Step |
Hyp |
Ref |
Expression |
1 |
|
eldiophelnn0 |
⊢ ( 𝐴 ∈ ( Dioph ‘ 𝑁 ) → 𝑁 ∈ ℕ0 ) |
2 |
|
nnex |
⊢ ℕ ∈ V |
3 |
2
|
jctr |
⊢ ( 𝑁 ∈ ℕ0 → ( 𝑁 ∈ ℕ0 ∧ ℕ ∈ V ) ) |
4 |
|
1z |
⊢ 1 ∈ ℤ |
5 |
|
nnuz |
⊢ ℕ = ( ℤ≥ ‘ 1 ) |
6 |
5
|
uzinf |
⊢ ( 1 ∈ ℤ → ¬ ℕ ∈ Fin ) |
7 |
4 6
|
ax-mp |
⊢ ¬ ℕ ∈ Fin |
8 |
|
elfznn |
⊢ ( 𝑎 ∈ ( 1 ... 𝑁 ) → 𝑎 ∈ ℕ ) |
9 |
8
|
ssriv |
⊢ ( 1 ... 𝑁 ) ⊆ ℕ |
10 |
7 9
|
pm3.2i |
⊢ ( ¬ ℕ ∈ Fin ∧ ( 1 ... 𝑁 ) ⊆ ℕ ) |
11 |
|
eldioph2b |
⊢ ( ( ( 𝑁 ∈ ℕ0 ∧ ℕ ∈ V ) ∧ ( ¬ ℕ ∈ Fin ∧ ( 1 ... 𝑁 ) ⊆ ℕ ) ) → ( 𝐴 ∈ ( Dioph ‘ 𝑁 ) ↔ ∃ 𝑎 ∈ ( mzPoly ‘ ℕ ) 𝐴 = { 𝑏 ∣ ∃ 𝑑 ∈ ( ℕ0 ↑m ℕ ) ( 𝑏 = ( 𝑑 ↾ ( 1 ... 𝑁 ) ) ∧ ( 𝑎 ‘ 𝑑 ) = 0 ) } ) ) |
12 |
|
eldioph2b |
⊢ ( ( ( 𝑁 ∈ ℕ0 ∧ ℕ ∈ V ) ∧ ( ¬ ℕ ∈ Fin ∧ ( 1 ... 𝑁 ) ⊆ ℕ ) ) → ( 𝐵 ∈ ( Dioph ‘ 𝑁 ) ↔ ∃ 𝑐 ∈ ( mzPoly ‘ ℕ ) 𝐵 = { 𝑏 ∣ ∃ 𝑑 ∈ ( ℕ0 ↑m ℕ ) ( 𝑏 = ( 𝑑 ↾ ( 1 ... 𝑁 ) ) ∧ ( 𝑐 ‘ 𝑑 ) = 0 ) } ) ) |
13 |
11 12
|
anbi12d |
⊢ ( ( ( 𝑁 ∈ ℕ0 ∧ ℕ ∈ V ) ∧ ( ¬ ℕ ∈ Fin ∧ ( 1 ... 𝑁 ) ⊆ ℕ ) ) → ( ( 𝐴 ∈ ( Dioph ‘ 𝑁 ) ∧ 𝐵 ∈ ( Dioph ‘ 𝑁 ) ) ↔ ( ∃ 𝑎 ∈ ( mzPoly ‘ ℕ ) 𝐴 = { 𝑏 ∣ ∃ 𝑑 ∈ ( ℕ0 ↑m ℕ ) ( 𝑏 = ( 𝑑 ↾ ( 1 ... 𝑁 ) ) ∧ ( 𝑎 ‘ 𝑑 ) = 0 ) } ∧ ∃ 𝑐 ∈ ( mzPoly ‘ ℕ ) 𝐵 = { 𝑏 ∣ ∃ 𝑑 ∈ ( ℕ0 ↑m ℕ ) ( 𝑏 = ( 𝑑 ↾ ( 1 ... 𝑁 ) ) ∧ ( 𝑐 ‘ 𝑑 ) = 0 ) } ) ) ) |
14 |
3 10 13
|
sylancl |
⊢ ( 𝑁 ∈ ℕ0 → ( ( 𝐴 ∈ ( Dioph ‘ 𝑁 ) ∧ 𝐵 ∈ ( Dioph ‘ 𝑁 ) ) ↔ ( ∃ 𝑎 ∈ ( mzPoly ‘ ℕ ) 𝐴 = { 𝑏 ∣ ∃ 𝑑 ∈ ( ℕ0 ↑m ℕ ) ( 𝑏 = ( 𝑑 ↾ ( 1 ... 𝑁 ) ) ∧ ( 𝑎 ‘ 𝑑 ) = 0 ) } ∧ ∃ 𝑐 ∈ ( mzPoly ‘ ℕ ) 𝐵 = { 𝑏 ∣ ∃ 𝑑 ∈ ( ℕ0 ↑m ℕ ) ( 𝑏 = ( 𝑑 ↾ ( 1 ... 𝑁 ) ) ∧ ( 𝑐 ‘ 𝑑 ) = 0 ) } ) ) ) |
