Mathbox for Stefan O'Rear < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  eldioph2b Structured version   Visualization version   GIF version

Theorem eldioph2b 39875
 Description: While Diophantine sets were defined to have a finite number of witness variables consequtively following the observable variables, this is not necessary; they can equivalently be taken to use any witness set (𝑆 ∖ (1...𝑁)). For instance, in diophin 39884 we use this to take the two input sets to have disjoint witness sets. (Contributed by Stefan O'Rear, 8-Oct-2014.)
Assertion
Ref Expression
eldioph2b (((𝑁 ∈ ℕ0𝑆 ∈ V) ∧ (¬ 𝑆 ∈ Fin ∧ (1...𝑁) ⊆ 𝑆)) → (𝐴 ∈ (Dioph‘𝑁) ↔ ∃𝑝 ∈ (mzPoly‘𝑆)𝐴 = {𝑡 ∣ ∃𝑢 ∈ (ℕ0m 𝑆)(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝𝑢) = 0)}))
Distinct variable groups:   𝐴,𝑝   𝑢,𝑁,𝑡,𝑝   𝑢,𝑆,𝑡,𝑝
Allowed substitution hints:   𝐴(𝑢,𝑡)

Proof of Theorem eldioph2b
Dummy variables 𝑎 𝑏 𝑐 𝑑 𝑒 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eldiophb 39869 . . 3 (𝐴 ∈ (Dioph‘𝑁) ↔ (𝑁 ∈ ℕ0 ∧ ∃𝑎 ∈ (ℤ𝑁)∃𝑏 ∈ (mzPoly‘(1...𝑎))𝐴 = {𝑡 ∣ ∃𝑑 ∈ (ℕ0m (1...𝑎))(𝑡 = (𝑑 ↾ (1...𝑁)) ∧ (𝑏𝑑) = 0)}))
2 simp-5r 785 . . . . . . . . 9 ((((((𝑁 ∈ ℕ0𝑆 ∈ V) ∧ (¬ 𝑆 ∈ Fin ∧ (1...𝑁) ⊆ 𝑆)) ∧ (𝑎 ∈ (ℤ𝑁) ∧ 𝑏 ∈ (mzPoly‘(1...𝑎)))) ∧ 𝑐 ∈ V) ∧ (𝑐:(1...𝑎)–1-1𝑆 ∧ (𝑐 ↾ (1...𝑁)) = ( I ↾ (1...𝑁)))) → 𝑆 ∈ V)
3 simprr 772 . . . . . . . . . 10 ((((𝑁 ∈ ℕ0𝑆 ∈ V) ∧ (¬ 𝑆 ∈ Fin ∧ (1...𝑁) ⊆ 𝑆)) ∧ (𝑎 ∈ (ℤ𝑁) ∧ 𝑏 ∈ (mzPoly‘(1...𝑎)))) → 𝑏 ∈ (mzPoly‘(1...𝑎)))
43ad2antrr 725 . . . . . . . . 9 ((((((𝑁 ∈ ℕ0𝑆 ∈ V) ∧ (¬ 𝑆 ∈ Fin ∧ (1...𝑁) ⊆ 𝑆)) ∧ (𝑎 ∈ (ℤ𝑁) ∧ 𝑏 ∈ (mzPoly‘(1...𝑎)))) ∧ 𝑐 ∈ V) ∧ (𝑐:(1...𝑎)–1-1𝑆 ∧ (𝑐 ↾ (1...𝑁)) = ( I ↾ (1...𝑁)))) → 𝑏 ∈ (mzPoly‘(1...𝑎)))
5 simprl 770 . . . . . . . . . 10 ((((((𝑁 ∈ ℕ0𝑆 ∈ V) ∧ (¬ 𝑆 ∈ Fin ∧ (1...𝑁) ⊆ 𝑆)) ∧ (𝑎 ∈ (ℤ𝑁) ∧ 𝑏 ∈ (mzPoly‘(1...𝑎)))) ∧ 𝑐 ∈ V) ∧ (𝑐:(1...𝑎)–1-1𝑆 ∧ (𝑐 ↾ (1...𝑁)) = ( I ↾ (1...𝑁)))) → 𝑐:(1...𝑎)–1-1𝑆)
