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Theorem diophun 41082
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 41073 . . 3 (𝐴 ∈ (Dioph‘𝑁) → 𝑁 ∈ ℕ0)
2 nnex 12159 . . . . . 6 ℕ ∈ V
32jctr 525 . . . . 5 (𝑁 ∈ ℕ0 → (𝑁 ∈ ℕ0 ∧ ℕ ∈ V))
4 1z 12533 . . . . . . 7 1 ∈ ℤ
5 nnuz 12806 . . . . . . . 8 ℕ = (ℤ‘1)
65uzinf 13870 . . . . . . 7 (1 ∈ ℤ → ¬ ℕ ∈ Fin)
74, 6ax-mp 5 . . . . . 6 ¬ ℕ ∈ Fin
8 elfznn 13470 . . . . . . 7 (𝑎 ∈ (1...𝑁) → 𝑎 ∈ ℕ)
98ssriv 3948 . . . . . 6 (1...𝑁) ⊆ ℕ
107, 9pm3.2i 471 . . . . 5 (¬ ℕ ∈ Fin ∧ (1...𝑁) ⊆ ℕ)
11 eldioph2b 41072 . . . . . 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 41072 . . . . . 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 3217 . . . . 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 4258 . . . . . . . . 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 3125 . . . . . . . . . . 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 1006 . . . . . . . . . . . . 13 ((𝑏 = (𝑑 ↾ (1...𝑁)) ∧ ((𝑎𝑑) = 0 ∨ (𝑐𝑑) = 0)) ↔ ((𝑏 = (𝑑 ↾ (1...𝑁)) ∧ (𝑎𝑑) = 0) ∨ (𝑏 = (𝑑 ↾ (1...𝑁)) ∧ (𝑐𝑑) = 0)))
19 zex 12508 . . . . . . . . . . . . . . . . . . . 20 ℤ ∈ V
20 nn0ssz 12522 . . . . . . . . . . . . . . . . . . . 20 0 ⊆ ℤ
21 mapss 8827 . . . . . . . . . . . . . . . . . . . 20 ((ℤ ∈ V ∧ ℕ0 ⊆ ℤ) → (ℕ0m ℕ) ⊆ (ℤ ↑m ℕ))
2219, 20, 21mp2an 690 . . . . . . . . . . . . . . . . . . 19 (ℕ0m ℕ) ⊆ (ℤ ↑m ℕ)
2322sseli 3940 . . . . . . . . . . . . . . . . . 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 6842 . . . . . . . . . . . . . . . . . . 19 (𝑒 = 𝑑 → (𝑎𝑒) = (𝑎𝑑))
26 fveq2 6842 . . . . . . . . . . . . . . . . . . 19 (𝑒 = 𝑑 → (𝑐𝑒) = (𝑐𝑑))
2725, 26oveq12d 7375 . . . . . . . . . . . . . . . . . 18 (𝑒 = 𝑑 → ((𝑎𝑒) · (𝑐𝑒)) = ((𝑎𝑑) · (𝑐𝑑)))
28 eqid 2736 . . . . . . . . . . . . . . . . . 18 (𝑒 ∈ (ℤ ↑m ℕ) ↦ ((𝑎𝑒) · (𝑐𝑒))) = (𝑒 ∈ (ℤ ↑m ℕ) ↦ ((𝑎𝑒) · (𝑐𝑒)))
29 ovex 7390 . . . . . . . . . . . . . . . . . 18 ((𝑎𝑑) · (𝑐𝑑)) ∈ V
3027, 28, 29fvmpt 6948 . . . . . . . . . . . . . . . . 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 2738 . . . . . . . . . . . . . . 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 775 . . . . . . . . . . . . . . . . . . 19 (((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘ℕ) ∧ 𝑐 ∈ (mzPoly‘ℕ))) ∧ 𝑑 ∈ (ℕ0m ℕ)) → 𝑎 ∈ (mzPoly‘ℕ))
34 mzpf 41045 . . . . . . . . . . . . . . . . . . 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 7036 . . . . . . . . . . . . . . . . 17 (((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘ℕ) ∧ 𝑐 ∈ (mzPoly‘ℕ))) ∧ 𝑑 ∈ (ℕ0m ℕ)) → (𝑎𝑑) ∈ ℤ)
3736zcnd 12608 . . . . . . . . . . . . . . . 16 (((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘ℕ) ∧ 𝑐 ∈ (mzPoly‘ℕ))) ∧ 𝑑 ∈ (ℕ0m ℕ)) → (𝑎𝑑) ∈ ℂ)
38 simplrr 776 . . . . . . . . . . . . . . . . . . 19 (((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘ℕ) ∧ 𝑐 ∈ (mzPoly‘ℕ))) ∧ 𝑑 ∈ (ℕ0m ℕ)) → 𝑐 ∈ (mzPoly‘ℕ))
39 mzpf 41045 . . . . . . . . . . . . . . . . . . 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 7036 . . . . . . . . . . . . . . . . 17 (((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘ℕ) ∧ 𝑐 ∈ (mzPoly‘ℕ))) ∧ 𝑑 ∈ (ℕ0m ℕ)) → (𝑐𝑑) ∈ ℤ)
4241zcnd 12608 . . . . . . . . . . . . . . . 16 (((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘ℕ) ∧ 𝑐 ∈ (mzPoly‘ℕ))) ∧ 𝑑 ∈ (ℕ0m ℕ)) → (𝑐𝑑) ∈ ℂ)
4337, 42mul0ord 11805 . . . . . . . . . . . . . . 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 3173 . . . . . . . . . . 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 2805 . . . . . . . . 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 2788 . . . . . . . 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 769 . . . . . . . . . . . . 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 6910 . . . . . . . . . . 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 771 . . . . . . . . . . . . 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 6910 . . . . . . . . . . 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 41051 . . . . . . . . . 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 41071 . . . . . . . . 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 1371 . . . . . . . 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 4118 . . . . . . . 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 3205 . . . . 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, 70biimtrrid 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 667 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 845   = wceq 1541  wcel 2106  {cab 2713  wrex 3073  Vcvv 3445  cun 3908  wss 3910  cmpt 5188  cres 5635  wf 6492  cfv 6496  (class class class)co 7357  m cmap 8765  Fincfn 8883  0cc0 11051  1c1 11052   · cmul 11056  cn 12153  0cn0 12413  cz 12499  ...cfz 13424  mzPolycmzp 41031  Diophcdioph 41064
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-inf2 9577  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-of 7617  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-oadd 8416  df-er 8648  df-map 8767  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-dju 9837  df-card 9875  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-nn 12154  df-n0 12414  df-z 12500  df-uz 12764  df-fz 13425  df-hash 14231  df-mzpcl 41032  df-mzp 41033  df-dioph 41065
This theorem is referenced by:  orrabdioph  41090
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