15 |
|
reeanv |
⊢ ( ∃ 𝑎 ∈ ( mzPoly ‘ ℕ ) ∃ 𝑐 ∈ ( mzPoly ‘ ℕ ) ( 𝐴 = { 𝑏 ∣ ∃ 𝑑 ∈ ( ℕ0 ↑m ℕ ) ( 𝑏 = ( 𝑑 ↾ ( 1 ... 𝑁 ) ) ∧ ( 𝑎 ‘ 𝑑 ) = 0 ) } ∧ 𝐵 = { 𝑏 ∣ ∃ 𝑑 ∈ ( ℕ0 ↑m ℕ ) ( 𝑏 = ( 𝑑 ↾ ( 1 ... 𝑁 ) ) ∧ ( 𝑐 ‘ 𝑑 ) = 0 ) } ) ↔ ( ∃ 𝑎 ∈ ( mzPoly ‘ ℕ ) 𝐴 = { 𝑏 ∣ ∃ 𝑑 ∈ ( ℕ0 ↑m ℕ ) ( 𝑏 = ( 𝑑 ↾ ( 1 ... 𝑁 ) ) ∧ ( 𝑎 ‘ 𝑑 ) = 0 ) } ∧ ∃ 𝑐 ∈ ( mzPoly ‘ ℕ ) 𝐵 = { 𝑏 ∣ ∃ 𝑑 ∈ ( ℕ0 ↑m ℕ ) ( 𝑏 = ( 𝑑 ↾ ( 1 ... 𝑁 ) ) ∧ ( 𝑐 ‘ 𝑑 ) = 0 ) } ) ) |
16 |
|
unab |
⊢ ( { 𝑏 ∣ ∃ 𝑑 ∈ ( ℕ0 ↑m ℕ ) ( 𝑏 = ( 𝑑 ↾ ( 1 ... 𝑁 ) ) ∧ ( 𝑎 ‘ 𝑑 ) = 0 ) } ∪ { 𝑏 ∣ ∃ 𝑑 ∈ ( ℕ0 ↑m ℕ ) ( 𝑏 = ( 𝑑 ↾ ( 1 ... 𝑁 ) ) ∧ ( 𝑐 ‘ 𝑑 ) = 0 ) } ) = { 𝑏 ∣ ( ∃ 𝑑 ∈ ( ℕ0 ↑m ℕ ) ( 𝑏 = ( 𝑑 ↾ ( 1 ... 𝑁 ) ) ∧ ( 𝑎 ‘ 𝑑 ) = 0 ) ∨ ∃ 𝑑 ∈ ( ℕ0 ↑m ℕ ) ( 𝑏 = ( 𝑑 ↾ ( 1 ... 𝑁 ) ) ∧ ( 𝑐 ‘ 𝑑 ) = 0 ) ) } |
17 |
|
r19.43 |
⊢ ( ∃ 𝑑 ∈ ( ℕ0 ↑m ℕ ) ( ( 𝑏 = ( 𝑑 ↾ ( 1 ... 𝑁 ) ) ∧ ( 𝑎 ‘ 𝑑 ) = 0 ) ∨ ( 𝑏 = ( 𝑑 ↾ ( 1 ... 𝑁 ) ) ∧ ( 𝑐 ‘ 𝑑 ) = 0 ) ) ↔ ( ∃ 𝑑 ∈ ( ℕ0 ↑m ℕ ) ( 𝑏 = ( 𝑑 ↾ ( 1 ... 𝑁 ) ) ∧ ( 𝑎 ‘ 𝑑 ) = 0 ) ∨ ∃ 𝑑 ∈ ( ℕ0 ↑m ℕ ) ( 𝑏 = ( 𝑑 ↾ ( 1 ... 𝑁 ) ) ∧ ( 𝑐 ‘ 𝑑 ) = 0 ) ) ) |
18 |
|
andi |
⊢ ( ( 𝑏 = ( 𝑑 ↾ ( 1 ... 𝑁 ) ) ∧ ( ( 𝑎 ‘ 𝑑 ) = 0 ∨ ( 𝑐 ‘ 𝑑 ) = 0 ) ) ↔ ( ( 𝑏 = ( 𝑑 ↾ ( 1 ... 𝑁 ) ) ∧ ( 𝑎 ‘ 𝑑 ) = 0 ) ∨ ( 𝑏 = ( 𝑑 ↾ ( 1 ... 𝑁 ) ) ∧ ( 𝑐 ‘ 𝑑 ) = 0 ) ) ) |
19 |
|
zex |
⊢ ℤ ∈ V |
20 |
|
nn0ssz |
⊢ ℕ0 ⊆ ℤ |
21 |
|
mapss |
⊢ ( ( ℤ ∈ V ∧ ℕ0 ⊆ ℤ ) → ( ℕ0 ↑m ℕ ) ⊆ ( ℤ ↑m ℕ ) ) |
22 |
19 20 21
|
mp2an |
⊢ ( ℕ0 ↑m ℕ ) ⊆ ( ℤ ↑m ℕ ) |
23 |
22
|
sseli |
⊢ ( 𝑑 ∈ ( ℕ0 ↑m ℕ ) → 𝑑 ∈ ( ℤ ↑m ℕ ) ) |
