Theorem diophun 43229

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)) → (𝐴 ∪ 𝐵) ∈ (Dioph‘𝑁))

Proof of Theorem diophun

Dummy variables 𝑎 𝑏 𝑐 𝑑 𝑒 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eldiophelnn0 43220	. . 3 ⊢ (𝐴 ∈ (Dioph‘𝑁) → 𝑁 ∈ ℕ0)
2		nnex 12178	. . . . . 6 ⊢ ℕ ∈ V
3	2	jctr 529	. . . . 5 ⊢ (𝑁 ∈ ℕ0 → (𝑁 ∈ ℕ0 ∧ ℕ ∈ V))
4		1z 12555	. . . . . . 7 ⊢ 1 ∈ ℤ
5		nnuz 12825	. . . . . . . 8 ⊢ ℕ = (ℤ≥‘1)
6	5	uzinf 13925	. . . . . . 7 ⊢ (1 ∈ ℤ → ¬ ℕ ∈ Fin)
7	4, 6	ax-mp 5	. . . . . 6 ⊢ ¬ ℕ ∈ Fin
8		elfznn 13505	. . . . . . 7 ⊢ (𝑎 ∈ (1...𝑁) → 𝑎 ∈ ℕ)
9	8	ssriv 3926	. . . . . 6 ⊢ (1...𝑁) ⊆ ℕ
10	7, 9	pm3.2i 471	. . . . 5 ⊢ (¬ ℕ ∈ Fin ∧ (1...𝑁) ⊆ ℕ)
11		eldioph2b 43219	. . . . . 6 ⊢ (((𝑁 ∈ ℕ0 ∧ ℕ ∈ V) ∧ (¬ ℕ ∈ Fin ∧ (1...𝑁) ⊆ ℕ)) → (𝐴 ∈ (Dioph‘𝑁) ↔ ∃𝑎 ∈ (mzPoly‘ℕ)𝐴 = {𝑏 ∣ ∃𝑑 ∈ (ℕ0 ↑m ℕ)(𝑏 = (𝑑 ↾ (1...𝑁)) ∧ (𝑎‘𝑑) = 0)}))
12		eldioph2b 43219	. . . . . 6 ⊢ (((𝑁 ∈ ℕ0 ∧ ℕ ∈ V) ∧ (¬ ℕ ∈ Fin ∧ (1...𝑁) ⊆ ℕ)) → (𝐵 ∈ (Dioph‘𝑁) ↔ ∃𝑐 ∈ (mzPoly‘ℕ)𝐵 = {𝑏 ∣ ∃𝑑 ∈ (ℕ0 ↑m ℕ)(𝑏 = (𝑑 ↾ (1...𝑁)) ∧ (𝑐‘𝑑) = 0)}))
13	11, 12	anbi12d 638	. . . . 5 ⊢ (((𝑁 ∈ ℕ0 ∧ ℕ ∈ V) ∧ (¬ ℕ ∈ Fin ∧ (1...𝑁) ⊆ ℕ)) → ((𝐴 ∈ (Dioph‘𝑁) ∧ 𝐵 ∈ (Dioph‘𝑁)) ↔ (∃𝑎 ∈ (mzPoly‘ℕ)𝐴 = {𝑏 ∣ ∃𝑑 ∈ (ℕ0 ↑m ℕ)(𝑏 = (𝑑 ↾ (1...𝑁)) ∧ (𝑎‘𝑑) = 0)} ∧ ∃𝑐 ∈ (mzPoly‘ℕ)𝐵 = {𝑏 ∣ ∃𝑑 ∈ (ℕ0 ↑m ℕ)(𝑏 = (𝑑 ↾ (1...𝑁)) ∧ (𝑐‘𝑑) = 0)})))
14	3, 10, 13	sylancl 592	. . . 4 ⊢ (𝑁 ∈ ℕ0 → ((𝐴 ∈ (Dioph‘𝑁) ∧ 𝐵 ∈ (Dioph‘𝑁)) ↔ (∃𝑎 ∈ (mzPoly‘ℕ)𝐴 = {𝑏 ∣ ∃𝑑 ∈ (ℕ0 ↑m ℕ)(𝑏 = (𝑑 ↾ (1...𝑁)) ∧ (𝑎‘𝑑) = 0)} ∧ ∃𝑐 ∈ (mzPoly‘ℕ)𝐵 = {𝑏 ∣ ∃𝑑 ∈ (ℕ0 ↑m ℕ)(𝑏 = (𝑑 ↾ (1...𝑁)) ∧ (𝑐‘𝑑) = 0)})))
