Theorem eldioph2b 43522

Description: While Diophantine sets were defined to have a finite number of witness variables consequtively following the observable variables, this is not necessary; they can equivalently be taken to use any witness set (𝑆 ∖ (1...𝑁)). For instance, in diophin 43531 we use this to take the two input sets to have disjoint witness sets. (Contributed by Stefan O'Rear, 8-Oct-2014.)

Assertion

Ref	Expression
eldioph2b	⊢ (((𝑁 ∈ ℕ0 ∧ 𝑆 ∈ V) ∧ (¬ 𝑆 ∈ Fin ∧ (1...𝑁) ⊆ 𝑆)) → (𝐴 ∈ (Dioph‘𝑁) ↔ ∃𝑝 ∈ (mzPoly‘𝑆)𝐴 = {𝑡 ∣ ∃𝑢 ∈ (ℕ0 ↑m 𝑆)(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝‘𝑢) = 0)}))

Distinct variable groups:   𝐴,𝑝   𝑢,𝑁,𝑡,𝑝   𝑢,𝑆,𝑡,𝑝

Allowed substitution hints:   𝐴(𝑢, 𝑡)

Proof of Theorem eldioph2b

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

Step	Hyp	Ref	Expression
1		eldiophb 43516	. . 3 ⊢ (𝐴 ∈ (Dioph‘𝑁) ↔ (𝑁 ∈ ℕ0 ∧ ∃𝑎 ∈ (ℤ≥‘𝑁)∃𝑏 ∈ (mzPoly‘(1...𝑎))𝐴 = {𝑡 ∣ ∃𝑑 ∈ (ℕ0 ↑m (1...𝑎))(𝑡 = (𝑑 ↾ (1...𝑁)) ∧ (𝑏‘𝑑) = 0)}))
2		simp-5r 797	. . . . . . . . 9 ⊢ ((((((𝑁 ∈ ℕ0 ∧ 𝑆 ∈ V) ∧ (¬ 𝑆 ∈ Fin ∧ (1...𝑁) ⊆ 𝑆)) ∧ (𝑎 ∈ (ℤ≥‘𝑁) ∧ 𝑏 ∈ (mzPoly‘(1...𝑎)))) ∧ 𝑐 ∈ V) ∧ (𝑐:(1...𝑎)–1-1→𝑆 ∧ (𝑐 ↾ (1...𝑁)) = ( I ↾ (1...𝑁)))) → 𝑆 ∈ V)
3		simprr 784	. . . . . . . . . 10 ⊢ ((((𝑁 ∈ ℕ0 ∧ 𝑆 ∈ V) ∧ (¬ 𝑆 ∈ Fin ∧ (1...𝑁) ⊆ 𝑆)) ∧ (𝑎 ∈ (ℤ≥‘𝑁) ∧ 𝑏 ∈ (mzPoly‘(1...𝑎)))) → 𝑏 ∈ (mzPoly‘(1...𝑎)))
4	3	ad2antrr 738	. . . . . . . . 9 ⊢ ((((((𝑁 ∈ ℕ0 ∧ 𝑆 ∈ V) ∧ (¬ 𝑆 ∈ Fin ∧ (1...𝑁) ⊆ 𝑆)) ∧ (𝑎 ∈ (ℤ≥‘𝑁) ∧ 𝑏 ∈ (mzPoly‘(1...𝑎)))) ∧ 𝑐 ∈ V) ∧ (𝑐:(1...𝑎)–1-1→𝑆 ∧ (𝑐 ↾ (1...𝑁)) = ( I ↾ (1...𝑁)))) → 𝑏 ∈ (mzPoly‘(1...𝑎)))
5		simprl 782	. . . . . . . . . 10 ⊢ ((((((𝑁 ∈ ℕ0 ∧ 𝑆 ∈ V) ∧ (¬ 𝑆 ∈ Fin ∧ (1...𝑁) ⊆ 𝑆)) ∧ (𝑎 ∈ (ℤ≥‘𝑁) ∧ 𝑏 ∈ (mzPoly‘(1...𝑎)))) ∧ 𝑐 ∈ V) ∧ (𝑐:(1...𝑎)–1-1→𝑆 ∧ (𝑐 ↾ (1...𝑁)) = ( I ↾ (1...𝑁)))) → 𝑐:(1...𝑎)–1-1→𝑆)