24 |
23
|
adantl |
⊢ ( ( ( 𝑁 ∈ ℕ0 ∧ ( 𝑎 ∈ ( mzPoly ‘ ℕ ) ∧ 𝑐 ∈ ( mzPoly ‘ ℕ ) ) ) ∧ 𝑑 ∈ ( ℕ0 ↑m ℕ ) ) → 𝑑 ∈ ( ℤ ↑m ℕ ) ) |
25 |
|
fveq2 |
⊢ ( 𝑒 = 𝑑 → ( 𝑎 ‘ 𝑒 ) = ( 𝑎 ‘ 𝑑 ) ) |
26 |
|
fveq2 |
⊢ ( 𝑒 = 𝑑 → ( 𝑐 ‘ 𝑒 ) = ( 𝑐 ‘ 𝑑 ) ) |
27 |
25 26
|
oveq12d |
⊢ ( 𝑒 = 𝑑 → ( ( 𝑎 ‘ 𝑒 ) · ( 𝑐 ‘ 𝑒 ) ) = ( ( 𝑎 ‘ 𝑑 ) · ( 𝑐 ‘ 𝑑 ) ) ) |
28 |
|
eqid |
⊢ ( 𝑒 ∈ ( ℤ ↑m ℕ ) ↦ ( ( 𝑎 ‘ 𝑒 ) · ( 𝑐 ‘ 𝑒 ) ) ) = ( 𝑒 ∈ ( ℤ ↑m ℕ ) ↦ ( ( 𝑎 ‘ 𝑒 ) · ( 𝑐 ‘ 𝑒 ) ) ) |
29 |
|
ovex |
⊢ ( ( 𝑎 ‘ 𝑑 ) · ( 𝑐 ‘ 𝑑 ) ) ∈ V |
30 |
27 28 29
|
fvmpt |
⊢ ( 𝑑 ∈ ( ℤ ↑m ℕ ) → ( ( 𝑒 ∈ ( ℤ ↑m ℕ ) ↦ ( ( 𝑎 ‘ 𝑒 ) · ( 𝑐 ‘ 𝑒 ) ) ) ‘ 𝑑 ) = ( ( 𝑎 ‘ 𝑑 ) · ( 𝑐 ‘ 𝑑 ) ) ) |
31 |
24 30
|
syl |
⊢ ( ( ( 𝑁 ∈ ℕ0 ∧ ( 𝑎 ∈ ( mzPoly ‘ ℕ ) ∧ 𝑐 ∈ ( mzPoly ‘ ℕ ) ) ) ∧ 𝑑 ∈ ( ℕ0 ↑m ℕ ) ) → ( ( 𝑒 ∈ ( ℤ ↑m ℕ ) ↦ ( ( 𝑎 ‘ 𝑒 ) · ( 𝑐 ‘ 𝑒 ) ) ) ‘ 𝑑 ) = ( ( 𝑎 ‘ 𝑑 ) · ( 𝑐 ‘ 𝑑 ) ) ) |
32 |
31
|
eqeq1d |
⊢ ( ( ( 𝑁 ∈ ℕ0 ∧ ( 𝑎 ∈ ( mzPoly ‘ ℕ ) ∧ 𝑐 ∈ ( mzPoly ‘ ℕ ) ) ) ∧ 𝑑 ∈ ( ℕ0 ↑m ℕ ) ) → ( ( ( 𝑒 ∈ ( ℤ ↑m ℕ ) ↦ ( ( 𝑎 ‘ 𝑒 ) · ( 𝑐 ‘ 𝑒 ) ) ) ‘ 𝑑 ) = 0 ↔ ( ( 𝑎 ‘ 𝑑 ) · ( 𝑐 ‘ 𝑑 ) ) = 0 ) ) |
33 |
|
simplrl |
⊢ ( ( ( 𝑁 ∈ ℕ0 ∧ ( 𝑎 ∈ ( mzPoly ‘ ℕ ) ∧ 𝑐 ∈ ( mzPoly ‘ ℕ ) ) ) ∧ 𝑑 ∈ ( ℕ0 ↑m ℕ ) ) → 𝑎 ∈ ( mzPoly ‘ ℕ ) ) |
34 |
|
mzpf |
⊢ ( 𝑎 ∈ ( mzPoly ‘ ℕ ) → 𝑎 : ( ℤ ↑m ℕ ) ⟶ ℤ ) |
35 |
33 34
|
syl |
⊢ ( ( ( 𝑁 ∈ ℕ0 ∧ ( 𝑎 ∈ ( mzPoly ‘ ℕ ) ∧ 𝑐 ∈ ( mzPoly ‘ ℕ ) ) ) ∧ 𝑑 ∈ ( ℕ0 ↑m ℕ ) ) → 𝑎 : ( ℤ ↑m ℕ ) ⟶ ℤ ) |
36 |
35 24
|
ffvelrnd |
⊢ ( ( ( 𝑁 ∈ ℕ0 ∧ ( 𝑎 ∈ ( mzPoly ‘ ℕ ) ∧ 𝑐 ∈ ( mzPoly ‘ ℕ ) ) ) ∧ 𝑑 ∈ ( ℕ0 ↑m ℕ ) ) → ( 𝑎 ‘ 𝑑 ) ∈ ℤ ) |
