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Theorem eldioph2b 37828
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 37838 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 ∈ ℕ0𝑆 ∈ V) ∧ (¬ 𝑆 ∈ Fin ∧ (1...𝑁) ⊆ 𝑆)) → (𝐴 ∈ (Dioph‘𝑁) ↔ ∃𝑝 ∈ (mzPoly‘𝑆)𝐴 = {𝑡 ∣ ∃𝑢 ∈ (ℕ0𝑚 𝑆)(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝𝑢) = 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.
StepHypRef Expression
1 eldiophb 37822 . . 3 (𝐴 ∈ (Dioph‘𝑁) ↔ (𝑁 ∈ ℕ0 ∧ ∃𝑎 ∈ (ℤ𝑁)∃𝑏 ∈ (mzPoly‘(1...𝑎))𝐴 = {𝑡 ∣ ∃𝑑 ∈ (ℕ0𝑚 (1...𝑎))(𝑡 = (𝑑 ↾ (1...𝑁)) ∧ (𝑏𝑑) = 0)}))
2 simp-5r 831 . . . . . . . . 9 ((((((𝑁 ∈ ℕ0𝑆 ∈ V) ∧ (¬ 𝑆 ∈ Fin ∧ (1...𝑁) ⊆ 𝑆)) ∧ (𝑎 ∈ (ℤ𝑁) ∧ 𝑏 ∈ (mzPoly‘(1...𝑎)))) ∧ 𝑐 ∈ V) ∧ (𝑐:(1...𝑎)–1-1𝑆 ∧ (𝑐 ↾ (1...𝑁)) = ( I ↾ (1...𝑁)))) → 𝑆 ∈ V)
3 simprr 813 . . . . . . . . . 10 ((((𝑁 ∈ ℕ0𝑆 ∈ V) ∧ (¬ 𝑆 ∈ Fin ∧ (1...𝑁) ⊆ 𝑆)) ∧ (𝑎 ∈ (ℤ𝑁) ∧ 𝑏 ∈ (mzPoly‘(1...𝑎)))) → 𝑏 ∈ (mzPoly‘(1...𝑎)))
43ad2antrr 764 . . . . . . . . 9 ((((((𝑁 ∈ ℕ0𝑆 ∈ V) ∧ (¬ 𝑆 ∈ Fin ∧ (1...𝑁) ⊆ 𝑆)) ∧ (𝑎 ∈ (ℤ𝑁) ∧ 𝑏 ∈ (mzPoly‘(1...𝑎)))) ∧ 𝑐 ∈ V) ∧ (𝑐:(1...𝑎)–1-1𝑆 ∧ (𝑐 ↾ (1...𝑁)) = ( I ↾ (1...𝑁)))) → 𝑏 ∈ (mzPoly‘(1...𝑎)))
5 simprl 811 . . . . . . . . . 10 ((((((𝑁 ∈ ℕ0𝑆 ∈ V) ∧ (¬ 𝑆 ∈ Fin ∧ (1...𝑁) ⊆ 𝑆)) ∧ (𝑎 ∈ (ℤ𝑁) ∧ 𝑏 ∈ (mzPoly‘(1...𝑎)))) ∧ 𝑐 ∈ V) ∧ (𝑐:(1...𝑎)–1-1𝑆 ∧ (𝑐 ↾ (1...𝑁)) = ( I ↾ (1...𝑁)))) → 𝑐:(1...𝑎)–1-1𝑆)
6 f1f 6262 . . . . . . . . . 10 (𝑐:(1...𝑎)–1-1𝑆𝑐:(1...𝑎)⟶𝑆)
75, 6syl 17 . . . . . . . . 9 ((((((𝑁 ∈ ℕ0𝑆 ∈ V) ∧ (¬ 𝑆 ∈ Fin ∧ (1...𝑁) ⊆ 𝑆)) ∧ (𝑎 ∈ (ℤ𝑁) ∧ 𝑏 ∈ (mzPoly‘(1...𝑎)))) ∧ 𝑐 ∈ V) ∧ (𝑐:(1...𝑎)–1-1𝑆 ∧ (𝑐 ↾ (1...𝑁)) = ( I ↾ (1...𝑁)))) → 𝑐:(1...𝑎)⟶𝑆)
8 mzprename 37814 . . . . . . . . 9 ((𝑆 ∈ V ∧ 𝑏 ∈ (mzPoly‘(1...𝑎)) ∧ 𝑐:(1...𝑎)⟶𝑆) → (𝑒 ∈ (ℤ ↑𝑚 𝑆) ↦ (𝑏‘(𝑒𝑐))) ∈ (mzPoly‘𝑆))
