eldioph2b - Mathbox for Stefan O'Rear

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Theorem eldioph2b 41486

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 41495 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 41480	. . 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 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)))) → 𝑆 ∈ V)
3		simprr 771	. . . . . . . . . 10 ⊢ ((((𝑁 ∈ ℕ₀ ∧ 𝑆 ∈ V) ∧ (¬ 𝑆 ∈ Fin ∧ (1...𝑁) ⊆ 𝑆)) ∧ (𝑎 ∈ (ℤ_≥‘𝑁) ∧ 𝑏 ∈ (mzPoly‘(1...𝑎)))) → 𝑏 ∈ (mzPoly‘(1...𝑎)))
4	3	ad2antrr 724	. . . . . . . . 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 769	. . . . . . . . . 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 6784	. . . . . . . . . 10 ⊢ (𝑐:(1...𝑎)–1-1→𝑆 → 𝑐:(1...𝑎)⟶𝑆)
7	5, 6	syl 17	. . . . . . . . 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 41472	. . . . . . . . 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 1371	. . . . . . . 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 771	. . . . . . . . 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 41482	. . . . . . . . . 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 2738	. . . . . . . . 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 1371	. . . . . . . 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 6887	. . . . . . . . . . . . 13 ⊢ (𝑝 = (𝑒 ∈ (ℤ ↑_m 𝑆) ↦ (𝑏‘(𝑒 ∘ 𝑐))) → (𝑝‘𝑢) = ((𝑒 ∈ (ℤ ↑_m 𝑆) ↦ (𝑏‘(𝑒 ∘ 𝑐)))‘𝑢))
15	14	eqeq1d 2734	. . . . . . . . . . . 12 ⊢ (𝑝 = (𝑒 ∈ (ℤ ↑_m 𝑆) ↦ (𝑏‘(𝑒 ∘ 𝑐))) → ((𝑝‘𝑢) = 0 ↔ ((𝑒 ∈ (ℤ ↑_m 𝑆) ↦ (𝑏‘(𝑒 ∘ 𝑐)))‘𝑢) = 0))
16	15	anbi2d 629	. . . . . . . . . . 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 3178	. . . . . . . . . 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 2801	. . . . . . . . 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 3632	. . . . . . . 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 584	. . . . . . 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 773	. . . . . . . . 9 ⊢ ((((𝑁 ∈ ℕ₀ ∧ 𝑆 ∈ V) ∧ (¬ 𝑆 ∈ Fin ∧ (1...𝑁) ⊆ 𝑆)) ∧ (𝑎 ∈ (ℤ_≥‘𝑁) ∧ 𝑏 ∈ (mzPoly‘(1...𝑎)))) → 𝑁 ∈ ℕ₀)
22		simplrl 775	. . . . . . . . 9 ⊢ ((((𝑁 ∈ ℕ₀ ∧ 𝑆 ∈ V) ∧ (¬ 𝑆 ∈ Fin ∧ (1...𝑁) ⊆ 𝑆)) ∧ (𝑎 ∈ (ℤ_≥‘𝑁) ∧ 𝑏 ∈ (mzPoly‘(1...𝑎)))) → ¬ 𝑆 ∈ Fin)
23		simplrr 776	. . . . . . . . 9 ⊢ ((((𝑁 ∈ ℕ₀ ∧ 𝑆 ∈ V) ∧ (¬ 𝑆 ∈ Fin ∧ (1...𝑁) ⊆ 𝑆)) ∧ (𝑎 ∈ (ℤ_≥‘𝑁) ∧ 𝑏 ∈ (mzPoly‘(1...𝑎)))) → (1...𝑁) ⊆ 𝑆)
24		simprl 769	. . . . . . . . 9 ⊢ ((((𝑁 ∈ ℕ₀ ∧ 𝑆 ∈ V) ∧ (¬ 𝑆 ∈ Fin ∧ (1...𝑁) ⊆ 𝑆)) ∧ (𝑎 ∈ (ℤ_≥‘𝑁) ∧ 𝑏 ∈ (mzPoly‘(1...𝑎)))) → 𝑎 ∈ (ℤ_≥‘𝑁))
25		eldioph2lem2 41484	. . . . . . . . 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 837	. . . . . . . 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 3499	. . . . . . . 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 233	. . . . . . 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 3162	. . . . . 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 2736	. . . . . . 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 3178	. . . . . 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 246	. . . . 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 3211	. . . 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 491	. . 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 241	. 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 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)}) → 𝐴 = {𝑡 ∣ ∃𝑢 ∈ (ℕ₀ ↑_m 𝑆)(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝‘𝑢) = 0)})
37		simplll 773	. . . . . 6 ⊢ ((((𝑁 ∈ ℕ₀ ∧ 𝑆 ∈ V) ∧ (¬ 𝑆 ∈ Fin ∧ (1...𝑁) ⊆ 𝑆)) ∧ 𝑝 ∈ (mzPoly‘𝑆)) → 𝑁 ∈ ℕ₀)
38		simpllr 774	. . . . . 6 ⊢ ((((𝑁 ∈ ℕ₀ ∧ 𝑆 ∈ V) ∧ (¬ 𝑆 ∈ Fin ∧ (1...𝑁) ⊆ 𝑆)) ∧ 𝑝 ∈ (mzPoly‘𝑆)) → 𝑆 ∈ V)
39		simplrr 776	. . . . . 6 ⊢ ((((𝑁 ∈ ℕ₀ ∧ 𝑆 ∈ V) ∧ (¬ 𝑆 ∈ Fin ∧ (1...𝑁) ⊆ 𝑆)) ∧ 𝑝 ∈ (mzPoly‘𝑆)) → (1...𝑁) ⊆ 𝑆)
40		simpr 485	. . . . . 6 ⊢ ((((𝑁 ∈ ℕ₀ ∧ 𝑆 ∈ V) ∧ (¬ 𝑆 ∈ Fin ∧ (1...𝑁) ⊆ 𝑆)) ∧ 𝑝 ∈ (mzPoly‘𝑆)) → 𝑝 ∈ (mzPoly‘𝑆))
41		eldioph2 41485	. . . . . 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 1375	. . . . 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 481	. . . 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 2833	. . 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 3157	. 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 211	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)}))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 396 ∧ w3a 1087 = wceq 1541 ∃wex 1781 ∈ wcel 2106 {cab 2709 ∃wrex 3070 Vcvv 3474 ⊆ wss 3947 ↦ cmpt 5230 I cid 5572 ↾ cres 5677 ∘ ccom 5679 ⟶wf 6536 –1-1→wf1 6537 ‘cfv 6540 (class class class)co 7405 ↑_m cmap 8816 Fincfn 8935 0cc0 11106 1c1 11107 ℕ₀cn0 12468 ℤcz 12554 ℤ_≥cuz 12818 ...cfz 13480 mzPolycmzp 41445 Diophcdioph 41478

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 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 theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-of 7666 df-om 7852 df-1st 7971 df-2nd 7972 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-oadd 8466 df-er 8699 df-map 8818 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-dju 9892 df-card 9930 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-nn 12209 df-n0 12469 df-z 12555 df-uz 12819 df-fz 13481 df-hash 14287 df-mzpcl 41446 df-mzp 41447 df-dioph 41479

This theorem is referenced by: eldioph3b 41488 diophin 41495 diophun 41496 eldioph4b 41534

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