37 |
36
|
zcnd |
⊢ ( ( ( 𝑁 ∈ ℕ0 ∧ ( 𝑎 ∈ ( mzPoly ‘ ℕ ) ∧ 𝑐 ∈ ( mzPoly ‘ ℕ ) ) ) ∧ 𝑑 ∈ ( ℕ0 ↑m ℕ ) ) → ( 𝑎 ‘ 𝑑 ) ∈ ℂ ) |
38 |
|
simplrr |
⊢ ( ( ( 𝑁 ∈ ℕ0 ∧ ( 𝑎 ∈ ( mzPoly ‘ ℕ ) ∧ 𝑐 ∈ ( mzPoly ‘ ℕ ) ) ) ∧ 𝑑 ∈ ( ℕ0 ↑m ℕ ) ) → 𝑐 ∈ ( mzPoly ‘ ℕ ) ) |
39 |
|
mzpf |
⊢ ( 𝑐 ∈ ( mzPoly ‘ ℕ ) → 𝑐 : ( ℤ ↑m ℕ ) ⟶ ℤ ) |
40 |
38 39
|
syl |
⊢ ( ( ( 𝑁 ∈ ℕ0 ∧ ( 𝑎 ∈ ( mzPoly ‘ ℕ ) ∧ 𝑐 ∈ ( mzPoly ‘ ℕ ) ) ) ∧ 𝑑 ∈ ( ℕ0 ↑m ℕ ) ) → 𝑐 : ( ℤ ↑m ℕ ) ⟶ ℤ ) |
41 |
40 24
|
ffvelrnd |
⊢ ( ( ( 𝑁 ∈ ℕ0 ∧ ( 𝑎 ∈ ( mzPoly ‘ ℕ ) ∧ 𝑐 ∈ ( mzPoly ‘ ℕ ) ) ) ∧ 𝑑 ∈ ( ℕ0 ↑m ℕ ) ) → ( 𝑐 ‘ 𝑑 ) ∈ ℤ ) |
42 |
41
|
zcnd |
⊢ ( ( ( 𝑁 ∈ ℕ0 ∧ ( 𝑎 ∈ ( mzPoly ‘ ℕ ) ∧ 𝑐 ∈ ( mzPoly ‘ ℕ ) ) ) ∧ 𝑑 ∈ ( ℕ0 ↑m ℕ ) ) → ( 𝑐 ‘ 𝑑 ) ∈ ℂ ) |
43 |
37 42
|
mul0ord |
⊢ ( ( ( 𝑁 ∈ ℕ0 ∧ ( 𝑎 ∈ ( mzPoly ‘ ℕ ) ∧ 𝑐 ∈ ( mzPoly ‘ ℕ ) ) ) ∧ 𝑑 ∈ ( ℕ0 ↑m ℕ ) ) → ( ( ( 𝑎 ‘ 𝑑 ) · ( 𝑐 ‘ 𝑑 ) ) = 0 ↔ ( ( 𝑎 ‘ 𝑑 ) = 0 ∨ ( 𝑐 ‘ 𝑑 ) = 0 ) ) ) |
44 |
32 43
|
bitr2d |
⊢ ( ( ( 𝑁 ∈ ℕ0 ∧ ( 𝑎 ∈ ( mzPoly ‘ ℕ ) ∧ 𝑐 ∈ ( mzPoly ‘ ℕ ) ) ) ∧ 𝑑 ∈ ( ℕ0 ↑m ℕ ) ) → ( ( ( 𝑎 ‘ 𝑑 ) = 0 ∨ ( 𝑐 ‘ 𝑑 ) = 0 ) ↔ ( ( 𝑒 ∈ ( ℤ ↑m ℕ ) ↦ ( ( 𝑎 ‘ 𝑒 ) · ( 𝑐 ‘ 𝑒 ) ) ) ‘ 𝑑 ) = 0 ) ) |
45 |
44
|
anbi2d |
⊢ ( ( ( 𝑁 ∈ ℕ0 ∧ ( 𝑎 ∈ ( mzPoly ‘ ℕ ) ∧ 𝑐 ∈ ( mzPoly ‘ ℕ ) ) ) ∧ 𝑑 ∈ ( ℕ0 ↑m ℕ ) ) → ( ( 𝑏 = ( 𝑑 ↾ ( 1 ... 𝑁 ) ) ∧ ( ( 𝑎 ‘ 𝑑 ) = 0 ∨ ( 𝑐 ‘ 𝑑 ) = 0 ) ) ↔ ( 𝑏 = ( 𝑑 ↾ ( 1 ... 𝑁 ) ) ∧ ( ( 𝑒 ∈ ( ℤ ↑m ℕ ) ↦ ( ( 𝑎 ‘ 𝑒 ) · ( 𝑐 ‘ 𝑒 ) ) ) ‘ 𝑑 ) = 0 ) ) ) |
46 |
18 45
|
bitr3id |
