Theorem diophun 36355

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 36345	. . 3 ⊢ (𝐴 ∈ (Dioph‘𝑁) → 𝑁 ∈ ℕ0)
2		nnex 10903	. . . . . 6 ⊢ ℕ ∈ V
3	2	jctr 563	. . . . 5 ⊢ (𝑁 ∈ ℕ0 → (𝑁 ∈ ℕ0 ∧ ℕ ∈ V))
4		1z 11284	. . . . . . 7 ⊢ 1 ∈ ℤ
5		nnuz 11599	. . . . . . . 8 ⊢ ℕ = (ℤ≥‘1)
6	5	uzinf 12626	. . . . . . 7 ⊢ (1 ∈ ℤ → ¬ ℕ ∈ Fin)
7	4, 6	ax-mp 5	. . . . . 6 ⊢ ¬ ℕ ∈ Fin
8		elfznn 12241	. . . . . . 7 ⊢ (𝑎 ∈ (1...𝑁) → 𝑎 ∈ ℕ)
9	8	ssriv 3572	. . . . . 6 ⊢ (1...𝑁) ⊆ ℕ
10	7, 9	pm3.2i 470	. . . . 5 ⊢ (¬ ℕ ∈ Fin ∧ (1...𝑁) ⊆ ℕ)
11		eldioph2b 36344	. . . . . 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 36344	. . . . . 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 743	. . . . 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 693	. . . 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 3086	. . . . 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 3853	. . . . . . . . 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 3074	. . . . . . . . . . 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 907	. . . . . . . . . . . . 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 11263	. . . . . . . . . . . . . . . . . . . 20 ⊢ ℤ ∈ V
20		nn0ssz 11275	. . . . . . . . . . . . . . . . . . . 20 ⊢ ℕ0 ⊆ ℤ
21		mapss 7786	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((ℤ ∈ V ∧ ℕ0 ⊆ ℤ) → (ℕ0 ↑𝑚 ℕ) ⊆ (ℤ ↑𝑚 ℕ))
22	19, 20, 21	mp2an 704	. . . . . . . . . . . . . . . . . . 19 ⊢ (ℕ0 ↑𝑚 ℕ) ⊆ (ℤ ↑𝑚 ℕ)
23	22	sseli 3564	. . . . . . . . . . . . . . . . . 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 6103	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑒 = 𝑑 → (𝑎‘𝑒) = (𝑎‘𝑑))
26		fveq2 6103	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑒 = 𝑑 → (𝑐‘𝑒) = (𝑐‘𝑑))
27	25, 26	oveq12d 6567	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑒 = 𝑑 → ((𝑎‘𝑒) · (𝑐‘𝑒)) = ((𝑎‘𝑑) · (𝑐‘𝑑)))
28		eqid 2610	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑒 ∈ (ℤ ↑𝑚 ℕ) ↦ ((𝑎‘𝑒) · (𝑐‘𝑒))) = (𝑒 ∈ (ℤ ↑𝑚 ℕ) ↦ ((𝑎‘𝑒) · (𝑐‘𝑒)))
29		ovex 6577	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑎‘𝑑) · (𝑐‘𝑑)) ∈ V
30	27, 28, 29	fvmpt 6191	. . . . . . . . . . . . . . . . 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 2612	. . . . . . . . . . . . . . 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 796	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘ℕ) ∧ 𝑐 ∈ (mzPoly‘ℕ))) ∧ 𝑑 ∈ (ℕ0 ↑𝑚 ℕ)) → 𝑎 ∈ (mzPoly‘ℕ))
34		mzpf 36317	. . . . . . . . . . . . . . . . . . 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 6268	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘ℕ) ∧ 𝑐 ∈ (mzPoly‘ℕ))) ∧ 𝑑 ∈ (ℕ0 ↑𝑚 ℕ)) → (𝑎‘𝑑) ∈ ℤ)
37	36	zcnd 11359	. . . . . . . . . . . . . . . 16 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘ℕ) ∧ 𝑐 ∈ (mzPoly‘ℕ))) ∧ 𝑑 ∈ (ℕ0 ↑𝑚 ℕ)) → (𝑎‘𝑑) ∈ ℂ)
38		simplrr 797	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘ℕ) ∧ 𝑐 ∈ (mzPoly‘ℕ))) ∧ 𝑑 ∈ (ℕ0 ↑𝑚 ℕ)) → 𝑐 ∈ (mzPoly‘ℕ))
39		mzpf 36317	. . . . . . . . . . . . . . . . . . 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 6268	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘ℕ) ∧ 𝑐 ∈ (mzPoly‘ℕ))) ∧ 𝑑 ∈ (ℕ0 ↑𝑚 ℕ)) → (𝑐‘𝑑) ∈ ℤ)
42	41	zcnd 11359	. . . . . . . . . . . . . . . 16 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘ℕ) ∧ 𝑐 ∈ (mzPoly‘ℕ))) ∧ 𝑑 ∈ (ℕ0 ↑𝑚 ℕ)) → (𝑐‘𝑑) ∈ ℂ)
