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Theorem diophun 39567
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 39558 . . 3 (𝐴 ∈ (Dioph‘𝑁) → 𝑁 ∈ ℕ0)
2 nnex 11636 . . . . . 6 ℕ ∈ V
32jctr 528 . . . . 5 (𝑁 ∈ ℕ0 → (𝑁 ∈ ℕ0 ∧ ℕ ∈ V))
4 1z 12005 . . . . . . 7 1 ∈ ℤ
5 nnuz 12274 . . . . . . . 8 ℕ = (ℤ‘1)
65uzinf 13333 . . . . . . 7 (1 ∈ ℤ → ¬ ℕ ∈ Fin)
74, 6ax-mp 5 . . . . . 6 ¬ ℕ ∈ Fin
8 elfznn 12936 . . . . . . 7 (𝑎 ∈ (1...𝑁) → 𝑎 ∈ ℕ)
98ssriv 3956 . . . . . 6 (1...𝑁) ⊆ ℕ
107, 9pm3.2i 474 . . . . 5 (¬ ℕ ∈ Fin ∧ (1...𝑁) ⊆ ℕ)
11 eldioph2b 39557 . . . . . 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 39557 . . . . . 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 633 . . . . 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 589 . . . 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 3359 . . . . 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 4254 . . . . . . . . 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 3343 . . . . . . . . . . 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 11983 . . . . . . . . . . . . . . . . . . . 20 ℤ ∈ V
20 nn0ssz 11996 . . . . . . . . . . . . . . . . . . . 20 0 ⊆ ℤ
21 mapss 8443 . . . . . . . . . . . . . . . . . . . 20 ((ℤ ∈ V ∧ ℕ0 ⊆ ℤ) → (ℕ0m ℕ) ⊆ (ℤ ↑m ℕ))
2219, 20, 21mp2an 691 . . . . . . . . . . . . . . . . . . 19 (ℕ0m ℕ) ⊆ (ℤ ↑m ℕ)
2322sseli 3948 . . . . . . . . . . . . . . . . . 18 (𝑑 ∈ (ℕ0m ℕ) → 𝑑 ∈ (ℤ ↑m ℕ))
2423adantl 485 . . . . . . . . . . . . . . . . 17 (((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘ℕ) ∧ 𝑐 ∈ (mzPoly‘ℕ))) ∧ 𝑑 ∈ (ℕ0m ℕ)) → 𝑑 ∈ (ℤ ↑m ℕ))
25 fveq2 6658 . . . . . . . . . . . . . . . . . . 19 (𝑒 = 𝑑 → (𝑎𝑒) = (𝑎𝑑))
26 fveq2 6658 . . . . . . . . . . . . . . . . . . 19 (𝑒 = 𝑑 → (𝑐𝑒) = (𝑐𝑑))
2725, 26oveq12d 7163 . . . . . . . . . . . . . . . . . 18 (𝑒 = 𝑑 → ((𝑎𝑒) · (𝑐𝑒)) = ((𝑎𝑑) · (𝑐𝑑)))
28 eqid 2824 . . . . . . . . . . . . . . . . . 18 (𝑒 ∈ (ℤ ↑m ℕ) ↦ ((𝑎𝑒) · (𝑐𝑒))) = (𝑒 ∈ (ℤ ↑m ℕ) ↦ ((𝑎𝑒) · (𝑐𝑒)))
29 ovex 7178 . . . . . . . . . . . . . . . . . 18 ((𝑎𝑑) · (𝑐𝑑)) ∈ V
3027, 28, 29fvmpt 6756 . . . . . . . . . . . . . . . . 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 2826 . . . . . . . . . . . . . . 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 776 . . . . . . . . . . . . . . . . . . 19 (((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘ℕ) ∧ 𝑐 ∈ (mzPoly‘ℕ))) ∧ 𝑑 ∈ (ℕ0m ℕ)) → 𝑎 ∈ (mzPoly‘ℕ))
34 mzpf 39530 . . . . . . . . . . . . . . . . . . 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, 24ffvelrnd 6840 . . . . . . . . . . . . . . . . 17 (((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘ℕ) ∧ 𝑐 ∈ (mzPoly‘ℕ))) ∧ 𝑑 ∈ (ℕ0m ℕ)) → (𝑎𝑑) ∈ ℤ)
3736zcnd 12081 . . . . . . . . . . . . . . . 16 (((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘ℕ) ∧ 𝑐 ∈ (mzPoly‘ℕ))) ∧ 𝑑 ∈ (ℕ0m ℕ)) → (𝑎𝑑) ∈ ℂ)
38 simplrr 777 . . . . . . . . . . . . . . . . . . 19 (((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘ℕ) ∧ 𝑐 ∈ (mzPoly‘ℕ))) ∧ 𝑑 ∈ (ℕ0m ℕ)) → 𝑐 ∈ (mzPoly‘ℕ))
39 mzpf 39530 . . . . . . . . . . . . . . . . . . 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, 24ffvelrnd 6840 . . . . . . . . . . . . . . . . 17 (((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘ℕ) ∧ 𝑐 ∈ (mzPoly‘ℕ))) ∧ 𝑑 ∈ (ℕ0m ℕ)) → (𝑐𝑑) ∈ ℤ)
4241zcnd 12081 . . . . . . . . . . . . . . . 16 (((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘ℕ) ∧ 𝑐 ∈ (mzPoly‘ℕ))) ∧ 𝑑 ∈ (ℕ0m ℕ)) → (𝑐𝑑) ∈ ℂ)
4337, 42mul0ord 11282 . . . . . . . . . . . . . . 15 (((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘ℕ) ∧ 𝑐 ∈ (mzPoly‘ℕ))) ∧ 𝑑 ∈ (ℕ0m ℕ)) → (((𝑎𝑑) · (𝑐𝑑)) = 0 ↔ ((𝑎𝑑) = 0 ∨ (𝑐𝑑) = 0)))
4432, 43bitr2d 283 . . . . . . . . . . . . . 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 631 . . . . . . . . . . . . 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 288 . . . . . . . . . . . 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 3289 . . . . . . . . . . 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 288 . . . . . . . . . 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 2888 . . . . . . . . 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, 49syl5eq 2871 . . . . . . . 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 486 . . . . . . . . 9 ((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘ℕ) ∧ 𝑐 ∈ (mzPoly‘ℕ))) → 𝑁 ∈ ℕ0)