⊢ ( ( ( 𝑁 ∈ ℕ0 ∧ ( 𝑎 ∈ ( mzPoly ‘ ℕ ) ∧ 𝑐 ∈ ( mzPoly ‘ ℕ ) ) ) ∧ 𝑑 ∈ ( ℕ0 ↑m ℕ ) ) → ( ( ( 𝑏 = ( 𝑑 ↾ ( 1 ... 𝑁 ) ) ∧ ( 𝑎 ‘ 𝑑 ) = 0 ) ∨ ( 𝑏 = ( 𝑑 ↾ ( 1 ... 𝑁 ) ) ∧ ( 𝑐 ‘ 𝑑 ) = 0 ) ) ↔ ( 𝑏 = ( 𝑑 ↾ ( 1 ... 𝑁 ) ) ∧ ( ( 𝑒 ∈ ( ℤ ↑m ℕ ) ↦ ( ( 𝑎 ‘ 𝑒 ) · ( 𝑐 ‘ 𝑒 ) ) ) ‘ 𝑑 ) = 0 ) ) ) |
47 |
46
|
rexbidva |
⊢ ( ( 𝑁 ∈ ℕ0 ∧ ( 𝑎 ∈ ( mzPoly ‘ ℕ ) ∧ 𝑐 ∈ ( mzPoly ‘ ℕ ) ) ) → ( ∃ 𝑑 ∈ ( ℕ0 ↑m ℕ ) ( ( 𝑏 = ( 𝑑 ↾ ( 1 ... 𝑁 ) ) ∧ ( 𝑎 ‘ 𝑑 ) = 0 ) ∨ ( 𝑏 = ( 𝑑 ↾ ( 1 ... 𝑁 ) ) ∧ ( 𝑐 ‘ 𝑑 ) = 0 ) ) ↔ ∃ 𝑑 ∈ ( ℕ0 ↑m ℕ ) ( 𝑏 = ( 𝑑 ↾ ( 1 ... 𝑁 ) ) ∧ ( ( 𝑒 ∈ ( ℤ ↑m ℕ ) ↦ ( ( 𝑎 ‘ 𝑒 ) · ( 𝑐 ‘ 𝑒 ) ) ) ‘ 𝑑 ) = 0 ) ) ) |
48 |
17 47
|
bitr3id |
⊢ ( ( 𝑁 ∈ ℕ0 ∧ ( 𝑎 ∈ ( mzPoly ‘ ℕ ) ∧ 𝑐 ∈ ( mzPoly ‘ ℕ ) ) ) → ( ( ∃ 𝑑 ∈ ( ℕ0 ↑m ℕ ) ( 𝑏 = ( 𝑑 ↾ ( 1 ... 𝑁 ) ) ∧ ( 𝑎 ‘ 𝑑 ) = 0 ) ∨ ∃ 𝑑 ∈ ( ℕ0 ↑m ℕ ) ( 𝑏 = ( 𝑑 ↾ ( 1 ... 𝑁 ) ) ∧ ( 𝑐 ‘ 𝑑 ) = 0 ) ) ↔ ∃ 𝑑 ∈ ( ℕ0 ↑m ℕ ) ( 𝑏 = ( 𝑑 ↾ ( 1 ... 𝑁 ) ) ∧ ( ( 𝑒 ∈ ( ℤ ↑m ℕ ) ↦ ( ( 𝑎 ‘ 𝑒 ) · ( 𝑐 ‘ 𝑒 ) ) ) ‘ 𝑑 ) = 0 ) ) ) |
49 |
48
|
abbidv |
⊢ ( ( 𝑁 ∈ ℕ0 ∧ ( 𝑎 ∈ ( mzPoly ‘ ℕ ) ∧ 𝑐 ∈ ( mzPoly ‘ ℕ ) ) ) → { 𝑏 ∣ ( ∃ 𝑑 ∈ ( ℕ0 ↑m ℕ ) ( 𝑏 = ( 𝑑 ↾ ( 1 ... 𝑁 ) ) ∧ ( 𝑎 ‘ 𝑑 ) = 0 ) ∨ ∃ 𝑑 ∈ ( ℕ0 ↑m ℕ ) ( 𝑏 = ( 𝑑 ↾ ( 1 ... 𝑁 ) ) ∧ ( 𝑐 ‘ 𝑑 ) = 0 ) ) } = { 𝑏 ∣ ∃ 𝑑 ∈ ( ℕ0 ↑m ℕ ) ( 𝑏 = ( 𝑑 ↾ ( 1 ... 𝑁 ) ) ∧ ( ( 𝑒 ∈ ( ℤ ↑m ℕ ) ↦ ( ( 𝑎 ‘ 𝑒 ) · ( 𝑐 ‘ 𝑒 ) ) ) ‘ 𝑑 ) = 0 ) } ) |
50 |
16 49
|
syl5eq |
⊢ ( ( 𝑁 ∈ ℕ0 ∧ ( 𝑎 ∈ ( mzPoly ‘ ℕ ) ∧ 𝑐 ∈ ( mzPoly ‘ ℕ ) ) ) → ( { 𝑏 ∣ ∃ 𝑑 ∈ ( ℕ0 ↑m ℕ ) ( 𝑏 = ( 𝑑 ↾ ( 1 ... 𝑁 ) ) ∧ ( 𝑎 ‘ 𝑑 ) = 0 ) } ∪ { 𝑏 ∣ ∃ 𝑑 ∈ ( ℕ0 ↑m ℕ ) ( 𝑏 = ( 𝑑 ↾ ( 1 ... 𝑁 ) ) ∧ ( 𝑐 ‘ 𝑑 ) = 0 ) } ) = { 𝑏 ∣ ∃ 𝑑 ∈ ( ℕ0 ↑m ℕ ) ( 𝑏 = ( 𝑑 ↾ ( 1 ... 𝑁 ) ) ∧ ( ( 𝑒 ∈ ( ℤ ↑m ℕ ) ↦ ( ( 𝑎 ‘ 𝑒 ) · ( 𝑐 ‘ 𝑒 ) ) ) ‘ 𝑑 ) = 0 ) } ) |