6 f1f 6557 . . . . . . . . . 10 (𝑐:(1...𝑎)–1-1𝑆𝑐:(1...𝑎)⟶𝑆)
75, 6syl 17 . . . . . . . . 9 ((((((𝑁 ∈ ℕ0𝑆 ∈ V) ∧ (¬ 𝑆 ∈ Fin ∧ (1...𝑁) ⊆ 𝑆)) ∧ (𝑎 ∈ (ℤ𝑁) ∧ 𝑏 ∈ (mzPoly‘(1...𝑎)))) ∧ 𝑐 ∈ V) ∧ (𝑐:(1...𝑎)–1-1𝑆 ∧ (𝑐 ↾ (1...𝑁)) = ( I ↾ (1...𝑁)))) → 𝑐:(1...𝑎)⟶𝑆)
8 mzprename 39861 . . . . . . . . 9 ((𝑆 ∈ V ∧ 𝑏 ∈ (mzPoly‘(1...𝑎)) ∧ 𝑐:(1...𝑎)⟶𝑆) → (𝑒 ∈ (ℤ ↑m 𝑆) ↦ (𝑏‘(𝑒𝑐))) ∈ (mzPoly‘𝑆))
92, 4, 7, 8syl3anc 1368 . . . . . . . 8 ((((((𝑁 ∈ ℕ0𝑆 ∈ V) ∧ (¬ 𝑆 ∈ Fin ∧ (1...𝑁) ⊆ 𝑆)) ∧ (𝑎 ∈ (ℤ𝑁) ∧ 𝑏 ∈ (mzPoly‘(1...𝑎)))) ∧ 𝑐 ∈ V) ∧ (𝑐:(1...𝑎)–1-1𝑆 ∧ (𝑐 ↾ (1...𝑁)) = ( I ↾ (1...𝑁)))) → (𝑒 ∈ (ℤ ↑m 𝑆) ↦ (𝑏‘(𝑒𝑐))) ∈ (mzPoly‘𝑆))
10 simprr 772 . . . . . . . . 9 ((((((𝑁 ∈ ℕ0𝑆 ∈ V) ∧ (¬ 𝑆 ∈ Fin ∧ (1...𝑁) ⊆ 𝑆)) ∧ (𝑎 ∈ (ℤ𝑁) ∧ 𝑏 ∈ (mzPoly‘(1...𝑎)))) ∧ 𝑐 ∈ V) ∧ (𝑐:(1...𝑎)–1-1𝑆 ∧ (𝑐 ↾ (1...𝑁)) = ( I ↾ (1...𝑁)))) → (𝑐 ↾ (1...𝑁)) = ( I ↾ (1...𝑁)))
11 diophrw 39871 . . . . . . . . . 10 ((𝑆 ∈ V ∧ 𝑐:(1...𝑎)–1-1𝑆 ∧ (𝑐 ↾ (1...𝑁)) = ( I ↾ (1...𝑁))) → {𝑡 ∣ ∃𝑢 ∈ (ℕ0m 𝑆)(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ ((𝑒 ∈ (ℤ ↑m 𝑆) ↦ (𝑏‘(𝑒𝑐)))‘𝑢) = 0)} = {𝑡 ∣ ∃𝑑 ∈ (ℕ0m (1...𝑎))(𝑡 = (𝑑 ↾ (1...𝑁)) ∧ (𝑏𝑑) = 0)})
1211eqcomd 2804 . . . . . . . . 9 ((𝑆 ∈ V ∧ 𝑐:(1...𝑎)–1-1𝑆 ∧ (𝑐 ↾ (1...𝑁)) = ( I ↾ (1...𝑁))) → {𝑡 ∣ ∃𝑑 ∈ (ℕ0m (1...𝑎))(𝑡 = (𝑑 ↾ (1...𝑁)) ∧ (𝑏𝑑) = 0)} = {𝑡 ∣ ∃𝑢 ∈ (ℕ0m 𝑆)(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ ((𝑒 ∈ (ℤ ↑m 𝑆) ↦ (𝑏‘(𝑒𝑐)))‘𝑢) = 0)})
132, 5, 10, 12syl3anc 1368 . . . . . . . 8 ((((((𝑁 ∈ ℕ0𝑆 ∈ V) ∧ (¬ 𝑆 ∈ Fin ∧ (1...𝑁) ⊆ 𝑆)) ∧ (𝑎 ∈ (ℤ𝑁) ∧ 𝑏 ∈ (mzPoly‘(1...𝑎)))) ∧ 𝑐 ∈ V) ∧ (𝑐:(1...𝑎)–1-1𝑆 ∧ (𝑐 ↾ (1...𝑁)) = ( I ↾ (1...𝑁)))) → {𝑡 ∣ ∃𝑑 ∈ (ℕ0m (1...𝑎))(𝑡 = (𝑑 ↾ (1...𝑁)) ∧ (𝑏𝑑) = 0)} = {𝑡 ∣ ∃𝑢 ∈ (ℕ0m 𝑆)(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ ((𝑒 ∈ (ℤ ↑m 𝑆) ↦ (𝑏‘(𝑒𝑐)))‘𝑢) = 0)})
