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

Theorem diophun 40811
Description: If two sets are Diophantine, so is their union. (Contributed by Stefan O'Rear, 9-Oct-2014.) (Revised by Stefan O'Rear, 6-May-2015.)
Assertion
Ref Expression
diophun ((𝐴 ∈ (Dioph‘𝑁) ∧ 𝐵 ∈ (Dioph‘𝑁)) → (𝐴𝐵) ∈ (Dioph‘𝑁))

Proof of Theorem diophun
Dummy variables 𝑎 𝑏 𝑐 𝑑 𝑒 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eldiophelnn0 40802 . . 3 (𝐴 ∈ (Dioph‘𝑁) → 𝑁 ∈ ℕ0)
2 nnex 12059 . . . . . 6 ℕ ∈ V
32jctr 525 . . . . 5 (𝑁 ∈ ℕ0 → (𝑁 ∈ ℕ0 ∧ ℕ ∈ V))
4 1z 12430 . . . . . . 7 1 ∈ ℤ
5 nnuz 12701 . . . . . . . 8 ℕ = (ℤ‘1)
65uzinf 13765 . . . . . . 7 (1 ∈ ℤ → ¬ ℕ ∈ Fin)
74, 6ax-mp 5 . . . . . 6 ¬ ℕ ∈ Fin
8 elfznn 13365 . . . . . . 7 (𝑎 ∈ (1...𝑁) → 𝑎 ∈ ℕ)
98ssriv 3935 . . . . . 6 (1...𝑁) ⊆ ℕ
107, 9pm3.2i 471 . . . . 5 (¬ ℕ ∈ Fin ∧ (1...𝑁) ⊆ ℕ)
11 eldioph2b 40801 . . . . . 6 (((𝑁 ∈ ℕ0 ∧ ℕ ∈ V) ∧ (¬ ℕ ∈ Fin ∧ (1...𝑁) ⊆ ℕ)) → (𝐴 ∈ (Dioph‘𝑁) ↔ ∃𝑎 ∈ (mzPoly‘ℕ)𝐴 = {𝑏 ∣ ∃𝑑 ∈ (ℕ0m ℕ)(𝑏 = (𝑑 ↾ (1...𝑁)) ∧ (𝑎𝑑) = 0)}))
12 eldioph2b 40801 . . . . . 6 (((𝑁 ∈ ℕ0 ∧ ℕ ∈ V) ∧ (¬ ℕ ∈ Fin ∧ (1...𝑁) ⊆ ℕ)) → (𝐵 ∈ (Dioph‘𝑁) ↔ ∃𝑐 ∈ (mzPoly‘ℕ)𝐵 = {𝑏 ∣ ∃𝑑 ∈ (ℕ0m ℕ)(𝑏 = (𝑑 ↾ (1...𝑁)) ∧ (𝑐𝑑) = 0)}))
1311, 12anbi12d 631 . . . . 5 (((𝑁 ∈ ℕ0 ∧ ℕ ∈ V) ∧ (¬ ℕ ∈ Fin ∧ (1...𝑁) ⊆ ℕ)) → ((𝐴 ∈ (Dioph‘𝑁) ∧ 𝐵 ∈ (Dioph‘𝑁)) ↔ (∃𝑎 ∈ (mzPoly‘ℕ)𝐴 = {𝑏 ∣ ∃𝑑 ∈ (ℕ0m ℕ)(𝑏 = (𝑑 ↾ (1...𝑁)) ∧ (𝑎𝑑) = 0)} ∧ ∃𝑐 ∈ (mzPoly‘ℕ)𝐵 = {𝑏 ∣ ∃𝑑 ∈ (ℕ0m ℕ)(𝑏 = (𝑑 ↾ (1...𝑁)) ∧ (𝑐𝑑) = 0)})))
143, 10, 13sylancl 586 . . . 4 (𝑁 ∈ ℕ0 → ((𝐴 ∈ (Dioph‘𝑁) ∧ 𝐵 ∈ (Dioph‘𝑁)) ↔ (∃𝑎 ∈ (mzPoly‘ℕ)𝐴 = {𝑏 ∣ ∃𝑑 ∈ (ℕ0m ℕ)(𝑏 = (𝑑 ↾ (1...𝑁)) ∧ (𝑎𝑑) = 0)} ∧ ∃𝑐 ∈ (mzPoly‘ℕ)𝐵 = {𝑏 ∣ ∃𝑑 ∈ (ℕ0m ℕ)(𝑏 = (𝑑 ↾ (1...𝑁)) ∧ (𝑐𝑑) = 0)})))