51 |
|
simpl |
⊢ ( ( 𝑁 ∈ ℕ0 ∧ ( 𝑎 ∈ ( mzPoly ‘ ℕ ) ∧ 𝑐 ∈ ( mzPoly ‘ ℕ ) ) ) → 𝑁 ∈ ℕ0 ) |
52 |
2 9
|
pm3.2i |
⊢ ( ℕ ∈ V ∧ ( 1 ... 𝑁 ) ⊆ ℕ ) |
53 |
52
|
a1i |
⊢ ( ( 𝑁 ∈ ℕ0 ∧ ( 𝑎 ∈ ( mzPoly ‘ ℕ ) ∧ 𝑐 ∈ ( mzPoly ‘ ℕ ) ) ) → ( ℕ ∈ V ∧ ( 1 ... 𝑁 ) ⊆ ℕ ) ) |
54 |
|
simprl |
⊢ ( ( 𝑁 ∈ ℕ0 ∧ ( 𝑎 ∈ ( mzPoly ‘ ℕ ) ∧ 𝑐 ∈ ( mzPoly ‘ ℕ ) ) ) → 𝑎 ∈ ( mzPoly ‘ ℕ ) ) |
55 |
54 34
|
syl |
⊢ ( ( 𝑁 ∈ ℕ0 ∧ ( 𝑎 ∈ ( mzPoly ‘ ℕ ) ∧ 𝑐 ∈ ( mzPoly ‘ ℕ ) ) ) → 𝑎 : ( ℤ ↑m ℕ ) ⟶ ℤ ) |
56 |
55
|
feqmptd |
⊢ ( ( 𝑁 ∈ ℕ0 ∧ ( 𝑎 ∈ ( mzPoly ‘ ℕ ) ∧ 𝑐 ∈ ( mzPoly ‘ ℕ ) ) ) → 𝑎 = ( 𝑒 ∈ ( ℤ ↑m ℕ ) ↦ ( 𝑎 ‘ 𝑒 ) ) ) |
57 |
56 54
|
eqeltrrd |
⊢ ( ( 𝑁 ∈ ℕ0 ∧ ( 𝑎 ∈ ( mzPoly ‘ ℕ ) ∧ 𝑐 ∈ ( mzPoly ‘ ℕ ) ) ) → ( 𝑒 ∈ ( ℤ ↑m ℕ ) ↦ ( 𝑎 ‘ 𝑒 ) ) ∈ ( mzPoly ‘ ℕ ) ) |
58 |
|
simprr |
⊢ ( ( 𝑁 ∈ ℕ0 ∧ ( 𝑎 ∈ ( mzPoly ‘ ℕ ) ∧ 𝑐 ∈ ( mzPoly ‘ ℕ ) ) ) → 𝑐 ∈ ( mzPoly ‘ ℕ ) ) |
59 |
58 39
|
syl |
⊢ ( ( 𝑁 ∈ ℕ0 ∧ ( 𝑎 ∈ ( mzPoly ‘ ℕ ) ∧ 𝑐 ∈ ( mzPoly ‘ ℕ ) ) ) → 𝑐 : ( ℤ ↑m ℕ ) ⟶ ℤ ) |
60 |
59
|
feqmptd |
⊢ ( ( 𝑁 ∈ ℕ0 ∧ ( 𝑎 ∈ ( mzPoly ‘ ℕ ) ∧ 𝑐 ∈ ( mzPoly ‘ ℕ ) ) ) → 𝑐 = ( 𝑒 ∈ ( ℤ ↑m ℕ ) ↦ ( 𝑐 ‘ 𝑒 ) ) ) |
61 |
60 58
|
eqeltrrd |
⊢ ( ( 𝑁 ∈ ℕ0 ∧ ( 𝑎 ∈ ( mzPoly ‘ ℕ ) ∧ 𝑐 ∈ ( mzPoly ‘ ℕ ) ) ) → ( 𝑒 ∈ ( ℤ ↑m ℕ ) ↦ ( 𝑐 ‘ 𝑒 ) ) ∈ ( mzPoly ‘ ℕ ) ) |
62 |
|
mzpmulmpt |
