Step |
Hyp |
Ref |
Expression |
1 |
|
eldiophelnn0 |
⊢ ( 𝐴 ∈ ( Dioph ‘ 𝑁 ) → 𝑁 ∈ ℕ0 ) |
2 |
|
id |
⊢ ( 𝑁 ∈ ℕ0 → 𝑁 ∈ ℕ0 ) |
3 |
|
zex |
⊢ ℤ ∈ V |
4 |
|
difexg |
⊢ ( ℤ ∈ V → ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ∈ V ) |
5 |
3 4
|
mp1i |
⊢ ( 𝑁 ∈ ℕ0 → ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ∈ V ) |
6 |
|
ominf |
⊢ ¬ ω ∈ Fin |
7 |
|
nn0z |
⊢ ( 𝑁 ∈ ℕ0 → 𝑁 ∈ ℤ ) |
8 |
|
lzenom |
⊢ ( 𝑁 ∈ ℤ → ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ≈ ω ) |
9 |
|
enfi |
⊢ ( ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ≈ ω → ( ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ∈ Fin ↔ ω ∈ Fin ) ) |
10 |
7 8 9
|
3syl |
⊢ ( 𝑁 ∈ ℕ0 → ( ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ∈ Fin ↔ ω ∈ Fin ) ) |
11 |
6 10
|
mtbiri |
⊢ ( 𝑁 ∈ ℕ0 → ¬ ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ∈ Fin ) |
12 |
|
fz1eqin |
⊢ ( 𝑁 ∈ ℕ0 → ( 1 ... 𝑁 ) = ( ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ∩ ℕ ) ) |
13 |
|
inss1 |
⊢ ( ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ∩ ℕ ) ⊆ ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) |
14 |
12 13
|
eqsstrdi |
⊢ ( 𝑁 ∈ ℕ0 → ( 1 ... 𝑁 ) ⊆ ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) |
15 |
|
eldioph2b |
⊢ ( ( ( 𝑁 ∈ ℕ0 ∧ ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ∈ V ) ∧ ( ¬ ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ∈ Fin ∧ ( 1 ... 𝑁 ) ⊆ ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) ) → ( 𝐴 ∈ ( Dioph ‘ 𝑁 ) ↔ ∃ 𝑎 ∈ ( mzPoly ‘ ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) 𝐴 = { 𝑐 ∣ ∃ 𝑑 ∈ ( ℕ0 ↑m ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) ( 𝑐 = ( 𝑑 ↾ ( 1 ... 𝑁 ) ) ∧ ( 𝑎 ‘ 𝑑 ) = 0 ) } ) ) |
16 |
2 5 11 14 15
|
syl22anc |
⊢ ( 𝑁 ∈ ℕ0 → ( 𝐴 ∈ ( Dioph ‘ 𝑁 ) ↔ ∃ 𝑎 ∈ ( mzPoly ‘ ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) 𝐴 = { 𝑐 ∣ ∃ 𝑑 ∈ ( ℕ0 ↑m ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) ( 𝑐 = ( 𝑑 ↾ ( 1 ... 𝑁 ) ) ∧ ( 𝑎 ‘ 𝑑 ) = 0 ) } ) ) |
17 |
|
nnex |
⊢ ℕ ∈ V |
18 |
17
|
a1i |
⊢ ( 𝑁 ∈ ℕ0 → ℕ ∈ V ) |
19 |
|
1z |
⊢ 1 ∈ ℤ |
20 |
|
nnuz |
⊢ ℕ = ( ℤ≥ ‘ 1 ) |
21 |
20
|
uzinf |
⊢ ( 1 ∈ ℤ → ¬ ℕ ∈ Fin ) |
22 |
19 21
|
mp1i |
⊢ ( 𝑁 ∈ ℕ0 → ¬ ℕ ∈ Fin ) |
23 |
|
elfznn |
⊢ ( 𝑎 ∈ ( 1 ... 𝑁 ) → 𝑎 ∈ ℕ ) |
24 |
23
|
ssriv |
⊢ ( 1 ... 𝑁 ) ⊆ ℕ |
25 |
24
|
a1i |
⊢ ( 𝑁 ∈ ℕ0 → ( 1 ... 𝑁 ) ⊆ ℕ ) |
26 |
|
eldioph2b |
⊢ ( ( ( 𝑁 ∈ ℕ0 ∧ ℕ ∈ V ) ∧ ( ¬ ℕ ∈ Fin ∧ ( 1 ... 𝑁 ) ⊆ ℕ ) ) → ( 𝐵 ∈ ( Dioph ‘ 𝑁 ) ↔ ∃ 𝑏 ∈ ( mzPoly ‘ ℕ ) 𝐵 = { 𝑐 ∣ ∃ 𝑒 ∈ ( ℕ0 ↑m ℕ ) ( 𝑐 = ( 𝑒 ↾ ( 1 ... 𝑁 ) ) ∧ ( 𝑏 ‘ 𝑒 ) = 0 ) } ) ) |
27 |
2 18 22 25 26
|
syl22anc |
⊢ ( 𝑁 ∈ ℕ0 → ( 𝐵 ∈ ( Dioph ‘ 𝑁 ) ↔ ∃ 𝑏 ∈ ( mzPoly ‘ ℕ ) 𝐵 = { 𝑐 ∣ ∃ 𝑒 ∈ ( ℕ0 ↑m ℕ ) ( 𝑐 = ( 𝑒 ↾ ( 1 ... 𝑁 ) ) ∧ ( 𝑏 ‘ 𝑒 ) = 0 ) } ) ) |
28 |
16 27
|
anbi12d |
⊢ ( 𝑁 ∈ ℕ0 → ( ( 𝐴 ∈ ( Dioph ‘ 𝑁 ) ∧ 𝐵 ∈ ( Dioph ‘ 𝑁 ) ) ↔ ( ∃ 𝑎 ∈ ( mzPoly ‘ ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) 𝐴 = { 𝑐 ∣ ∃ 𝑑 ∈ ( ℕ0 ↑m ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) ( 𝑐 = ( 𝑑 ↾ ( 1 ... 𝑁 ) ) ∧ ( 𝑎 ‘ 𝑑 ) = 0 ) } ∧ ∃ 𝑏 ∈ ( mzPoly ‘ ℕ ) 𝐵 = { 𝑐 ∣ ∃ 𝑒 ∈ ( ℕ0 ↑m ℕ ) ( 𝑐 = ( 𝑒 ↾ ( 1 ... 𝑁 ) ) ∧ ( 𝑏 ‘ 𝑒 ) = 0 ) } ) ) ) |
29 |
|
reeanv |
⊢ ( ∃ 𝑎 ∈ ( mzPoly ‘ ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) ∃ 𝑏 ∈ ( mzPoly ‘ ℕ ) ( 𝐴 = { 𝑐 ∣ ∃ 𝑑 ∈ ( ℕ0 ↑m ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) ( 𝑐 = ( 𝑑 ↾ ( 1 ... 𝑁 ) ) ∧ ( 𝑎 ‘ 𝑑 ) = 0 ) } ∧ 𝐵 = { 𝑐 ∣ ∃ 𝑒 ∈ ( ℕ0 ↑m ℕ ) ( 𝑐 = ( 𝑒 ↾ ( 1 ... 𝑁 ) ) ∧ ( 𝑏 ‘ 𝑒 ) = 0 ) } ) ↔ ( ∃ 𝑎 ∈ ( mzPoly ‘ ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) 𝐴 = { 𝑐 ∣ ∃ 𝑑 ∈ ( ℕ0 ↑m ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) ( 𝑐 = ( 𝑑 ↾ ( 1 ... 𝑁 ) ) ∧ ( 𝑎 ‘ 𝑑 ) = 0 ) } ∧ ∃ 𝑏 ∈ ( mzPoly ‘ ℕ ) 𝐵 = { 𝑐 ∣ ∃ 𝑒 ∈ ( ℕ0 ↑m ℕ ) ( 𝑐 = ( 𝑒 ↾ ( 1 ... 𝑁 ) ) ∧ ( 𝑏 ‘ 𝑒 ) = 0 ) } ) ) |
30 |
|
inab |
⊢ ( { 𝑐 ∣ ∃ 𝑑 ∈ ( ℕ0 ↑m ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) ( 𝑐 = ( 𝑑 ↾ ( 1 ... 𝑁 ) ) ∧ ( 𝑎 ‘ 𝑑 ) = 0 ) } ∩ { 𝑐 ∣ ∃ 𝑒 ∈ ( ℕ0 ↑m ℕ ) ( 𝑐 = ( 𝑒 ↾ ( 1 ... 𝑁 ) ) ∧ ( 𝑏 ‘ 𝑒 ) = 0 ) } ) = { 𝑐 ∣ ( ∃ 𝑑 ∈ ( ℕ0 ↑m ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) ( 𝑐 = ( 𝑑 ↾ ( 1 ... 𝑁 ) ) ∧ ( 𝑎 ‘ 𝑑 ) = 0 ) ∧ ∃ 𝑒 ∈ ( ℕ0 ↑m ℕ ) ( 𝑐 = ( 𝑒 ↾ ( 1 ... 𝑁 ) ) ∧ ( 𝑏 ‘ 𝑒 ) = 0 ) ) } |
31 |
|
reeanv |
⊢ ( ∃ 𝑑 ∈ ( ℕ0 ↑m ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) ∃ 𝑒 ∈ ( ℕ0 ↑m ℕ ) ( ( 𝑐 = ( 𝑑 ↾ ( 1 ... 𝑁 ) ) ∧ ( 𝑎 ‘ 𝑑 ) = 0 ) ∧ ( 𝑐 = ( 𝑒 ↾ ( 1 ... 𝑁 ) ) ∧ ( 𝑏 ‘ 𝑒 ) = 0 ) ) ↔ ( ∃ 𝑑 ∈ ( ℕ0 ↑m ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) ( 𝑐 = ( 𝑑 ↾ ( 1 ... 𝑁 ) ) ∧ ( 𝑎 ‘ 𝑑 ) = 0 ) ∧ ∃ 𝑒 ∈ ( ℕ0 ↑m ℕ ) ( 𝑐 = ( 𝑒 ↾ ( 1 ... 𝑁 ) ) ∧ ( 𝑏 ‘ 𝑒 ) = 0 ) ) ) |
32 |
|
simplrl |
⊢ ( ( ( ( 𝑁 ∈ ℕ0 ∧ ( 𝑎 ∈ ( mzPoly ‘ ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) ∧ 𝑏 ∈ ( mzPoly ‘ ℕ ) ) ) ∧ ( 𝑑 ∈ ( ℕ0 ↑m ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) ∧ 𝑒 ∈ ( ℕ0 ↑m ℕ ) ) ) ∧ ( ( 𝑐 = ( 𝑑 ↾ ( 1 ... 𝑁 ) ) ∧ ( 𝑎 ‘ 𝑑 ) = 0 ) ∧ ( 𝑐 = ( 𝑒 ↾ ( 1 ... 𝑁 ) ) ∧ ( 𝑏 ‘ 𝑒 ) = 0 ) ) ) → 𝑑 ∈ ( ℕ0 ↑m ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) ) |
33 |
|
simplrr |
⊢ ( ( ( ( 𝑁 ∈ ℕ0 ∧ ( 𝑎 ∈ ( mzPoly ‘ ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) ∧ 𝑏 ∈ ( mzPoly ‘ ℕ ) ) ) ∧ ( 𝑑 ∈ ( ℕ0 ↑m ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) ∧ 𝑒 ∈ ( ℕ0 ↑m ℕ ) ) ) ∧ ( ( 𝑐 = ( 𝑑 ↾ ( 1 ... 𝑁 ) ) ∧ ( 𝑎 ‘ 𝑑 ) = 0 ) ∧ ( 𝑐 = ( 𝑒 ↾ ( 1 ... 𝑁 ) ) ∧ ( 𝑏 ‘ 𝑒 ) = 0 ) ) ) → 𝑒 ∈ ( ℕ0 ↑m ℕ ) ) |
34 |
12
|
eqcomd |
⊢ ( 𝑁 ∈ ℕ0 → ( ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ∩ ℕ ) = ( 1 ... 𝑁 ) ) |
35 |
34
|
reseq2d |
⊢ ( 𝑁 ∈ ℕ0 → ( 𝑑 ↾ ( ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ∩ ℕ ) ) = ( 𝑑 ↾ ( 1 ... 𝑁 ) ) ) |
36 |
35
|
ad3antrrr |
⊢ ( ( ( ( 𝑁 ∈ ℕ0 ∧ ( 𝑎 ∈ ( mzPoly ‘ ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) ∧ 𝑏 ∈ ( mzPoly ‘ ℕ ) ) ) ∧ ( 𝑑 ∈ ( ℕ0 ↑m ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) ∧ 𝑒 ∈ ( ℕ0 ↑m ℕ ) ) ) ∧ ( ( 𝑐 = ( 𝑑 ↾ ( 1 ... 𝑁 ) ) ∧ ( 𝑎 ‘ 𝑑 ) = 0 ) ∧ ( 𝑐 = ( 𝑒 ↾ ( 1 ... 𝑁 ) ) ∧ ( 𝑏 ‘ 𝑒 ) = 0 ) ) ) → ( 𝑑 ↾ ( ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ∩ ℕ ) ) = ( 𝑑 ↾ ( 1 ... 𝑁 ) ) ) |
37 |
34
|
reseq2d |
⊢ ( 𝑁 ∈ ℕ0 → ( 𝑒 ↾ ( ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ∩ ℕ ) ) = ( 𝑒 ↾ ( 1 ... 𝑁 ) ) ) |
38 |
37
|
ad3antrrr |