6		f1f 6774	. . . . . . . . . 10 ⊢ (𝑐:(1...𝑎)–1-1→𝑆 → 𝑐:(1...𝑎)⟶𝑆)
7	5, 6	syl 18	. . . . . . . . 9 ⊢ ((((((𝑁 ∈ ℕ0 ∧ 𝑆 ∈ V) ∧ (¬ 𝑆 ∈ Fin ∧ (1...𝑁) ⊆ 𝑆)) ∧ (𝑎 ∈ (ℤ≥‘𝑁) ∧ 𝑏 ∈ (mzPoly‘(1...𝑎)))) ∧ 𝑐 ∈ V) ∧ (𝑐:(1...𝑎)–1-1→𝑆 ∧ (𝑐 ↾ (1...𝑁)) = ( I ↾ (1...𝑁)))) → 𝑐:(1...𝑎)⟶𝑆)
8		mzprename 43508	. . . . . . . . 9 ⊢ ((𝑆 ∈ V ∧ 𝑏 ∈ (mzPoly‘(1...𝑎)) ∧ 𝑐:(1...𝑎)⟶𝑆) → (𝑒 ∈ (ℤ ↑m 𝑆) ↦ (𝑏‘(𝑒 ∘ 𝑐))) ∈ (mzPoly‘𝑆))
9	2, 4, 7, 8	syl3anc 1397	. . . . . . . 8 ⊢ ((((((𝑁 ∈ ℕ0 ∧ 𝑆 ∈ V) ∧ (¬ 𝑆 ∈ Fin ∧ (1...𝑁) ⊆ 𝑆)) ∧ (𝑎 ∈ (ℤ≥‘𝑁) ∧ 𝑏 ∈ (mzPoly‘(1...𝑎)))) ∧ 𝑐 ∈ V) ∧ (𝑐:(1...𝑎)–1-1→𝑆 ∧ (𝑐 ↾ (1...𝑁)) = ( I ↾ (1...𝑁)))) → (𝑒 ∈ (ℤ ↑m 𝑆) ↦ (𝑏‘(𝑒 ∘ 𝑐))) ∈ (mzPoly‘𝑆))
10		simprr 784	. . . . . . . . 9 ⊢ ((((((𝑁 ∈ ℕ0 ∧ 𝑆 ∈ V) ∧ (¬ 𝑆 ∈ Fin ∧ (1...𝑁) ⊆ 𝑆)) ∧ (𝑎 ∈ (ℤ≥‘𝑁) ∧ 𝑏 ∈ (mzPoly‘(1...𝑎)))) ∧ 𝑐 ∈ V) ∧ (𝑐:(1...𝑎)–1-1→𝑆 ∧ (𝑐 ↾ (1...𝑁)) = ( I ↾ (1...𝑁)))) → (𝑐 ↾ (1...𝑁)) = ( I ↾ (1...𝑁)))
11		diophrw 43518	. . . . . . . . . 10 ⊢ ((𝑆 ∈ V ∧ 𝑐:(1...𝑎)–1-1→𝑆 ∧ (𝑐 ↾ (1...𝑁)) = ( I ↾ (1...𝑁))) → {𝑡 ∣ ∃𝑢 ∈ (ℕ0 ↑m 𝑆)(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ ((𝑒 ∈ (ℤ ↑m 𝑆) ↦ (𝑏‘(𝑒 ∘ 𝑐)))‘𝑢) = 0)} = {𝑡 ∣ ∃𝑑 ∈ (ℕ0 ↑m (1...𝑎))(𝑡 = (𝑑 ↾ (1...𝑁)) ∧ (𝑏‘𝑑) = 0)})
12	11	eqcomd 2768	. . . . . . . . 9 ⊢ ((𝑆 ∈ V ∧ 𝑐:(1...𝑎)–1-1→𝑆 ∧ (𝑐 ↾ (1...𝑁)) = ( I ↾ (1...𝑁))) → {𝑡 ∣ ∃𝑑 ∈ (ℕ0 ↑m (1...𝑎))(𝑡 = (𝑑 ↾ (1...𝑁)) ∧ (𝑏‘𝑑) = 0)} = {𝑡 ∣ ∃𝑢 ∈ (ℕ0 ↑m 𝑆)(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ ((𝑒 ∈ (ℤ ↑m 𝑆) ↦ (𝑏‘(𝑒 ∘ 𝑐)))‘𝑢) = 0)})