15		reeanv 3212	. . . . 5 ⊢ (∃𝑎 ∈ (mzPoly‘ℕ)∃𝑐 ∈ (mzPoly‘ℕ)(𝐴 = {𝑏 ∣ ∃𝑑 ∈ (ℕ0 ↑m ℕ)(𝑏 = (𝑑 ↾ (1...𝑁)) ∧ (𝑎‘𝑑) = 0)} ∧ 𝐵 = {𝑏 ∣ ∃𝑑 ∈ (ℕ0 ↑m ℕ)(𝑏 = (𝑑 ↾ (1...𝑁)) ∧ (𝑐‘𝑑) = 0)}) ↔ (∃𝑎 ∈ (mzPoly‘ℕ)𝐴 = {𝑏 ∣ ∃𝑑 ∈ (ℕ0 ↑m ℕ)(𝑏 = (𝑑 ↾ (1...𝑁)) ∧ (𝑎‘𝑑) = 0)} ∧ ∃𝑐 ∈ (mzPoly‘ℕ)𝐵 = {𝑏 ∣ ∃𝑑 ∈ (ℕ0 ↑m ℕ)(𝑏 = (𝑑 ↾ (1...𝑁)) ∧ (𝑐‘𝑑) = 0)}))
16		unab 4243	. . . . . . . . 9 ⊢ ({𝑏 ∣ ∃𝑑 ∈ (ℕ0 ↑m ℕ)(𝑏 = (𝑑 ↾ (1...𝑁)) ∧ (𝑎‘𝑑) = 0)} ∪ {𝑏 ∣ ∃𝑑 ∈ (ℕ0 ↑m ℕ)(𝑏 = (𝑑 ↾ (1...𝑁)) ∧ (𝑐‘𝑑) = 0)}) = {𝑏 ∣ (∃𝑑 ∈ (ℕ0 ↑m ℕ)(𝑏 = (𝑑 ↾ (1...𝑁)) ∧ (𝑎‘𝑑) = 0) ∨ ∃𝑑 ∈ (ℕ0 ↑m ℕ)(𝑏 = (𝑑 ↾ (1...𝑁)) ∧ (𝑐‘𝑑) = 0))}
17		r19.43 3108	. . . . . . . . . . 11 ⊢ (∃𝑑 ∈ (ℕ0 ↑m ℕ)((𝑏 = (𝑑 ↾ (1...𝑁)) ∧ (𝑎‘𝑑) = 0) ∨ (𝑏 = (𝑑 ↾ (1...𝑁)) ∧ (𝑐‘𝑑) = 0)) ↔ (∃𝑑 ∈ (ℕ0 ↑m ℕ)(𝑏 = (𝑑 ↾ (1...𝑁)) ∧ (𝑎‘𝑑) = 0) ∨ ∃𝑑 ∈ (ℕ0 ↑m ℕ)(𝑏 = (𝑑 ↾ (1...𝑁)) ∧ (𝑐‘𝑑) = 0)))
18		andi 1015	. . . . . . . . . . . . 13 ⊢ ((𝑏 = (𝑑 ↾ (1...𝑁)) ∧ ((𝑎‘𝑑) = 0 ∨ (𝑐‘𝑑) = 0)) ↔ ((𝑏 = (𝑑 ↾ (1...𝑁)) ∧ (𝑎‘𝑑) = 0) ∨ (𝑏 = (𝑑 ↾ (1...𝑁)) ∧ (𝑐‘𝑑) = 0)))
19		zex 12531	. . . . . . . . . . . . . . . . . . . 20 ⊢ ℤ ∈ V
20		nn0ssz 12545	. . . . . . . . . . . . . . . . . . . 20 ⊢ ℕ0 ⊆ ℤ
21		mapss 8834	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((ℤ ∈ V ∧ ℕ0 ⊆ ℤ) → (ℕ0 ↑m ℕ) ⊆ (ℤ ↑m ℕ))
22	19, 20, 21	mp2an 698	. . . . . . . . . . . . . . . . . . 19 ⊢ (ℕ0 ↑m ℕ) ⊆ (ℤ ↑m ℕ)
23	22	sseli 3918	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑑 ∈ (ℕ0 ↑m ℕ) → 𝑑 ∈ (ℤ ↑m ℕ))
24	23	adantl 482	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘ℕ) ∧ 𝑐 ∈ (mzPoly‘ℕ))) ∧ 𝑑 ∈ (ℕ0 ↑m ℕ)) → 𝑑 ∈ (ℤ ↑m ℕ))
25		fveq2 6834	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑒 = 𝑑 → (𝑎‘𝑒) = (𝑎‘𝑑))
26		fveq2 6834	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑒 = 𝑑 → (𝑐‘𝑒) = (𝑐‘𝑑))
27	25, 26	oveq12d 7381	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑒 = 𝑑 → ((𝑎‘𝑒) · (𝑐‘𝑒)) = ((𝑎‘𝑑) · (𝑐‘𝑑)))
