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Theorem diophun 42761
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 42752 . . 3 (𝐴 ∈ (Dioph‘𝑁) → 𝑁 ∈ ℕ0)
2 nnex 12270 . . . . . 6 ℕ ∈ V
32jctr 524 . . . . 5 (𝑁 ∈ ℕ0 → (𝑁 ∈ ℕ0 ∧ ℕ ∈ V))
4 1z 12645 . . . . . . 7 1 ∈ ℤ
5 nnuz 12919 . . . . . . . 8 ℕ = (ℤ‘1)
65uzinf 14003 . . . . . . 7 (1 ∈ ℤ → ¬ ℕ ∈ Fin)
74, 6ax-mp 5 . . . . . 6 ¬ ℕ ∈ Fin
8 elfznn 13590 . . . . . . 7 (𝑎 ∈ (1...𝑁) → 𝑎 ∈ ℕ)
98ssriv 3999 . . . . . 6 (1...𝑁) ⊆ ℕ
107, 9pm3.2i 470 . . . . 5 (¬ ℕ ∈ Fin ∧ (1...𝑁) ⊆ ℕ)
11 eldioph2b 42751 . . . . . 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 42751 . . . . . 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 632 . . . . 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 3227 . . . . 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 4314 . . . . . . . . 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 3120 . . . . . . . . . . 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 1009 . . . . . . . . . . . . 13 ((𝑏 = (𝑑 ↾ (1...𝑁)) ∧ ((𝑎𝑑) = 0 ∨ (𝑐𝑑) = 0)) ↔ ((𝑏 = (𝑑 ↾ (1...𝑁)) ∧ (𝑎𝑑) = 0) ∨ (𝑏 = (𝑑 ↾ (1...𝑁)) ∧ (𝑐𝑑) = 0)))
19 zex 12620 . . . . . . . . . . . . . . . . . . . 20 ℤ ∈ V
20 nn0ssz 12634 . . . . . . . . . . . . . . . . . . . 20 0 ⊆ ℤ
21 mapss 8928 . . . . . . . . . . . . . . . . . . . 20 ((ℤ ∈ V ∧ ℕ0 ⊆ ℤ) → (ℕ0m ℕ) ⊆ (ℤ ↑m ℕ))
2219, 20, 21mp2an 692 . . . . . . . . . . . . . . . . . . 19 (ℕ0m ℕ) ⊆ (ℤ ↑m ℕ)
2322sseli 3991 . . . . . . . . . . . . . . . . . 18 (𝑑 ∈ (ℕ0m ℕ) → 𝑑 ∈ (ℤ ↑m ℕ))
2423adantl 481 . . . . . . . . . . . . . . . . 17 (((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘ℕ) ∧ 𝑐 ∈ (mzPoly‘ℕ))) ∧ 𝑑 ∈ (ℕ0m ℕ)) → 𝑑 ∈ (ℤ ↑m ℕ))
25 fveq2 6907 . . . . . . . . . . . . . . . . . . 19 (𝑒 = 𝑑 → (𝑎𝑒) = (𝑎𝑑))
26 fveq2 6907 . . . . . . . . . . . . . . . . . . 19 (𝑒 = 𝑑 → (𝑐𝑒) = (𝑐𝑑))
2725, 26oveq12d 7449 . . . . . . . . . . . . . . . . . 18 (𝑒 = 𝑑 → ((𝑎𝑒) · (𝑐𝑒)) = ((𝑎𝑑) · (𝑐𝑑)))
28 eqid 2735 . . . . . . . . . . . . . . . . . 18 (𝑒 ∈ (ℤ ↑m ℕ) ↦ ((𝑎𝑒) · (𝑐𝑒))) = (𝑒 ∈ (ℤ ↑m ℕ) ↦ ((𝑎𝑒) · (𝑐𝑒)))
29 ovex 7464 . . . . . . . . . . . . . . . . . 18 ((𝑎𝑑) · (𝑐𝑑)) ∈ V
3027, 28, 29fvmpt 7016 . . . . . . . . . . . . . . . . 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 2737 . . . . . . . . . . . . . . 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 777 . . . . . . . . . . . . . . . . . . 19 (((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘ℕ) ∧ 𝑐 ∈ (mzPoly‘ℕ))) ∧ 𝑑 ∈ (ℕ0m ℕ)) → 𝑎 ∈ (mzPoly‘ℕ))
34 mzpf 42724 . . . . . . . . . . . . . . . . . . 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 7105 . . . . . . . . . . . . . . . . 17 (((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘ℕ) ∧ 𝑐 ∈ (mzPoly‘ℕ))) ∧ 𝑑 ∈ (ℕ0m ℕ)) → (𝑎𝑑) ∈ ℤ)
3736zcnd 12721 . . . . . . . . . . . . . . . 16 (((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘ℕ) ∧ 𝑐 ∈ (mzPoly‘ℕ))) ∧ 𝑑 ∈ (ℕ0m ℕ)) → (𝑎𝑑) ∈ ℂ)
38 simplrr 778 . . . . . . . . . . . . . . . . . . 19 (((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘ℕ) ∧ 𝑐 ∈ (mzPoly‘ℕ))) ∧ 𝑑 ∈ (ℕ0m ℕ)) → 𝑐 ∈ (mzPoly‘ℕ))
39 mzpf 42724 . . . . . . . . . . . . . . . . . . 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 7105 . . . . . . . . . . . . . . . . 17 (((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘ℕ) ∧ 𝑐 ∈ (mzPoly‘ℕ))) ∧ 𝑑 ∈ (ℕ0m ℕ)) → (𝑐𝑑) ∈ ℤ)
4241zcnd 12721 . . . . . . . . . . . . . . . 16 (((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘ℕ) ∧ 𝑐 ∈ (mzPoly‘ℕ))) ∧ 𝑑 ∈ (ℕ0m ℕ)) → (𝑐𝑑) ∈ ℂ)
