀)) |
| 37 | | incom 4209 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((ℤ
∖ ℕ) ∩ (1...𝑀)) = ((1...𝑀) ∩ (ℤ ∖
ℕ)) |
| 38 | | fz1ssnn 13595 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢
(1...𝑀) ⊆
ℕ |
| 39 | | disjdif 4472 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (ℕ
∩ (ℤ ∖ ℕ)) = ∅ |
| 40 | | ssdisj 4460 |
. . . . . . . . . . . . . . . . . . . . . . . 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 42764 |
. . . . . . . . . . . . . . . . . . . . 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 4168 |
. . . . . . . . . . . . . . . . . . 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 | eqtrid 2789 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝐹:(1...𝑁)⟶(1...𝑀) ∧ 𝑎 ∈ (ℕ0
↑m (1...𝑀))
∧ 𝑑 ∈
(ℕ0 ↑m (ℤ ∖ ℕ))) →
((𝑎 ∪ 𝑑) ∘ 𝐹) = ((𝑎 ∘ 𝐹) ∪ ∅)) |
| 48 | | un0 4394 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑎 ∘ 𝐹) ∪ ∅) = (𝑎 ∘ 𝐹) |
| 49 | 47, 48 | eqtrdi 2793 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝐹:(1...𝑁)⟶(1...𝑀) ∧ 𝑎 ∈ (ℕ0
↑m (1...𝑀))
∧ 𝑑 ∈
(ℕ0 ↑m (ℤ ∖ ℕ))) →
((𝑎 ∪ 𝑑) ∘ 𝐹) = (𝑎 ∘ 𝐹)) |
| 50 | | coundir 6268 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑎 ∪ 𝑑) ∘ ( I ↾ (ℤ ∖
ℕ))) = ((𝑎 ∘ (
I ↾ (ℤ ∖ ℕ))) ∪ (𝑑 ∘ ( I ↾ (ℤ ∖
ℕ)))) |
| 51 | | elmapi 8889 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑎 ∈ (ℕ0
↑m (1...𝑀))
→ 𝑎:(1...𝑀)⟶ℕ0) |
| 52 | 51 | 3ad2ant2 1135 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝐹:(1...𝑁)⟶(1...𝑀) ∧ 𝑎 ∈ (ℕ0
↑m (1...𝑀))
∧ 𝑑 ∈
(ℕ0 ↑m (ℤ ∖ ℕ))) →
𝑎:(1...𝑀)⟶ℕ0) |
| 53 | | f1oi 6886 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ( I
↾ (ℤ ∖ ℕ)):(ℤ ∖ ℕ)–1-1-onto→(ℤ ∖ ℕ) |
| 54 | | f1of 6848 |
. . . . . . . . . . . . . . . . . . . . . . 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 42764 |
. . . . . . . . . . . . . . . . . . . . . 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 6284 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑑 ∘ ( I ↾ (ℤ
∖ ℕ))) = (𝑑
↾ (ℤ ∖ ℕ)) |
| 60 | | ffn 6736 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑑:(ℤ ∖
ℕ)⟶ℕ0 → 𝑑 Fn (ℤ ∖
ℕ)) |
| 61 | | fnresdm 6687 |
. . . . . . . . . . . . . . . . . . . . . . 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 | eqtrid 2789 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑑 ∈ (ℕ0
↑m (ℤ ∖ ℕ)) → (𝑑 ∘ ( I ↾ (ℤ ∖
ℕ))) = 𝑑) |
| 64 | 63 | 3ad2ant3 1136 |
. . . . . . . . . . . . . . . . . . . 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 4169 |
. . . . . . . . . . . . . . . . . . 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 | eqtrid 2789 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝐹:(1...𝑁)⟶(1...𝑀) ∧ 𝑎 ∈ (ℕ0
↑m (1...𝑀))
∧ 𝑑 ∈
(ℕ0 ↑m (ℤ ∖ ℕ))) →
((𝑎 ∪ 𝑑) ∘ ( I ↾ (ℤ
∖ ℕ))) = (∅ ∪ 𝑑)) |
| 67 | | uncom 4158 |
. . . . . . . . . . . . . . . . . . 19
⊢ (∅
∪ 𝑑) = (𝑑 ∪ ∅) |
| 68 | | un0 4394 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑑 ∪ ∅) = 𝑑 |
| 69 | 67, 68 | eqtri 2765 |
. . . . . . . . . . . . . . . . . 18
⊢ (∅
∪ 𝑑) = 𝑑 |
| 70 | 66, 69 | eqtrdi 2793 |
. . . . . . . . . . . . . . . . 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 4169 |
. . . . . . . . . . . . . . . 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 2790 |
. . . . . . . . . . . . . . 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 6910 |
. . . . . . . . . . . . 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 12636 |
. . . . . . . . . . . . . . . . 17
⊢
ℕ0 ⊆ ℤ |
| 76 | | mapss 8929 |
. . . . . . . . . . . . . . . . 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 5994 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑎 ↾ ((1...𝑀) ∩ (ℤ ∖ ℕ))) = (𝑎 ↾
∅) |