⊢ ( ( ( ( 𝑁 ∈ ℕ0 ∧ ( 𝑎 ∈ ( mzPoly ‘ ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) ∧ 𝑏 ∈ ( mzPoly ‘ ℕ ) ) ) ∧ ( 𝑑 ∈ ( ℕ0 ↑m ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) ∧ 𝑒 ∈ ( ℕ0 ↑m ℕ ) ) ) ∧ ( ( 𝑐 = ( 𝑑 ↾ ( 1 ... 𝑁 ) ) ∧ ( 𝑎 ‘ 𝑑 ) = 0 ) ∧ ( 𝑐 = ( 𝑒 ↾ ( 1 ... 𝑁 ) ) ∧ ( 𝑏 ‘ 𝑒 ) = 0 ) ) ) → ( 𝑒 ↾ ( ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ∩ ℕ ) ) = ( 𝑒 ↾ ( 1 ... 𝑁 ) ) ) |
39 |
|
simprrl |
⊢ ( ( ( ( 𝑁 ∈ ℕ0 ∧ ( 𝑎 ∈ ( mzPoly ‘ ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) ∧ 𝑏 ∈ ( mzPoly ‘ ℕ ) ) ) ∧ ( 𝑑 ∈ ( ℕ0 ↑m ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) ∧ 𝑒 ∈ ( ℕ0 ↑m ℕ ) ) ) ∧ ( ( 𝑐 = ( 𝑑 ↾ ( 1 ... 𝑁 ) ) ∧ ( 𝑎 ‘ 𝑑 ) = 0 ) ∧ ( 𝑐 = ( 𝑒 ↾ ( 1 ... 𝑁 ) ) ∧ ( 𝑏 ‘ 𝑒 ) = 0 ) ) ) → 𝑐 = ( 𝑒 ↾ ( 1 ... 𝑁 ) ) ) |
40 |
|
simprll |
⊢ ( ( ( ( 𝑁 ∈ ℕ0 ∧ ( 𝑎 ∈ ( mzPoly ‘ ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) ∧ 𝑏 ∈ ( mzPoly ‘ ℕ ) ) ) ∧ ( 𝑑 ∈ ( ℕ0 ↑m ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) ∧ 𝑒 ∈ ( ℕ0 ↑m ℕ ) ) ) ∧ ( ( 𝑐 = ( 𝑑 ↾ ( 1 ... 𝑁 ) ) ∧ ( 𝑎 ‘ 𝑑 ) = 0 ) ∧ ( 𝑐 = ( 𝑒 ↾ ( 1 ... 𝑁 ) ) ∧ ( 𝑏 ‘ 𝑒 ) = 0 ) ) ) → 𝑐 = ( 𝑑 ↾ ( 1 ... 𝑁 ) ) ) |
41 |
38 39 40
|
3eqtr2d |
⊢ ( ( ( ( 𝑁 ∈ ℕ0 ∧ ( 𝑎 ∈ ( mzPoly ‘ ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) ∧ 𝑏 ∈ ( mzPoly ‘ ℕ ) ) ) ∧ ( 𝑑 ∈ ( ℕ0 ↑m ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) ∧ 𝑒 ∈ ( ℕ0 ↑m ℕ ) ) ) ∧ ( ( 𝑐 = ( 𝑑 ↾ ( 1 ... 𝑁 ) ) ∧ ( 𝑎 ‘ 𝑑 ) = 0 ) ∧ ( 𝑐 = ( 𝑒 ↾ ( 1 ... 𝑁 ) ) ∧ ( 𝑏 ‘ 𝑒 ) = 0 ) ) ) → ( 𝑒 ↾ ( ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ∩ ℕ ) ) = ( 𝑑 ↾ ( 1 ... 𝑁 ) ) ) |
42 |
36 41
|
eqtr4d |
⊢ ( ( ( ( 𝑁 ∈ ℕ0 ∧ ( 𝑎 ∈ ( mzPoly ‘ ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) ∧ 𝑏 ∈ ( mzPoly ‘ ℕ ) ) ) ∧ ( 𝑑 ∈ ( ℕ0 ↑m ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) ∧ 𝑒 ∈ ( ℕ0 ↑m ℕ ) ) ) ∧ ( ( 𝑐 = ( 𝑑 ↾ ( 1 ... 𝑁 ) ) ∧ ( 𝑎 ‘ 𝑑 ) = 0 ) ∧ ( 𝑐 = ( 𝑒 ↾ ( 1 ... 𝑁 ) ) ∧ ( 𝑏 ‘ 𝑒 ) = 0 ) ) ) → ( 𝑑 ↾ ( ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ∩ ℕ ) ) = ( 𝑒 ↾ ( ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ∩ ℕ ) ) ) |
43 |
|
elmapresaun |
⊢ ( ( 𝑑 ∈ ( ℕ0 ↑m ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) ∧ 𝑒 ∈ ( ℕ0 ↑m ℕ ) ∧ ( 𝑑 ↾ ( ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ∩ ℕ ) ) = ( 𝑒 ↾ ( ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ∩ ℕ ) ) ) → ( 𝑑 ∪ 𝑒 ) ∈ ( ℕ0 ↑m ( ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ∪ ℕ ) ) ) |
44 |
32 33 42 43
|
syl3anc |
⊢ ( ( ( ( 𝑁 ∈ ℕ0 ∧ ( 𝑎 ∈ ( mzPoly ‘ ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) ∧ 𝑏 ∈ ( mzPoly ‘ ℕ ) ) ) ∧ ( 𝑑 ∈ ( ℕ0 ↑m ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) ∧ 𝑒 ∈ ( ℕ0 ↑m ℕ ) ) ) ∧ ( ( 𝑐 = ( 𝑑 ↾ ( 1 ... 𝑁 ) ) ∧ ( 𝑎 ‘ 𝑑 ) = 0 ) ∧ ( 𝑐 = ( 𝑒 ↾ ( 1 ... 𝑁 ) ) ∧ ( 𝑏 ‘ 𝑒 ) = 0 ) ) ) → ( 𝑑 ∪ 𝑒 ) ∈ ( ℕ0 ↑m ( ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ∪ ℕ ) ) ) |
45 |
20
|
uneq2i |
⊢ ( ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ∪ ℕ ) = ( ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ∪ ( ℤ≥ ‘ 1 ) ) |
46 |
19
|
a1i |
⊢ ( 𝑁 ∈ ℕ0 → 1 ∈ ℤ ) |
47 |
|
nn0p1nn |
⊢ ( 𝑁 ∈ ℕ0 → ( 𝑁 + 1 ) ∈ ℕ ) |
48 |
47
|
nnge1d |
⊢ ( 𝑁 ∈ ℕ0 → 1 ≤ ( 𝑁 + 1 ) ) |
49 |
|
lzunuz |
⊢ ( ( 𝑁 ∈ ℤ ∧ 1 ∈ ℤ ∧ 1 ≤ ( 𝑁 + 1 ) ) → ( ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ∪ ( ℤ≥ ‘ 1 ) ) = ℤ ) |
50 |
7 46 48 49
|
syl3anc |
⊢ ( 𝑁 ∈ ℕ0 → ( ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ∪ ( ℤ≥ ‘ 1 ) ) = ℤ ) |
51 |
45 50
|
syl5eq |
⊢ ( 𝑁 ∈ ℕ0 → ( ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ∪ ℕ ) = ℤ ) |
52 |
51
|
oveq2d |
⊢ ( 𝑁 ∈ ℕ0 → ( ℕ0 ↑m ( ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ∪ ℕ ) ) = ( ℕ0 ↑m ℤ ) ) |
53 |
52
|
ad3antrrr |
⊢ ( ( ( ( 𝑁 ∈ ℕ0 ∧ ( 𝑎 ∈ ( mzPoly ‘ ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) ∧ 𝑏 ∈ ( mzPoly ‘ ℕ ) ) ) ∧ ( 𝑑 ∈ ( ℕ0 ↑m ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) ∧ 𝑒 ∈ ( ℕ0 ↑m ℕ ) ) ) ∧ ( ( 𝑐 = ( 𝑑 ↾ ( 1 ... 𝑁 ) ) ∧ ( 𝑎 ‘ 𝑑 ) = 0 ) ∧ ( 𝑐 = ( 𝑒 ↾ ( 1 ... 𝑁 ) ) ∧ ( 𝑏 ‘ 𝑒 ) = 0 ) ) ) → ( ℕ0 ↑m ( ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ∪ ℕ ) ) = ( ℕ0 ↑m ℤ ) ) |
54 |
44 53
|
eleqtrd |
⊢ ( ( ( ( 𝑁 ∈ ℕ0 ∧ ( 𝑎 ∈ ( mzPoly ‘ ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) ∧ 𝑏 ∈ ( mzPoly ‘ ℕ ) ) ) ∧ ( 𝑑 ∈ ( ℕ0 ↑m ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) ∧ 𝑒 ∈ ( ℕ0 ↑m ℕ ) ) ) ∧ ( ( 𝑐 = ( 𝑑 ↾ ( 1 ... 𝑁 ) ) ∧ ( 𝑎 ‘ 𝑑 ) = 0 ) ∧ ( 𝑐 = ( 𝑒 ↾ ( 1 ... 𝑁 ) ) ∧ ( 𝑏 ‘ 𝑒 ) = 0 ) ) ) → ( 𝑑 ∪ 𝑒 ) ∈ ( ℕ0 ↑m ℤ ) ) |
55 |
|
unidm |
⊢ ( 𝑐 ∪ 𝑐 ) = 𝑐 |
56 |
40 39
|
uneq12d |
⊢ ( ( ( ( 𝑁 ∈ ℕ0 ∧ ( 𝑎 ∈ ( mzPoly ‘ ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) ∧ 𝑏 ∈ ( mzPoly ‘ ℕ ) ) ) ∧ ( 𝑑 ∈ ( ℕ0 ↑m ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) ∧ 𝑒 ∈ ( ℕ0 ↑m ℕ ) ) ) ∧ ( ( 𝑐 = ( 𝑑 ↾ ( 1 ... 𝑁 ) ) ∧ ( 𝑎 ‘ 𝑑 ) = 0 ) ∧ ( 𝑐 = ( 𝑒 ↾ ( 1 ... 𝑁 ) ) ∧ ( 𝑏 ‘ 𝑒 ) = 0 ) ) ) → ( 𝑐 ∪ 𝑐 ) = ( ( 𝑑 ↾ ( 1 ... 𝑁 ) ) ∪ ( 𝑒 ↾ ( 1 ... 𝑁 ) ) ) ) |
57 |
55 56
|
eqtr3id |
⊢ ( ( ( ( 𝑁 ∈ ℕ0 ∧ ( 𝑎 ∈ ( mzPoly ‘ ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) ∧ 𝑏 ∈ ( mzPoly ‘ ℕ ) ) ) ∧ ( 𝑑 ∈ ( ℕ0 ↑m ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) ∧ 𝑒 ∈ ( ℕ0 ↑m ℕ ) ) ) ∧ ( ( 𝑐 = ( 𝑑 ↾ ( 1 ... 𝑁 ) ) ∧ ( 𝑎 ‘ 𝑑 ) = 0 ) ∧ ( 𝑐 = ( 𝑒 ↾ ( 1 ... 𝑁 ) ) ∧ ( 𝑏 ‘ 𝑒 ) = 0 ) ) ) → 𝑐 = ( ( 𝑑 ↾ ( 1 ... 𝑁 ) ) ∪ ( 𝑒 ↾ ( 1 ... 𝑁 ) ) ) ) |
58 |
|
resundir |
⊢ ( ( 𝑑 ∪ 𝑒 ) ↾ ( 1 ... 𝑁 ) ) = ( ( 𝑑 ↾ ( 1 ... 𝑁 ) ) ∪ ( 𝑒 ↾ ( 1 ... 𝑁 ) ) ) |
59 |
57 58
|
eqtr4di |
⊢ ( ( ( ( 𝑁 ∈ ℕ0 ∧ ( 𝑎 ∈ ( mzPoly ‘ ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) ∧ 𝑏 ∈ ( mzPoly ‘ ℕ ) ) ) ∧ ( 𝑑 ∈ ( ℕ0 ↑m ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) ∧ 𝑒 ∈ ( ℕ0 ↑m ℕ ) ) ) ∧ ( ( 𝑐 = ( 𝑑 ↾ ( 1 ... 𝑁 ) ) ∧ ( 𝑎 ‘ 𝑑 ) = 0 ) ∧ ( 𝑐 = ( 𝑒 ↾ ( 1 ... 𝑁 ) ) ∧ ( 𝑏 ‘ 𝑒 ) = 0 ) ) ) → 𝑐 = ( ( 𝑑 ∪ 𝑒 ) ↾ ( 1 ... 𝑁 ) ) ) |
60 |
|
uncom |
⊢ ( 𝑑 ∪ 𝑒 ) = ( 𝑒 ∪ 𝑑 ) |
61 |
60
|
reseq1i |
⊢ ( ( 𝑑 ∪ 𝑒 ) ↾ ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) = ( ( 𝑒 ∪ 𝑑 ) ↾ ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) |
62 |
|
incom |
⊢ ( ℕ ∩ ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) = ( ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ∩ ℕ ) |
63 |
62 34
|
syl5eq |
⊢ ( 𝑁 ∈ ℕ0 → ( ℕ ∩ ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) = ( 1 ... 𝑁 ) ) |
64 |
63
|
reseq2d |
⊢ ( 𝑁 ∈ ℕ0 → ( 𝑒 ↾ ( ℕ ∩ ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) ) = ( 𝑒 ↾ ( 1 ... 𝑁 ) ) ) |
65 |
64
|
ad3antrrr |
