Theorem diophun 43011

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 43002	. . 3 ⊢ (𝐴 ∈ (Dioph‘𝑁) → 𝑁 ∈ ℕ0)
2		nnex 12151	. . . . . 6 ⊢ ℕ ∈ V
3	2	jctr 524	. . . . 5 ⊢ (𝑁 ∈ ℕ0 → (𝑁 ∈ ℕ0 ∧ ℕ ∈ V))
4		1z 12521	. . . . . . 7 ⊢ 1 ∈ ℤ
5		nnuz 12790	. . . . . . . 8 ⊢ ℕ = (ℤ≥‘1)
6	5	uzinf 13888	. . . . . . 7 ⊢ (1 ∈ ℤ → ¬ ℕ ∈ Fin)
7	4, 6	ax-mp 5	. . . . . 6 ⊢ ¬ ℕ ∈ Fin
8		elfznn 13469	. . . . . . 7 ⊢ (𝑎 ∈ (1...𝑁) → 𝑎 ∈ ℕ)
9	8	ssriv 3937	. . . . . 6 ⊢ (1...𝑁) ⊆ ℕ
10	7, 9	pm3.2i 470	. . . . 5 ⊢ (¬ ℕ ∈ Fin ∧ (1...𝑁) ⊆ ℕ)
11		eldioph2b 43001	. . . . . 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 43001	. . . . . 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 632	. . . . 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 586	. . . 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 3208	. . . . 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 4260	. . . . . . . . 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 3104	. . . . . . . . . . 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 1009	. . . . . . . . . . . . 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 12497	. . . . . . . . . . . . . . . . . . . 20 ⊢ ℤ ∈ V
20		nn0ssz 12511	. . . . . . . . . . . . . . . . . . . 20 ⊢ ℕ0 ⊆ ℤ
21		mapss 8827	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((ℤ ∈ V ∧ ℕ0 ⊆ ℤ) → (ℕ0 ↑m ℕ) ⊆ (ℤ ↑m ℕ))
22	19, 20, 21	mp2an 692	. . . . . . . . . . . . . . . . . . 19 ⊢ (ℕ0 ↑m ℕ) ⊆ (ℤ ↑m ℕ)
23	22	sseli 3929	. . . . . . . . . . . . . . . . . 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 6834	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑒 = 𝑑 → (𝑎‘𝑒) = (𝑎‘𝑑))
26		fveq2 6834	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑒 = 𝑑 → (𝑐‘𝑒) = (𝑐‘𝑑))
27	25, 26	oveq12d 7376	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑒 = 𝑑 → ((𝑎‘𝑒) · (𝑐‘𝑒)) = ((𝑎‘𝑑) · (𝑐‘𝑑)))
28		eqid 2736	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑒 ∈ (ℤ ↑m ℕ) ↦ ((𝑎‘𝑒) · (𝑐‘𝑒))) = (𝑒 ∈ (ℤ ↑m ℕ) ↦ ((𝑎‘𝑒) · (𝑐‘𝑒)))
29		ovex 7391	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑎‘𝑑) · (𝑐‘𝑑)) ∈ V
30	27, 28, 29	fvmpt 6941	. . . . . . . . . . . . . . . . 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 2738	. . . . . . . . . . . . . . 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 776	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘ℕ) ∧ 𝑐 ∈ (mzPoly‘ℕ))) ∧ 𝑑 ∈ (ℕ0 ↑m ℕ)) → 𝑎 ∈ (mzPoly‘ℕ))
34		mzpf 42974	. . . . . . . . . . . . . . . . . . 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 7030	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘ℕ) ∧ 𝑐 ∈ (mzPoly‘ℕ))) ∧ 𝑑 ∈ (ℕ0 ↑m ℕ)) → (𝑎‘𝑑) ∈ ℤ)
37	36	zcnd 12597	. . . . . . . . . . . . . . . 16 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘ℕ) ∧ 𝑐 ∈ (mzPoly‘ℕ))) ∧ 𝑑 ∈ (ℕ0 ↑m ℕ)) → (𝑎‘𝑑) ∈ ℂ)
38		simplrr 777	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘ℕ) ∧ 𝑐 ∈ (mzPoly‘ℕ))) ∧ 𝑑 ∈ (ℕ0 ↑m ℕ)) → 𝑐 ∈ (mzPoly‘ℕ))
39		mzpf 42974	. . . . . . . . . . . . . . . . . . 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 7030	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘ℕ) ∧ 𝑐 ∈ (mzPoly‘ℕ))) ∧ 𝑑 ∈ (ℕ0 ↑m ℕ)) → (𝑐‘𝑑) ∈ ℤ)
42	41	zcnd 12597	. . . . . . . . . . . . . . . 16 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘ℕ) ∧ 𝑐 ∈ (mzPoly‘ℕ))) ∧ 𝑑 ∈ (ℕ0 ↑m ℕ)) → (𝑐‘𝑑) ∈ ℂ)
43	37, 42	mul0ord 11785	. . . . . . . . . . . . . . 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 280	. . . . . . . . . . . . . 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 630	. . . . . . . . . . . . 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 285	. . . . . . . . . . . 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 3158	. . . . . . . . . . 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 285	. . . . . . . . . 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 2802	. . . . . . . . 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 2783	. . . . . . . 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 770	. . . . . . . . . . . . 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 2837	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘ℕ) ∧ 𝑐 ∈ (mzPoly‘ℕ))) → (𝑒 ∈ (ℤ ↑m ℕ) ↦ (𝑎‘𝑒)) ∈ (mzPoly‘ℕ))
58		simprr 772	. . . . . . . . . . . . 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 2837	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘ℕ) ∧ 𝑐 ∈ (mzPoly‘ℕ))) → (𝑒 ∈ (ℤ ↑m ℕ) ↦ (𝑐‘𝑒)) ∈ (mzPoly‘ℕ))
62		mzpmulmpt 42980	. . . . . . . . . 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 584	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘ℕ) ∧ 𝑐 ∈ (mzPoly‘ℕ))) → (𝑒 ∈ (ℤ ↑m ℕ) ↦ ((𝑎‘𝑒) · (𝑐‘𝑒))) ∈ (mzPoly‘ℕ))
64		eldioph2 43000	. . . . . . . . 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 1373	. . . . . . . 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 2836	. . . . . . 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 4115	. . . . . . . 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 2821	. . . . . . 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 247	. . . . . 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 3193	. . . . 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 243	. . . 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 240	. . 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 669	1 ⊢ ((𝐴 ∈ (Dioph‘𝑁) ∧ 𝐵 ∈ (Dioph‘𝑁)) → (𝐴 ∪ 𝐵) ∈ (Dioph‘𝑁))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 847   = wceq 1541   ∈ wcel 2113  {cab 2714  ∃wrex 3060  Vcvv 3440   ∪ cun 3899   ⊆ wss 3901   ↦ cmpt 5179   ↾ cres 5626  ⟶wf 6488  ‘cfv 6492  (class class class)co 7358   ↑_m cmap 8763  Fincfn 8883  0cc0 11026  1c1 11027   · cmul 11031  ℕcn 12145  ℕ₀cn0 12401  ℤcz 12488  ...cfz 13423  mzPolycmzp 42960  Diophcdioph 42993