14 fveq1 6654 . . . . . . . . . . . . 13 (𝑝 = (𝑒 ∈ (ℤ ↑m 𝑆) ↦ (𝑏‘(𝑒𝑐))) → (𝑝𝑢) = ((𝑒 ∈ (ℤ ↑m 𝑆) ↦ (𝑏‘(𝑒𝑐)))‘𝑢))
1514eqeq1d 2800 . . . . . . . . . . . 12 (𝑝 = (𝑒 ∈ (ℤ ↑m 𝑆) ↦ (𝑏‘(𝑒𝑐))) → ((𝑝𝑢) = 0 ↔ ((𝑒 ∈ (ℤ ↑m 𝑆) ↦ (𝑏‘(𝑒𝑐)))‘𝑢) = 0))
1615anbi2d 631 . . . . . . . . . . 11 (𝑝 = (𝑒 ∈ (ℤ ↑m 𝑆) ↦ (𝑏‘(𝑒𝑐))) → ((𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝𝑢) = 0) ↔ (𝑡 = (𝑢 ↾ (1...𝑁)) ∧ ((𝑒 ∈ (ℤ ↑m 𝑆) ↦ (𝑏‘(𝑒𝑐)))‘𝑢) = 0)))
1716rexbidv 3257 . . . . . . . . . 10 (𝑝 = (𝑒 ∈ (ℤ ↑m 𝑆) ↦ (𝑏‘(𝑒𝑐))) → (∃𝑢 ∈ (ℕ0m 𝑆)(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝𝑢) = 0) ↔ ∃𝑢 ∈ (ℕ0m 𝑆)(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ ((𝑒 ∈ (ℤ ↑m 𝑆) ↦ (𝑏‘(𝑒𝑐)))‘𝑢) = 0)))
1817abbidv 2862 . . . . . . . . 9 (𝑝 = (𝑒 ∈ (ℤ ↑m 𝑆) ↦ (𝑏‘(𝑒𝑐))) → {𝑡 ∣ ∃𝑢 ∈ (ℕ0m 𝑆)(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝𝑢) = 0)} = {𝑡 ∣ ∃𝑢 ∈ (ℕ0m 𝑆)(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ ((𝑒 ∈ (ℤ ↑m 𝑆) ↦ (𝑏‘(𝑒𝑐)))‘𝑢) = 0)})
1918rspceeqv 3587 . . . . . . . 8 (((𝑒 ∈ (ℤ ↑m 𝑆) ↦ (𝑏‘(𝑒𝑐))) ∈ (mzPoly‘𝑆) ∧ {𝑡 ∣ ∃𝑑 ∈ (ℕ0m (1...𝑎))(𝑡 = (𝑑 ↾ (1...𝑁)) ∧ (𝑏𝑑) = 0)} = {𝑡 ∣ ∃𝑢 ∈ (ℕ0m 𝑆)(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ ((𝑒 ∈ (ℤ ↑m 𝑆) ↦ (𝑏‘(𝑒𝑐)))‘𝑢) = 0)}) → ∃𝑝 ∈ (mzPoly‘𝑆){𝑡 ∣ ∃𝑑 ∈ (ℕ0m (1...𝑎))(𝑡 = (𝑑 ↾ (1...𝑁)) ∧ (𝑏𝑑) = 0)} = {𝑡 ∣ ∃𝑢 ∈ (ℕ0m 𝑆)(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝𝑢) = 0)})
209, 13, 19syl2anc 587 . . . . . . 7 ((((((𝑁 ∈ ℕ0𝑆 ∈ V) ∧ (¬ 𝑆 ∈ Fin ∧ (1...𝑁) ⊆ 𝑆)) ∧ (𝑎 ∈ (ℤ𝑁) ∧ 𝑏 ∈ (mzPoly‘(1...𝑎)))) ∧ 𝑐 ∈ V) ∧ (𝑐:(1...𝑎)–1-1𝑆 ∧ (𝑐 ↾ (1...𝑁)) = ( I ↾ (1...𝑁)))) → ∃𝑝 ∈ (mzPoly‘𝑆){𝑡 ∣ ∃𝑑 ∈ (ℕ0m (1...𝑎))(𝑡 = (𝑑 ↾ (1...𝑁)) ∧ (𝑏𝑑) = 0)} = {𝑡 ∣ ∃𝑢 ∈ (ℕ0m 𝑆)(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝𝑢) = 0)})
21 simplll 774 . . . . . . . . 9 ((((𝑁 ∈ ℕ0𝑆 ∈ V) ∧ (¬ 𝑆 ∈ Fin ∧ (1...𝑁) ⊆ 𝑆)) ∧ (𝑎 ∈ (ℤ𝑁) ∧ 𝑏 ∈ (mzPoly‘(1...𝑎)))) → 𝑁 ∈ ℕ0)