4337, 42mul0ord 11911 . . . . . . . . . . . . . . 15 (((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘ℕ) ∧ 𝑐 ∈ (mzPoly‘ℕ))) ∧ 𝑑 ∈ (ℕ0m ℕ)) → (((𝑎𝑑) · (𝑐𝑑)) = 0 ↔ ((𝑎𝑑) = 0 ∨ (𝑐𝑑) = 0)))
4432, 43bitr2d 280 . . . . . . . . . . . . . 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 630 . . . . . . . . . . . . 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 285 . . . . . . . . . . . 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 3175 . . . . . . . . . . 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 285 . . . . . . . . . 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 2787 . . . . . . . 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 482 . . . . . . . . 9 ((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘ℕ) ∧ 𝑐 ∈ (mzPoly‘ℕ))) → 𝑁 ∈ ℕ0)
522, 9pm3.2i 470 . . . . . . . . . 10 (ℕ ∈ V ∧ (1...𝑁) ⊆ ℕ)
5352a1i 11 . . . . . . . . 9 ((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘ℕ) ∧ 𝑐 ∈ (mzPoly‘ℕ))) → (ℕ ∈ V ∧ (1...𝑁) ⊆ ℕ))
54 simprl 771 . . . . . . . . . . . . 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 6977 . . . . . . . . . . 11 ((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘ℕ) ∧ 𝑐 ∈ (mzPoly‘ℕ))) → 𝑎 = (𝑒 ∈ (ℤ ↑m ℕ) ↦ (𝑎𝑒)))
5756, 54eqeltrrd 2840 . . . . . . . . . 10 ((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘ℕ) ∧ 𝑐 ∈ (mzPoly‘ℕ))) → (𝑒 ∈ (ℤ ↑m ℕ) ↦ (𝑎𝑒)) ∈ (mzPoly‘ℕ))
58 simprr 773 . . . . . . . . . . . . 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 6977 . . . . . . . . . . 11 ((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘ℕ) ∧ 𝑐 ∈ (mzPoly‘ℕ))) → 𝑐 = (𝑒 ∈ (ℤ ↑m ℕ) ↦ (𝑐𝑒)))
6160, 58eqeltrrd 2840 . . . . . . . . . 10 ((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘ℕ) ∧ 𝑐 ∈ (mzPoly‘ℕ))) → (𝑒 ∈ (ℤ ↑m ℕ) ↦ (𝑐𝑒)) ∈ (mzPoly‘ℕ))
62 mzpmulmpt 42730 . . . . . . . . . 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 42750 . . . . . . . . 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 2839 . . . . . . 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 4173 . . . . . . . 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 2824 . . . . . . 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 247 . . . . . 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 3211 . . . . 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 243 . . . 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 240 . . 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 669 1 ((𝐴 ∈ (Dioph‘𝑁) ∧ 𝐵 ∈ (Dioph‘𝑁)) → (𝐴𝐵) ∈ (Dioph‘𝑁))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395  wo 847   = wceq 1537  wcel 2106  {cab 2712  wrex 3068  Vcvv 3478  cun 3961  wss 3963  cmpt 5231  cres 5691  wf 6559  cfv 6563  (class class class)co 7431  m cmap 8865  Fincfn 8984  0cc0 11153  1c1 11154   · cmul 11158  cn 12264  0cn0 12524  cz 12611  ...cfz 13544  mzPolycmzp 42710  Diophcdioph 42743
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-inf2 9679  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-of 7697  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-oadd 8509  df-er 8744  df-map 8867  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-dju 9939  df-card 9977  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-nn 12265  df-n0 12525  df-z 12612  df-uz 12877  df-fz 13545  df-hash 14367  df-mzpcl 42711  df-mzp 42712  df-dioph 42744
This theorem is referenced by:  orrabdioph  42769
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