Theorem diophun 40511

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 40502	. . 3 ⊢ (𝐴 ∈ (Dioph‘𝑁) → 𝑁 ∈ ℕ0)
2		nnex 11909	. . . . . 6 ⊢ ℕ ∈ V
3	2	jctr 524	. . . . 5 ⊢ (𝑁 ∈ ℕ0 → (𝑁 ∈ ℕ0 ∧ ℕ ∈ V))
4		1z 12280	. . . . . . 7 ⊢ 1 ∈ ℤ
5		nnuz 12550	. . . . . . . 8 ⊢ ℕ = (ℤ≥‘1)
6	5	uzinf 13613	. . . . . . 7 ⊢ (1 ∈ ℤ → ¬ ℕ ∈ Fin)
7	4, 6	ax-mp 5	. . . . . 6 ⊢ ¬ ℕ ∈ Fin
8		elfznn 13214	. . . . . . 7 ⊢ (𝑎 ∈ (1...𝑁) → 𝑎 ∈ ℕ)
9	8	ssriv 3921	. . . . . 6 ⊢ (1...𝑁) ⊆ ℕ
10	7, 9	pm3.2i 470	. . . . 5 ⊢ (¬ ℕ ∈ Fin ∧ (1...𝑁) ⊆ ℕ)
11		eldioph2b 40501	. . . . . 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 40501	. . . . . 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 630	. . . . 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 585	. . . 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 3292	. . . . 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 4229	. . . . . . . . 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 3277	. . . . . . . . . . 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 1004	. . . . . . . . . . . . 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 12258	. . . . . . . . . . . . . . . . . . . 20 ⊢ ℤ ∈ V
20		nn0ssz 12271	. . . . . . . . . . . . . . . . . . . 20 ⊢ ℕ0 ⊆ ℤ
21		mapss 8635	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((ℤ ∈ V ∧ ℕ0 ⊆ ℤ) → (ℕ0 ↑m ℕ) ⊆ (ℤ ↑m ℕ))
22	19, 20, 21	mp2an 688	. . . . . . . . . . . . . . . . . . 19 ⊢ (ℕ0 ↑m ℕ) ⊆ (ℤ ↑m ℕ)
23	22	sseli 3913	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑑 ∈ (ℕ0 ↑m ℕ) → 𝑑 ∈ (ℤ ↑m ℕ))
24	23	adantl 481	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘ℕ) ∧ 𝑐 ∈ (mzPoly‘ℕ))) ∧ 𝑑 ∈ (ℕ0 ↑m ℕ)) → 𝑑 ∈ (ℤ ↑m ℕ))
25		fveq2 6756	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑒 = 𝑑 → (𝑎‘𝑒) = (𝑎‘𝑑))
26		fveq2 6756	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑒 = 𝑑 → (𝑐‘𝑒) = (𝑐‘𝑑))
27	25, 26	oveq12d 7273	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑒 = 𝑑 → ((𝑎‘𝑒) · (𝑐‘𝑒)) = ((𝑎‘𝑑) · (𝑐‘𝑑)))
28		eqid 2738	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑒 ∈ (ℤ ↑m ℕ) ↦ ((𝑎‘𝑒) · (𝑐‘𝑒))) = (𝑒 ∈ (ℤ ↑m ℕ) ↦ ((𝑎‘𝑒) · (𝑐‘𝑒)))
29		ovex 7288	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑎‘𝑑) · (𝑐‘𝑑)) ∈ V
30	27, 28, 29	fvmpt 6857	. . . . . . . . . . . . . . . . 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 2740	. . . . . . . . . . . . . . 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 773	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘ℕ) ∧ 𝑐 ∈ (mzPoly‘ℕ))) ∧ 𝑑 ∈ (ℕ0 ↑m ℕ)) → 𝑎 ∈ (mzPoly‘ℕ))
34		mzpf 40474	. . . . . . . . . . . . . . . . . . 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	ffvelrnd 6944	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘ℕ) ∧ 𝑐 ∈ (mzPoly‘ℕ))) ∧ 𝑑 ∈ (ℕ0 ↑m ℕ)) → (𝑎‘𝑑) ∈ ℤ)
37	36	zcnd 12356	. . . . . . . . . . . . . . . 16 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘ℕ) ∧ 𝑐 ∈ (mzPoly‘ℕ))) ∧ 𝑑 ∈ (ℕ0 ↑m ℕ)) → (𝑎‘𝑑) ∈ ℂ)
38		simplrr 774	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘ℕ) ∧ 𝑐 ∈ (mzPoly‘ℕ))) ∧ 𝑑 ∈ (ℕ0 ↑m ℕ)) → 𝑐 ∈ (mzPoly‘ℕ))
39		mzpf 40474	. . . . . . . . . . . . . . . . . . 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	ffvelrnd 6944	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘ℕ) ∧ 𝑐 ∈ (mzPoly‘ℕ))) ∧ 𝑑 ∈ (ℕ0 ↑m ℕ)) → (𝑐‘𝑑) ∈ ℤ)
42	41	zcnd 12356	. . . . . . . . . . . . . . . 16 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘ℕ) ∧ 𝑐 ∈ (mzPoly‘ℕ))) ∧ 𝑑 ∈ (ℕ0 ↑m ℕ)) → (𝑐‘𝑑) ∈ ℂ)
43	37, 42	mul0ord 11555	. . . . . . . . . . . . . . 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 279	. . . . . . . . . . . . . 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 628	. . . . . . . . . . . . 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 284	. . . . . . . . . . . 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 3224	. . . . . . . . . . 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 284	. . . . . . . . . 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 2808	. . . . . . . . 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	syl5eq 2791	. . . . . . . 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 482	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘ℕ) ∧ 𝑐 ∈ (mzPoly‘ℕ))) → 𝑁 ∈ ℕ0)
52	2, 9	pm3.2i 470	. . . . . . . . . 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 767	. . . . . . . . . . . . 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 6819	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘ℕ) ∧ 𝑐 ∈ (mzPoly‘ℕ))) → 𝑎 = (𝑒 ∈ (ℤ ↑m ℕ) ↦ (𝑎‘𝑒)))
57	56, 54	eqeltrrd 2840	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘ℕ) ∧ 𝑐 ∈ (mzPoly‘ℕ))) → (𝑒 ∈ (ℤ ↑m ℕ) ↦ (𝑎‘𝑒)) ∈ (mzPoly‘ℕ))
58		simprr 769	. . . . . . . . . . . . 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 6819	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘ℕ) ∧ 𝑐 ∈ (mzPoly‘ℕ))) → 𝑐 = (𝑒 ∈ (ℤ ↑m ℕ) ↦ (𝑐‘𝑒)))
61	60, 58	eqeltrrd 2840	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘ℕ) ∧ 𝑐 ∈ (mzPoly‘ℕ))) → (𝑒 ∈ (ℤ ↑m ℕ) ↦ (𝑐‘𝑒)) ∈ (mzPoly‘ℕ))
62		mzpmulmpt 40480	. . . . . . . . . 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 583	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘ℕ) ∧ 𝑐 ∈ (mzPoly‘ℕ))) → (𝑒 ∈ (ℤ ↑m ℕ) ↦ ((𝑎‘𝑒) · (𝑐‘𝑒))) ∈ (mzPoly‘ℕ))
64		eldioph2 40500	. . . . . . . . 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 1369	. . . . . . . 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 2839	. . . . . . 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 4088	. . . . . . . 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 2823	. . . . . . 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 246	. . . . . 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 3222	. . . . 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	syl5bir 242	. . . 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 239	. . 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 665	1 ⊢ ((𝐴 ∈ (Dioph‘𝑁) ∧ 𝐵 ∈ (Dioph‘𝑁)) → (𝐴 ∪ 𝐵) ∈ (Dioph‘𝑁))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 395   ∨ wo 843   = wceq 1539   ∈ wcel 2108  {cab 2715  ∃wrex 3064  Vcvv 3422   ∪ cun 3881   ⊆ wss 3883   ↦ cmpt 5153   ↾ cres 5582  ⟶wf 6414  ‘cfv 6418  (class class class)co 7255   ↑_m cmap 8573  Fincfn 8691  0cc0 10802  1c1 10803   · cmul 10807  ℕcn 11903  ℕ₀cn0 12163  ℤcz 12249  ...cfz 13168  mzPolycmzp 40460  Diophcdioph 40493