22 simplrl 776 . . . . . . . . 9 ((((𝑁 ∈ ℕ0𝑆 ∈ V) ∧ (¬ 𝑆 ∈ Fin ∧ (1...𝑁) ⊆ 𝑆)) ∧ (𝑎 ∈ (ℤ𝑁) ∧ 𝑏 ∈ (mzPoly‘(1...𝑎)))) → ¬ 𝑆 ∈ Fin)
23 simplrr 777 . . . . . . . . 9 ((((𝑁 ∈ ℕ0𝑆 ∈ V) ∧ (¬ 𝑆 ∈ Fin ∧ (1...𝑁) ⊆ 𝑆)) ∧ (𝑎 ∈ (ℤ𝑁) ∧ 𝑏 ∈ (mzPoly‘(1...𝑎)))) → (1...𝑁) ⊆ 𝑆)
24 simprl 770 . . . . . . . . 9 ((((𝑁 ∈ ℕ0𝑆 ∈ V) ∧ (¬ 𝑆 ∈ Fin ∧ (1...𝑁) ⊆ 𝑆)) ∧ (𝑎 ∈ (ℤ𝑁) ∧ 𝑏 ∈ (mzPoly‘(1...𝑎)))) → 𝑎 ∈ (ℤ𝑁))
25 eldioph2lem2 39873 . . . . . . . . 9 (((𝑁 ∈ ℕ0 ∧ ¬ 𝑆 ∈ Fin) ∧ ((1...𝑁) ⊆ 𝑆𝑎 ∈ (ℤ𝑁))) → ∃𝑐(𝑐:(1...𝑎)–1-1𝑆 ∧ (𝑐 ↾ (1...𝑁)) = ( I ↾ (1...𝑁))))
2621, 22, 23, 24, 25syl22anc 837 . . . . . . . 8 ((((𝑁 ∈ ℕ0𝑆 ∈ V) ∧ (¬ 𝑆 ∈ Fin ∧ (1...𝑁) ⊆ 𝑆)) ∧ (𝑎 ∈ (ℤ𝑁) ∧ 𝑏 ∈ (mzPoly‘(1...𝑎)))) → ∃𝑐(𝑐:(1...𝑎)–1-1𝑆 ∧ (𝑐 ↾ (1...𝑁)) = ( I ↾ (1...𝑁))))
27 rexv 3468 . . . . . . . 8 (∃𝑐 ∈ V (𝑐:(1...𝑎)–1-1𝑆 ∧ (𝑐 ↾ (1...𝑁)) = ( I ↾ (1...𝑁))) ↔ ∃𝑐(𝑐:(1...𝑎)–1-1𝑆 ∧ (𝑐 ↾ (1...𝑁)) = ( I ↾ (1...𝑁))))
2826, 27sylibr 237 . . . . . . 7 ((((𝑁 ∈ ℕ0𝑆 ∈ V) ∧ (¬ 𝑆 ∈ Fin ∧ (1...𝑁) ⊆ 𝑆)) ∧ (𝑎 ∈ (ℤ𝑁) ∧ 𝑏 ∈ (mzPoly‘(1...𝑎)))) → ∃𝑐 ∈ V (𝑐:(1...𝑎)–1-1𝑆 ∧ (𝑐 ↾ (1...𝑁)) = ( I ↾ (1...𝑁))))
2920, 28r19.29a 3249 . . . . . 6 ((((𝑁 ∈ ℕ0𝑆 ∈ V) ∧ (¬ 𝑆 ∈ Fin ∧ (1...𝑁) ⊆ 𝑆)) ∧ (𝑎 ∈ (ℤ𝑁) ∧ 𝑏 ∈ (mzPoly‘(1...𝑎)))) → ∃𝑝 ∈ (mzPoly‘𝑆){𝑡 ∣ ∃𝑑 ∈ (ℕ0m (1...𝑎))(𝑡 = (𝑑 ↾ (1...𝑁)) ∧ (𝑏𝑑) = 0)} = {𝑡 ∣ ∃𝑢 ∈ (ℕ0m 𝑆)(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝𝑢) = 0)})
30 eqeq1 2802 . . . . . . 7 (𝐴 = {𝑡 ∣ ∃𝑑 ∈ (ℕ0m (1...𝑎))(𝑡 = (𝑑 ↾ (1...𝑁)) ∧ (𝑏𝑑) = 0)} → (𝐴 = {𝑡 ∣ ∃𝑢 ∈ (ℕ0m 𝑆)(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝𝑢) = 0)} ↔ {𝑡 ∣ ∃𝑑 ∈ (ℕ0m (1...𝑎))(𝑡 = (𝑑 ↾ (1...𝑁)) ∧ (𝑏𝑑) = 0)} = {𝑡 ∣ ∃𝑢 ∈ (ℕ0m 𝑆)(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝𝑢) = 0)}))
