Theorem diophun 43618

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 43609	. . 3 ⊢ (𝐴 ∈ (Dioph‘𝑁) → 𝑁 ∈ ℕ0)
2		nnex 12263	. . . . . 6 ⊢ ℕ ∈ V
3	2	jctr 534	. . . . 5 ⊢ (𝑁 ∈ ℕ0 → (𝑁 ∈ ℕ0 ∧ ℕ ∈ V))
4		1z 12648	. . . . . . 7 ⊢ 1 ∈ ℤ
5		nnuz 12926	. . . . . . . 8 ⊢ ℕ = (ℤ≥‘1)
6	5	uzinf 14029	. . . . . . 7 ⊢ (1 ∈ ℤ → ¬ ℕ ∈ Fin)
7	4, 6	ax-mp 5	. . . . . 6 ⊢ ¬ ℕ ∈ Fin
8		elfznn 13608	. . . . . . 7 ⊢ (𝑎 ∈ (1...𝑁) → 𝑎 ∈ ℕ)
9	8	ssriv 3935	. . . . . 6 ⊢ (1...𝑁) ⊆ ℕ
10	7, 9	pm3.2i 476	. . . . 5 ⊢ (¬ ℕ ∈ Fin ∧ (1...𝑁) ⊆ ℕ)
11		eldioph2b 43608	. . . . . 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 43608	. . . . . 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 644	. . . . 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 598	. . . 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 3234	. . . . 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 4254	. . . . . . . . 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 3130	. . . . . . . . . . 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 1025	. . . . . . . . . . . . 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 12624	. . . . . . . . . . . . . . . . . . . 20 ⊢ ℤ ∈ V
20		nn0ssz 12638	. . . . . . . . . . . . . . . . . . . 20 ⊢ ℕ0 ⊆ ℤ
21		mapss 8896	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((ℤ ∈ V ∧ ℕ0 ⊆ ℤ) → (ℕ0 ↑m ℕ) ⊆ (ℤ ↑m ℕ))
22	19, 20, 21	mp2an 705	. . . . . . . . . . . . . . . . . . 19 ⊢ (ℕ0 ↑m ℕ) ⊆ (ℤ ↑m ℕ)
23	22	sseli 3927	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑑 ∈ (ℕ0 ↑m ℕ) → 𝑑 ∈ (ℤ ↑m ℕ))
24	23	adantl 487	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘ℕ) ∧ 𝑐 ∈ (mzPoly‘ℕ))) ∧ 𝑑 ∈ (ℕ0 ↑m ℕ)) → 𝑑 ∈ (ℤ ↑m ℕ))
25		fveq2 6878	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑒 = 𝑑 → (𝑎‘𝑒) = (𝑎‘𝑑))
26		fveq2 6878	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑒 = 𝑑 → (𝑐‘𝑒) = (𝑐‘𝑑))
27	25, 26	oveq12d 7431	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑒 = 𝑑 → ((𝑎‘𝑒) · (𝑐‘𝑒)) = ((𝑎‘𝑑) · (𝑐‘𝑑)))
28		eqid 2760	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑒 ∈ (ℤ ↑m ℕ) ↦ ((𝑎‘𝑒) · (𝑐‘𝑒))) = (𝑒 ∈ (ℤ ↑m ℕ) ↦ ((𝑎‘𝑒) · (𝑐‘𝑒)))
29		ovex 7446	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑎‘𝑑) · (𝑐‘𝑑)) ∈ V
30	27, 28, 29	fvmpt 6986	. . . . . . . . . . . . . . . . 17 ⊢ (𝑑 ∈ (ℤ ↑m ℕ) → ((𝑒 ∈ (ℤ ↑m ℕ) ↦ ((𝑎‘𝑒) · (𝑐‘𝑒)))‘𝑑) = ((𝑎‘𝑑) · (𝑐‘𝑑)))
31	24, 30	syl 18	. . . . . . . . . . . . . . . 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 2762	. . . . . . . . . . . . . . 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 789	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘ℕ) ∧ 𝑐 ∈ (mzPoly‘ℕ))) ∧ 𝑑 ∈ (ℕ0 ↑m ℕ)) → 𝑎 ∈ (mzPoly‘ℕ))
34		mzpf 43581	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑎 ∈ (mzPoly‘ℕ) → 𝑎:(ℤ ↑m ℕ)⟶ℤ)