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566  ax-inf2 9329  ax-cnex 10858  ax-resscn 10859  ax-1cn 10860  ax-icn 10861  ax-addcl 10862  ax-addrcl 10863  ax-mulcl 10864  ax-mulrcl 10865  ax-mulcom 10866  ax-addass 10867  ax-mulass 10868  ax-distr 10869  ax-i2m1 10870  ax-1ne0 10871  ax-1rid 10872  ax-rnegex 10873  ax-rrecex 10874  ax-cnre 10875  ax-pre-lttri 10876  ax-pre-lttrn 10877  ax-pre-ltadd 10878  ax-pre-mulgt0 10879

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-nel 3049  df-ral 3068  df-rex 3069  df-reu 3070  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-int 4877  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-pred 6191  df-ord 6254  df-on 6255  df-lim 6256  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-riota 7212  df-ov 7258  df-oprab 7259  df-mpo 7260  df-of 7511  df-om 7688  df-1st 7804  df-2nd 7805  df-frecs 8068  df-wrecs 8099  df-recs 8173  df-rdg 8212  df-1o 8267  df-oadd 8271  df-er 8456  df-map 8575  df-en 8692  df-dom 8693  df-sdom 8694  df-fin 8695  df-dju 9590  df-card 9628  df-pnf 10942  df-mnf 10943  df-xr 10944  df-ltxr 10945  df-le 10946  df-sub 11137  df-neg 11138  df-nn 11904  df-n0 12164  df-z 12250  df-uz 12512  df-fz 13169  df-hash 13973  df-mzpcl 40461  df-mzp 40462  df-dioph 40494

This theorem is referenced by:  orrabdioph  40519