⊢ ( ( ( ( 𝑁 ∈ ℕ0 ∧ ( 𝑎 ∈ ( mzPoly ‘ ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) ∧ 𝑏 ∈ ( mzPoly ‘ ℕ ) ) ) ∧ ( 𝑑 ∈ ( ℕ0 ↑m ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) ∧ 𝑒 ∈ ( ℕ0 ↑m ℕ ) ) ) ∧ ( ( 𝑐 = ( 𝑑 ↾ ( 1 ... 𝑁 ) ) ∧ ( 𝑎 ‘ 𝑑 ) = 0 ) ∧ ( 𝑐 = ( 𝑒 ↾ ( 1 ... 𝑁 ) ) ∧ ( 𝑏 ‘ 𝑒 ) = 0 ) ) ) → ( 𝑒 ↾ ( ℕ ∩ ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) ) = ( 𝑒 ↾ ( 1 ... 𝑁 ) ) ) |
66 |
63
|
reseq2d |
⊢ ( 𝑁 ∈ ℕ0 → ( 𝑑 ↾ ( ℕ ∩ ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) ) = ( 𝑑 ↾ ( 1 ... 𝑁 ) ) ) |
67 |
66
|
ad3antrrr |
⊢ ( ( ( ( 𝑁 ∈ ℕ0 ∧ ( 𝑎 ∈ ( mzPoly ‘ ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) ∧ 𝑏 ∈ ( mzPoly ‘ ℕ ) ) ) ∧ ( 𝑑 ∈ ( ℕ0 ↑m ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) ∧ 𝑒 ∈ ( ℕ0 ↑m ℕ ) ) ) ∧ ( ( 𝑐 = ( 𝑑 ↾ ( 1 ... 𝑁 ) ) ∧ ( 𝑎 ‘ 𝑑 ) = 0 ) ∧ ( 𝑐 = ( 𝑒 ↾ ( 1 ... 𝑁 ) ) ∧ ( 𝑏 ‘ 𝑒 ) = 0 ) ) ) → ( 𝑑 ↾ ( ℕ ∩ ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) ) = ( 𝑑 ↾ ( 1 ... 𝑁 ) ) ) |
68 |
67 40 39
|
3eqtr2d |
⊢ ( ( ( ( 𝑁 ∈ ℕ0 ∧ ( 𝑎 ∈ ( mzPoly ‘ ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) ∧ 𝑏 ∈ ( mzPoly ‘ ℕ ) ) ) ∧ ( 𝑑 ∈ ( ℕ0 ↑m ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) ∧ 𝑒 ∈ ( ℕ0 ↑m ℕ ) ) ) ∧ ( ( 𝑐 = ( 𝑑 ↾ ( 1 ... 𝑁 ) ) ∧ ( 𝑎 ‘ 𝑑 ) = 0 ) ∧ ( 𝑐 = ( 𝑒 ↾ ( 1 ... 𝑁 ) ) ∧ ( 𝑏 ‘ 𝑒 ) = 0 ) ) ) → ( 𝑑 ↾ ( ℕ ∩ ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) ) = ( 𝑒 ↾ ( 1 ... 𝑁 ) ) ) |
69 |
65 68
|
eqtr4d |
⊢ ( ( ( ( 𝑁 ∈ ℕ0 ∧ ( 𝑎 ∈ ( mzPoly ‘ ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) ∧ 𝑏 ∈ ( mzPoly ‘ ℕ ) ) ) ∧ ( 𝑑 ∈ ( ℕ0 ↑m ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) ∧ 𝑒 ∈ ( ℕ0 ↑m ℕ ) ) ) ∧ ( ( 𝑐 = ( 𝑑 ↾ ( 1 ... 𝑁 ) ) ∧ ( 𝑎 ‘ 𝑑 ) = 0 ) ∧ ( 𝑐 = ( 𝑒 ↾ ( 1 ... 𝑁 ) ) ∧ ( 𝑏 ‘ 𝑒 ) = 0 ) ) ) → ( 𝑒 ↾ ( ℕ ∩ ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) ) = ( 𝑑 ↾ ( ℕ ∩ ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) ) ) |
70 |
|
elmapresaunres2 |
⊢ ( ( 𝑒 ∈ ( ℕ0 ↑m ℕ ) ∧ 𝑑 ∈ ( ℕ0 ↑m ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) ∧ ( 𝑒 ↾ ( ℕ ∩ ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) ) = ( 𝑑 ↾ ( ℕ ∩ ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) ) ) → ( ( 𝑒 ∪ 𝑑 ) ↾ ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) = 𝑑 ) |
71 |
33 32 69 70
|
syl3anc |
⊢ ( ( ( ( 𝑁 ∈ ℕ0 ∧ ( 𝑎 ∈ ( mzPoly ‘ ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) ∧ 𝑏 ∈ ( mzPoly ‘ ℕ ) ) ) ∧ ( 𝑑 ∈ ( ℕ0 ↑m ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) ∧ 𝑒 ∈ ( ℕ0 ↑m ℕ ) ) ) ∧ ( ( 𝑐 = ( 𝑑 ↾ ( 1 ... 𝑁 ) ) ∧ ( 𝑎 ‘ 𝑑 ) = 0 ) ∧ ( 𝑐 = ( 𝑒 ↾ ( 1 ... 𝑁 ) ) ∧ ( 𝑏 ‘ 𝑒 ) = 0 ) ) ) → ( ( 𝑒 ∪ 𝑑 ) ↾ ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) = 𝑑 ) |
72 |
61 71
|
syl5eq |
⊢ ( ( ( ( 𝑁 ∈ ℕ0 ∧ ( 𝑎 ∈ ( mzPoly ‘ ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) ∧ 𝑏 ∈ ( mzPoly ‘ ℕ ) ) ) ∧ ( 𝑑 ∈ ( ℕ0 ↑m ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) ∧ 𝑒 ∈ ( ℕ0 ↑m ℕ ) ) ) ∧ ( ( 𝑐 = ( 𝑑 ↾ ( 1 ... 𝑁 ) ) ∧ ( 𝑎 ‘ 𝑑 ) = 0 ) ∧ ( 𝑐 = ( 𝑒 ↾ ( 1 ... 𝑁 ) ) ∧ ( 𝑏 ‘ 𝑒 ) = 0 ) ) ) → ( ( 𝑑 ∪ 𝑒 ) ↾ ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) = 𝑑 ) |
73 |
72
|
fveq2d |
⊢ ( ( ( ( 𝑁 ∈ ℕ0 ∧ ( 𝑎 ∈ ( mzPoly ‘ ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) ∧ 𝑏 ∈ ( mzPoly ‘ ℕ ) ) ) ∧ ( 𝑑 ∈ ( ℕ0 ↑m ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) ∧ 𝑒 ∈ ( ℕ0 ↑m ℕ ) ) ) ∧ ( ( 𝑐 = ( 𝑑 ↾ ( 1 ... 𝑁 ) ) ∧ ( 𝑎 ‘ 𝑑 ) = 0 ) ∧ ( 𝑐 = ( 𝑒 ↾ ( 1 ... 𝑁 ) ) ∧ ( 𝑏 ‘ 𝑒 ) = 0 ) ) ) → ( 𝑎 ‘ ( ( 𝑑 ∪ 𝑒 ) ↾ ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) ) = ( 𝑎 ‘ 𝑑 ) ) |
74 |
|
simprlr |
⊢ ( ( ( ( 𝑁 ∈ ℕ0 ∧ ( 𝑎 ∈ ( mzPoly ‘ ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) ∧ 𝑏 ∈ ( mzPoly ‘ ℕ ) ) ) ∧ ( 𝑑 ∈ ( ℕ0 ↑m ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) ∧ 𝑒 ∈ ( ℕ0 ↑m ℕ ) ) ) ∧ ( ( 𝑐 = ( 𝑑 ↾ ( 1 ... 𝑁 ) ) ∧ ( 𝑎 ‘ 𝑑 ) = 0 ) ∧ ( 𝑐 = ( 𝑒 ↾ ( 1 ... 𝑁 ) ) ∧ ( 𝑏 ‘ 𝑒 ) = 0 ) ) ) → ( 𝑎 ‘ 𝑑 ) = 0 ) |
75 |
73 74
|
eqtrd |
⊢ ( ( ( ( 𝑁 ∈ ℕ0 ∧ ( 𝑎 ∈ ( mzPoly ‘ ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) ∧ 𝑏 ∈ ( mzPoly ‘ ℕ ) ) ) ∧ ( 𝑑 ∈ ( ℕ0 ↑m ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) ∧ 𝑒 ∈ ( ℕ0 ↑m ℕ ) ) ) ∧ ( ( 𝑐 = ( 𝑑 ↾ ( 1 ... 𝑁 ) ) ∧ ( 𝑎 ‘ 𝑑 ) = 0 ) ∧ ( 𝑐 = ( 𝑒 ↾ ( 1 ... 𝑁 ) ) ∧ ( 𝑏 ‘ 𝑒 ) = 0 ) ) ) → ( 𝑎 ‘ ( ( 𝑑 ∪ 𝑒 ) ↾ ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) ) = 0 ) |
76 |
|
elmapresaunres2 |
⊢ ( ( 𝑑 ∈ ( ℕ0 ↑m ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) ∧ 𝑒 ∈ ( ℕ0 ↑m ℕ ) ∧ ( 𝑑 ↾ ( ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ∩ ℕ ) ) = ( 𝑒 ↾ ( ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ∩ ℕ ) ) ) → ( ( 𝑑 ∪ 𝑒 ) ↾ ℕ ) = 𝑒 ) |
77 |
32 33 42 76
|
syl3anc |
⊢ ( ( ( ( 𝑁 ∈ ℕ0 ∧ ( 𝑎 ∈ ( mzPoly ‘ ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) ∧ 𝑏 ∈ ( mzPoly ‘ ℕ ) ) ) ∧ ( 𝑑 ∈ ( ℕ0 ↑m ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) ∧ 𝑒 ∈ ( ℕ0 ↑m ℕ ) ) ) ∧ ( ( 𝑐 = ( 𝑑 ↾ ( 1 ... 𝑁 ) ) ∧ ( 𝑎 ‘ 𝑑 ) = 0 ) ∧ ( 𝑐 = ( 𝑒 ↾ ( 1 ... 𝑁 ) ) ∧ ( 𝑏 ‘ 𝑒 ) = 0 ) ) ) → ( ( 𝑑 ∪ 𝑒 ) ↾ ℕ ) = 𝑒 ) |
78 |
77
|
fveq2d |
⊢ ( ( ( ( 𝑁 ∈ ℕ0 ∧ ( 𝑎 ∈ ( mzPoly ‘ ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) ∧ 𝑏 ∈ ( mzPoly ‘ ℕ ) ) ) ∧ ( 𝑑 ∈ ( ℕ0 ↑m ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) ∧ 𝑒 ∈ ( ℕ0 ↑m ℕ ) ) ) ∧ ( ( 𝑐 = ( 𝑑 ↾ ( 1 ... 𝑁 ) ) ∧ ( 𝑎 ‘ 𝑑 ) = 0 ) ∧ ( 𝑐 = ( 𝑒 ↾ ( 1 ... 𝑁 ) ) ∧ ( 𝑏 ‘ 𝑒 ) = 0 ) ) ) → ( 𝑏 ‘ ( ( 𝑑 ∪ 𝑒 ) ↾ ℕ ) ) = ( 𝑏 ‘ 𝑒 ) ) |
79 |
|
simprrr |
⊢ ( ( ( ( 𝑁 ∈ ℕ0 ∧ ( 𝑎 ∈ ( mzPoly ‘ ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) ∧ 𝑏 ∈ ( mzPoly ‘ ℕ ) ) ) ∧ ( 𝑑 ∈ ( ℕ0 ↑m ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) ∧ 𝑒 ∈ ( ℕ0 ↑m ℕ ) ) ) ∧ ( ( 𝑐 = ( 𝑑 ↾ ( 1 ... 𝑁 ) ) ∧ ( 𝑎 ‘ 𝑑 ) = 0 ) ∧ ( 𝑐 = ( 𝑒 ↾ ( 1 ... 𝑁 ) ) ∧ ( 𝑏 ‘ 𝑒 ) = 0 ) ) ) → ( 𝑏 ‘ 𝑒 ) = 0 ) |
80 |
78 79
|
eqtrd |
⊢ ( ( ( ( 𝑁 ∈ ℕ0 ∧ ( 𝑎 ∈ ( mzPoly ‘ ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) ∧ 𝑏 ∈ ( mzPoly ‘ ℕ ) ) ) ∧ ( 𝑑 ∈ ( ℕ0 ↑m ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) ∧ 𝑒 ∈ ( ℕ0 ↑m ℕ ) ) ) ∧ ( ( 𝑐 = ( 𝑑 ↾ ( 1 ... 𝑁 ) ) ∧ ( 𝑎 ‘ 𝑑 ) = 0 ) ∧ ( 𝑐 = ( 𝑒 ↾ ( 1 ... 𝑁 ) ) ∧ ( 𝑏 ‘ 𝑒 ) = 0 ) ) ) → ( 𝑏 ‘ ( ( 𝑑 ∪ 𝑒 ) ↾ ℕ ) ) = 0 ) |
81 |
59 75 80
|
jca32 |
⊢ ( ( ( ( 𝑁 ∈ ℕ0 ∧ ( 𝑎 ∈ ( mzPoly ‘ ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) ∧ 𝑏 ∈ ( mzPoly ‘ ℕ ) ) ) ∧ ( 𝑑 ∈ ( ℕ0 ↑m ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) ∧ 𝑒 ∈ ( ℕ0 ↑m ℕ ) ) ) ∧ ( ( 𝑐 = ( 𝑑 ↾ ( 1 ... 𝑁 ) ) ∧ ( 𝑎 ‘ 𝑑 ) = 0 ) ∧ ( 𝑐 = ( 𝑒 ↾ ( 1 ... 𝑁 ) ) ∧ ( 𝑏 ‘ 𝑒 ) = 0 ) ) ) → ( 𝑐 = ( ( 𝑑 ∪ 𝑒 ) ↾ ( 1 ... 𝑁 ) ) ∧ ( ( 𝑎 ‘ ( ( 𝑑 ∪ 𝑒 ) ↾ ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) ) = 0 ∧ ( 𝑏 ‘ ( ( 𝑑 ∪ 𝑒 ) ↾ ℕ ) ) = 0 ) ) ) |