28		eqid 2740	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑒 ∈ (ℤ ↑m ℕ) ↦ ((𝑎‘𝑒) · (𝑐‘𝑒))) = (𝑒 ∈ (ℤ ↑m ℕ) ↦ ((𝑎‘𝑒) · (𝑐‘𝑒)))
29		ovex 7396	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑎‘𝑑) · (𝑐‘𝑑)) ∈ V
30	27, 28, 29	fvmpt 6942	. . . . . . . . . . . . . . . . 17 ⊢ (𝑑 ∈ (ℤ ↑m ℕ) → ((𝑒 ∈ (ℤ ↑m ℕ) ↦ ((𝑎‘𝑒) · (𝑐‘𝑒)))‘𝑑) = ((𝑎‘𝑑) · (𝑐‘𝑑)))
31	24, 30	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘ℕ) ∧ 𝑐 ∈ (mzPoly‘ℕ))) ∧ 𝑑 ∈ (ℕ0 ↑m ℕ)) → ((𝑒 ∈ (ℤ ↑m ℕ) ↦ ((𝑎‘𝑒) · (𝑐‘𝑒)))‘𝑑) = ((𝑎‘𝑑) · (𝑐‘𝑑)))
32	31	eqeq1d 2742	. . . . . . . . . . . . . . 15 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘ℕ) ∧ 𝑐 ∈ (mzPoly‘ℕ))) ∧ 𝑑 ∈ (ℕ0 ↑m ℕ)) → (((𝑒 ∈ (ℤ ↑m ℕ) ↦ ((𝑎‘𝑒) · (𝑐‘𝑒)))‘𝑑) = 0 ↔ ((𝑎‘𝑑) · (𝑐‘𝑑)) = 0))
33		simplrl 782	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘ℕ) ∧ 𝑐 ∈ (mzPoly‘ℕ))) ∧ 𝑑 ∈ (ℕ0 ↑m ℕ)) → 𝑎 ∈ (mzPoly‘ℕ))
34		mzpf 43192	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑎 ∈ (mzPoly‘ℕ) → 𝑎:(ℤ ↑m ℕ)⟶ℤ)
35	33, 34	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘ℕ) ∧ 𝑐 ∈ (mzPoly‘ℕ))) ∧ 𝑑 ∈ (ℕ0 ↑m ℕ)) → 𝑎:(ℤ ↑m ℕ)⟶ℤ)
36	35, 24	ffvelcdmd 7033	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘ℕ) ∧ 𝑐 ∈ (mzPoly‘ℕ))) ∧ 𝑑 ∈ (ℕ0 ↑m ℕ)) → (𝑎‘𝑑) ∈ ℤ)
37	36	zcnd 12632	. . . . . . . . . . . . . . . 16 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘ℕ) ∧ 𝑐 ∈ (mzPoly‘ℕ))) ∧ 𝑑 ∈ (ℕ0 ↑m ℕ)) → (𝑎‘𝑑) ∈ ℂ)
38		simplrr 783	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘ℕ) ∧ 𝑐 ∈ (mzPoly‘ℕ))) ∧ 𝑑 ∈ (ℕ0 ↑m ℕ)) → 𝑐 ∈ (mzPoly‘ℕ))
39		mzpf 43192	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑐 ∈ (mzPoly‘ℕ) → 𝑐:(ℤ ↑m ℕ)⟶ℤ)