15 reeanv 3214 . . . . 5 (∃𝑎 ∈ (mzPoly‘ℕ)∃𝑐 ∈ (mzPoly‘ℕ)(𝐴 = {𝑏 ∣ ∃𝑑 ∈ (ℕ0m ℕ)(𝑏 = (𝑑 ↾ (1...𝑁)) ∧ (𝑎𝑑) = 0)} ∧ 𝐵 = {𝑏 ∣ ∃𝑑 ∈ (ℕ0m ℕ)(𝑏 = (𝑑 ↾ (1...𝑁)) ∧ (𝑐𝑑) = 0)}) ↔ (∃𝑎 ∈ (mzPoly‘ℕ)𝐴 = {𝑏 ∣ ∃𝑑 ∈ (ℕ0m ℕ)(𝑏 = (𝑑 ↾ (1...𝑁)) ∧ (𝑎𝑑) = 0)} ∧ ∃𝑐 ∈ (mzPoly‘ℕ)𝐵 = {𝑏 ∣ ∃𝑑 ∈ (ℕ0m ℕ)(𝑏 = (𝑑 ↾ (1...𝑁)) ∧ (𝑐𝑑) = 0)}))
16 unab 4243 . . . . . . . . 9 ({𝑏 ∣ ∃𝑑 ∈ (ℕ0m ℕ)(𝑏 = (𝑑 ↾ (1...𝑁)) ∧ (𝑎𝑑) = 0)} ∪ {𝑏 ∣ ∃𝑑 ∈ (ℕ0m ℕ)(𝑏 = (𝑑 ↾ (1...𝑁)) ∧ (𝑐𝑑) = 0)}) = {𝑏 ∣ (∃𝑑 ∈ (ℕ0m ℕ)(𝑏 = (𝑑 ↾ (1...𝑁)) ∧ (𝑎𝑑) = 0) ∨ ∃𝑑 ∈ (ℕ0m ℕ)(𝑏 = (𝑑 ↾ (1...𝑁)) ∧ (𝑐𝑑) = 0))}
17 r19.43 3122 . . . . . . . . . . 11 (∃𝑑 ∈ (ℕ0m ℕ)((𝑏 = (𝑑 ↾ (1...𝑁)) ∧ (𝑎𝑑) = 0) ∨ (𝑏 = (𝑑 ↾ (1...𝑁)) ∧ (𝑐𝑑) = 0)) ↔ (∃𝑑 ∈ (ℕ0m ℕ)(𝑏 = (𝑑 ↾ (1...𝑁)) ∧ (𝑎𝑑) = 0) ∨ ∃𝑑 ∈ (ℕ0m ℕ)(𝑏 = (𝑑 ↾ (1...𝑁)) ∧ (𝑐𝑑) = 0)))
18 andi 1005 . . . . . . . . . . . . 13 ((𝑏 = (𝑑 ↾ (1...𝑁)) ∧ ((𝑎𝑑) = 0 ∨ (𝑐𝑑) = 0)) ↔ ((𝑏 = (𝑑 ↾ (1...𝑁)) ∧ (𝑎𝑑) = 0) ∨ (𝑏 = (𝑑 ↾ (1...𝑁)) ∧ (𝑐𝑑) = 0)))
19 zex 12408 . . . . . . . . . . . . . . . . . . . 20 ℤ ∈ V
20 nn0ssz 12421 . . . . . . . . . . . . . . . . . . . 20 0 ⊆ ℤ
21 mapss 8727 . . . . . . . . . . . . . . . . . . . 20 ((ℤ ∈ V ∧ ℕ0 ⊆ ℤ) → (ℕ0m ℕ) ⊆ (ℤ ↑m ℕ))
2219, 20, 21mp2an 689 . . . . . . . . . . . . . . . . . . 19 (ℕ0m ℕ) ⊆ (ℤ ↑m ℕ)
2322sseli 3927 . . . . . . . . . . . . . . . . . 18 (𝑑 ∈ (ℕ0m ℕ) → 𝑑 ∈ (ℤ ↑m ℕ))
2423adantl 482 . . . . . . . . . . . . . . . . 17 (((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘ℕ) ∧ 𝑐 ∈ (mzPoly‘ℕ))) ∧ 𝑑 ∈ (ℕ0m ℕ)) → 𝑑 ∈ (ℤ ↑m ℕ))
25 fveq2 6812 . . . . . . . . . . . . . . . . . . 19 (𝑒 = 𝑑 → (𝑎𝑒) = (𝑎𝑑))
26 fveq2 6812 . . . . . . . . . . . . . . . . . . 19 (𝑒 = 𝑑 → (𝑐𝑒) = (𝑐𝑑))
2725, 26oveq12d 7335 . . . . . . . . . . . . . . . . . 18 (𝑒 = 𝑑 → ((𝑎𝑒) · (𝑐𝑒)) = ((𝑎𝑑) · (𝑐𝑑)))
28 eqid 2737 . . . . . . . . . . . . . . . . . 18 (𝑒 ∈ (ℤ ↑m ℕ) ↦ ((𝑎𝑒) · (𝑐𝑒))) = (𝑒 ∈ (ℤ ↑m ℕ) ↦ ((𝑎𝑒) · (𝑐𝑒)))