⊢ ( ( ( 𝑒 ∈ ( ℤ ↑m ℕ ) ↦ ( 𝑎 ‘ 𝑒 ) ) ∈ ( mzPoly ‘ ℕ ) ∧ ( 𝑒 ∈ ( ℤ ↑m ℕ ) ↦ ( 𝑐 ‘ 𝑒 ) ) ∈ ( mzPoly ‘ ℕ ) ) → ( 𝑒 ∈ ( ℤ ↑m ℕ ) ↦ ( ( 𝑎 ‘ 𝑒 ) · ( 𝑐 ‘ 𝑒 ) ) ) ∈ ( mzPoly ‘ ℕ ) ) |
63 |
57 61 62
|
syl2anc |
⊢ ( ( 𝑁 ∈ ℕ0 ∧ ( 𝑎 ∈ ( mzPoly ‘ ℕ ) ∧ 𝑐 ∈ ( mzPoly ‘ ℕ ) ) ) → ( 𝑒 ∈ ( ℤ ↑m ℕ ) ↦ ( ( 𝑎 ‘ 𝑒 ) · ( 𝑐 ‘ 𝑒 ) ) ) ∈ ( mzPoly ‘ ℕ ) ) |
64 |
|
eldioph2 |
⊢ ( ( 𝑁 ∈ ℕ0 ∧ ( ℕ ∈ V ∧ ( 1 ... 𝑁 ) ⊆ ℕ ) ∧ ( 𝑒 ∈ ( ℤ ↑m ℕ ) ↦ ( ( 𝑎 ‘ 𝑒 ) · ( 𝑐 ‘ 𝑒 ) ) ) ∈ ( mzPoly ‘ ℕ ) ) → { 𝑏 ∣ ∃ 𝑑 ∈ ( ℕ0 ↑m ℕ ) ( 𝑏 = ( 𝑑 ↾ ( 1 ... 𝑁 ) ) ∧ ( ( 𝑒 ∈ ( ℤ ↑m ℕ ) ↦ ( ( 𝑎 ‘ 𝑒 ) · ( 𝑐 ‘ 𝑒 ) ) ) ‘ 𝑑 ) = 0 ) } ∈ ( Dioph ‘ 𝑁 ) ) |
65 |
51 53 63 64
|
syl3anc |
⊢ ( ( 𝑁 ∈ ℕ0 ∧ ( 𝑎 ∈ ( mzPoly ‘ ℕ ) ∧ 𝑐 ∈ ( mzPoly ‘ ℕ ) ) ) → { 𝑏 ∣ ∃ 𝑑 ∈ ( ℕ0 ↑m ℕ ) ( 𝑏 = ( 𝑑 ↾ ( 1 ... 𝑁 ) ) ∧ ( ( 𝑒 ∈ ( ℤ ↑m ℕ ) ↦ ( ( 𝑎 ‘ 𝑒 ) · ( 𝑐 ‘ 𝑒 ) ) ) ‘ 𝑑 ) = 0 ) } ∈ ( Dioph ‘ 𝑁 ) ) |
66 |
50 65
|
eqeltrd |
⊢ ( ( 𝑁 ∈ ℕ0 ∧ ( 𝑎 ∈ ( mzPoly ‘ ℕ ) ∧ 𝑐 ∈ ( mzPoly ‘ ℕ ) ) ) → ( { 𝑏 ∣ ∃ 𝑑 ∈ ( ℕ0 ↑m ℕ ) ( 𝑏 = ( 𝑑 ↾ ( 1 ... 𝑁 ) ) ∧ ( 𝑎 ‘ 𝑑 ) = 0 ) } ∪ { 𝑏 ∣ ∃ 𝑑 ∈ ( ℕ0 ↑m ℕ ) ( 𝑏 = ( 𝑑 ↾ ( 1 ... 𝑁 ) ) ∧ ( 𝑐 ‘ 𝑑 ) = 0 ) } ) ∈ ( Dioph ‘ 𝑁 ) ) |
67 |
|
uneq12 |
⊢ ( ( 𝐴 = { 𝑏 ∣ ∃ 𝑑 ∈ ( ℕ0 ↑m ℕ ) ( 𝑏 = ( 𝑑 ↾ ( 1 ... 𝑁 ) ) ∧ ( 𝑎 ‘ 𝑑 ) = 0 ) } ∧ 𝐵 = { 𝑏 ∣ ∃ 𝑑 ∈ ( ℕ0 ↑m ℕ ) ( 𝑏 = ( 𝑑 ↾ ( 1 ... 𝑁 ) ) ∧ ( 𝑐 ‘ 𝑑 ) = 0 ) } ) → ( 𝐴 ∪ 𝐵 ) = ( { 𝑏 ∣ ∃ 𝑑 ∈ ( ℕ0 ↑m ℕ ) ( 𝑏 = ( 𝑑 ↾ ( 1 ... 𝑁 ) ) ∧ ( 𝑎 ‘ 𝑑 ) = 0 ) } ∪ { 𝑏 ∣ ∃ 𝑑 ∈ ( ℕ0 ↑m ℕ ) ( 𝑏 = ( 𝑑 ↾ ( 1 ... 𝑁 ) ) ∧ ( 𝑐 ‘ 𝑑 ) = 0 ) } ) ) |
68 |
67
|
eleq1d |