40	38, 39	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘ℕ) ∧ 𝑐 ∈ (mzPoly‘ℕ))) ∧ 𝑑 ∈ (ℕ0 ↑m ℕ)) → 𝑐:(ℤ ↑m ℕ)⟶ℤ)
41	40, 24	ffvelcdmd 7033	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘ℕ) ∧ 𝑐 ∈ (mzPoly‘ℕ))) ∧ 𝑑 ∈ (ℕ0 ↑m ℕ)) → (𝑐‘𝑑) ∈ ℤ)
42	41	zcnd 12632	. . . . . . . . . . . . . . . 16 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘ℕ) ∧ 𝑐 ∈ (mzPoly‘ℕ))) ∧ 𝑑 ∈ (ℕ0 ↑m ℕ)) → (𝑐‘𝑑) ∈ ℂ)
43	37, 42	mul0ord 11796	. . . . . . . . . . . . . . 15 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘ℕ) ∧ 𝑐 ∈ (mzPoly‘ℕ))) ∧ 𝑑 ∈ (ℕ0 ↑m ℕ)) → (((𝑎‘𝑑) · (𝑐‘𝑑)) = 0 ↔ ((𝑎‘𝑑) = 0 ∨ (𝑐‘𝑑) = 0)))
44	32, 43	bitr2d 281	. . . . . . . . . . . . . 14 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘ℕ) ∧ 𝑐 ∈ (mzPoly‘ℕ))) ∧ 𝑑 ∈ (ℕ0 ↑m ℕ)) → (((𝑎‘𝑑) = 0 ∨ (𝑐‘𝑑) = 0) ↔ ((𝑒 ∈ (ℤ ↑m ℕ) ↦ ((𝑎‘𝑒) · (𝑐‘𝑒)))‘𝑑) = 0))
45	44	anbi2d 636	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘ℕ) ∧ 𝑐 ∈ (mzPoly‘ℕ))) ∧ 𝑑 ∈ (ℕ0 ↑m ℕ)) → ((𝑏 = (𝑑 ↾ (1...𝑁)) ∧ ((𝑎‘𝑑) = 0 ∨ (𝑐‘𝑑) = 0)) ↔ (𝑏 = (𝑑 ↾ (1...𝑁)) ∧ ((𝑒 ∈ (ℤ ↑m ℕ) ↦ ((𝑎‘𝑒) · (𝑐‘𝑒)))‘𝑑) = 0)))
46	18, 45	bitr3id 286	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘ℕ) ∧ 𝑐 ∈ (mzPoly‘ℕ))) ∧ 𝑑 ∈ (ℕ0 ↑m ℕ)) → (((𝑏 = (𝑑 ↾ (1...𝑁)) ∧ (𝑎‘𝑑) = 0) ∨ (𝑏 = (𝑑 ↾ (1...𝑁)) ∧ (𝑐‘𝑑) = 0)) ↔ (𝑏 = (𝑑 ↾ (1...𝑁)) ∧ ((𝑒 ∈ (ℤ ↑m ℕ) ↦ ((𝑎‘𝑒) · (𝑐‘𝑒)))‘𝑑) = 0)))
47	46	rexbidva 3162	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘ℕ) ∧ 𝑐 ∈ (mzPoly‘ℕ))) → (∃𝑑 ∈ (ℕ0 ↑m ℕ)((𝑏 = (𝑑 ↾ (1...𝑁)) ∧ (𝑎‘𝑑) = 0) ∨ (𝑏 = (𝑑 ↾ (1...𝑁)) ∧ (𝑐‘𝑑) = 0)) ↔ ∃𝑑 ∈ (ℕ0 ↑m ℕ)(𝑏 = (𝑑 ↾ (1...𝑁)) ∧ ((𝑒 ∈ (ℤ ↑m ℕ) ↦ ((𝑎‘𝑒) · (𝑐‘𝑒)))‘𝑑) = 0)))