13	2, 5, 10, 12	syl3anc 1397	. . . . . . . 8 ⊢ ((((((𝑁 ∈ ℕ0 ∧ 𝑆 ∈ V) ∧ (¬ 𝑆 ∈ Fin ∧ (1...𝑁) ⊆ 𝑆)) ∧ (𝑎 ∈ (ℤ≥‘𝑁) ∧ 𝑏 ∈ (mzPoly‘(1...𝑎)))) ∧ 𝑐 ∈ V) ∧ (𝑐:(1...𝑎)–1-1→𝑆 ∧ (𝑐 ↾ (1...𝑁)) = ( I ↾ (1...𝑁)))) → {𝑡 ∣ ∃𝑑 ∈ (ℕ0 ↑m (1...𝑎))(𝑡 = (𝑑 ↾ (1...𝑁)) ∧ (𝑏‘𝑑) = 0)} = {𝑡 ∣ ∃𝑢 ∈ (ℕ0 ↑m 𝑆)(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ ((𝑒 ∈ (ℤ ↑m 𝑆) ↦ (𝑏‘(𝑒 ∘ 𝑐)))‘𝑢) = 0)})
14		fveq1 6880	. . . . . . . . . . . . 13 ⊢ (𝑝 = (𝑒 ∈ (ℤ ↑m 𝑆) ↦ (𝑏‘(𝑒 ∘ 𝑐))) → (𝑝‘𝑢) = ((𝑒 ∈ (ℤ ↑m 𝑆) ↦ (𝑏‘(𝑒 ∘ 𝑐)))‘𝑢))
15	14	eqeq1d 2764	. . . . . . . . . . . 12 ⊢ (𝑝 = (𝑒 ∈ (ℤ ↑m 𝑆) ↦ (𝑏‘(𝑒 ∘ 𝑐))) → ((𝑝‘𝑢) = 0 ↔ ((𝑒 ∈ (ℤ ↑m 𝑆) ↦ (𝑏‘(𝑒 ∘ 𝑐)))‘𝑢) = 0))
16	15	anbi2d 641	. . . . . . . . . . 11 ⊢ (𝑝 = (𝑒 ∈ (ℤ ↑m 𝑆) ↦ (𝑏‘(𝑒 ∘ 𝑐))) → ((𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝‘𝑢) = 0) ↔ (𝑡 = (𝑢 ↾ (1...𝑁)) ∧ ((𝑒 ∈ (ℤ ↑m 𝑆) ↦ (𝑏‘(𝑒 ∘ 𝑐)))‘𝑢) = 0)))
17	16	rexbidv 3188	. . . . . . . . . 10 ⊢ (𝑝 = (𝑒 ∈ (ℤ ↑m 𝑆) ↦ (𝑏‘(𝑒 ∘ 𝑐))) → (∃𝑢 ∈ (ℕ0 ↑m 𝑆)(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝‘𝑢) = 0) ↔ ∃𝑢 ∈ (ℕ0 ↑m 𝑆)(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ ((𝑒 ∈ (ℤ ↑m 𝑆) ↦ (𝑏‘(𝑒 ∘ 𝑐)))‘𝑢) = 0)))
18	17	abbidv 2828	. . . . . . . . 9 ⊢ (𝑝 = (𝑒 ∈ (ℤ ↑m 𝑆) ↦ (𝑏‘(𝑒 ∘ 𝑐))) → {𝑡 ∣ ∃𝑢 ∈ (ℕ0 ↑m 𝑆)(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝‘𝑢) = 0)} = {𝑡 ∣ ∃𝑢 ∈ (ℕ0 ↑m 𝑆)(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ ((𝑒 ∈ (ℤ ↑m 𝑆) ↦ (𝑏‘(𝑒 ∘ 𝑐)))‘𝑢) = 0)})
19	18	rspceeqv 3603	. . . . . . . 8 ⊢ (((𝑒 ∈ (ℤ ↑m 𝑆) ↦ (𝑏‘(𝑒 ∘ 𝑐))) ∈ (mzPoly‘𝑆) ∧ {𝑡 ∣ ∃𝑑 ∈ (ℕ0 ↑m (1...𝑎))(𝑡 = (𝑑 ↾ (1...𝑁)) ∧ (𝑏‘𝑑) = 0)} = {𝑡 ∣ ∃𝑢 ∈ (ℕ0 ↑m 𝑆)(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ ((𝑒 ∈ (ℤ ↑m 𝑆) ↦ (𝑏‘(𝑒 ∘ 𝑐)))‘𝑢) = 0)}) → ∃𝑝 ∈ (mzPoly‘𝑆){𝑡 ∣ ∃𝑑 ∈ (ℕ0 ↑m (1...𝑎))(𝑡 = (𝑑 ↾ (1...𝑁)) ∧ (𝑏‘𝑑) = 0)} = {𝑡 ∣ ∃𝑢 ∈ (ℕ0 ↑m 𝑆)(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝‘𝑢) = 0)})