3130rexbidv 3257 . . . . . 6 (𝐴 = {𝑡 ∣ ∃𝑑 ∈ (ℕ0m (1...𝑎))(𝑡 = (𝑑 ↾ (1...𝑁)) ∧ (𝑏𝑑) = 0)} → (∃𝑝 ∈ (mzPoly‘𝑆)𝐴 = {𝑡 ∣ ∃𝑢 ∈ (ℕ0m 𝑆)(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝𝑢) = 0)} ↔ ∃𝑝 ∈ (mzPoly‘𝑆){𝑡 ∣ ∃𝑑 ∈ (ℕ0m (1...𝑎))(𝑡 = (𝑑 ↾ (1...𝑁)) ∧ (𝑏𝑑) = 0)} = {𝑡 ∣ ∃𝑢 ∈ (ℕ0m 𝑆)(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝𝑢) = 0)}))
3229, 31syl5ibrcom 250 . . . . 5 ((((𝑁 ∈ ℕ0𝑆 ∈ V) ∧ (¬ 𝑆 ∈ Fin ∧ (1...𝑁) ⊆ 𝑆)) ∧ (𝑎 ∈ (ℤ𝑁) ∧ 𝑏 ∈ (mzPoly‘(1...𝑎)))) → (𝐴 = {𝑡 ∣ ∃𝑑 ∈ (ℕ0m (1...𝑎))(𝑡 = (𝑑 ↾ (1...𝑁)) ∧ (𝑏𝑑) = 0)} → ∃𝑝 ∈ (mzPoly‘𝑆)𝐴 = {𝑡 ∣ ∃𝑢 ∈ (ℕ0m 𝑆)(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝𝑢) = 0)}))
3332rexlimdvva 3254 . . . 4 (((𝑁 ∈ ℕ0𝑆 ∈ V) ∧ (¬ 𝑆 ∈ Fin ∧ (1...𝑁) ⊆ 𝑆)) → (∃𝑎 ∈ (ℤ𝑁)∃𝑏 ∈ (mzPoly‘(1...𝑎))𝐴 = {𝑡 ∣ ∃𝑑 ∈ (ℕ0m (1...𝑎))(𝑡 = (𝑑 ↾ (1...𝑁)) ∧ (𝑏𝑑) = 0)} → ∃𝑝 ∈ (mzPoly‘𝑆)𝐴 = {𝑡 ∣ ∃𝑢 ∈ (ℕ0m 𝑆)(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝𝑢) = 0)}))
3433adantld 494 . . 3 (((𝑁 ∈ ℕ0𝑆 ∈ V) ∧ (¬ 𝑆 ∈ Fin ∧ (1...𝑁) ⊆ 𝑆)) → ((𝑁 ∈ ℕ0 ∧ ∃𝑎 ∈ (ℤ𝑁)∃𝑏 ∈ (mzPoly‘(1...𝑎))𝐴 = {𝑡 ∣ ∃𝑑 ∈ (ℕ0m (1...𝑎))(𝑡 = (𝑑 ↾ (1...𝑁)) ∧ (𝑏𝑑) = 0)}) → ∃𝑝 ∈ (mzPoly‘𝑆)𝐴 = {𝑡 ∣ ∃𝑢 ∈ (ℕ0m 𝑆)(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝𝑢) = 0)}))
351, 34syl5bi 245 . 2 (((𝑁 ∈ ℕ0𝑆 ∈ V) ∧ (¬ 𝑆 ∈ Fin ∧ (1...𝑁) ⊆ 𝑆)) → (𝐴 ∈ (Dioph‘𝑁) → ∃𝑝 ∈ (mzPoly‘𝑆)𝐴 = {𝑡 ∣ ∃𝑢 ∈ (ℕ0m 𝑆)(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝𝑢) = 0)}))
36 simpr 488 . . . 4 (((((𝑁 ∈ ℕ0𝑆 ∈ V) ∧ (¬ 𝑆 ∈ Fin ∧ (1...𝑁) ⊆ 𝑆)) ∧ 𝑝 ∈ (mzPoly‘𝑆)) ∧ 𝐴 = {𝑡 ∣ ∃𝑢 ∈ (ℕ0m 𝑆)(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝𝑢) = 0)}) → 𝐴 = {𝑡 ∣ ∃𝑢 ∈ (ℕ0m 𝑆)(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝𝑢) = 0)})
37 simplll 774 . . . . . 6 ((((𝑁 ∈ ℕ0𝑆 ∈ V) ∧ (¬ 𝑆 ∈ Fin ∧ (1...𝑁) ⊆ 𝑆)) ∧ 𝑝 ∈ (mzPoly‘𝑆)) → 𝑁 ∈ ℕ0)
38 simpllr 775 . . . . . 6 ((((𝑁 ∈ ℕ0𝑆 ∈ V) ∧ (¬ 𝑆 ∈ Fin ∧ (1...𝑁) ⊆ 𝑆)) ∧ 𝑝 ∈ (mzPoly‘𝑆)) → 𝑆 ∈ V)