48	17, 47	bitr3id 286	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘ℕ) ∧ 𝑐 ∈ (mzPoly‘ℕ))) → ((∃𝑑 ∈ (ℕ0 ↑m ℕ)(𝑏 = (𝑑 ↾ (1...𝑁)) ∧ (𝑎‘𝑑) = 0) ∨ ∃𝑑 ∈ (ℕ0 ↑m ℕ)(𝑏 = (𝑑 ↾ (1...𝑁)) ∧ (𝑐‘𝑑) = 0)) ↔ ∃𝑑 ∈ (ℕ0 ↑m ℕ)(𝑏 = (𝑑 ↾ (1...𝑁)) ∧ ((𝑒 ∈ (ℤ ↑m ℕ) ↦ ((𝑎‘𝑒) · (𝑐‘𝑒)))‘𝑑) = 0)))
49	48	abbidv 2806	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘ℕ) ∧ 𝑐 ∈ (mzPoly‘ℕ))) → {𝑏 ∣ (∃𝑑 ∈ (ℕ0 ↑m ℕ)(𝑏 = (𝑑 ↾ (1...𝑁)) ∧ (𝑎‘𝑑) = 0) ∨ ∃𝑑 ∈ (ℕ0 ↑m ℕ)(𝑏 = (𝑑 ↾ (1...𝑁)) ∧ (𝑐‘𝑑) = 0))} = {𝑏 ∣ ∃𝑑 ∈ (ℕ0 ↑m ℕ)(𝑏 = (𝑑 ↾ (1...𝑁)) ∧ ((𝑒 ∈ (ℤ ↑m ℕ) ↦ ((𝑎‘𝑒) · (𝑐‘𝑒)))‘𝑑) = 0)})
50	16, 49	eqtrid 2787	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘ℕ) ∧ 𝑐 ∈ (mzPoly‘ℕ))) → ({𝑏 ∣ ∃𝑑 ∈ (ℕ0 ↑m ℕ)(𝑏 = (𝑑 ↾ (1...𝑁)) ∧ (𝑎‘𝑑) = 0)} ∪ {𝑏 ∣ ∃𝑑 ∈ (ℕ0 ↑m ℕ)(𝑏 = (𝑑 ↾ (1...𝑁)) ∧ (𝑐‘𝑑) = 0)}) = {𝑏 ∣ ∃𝑑 ∈ (ℕ0 ↑m ℕ)(𝑏 = (𝑑 ↾ (1...𝑁)) ∧ ((𝑒 ∈ (ℤ ↑m ℕ) ↦ ((𝑎‘𝑒) · (𝑐‘𝑒)))‘𝑑) = 0)})
51		simpl 483	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘ℕ) ∧ 𝑐 ∈ (mzPoly‘ℕ))) → 𝑁 ∈ ℕ0)
52	2, 9	pm3.2i 471	. . . . . . . . . 10 ⊢ (ℕ ∈ V ∧ (1...𝑁) ⊆ ℕ)
53	52	a1i 11	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘ℕ) ∧ 𝑐 ∈ (mzPoly‘ℕ))) → (ℕ ∈ V ∧ (1...𝑁) ⊆ ℕ))
54		simprl 776	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘ℕ) ∧ 𝑐 ∈ (mzPoly‘ℕ))) → 𝑎 ∈ (mzPoly‘ℕ))
55	54, 34	syl 17	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘ℕ) ∧ 𝑐 ∈ (mzPoly‘ℕ))) → 𝑎:(ℤ ↑m ℕ)⟶ℤ)
56	55	feqmptd 6902	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘ℕ) ∧ 𝑐 ∈ (mzPoly‘ℕ))) → 𝑎 = (𝑒 ∈ (ℤ ↑m ℕ) ↦ (𝑎‘𝑒)))
57	56, 54	eqeltrrd 2841	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘ℕ) ∧ 𝑐 ∈ (mzPoly‘ℕ))) → (𝑒 ∈ (ℤ ↑m ℕ) ↦ (𝑎‘𝑒)) ∈ (mzPoly‘ℕ))
58		simprr 778	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘ℕ) ∧ 𝑐 ∈ (mzPoly‘ℕ))) → 𝑐 ∈ (mzPoly‘ℕ))
