eldioph2b - Mathbox for Stefan O'Rear

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Theorem eldioph2b 42079

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 42088 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 42073	. . 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 783	. . . . . . . . 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 770	. . . . . . . . . 10 ⊢ ((((𝑁 ∈ ℕ₀ ∧ 𝑆 ∈ V) ∧ (¬ 𝑆 ∈ Fin ∧ (1...𝑁) ⊆ 𝑆)) ∧ (𝑎 ∈ (ℤ_≥‘𝑁) ∧ 𝑏 ∈ (mzPoly‘(1...𝑎)))) → 𝑏 ∈ (mzPoly‘(1...𝑎)))
4	3	ad2antrr 723	. . . . . . . . 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 768	. . . . . . . . . 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 6781	. . . . . . . . . 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 42065	. . . . . . . . 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 1368	. . . . . . . 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 770	. . . . . . . . 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 42075	. . . . . . . . . 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 2732	. . . . . . . . 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 1368	. . . . . . . 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 6884	. . . . . . . . . . . . 13 ⊢ (𝑝 = (𝑒 ∈ (ℤ ↑_m 𝑆) ↦ (𝑏‘(𝑒 ∘ 𝑐))) → (𝑝‘𝑢) = ((𝑒 ∈ (ℤ ↑_m 𝑆) ↦ (𝑏‘(𝑒 ∘ 𝑐)))‘𝑢))
15	14	eqeq1d 2728	. . . . . . . . . . . 12 ⊢ (𝑝 = (𝑒 ∈ (ℤ ↑_m 𝑆) ↦ (𝑏‘(𝑒 ∘ 𝑐))) → ((𝑝‘𝑢) = 0 ↔ ((𝑒 ∈ (ℤ ↑_m 𝑆) ↦ (𝑏‘(𝑒 ∘ 𝑐)))‘𝑢) = 0))
16	15	anbi2d 628	. . . . . . . . . . 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 3172	. . . . . . . . . 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 2795	. . . . . . . . 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 3628	. . . . . . . 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 583	. . . . . . 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 772	. . . . . . . . 9 ⊢ ((((𝑁 ∈ ℕ₀ ∧ 𝑆 ∈ V) ∧ (¬ 𝑆 ∈ Fin ∧ (1...𝑁) ⊆ 𝑆)) ∧ (𝑎 ∈ (ℤ_≥‘𝑁) ∧ 𝑏 ∈ (mzPoly‘(1...𝑎)))) → 𝑁 ∈ ℕ₀)
22		simplrl 774	. . . . . . . . 9 ⊢ ((((𝑁 ∈ ℕ₀ ∧ 𝑆 ∈ V) ∧ (¬ 𝑆 ∈ Fin ∧ (1...𝑁) ⊆ 𝑆)) ∧ (𝑎 ∈ (ℤ_≥‘𝑁) ∧ 𝑏 ∈ (mzPoly‘(1...𝑎)))) → ¬ 𝑆 ∈ Fin)
23		simplrr 775	. . . . . . . . 9 ⊢ ((((𝑁 ∈ ℕ₀ ∧ 𝑆 ∈ V) ∧ (¬ 𝑆 ∈ Fin ∧ (1...𝑁) ⊆ 𝑆)) ∧ (𝑎 ∈ (ℤ_≥‘𝑁) ∧ 𝑏 ∈ (mzPoly‘(1...𝑎)))) → (1...𝑁) ⊆ 𝑆)
24		simprl 768	. . . . . . . . 9 ⊢ ((((𝑁 ∈ ℕ₀ ∧ 𝑆 ∈ V) ∧ (¬ 𝑆 ∈ Fin ∧ (1...𝑁) ⊆ 𝑆)) ∧ (𝑎 ∈ (ℤ_≥‘𝑁) ∧ 𝑏 ∈ (mzPoly‘(1...𝑎)))) → 𝑎 ∈ (ℤ_≥‘𝑁))
25		eldioph2lem2 42077	. . . . . . . . 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 836	. . . . . . . 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 3494	. . . . . . . 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 3156	. . . . . 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 2730	. . . . . . 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 3172	. . . . . 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 3205	. . . 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 490	. . 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 484	. . . 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 772	. . . . . 6 ⊢ ((((𝑁 ∈ ℕ₀ ∧ 𝑆 ∈ V) ∧ (¬ 𝑆 ∈ Fin ∧ (1...𝑁) ⊆ 𝑆)) ∧ 𝑝 ∈ (mzPoly‘𝑆)) → 𝑁 ∈ ℕ₀)
38		simpllr 773	. . . . . 6 ⊢ ((((𝑁 ∈ ℕ₀ ∧ 𝑆 ∈ V) ∧ (¬ 𝑆 ∈ Fin ∧ (1...𝑁) ⊆ 𝑆)) ∧ 𝑝 ∈ (mzPoly‘𝑆)) → 𝑆 ∈ V)
39		simplrr 775	. . . . . 6 ⊢ ((((𝑁 ∈ ℕ₀ ∧ 𝑆 ∈ V) ∧ (¬ 𝑆 ∈ Fin ∧ (1...𝑁) ⊆ 𝑆)) ∧ 𝑝 ∈ (mzPoly‘𝑆)) → (1...𝑁) ⊆ 𝑆)
40		simpr 484	. . . . . 6 ⊢ ((((𝑁 ∈ ℕ₀ ∧ 𝑆 ∈ V) ∧ (¬ 𝑆 ∈ Fin ∧ (1...𝑁) ⊆ 𝑆)) ∧ 𝑝 ∈ (mzPoly‘𝑆)) → 𝑝 ∈ (mzPoly‘𝑆))
41		eldioph2 42078	. . . . . 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 1372	. . . . 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 480	. . . 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 2827	. . 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 3151	. 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 395 ∧ w3a 1084 = wceq 1533 ∃wex 1773 ∈ wcel 2098 {cab 2703 ∃wrex 3064 Vcvv 3468 ⊆ wss 3943 ↦ cmpt 5224 I cid 5566 ↾ cres 5671 ∘ ccom 5673 ⟶wf 6533 –1-1→wf1 6534 ‘cfv 6537 (class class class)co 7405 ↑_m cmap 8822 Fincfn 8941 0cc0 11112 1c1 11113 ℕ₀cn0 12476 ℤcz 12562 ℤ_≥cuz 12826 ...cfz 13490 mzPolycmzp 42038 Diophcdioph 42071

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-int 4944 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6294 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-of 7667 df-om 7853 df-1st 7974 df-2nd 7975 df-frecs 8267 df-wrecs 8298 df-recs 8372 df-rdg 8411 df-1o 8467 df-oadd 8471 df-er 8705 df-map 8824 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-dju 9898 df-card 9936 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-nn 12217 df-n0 12477 df-z 12563 df-uz 12827 df-fz 13491 df-hash 14296 df-mzpcl 42039 df-mzp 42040 df-dioph 42072

This theorem is referenced by: eldioph3b 42081 diophin 42088 diophun 42089 eldioph4b 42127

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