43	37, 42	mul0ord 10556	. . . . . . . . . . . . . . 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 268	. . . . . . . . . . . . . 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 736	. . . . . . . . . . . . 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	syl5bbr 273	. . . . . . . . . . . 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 3031	. . . . . . . . . . 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	syl5bbr 273	. . . . . . . . . 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 2728	. . . . . . . . 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 2656	. . . . . . . 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 472	. . . . . . . . 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 790	. . . . . . . . . . . . 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 6159	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘ℕ) ∧ 𝑐 ∈ (mzPoly‘ℕ))) → 𝑎 = (𝑒 ∈ (ℤ ↑𝑚 ℕ) ↦ (𝑎‘𝑒)))
57	56, 54	eqeltrrd 2689	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘ℕ) ∧ 𝑐 ∈ (mzPoly‘ℕ))) → (𝑒 ∈ (ℤ ↑𝑚 ℕ) ↦ (𝑎‘𝑒)) ∈ (mzPoly‘ℕ))
58		simprr 792	. . . . . . . . . . . . 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 6159	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘ℕ) ∧ 𝑐 ∈ (mzPoly‘ℕ))) → 𝑐 = (𝑒 ∈ (ℤ ↑𝑚 ℕ) ↦ (𝑐‘𝑒)))
61	60, 58	eqeltrrd 2689	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘ℕ) ∧ 𝑐 ∈ (mzPoly‘ℕ))) → (𝑒 ∈ (ℤ ↑𝑚 ℕ) ↦ (𝑐‘𝑒)) ∈ (mzPoly‘ℕ))
62		mzpmulmpt 36323	. . . . . . . . . 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 691	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘ℕ) ∧ 𝑐 ∈ (mzPoly‘ℕ))) → (𝑒 ∈ (ℤ ↑𝑚 ℕ) ↦ ((𝑎‘𝑒) · (𝑐‘𝑒))) ∈ (mzPoly‘ℕ))
64		eldioph2 36343	. . . . . . . . 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 1318	. . . . . . . 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 2688	. . . . . . 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 3724	. . . . . . . 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 2672	. . . . . . 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 236	. . . . . 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 3020	. . . . 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 232	. . . 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 229	. . 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 854	1 ⊢ ((𝐴 ∈ (Dioph‘𝑁) ∧ 𝐵 ∈ (Dioph‘𝑁)) → (𝐴 ∪ 𝐵) ∈ (Dioph‘𝑁))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∨ wo 382   ∧ wa 383   = wceq 1475   ∈ wcel 1977  {cab 2596  ∃wrex 2897  Vcvv 3173   ∪ cun 3538   ⊆ wss 3540   ↦ cmpt 4643   ↾ cres 5040  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549   ↑_𝑚 cmap 7744  Fincfn 7841  0cc0 9815  1c1 9816   · cmul 9820  ℕcn 10897  ℕ₀cn0 11169  ℤcz 11254  ...cfz 12197  mzPolycmzp 36303  Diophcdioph 36336

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-inf2 8421  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-of 6795  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-map 7746  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-card 8648  df-cda 8873  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-n0 11170  df-z 11255  df-uz 11564  df-fz 12198  df-hash 12980  df-mzpcl 36304  df-mzp 36305  df-dioph 36337

This theorem is referenced by:  orrabdioph  36363