29 ovex 7350 . . . . . . . . . . . . . . . . . 18 ((𝑎𝑑) · (𝑐𝑑)) ∈ V
3027, 28, 29fvmpt 6915 . . . . . . . . . . . . . . . . 17 (𝑑 ∈ (ℤ ↑m ℕ) → ((𝑒 ∈ (ℤ ↑m ℕ) ↦ ((𝑎𝑒) · (𝑐𝑒)))‘𝑑) = ((𝑎𝑑) · (𝑐𝑑)))
3124, 30syl 17 . . . . . . . . . . . . . . . 16 (((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘ℕ) ∧ 𝑐 ∈ (mzPoly‘ℕ))) ∧ 𝑑 ∈ (ℕ0m ℕ)) → ((𝑒 ∈ (ℤ ↑m ℕ) ↦ ((𝑎𝑒) · (𝑐𝑒)))‘𝑑) = ((𝑎𝑑) · (𝑐𝑑)))
3231eqeq1d 2739 . . . . . . . . . . . . . . 15 (((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘ℕ) ∧ 𝑐 ∈ (mzPoly‘ℕ))) ∧ 𝑑 ∈ (ℕ0m ℕ)) → (((𝑒 ∈ (ℤ ↑m ℕ) ↦ ((𝑎𝑒) · (𝑐𝑒)))‘𝑑) = 0 ↔ ((𝑎𝑑) · (𝑐𝑑)) = 0))
33 simplrl 774 . . . . . . . . . . . . . . . . . . 19 (((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘ℕ) ∧ 𝑐 ∈ (mzPoly‘ℕ))) ∧ 𝑑 ∈ (ℕ0m ℕ)) → 𝑎 ∈ (mzPoly‘ℕ))
34 mzpf 40774 . . . . . . . . . . . . . . . . . . 19 (𝑎 ∈ (mzPoly‘ℕ) → 𝑎:(ℤ ↑m ℕ)⟶ℤ)
3533, 34syl 17 . . . . . . . . . . . . . . . . . 18 (((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘ℕ) ∧ 𝑐 ∈ (mzPoly‘ℕ))) ∧ 𝑑 ∈ (ℕ0m ℕ)) → 𝑎:(ℤ ↑m ℕ)⟶ℤ)
3635, 24ffvelcdmd 7002 . . . . . . . . . . . . . . . . 17 (((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘ℕ) ∧ 𝑐 ∈ (mzPoly‘ℕ))) ∧ 𝑑 ∈ (ℕ0m ℕ)) → (𝑎𝑑) ∈ ℤ)
3736zcnd 12507 . . . . . . . . . . . . . . . 16 (((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘ℕ) ∧ 𝑐 ∈ (mzPoly‘ℕ))) ∧ 𝑑 ∈ (ℕ0m ℕ)) → (𝑎𝑑) ∈ ℂ)
38 simplrr 775 . . . . . . . . . . . . . . . . . . 19 (((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘ℕ) ∧ 𝑐 ∈ (mzPoly‘ℕ))) ∧ 𝑑 ∈ (ℕ0m ℕ)) → 𝑐 ∈ (mzPoly‘ℕ))
39 mzpf 40774 . . . . . . . . . . . . . . . . . . 19 (𝑐 ∈ (mzPoly‘ℕ) → 𝑐:(ℤ ↑m ℕ)⟶ℤ)
4038, 39syl 17 . . . . . . . . . . . . . . . . . 18 (((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘ℕ) ∧ 𝑐 ∈ (mzPoly‘ℕ))) ∧ 𝑑 ∈ (ℕ0m ℕ)) → 𝑐:(ℤ ↑m ℕ)⟶ℤ)