82 |
|
reseq1 |
⊢ ( 𝑓 = ( 𝑑 ∪ 𝑒 ) → ( 𝑓 ↾ ( 1 ... 𝑁 ) ) = ( ( 𝑑 ∪ 𝑒 ) ↾ ( 1 ... 𝑁 ) ) ) |
83 |
82
|
eqeq2d |
⊢ ( 𝑓 = ( 𝑑 ∪ 𝑒 ) → ( 𝑐 = ( 𝑓 ↾ ( 1 ... 𝑁 ) ) ↔ 𝑐 = ( ( 𝑑 ∪ 𝑒 ) ↾ ( 1 ... 𝑁 ) ) ) ) |
84 |
|
reseq1 |
⊢ ( 𝑓 = ( 𝑑 ∪ 𝑒 ) → ( 𝑓 ↾ ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) = ( ( 𝑑 ∪ 𝑒 ) ↾ ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) ) |
85 |
84
|
fveqeq2d |
⊢ ( 𝑓 = ( 𝑑 ∪ 𝑒 ) → ( ( 𝑎 ‘ ( 𝑓 ↾ ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) ) = 0 ↔ ( 𝑎 ‘ ( ( 𝑑 ∪ 𝑒 ) ↾ ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) ) = 0 ) ) |
86 |
|
reseq1 |
⊢ ( 𝑓 = ( 𝑑 ∪ 𝑒 ) → ( 𝑓 ↾ ℕ ) = ( ( 𝑑 ∪ 𝑒 ) ↾ ℕ ) ) |
87 |
86
|
fveqeq2d |
⊢ ( 𝑓 = ( 𝑑 ∪ 𝑒 ) → ( ( 𝑏 ‘ ( 𝑓 ↾ ℕ ) ) = 0 ↔ ( 𝑏 ‘ ( ( 𝑑 ∪ 𝑒 ) ↾ ℕ ) ) = 0 ) ) |
88 |
85 87
|
anbi12d |
⊢ ( 𝑓 = ( 𝑑 ∪ 𝑒 ) → ( ( ( 𝑎 ‘ ( 𝑓 ↾ ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) ) = 0 ∧ ( 𝑏 ‘ ( 𝑓 ↾ ℕ ) ) = 0 ) ↔ ( ( 𝑎 ‘ ( ( 𝑑 ∪ 𝑒 ) ↾ ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) ) = 0 ∧ ( 𝑏 ‘ ( ( 𝑑 ∪ 𝑒 ) ↾ ℕ ) ) = 0 ) ) ) |
89 |
83 88
|
anbi12d |
⊢ ( 𝑓 = ( 𝑑 ∪ 𝑒 ) → ( ( 𝑐 = ( 𝑓 ↾ ( 1 ... 𝑁 ) ) ∧ ( ( 𝑎 ‘ ( 𝑓 ↾ ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) ) = 0 ∧ ( 𝑏 ‘ ( 𝑓 ↾ ℕ ) ) = 0 ) ) ↔ ( 𝑐 = ( ( 𝑑 ∪ 𝑒 ) ↾ ( 1 ... 𝑁 ) ) ∧ ( ( 𝑎 ‘ ( ( 𝑑 ∪ 𝑒 ) ↾ ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) ) = 0 ∧ ( 𝑏 ‘ ( ( 𝑑 ∪ 𝑒 ) ↾ ℕ ) ) = 0 ) ) ) ) |
90 |
89
|
rspcev |
⊢ ( ( ( 𝑑 ∪ 𝑒 ) ∈ ( ℕ0 ↑m ℤ ) ∧ ( 𝑐 = ( ( 𝑑 ∪ 𝑒 ) ↾ ( 1 ... 𝑁 ) ) ∧ ( ( 𝑎 ‘ ( ( 𝑑 ∪ 𝑒 ) ↾ ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) ) = 0 ∧ ( 𝑏 ‘ ( ( 𝑑 ∪ 𝑒 ) ↾ ℕ ) ) = 0 ) ) ) → ∃ 𝑓 ∈ ( ℕ0 ↑m ℤ ) ( 𝑐 = ( 𝑓 ↾ ( 1 ... 𝑁 ) ) ∧ ( ( 𝑎 ‘ ( 𝑓 ↾ ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) ) = 0 ∧ ( 𝑏 ‘ ( 𝑓 ↾ ℕ ) ) = 0 ) ) ) |
91 |
54 81 90
|
syl2anc |
⊢ ( ( ( ( 𝑁 ∈ ℕ0 ∧ ( 𝑎 ∈ ( mzPoly ‘ ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) ∧ 𝑏 ∈ ( mzPoly ‘ ℕ ) ) ) ∧ ( 𝑑 ∈ ( ℕ0 ↑m ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) ∧ 𝑒 ∈ ( ℕ0 ↑m ℕ ) ) ) ∧ ( ( 𝑐 = ( 𝑑 ↾ ( 1 ... 𝑁 ) ) ∧ ( 𝑎 ‘ 𝑑 ) = 0 ) ∧ ( 𝑐 = ( 𝑒 ↾ ( 1 ... 𝑁 ) ) ∧ ( 𝑏 ‘ 𝑒 ) = 0 ) ) ) → ∃ 𝑓 ∈ ( ℕ0 ↑m ℤ ) ( 𝑐 = ( 𝑓 ↾ ( 1 ... 𝑁 ) ) ∧ ( ( 𝑎 ‘ ( 𝑓 ↾ ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) ) = 0 ∧ ( 𝑏 ‘ ( 𝑓 ↾ ℕ ) ) = 0 ) ) ) |
92 |
91
|
ex |
⊢ ( ( ( 𝑁 ∈ ℕ0 ∧ ( 𝑎 ∈ ( mzPoly ‘ ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) ∧ 𝑏 ∈ ( mzPoly ‘ ℕ ) ) ) ∧ ( 𝑑 ∈ ( ℕ0 ↑m ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) ∧ 𝑒 ∈ ( ℕ0 ↑m ℕ ) ) ) → ( ( ( 𝑐 = ( 𝑑 ↾ ( 1 ... 𝑁 ) ) ∧ ( 𝑎 ‘ 𝑑 ) = 0 ) ∧ ( 𝑐 = ( 𝑒 ↾ ( 1 ... 𝑁 ) ) ∧ ( 𝑏 ‘ 𝑒 ) = 0 ) ) → ∃ 𝑓 ∈ ( ℕ0 ↑m ℤ ) ( 𝑐 = ( 𝑓 ↾ ( 1 ... 𝑁 ) ) ∧ ( ( 𝑎 ‘ ( 𝑓 ↾ ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) ) = 0 ∧ ( 𝑏 ‘ ( 𝑓 ↾ ℕ ) ) = 0 ) ) ) ) |
93 |
92
|
rexlimdvva |
⊢ ( ( 𝑁 ∈ ℕ0 ∧ ( 𝑎 ∈ ( mzPoly ‘ ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) ∧ 𝑏 ∈ ( mzPoly ‘ ℕ ) ) ) → ( ∃ 𝑑 ∈ ( ℕ0 ↑m ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) ∃ 𝑒 ∈ ( ℕ0 ↑m ℕ ) ( ( 𝑐 = ( 𝑑 ↾ ( 1 ... 𝑁 ) ) ∧ ( 𝑎 ‘ 𝑑 ) = 0 ) ∧ ( 𝑐 = ( 𝑒 ↾ ( 1 ... 𝑁 ) ) ∧ ( 𝑏 ‘ 𝑒 ) = 0 ) ) → ∃ 𝑓 ∈ ( ℕ0 ↑m ℤ ) ( 𝑐 = ( 𝑓 ↾ ( 1 ... 𝑁 ) ) ∧ ( ( 𝑎 ‘ ( 𝑓 ↾ ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) ) = 0 ∧ ( 𝑏 ‘ ( 𝑓 ↾ ℕ ) ) = 0 ) ) ) ) |
94 |
|
simpr |
⊢ ( ( ( 𝑁 ∈ ℕ0 ∧ ( 𝑎 ∈ ( mzPoly ‘ ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) ∧ 𝑏 ∈ ( mzPoly ‘ ℕ ) ) ) ∧ 𝑓 ∈ ( ℕ0 ↑m ℤ ) ) → 𝑓 ∈ ( ℕ0 ↑m ℤ ) ) |
95 |
|
difss |
⊢ ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ⊆ ℤ |
96 |
|
elmapssres |
⊢ ( ( 𝑓 ∈ ( ℕ0 ↑m ℤ ) ∧ ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ⊆ ℤ ) → ( 𝑓 ↾ ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) ∈ ( ℕ0 ↑m ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) ) |
97 |
94 95 96
|
sylancl |
⊢ ( ( ( 𝑁 ∈ ℕ0 ∧ ( 𝑎 ∈ ( mzPoly ‘ ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) ∧ 𝑏 ∈ ( mzPoly ‘ ℕ ) ) ) ∧ 𝑓 ∈ ( ℕ0 ↑m ℤ ) ) → ( 𝑓 ↾ ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) ∈ ( ℕ0 ↑m ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) ) |
98 |
97
|
adantr |
⊢ ( ( ( ( 𝑁 ∈ ℕ0 ∧ ( 𝑎 ∈ ( mzPoly ‘ ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) ∧ 𝑏 ∈ ( mzPoly ‘ ℕ ) ) ) ∧ 𝑓 ∈ ( ℕ0 ↑m ℤ ) ) ∧ ( 𝑐 = ( 𝑓 ↾ ( 1 ... 𝑁 ) ) ∧ ( ( 𝑎 ‘ ( 𝑓 ↾ ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) ) = 0 ∧ ( 𝑏 ‘ ( 𝑓 ↾ ℕ ) ) = 0 ) ) ) → ( 𝑓 ↾ ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) ∈ ( ℕ0 ↑m ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) ) |
99 |
|
nnssz |
⊢ ℕ ⊆ ℤ |
100 |
|
elmapssres |
⊢ ( ( 𝑓 ∈ ( ℕ0 ↑m ℤ ) ∧ ℕ ⊆ ℤ ) → ( 𝑓 ↾ ℕ ) ∈ ( ℕ0 ↑m ℕ ) ) |
101 |
94 99 100
|
sylancl |
⊢ ( ( ( 𝑁 ∈ ℕ0 ∧ ( 𝑎 ∈ ( mzPoly ‘ ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) ∧ 𝑏 ∈ ( mzPoly ‘ ℕ ) ) ) ∧ 𝑓 ∈ ( ℕ0 ↑m ℤ ) ) → ( 𝑓 ↾ ℕ ) ∈ ( ℕ0 ↑m ℕ ) ) |
102 |
101
|
adantr |
⊢ ( ( ( ( 𝑁 ∈ ℕ0 ∧ ( 𝑎 ∈ ( mzPoly ‘ ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) ∧ 𝑏 ∈ ( mzPoly ‘ ℕ ) ) ) ∧ 𝑓 ∈ ( ℕ0 ↑m ℤ ) ) ∧ ( 𝑐 = ( 𝑓 ↾ ( 1 ... 𝑁 ) ) ∧ ( ( 𝑎 ‘ ( 𝑓 ↾ ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) ) = 0 ∧ ( 𝑏 ‘ ( 𝑓 ↾ ℕ ) ) = 0 ) ) ) → ( 𝑓 ↾ ℕ ) ∈ ( ℕ0 ↑m ℕ ) ) |
103 |
|
simprl |
⊢ ( ( ( ( 𝑁 ∈ ℕ0 ∧ ( 𝑎 ∈ ( mzPoly ‘ ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) ∧ 𝑏 ∈ ( mzPoly ‘ ℕ ) ) ) ∧ 𝑓 ∈ ( ℕ0 ↑m ℤ ) ) ∧ ( 𝑐 = ( 𝑓 ↾ ( 1 ... 𝑁 ) ) ∧ ( ( 𝑎 ‘ ( 𝑓 ↾ ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) ) = 0 ∧ ( 𝑏 ‘ ( 𝑓 ↾ ℕ ) ) = 0 ) ) ) → 𝑐 = ( 𝑓 ↾ ( 1 ... 𝑁 ) ) ) |
104 |
14
|
ad3antrrr |
⊢ ( ( ( ( 𝑁 ∈ ℕ0 ∧ ( 𝑎 ∈ ( mzPoly ‘ ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) ∧ 𝑏 ∈ ( mzPoly ‘ ℕ ) ) ) ∧ 𝑓 ∈ ( ℕ0 ↑m ℤ ) ) ∧ ( 𝑐 = ( 𝑓 ↾ ( 1 ... 𝑁 ) ) ∧ ( ( 𝑎 ‘ ( 𝑓 ↾ ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) ) = 0 ∧ ( 𝑏 ‘ ( 𝑓 ↾ ℕ ) ) = 0 ) ) ) → ( 1 ... 𝑁 ) ⊆ ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) |
105 |
104
|
resabs1d |
⊢ ( ( ( ( 𝑁 ∈ ℕ0 ∧ ( 𝑎 ∈ ( mzPoly ‘ ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) ∧ 𝑏 ∈ ( mzPoly ‘ ℕ ) ) ) ∧ 𝑓 ∈ ( ℕ0 ↑m ℤ ) ) ∧ ( 𝑐 = ( 𝑓 ↾ ( 1 ... 𝑁 ) ) ∧ ( ( 𝑎 ‘ ( 𝑓 ↾ ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) ) = 0 ∧ ( 𝑏 ‘ ( 𝑓 ↾ ℕ ) ) = 0 ) ) ) → ( ( 𝑓 ↾ ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) ↾ ( 1 ... 𝑁 ) ) = ( 𝑓 ↾ ( 1 ... 𝑁 ) ) ) |