39 simplrr 777 . . . . . 6 ((((𝑁 ∈ ℕ0𝑆 ∈ V) ∧ (¬ 𝑆 ∈ Fin ∧ (1...𝑁) ⊆ 𝑆)) ∧ 𝑝 ∈ (mzPoly‘𝑆)) → (1...𝑁) ⊆ 𝑆)
40 simpr 488 . . . . . 6 ((((𝑁 ∈ ℕ0𝑆 ∈ V) ∧ (¬ 𝑆 ∈ Fin ∧ (1...𝑁) ⊆ 𝑆)) ∧ 𝑝 ∈ (mzPoly‘𝑆)) → 𝑝 ∈ (mzPoly‘𝑆))
41 eldioph2 39874 . . . . . 6 ((𝑁 ∈ ℕ0 ∧ (𝑆 ∈ V ∧ (1...𝑁) ⊆ 𝑆) ∧ 𝑝 ∈ (mzPoly‘𝑆)) → {𝑡 ∣ ∃𝑢 ∈ (ℕ0m 𝑆)(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝𝑢) = 0)} ∈ (Dioph‘𝑁))
4237, 38, 39, 40, 41syl121anc 1372 . . . . 5 ((((𝑁 ∈ ℕ0𝑆 ∈ V) ∧ (¬ 𝑆 ∈ Fin ∧ (1...𝑁) ⊆ 𝑆)) ∧ 𝑝 ∈ (mzPoly‘𝑆)) → {𝑡 ∣ ∃𝑢 ∈ (ℕ0m 𝑆)(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝𝑢) = 0)} ∈ (Dioph‘𝑁))
4342adantr 484 . . . 4 (((((𝑁 ∈ ℕ0𝑆 ∈ V) ∧ (¬ 𝑆 ∈ Fin ∧ (1...𝑁) ⊆ 𝑆)) ∧ 𝑝 ∈ (mzPoly‘𝑆)) ∧ 𝐴 = {𝑡 ∣ ∃𝑢 ∈ (ℕ0m 𝑆)(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝𝑢) = 0)}) → {𝑡 ∣ ∃𝑢 ∈ (ℕ0m 𝑆)(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝𝑢) = 0)} ∈ (Dioph‘𝑁))
4436, 43eqeltrd 2890 . . 3 (((((𝑁 ∈ ℕ0𝑆 ∈ V) ∧ (¬ 𝑆 ∈ Fin ∧ (1...𝑁) ⊆ 𝑆)) ∧ 𝑝 ∈ (mzPoly‘𝑆)) ∧ 𝐴 = {𝑡 ∣ ∃𝑢 ∈ (ℕ0m 𝑆)(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝𝑢) = 0)}) → 𝐴 ∈ (Dioph‘𝑁))
4544rexlimdva2 3247 . 2 (((𝑁 ∈ ℕ0𝑆 ∈ V) ∧ (¬ 𝑆 ∈ Fin ∧ (1...𝑁) ⊆ 𝑆)) → (∃𝑝 ∈ (mzPoly‘𝑆)𝐴 = {𝑡 ∣ ∃𝑢 ∈ (ℕ0m 𝑆)(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝𝑢) = 0)} → 𝐴 ∈ (Dioph‘𝑁)))
4635, 45impbid 215 1 (((𝑁 ∈ ℕ0𝑆 ∈ V) ∧ (¬ 𝑆 ∈ Fin ∧ (1...𝑁) ⊆ 𝑆)) → (𝐴 ∈ (Dioph‘𝑁) ↔ ∃𝑝 ∈ (mzPoly‘𝑆)𝐴 = {𝑡 ∣ ∃𝑢 ∈ (ℕ0m 𝑆)(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝𝑢) = 0)}))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1084   = wceq 1538  ∃wex 1781   ∈ wcel 2111  {cab 2776  ∃wrex 3107  Vcvv 3442   ⊆ wss 3883   ↦ cmpt 5114   I cid 5428   ↾ cres 5525   ∘ ccom 5527  ⟶wf 6328  –1-1→wf1 6329  ‘cfv 6332  (class class class)co 7145   ↑m cmap 8407  Fincfn 8510  0cc0 10544  1c1 10545  ℕ0cn0 11903  ℤcz 11989  ℤ≥cuz 12251  ...cfz 12905  