20	9, 13, 19	syl2anc 595	. . . . . . 7 ⊢ ((((((𝑁 ∈ ℕ0 ∧ 𝑆 ∈ V) ∧ (¬ 𝑆 ∈ Fin ∧ (1...𝑁) ⊆ 𝑆)) ∧ (𝑎 ∈ (ℤ≥‘𝑁) ∧ 𝑏 ∈ (mzPoly‘(1...𝑎)))) ∧ 𝑐 ∈ V) ∧ (𝑐:(1...𝑎)–1-1→𝑆 ∧ (𝑐 ↾ (1...𝑁)) = ( I ↾ (1...𝑁)))) → ∃𝑝 ∈ (mzPoly‘𝑆){𝑡 ∣ ∃𝑑 ∈ (ℕ0 ↑m (1...𝑎))(𝑡 = (𝑑 ↾ (1...𝑁)) ∧ (𝑏‘𝑑) = 0)} = {𝑡 ∣ ∃𝑢 ∈ (ℕ0 ↑m 𝑆)(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝‘𝑢) = 0)})
21		simplll 786	. . . . . . . . 9 ⊢ ((((𝑁 ∈ ℕ0 ∧ 𝑆 ∈ V) ∧ (¬ 𝑆 ∈ Fin ∧ (1...𝑁) ⊆ 𝑆)) ∧ (𝑎 ∈ (ℤ≥‘𝑁) ∧ 𝑏 ∈ (mzPoly‘(1...𝑎)))) → 𝑁 ∈ ℕ0)
22		simplrl 788	. . . . . . . . 9 ⊢ ((((𝑁 ∈ ℕ0 ∧ 𝑆 ∈ V) ∧ (¬ 𝑆 ∈ Fin ∧ (1...𝑁) ⊆ 𝑆)) ∧ (𝑎 ∈ (ℤ≥‘𝑁) ∧ 𝑏 ∈ (mzPoly‘(1...𝑎)))) → ¬ 𝑆 ∈ Fin)
23		simplrr 789	. . . . . . . . 9 ⊢ ((((𝑁 ∈ ℕ0 ∧ 𝑆 ∈ V) ∧ (¬ 𝑆 ∈ Fin ∧ (1...𝑁) ⊆ 𝑆)) ∧ (𝑎 ∈ (ℤ≥‘𝑁) ∧ 𝑏 ∈ (mzPoly‘(1...𝑎)))) → (1...𝑁) ⊆ 𝑆)
24		simprl 782	. . . . . . . . 9 ⊢ ((((𝑁 ∈ ℕ0 ∧ 𝑆 ∈ V) ∧ (¬ 𝑆 ∈ Fin ∧ (1...𝑁) ⊆ 𝑆)) ∧ (𝑎 ∈ (ℤ≥‘𝑁) ∧ 𝑏 ∈ (mzPoly‘(1...𝑎)))) → 𝑎 ∈ (ℤ≥‘𝑁))
25		eldioph2lem2 43520	. . . . . . . . 9 ⊢ (((𝑁 ∈ ℕ0 ∧ ¬ 𝑆 ∈ Fin) ∧ ((1...𝑁) ⊆ 𝑆 ∧ 𝑎 ∈ (ℤ≥‘𝑁))) → ∃𝑐(𝑐:(1...𝑎)–1-1→𝑆 ∧ (𝑐 ↾ (1...𝑁)) = ( I ↾ (1...𝑁))))
26	21, 22, 23, 24, 25	syl22anc 851	. . . . . . . 8 ⊢ ((((𝑁 ∈ ℕ0 ∧ 𝑆 ∈ V) ∧ (¬ 𝑆 ∈ Fin ∧ (1...𝑁) ⊆ 𝑆)) ∧ (𝑎 ∈ (ℤ≥‘𝑁) ∧ 𝑏 ∈ (mzPoly‘(1...𝑎)))) → ∃𝑐(𝑐:(1...𝑎)–1-1→𝑆 ∧ (𝑐 ↾ (1...𝑁)) = ( I ↾ (1...𝑁))))
27		rexv 3481	. . . . . . . 8 ⊢ (∃𝑐 ∈ V (𝑐:(1...𝑎)–1-1→𝑆 ∧ (𝑐 ↾ (1...𝑁)) = ( I ↾ (1...𝑁))) ↔ ∃𝑐(𝑐:(1...𝑎)–1-1→𝑆 ∧ (𝑐 ↾ (1...𝑁)) = ( I ↾ (1...𝑁))))
28	26, 27	sylibr 237	. . . . . . 7 ⊢ ((((𝑁 ∈ ℕ0 ∧ 𝑆 ∈ V) ∧ (¬ 𝑆 ∈ Fin ∧ (1...𝑁) ⊆ 𝑆)) ∧ (𝑎 ∈ (ℤ≥‘𝑁) ∧ 𝑏 ∈ (mzPoly‘(1...𝑎)))) → ∃𝑐 ∈ V (𝑐:(1...𝑎)–1-1→𝑆 ∧ (𝑐 ↾ (1...𝑁)) = ( I ↾ (1...𝑁))))