106 |
103 105
|
eqtr4d |
⊢ ( ( ( ( 𝑁 ∈ ℕ0 ∧ ( 𝑎 ∈ ( mzPoly ‘ ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) ∧ 𝑏 ∈ ( mzPoly ‘ ℕ ) ) ) ∧ 𝑓 ∈ ( ℕ0 ↑m ℤ ) ) ∧ ( 𝑐 = ( 𝑓 ↾ ( 1 ... 𝑁 ) ) ∧ ( ( 𝑎 ‘ ( 𝑓 ↾ ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) ) = 0 ∧ ( 𝑏 ‘ ( 𝑓 ↾ ℕ ) ) = 0 ) ) ) → 𝑐 = ( ( 𝑓 ↾ ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) ↾ ( 1 ... 𝑁 ) ) ) |
107 |
|
simprrl |
⊢ ( ( ( ( 𝑁 ∈ ℕ0 ∧ ( 𝑎 ∈ ( mzPoly ‘ ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) ∧ 𝑏 ∈ ( mzPoly ‘ ℕ ) ) ) ∧ 𝑓 ∈ ( ℕ0 ↑m ℤ ) ) ∧ ( 𝑐 = ( 𝑓 ↾ ( 1 ... 𝑁 ) ) ∧ ( ( 𝑎 ‘ ( 𝑓 ↾ ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) ) = 0 ∧ ( 𝑏 ‘ ( 𝑓 ↾ ℕ ) ) = 0 ) ) ) → ( 𝑎 ‘ ( 𝑓 ↾ ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) ) = 0 ) |
108 |
106 107
|
jca |
⊢ ( ( ( ( 𝑁 ∈ ℕ0 ∧ ( 𝑎 ∈ ( mzPoly ‘ ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) ∧ 𝑏 ∈ ( mzPoly ‘ ℕ ) ) ) ∧ 𝑓 ∈ ( ℕ0 ↑m ℤ ) ) ∧ ( 𝑐 = ( 𝑓 ↾ ( 1 ... 𝑁 ) ) ∧ ( ( 𝑎 ‘ ( 𝑓 ↾ ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) ) = 0 ∧ ( 𝑏 ‘ ( 𝑓 ↾ ℕ ) ) = 0 ) ) ) → ( 𝑐 = ( ( 𝑓 ↾ ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) ↾ ( 1 ... 𝑁 ) ) ∧ ( 𝑎 ‘ ( 𝑓 ↾ ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) ) = 0 ) ) |
109 |
|
resabs1 |
⊢ ( ( 1 ... 𝑁 ) ⊆ ℕ → ( ( 𝑓 ↾ ℕ ) ↾ ( 1 ... 𝑁 ) ) = ( 𝑓 ↾ ( 1 ... 𝑁 ) ) ) |
110 |
24 109
|
mp1i |
⊢ ( ( ( ( 𝑁 ∈ ℕ0 ∧ ( 𝑎 ∈ ( mzPoly ‘ ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) ∧ 𝑏 ∈ ( mzPoly ‘ ℕ ) ) ) ∧ 𝑓 ∈ ( ℕ0 ↑m ℤ ) ) ∧ ( 𝑐 = ( 𝑓 ↾ ( 1 ... 𝑁 ) ) ∧ ( ( 𝑎 ‘ ( 𝑓 ↾ ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) ) = 0 ∧ ( 𝑏 ‘ ( 𝑓 ↾ ℕ ) ) = 0 ) ) ) → ( ( 𝑓 ↾ ℕ ) ↾ ( 1 ... 𝑁 ) ) = ( 𝑓 ↾ ( 1 ... 𝑁 ) ) ) |
111 |
103 110
|
eqtr4d |
⊢ ( ( ( ( 𝑁 ∈ ℕ0 ∧ ( 𝑎 ∈ ( mzPoly ‘ ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) ∧ 𝑏 ∈ ( mzPoly ‘ ℕ ) ) ) ∧ 𝑓 ∈ ( ℕ0 ↑m ℤ ) ) ∧ ( 𝑐 = ( 𝑓 ↾ ( 1 ... 𝑁 ) ) ∧ ( ( 𝑎 ‘ ( 𝑓 ↾ ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) ) = 0 ∧ ( 𝑏 ‘ ( 𝑓 ↾ ℕ ) ) = 0 ) ) ) → 𝑐 = ( ( 𝑓 ↾ ℕ ) ↾ ( 1 ... 𝑁 ) ) ) |
112 |
|
simprrr |
⊢ ( ( ( ( 𝑁 ∈ ℕ0 ∧ ( 𝑎 ∈ ( mzPoly ‘ ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) ∧ 𝑏 ∈ ( mzPoly ‘ ℕ ) ) ) ∧ 𝑓 ∈ ( ℕ0 ↑m ℤ ) ) ∧ ( 𝑐 = ( 𝑓 ↾ ( 1 ... 𝑁 ) ) ∧ ( ( 𝑎 ‘ ( 𝑓 ↾ ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) ) = 0 ∧ ( 𝑏 ‘ ( 𝑓 ↾ ℕ ) ) = 0 ) ) ) → ( 𝑏 ‘ ( 𝑓 ↾ ℕ ) ) = 0 ) |
113 |
108 111 112
|
jca32 |
⊢ ( ( ( ( 𝑁 ∈ ℕ0 ∧ ( 𝑎 ∈ ( mzPoly ‘ ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) ∧ 𝑏 ∈ ( mzPoly ‘ ℕ ) ) ) ∧ 𝑓 ∈ ( ℕ0 ↑m ℤ ) ) ∧ ( 𝑐 = ( 𝑓 ↾ ( 1 ... 𝑁 ) ) ∧ ( ( 𝑎 ‘ ( 𝑓 ↾ ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) ) = 0 ∧ ( 𝑏 ‘ ( 𝑓 ↾ ℕ ) ) = 0 ) ) ) → ( ( 𝑐 = ( ( 𝑓 ↾ ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) ↾ ( 1 ... 𝑁 ) ) ∧ ( 𝑎 ‘ ( 𝑓 ↾ ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) ) = 0 ) ∧ ( 𝑐 = ( ( 𝑓 ↾ ℕ ) ↾ ( 1 ... 𝑁 ) ) ∧ ( 𝑏 ‘ ( 𝑓 ↾ ℕ ) ) = 0 ) ) ) |
114 |
|
reseq1 |
⊢ ( 𝑑 = ( 𝑓 ↾ ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) → ( 𝑑 ↾ ( 1 ... 𝑁 ) ) = ( ( 𝑓 ↾ ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) ↾ ( 1 ... 𝑁 ) ) ) |
115 |
114
|
eqeq2d |
⊢ ( 𝑑 = ( 𝑓 ↾ ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) → ( 𝑐 = ( 𝑑 ↾ ( 1 ... 𝑁 ) ) ↔ 𝑐 = ( ( 𝑓 ↾ ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) ↾ ( 1 ... 𝑁 ) ) ) ) |
116 |
|
fveqeq2 |
⊢ ( 𝑑 = ( 𝑓 ↾ ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) → ( ( 𝑎 ‘ 𝑑 ) = 0 ↔ ( 𝑎 ‘ ( 𝑓 ↾ ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) ) = 0 ) ) |
117 |
115 116
|
anbi12d |
⊢ ( 𝑑 = ( 𝑓 ↾ ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) → ( ( 𝑐 = ( 𝑑 ↾ ( 1 ... 𝑁 ) ) ∧ ( 𝑎 ‘ 𝑑 ) = 0 ) ↔ ( 𝑐 = ( ( 𝑓 ↾ ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) ↾ ( 1 ... 𝑁 ) ) ∧ ( 𝑎 ‘ ( 𝑓 ↾ ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) ) = 0 ) ) ) |
118 |
117
|
anbi1d |
⊢ ( 𝑑 = ( 𝑓 ↾ ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) → ( ( ( 𝑐 = ( 𝑑 ↾ ( 1 ... 𝑁 ) ) ∧ ( 𝑎 ‘ 𝑑 ) = 0 ) ∧ ( 𝑐 = ( 𝑒 ↾ ( 1 ... 𝑁 ) ) ∧ ( 𝑏 ‘ 𝑒 ) = 0 ) ) ↔ ( ( 𝑐 = ( ( 𝑓 ↾ ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) ↾ ( 1 ... 𝑁 ) ) ∧ ( 𝑎 ‘ ( 𝑓 ↾ ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) ) = 0 ) ∧ ( 𝑐 = ( 𝑒 ↾ ( 1 ... 𝑁 ) ) ∧ ( 𝑏 ‘ 𝑒 ) = 0 ) ) ) ) |
119 |
|
reseq1 |
⊢ ( 𝑒 = ( 𝑓 ↾ ℕ ) → ( 𝑒 ↾ ( 1 ... 𝑁 ) ) = ( ( 𝑓 ↾ ℕ ) ↾ ( 1 ... 𝑁 ) ) ) |
120 |
119
|
eqeq2d |
⊢ ( 𝑒 = ( 𝑓 ↾ ℕ ) → ( 𝑐 = ( 𝑒 ↾ ( 1 ... 𝑁 ) ) ↔ 𝑐 = ( ( 𝑓 ↾ ℕ ) ↾ ( 1 ... 𝑁 ) ) ) ) |
121 |
|
fveqeq2 |
⊢ ( 𝑒 = ( 𝑓 ↾ ℕ ) → ( ( 𝑏 ‘ 𝑒 ) = 0 ↔ ( 𝑏 ‘ ( 𝑓 ↾ ℕ ) ) = 0 ) ) |
122 |
120 121
|
anbi12d |
⊢ ( 𝑒 = ( 𝑓 ↾ ℕ ) → ( ( 𝑐 = ( 𝑒 ↾ ( 1 ... 𝑁 ) ) ∧ ( 𝑏 ‘ 𝑒 ) = 0 ) ↔ ( 𝑐 = ( ( 𝑓 ↾ ℕ ) ↾ ( 1 ... 𝑁 ) ) ∧ ( 𝑏 ‘ ( 𝑓 ↾ ℕ ) ) = 0 ) ) ) |
123 |
122
|
anbi2d |
⊢ ( 𝑒 = ( 𝑓 ↾ ℕ ) → ( ( ( 𝑐 = ( ( 𝑓 ↾ ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) ↾ ( 1 ... 𝑁 ) ) ∧ ( 𝑎 ‘ ( 𝑓 ↾ ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) ) = 0 ) ∧ ( 𝑐 = ( 𝑒 ↾ ( 1 ... 𝑁 ) ) ∧ ( 𝑏 ‘ 𝑒 ) = 0 ) ) ↔ ( ( 𝑐 = ( ( 𝑓 ↾ ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) ↾ ( 1 ... 𝑁 ) ) ∧ ( 𝑎 ‘ ( 𝑓 ↾ ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) ) = 0 ) ∧ ( 𝑐 = ( ( 𝑓 ↾ ℕ ) ↾ ( 1 ... 𝑁 ) ) ∧ ( 𝑏 ‘ ( 𝑓 ↾ ℕ ) ) = 0 ) ) ) ) |
124 |
118 123
|
rspc2ev |
⊢ ( ( ( 𝑓 ↾ ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) ∈ ( ℕ0 ↑m ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) ∧ ( 𝑓 ↾ ℕ ) ∈ ( ℕ0 ↑m ℕ ) ∧ ( ( 𝑐 = ( ( 𝑓 ↾ ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) ↾ ( 1 ... 𝑁 ) ) ∧ ( 𝑎 ‘ ( 𝑓 ↾ ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) ) = 0 ) ∧ ( 𝑐 = ( ( 𝑓 ↾ ℕ ) ↾ ( 1 ... 𝑁 ) ) ∧ ( 𝑏 ‘ ( 𝑓 ↾ ℕ ) ) = 0 ) ) ) → ∃ 𝑑 ∈ ( ℕ0 ↑m ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) ∃ 𝑒 ∈ ( ℕ0 ↑m ℕ ) ( ( 𝑐 = ( 𝑑 ↾ ( 1 ... 𝑁 ) ) ∧ ( 𝑎 ‘ 𝑑 ) = 0 ) ∧ ( 𝑐 = ( 𝑒 ↾ ( 1 ... 𝑁 ) ) ∧ ( 𝑏 ‘ 𝑒 ) = 0 ) ) ) |
125 |
98 102 113 124
|
syl3anc |
⊢ ( ( ( ( 𝑁 ∈ ℕ0 ∧ ( 𝑎 ∈ ( mzPoly ‘ ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) ∧ 𝑏 ∈ ( mzPoly ‘ ℕ ) ) ) ∧ 𝑓 ∈ ( ℕ0 ↑m ℤ ) ) ∧ ( 𝑐 = ( 𝑓 ↾ ( 1 ... 𝑁 ) ) ∧ ( ( 𝑎 ‘ ( 𝑓 ↾ ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) ) = 0 ∧ ( 𝑏 ‘ ( 𝑓 ↾ ℕ ) ) = 0 ) ) ) → ∃ 𝑑 ∈ ( ℕ0 ↑m ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) ∃ 𝑒 ∈ ( ℕ0 ↑m ℕ ) ( ( 𝑐 = ( 𝑑 ↾ ( 1 ... 𝑁 ) ) ∧ ( 𝑎 ‘ 𝑑 ) = 0 ) ∧ ( 𝑐 = ( 𝑒 ↾ ( 1 ... 𝑁 ) ) ∧ ( 𝑏 ‘ 𝑒 ) = 0 ) ) ) |