59	58, 39	syl 17	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘ℕ) ∧ 𝑐 ∈ (mzPoly‘ℕ))) → 𝑐:(ℤ ↑m ℕ)⟶ℤ)
60	59	feqmptd 6902	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘ℕ) ∧ 𝑐 ∈ (mzPoly‘ℕ))) → 𝑐 = (𝑒 ∈ (ℤ ↑m ℕ) ↦ (𝑐‘𝑒)))
61	60, 58	eqeltrrd 2841	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘ℕ) ∧ 𝑐 ∈ (mzPoly‘ℕ))) → (𝑒 ∈ (ℤ ↑m ℕ) ↦ (𝑐‘𝑒)) ∈ (mzPoly‘ℕ))
62		mzpmulmpt 43198	. . . . . . . . . 10 ⊢ (((𝑒 ∈ (ℤ ↑m ℕ) ↦ (𝑎‘𝑒)) ∈ (mzPoly‘ℕ) ∧ (𝑒 ∈ (ℤ ↑m ℕ) ↦ (𝑐‘𝑒)) ∈ (mzPoly‘ℕ)) → (𝑒 ∈ (ℤ ↑m ℕ) ↦ ((𝑎‘𝑒) · (𝑐‘𝑒))) ∈ (mzPoly‘ℕ))
63	57, 61, 62	syl2anc 590	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘ℕ) ∧ 𝑐 ∈ (mzPoly‘ℕ))) → (𝑒 ∈ (ℤ ↑m ℕ) ↦ ((𝑎‘𝑒) · (𝑐‘𝑒))) ∈ (mzPoly‘ℕ))
64		eldioph2 43218	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ0 ∧ (ℕ ∈ V ∧ (1...𝑁) ⊆ ℕ) ∧ (𝑒 ∈ (ℤ ↑m ℕ) ↦ ((𝑎‘𝑒) · (𝑐‘𝑒))) ∈ (mzPoly‘ℕ)) → {𝑏 ∣ ∃𝑑 ∈ (ℕ0 ↑m ℕ)(𝑏 = (𝑑 ↾ (1...𝑁)) ∧ ((𝑒 ∈ (ℤ ↑m ℕ) ↦ ((𝑎‘𝑒) · (𝑐‘𝑒)))‘𝑑) = 0)} ∈ (Dioph‘𝑁))
65	51, 53, 63, 64	syl3anc 1379	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘ℕ) ∧ 𝑐 ∈ (mzPoly‘ℕ))) → {𝑏 ∣ ∃𝑑 ∈ (ℕ0 ↑m ℕ)(𝑏 = (𝑑 ↾ (1...𝑁)) ∧ ((𝑒 ∈ (ℤ ↑m ℕ) ↦ ((𝑎‘𝑒) · (𝑐‘𝑒)))‘𝑑) = 0)} ∈ (Dioph‘𝑁))
66	50, 65	eqeltrd 2840	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘ℕ) ∧ 𝑐 ∈ (mzPoly‘ℕ))) → ({𝑏 ∣ ∃𝑑 ∈ (ℕ0 ↑m ℕ)(𝑏 = (𝑑 ↾ (1...𝑁)) ∧ (𝑎‘𝑑) = 0)} ∪ {𝑏 ∣ ∃𝑑 ∈ (ℕ0 ↑m ℕ)(𝑏 = (𝑑 ↾ (1...𝑁)) ∧ (𝑐‘𝑑) = 0)}) ∈ (Dioph‘𝑁))
67		uneq12 4100	. . . . . . . 8 ⊢ ((𝐴 = {𝑏 ∣ ∃𝑑 ∈ (ℕ0 ↑m ℕ)(𝑏 = (𝑑 ↾ (1...𝑁)) ∧ (𝑎‘𝑑) = 0)} ∧ 𝐵 = {𝑏 ∣ ∃𝑑 ∈ (ℕ0 ↑m ℕ)(𝑏 = (𝑑 ↾ (1...𝑁)) ∧ (𝑐‘𝑑) = 0)}) → (𝐴 ∪ 𝐵) = ({𝑏 ∣ ∃𝑑 ∈ (ℕ0 ↑m ℕ)(𝑏 = (𝑑 ↾ (1...𝑁)) ∧ (𝑎‘𝑑) = 0)} ∪ {𝑏 ∣ ∃𝑑 ∈ (ℕ0 ↑m ℕ)(𝑏 = (𝑑 ↾ (1...𝑁)) ∧ (𝑐‘𝑑) = 0)}))