4140, 24ffvelcdmd 7002 . . . . . . . . . . . . . . . . 17 (((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘ℕ) ∧ 𝑐 ∈ (mzPoly‘ℕ))) ∧ 𝑑 ∈ (ℕ0m ℕ)) → (𝑐𝑑) ∈ ℤ)
4241zcnd 12507 . . . . . . . . . . . . . . . 16 (((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘ℕ) ∧ 𝑐 ∈ (mzPoly‘ℕ))) ∧ 𝑑 ∈ (ℕ0m ℕ)) → (𝑐𝑑) ∈ ℂ)
4337, 42mul0ord 11705 . . . . . . . . . . . . . . 15 (((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘ℕ) ∧ 𝑐 ∈ (mzPoly‘ℕ))) ∧ 𝑑 ∈ (ℕ0m ℕ)) → (((𝑎𝑑) · (𝑐𝑑)) = 0 ↔ ((𝑎𝑑) = 0 ∨ (𝑐𝑑) = 0)))
4432, 43bitr2d 279 . . . . . . . . . . . . . 14 (((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘ℕ) ∧ 𝑐 ∈ (mzPoly‘ℕ))) ∧ 𝑑 ∈ (ℕ0m ℕ)) → (((𝑎𝑑) = 0 ∨ (𝑐𝑑) = 0) ↔ ((𝑒 ∈ (ℤ ↑m ℕ) ↦ ((𝑎𝑒) · (𝑐𝑒)))‘𝑑) = 0))
4544anbi2d 629 . . . . . . . . . . . . 13 (((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘ℕ) ∧ 𝑐 ∈ (mzPoly‘ℕ))) ∧ 𝑑 ∈ (ℕ0m ℕ)) → ((𝑏 = (𝑑 ↾ (1...𝑁)) ∧ ((𝑎𝑑) = 0 ∨ (𝑐𝑑) = 0)) ↔ (𝑏 = (𝑑 ↾ (1...𝑁)) ∧ ((𝑒 ∈ (ℤ ↑m ℕ) ↦ ((𝑎𝑒) · (𝑐𝑒)))‘𝑑) = 0)))
4618, 45bitr3id 284 . . . . . . . . . . . 12 (((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘ℕ) ∧ 𝑐 ∈ (mzPoly‘ℕ))) ∧ 𝑑 ∈ (ℕ0m ℕ)) → (((𝑏 = (𝑑 ↾ (1...𝑁)) ∧ (𝑎𝑑) = 0) ∨ (𝑏 = (𝑑 ↾ (1...𝑁)) ∧ (𝑐𝑑) = 0)) ↔ (𝑏 = (𝑑 ↾ (1...𝑁)) ∧ ((𝑒 ∈ (ℤ ↑m ℕ) ↦ ((𝑎𝑒) · (𝑐𝑒)))‘𝑑) = 0)))
4746rexbidva 3170 . . . . . . . . . . 11 ((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘ℕ) ∧ 𝑐 ∈ (mzPoly‘ℕ))) → (∃𝑑 ∈ (ℕ0m ℕ)((𝑏 = (𝑑 ↾ (1...𝑁)) ∧ (𝑎𝑑) = 0) ∨ (𝑏 = (𝑑 ↾ (1...𝑁)) ∧ (𝑐𝑑) = 0)) ↔ ∃𝑑 ∈ (ℕ0m ℕ)(𝑏 = (𝑑 ↾ (1...𝑁)) ∧ ((𝑒 ∈ (ℤ ↑m ℕ) ↦ ((𝑎𝑒) · (𝑐𝑒)))‘𝑑) = 0)))