29	20, 28	r19.29a 3172	. . . . . 6 ⊢ ((((𝑁 ∈ ℕ0 ∧ 𝑆 ∈ V) ∧ (¬ 𝑆 ∈ Fin ∧ (1...𝑁) ⊆ 𝑆)) ∧ (𝑎 ∈ (ℤ≥‘𝑁) ∧ 𝑏 ∈ (mzPoly‘(1...𝑎)))) → ∃𝑝 ∈ (mzPoly‘𝑆){𝑡 ∣ ∃𝑑 ∈ (ℕ0 ↑m (1...𝑎))(𝑡 = (𝑑 ↾ (1...𝑁)) ∧ (𝑏‘𝑑) = 0)} = {𝑡 ∣ ∃𝑢 ∈ (ℕ0 ↑m 𝑆)(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝‘𝑢) = 0)})
30		eqeq1 2766	. . . . . . 7 ⊢ (𝐴 = {𝑡 ∣ ∃𝑑 ∈ (ℕ0 ↑m (1...𝑎))(𝑡 = (𝑑 ↾ (1...𝑁)) ∧ (𝑏‘𝑑) = 0)} → (𝐴 = {𝑡 ∣ ∃𝑢 ∈ (ℕ0 ↑m 𝑆)(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝‘𝑢) = 0)} ↔ {𝑡 ∣ ∃𝑑 ∈ (ℕ0 ↑m (1...𝑎))(𝑡 = (𝑑 ↾ (1...𝑁)) ∧ (𝑏‘𝑑) = 0)} = {𝑡 ∣ ∃𝑢 ∈ (ℕ0 ↑m 𝑆)(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝‘𝑢) = 0)}))
31	30	rexbidv 3188	. . . . . 6 ⊢ (𝐴 = {𝑡 ∣ ∃𝑑 ∈ (ℕ0 ↑m (1...𝑎))(𝑡 = (𝑑 ↾ (1...𝑁)) ∧ (𝑏‘𝑑) = 0)} → (∃𝑝 ∈ (mzPoly‘𝑆)𝐴 = {𝑡 ∣ ∃𝑢 ∈ (ℕ0 ↑m 𝑆)(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝‘𝑢) = 0)} ↔ ∃𝑝 ∈ (mzPoly‘𝑆){𝑡 ∣ ∃𝑑 ∈ (ℕ0 ↑m (1...𝑎))(𝑡 = (𝑑 ↾ (1...𝑁)) ∧ (𝑏‘𝑑) = 0)} = {𝑡 ∣ ∃𝑢 ∈ (ℕ0 ↑m 𝑆)(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝‘𝑢) = 0)}))
32	29, 31	syl5ibrcom 250	. . . . 5 ⊢ ((((𝑁 ∈ ℕ0 ∧ 𝑆 ∈ V) ∧ (¬ 𝑆 ∈ Fin ∧ (1...𝑁) ⊆ 𝑆)) ∧ (𝑎 ∈ (ℤ≥‘𝑁) ∧ 𝑏 ∈ (mzPoly‘(1...𝑎)))) → (𝐴 = {𝑡 ∣ ∃𝑑 ∈ (ℕ0 ↑m (1...𝑎))(𝑡 = (𝑑 ↾ (1...𝑁)) ∧ (𝑏‘𝑑) = 0)} → ∃𝑝 ∈ (mzPoly‘𝑆)𝐴 = {𝑡 ∣ ∃𝑢 ∈ (ℕ0 ↑m 𝑆)(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝‘𝑢) = 0)}))
33	32	rexlimdvva 3221	. . . 4 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑆 ∈ V) ∧ (¬ 𝑆 ∈ Fin ∧ (1...𝑁) ⊆ 𝑆)) → (∃𝑎 ∈ (ℤ≥‘𝑁)∃𝑏 ∈ (mzPoly‘(1...𝑎))𝐴 = {𝑡 ∣ ∃𝑑 ∈ (ℕ0 ↑m (1...𝑎))(𝑡 = (𝑑 ↾ (1...𝑁)) ∧ (𝑏‘𝑑) = 0)} → ∃𝑝 ∈ (mzPoly‘𝑆)𝐴 = {𝑡 ∣ ∃𝑢 ∈ (ℕ0 ↑m 𝑆)(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝‘𝑢) = 0)}))