68	67	eleq1d 2825	. . . . . . 7 ⊢ ((𝐴 = {𝑏 ∣ ∃𝑑 ∈ (ℕ0 ↑m ℕ)(𝑏 = (𝑑 ↾ (1...𝑁)) ∧ (𝑎‘𝑑) = 0)} ∧ 𝐵 = {𝑏 ∣ ∃𝑑 ∈ (ℕ0 ↑m ℕ)(𝑏 = (𝑑 ↾ (1...𝑁)) ∧ (𝑐‘𝑑) = 0)}) → ((𝐴 ∪ 𝐵) ∈ (Dioph‘𝑁) ↔ ({𝑏 ∣ ∃𝑑 ∈ (ℕ0 ↑m ℕ)(𝑏 = (𝑑 ↾ (1...𝑁)) ∧ (𝑎‘𝑑) = 0)} ∪ {𝑏 ∣ ∃𝑑 ∈ (ℕ0 ↑m ℕ)(𝑏 = (𝑑 ↾ (1...𝑁)) ∧ (𝑐‘𝑑) = 0)}) ∈ (Dioph‘𝑁)))
69	66, 68	syl5ibrcom 248	. . . . . 6 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘ℕ) ∧ 𝑐 ∈ (mzPoly‘ℕ))) → ((𝐴 = {𝑏 ∣ ∃𝑑 ∈ (ℕ0 ↑m ℕ)(𝑏 = (𝑑 ↾ (1...𝑁)) ∧ (𝑎‘𝑑) = 0)} ∧ 𝐵 = {𝑏 ∣ ∃𝑑 ∈ (ℕ0 ↑m ℕ)(𝑏 = (𝑑 ↾ (1...𝑁)) ∧ (𝑐‘𝑑) = 0)}) → (𝐴 ∪ 𝐵) ∈ (Dioph‘𝑁)))
70	69	rexlimdvva 3197	. . . . 5 ⊢ (𝑁 ∈ ℕ0 → (∃𝑎 ∈ (mzPoly‘ℕ)∃𝑐 ∈ (mzPoly‘ℕ)(𝐴 = {𝑏 ∣ ∃𝑑 ∈ (ℕ0 ↑m ℕ)(𝑏 = (𝑑 ↾ (1...𝑁)) ∧ (𝑎‘𝑑) = 0)} ∧ 𝐵 = {𝑏 ∣ ∃𝑑 ∈ (ℕ0 ↑m ℕ)(𝑏 = (𝑑 ↾ (1...𝑁)) ∧ (𝑐‘𝑑) = 0)}) → (𝐴 ∪ 𝐵) ∈ (Dioph‘𝑁)))
71	15, 70	biimtrrid 244	. . . 4 ⊢ (𝑁 ∈ ℕ0 → ((∃𝑎 ∈ (mzPoly‘ℕ)𝐴 = {𝑏 ∣ ∃𝑑 ∈ (ℕ0 ↑m ℕ)(𝑏 = (𝑑 ↾ (1...𝑁)) ∧ (𝑎‘𝑑) = 0)} ∧ ∃𝑐 ∈ (mzPoly‘ℕ)𝐵 = {𝑏 ∣ ∃𝑑 ∈ (ℕ0 ↑m ℕ)(𝑏 = (𝑑 ↾ (1...𝑁)) ∧ (𝑐‘𝑑) = 0)}) → (𝐴 ∪ 𝐵) ∈ (Dioph‘𝑁)))
72	14, 71	sylbid 241	. . 3 ⊢ (𝑁 ∈ ℕ0 → ((𝐴 ∈ (Dioph‘𝑁) ∧ 𝐵 ∈ (Dioph‘𝑁)) → (𝐴 ∪ 𝐵) ∈ (Dioph‘𝑁)))
73	1, 72	syl 17	. 2 ⊢ (𝐴 ∈ (Dioph‘𝑁) → ((𝐴 ∈ (Dioph‘𝑁) ∧ 𝐵 ∈ (Dioph‘𝑁)) → (𝐴 ∪ 𝐵) ∈ (Dioph‘𝑁)))
74	73	anabsi5 675	1 ⊢ ((𝐴 ∈ (Dioph‘𝑁) ∧ 𝐵 ∈ (Dioph‘𝑁)) → (𝐴 ∪ 𝐵) ∈ (Dioph‘𝑁))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 207   ∧ wa 396   ∨ wo 853   = wceq 1547   ∈ wcel 2119  {cab 2718  ∃wrex 3064  Vcvv 3432   ∪ cun 3888   ⊆ wss 3890   ↦ cmpt 5160   ↾ cres 5627  ⟶wf 6488  ‘cfv 6492  (class class class)co 7363   ↑_m cmap 8770  Fincfn 8890  0cc0 11036  1c1 11037   · cmul 11041  ℕcn 12172  ℕ₀cn0 12435  ℤcz 12522  ...cfz 13459  mzPolycmzp 43178  Diophcdioph 43211