4817, 47bitr3id 284 . . . . . . . . . 10 ((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘ℕ) ∧ 𝑐 ∈ (mzPoly‘ℕ))) → ((∃𝑑 ∈ (ℕ0m ℕ)(𝑏 = (𝑑 ↾ (1...𝑁)) ∧ (𝑎𝑑) = 0) ∨ ∃𝑑 ∈ (ℕ0m ℕ)(𝑏 = (𝑑 ↾ (1...𝑁)) ∧ (𝑐𝑑) = 0)) ↔ ∃𝑑 ∈ (ℕ0m ℕ)(𝑏 = (𝑑 ↾ (1...𝑁)) ∧ ((𝑒 ∈ (ℤ ↑m ℕ) ↦ ((𝑎𝑒) · (𝑐𝑒)))‘𝑑) = 0)))
4948abbidv 2806 . . . . . . . . 9 ((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘ℕ) ∧ 𝑐 ∈ (mzPoly‘ℕ))) → {𝑏 ∣ (∃𝑑 ∈ (ℕ0m ℕ)(𝑏 = (𝑑 ↾ (1...𝑁)) ∧ (𝑎𝑑) = 0) ∨ ∃𝑑 ∈ (ℕ0m ℕ)(𝑏 = (𝑑 ↾ (1...𝑁)) ∧ (𝑐𝑑) = 0))} = {𝑏 ∣ ∃𝑑 ∈ (ℕ0m ℕ)(𝑏 = (𝑑 ↾ (1...𝑁)) ∧ ((𝑒 ∈ (ℤ ↑m ℕ) ↦ ((𝑎𝑒) · (𝑐𝑒)))‘𝑑) = 0)})
5016, 49eqtrid 2789 . . . . . . . 8 ((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘ℕ) ∧ 𝑐 ∈ (mzPoly‘ℕ))) → ({𝑏 ∣ ∃𝑑 ∈ (ℕ0m ℕ)(𝑏 = (𝑑 ↾ (1...𝑁)) ∧ (𝑎𝑑) = 0)} ∪ {𝑏 ∣ ∃𝑑 ∈ (ℕ0m ℕ)(𝑏 = (𝑑 ↾ (1...𝑁)) ∧ (𝑐𝑑) = 0)}) = {𝑏 ∣ ∃𝑑 ∈ (ℕ0m ℕ)(𝑏 = (𝑑 ↾ (1...𝑁)) ∧ ((𝑒 ∈ (ℤ ↑m ℕ) ↦ ((𝑎𝑒) · (𝑐𝑒)))‘𝑑) = 0)})
51 simpl 483 . . . . . . . . 9 ((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘ℕ) ∧ 𝑐 ∈ (mzPoly‘ℕ))) → 𝑁 ∈ ℕ0)
522, 9pm3.2i 471 . . . . . . . . . 10 (ℕ ∈ V ∧ (1...𝑁) ⊆ ℕ)
5352a1i 11 . . . . . . . . 9 ((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘ℕ) ∧ 𝑐 ∈ (mzPoly‘ℕ))) → (ℕ ∈ V ∧ (1...𝑁) ⊆ ℕ))
54 simprl 768 . . . . . . . . . . . . 13 ((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘ℕ) ∧ 𝑐 ∈ (mzPoly‘ℕ))) → 𝑎 ∈ (mzPoly‘ℕ))
5554, 34syl 17 . . . . . . . . . . . 12 ((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘ℕ) ∧ 𝑐 ∈ (mzPoly‘ℕ))) → 𝑎:(ℤ ↑m ℕ)⟶ℤ)
5655feqmptd 6877 . . . . . . . . . . 11 ((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘ℕ) ∧ 𝑐 ∈ (mzPoly‘ℕ))) → 𝑎 = (𝑒 ∈ (ℤ ↑m ℕ) ↦ (𝑎𝑒)))
5756, 54eqeltrrd 2839 . . . . . . . . . 10 ((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘ℕ) ∧ 𝑐 ∈ (mzPoly‘ℕ))) → (𝑒 ∈ (ℤ ↑m ℕ) ↦ (𝑎𝑒)) ∈ (mzPoly‘ℕ))
58 simprr 770 . . . . . . . . . . . . 13 ((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘ℕ) ∧ 𝑐 ∈ (mzPoly‘ℕ))) → 𝑐 ∈ (mzPoly‘ℕ))
