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Theorem diophun 37857
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 37847 . . 3 (𝐴 ∈ (Dioph‘𝑁) → 𝑁 ∈ ℕ0)
2 nnex 11238 . . . . . 6 ℕ ∈ V
32jctr 566 . . . . 5 (𝑁 ∈ ℕ0 → (𝑁 ∈ ℕ0 ∧ ℕ ∈ V))
4 1z 11619 . . . . . . 7 1 ∈ ℤ
5 nnuz 11936 . . . . . . . 8 ℕ = (ℤ‘1)
65uzinf 12978 . . . . . . 7 (1 ∈ ℤ → ¬ ℕ ∈ Fin)
74, 6ax-mp 5 . . . . . 6 ¬ ℕ ∈ Fin
8 elfznn 12583 . . . . . . 7 (𝑎 ∈ (1...𝑁) → 𝑎 ∈ ℕ)
98ssriv 3748 . . . . . 6 (1...𝑁) ⊆ ℕ
107, 9pm3.2i 470 . . . . 5 (¬ ℕ ∈ Fin ∧ (1...𝑁) ⊆ ℕ)
11 eldioph2b 37846 . . . . . 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 37846 . . . . . 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 749 . . . . 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 697 . . . 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 3245 . . . . 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 4037 . . . . . . . . 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 3231 . . . . . . . . . . 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 947 . . . . . . . . . . . . 13 ((𝑏 = (𝑑 ↾ (1...𝑁)) ∧ ((𝑎𝑑) = 0 ∨ (𝑐𝑑) = 0)) ↔ ((𝑏 = (𝑑 ↾ (1...𝑁)) ∧ (𝑎𝑑) = 0) ∨ (𝑏 = (𝑑 ↾ (1...𝑁)) ∧ (𝑐𝑑) = 0)))
19 zex 11598 . . . . . . . . . . . . . . . . . . . 20 ℤ ∈ V
20 nn0ssz 11610 . . . . . . . . . . . . . . . . . . . 20 0 ⊆ ℤ
21 mapss 8068 . . . . . . . . . . . . . . . . . . . 20 ((ℤ ∈ V ∧ ℕ0 ⊆ ℤ) → (ℕ0𝑚 ℕ) ⊆ (ℤ ↑𝑚 ℕ))
2219, 20, 21mp2an 710 . . . . . . . . . . . . . . . . . . 19 (ℕ0𝑚 ℕ) ⊆ (ℤ ↑𝑚 ℕ)
2322sseli 3740 . . . . . . . . . . . . . . . . . 18 (𝑑 ∈ (ℕ0𝑚 ℕ) → 𝑑 ∈ (ℤ ↑𝑚 ℕ))
2423adantl 473 . . . . . . . . . . . . . . . . 17 (((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘ℕ) ∧ 𝑐 ∈ (mzPoly‘ℕ))) ∧ 𝑑 ∈ (ℕ0𝑚 ℕ)) → 𝑑 ∈ (ℤ ↑𝑚 ℕ))
25 fveq2 6353 . . . . . . . . . . . . . . . . . . 19 (𝑒 = 𝑑 → (𝑎𝑒) = (𝑎𝑑))
26 fveq2 6353 . . . . . . . . . . . . . . . . . . 19 (𝑒 = 𝑑 → (𝑐𝑒) = (𝑐𝑑))
2725, 26oveq12d 6832 . . . . . . . . . . . . . . . . . 18 (𝑒 = 𝑑 → ((𝑎𝑒) · (𝑐𝑒)) = ((𝑎𝑑) · (𝑐𝑑)))
28 eqid 2760 . . . . . . . . . . . . . . . . . 18 (𝑒 ∈ (ℤ ↑𝑚 ℕ) ↦ ((𝑎𝑒) · (𝑐𝑒))) = (𝑒 ∈ (ℤ ↑𝑚 ℕ) ↦ ((𝑎𝑒) · (𝑐𝑒)))
29 ovex 6842 . . . . . . . . . . . . . . . . . 18 ((𝑎𝑑) · (𝑐𝑑)) ∈ V
3027, 28, 29fvmpt 6445 . . . . . . . . . . . . . . . . 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 2762 . . . . . . . . . . . . . . 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 819 . . . . . . . . . . . . . . . . . . 19 (((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘ℕ) ∧ 𝑐 ∈ (mzPoly‘ℕ))) ∧ 𝑑 ∈ (ℕ0𝑚 ℕ)) → 𝑎 ∈ (mzPoly‘ℕ))
34 mzpf 37819 . . . . . . . . . . . . . . . . . . 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 6524 . . . . . . . . . . . . . . . . 17 (((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘ℕ) ∧ 𝑐 ∈ (mzPoly‘ℕ))) ∧ 𝑑 ∈ (ℕ0𝑚 ℕ)) → (𝑎𝑑) ∈ ℤ)
3736zcnd 11695 . . . . . . . . . . . . . . . 16 (((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘ℕ) ∧ 𝑐 ∈ (mzPoly‘ℕ))) ∧ 𝑑 ∈ (ℕ0𝑚 ℕ)) → (𝑎𝑑) ∈ ℂ)
38 simplrr 820 . . . . . . . . . . . . . . . . . . 19 (((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘ℕ) ∧ 𝑐 ∈ (mzPoly‘ℕ))) ∧ 𝑑 ∈ (ℕ0𝑚 ℕ)) → 𝑐 ∈ (mzPoly‘ℕ))
39 mzpf 37819 . . . . . . . . . . . . . . . . . . 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 6524 . . . . . . . . . . . . . . . . 17 (((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘ℕ) ∧ 𝑐 ∈ (mzPoly‘ℕ))) ∧ 𝑑 ∈ (ℕ0𝑚 ℕ)) → (𝑐𝑑) ∈ ℤ)
4241zcnd 11695 . . . . . . . . . . . . . . . 16 (((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘ℕ) ∧ 𝑐 ∈ (mzPoly‘ℕ))) ∧ 𝑑 ∈ (ℕ0𝑚 ℕ)) → (𝑐𝑑) ∈ ℂ)