35	33, 34	syl 18	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘ℕ) ∧ 𝑐 ∈ (mzPoly‘ℕ))) ∧ 𝑑 ∈ (ℕ0 ↑m ℕ)) → 𝑎:(ℤ ↑m ℕ)⟶ℤ)
36	35, 24	ffvelcdmd 7078	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘ℕ) ∧ 𝑐 ∈ (mzPoly‘ℕ))) ∧ 𝑑 ∈ (ℕ0 ↑m ℕ)) → (𝑎‘𝑑) ∈ ℤ)
37	36	zcnd 12726	. . . . . . . . . . . . . . . 16 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘ℕ) ∧ 𝑐 ∈ (mzPoly‘ℕ))) ∧ 𝑑 ∈ (ℕ0 ↑m ℕ)) → (𝑎‘𝑑) ∈ ℂ)
38		simplrr 790	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘ℕ) ∧ 𝑐 ∈ (mzPoly‘ℕ))) ∧ 𝑑 ∈ (ℕ0 ↑m ℕ)) → 𝑐 ∈ (mzPoly‘ℕ))
39		mzpf 43581	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑐 ∈ (mzPoly‘ℕ) → 𝑐:(ℤ ↑m ℕ)⟶ℤ)
40	38, 39	syl 18	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘ℕ) ∧ 𝑐 ∈ (mzPoly‘ℕ))) ∧ 𝑑 ∈ (ℕ0 ↑m ℕ)) → 𝑐:(ℤ ↑m ℕ)⟶ℤ)
41	40, 24	ffvelcdmd 7078	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘ℕ) ∧ 𝑐 ∈ (mzPoly‘ℕ))) ∧ 𝑑 ∈ (ℕ0 ↑m ℕ)) → (𝑐‘𝑑) ∈ ℤ)
42	41	zcnd 12726	. . . . . . . . . . . . . . . 16 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘ℕ) ∧ 𝑐 ∈ (mzPoly‘ℕ))) ∧ 𝑑 ∈ (ℕ0 ↑m ℕ)) → (𝑐‘𝑑) ∈ ℂ)
43	37, 42	mul0ord 11886	. . . . . . . . . . . . . . 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 283	. . . . . . . . . . . . . 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 642	. . . . . . . . . . . . 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 288	. . . . . . . . . . . 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 3184	. . . . . . . . . . 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 288	. . . . . . . . . 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 2826	. . . . . . . . 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 2807	. . . . . . . 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 488	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘ℕ) ∧ 𝑐 ∈ (mzPoly‘ℕ))) → 𝑁 ∈ ℕ0)
52	2, 9	pm3.2i 476	. . . . . . . . . 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 783	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘ℕ) ∧ 𝑐 ∈ (mzPoly‘ℕ))) → 𝑎 ∈ (mzPoly‘ℕ))
55	54, 34	syl 18	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘ℕ) ∧ 𝑐 ∈ (mzPoly‘ℕ))) → 𝑎:(ℤ ↑m ℕ)⟶ℤ)
56	55	feqmptd 6946	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘ℕ) ∧ 𝑐 ∈ (mzPoly‘ℕ))) → 𝑎 = (𝑒 ∈ (ℤ ↑m ℕ) ↦ (𝑎‘𝑒)))
57	56, 54	eqeltrrd 2861	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘ℕ) ∧ 𝑐 ∈ (mzPoly‘ℕ))) → (𝑒 ∈ (ℤ ↑m ℕ) ↦ (𝑎‘𝑒)) ∈ (mzPoly‘ℕ))
58		simprr 785	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘ℕ) ∧ 𝑐 ∈ (mzPoly‘ℕ))) → 𝑐 ∈ (mzPoly‘ℕ))
59	58, 39	syl 18	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘ℕ) ∧ 𝑐 ∈ (mzPoly‘ℕ))) → 𝑐:(ℤ ↑m ℕ)⟶ℤ)
60	59	feqmptd 6946	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘ℕ) ∧ 𝑐 ∈ (mzPoly‘ℕ))) → 𝑐 = (𝑒 ∈ (ℤ ↑m ℕ) ↦ (𝑐‘𝑒)))
61	60, 58	eqeltrrd 2861	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘ℕ) ∧ 𝑐 ∈ (mzPoly‘ℕ))) → (𝑒 ∈ (ℤ ↑m ℕ) ↦ (𝑐‘𝑒)) ∈ (mzPoly‘ℕ))