126 |
125
|
rexlimdva2 |
⊢ ( ( 𝑁 ∈ ℕ0 ∧ ( 𝑎 ∈ ( mzPoly ‘ ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) ∧ 𝑏 ∈ ( mzPoly ‘ ℕ ) ) ) → ( ∃ 𝑓 ∈ ( ℕ0 ↑m ℤ ) ( 𝑐 = ( 𝑓 ↾ ( 1 ... 𝑁 ) ) ∧ ( ( 𝑎 ‘ ( 𝑓 ↾ ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) ) = 0 ∧ ( 𝑏 ‘ ( 𝑓 ↾ ℕ ) ) = 0 ) ) → ∃ 𝑑 ∈ ( ℕ0 ↑m ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) ∃ 𝑒 ∈ ( ℕ0 ↑m ℕ ) ( ( 𝑐 = ( 𝑑 ↾ ( 1 ... 𝑁 ) ) ∧ ( 𝑎 ‘ 𝑑 ) = 0 ) ∧ ( 𝑐 = ( 𝑒 ↾ ( 1 ... 𝑁 ) ) ∧ ( 𝑏 ‘ 𝑒 ) = 0 ) ) ) ) |
127 |
93 126
|
impbid |
⊢ ( ( 𝑁 ∈ ℕ0 ∧ ( 𝑎 ∈ ( mzPoly ‘ ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) ∧ 𝑏 ∈ ( mzPoly ‘ ℕ ) ) ) → ( ∃ 𝑑 ∈ ( ℕ0 ↑m ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) ∃ 𝑒 ∈ ( ℕ0 ↑m ℕ ) ( ( 𝑐 = ( 𝑑 ↾ ( 1 ... 𝑁 ) ) ∧ ( 𝑎 ‘ 𝑑 ) = 0 ) ∧ ( 𝑐 = ( 𝑒 ↾ ( 1 ... 𝑁 ) ) ∧ ( 𝑏 ‘ 𝑒 ) = 0 ) ) ↔ ∃ 𝑓 ∈ ( ℕ0 ↑m ℤ ) ( 𝑐 = ( 𝑓 ↾ ( 1 ... 𝑁 ) ) ∧ ( ( 𝑎 ‘ ( 𝑓 ↾ ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) ) = 0 ∧ ( 𝑏 ‘ ( 𝑓 ↾ ℕ ) ) = 0 ) ) ) ) |
128 |
|
simplrl |
⊢ ( ( ( 𝑁 ∈ ℕ0 ∧ ( 𝑎 ∈ ( mzPoly ‘ ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) ∧ 𝑏 ∈ ( mzPoly ‘ ℕ ) ) ) ∧ 𝑓 ∈ ( ℕ0 ↑m ℤ ) ) → 𝑎 ∈ ( mzPoly ‘ ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) ) |
129 |
|
mzpf |
⊢ ( 𝑎 ∈ ( mzPoly ‘ ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) → 𝑎 : ( ℤ ↑m ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) ⟶ ℤ ) |
130 |
128 129
|
syl |
⊢ ( ( ( 𝑁 ∈ ℕ0 ∧ ( 𝑎 ∈ ( mzPoly ‘ ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) ∧ 𝑏 ∈ ( mzPoly ‘ ℕ ) ) ) ∧ 𝑓 ∈ ( ℕ0 ↑m ℤ ) ) → 𝑎 : ( ℤ ↑m ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) ⟶ ℤ ) |
131 |
|
nn0ssz |
⊢ ℕ0 ⊆ ℤ |
132 |
|
mapss |
⊢ ( ( ℤ ∈ V ∧ ℕ0 ⊆ ℤ ) → ( ℕ0 ↑m ℤ ) ⊆ ( ℤ ↑m ℤ ) ) |
133 |
3 131 132
|
mp2an |
⊢ ( ℕ0 ↑m ℤ ) ⊆ ( ℤ ↑m ℤ ) |
134 |
133
|
sseli |
⊢ ( 𝑓 ∈ ( ℕ0 ↑m ℤ ) → 𝑓 ∈ ( ℤ ↑m ℤ ) ) |
135 |
|
elmapssres |
⊢ ( ( 𝑓 ∈ ( ℤ ↑m ℤ ) ∧ ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ⊆ ℤ ) → ( 𝑓 ↾ ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) ∈ ( ℤ ↑m ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) ) |
136 |
134 95 135
|
sylancl |
⊢ ( 𝑓 ∈ ( ℕ0 ↑m ℤ ) → ( 𝑓 ↾ ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) ∈ ( ℤ ↑m ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) ) |
137 |
136
|
adantl |
⊢ ( ( ( 𝑁 ∈ ℕ0 ∧ ( 𝑎 ∈ ( mzPoly ‘ ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) ∧ 𝑏 ∈ ( mzPoly ‘ ℕ ) ) ) ∧ 𝑓 ∈ ( ℕ0 ↑m ℤ ) ) → ( 𝑓 ↾ ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) ∈ ( ℤ ↑m ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) ) |
138 |
130 137
|
ffvelrnd |
⊢ ( ( ( 𝑁 ∈ ℕ0 ∧ ( 𝑎 ∈ ( mzPoly ‘ ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) ∧ 𝑏 ∈ ( mzPoly ‘ ℕ ) ) ) ∧ 𝑓 ∈ ( ℕ0 ↑m ℤ ) ) → ( 𝑎 ‘ ( 𝑓 ↾ ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) ) ∈ ℤ ) |
139 |
138
|
zred |
⊢ ( ( ( 𝑁 ∈ ℕ0 ∧ ( 𝑎 ∈ ( mzPoly ‘ ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) ∧ 𝑏 ∈ ( mzPoly ‘ ℕ ) ) ) ∧ 𝑓 ∈ ( ℕ0 ↑m ℤ ) ) → ( 𝑎 ‘ ( 𝑓 ↾ ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) ) ∈ ℝ ) |
140 |
|
simplrr |
⊢ ( ( ( 𝑁 ∈ ℕ0 ∧ ( 𝑎 ∈ ( mzPoly ‘ ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) ∧ 𝑏 ∈ ( mzPoly ‘ ℕ ) ) ) ∧ 𝑓 ∈ ( ℕ0 ↑m ℤ ) ) → 𝑏 ∈ ( mzPoly ‘ ℕ ) ) |
141 |
|
mzpf |
⊢ ( 𝑏 ∈ ( mzPoly ‘ ℕ ) → 𝑏 : ( ℤ ↑m ℕ ) ⟶ ℤ ) |
142 |
140 141
|
syl |
⊢ ( ( ( 𝑁 ∈ ℕ0 ∧ ( 𝑎 ∈ ( mzPoly ‘ ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) ∧ 𝑏 ∈ ( mzPoly ‘ ℕ ) ) ) ∧ 𝑓 ∈ ( ℕ0 ↑m ℤ ) ) → 𝑏 : ( ℤ ↑m ℕ ) ⟶ ℤ ) |
143 |
|
elmapssres |
⊢ ( ( 𝑓 ∈ ( ℤ ↑m ℤ ) ∧ ℕ ⊆ ℤ ) → ( 𝑓 ↾ ℕ ) ∈ ( ℤ ↑m ℕ ) ) |
144 |
134 99 143
|
sylancl |
⊢ ( 𝑓 ∈ ( ℕ0 ↑m ℤ ) → ( 𝑓 ↾ ℕ ) ∈ ( ℤ ↑m ℕ ) ) |
145 |
144
|
adantl |
⊢ ( ( ( 𝑁 ∈ ℕ0 ∧ ( 𝑎 ∈ ( mzPoly ‘ ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) ∧ 𝑏 ∈ ( mzPoly ‘ ℕ ) ) ) ∧ 𝑓 ∈ ( ℕ0 ↑m ℤ ) ) → ( 𝑓 ↾ ℕ ) ∈ ( ℤ ↑m ℕ ) ) |
146 |
142 145
|
ffvelrnd |
⊢ ( ( ( 𝑁 ∈ ℕ0 ∧ ( 𝑎 ∈ ( mzPoly ‘ ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) ∧ 𝑏 ∈ ( mzPoly ‘ ℕ ) ) ) ∧ 𝑓 ∈ ( ℕ0 ↑m ℤ ) ) → ( 𝑏 ‘ ( 𝑓 ↾ ℕ ) ) ∈ ℤ ) |
147 |
146
|
zred |
⊢ ( ( ( 𝑁 ∈ ℕ0 ∧ ( 𝑎 ∈ ( mzPoly ‘ ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) ∧ 𝑏 ∈ ( mzPoly ‘ ℕ ) ) ) ∧ 𝑓 ∈ ( ℕ0 ↑m ℤ ) ) → ( 𝑏 ‘ ( 𝑓 ↾ ℕ ) ) ∈ ℝ ) |
148 |
|
sumsqeq0 |
⊢ ( ( ( 𝑎 ‘ ( 𝑓 ↾ ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) ) ∈ ℝ ∧ ( 𝑏 ‘ ( 𝑓 ↾ ℕ ) ) ∈ ℝ ) → ( ( ( 𝑎 ‘ ( 𝑓 ↾ ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) ) = 0 ∧ ( 𝑏 ‘ ( 𝑓 ↾ ℕ ) ) = 0 ) ↔ ( ( ( 𝑎 ‘ ( 𝑓 ↾ ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) ) ↑ 2 ) + ( ( 𝑏 ‘ ( 𝑓 ↾ ℕ ) ) ↑ 2 ) ) = 0 ) ) |
149 |
139 147 148
|
syl2anc |
⊢ ( ( ( 𝑁 ∈ ℕ0 ∧ ( 𝑎 ∈ ( mzPoly ‘ ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) ∧ 𝑏 ∈ ( mzPoly ‘ ℕ ) ) ) ∧ 𝑓 ∈ ( ℕ0 ↑m ℤ ) ) → ( ( ( 𝑎 ‘ ( 𝑓 ↾ ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) ) = 0 ∧ ( 𝑏 ‘ ( 𝑓 ↾ ℕ ) ) = 0 ) ↔ ( ( ( 𝑎 ‘ ( 𝑓 ↾ ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) ) ↑ 2 ) + ( ( 𝑏 ‘ ( 𝑓 ↾ ℕ ) ) ↑ 2 ) ) = 0 ) ) |
150 |
134
|
adantl |
⊢ ( ( ( 𝑁 ∈ ℕ0 ∧ ( 𝑎 ∈ ( mzPoly ‘ ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) ∧ 𝑏 ∈ ( mzPoly ‘ ℕ ) ) ) ∧ 𝑓 ∈ ( ℕ0 ↑m ℤ ) ) → 𝑓 ∈ ( ℤ ↑m ℤ ) ) |
151 |
|
reseq1 |
⊢ ( 𝑔 = 𝑓 → ( 𝑔 ↾ ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) = ( 𝑓 ↾ ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) ) |
152 |
151
|
fveq2d |
⊢ ( 𝑔 = 𝑓 → ( 𝑎 ‘ ( 𝑔 ↾ ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) ) = ( 𝑎 ‘ ( 𝑓 ↾ ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) ) ) |
153 |
152
|
oveq1d |
⊢ ( 𝑔 = 𝑓 → ( ( 𝑎 ‘ ( 𝑔 ↾ ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) ) ↑ 2 ) = ( ( 𝑎 ‘ ( 𝑓 ↾ ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) ) ↑ 2 ) ) |
154 |
|
reseq1 |
⊢ ( 𝑔 = 𝑓 → ( 𝑔 ↾ ℕ ) = ( 𝑓 ↾ ℕ ) ) |
155 |
154
|
fveq2d |
⊢ ( 𝑔 = 𝑓 → ( 𝑏 ‘ ( 𝑔 ↾ ℕ ) ) = ( 𝑏 ‘ ( 𝑓 ↾ ℕ ) ) ) |
156 |
155
|
oveq1d |
⊢ ( 𝑔 = 𝑓 → ( ( 𝑏 ‘ ( 𝑔 ↾ ℕ ) ) ↑ 2 ) = ( ( 𝑏 ‘ ( 𝑓 ↾ ℕ ) ) ↑ 2 ) ) |
157 |
153 156
|
oveq12d |
⊢ ( 𝑔 = 𝑓 → ( ( ( 𝑎 ‘ ( 𝑔 ↾ ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) ) ↑ 2 ) + ( ( 𝑏 ‘ ( 𝑔 ↾ ℕ ) ) ↑ 2 ) ) = ( ( ( 𝑎 ‘ ( 𝑓 ↾ ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) ) ↑ 2 ) + ( ( 𝑏 ‘ ( 𝑓 ↾ ℕ ) ) ↑ 2 ) ) ) |
158 |
|
eqid |