| 79 | | res0 6001 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑎 ↾ ∅) =
∅ |
| 80 | 78, 79 | eqtri 2765 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑎 ↾ ((1...𝑀) ∩ (ℤ ∖ ℕ))) =
∅ |
| 81 | 41 | reseq2i 5994 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑑 ↾ ((1...𝑀) ∩ (ℤ ∖ ℕ))) = (𝑑 ↾
∅) |
| 82 | | res0 6001 |
. . . . . . . . . . . . . . . . . . 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 8920 |
. . . . . . . . . . . . . . . . . 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 4158 |
. . . . . . . . . . . . . . . . . . 19
⊢
((1...𝑀) ∪
(ℤ ∖ ℕ)) = ((ℤ ∖ ℕ) ∪ (1...𝑀)) |
| 87 | 86 | oveq2i 7442 |
. . . . . . . . . . . . . . . . . 18
⊢
(ℕ0 ↑m ((1...𝑀) ∪ (ℤ ∖ ℕ))) =
(ℕ0 ↑m ((ℤ ∖ ℕ) ∪
(1...𝑀))) |
| 88 | 85, 87 | eleqtrdi 2851 |
. . . . . . . . . . . . . . . . 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 | sselid 3981 |
. . . . . . . . . . . . . . 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 5868 |
. . . . . . . . . . . . . . . 16
⊢ (𝑒 = (𝑎 ∪ 𝑑) → (𝑒 ∘ (𝐹 ∪ ( I ↾ (ℤ ∖
ℕ)))) = ((𝑎 ∪
𝑑) ∘ (𝐹 ∪ ( I ↾ (ℤ
∖ ℕ))))) |
| 93 | 92 | fveq2d 6910 |
. . . . . . . . . . . . . . 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 6919 |
. . . . . . . . . . . . . . 15
⊢ (𝑏‘((𝑎 ∪ 𝑑) ∘ (𝐹 ∪ ( I ↾ (ℤ ∖
ℕ))))) ∈ V |
| 96 | 93, 94, 95 | fvmpt 7016 |
. . . . . . . . . . . . . 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 3177 |
. . . . . . . . . 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 279 |
. . . . . . . . 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 3443 |
. . . . . . . 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 7464 |
. . . . . . . . . . . 12
⊢
(1...𝑀) ∈
V |
| 105 | 3, 104 | unex 7764 |
. . . . . . . . . . 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 484 |
. . . . . . . . . 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 4209 |
. . . . . . . . . . . . . . 15
⊢ ((ℤ
∖ ℕ) ∩ (1...𝑁)) = ((1...𝑁) ∩ (ℤ ∖
ℕ)) |
| 111 | | fz1ssnn 13595 |
. . . . . . . . . . . . . . . 16
⊢
(1...𝑁) ⊆
ℕ |
| 112 | | ssdisj 4460 |
. . . . . . . . . . . . . . . 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 6770 |
. . . . . . . . . . . . 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 4158 |
. . . . . . . . . . . . 13
⊢ (( I
↾ (ℤ ∖ ℕ)) ∪ 𝐹) = (𝐹 ∪ ( I ↾ (ℤ ∖
ℕ))) |
| 119 | 118 | feq1i 6727 |
. . . . . . . . . . . 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 218 |
. . . . . . . . . . 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 42760 |
. . . . . . . . . 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 42823 |
. . . . . . . . 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 584 |
. . . . . . . 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 2841 |
. . . . . . 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 2830 |
. . . . . . . . 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 3444 |
. . . . . . . 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 2826 |
. . . . . . 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 247 |
. . . . . 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 3155 |
. . . . 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 453 |
. . . 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 | biimtrid 242 |
. . 3
⊢ ((𝑀 ∈ ℕ0
∧ 𝐹:(1...𝑁)⟶(1...𝑀)) → (𝑆 ∈ (Dioph‘𝑁) → {𝑎 ∈ (ℕ0
↑m (1...𝑀))
∣ (𝑎 ∘ 𝐹) ∈ 𝑆} ∈ (Dioph‘𝑀))) |
| 134 | 133 | impcom 407 |
. 2
⊢ ((𝑆 ∈ (Dioph‘𝑁) ∧ (𝑀 ∈ ℕ0 ∧ 𝐹:(1...𝑁)⟶(1...𝑀))) → {𝑎 ∈ (ℕ0
↑m (1...𝑀))
∣ (𝑎 ∘ 𝐹) ∈ 𝑆} ∈ (Dioph‘𝑀)) |
| 135 | 134 | 3impb 1115 |
1
⊢ ((𝑆 ∈ (Dioph‘𝑁) ∧ 𝑀 ∈ ℕ0 ∧ 𝐹:(1...𝑁)⟶(1...𝑀)) → {𝑎 ∈ (ℕ0
↑m (1...𝑀))
∣ (𝑎 ∘ 𝐹) ∈ 𝑆} ∈ (Dioph‘𝑀)) |