34	33	adantld 495	. . 3 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑆 ∈ V) ∧ (¬ 𝑆 ∈ Fin ∧ (1...𝑁) ⊆ 𝑆)) → ((𝑁 ∈ ℕ0 ∧ ∃𝑎 ∈ (ℤ≥‘𝑁)∃𝑏 ∈ (mzPoly‘(1...𝑎))𝐴 = {𝑡 ∣ ∃𝑑 ∈ (ℕ0 ↑m (1...𝑎))(𝑡 = (𝑑 ↾ (1...𝑁)) ∧ (𝑏‘𝑑) = 0)}) → ∃𝑝 ∈ (mzPoly‘𝑆)𝐴 = {𝑡 ∣ ∃𝑢 ∈ (ℕ0 ↑m 𝑆)(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝‘𝑢) = 0)}))
35	1, 34	biimtrid 245	. 2 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑆 ∈ V) ∧ (¬ 𝑆 ∈ Fin ∧ (1...𝑁) ⊆ 𝑆)) → (𝐴 ∈ (Dioph‘𝑁) → ∃𝑝 ∈ (mzPoly‘𝑆)𝐴 = {𝑡 ∣ ∃𝑢 ∈ (ℕ0 ↑m 𝑆)(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝‘𝑢) = 0)}))
36		simpr 489	. . . 4 ⊢ (((((𝑁 ∈ ℕ0 ∧ 𝑆 ∈ V) ∧ (¬ 𝑆 ∈ Fin ∧ (1...𝑁) ⊆ 𝑆)) ∧ 𝑝 ∈ (mzPoly‘𝑆)) ∧ 𝐴 = {𝑡 ∣ ∃𝑢 ∈ (ℕ0 ↑m 𝑆)(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝‘𝑢) = 0)}) → 𝐴 = {𝑡 ∣ ∃𝑢 ∈ (ℕ0 ↑m 𝑆)(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝‘𝑢) = 0)})
37		simplll 786	. . . . . 6 ⊢ ((((𝑁 ∈ ℕ0 ∧ 𝑆 ∈ V) ∧ (¬ 𝑆 ∈ Fin ∧ (1...𝑁) ⊆ 𝑆)) ∧ 𝑝 ∈ (mzPoly‘𝑆)) → 𝑁 ∈ ℕ0)
38		simpllr 787	. . . . . 6 ⊢ ((((𝑁 ∈ ℕ0 ∧ 𝑆 ∈ V) ∧ (¬ 𝑆 ∈ Fin ∧ (1...𝑁) ⊆ 𝑆)) ∧ 𝑝 ∈ (mzPoly‘𝑆)) → 𝑆 ∈ V)
39		simplrr 789	. . . . . 6 ⊢ ((((𝑁 ∈ ℕ0 ∧ 𝑆 ∈ V) ∧ (¬ 𝑆 ∈ Fin ∧ (1...𝑁) ⊆ 𝑆)) ∧ 𝑝 ∈ (mzPoly‘𝑆)) → (1...𝑁) ⊆ 𝑆)
40		simpr 489	. . . . . 6 ⊢ ((((𝑁 ∈ ℕ0 ∧ 𝑆 ∈ V) ∧ (¬ 𝑆 ∈ Fin ∧ (1...𝑁) ⊆ 𝑆)) ∧ 𝑝 ∈ (mzPoly‘𝑆)) → 𝑝 ∈ (mzPoly‘𝑆))
41		eldioph2 43521	. . . . . 6 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑆 ∈ V ∧ (1...𝑁) ⊆ 𝑆) ∧ 𝑝 ∈ (mzPoly‘𝑆)) → {𝑡 ∣ ∃𝑢 ∈ (ℕ0 ↑m 𝑆)(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝‘𝑢) = 0)} ∈ (Dioph‘𝑁))
42	37, 38, 39, 40, 41	syl121anc 1401	. . . . 5 ⊢ ((((𝑁 ∈ ℕ0 ∧ 𝑆 ∈ V) ∧ (¬ 𝑆 ∈ Fin ∧ (1...𝑁) ⊆ 𝑆)) ∧ 𝑝 ∈ (mzPoly‘𝑆)) → {𝑡 ∣ ∃𝑢 ∈ (ℕ0 ↑m 𝑆)(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝‘𝑢) = 0)} ∈ (Dioph‘𝑁))