522, 9pm3.2i 474 . . . . . . . . . 10 (ℕ ∈ V ∧ (1...𝑁) ⊆ ℕ)
5352a1i 11 . . . . . . . . 9 ((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘ℕ) ∧ 𝑐 ∈ (mzPoly‘ℕ))) → (ℕ ∈ V ∧ (1...𝑁) ⊆ ℕ))
54 simprl 770 . . . . . . . . . . . . 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 6721 . . . . . . . . . . 11 ((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘ℕ) ∧ 𝑐 ∈ (mzPoly‘ℕ))) → 𝑎 = (𝑒 ∈ (ℤ ↑m ℕ) ↦ (𝑎𝑒)))
5756, 54eqeltrrd 2917 . . . . . . . . . 10 ((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘ℕ) ∧ 𝑐 ∈ (mzPoly‘ℕ))) → (𝑒 ∈ (ℤ ↑m ℕ) ↦ (𝑎𝑒)) ∈ (mzPoly‘ℕ))
58 simprr 772 . . . . . . . . . . . . 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 6721 . . . . . . . . . . 11 ((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘ℕ) ∧ 𝑐 ∈ (mzPoly‘ℕ))) → 𝑐 = (𝑒 ∈ (ℤ ↑m ℕ) ↦ (𝑐𝑒)))
6160, 58eqeltrrd 2917 . . . . . . . . . 10 ((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘ℕ) ∧ 𝑐 ∈ (mzPoly‘ℕ))) → (𝑒 ∈ (ℤ ↑m ℕ) ↦ (𝑐𝑒)) ∈ (mzPoly‘ℕ))
62 mzpmulmpt 39536 . . . . . . . . . 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 587 . . . . . . . . 9 ((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘ℕ) ∧ 𝑐 ∈ (mzPoly‘ℕ))) → (𝑒 ∈ (ℤ ↑m ℕ) ↦ ((𝑎𝑒) · (𝑐𝑒))) ∈ (mzPoly‘ℕ))
64 eldioph2 39556 . . . . . . . . 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 1368 . . . . . . . 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 2916 . . . . . . 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 4119 . . . . . . . 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 2900 . . . . . . 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 250 . . . . . 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 3287 . . . . 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 246 . . . 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 243 . . 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 668 1 ((𝐴 ∈ (Dioph‘𝑁) ∧ 𝐵 ∈ (Dioph‘𝑁)) → (𝐴𝐵) ∈ (Dioph‘𝑁))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wa 399  wo 844   = wceq 1538  wcel 2115  {cab 2802  wrex 3134  Vcvv 3480  cun 3917  wss 3919  cmpt 5132  cres 5544  wf 6339  cfv 6343  (class class class)co 7145  m cmap 8396  Fincfn 8499  0cc0 10529  1c1 10530   · cmul 10534  cn 11630  0cn0 11890  cz 11974  ...cfz 12890  mzPolycmzp 39516  Diophcdioph 39549
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5176  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7451  ax-inf2 9095  ax-cnex 10585  ax-resscn 10586  ax-1cn 10587  ax-icn 10588  ax-addcl 10589  ax-addrcl 10590  ax-mulcl 10591  ax-mulrcl 10592  ax-mulcom 10593  ax-addass 10594  ax-mulass 10595  ax-distr 10596  ax-i2m1 10597  ax-1ne0 10598  ax-1rid 10599  ax-rnegex 10600  ax-rrecex 10601  ax-cnre 10602  ax-pre-lttri 10603  ax-pre-lttrn 10604  ax-pre-ltadd 10605  ax-pre-mulgt0 10606
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 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-nel 3119  df-ral 3138  df-rex 3139  df-reu 3140  df-rmo 3141  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-pss 3938  df-nul 4276  df-if 4450  df-pw 4523  df-sn 4550  df-pr 4552  df-tp 4554  df-op 4556  df-uni 4825  df-int 4863  df-iun 4907  df-br 5053  df-opab 5115  df-mpt 5133  df-tr 5159  df-id 5447  df-eprel 5452  df-po 5461  df-so 5462  df-fr 5501  df-we 5503  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-pred 6135  df-ord 6181  df-on 6182  df-lim 6183  df-suc 6184  df-iota 6302  df-fun 6345  df-fn 6346  df-f 6347  df-f1 6348  df-fo 6349  df-f1o 6350  df-fv 6351  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-of 7399  df-om 7571  df-1st 7679  df-2nd 7680  df-wrecs 7937  df-recs 7998  df-rdg 8036  df-1o 8092  df-oadd 8096  df-er 8279  df-map 8398  df-en 8500  df-dom 8501  df-sdom 8502  df-fin 8503  df-dju 9321  df-card 9359  df-pnf 10669  df-mnf 10670  df-xr 10671  df-ltxr 10672  df-le 10673  df-sub 10864  df-neg 10865  df-nn 11631  df-n0 11891  df-z 11975  df-uz 12237  df-fz 12891  df-hash 13692  df-mzpcl 39517  df-mzp 39518  df-dioph 39550
This theorem is referenced by:  orrabdioph  39575
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