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2712  ax-rep 5206  ax-sep 5225  ax-nul 5235  ax-pow 5301  ax-pr 5369  ax-un 7685  ax-inf2 9560  ax-cnex 11092  ax-resscn 11093  ax-1cn 11094  ax-icn 11095  ax-addcl 11096  ax-addrcl 11097  ax-mulcl 11098  ax-mulrcl 11099  ax-mulcom 11100  ax-addass 11101  ax-mulass 11102  ax-distr 11103  ax-i2m1 11104  ax-1ne0 11105  ax-1rid 11106  ax-rnegex 11107  ax-rrecex 11108  ax-cnre 11109  ax-pre-lttri 11110  ax-pre-lttrn 11111  ax-pre-ltadd 11112  ax-pre-mulgt0 11113

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2719  df-cleq 2732  df-clel 2815  df-nfc 2889  df-ne 2936  df-nel 3040  df-ral 3055  df-rex 3065  df-reu 3346  df-rab 3393  df-v 3434  df-sbc 3731  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4269  df-if 4462  df-pw 4538  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4846  df-int 4885  df-iun 4930  df-br 5080  df-opab 5142  df-mpt 5161  df-tr 5187  df-id 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-we 5580  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-riota 7320  df-ov 7366  df-oprab 7367  df-mpo 7368  df-of 7627  df-om 7814  df-1st 7938  df-2nd 7939  df-frecs 8228  df-wrecs 8259  df-recs 8308  df-rdg 8346  df-1o 8402  df-oadd 8406  df-er 8640  df-map 8772  df-en 8891  df-dom 8892  df-sdom 8893  df-fin 8894  df-dju 9823  df-card 9861  df-pnf 11179  df-mnf 11180  df-xr 11181  df-ltxr 11182  df-le 11183  df-sub 11377  df-neg 11378  df-nn 12173  df-n0 12436  df-z 12523  df-uz 12787  df-fz 13460  df-hash 14291  df-mzpcl 43179  df-mzp 43180  df-dioph 43212

This theorem is referenced by:  orrabdioph  43237