43	42	adantr 485	. . . 4 ⊢ (((((𝑁 ∈ ℕ0 ∧ 𝑆 ∈ V) ∧ (¬ 𝑆 ∈ Fin ∧ (1...𝑁) ⊆ 𝑆)) ∧ 𝑝 ∈ (mzPoly‘𝑆)) ∧ 𝐴 = {𝑡 ∣ ∃𝑢 ∈ (ℕ0 ↑m 𝑆)(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝‘𝑢) = 0)}) → {𝑡 ∣ ∃𝑢 ∈ (ℕ0 ↑m 𝑆)(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝‘𝑢) = 0)} ∈ (Dioph‘𝑁))
44	36, 43	eqeltrd 2862	. . 3 ⊢ (((((𝑁 ∈ ℕ0 ∧ 𝑆 ∈ V) ∧ (¬ 𝑆 ∈ Fin ∧ (1...𝑁) ⊆ 𝑆)) ∧ 𝑝 ∈ (mzPoly‘𝑆)) ∧ 𝐴 = {𝑡 ∣ ∃𝑢 ∈ (ℕ0 ↑m 𝑆)(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝‘𝑢) = 0)}) → 𝐴 ∈ (Dioph‘𝑁))
45	44	rexlimdva2 3167	. 2 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑆 ∈ V) ∧ (¬ 𝑆 ∈ Fin ∧ (1...𝑁) ⊆ 𝑆)) → (∃𝑝 ∈ (mzPoly‘𝑆)𝐴 = {𝑡 ∣ ∃𝑢 ∈ (ℕ0 ↑m 𝑆)(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝‘𝑢) = 0)} → 𝐴 ∈ (Dioph‘𝑁)))
46	35, 45	impbid 215	1 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑆 ∈ V) ∧ (¬ 𝑆 ∈ Fin ∧ (1...𝑁) ⊆ 𝑆)) → (𝐴 ∈ (Dioph‘𝑁) ↔ ∃𝑝 ∈ (mzPoly‘𝑆)𝐴 = {𝑡 ∣ ∃𝑢 ∈ (ℕ0 ↑m 𝑆)(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝‘𝑢) = 0)}))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1102   = wceq 1569  ∃wex 1808   ∈ wcel 2142  {cab 2740  ∃wrex 3088  Vcvv 3454   ⊆ wss 3904   ↦ cmpt 5191   I cid 5554   ↾ cres 5662   ∘ ccom 5664  ⟶wf 6532  –1-1→wf1 6533  ‘cfv 6536  (class class class)co 7412   ↑_m cmap 8822  Fincfn 8941  0cc0 11106  1c1 11107  ℕ₀cn0 12510  ℤcz 12597  ℤ_≥cuz 12868  ...cfz 13541  mzPolycmzp 43481  Diophcdioph 43514