5958, 39syl 17 . . . . . . . . . . . 12 ((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘ℕ) ∧ 𝑐 ∈ (mzPoly‘ℕ))) → 𝑐:(ℤ ↑m ℕ)⟶ℤ)
6059feqmptd 6877 . . . . . . . . . . 11 ((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘ℕ) ∧ 𝑐 ∈ (mzPoly‘ℕ))) → 𝑐 = (𝑒 ∈ (ℤ ↑m ℕ) ↦ (𝑐𝑒)))
6160, 58eqeltrrd 2839 . . . . . . . . . 10 ((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘ℕ) ∧ 𝑐 ∈ (mzPoly‘ℕ))) → (𝑒 ∈ (ℤ ↑m ℕ) ↦ (𝑐𝑒)) ∈ (mzPoly‘ℕ))
62 mzpmulmpt 40780 . . . . . . . . . 10 (((𝑒 ∈ (ℤ ↑m ℕ) ↦ (𝑎𝑒)) ∈ (mzPoly‘ℕ) ∧ (𝑒 ∈ (ℤ ↑m ℕ) ↦ (𝑐𝑒)) ∈ (mzPoly‘ℕ)) → (𝑒 ∈ (ℤ ↑m ℕ) ↦ ((𝑎𝑒) · (𝑐𝑒))) ∈ (mzPoly‘ℕ))
6357, 61, 62syl2anc 584 . . . . . . . . 9 ((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘ℕ) ∧ 𝑐 ∈ (mzPoly‘ℕ))) → (𝑒 ∈ (ℤ ↑m ℕ) ↦ ((𝑎𝑒) · (𝑐𝑒))) ∈ (mzPoly‘ℕ))
64 eldioph2 40800 . . . . . . . . 9 ((𝑁 ∈ ℕ0 ∧ (ℕ ∈ V ∧ (1...𝑁) ⊆ ℕ) ∧ (𝑒 ∈ (ℤ ↑m ℕ) ↦ ((𝑎𝑒) · (𝑐𝑒))) ∈ (mzPoly‘ℕ)) → {𝑏 ∣ ∃𝑑 ∈ (ℕ0m ℕ)(𝑏 = (𝑑 ↾ (1...𝑁)) ∧ ((𝑒 ∈ (ℤ ↑m ℕ) ↦ ((𝑎𝑒) · (𝑐𝑒)))‘𝑑) = 0)} ∈ (Dioph‘𝑁))
6551, 53, 63, 64syl3anc 1370 . . . . . . . 8 ((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘ℕ) ∧ 𝑐 ∈ (mzPoly‘ℕ))) → {𝑏 ∣ ∃𝑑 ∈ (ℕ0m ℕ)(𝑏 = (𝑑 ↾ (1...𝑁)) ∧ ((𝑒 ∈ (ℤ ↑m ℕ) ↦ ((𝑎𝑒) · (𝑐𝑒)))‘𝑑) = 0)} ∈ (Dioph‘𝑁))
6650, 65eqeltrd 2838 . . . . . . 7 ((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘ℕ) ∧ 𝑐 ∈ (mzPoly‘ℕ))) → ({𝑏 ∣ ∃𝑑 ∈ (ℕ0m ℕ)(𝑏 = (𝑑 ↾ (1...𝑁)) ∧ (𝑎𝑑) = 0)} ∪ {𝑏 ∣ ∃𝑑 ∈ (ℕ0m ℕ)(𝑏 = (𝑑 ↾ (1...𝑁)) ∧ (𝑐𝑑) = 0)}) ∈ (Dioph‘𝑁))
67 uneq12 4103 . . . . . . . 8 ((𝐴 = {𝑏 ∣ ∃𝑑 ∈ (ℕ0m ℕ)(𝑏 = (𝑑 ↾ (1...𝑁)) ∧ (𝑎𝑑) = 0)} ∧ 𝐵 = {𝑏 ∣ ∃𝑑 ∈ (ℕ0m ℕ)(𝑏 = (𝑑 ↾ (1...𝑁)) ∧ (𝑐𝑑) = 0)}) → (𝐴𝐵) = ({𝑏 ∣ ∃𝑑 ∈ (ℕ0m ℕ)(𝑏 = (𝑑 ↾ (1...𝑁)) ∧ (𝑎𝑑) = 0)} ∪ {𝑏 ∣ ∃𝑑 ∈ (ℕ0m ℕ)(𝑏 = (𝑑 ↾ (1...𝑁)) ∧ (𝑐𝑑) = 0)}))