⊢ ( 𝑔 ∈ ( ℤ ↑m ℤ ) ↦ ( ( ( 𝑎 ‘ ( 𝑔 ↾ ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) ) ↑ 2 ) + ( ( 𝑏 ‘ ( 𝑔 ↾ ℕ ) ) ↑ 2 ) ) ) = ( 𝑔 ∈ ( ℤ ↑m ℤ ) ↦ ( ( ( 𝑎 ‘ ( 𝑔 ↾ ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) ) ↑ 2 ) + ( ( 𝑏 ‘ ( 𝑔 ↾ ℕ ) ) ↑ 2 ) ) ) |
159 |
|
ovex |
⊢ ( ( ( 𝑎 ‘ ( 𝑓 ↾ ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) ) ↑ 2 ) + ( ( 𝑏 ‘ ( 𝑓 ↾ ℕ ) ) ↑ 2 ) ) ∈ V |
160 |
157 158 159
|
fvmpt |
⊢ ( 𝑓 ∈ ( ℤ ↑m ℤ ) → ( ( 𝑔 ∈ ( ℤ ↑m ℤ ) ↦ ( ( ( 𝑎 ‘ ( 𝑔 ↾ ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) ) ↑ 2 ) + ( ( 𝑏 ‘ ( 𝑔 ↾ ℕ ) ) ↑ 2 ) ) ) ‘ 𝑓 ) = ( ( ( 𝑎 ‘ ( 𝑓 ↾ ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) ) ↑ 2 ) + ( ( 𝑏 ‘ ( 𝑓 ↾ ℕ ) ) ↑ 2 ) ) ) |
161 |
150 160
|
syl |
⊢ ( ( ( 𝑁 ∈ ℕ0 ∧ ( 𝑎 ∈ ( mzPoly ‘ ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) ∧ 𝑏 ∈ ( mzPoly ‘ ℕ ) ) ) ∧ 𝑓 ∈ ( ℕ0 ↑m ℤ ) ) → ( ( 𝑔 ∈ ( ℤ ↑m ℤ ) ↦ ( ( ( 𝑎 ‘ ( 𝑔 ↾ ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) ) ↑ 2 ) + ( ( 𝑏 ‘ ( 𝑔 ↾ ℕ ) ) ↑ 2 ) ) ) ‘ 𝑓 ) = ( ( ( 𝑎 ‘ ( 𝑓 ↾ ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) ) ↑ 2 ) + ( ( 𝑏 ‘ ( 𝑓 ↾ ℕ ) ) ↑ 2 ) ) ) |
162 |
161
|
eqeq1d |
⊢ ( ( ( 𝑁 ∈ ℕ0 ∧ ( 𝑎 ∈ ( mzPoly ‘ ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) ∧ 𝑏 ∈ ( mzPoly ‘ ℕ ) ) ) ∧ 𝑓 ∈ ( ℕ0 ↑m ℤ ) ) → ( ( ( 𝑔 ∈ ( ℤ ↑m ℤ ) ↦ ( ( ( 𝑎 ‘ ( 𝑔 ↾ ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) ) ↑ 2 ) + ( ( 𝑏 ‘ ( 𝑔 ↾ ℕ ) ) ↑ 2 ) ) ) ‘ 𝑓 ) = 0 ↔ ( ( ( 𝑎 ‘ ( 𝑓 ↾ ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) ) ↑ 2 ) + ( ( 𝑏 ‘ ( 𝑓 ↾ ℕ ) ) ↑ 2 ) ) = 0 ) ) |
163 |
149 162
|
bitr4d |
⊢ ( ( ( 𝑁 ∈ ℕ0 ∧ ( 𝑎 ∈ ( mzPoly ‘ ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) ∧ 𝑏 ∈ ( mzPoly ‘ ℕ ) ) ) ∧ 𝑓 ∈ ( ℕ0 ↑m ℤ ) ) → ( ( ( 𝑎 ‘ ( 𝑓 ↾ ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) ) = 0 ∧ ( 𝑏 ‘ ( 𝑓 ↾ ℕ ) ) = 0 ) ↔ ( ( 𝑔 ∈ ( ℤ ↑m ℤ ) ↦ ( ( ( 𝑎 ‘ ( 𝑔 ↾ ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) ) ↑ 2 ) + ( ( 𝑏 ‘ ( 𝑔 ↾ ℕ ) ) ↑ 2 ) ) ) ‘ 𝑓 ) = 0 ) ) |
164 |
163
|
anbi2d |
⊢ ( ( ( 𝑁 ∈ ℕ0 ∧ ( 𝑎 ∈ ( mzPoly ‘ ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) ∧ 𝑏 ∈ ( mzPoly ‘ ℕ ) ) ) ∧ 𝑓 ∈ ( ℕ0 ↑m ℤ ) ) → ( ( 𝑐 = ( 𝑓 ↾ ( 1 ... 𝑁 ) ) ∧ ( ( 𝑎 ‘ ( 𝑓 ↾ ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) ) = 0 ∧ ( 𝑏 ‘ ( 𝑓 ↾ ℕ ) ) = 0 ) ) ↔ ( 𝑐 = ( 𝑓 ↾ ( 1 ... 𝑁 ) ) ∧ ( ( 𝑔 ∈ ( ℤ ↑m ℤ ) ↦ ( ( ( 𝑎 ‘ ( 𝑔 ↾ ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) ) ↑ 2 ) + ( ( 𝑏 ‘ ( 𝑔 ↾ ℕ ) ) ↑ 2 ) ) ) ‘ 𝑓 ) = 0 ) ) ) |
165 |
164
|
rexbidva |
⊢ ( ( 𝑁 ∈ ℕ0 ∧ ( 𝑎 ∈ ( mzPoly ‘ ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) ∧ 𝑏 ∈ ( mzPoly ‘ ℕ ) ) ) → ( ∃ 𝑓 ∈ ( ℕ0 ↑m ℤ ) ( 𝑐 = ( 𝑓 ↾ ( 1 ... 𝑁 ) ) ∧ ( ( 𝑎 ‘ ( 𝑓 ↾ ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) ) = 0 ∧ ( 𝑏 ‘ ( 𝑓 ↾ ℕ ) ) = 0 ) ) ↔ ∃ 𝑓 ∈ ( ℕ0 ↑m ℤ ) ( 𝑐 = ( 𝑓 ↾ ( 1 ... 𝑁 ) ) ∧ ( ( 𝑔 ∈ ( ℤ ↑m ℤ ) ↦ ( ( ( 𝑎 ‘ ( 𝑔 ↾ ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) ) ↑ 2 ) + ( ( 𝑏 ‘ ( 𝑔 ↾ ℕ ) ) ↑ 2 ) ) ) ‘ 𝑓 ) = 0 ) ) ) |
166 |
127 165
|
bitrd |
⊢ ( ( 𝑁 ∈ ℕ0 ∧ ( 𝑎 ∈ ( mzPoly ‘ ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) ∧ 𝑏 ∈ ( mzPoly ‘ ℕ ) ) ) → ( ∃ 𝑑 ∈ ( ℕ0 ↑m ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) ∃ 𝑒 ∈ ( ℕ0 ↑m ℕ ) ( ( 𝑐 = ( 𝑑 ↾ ( 1 ... 𝑁 ) ) ∧ ( 𝑎 ‘ 𝑑 ) = 0 ) ∧ ( 𝑐 = ( 𝑒 ↾ ( 1 ... 𝑁 ) ) ∧ ( 𝑏 ‘ 𝑒 ) = 0 ) ) ↔ ∃ 𝑓 ∈ ( ℕ0 ↑m ℤ ) ( 𝑐 = ( 𝑓 ↾ ( 1 ... 𝑁 ) ) ∧ ( ( 𝑔 ∈ ( ℤ ↑m ℤ ) ↦ ( ( ( 𝑎 ‘ ( 𝑔 ↾ ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) ) ↑ 2 ) + ( ( 𝑏 ‘ ( 𝑔 ↾ ℕ ) ) ↑ 2 ) ) ) ‘ 𝑓 ) = 0 ) ) ) |
167 |
31 166
|
bitr3id |
⊢ ( ( 𝑁 ∈ ℕ0 ∧ ( 𝑎 ∈ ( mzPoly ‘ ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) ∧ 𝑏 ∈ ( mzPoly ‘ ℕ ) ) ) → ( ( ∃ 𝑑 ∈ ( ℕ0 ↑m ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) ( 𝑐 = ( 𝑑 ↾ ( 1 ... 𝑁 ) ) ∧ ( 𝑎 ‘ 𝑑 ) = 0 ) ∧ ∃ 𝑒 ∈ ( ℕ0 ↑m ℕ ) ( 𝑐 = ( 𝑒 ↾ ( 1 ... 𝑁 ) ) ∧ ( 𝑏 ‘ 𝑒 ) = 0 ) ) ↔ ∃ 𝑓 ∈ ( ℕ0 ↑m ℤ ) ( 𝑐 = ( 𝑓 ↾ ( 1 ... 𝑁 ) ) ∧ ( ( 𝑔 ∈ ( ℤ ↑m ℤ ) ↦ ( ( ( 𝑎 ‘ ( 𝑔 ↾ ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) ) ↑ 2 ) + ( ( 𝑏 ‘ ( 𝑔 ↾ ℕ ) ) ↑ 2 ) ) ) ‘ 𝑓 ) = 0 ) ) ) |
168 |
167
|
abbidv |
⊢ ( ( 𝑁 ∈ ℕ0 ∧ ( 𝑎 ∈ ( mzPoly ‘ ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) ∧ 𝑏 ∈ ( mzPoly ‘ ℕ ) ) ) → { 𝑐 ∣ ( ∃ 𝑑 ∈ ( ℕ0 ↑m ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) ( 𝑐 = ( 𝑑 ↾ ( 1 ... 𝑁 ) ) ∧ ( 𝑎 ‘ 𝑑 ) = 0 ) ∧ ∃ 𝑒 ∈ ( ℕ0 ↑m ℕ ) ( 𝑐 = ( 𝑒 ↾ ( 1 ... 𝑁 ) ) ∧ ( 𝑏 ‘ 𝑒 ) = 0 ) ) } = { 𝑐 ∣ ∃ 𝑓 ∈ ( ℕ0 ↑m ℤ ) ( 𝑐 = ( 𝑓 ↾ ( 1 ... 𝑁 ) ) ∧ ( ( 𝑔 ∈ ( ℤ ↑m ℤ ) ↦ ( ( ( 𝑎 ‘ ( 𝑔 ↾ ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) ) ↑ 2 ) + ( ( 𝑏 ‘ ( 𝑔 ↾ ℕ ) ) ↑ 2 ) ) ) ‘ 𝑓 ) = 0 ) } ) |
169 |
30 168
|
syl5eq |
⊢ ( ( 𝑁 ∈ ℕ0 ∧ ( 𝑎 ∈ ( mzPoly ‘ ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) ∧ 𝑏 ∈ ( mzPoly ‘ ℕ ) ) ) → ( { 𝑐 ∣ ∃ 𝑑 ∈ ( ℕ0 ↑m ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) ( 𝑐 = ( 𝑑 ↾ ( 1 ... 𝑁 ) ) ∧ ( 𝑎 ‘ 𝑑 ) = 0 ) } ∩ { 𝑐 ∣ ∃ 𝑒 ∈ ( ℕ0 ↑m ℕ ) ( 𝑐 = ( 𝑒 ↾ ( 1 ... 𝑁 ) ) ∧ ( 𝑏 ‘ 𝑒 ) = 0 ) } ) = { 𝑐 ∣ ∃ 𝑓 ∈ ( ℕ0 ↑m ℤ ) ( 𝑐 = ( 𝑓 ↾ ( 1 ... 𝑁 ) ) ∧ ( ( 𝑔 ∈ ( ℤ ↑m ℤ ) ↦ ( ( ( 𝑎 ‘ ( 𝑔 ↾ ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) ) ↑ 2 ) + ( ( 𝑏 ‘ ( 𝑔 ↾ ℕ ) ) ↑ 2 ) ) ) ‘ 𝑓 ) = 0 ) } ) |
170 |
|
simpl |
⊢ ( ( 𝑁 ∈ ℕ0 ∧ ( 𝑎 ∈ ( mzPoly ‘ ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) ∧ 𝑏 ∈ ( mzPoly ‘ ℕ ) ) ) → 𝑁 ∈ ℕ0 ) |
171 |
|
fzssuz |
⊢ ( 1 ... 𝑁 ) ⊆ ( ℤ≥ ‘ 1 ) |
172 |
|
uzssz |
⊢ ( ℤ≥ ‘ 1 ) ⊆ ℤ |
173 |
171 172
|
sstri |
⊢ ( 1 ... 𝑁 ) ⊆ ℤ |
174 |
3 173
|
pm3.2i |
⊢ ( ℤ ∈ V ∧ ( 1 ... 𝑁 ) ⊆ ℤ ) |
175 |
174
|
a1i |
⊢ ( ( 𝑁 ∈ ℕ0 ∧ ( 𝑎 ∈ ( mzPoly ‘ ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) ∧ 𝑏 ∈ ( mzPoly ‘ ℕ ) ) ) → ( ℤ ∈ V ∧ ( 1 ... 𝑁 ) ⊆ ℤ ) ) |
176 |
3
|
a1i |
⊢ ( ( 𝑁 ∈ ℕ0 ∧ ( 𝑎 ∈ ( mzPoly ‘ ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) ∧ 𝑏 ∈ ( mzPoly ‘ ℕ ) ) ) → ℤ ∈ V ) |
177 |
95
|
a1i |
⊢ ( ( 𝑁 ∈ ℕ0 ∧ ( 𝑎 ∈ ( mzPoly ‘ ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) ∧ 𝑏 ∈ ( mzPoly ‘ ℕ ) ) ) → ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ⊆ ℤ ) |
178 |
|
simprl |
⊢ ( ( 𝑁 ∈ ℕ0 ∧ ( 𝑎 ∈ ( mzPoly ‘ ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) ∧ 𝑏 ∈ ( mzPoly ‘ ℕ ) ) ) → 𝑎 ∈ ( mzPoly ‘ ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) ) |
179 |
|
mzpresrename |