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-rep 5237  ax-sep 5256  ax-nul 5268  ax-pow 5335  ax-pr 5403  ax-un 7734  ax-cnex 11162  ax-resscn 11163  ax-1cn 11164  ax-icn 11165  ax-addcl 11166  ax-addrcl 11167  ax-mulcl 11168  ax-mulrcl 11169  ax-mulcom 11170  ax-addass 11171  ax-mulass 11172  ax-distr 11173  ax-i2m1 11174  ax-1ne0 11175  ax-1rid 11176  ax-rnegex 11177  ax-rrecex 11178  ax-cnre 11179  ax-pre-lttri 11180  ax-pre-lttrn 11181  ax-pre-ltadd 11182  ax-pre-mulgt0 11183

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1103  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-nf 1813  df-sb 2096  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-nel 3064  df-ral 3079  df-rex 3089  df-reu 3369  df-rab 3416  df-v 3456  df-sbc 3744  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4487  df-pw 4563  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-int 4912  df-iun 4957  df-br 5109  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5555  df-eprel 5560  df-po 5568  df-so 5569  df-fr 5613  df-we 5615  df-xp 5666  df-rel 5667  df-cnv 5668  df-co 5669  df-dm 5670  df-rn 5671  df-res 5672  df-ima 5673  df-pred 6302  df-ord 6363  df-on 6364  df-lim 6365  df-suc 6366  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7369  df-ov 7415  df-oprab 7416  df-mpo 7417  df-of 7676  df-om 7861  df-1st 7984  df-2nd 7985  df-frecs 8276  df-wrecs 8307  df-recs 8356  df-rdg 8395  df-1o 8451  df-oadd 8455  df-er 8692  df-map 8824  df-en 8942  df-dom 8943  df-sdom 8944  df-fin 8945  df-dju 9894  df-card 9932  df-pnf 11251  df-mnf 11252  df-xr 11253  df-ltxr 11254  df-le 11255  df-sub 11449  df-neg 11450  df-nn 12240  df-n0 12511  df-z 12598  df-uz 12869  df-fz 13542  df-hash 14374  df-mzpcl 43482  df-mzp 43483  df-dioph 43515

This theorem is used by:  eldioph3b  43524  diophin  43531  diophun  43532  eldioph4b  43566