6867eleq1d 2822 . . . . . . 7 ((𝐴 = {𝑏 ∣ ∃𝑑 ∈ (ℕ0m ℕ)(𝑏 = (𝑑 ↾ (1...𝑁)) ∧ (𝑎𝑑) = 0)} ∧ 𝐵 = {𝑏 ∣ ∃𝑑 ∈ (ℕ0m ℕ)(𝑏 = (𝑑 ↾ (1...𝑁)) ∧ (𝑐𝑑) = 0)}) → ((𝐴𝐵) ∈ (Dioph‘𝑁) ↔ ({𝑏 ∣ ∃𝑑 ∈ (ℕ0m ℕ)(𝑏 = (𝑑 ↾ (1...𝑁)) ∧ (𝑎𝑑) = 0)} ∪ {𝑏 ∣ ∃𝑑 ∈ (ℕ0m ℕ)(𝑏 = (𝑑 ↾ (1...𝑁)) ∧ (𝑐𝑑) = 0)}) ∈ (Dioph‘𝑁)))
6966, 68syl5ibrcom 246 . . . . . 6 ((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘ℕ) ∧ 𝑐 ∈ (mzPoly‘ℕ))) → ((𝐴 = {𝑏 ∣ ∃𝑑 ∈ (ℕ0m ℕ)(𝑏 = (𝑑 ↾ (1...𝑁)) ∧ (𝑎𝑑) = 0)} ∧ 𝐵 = {𝑏 ∣ ∃𝑑 ∈ (ℕ0m ℕ)(𝑏 = (𝑑 ↾ (1...𝑁)) ∧ (𝑐𝑑) = 0)}) → (𝐴𝐵) ∈ (Dioph‘𝑁)))
7069rexlimdvva 3202 . . . . 5 (𝑁 ∈ ℕ0 → (∃𝑎 ∈ (mzPoly‘ℕ)∃𝑐 ∈ (mzPoly‘ℕ)(𝐴 = {𝑏 ∣ ∃𝑑 ∈ (ℕ0m ℕ)(𝑏 = (𝑑 ↾ (1...𝑁)) ∧ (𝑎𝑑) = 0)} ∧ 𝐵 = {𝑏 ∣ ∃𝑑 ∈ (ℕ0m ℕ)(𝑏 = (𝑑 ↾ (1...𝑁)) ∧ (𝑐𝑑) = 0)}) → (𝐴𝐵) ∈ (Dioph‘𝑁)))
7115, 70syl5bir 242 . . . 4 (𝑁 ∈ ℕ0 → ((∃𝑎 ∈ (mzPoly‘ℕ)𝐴 = {𝑏 ∣ ∃𝑑 ∈ (ℕ0m ℕ)(𝑏 = (𝑑 ↾ (1...𝑁)) ∧ (𝑎𝑑) = 0)} ∧ ∃𝑐 ∈ (mzPoly‘ℕ)𝐵 = {𝑏 ∣ ∃𝑑 ∈ (ℕ0m ℕ)(𝑏 = (𝑑 ↾ (1...𝑁)) ∧ (𝑐𝑑) = 0)}) → (𝐴𝐵) ∈ (Dioph‘𝑁)))
7214, 71sylbid 239 . . 3 (𝑁 ∈ ℕ0 → ((𝐴 ∈ (Dioph‘𝑁) ∧ 𝐵 ∈ (Dioph‘𝑁)) → (𝐴𝐵) ∈ (Dioph‘𝑁)))
731, 72syl 17 . 2 (𝐴 ∈ (Dioph‘𝑁) → ((𝐴 ∈ (Dioph‘𝑁) ∧ 𝐵 ∈ (Dioph‘𝑁)) → (𝐴𝐵) ∈ (Dioph‘𝑁)))
7473anabsi5 666 1 ((𝐴 ∈ (Dioph‘𝑁) ∧ 𝐵 ∈ (Dioph‘𝑁)) → (𝐴𝐵) ∈ (Dioph‘𝑁))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 396  wo 844   = wceq 1540  wcel 2105  {cab 2714  wrex 3071  Vcvv 3441  cun 3895  wss 3897  cmpt 5170  cres 5610  wf 6462  cfv 6466  (class class class)co 7317  m cmap 8665  Fincfn 8783  0cc0 10951  1c1 10952   · cmul 10956  cn 12053  0cn0 12313  cz 12399  ...cfz 13319  mzPolycmzp 40760  Diophcdioph 40793