⊢ ( ( 𝐴 = { 𝑏 ∣ ∃ 𝑑 ∈ ( ℕ0 ↑m ℕ ) ( 𝑏 = ( 𝑑 ↾ ( 1 ... 𝑁 ) ) ∧ ( 𝑎 ‘ 𝑑 ) = 0 ) } ∧ 𝐵 = { 𝑏 ∣ ∃ 𝑑 ∈ ( ℕ0 ↑m ℕ ) ( 𝑏 = ( 𝑑 ↾ ( 1 ... 𝑁 ) ) ∧ ( 𝑐 ‘ 𝑑 ) = 0 ) } ) → ( ( 𝐴 ∪ 𝐵 ) ∈ ( Dioph ‘ 𝑁 ) ↔ ( { 𝑏 ∣ ∃ 𝑑 ∈ ( ℕ0 ↑m ℕ ) ( 𝑏 = ( 𝑑 ↾ ( 1 ... 𝑁 ) ) ∧ ( 𝑎 ‘ 𝑑 ) = 0 ) } ∪ { 𝑏 ∣ ∃ 𝑑 ∈ ( ℕ0 ↑m ℕ ) ( 𝑏 = ( 𝑑 ↾ ( 1 ... 𝑁 ) ) ∧ ( 𝑐 ‘ 𝑑 ) = 0 ) } ) ∈ ( Dioph ‘ 𝑁 ) ) ) |
69 |
66 68
|
syl5ibrcom |
⊢ ( ( 𝑁 ∈ ℕ0 ∧ ( 𝑎 ∈ ( mzPoly ‘ ℕ ) ∧ 𝑐 ∈ ( mzPoly ‘ ℕ ) ) ) → ( ( 𝐴 = { 𝑏 ∣ ∃ 𝑑 ∈ ( ℕ0 ↑m ℕ ) ( 𝑏 = ( 𝑑 ↾ ( 1 ... 𝑁 ) ) ∧ ( 𝑎 ‘ 𝑑 ) = 0 ) } ∧ 𝐵 = { 𝑏 ∣ ∃ 𝑑 ∈ ( ℕ0 ↑m ℕ ) ( 𝑏 = ( 𝑑 ↾ ( 1 ... 𝑁 ) ) ∧ ( 𝑐 ‘ 𝑑 ) = 0 ) } ) → ( 𝐴 ∪ 𝐵 ) ∈ ( Dioph ‘ 𝑁 ) ) ) |
70 |
69
|
rexlimdvva |
⊢ ( 𝑁 ∈ ℕ0 → ( ∃ 𝑎 ∈ ( mzPoly ‘ ℕ ) ∃ 𝑐 ∈ ( mzPoly ‘ ℕ ) ( 𝐴 = { 𝑏 ∣ ∃ 𝑑 ∈ ( ℕ0 ↑m ℕ ) ( 𝑏 = ( 𝑑 ↾ ( 1 ... 𝑁 ) ) ∧ ( 𝑎 ‘ 𝑑 ) = 0 ) } ∧ 𝐵 = { 𝑏 ∣ ∃ 𝑑 ∈ ( ℕ0 ↑m ℕ ) ( 𝑏 = ( 𝑑 ↾ ( 1 ... 𝑁 ) ) ∧ ( 𝑐 ‘ 𝑑 ) = 0 ) } ) → ( 𝐴 ∪ 𝐵 ) ∈ ( Dioph ‘ 𝑁 ) ) ) |
71 |
15 70
|
syl5bir |
⊢ ( 𝑁 ∈ ℕ0 → ( ( ∃ 𝑎 ∈ ( mzPoly ‘ ℕ ) 𝐴 = { 𝑏 ∣ ∃ 𝑑 ∈ ( ℕ0 ↑m ℕ ) ( 𝑏 = ( 𝑑 ↾ ( 1 ... 𝑁 ) ) ∧ ( 𝑎 ‘ 𝑑 ) = 0 ) } ∧ ∃ 𝑐 ∈ ( mzPoly ‘ ℕ ) 𝐵 = { 𝑏 ∣ ∃ 𝑑 ∈ ( ℕ0 ↑m ℕ ) ( 𝑏 = ( 𝑑 ↾ ( 1 ... 𝑁 ) ) ∧ ( 𝑐 ‘ 𝑑 ) = 0 ) } ) → ( 𝐴 ∪ 𝐵 ) ∈ ( Dioph ‘ 𝑁 ) ) ) |
72 |
14 71
|
sylbid |
⊢ ( 𝑁 ∈ ℕ0 → ( ( 𝐴 ∈ ( Dioph ‘ 𝑁 ) ∧ 𝐵 ∈ ( Dioph ‘ 𝑁 ) ) → ( 𝐴 ∪ 𝐵 ) ∈ ( Dioph ‘ 𝑁 ) ) ) |
73 |
1 72
|
syl |
⊢ ( 𝐴 ∈ ( Dioph ‘ 𝑁 ) → ( ( 𝐴 ∈ ( Dioph ‘ 𝑁 ) ∧ 𝐵 ∈ ( Dioph ‘ 𝑁 ) ) → ( 𝐴 ∪ 𝐵 ) ∈ ( Dioph ‘ 𝑁 ) ) ) |
74 |
73
|
anabsi5 |
⊢ ( ( 𝐴 ∈ ( Dioph ‘ 𝑁 ) ∧ 𝐵 ∈ ( Dioph ‘ 𝑁 ) ) → ( 𝐴 ∪ 𝐵 ) ∈ ( Dioph ‘ 𝑁 ) ) |