62		mzpmulmpt 43587	. . . . . . . . . 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 596	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘ℕ) ∧ 𝑐 ∈ (mzPoly‘ℕ))) → (𝑒 ∈ (ℤ ↑m ℕ) ↦ ((𝑎‘𝑒) · (𝑐‘𝑒))) ∈ (mzPoly‘ℕ))
64		eldioph2 43607	. . . . . . . . 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 1398	. . . . . . . 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 2860	. . . . . . 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 4110	. . . . . . . 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 2845	. . . . . . 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 250	. . . . . 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 3219	. . . . 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 246	. . . 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 243	. . 3 ⊢ (𝑁 ∈ ℕ0 → ((𝐴 ∈ (Dioph‘𝑁) ∧ 𝐵 ∈ (Dioph‘𝑁)) → (𝐴 ∪ 𝐵) ∈ (Dioph‘𝑁)))
73	1, 72	syl 18	. 2 ⊢ (𝐴 ∈ (Dioph‘𝑁) → ((𝐴 ∈ (Dioph‘𝑁) ∧ 𝐵 ∈ (Dioph‘𝑁)) → (𝐴 ∪ 𝐵) ∈ (Dioph‘𝑁)))
74	73	anabsi5 682	1 ⊢ ((𝐴 ∈ (Dioph‘𝑁) ∧ 𝐵 ∈ (Dioph‘𝑁)) → (𝐴 ∪ 𝐵) ∈ (Dioph‘𝑁))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861   = wceq 1570   ∈ wcel 2145  {cab 2738  ∃wrex 3086  Vcvv 3450   ∪ cun 3897   ⊆ wss 3899   ↦ cmpt 5186   ↾ cres 5657  ⟶wf 6529  ‘cfv 6533  (class class class)co 7413   ↑_m cmap 8826  Fincfn 8952  0cc0 11124  1c1 11125   · cmul 11129  ℕcn 12257  ℕ₀cn0 12528  ℤcz 12615  ...cfz 13561  mzPolycmzp 43567  Diophcdioph 43600

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-rep 5232  ax-sep 5251  ax-nul 5263  ax-pow 5330  ax-pr 5398  ax-un 7736  ax-inf2 9620  ax-cnex 11180  ax-resscn 11181  ax-1cn 11182  ax-icn 11183  ax-addcl 11184  ax-addrcl 11185  ax-mulcl 11186  ax-mulrcl 11187  ax-mulcom 11188  ax-addass 11189  ax-mulass 11190  ax-distr 11191  ax-i2m1 11192  ax-1ne0 11193  ax-1rid 11194  ax-rnegex 11195  ax-rrecex 11196  ax-cnre 11197  ax-pre-lttri 11198  ax-pre-lttrn 11199  ax-pre-ltadd 11200  ax-pre-mulgt0 11201

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-nel 3062  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-int 4908  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6299  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-riota 7370  df-ov 7416  df-oprab 7417  df-mpo 7418  df-of 7678  df-om 7863  df-1st 7986  df-2nd 7987  df-frecs 8280  df-wrecs 8311  df-recs 8360  df-rdg 8399  df-1o 8455  df-oadd 8459  df-er 8696  df-map 8828  df-en 8953  df-dom 8954  df-sdom 8955  df-fin 8956  df-dju 9906  df-card 9944  df-pnf 11269  df-mnf 11270  df-xr 11271  df-ltxr 11272  df-le 11273  df-sub 11467  df-neg 11468  df-nn 12258  df-n0 12529  df-z 12616  df-uz 12888  df-fz 13562  df-hash 14395  df-mzpcl 43568  df-mzp 43569  df-dioph 43601

This theorem is used by:  orrabdioph  43626