mzPolycmzp 39834  Diophcdioph 39867 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5158  ax-sep 5171  ax-nul 5178  ax-pow 5235  ax-pr 5299  ax-un 7454  ax-cnex 10600  ax-resscn 10601  ax-1cn 10602  ax-icn 10603  ax-addcl 10604  ax-addrcl 10605  ax-mulcl 10606  ax-mulrcl 10607  ax-mulcom 10608  ax-addass 10609  ax-mulass 10610  ax-distr 10611  ax-i2m1 10612  ax-1ne0 10613  ax-1rid 10614  ax-rnegex 10615  ax-rrecex 10616  ax-cnre 10617  ax-pre-lttri 10618  ax-pre-lttrn 10619  ax-pre-ltadd 10620  ax-pre-mulgt0 10621 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-nel 3092  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3444  df-sbc 3723  df-csb 3831  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-tp 4533  df-op 4535  df-uni 4805  df-int 4843  df-iun 4887  df-br 5035  df-opab 5097  df-mpt 5115  df-tr 5141  df-id 5429  df-eprel 5434  df-po 5442  df-so 5443  df-fr 5482  df-we 5484  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-pred 6123  df-ord 6169  df-on 6170  df-lim 6171  df-suc 6172  df-iota 6291  df-fun 6334  df-fn 6335  df-f 6336  df-f1 6337  df-fo 6338  df-f1o 6339  df-fv 6340  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-of 7400  df-om 7574  df-1st 7684  df-2nd 7685  df-wrecs 7948  df-recs 8009  df-rdg 8047  df-1o 8103  df-oadd 8107  df-er 8290  df-map 8409  df-en 8511  df-dom 8512  df-sdom 8513  df-fin 8514  df-dju 9332  df-card 9370  df-pnf 10684  df-mnf 10685  df-xr 10686  df-ltxr 10687  df-le 10688  df-sub 10879  df-neg 10880  df-nn 11644  df-n0 11904  df-z 11990  df-uz 12252  df-fz 12906  df-hash 13707  df-mzpcl 39835  df-mzp 39836  df-dioph 39868 This theorem is referenced by:  eldioph3b  39877  diophin  39884  diophun  39885  eldioph4b  39923
 Copyright terms: Public domain W3C validator