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-rep 5224  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680  ax-inf2 9550  ax-cnex 11082  ax-resscn 11083  ax-1cn 11084  ax-icn 11085  ax-addcl 11086  ax-addrcl 11087  ax-mulcl 11088  ax-mulrcl 11089  ax-mulcom 11090  ax-addass 11091  ax-mulass 11092  ax-distr 11093  ax-i2m1 11094  ax-1ne0 11095  ax-1rid 11096  ax-rnegex 11097  ax-rrecex 11098  ax-cnre 11099  ax-pre-lttri 11100  ax-pre-lttrn 11101  ax-pre-ltadd 11102  ax-pre-mulgt0 11103

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3061  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-pss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-int 4903  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-tr 5206  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  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 7315  df-ov 7361  df-oprab 7362  df-mpo 7363  df-of 7622  df-om 7809  df-1st 7933  df-2nd 7934  df-frecs 8223  df-wrecs 8254  df-recs 8303  df-rdg 8341  df-1o 8397  df-oadd 8401  df-er 8635  df-map 8765  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-dju 9813  df-card 9851  df-pnf 11168  df-mnf 11169  df-xr 11170  df-ltxr 11171  df-le 11172  df-sub 11366  df-neg 11367  df-nn 12146  df-n0 12402  df-z 12489  df-uz 12752  df-fz 13424  df-hash 14254  df-mzpcl 42961  df-mzp 42962  df-dioph 42994

This theorem is referenced by:  orrabdioph  43019