4337, 42mul0ord 10889 . . . . . . . . . . . . . . 15 (((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘ℕ) ∧ 𝑐 ∈ (mzPoly‘ℕ))) ∧ 𝑑 ∈ (ℕ0𝑚 ℕ)) → (((𝑎𝑑) · (𝑐𝑑)) = 0 ↔ ((𝑎𝑑) = 0 ∨ (𝑐𝑑) = 0)))
4432, 43bitr2d 269 . . . . . . . . . . . . . 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 742 . . . . . . . . . . . . 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, 45syl5bbr 274 . . . . . . . . . . . 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 3187 . . . . . . . . . . 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, 47syl5bbr 274 . . . . . . . . . 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 2879 . . . . . . . . 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 2806 . . . . . . . 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 474 . . . . . . . . 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 811 . . . . . . . . . . . . 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 6412 . . . . . . . . . . 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 813 . . . . . . . . . . . . 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 6412 . . . . . . . . . . 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 37825 . . . . . . . . . 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 696 . . . . . . . . 9 ((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘ℕ) ∧ 𝑐 ∈ (mzPoly‘ℕ))) → (𝑒 ∈ (ℤ ↑𝑚 ℕ) ↦ ((𝑎𝑒) · (𝑐𝑒))) ∈ (mzPoly‘ℕ))
64 eldioph2 37845 . . . . . . . . 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 1477 . . . . . . . 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 3905 . . . . . . . 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 237 . . . . . 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 3176 . . . . 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 233 . . . 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 230 . . 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 893 1 ((𝐴 ∈ (Dioph‘𝑁) ∧ 𝐵 ∈ (Dioph‘𝑁)) → (𝐴𝐵) ∈ (Dioph‘𝑁))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wo 382  wa 383   = wceq 1632  wcel 2139  {cab 2746  wrex 3051  Vcvv 3340  cun 3713  wss 3715  cmpt 4881  cres 5268  wf 6045  cfv 6049  (class class class)co 6814  𝑚 cmap 8025  Fincfn 8123  0cc0 10148  1c1 10149   · cmul 10153  cn 11232  0cn0 11504  cz 11589  ...cfz 12539  mzPolycmzp 37805  Diophcdioph 37838
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7115  ax-inf2 8713  ax-cnex 10204  ax-resscn 10205  ax-1cn 10206  ax-icn 10207  ax-addcl 10208  ax-addrcl 10209  ax-mulcl 10210  ax-mulrcl 10211  ax-mulcom 10212  ax-addass 10213  ax-mulass 10214  ax-distr 10215  ax-i2m1 10216  ax-1ne0 10217  ax-1rid 10218  ax-rnegex 10219  ax-rrecex 10220  ax-cnre 10221  ax-pre-lttri 10222  ax-pre-lttrn 10223  ax-pre-ltadd 10224  ax-pre-mulgt0 10225
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-nel 3036  df-ral 3055  df-rex 3056  df-reu 3057  df-rmo 3058  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-int 4628  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-riota 6775  df-ov 6817  df-oprab 6818  df-mpt2 6819  df-of 7063  df-om 7232  df-1st 7334  df-2nd 7335  df-wrecs 7577  df-recs 7638  df-rdg 7676  df-1o 7730  df-oadd 7734  df-er 7913  df-map 8027  df-en 8124  df-dom 8125  df-sdom 8126  df-fin 8127  df-card 8975  df-cda 9202  df-pnf 10288  df-mnf 10289  df-xr 10290  df-ltxr 10291  df-le 10292  df-sub 10480  df-neg 10481  df-nn 11233  df-n0 11505  df-z 11590  df-uz 11900  df-fz 12540  df-hash 13332  df-mzpcl 37806  df-mzp 37807  df-dioph 37839
This theorem is referenced by:  orrabdioph  37865
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