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2708  ax-rep 5224  ax-sep 5238  ax-nul 5245  ax-pow 5303  ax-pr 5367  ax-un 7630  ax-inf2 9477  ax-cnex 11007  ax-resscn 11008  ax-1cn 11009  ax-icn 11010  ax-addcl 11011  ax-addrcl 11012  ax-mulcl 11013  ax-mulrcl 11014  ax-mulcom 11015  ax-addass 11016  ax-mulass 11017  ax-distr 11018  ax-i2m1 11019  ax-1ne0 11020  ax-1rid 11021  ax-rnegex 11022  ax-rrecex 11023  ax-cnre 11024  ax-pre-lttri 11025  ax-pre-lttrn 11026  ax-pre-ltadd 11027  ax-pre-mulgt0 11028
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2887  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-reu 3351  df-rab 3405  df-v 3443  df-sbc 3727  df-csb 3843  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-pss 3916  df-nul 4268  df-if 4472  df-pw 4547  df-sn 4572  df-pr 4574  df-op 4578  df-uni 4851  df-int 4893  df-iun 4939  df-br 5088  df-opab 5150  df-mpt 5171  df-tr 5205  df-id 5507  df-eprel 5513  df-po 5521  df-so 5522  df-fr 5563  df-we 5565  df-xp 5614  df-rel 5615  df-cnv 5616  df-co 5617  df-dm 5618  df-rn 5619  df-res 5620  df-ima 5621  df-pred 6225  df-ord 6292  df-on 6293  df-lim 6294  df-suc 6295  df-iota 6418  df-fun 6468  df-fn 6469  df-f 6470  df-f1 6471  df-fo 6472  df-f1o 6473  df-fv 6474  df-riota 7274  df-ov 7320  df-oprab 7321  df-mpo 7322  df-of 7575  df-om 7760  df-1st 7878  df-2nd 7879  df-frecs 8146  df-wrecs 8177  df-recs 8251  df-rdg 8290  df-1o 8346  df-oadd 8350  df-er 8548  df-map 8667  df-en 8784  df-dom 8785  df-sdom 8786  df-fin 8787  df-dju 9737  df-card 9775  df-pnf 11091  df-mnf 11092  df-xr 11093  df-ltxr 11094  df-le 11095  df-sub 11287  df-neg 11288  df-nn 12054  df-n0 12314  df-z 12400  df-uz 12663  df-fz 13320  df-hash 14125  df-mzpcl 40761  df-mzp 40762  df-dioph 40794
This theorem is referenced by:  orrabdioph  40819
  Copyright terms: Public domain W3C validator