⊢ ( ( ℤ ∈ V ∧ ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ⊆ ℤ ∧ 𝑎 ∈ ( mzPoly ‘ ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) ) → ( 𝑔 ∈ ( ℤ ↑m ℤ ) ↦ ( 𝑎 ‘ ( 𝑔 ↾ ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) ) ) ∈ ( mzPoly ‘ ℤ ) ) |
180 |
176 177 178 179
|
syl3anc |
⊢ ( ( 𝑁 ∈ ℕ0 ∧ ( 𝑎 ∈ ( mzPoly ‘ ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) ∧ 𝑏 ∈ ( mzPoly ‘ ℕ ) ) ) → ( 𝑔 ∈ ( ℤ ↑m ℤ ) ↦ ( 𝑎 ‘ ( 𝑔 ↾ ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) ) ) ∈ ( mzPoly ‘ ℤ ) ) |
181 |
|
2nn0 |
⊢ 2 ∈ ℕ0 |
182 |
|
mzpexpmpt |
⊢ ( ( ( 𝑔 ∈ ( ℤ ↑m ℤ ) ↦ ( 𝑎 ‘ ( 𝑔 ↾ ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) ) ) ∈ ( mzPoly ‘ ℤ ) ∧ 2 ∈ ℕ0 ) → ( 𝑔 ∈ ( ℤ ↑m ℤ ) ↦ ( ( 𝑎 ‘ ( 𝑔 ↾ ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) ) ↑ 2 ) ) ∈ ( mzPoly ‘ ℤ ) ) |
183 |
180 181 182
|
sylancl |
⊢ ( ( 𝑁 ∈ ℕ0 ∧ ( 𝑎 ∈ ( mzPoly ‘ ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) ∧ 𝑏 ∈ ( mzPoly ‘ ℕ ) ) ) → ( 𝑔 ∈ ( ℤ ↑m ℤ ) ↦ ( ( 𝑎 ‘ ( 𝑔 ↾ ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) ) ↑ 2 ) ) ∈ ( mzPoly ‘ ℤ ) ) |
184 |
99
|
a1i |
⊢ ( ( 𝑁 ∈ ℕ0 ∧ ( 𝑎 ∈ ( mzPoly ‘ ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) ∧ 𝑏 ∈ ( mzPoly ‘ ℕ ) ) ) → ℕ ⊆ ℤ ) |
185 |
|
simprr |
⊢ ( ( 𝑁 ∈ ℕ0 ∧ ( 𝑎 ∈ ( mzPoly ‘ ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) ∧ 𝑏 ∈ ( mzPoly ‘ ℕ ) ) ) → 𝑏 ∈ ( mzPoly ‘ ℕ ) ) |
186 |
|
mzpresrename |
⊢ ( ( ℤ ∈ V ∧ ℕ ⊆ ℤ ∧ 𝑏 ∈ ( mzPoly ‘ ℕ ) ) → ( 𝑔 ∈ ( ℤ ↑m ℤ ) ↦ ( 𝑏 ‘ ( 𝑔 ↾ ℕ ) ) ) ∈ ( mzPoly ‘ ℤ ) ) |
187 |
176 184 185 186
|
syl3anc |
⊢ ( ( 𝑁 ∈ ℕ0 ∧ ( 𝑎 ∈ ( mzPoly ‘ ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) ∧ 𝑏 ∈ ( mzPoly ‘ ℕ ) ) ) → ( 𝑔 ∈ ( ℤ ↑m ℤ ) ↦ ( 𝑏 ‘ ( 𝑔 ↾ ℕ ) ) ) ∈ ( mzPoly ‘ ℤ ) ) |
188 |
|
mzpexpmpt |
⊢ ( ( ( 𝑔 ∈ ( ℤ ↑m ℤ ) ↦ ( 𝑏 ‘ ( 𝑔 ↾ ℕ ) ) ) ∈ ( mzPoly ‘ ℤ ) ∧ 2 ∈ ℕ0 ) → ( 𝑔 ∈ ( ℤ ↑m ℤ ) ↦ ( ( 𝑏 ‘ ( 𝑔 ↾ ℕ ) ) ↑ 2 ) ) ∈ ( mzPoly ‘ ℤ ) ) |
189 |
187 181 188
|
sylancl |
⊢ ( ( 𝑁 ∈ ℕ0 ∧ ( 𝑎 ∈ ( mzPoly ‘ ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) ∧ 𝑏 ∈ ( mzPoly ‘ ℕ ) ) ) → ( 𝑔 ∈ ( ℤ ↑m ℤ ) ↦ ( ( 𝑏 ‘ ( 𝑔 ↾ ℕ ) ) ↑ 2 ) ) ∈ ( mzPoly ‘ ℤ ) ) |
190 |
|
mzpaddmpt |
⊢ ( ( ( 𝑔 ∈ ( ℤ ↑m ℤ ) ↦ ( ( 𝑎 ‘ ( 𝑔 ↾ ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) ) ↑ 2 ) ) ∈ ( mzPoly ‘ ℤ ) ∧ ( 𝑔 ∈ ( ℤ ↑m ℤ ) ↦ ( ( 𝑏 ‘ ( 𝑔 ↾ ℕ ) ) ↑ 2 ) ) ∈ ( mzPoly ‘ ℤ ) ) → ( 𝑔 ∈ ( ℤ ↑m ℤ ) ↦ ( ( ( 𝑎 ‘ ( 𝑔 ↾ ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) ) ↑ 2 ) + ( ( 𝑏 ‘ ( 𝑔 ↾ ℕ ) ) ↑ 2 ) ) ) ∈ ( mzPoly ‘ ℤ ) ) |
191 |
183 189 190
|
syl2anc |
⊢ ( ( 𝑁 ∈ ℕ0 ∧ ( 𝑎 ∈ ( mzPoly ‘ ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) ∧ 𝑏 ∈ ( mzPoly ‘ ℕ ) ) ) → ( 𝑔 ∈ ( ℤ ↑m ℤ ) ↦ ( ( ( 𝑎 ‘ ( 𝑔 ↾ ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) ) ↑ 2 ) + ( ( 𝑏 ‘ ( 𝑔 ↾ ℕ ) ) ↑ 2 ) ) ) ∈ ( mzPoly ‘ ℤ ) ) |
192 |
|
eldioph2 |
⊢ ( ( 𝑁 ∈ ℕ0 ∧ ( ℤ ∈ V ∧ ( 1 ... 𝑁 ) ⊆ ℤ ) ∧ ( 𝑔 ∈ ( ℤ ↑m ℤ ) ↦ ( ( ( 𝑎 ‘ ( 𝑔 ↾ ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) ) ↑ 2 ) + ( ( 𝑏 ‘ ( 𝑔 ↾ ℕ ) ) ↑ 2 ) ) ) ∈ ( mzPoly ‘ ℤ ) ) → { 𝑐 ∣ ∃ 𝑓 ∈ ( ℕ0 ↑m ℤ ) ( 𝑐 = ( 𝑓 ↾ ( 1 ... 𝑁 ) ) ∧ ( ( 𝑔 ∈ ( ℤ ↑m ℤ ) ↦ ( ( ( 𝑎 ‘ ( 𝑔 ↾ ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) ) ↑ 2 ) + ( ( 𝑏 ‘ ( 𝑔 ↾ ℕ ) ) ↑ 2 ) ) ) ‘ 𝑓 ) = 0 ) } ∈ ( Dioph ‘ 𝑁 ) ) |
193 |
170 175 191 192
|
syl3anc |
⊢ ( ( 𝑁 ∈ ℕ0 ∧ ( 𝑎 ∈ ( mzPoly ‘ ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) ∧ 𝑏 ∈ ( mzPoly ‘ ℕ ) ) ) → { 𝑐 ∣ ∃ 𝑓 ∈ ( ℕ0 ↑m ℤ ) ( 𝑐 = ( 𝑓 ↾ ( 1 ... 𝑁 ) ) ∧ ( ( 𝑔 ∈ ( ℤ ↑m ℤ ) ↦ ( ( ( 𝑎 ‘ ( 𝑔 ↾ ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) ) ↑ 2 ) + ( ( 𝑏 ‘ ( 𝑔 ↾ ℕ ) ) ↑ 2 ) ) ) ‘ 𝑓 ) = 0 ) } ∈ ( Dioph ‘ 𝑁 ) ) |
194 |
169 193
|
eqeltrd |
⊢ ( ( 𝑁 ∈ ℕ0 ∧ ( 𝑎 ∈ ( mzPoly ‘ ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) ∧ 𝑏 ∈ ( mzPoly ‘ ℕ ) ) ) → ( { 𝑐 ∣ ∃ 𝑑 ∈ ( ℕ0 ↑m ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) ( 𝑐 = ( 𝑑 ↾ ( 1 ... 𝑁 ) ) ∧ ( 𝑎 ‘ 𝑑 ) = 0 ) } ∩ { 𝑐 ∣ ∃ 𝑒 ∈ ( ℕ0 ↑m ℕ ) ( 𝑐 = ( 𝑒 ↾ ( 1 ... 𝑁 ) ) ∧ ( 𝑏 ‘ 𝑒 ) = 0 ) } ) ∈ ( Dioph ‘ 𝑁 ) ) |
195 |
|
ineq12 |
⊢ ( ( 𝐴 = { 𝑐 ∣ ∃ 𝑑 ∈ ( ℕ0 ↑m ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) ( 𝑐 = ( 𝑑 ↾ ( 1 ... 𝑁 ) ) ∧ ( 𝑎 ‘ 𝑑 ) = 0 ) } ∧ 𝐵 = { 𝑐 ∣ ∃ 𝑒 ∈ ( ℕ0 ↑m ℕ ) ( 𝑐 = ( 𝑒 ↾ ( 1 ... 𝑁 ) ) ∧ ( 𝑏 ‘ 𝑒 ) = 0 ) } ) → ( 𝐴 ∩ 𝐵 ) = ( { 𝑐 ∣ ∃ 𝑑 ∈ ( ℕ0 ↑m ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) ( 𝑐 = ( 𝑑 ↾ ( 1 ... 𝑁 ) ) ∧ ( 𝑎 ‘ 𝑑 ) = 0 ) } ∩ { 𝑐 ∣ ∃ 𝑒 ∈ ( ℕ0 ↑m ℕ ) ( 𝑐 = ( 𝑒 ↾ ( 1 ... 𝑁 ) ) ∧ ( 𝑏 ‘ 𝑒 ) = 0 ) } ) ) |
196 |
195
|
eleq1d |
⊢ ( ( 𝐴 = { 𝑐 ∣ ∃ 𝑑 ∈ ( ℕ0 ↑m ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) ( 𝑐 = ( 𝑑 ↾ ( 1 ... 𝑁 ) ) ∧ ( 𝑎 ‘ 𝑑 ) = 0 ) } ∧ 𝐵 = { 𝑐 ∣ ∃ 𝑒 ∈ ( ℕ0 ↑m ℕ ) ( 𝑐 = ( 𝑒 ↾ ( 1 ... 𝑁 ) ) ∧ ( 𝑏 ‘ 𝑒 ) = 0 ) } ) → ( ( 𝐴 ∩ 𝐵 ) ∈ ( Dioph ‘ 𝑁 ) ↔ ( { 𝑐 ∣ ∃ 𝑑 ∈ ( ℕ0 ↑m ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) ( 𝑐 = ( 𝑑 ↾ ( 1 ... 𝑁 ) ) ∧ ( 𝑎 ‘ 𝑑 ) = 0 ) } ∩ { 𝑐 ∣ ∃ 𝑒 ∈ ( ℕ0 ↑m ℕ ) ( 𝑐 = ( 𝑒 ↾ ( 1 ... 𝑁 ) ) ∧ ( 𝑏 ‘ 𝑒 ) = 0 ) } ) ∈ ( Dioph ‘ 𝑁 ) ) ) |
197 |
194 196
|
syl5ibrcom |
⊢ ( ( 𝑁 ∈ ℕ0 ∧ ( 𝑎 ∈ ( mzPoly ‘ ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) ∧ 𝑏 ∈ ( mzPoly ‘ ℕ ) ) ) → ( ( 𝐴 = { 𝑐 ∣ ∃ 𝑑 ∈ ( ℕ0 ↑m ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) ( 𝑐 = ( 𝑑 ↾ ( 1 ... 𝑁 ) ) ∧ ( 𝑎 ‘ 𝑑 ) = 0 ) } ∧ 𝐵 = { 𝑐 ∣ ∃ 𝑒 ∈ ( ℕ0 ↑m ℕ ) ( 𝑐 = ( 𝑒 ↾ ( 1 ... 𝑁 ) ) ∧ ( 𝑏 ‘ 𝑒 ) = 0 ) } ) → ( 𝐴 ∩ 𝐵 ) ∈ ( Dioph ‘ 𝑁 ) ) ) |
198 |
197
|
rexlimdvva |
⊢ ( 𝑁 ∈ ℕ0 → ( ∃ 𝑎 ∈ ( mzPoly ‘ ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) ∃ 𝑏 ∈ ( mzPoly ‘ ℕ ) ( 𝐴 = { 𝑐 ∣ ∃ 𝑑 ∈ ( ℕ0 ↑m ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) ( 𝑐 = ( 𝑑 ↾ ( 1 ... 𝑁 ) ) ∧ ( 𝑎 ‘ 𝑑 ) = 0 ) } ∧ 𝐵 = { 𝑐 ∣ ∃ 𝑒 ∈ ( ℕ0 ↑m ℕ ) ( 𝑐 = ( 𝑒 ↾ ( 1 ... 𝑁 ) ) ∧ ( 𝑏 ‘ 𝑒 ) = 0 ) } ) → ( 𝐴 ∩ 𝐵 ) ∈ ( Dioph ‘ 𝑁 ) ) ) |
199 |
29 198
|
syl5bir |
⊢ ( 𝑁 ∈ ℕ0 → ( ( ∃ 𝑎 ∈ ( mzPoly ‘ ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) 𝐴 = { 𝑐 ∣ ∃ 𝑑 ∈ ( ℕ0 ↑m ( ℤ ∖ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) ( 𝑐 = ( 𝑑 ↾ ( 1 ... 𝑁 ) ) ∧ ( 𝑎 ‘ 𝑑 ) = 0 ) } ∧ ∃ 𝑏 ∈ ( mzPoly ‘ ℕ ) 𝐵 = { 𝑐 ∣ ∃ 𝑒 ∈ ( ℕ0 ↑m ℕ ) ( 𝑐 = ( 𝑒 ↾ ( 1 ... 𝑁 ) ) ∧ ( 𝑏 ‘ 𝑒 ) = 0 ) } ) → ( 𝐴 ∩ 𝐵 ) ∈ ( Dioph ‘ 𝑁 ) ) ) |
200 |
28 199
|
sylbid |
⊢ ( 𝑁 ∈ ℕ0 → ( ( 𝐴 ∈ ( Dioph ‘ 𝑁 ) ∧ 𝐵 ∈ ( Dioph ‘ 𝑁 ) ) → ( 𝐴 ∩ 𝐵 ) ∈ ( Dioph ‘ 𝑁 ) ) ) |
201 |
1 200
|
syl |
⊢ ( 𝐴 ∈ ( Dioph ‘ 𝑁 ) → ( ( 𝐴 ∈ ( Dioph ‘ 𝑁 ) ∧ 𝐵 ∈ ( Dioph ‘ 𝑁 ) ) → ( 𝐴 ∩ 𝐵 ) ∈ ( Dioph ‘ 𝑁 ) ) ) |
202 |
201
|
anabsi5 |
⊢ ( ( 𝐴 ∈ ( Dioph ‘ 𝑁 ) ∧ 𝐵 ∈ ( Dioph ‘ 𝑁 ) ) → ( 𝐴 ∩ 𝐵 ) ∈ ( Dioph ‘ 𝑁 ) ) |