92, 4, 7, 8syl3anc 1477 . . . . . . . 8 ((((((𝑁 ∈ ℕ0𝑆 ∈ V) ∧ (¬ 𝑆 ∈ Fin ∧ (1...𝑁) ⊆ 𝑆)) ∧ (𝑎 ∈ (ℤ𝑁) ∧ 𝑏 ∈ (mzPoly‘(1...𝑎)))) ∧ 𝑐 ∈ V) ∧ (𝑐:(1...𝑎)–1-1𝑆 ∧ (𝑐 ↾ (1...𝑁)) = ( I ↾ (1...𝑁)))) → (𝑒 ∈ (ℤ ↑𝑚 𝑆) ↦ (𝑏‘(𝑒𝑐))) ∈ (mzPoly‘𝑆))
10 simprr 813 . . . . . . . . 9 ((((((𝑁 ∈ ℕ0𝑆 ∈ V) ∧ (¬ 𝑆 ∈ Fin ∧ (1...𝑁) ⊆ 𝑆)) ∧ (𝑎 ∈ (ℤ𝑁) ∧ 𝑏 ∈ (mzPoly‘(1...𝑎)))) ∧ 𝑐 ∈ V) ∧ (𝑐:(1...𝑎)–1-1𝑆 ∧ (𝑐 ↾ (1...𝑁)) = ( I ↾ (1...𝑁)))) → (𝑐 ↾ (1...𝑁)) = ( I ↾ (1...𝑁)))
11 diophrw 37824 . . . . . . . . . 10 ((𝑆 ∈ V ∧ 𝑐:(1...𝑎)–1-1𝑆 ∧ (𝑐 ↾ (1...𝑁)) = ( I ↾ (1...𝑁))) → {𝑡 ∣ ∃𝑢 ∈ (ℕ0𝑚 𝑆)(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ ((𝑒 ∈ (ℤ ↑𝑚 𝑆) ↦ (𝑏‘(𝑒𝑐)))‘𝑢) = 0)} = {𝑡 ∣ ∃𝑑 ∈ (ℕ0𝑚 (1...𝑎))(𝑡 = (𝑑 ↾ (1...𝑁)) ∧ (𝑏𝑑) = 0)})
1211eqcomd 2766 . . . . . . . . 9 ((𝑆 ∈ V ∧ 𝑐:(1...𝑎)–1-1𝑆 ∧ (𝑐 ↾ (1...𝑁)) = ( I ↾ (1...𝑁))) → {𝑡 ∣ ∃𝑑 ∈ (ℕ0𝑚 (1...𝑎))(𝑡 = (𝑑 ↾ (1...𝑁)) ∧ (𝑏𝑑) = 0)} = {𝑡 ∣ ∃𝑢 ∈ (ℕ0𝑚 𝑆)(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ ((𝑒 ∈ (ℤ ↑𝑚 𝑆) ↦ (𝑏‘(𝑒𝑐)))‘𝑢) = 0)})
132, 5, 10, 12syl3anc 1477 . . . . . . . 8 ((((((𝑁 ∈ ℕ0𝑆 ∈ V) ∧ (¬ 𝑆 ∈ Fin ∧ (1...𝑁) ⊆ 𝑆)) ∧ (𝑎 ∈ (ℤ𝑁) ∧ 𝑏 ∈ (mzPoly‘(1...𝑎)))) ∧ 𝑐 ∈ V) ∧ (𝑐:(1...𝑎)–1-1𝑆 ∧ (𝑐 ↾ (1...𝑁)) = ( I ↾ (1...𝑁)))) → {𝑡 ∣ ∃𝑑 ∈ (ℕ0𝑚 (1...𝑎))(𝑡 = (𝑑 ↾ (1...𝑁)) ∧ (𝑏𝑑) = 0)} = {𝑡 ∣ ∃𝑢 ∈ (ℕ0𝑚 𝑆)(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ ((𝑒 ∈ (ℤ ↑𝑚 𝑆) ↦ (𝑏‘(𝑒𝑐)))‘𝑢) = 0)})
14 fveq1 6351 . . . . . . . . . . . . . 14 (𝑝 = (𝑒 ∈ (ℤ ↑𝑚 𝑆) ↦ (𝑏‘(𝑒𝑐))) → (𝑝𝑢) = ((𝑒 ∈ (ℤ ↑𝑚 𝑆) ↦ (𝑏‘(𝑒𝑐)))‘𝑢))
1514eqeq1d 2762 . . . . . . . . . . . . 13 (𝑝 = (𝑒 ∈ (ℤ ↑𝑚 𝑆) ↦ (𝑏‘(𝑒𝑐))) → ((𝑝𝑢) = 0 ↔ ((𝑒 ∈ (ℤ ↑𝑚 𝑆) ↦ (𝑏‘(𝑒𝑐)))‘𝑢) = 0))
1615anbi2d 742 . . . . . . . . . . . 12 (𝑝 = (𝑒 ∈ (ℤ ↑𝑚 𝑆) ↦ (𝑏‘(𝑒𝑐))) → ((𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝𝑢) = 0) ↔ (𝑡 = (𝑢 ↾ (1...𝑁)) ∧ ((𝑒 ∈ (ℤ ↑𝑚 𝑆) ↦ (𝑏‘(𝑒𝑐)))‘𝑢) = 0)))
1716rexbidv 3190 . . . . . . . . . . 11 (𝑝 = (𝑒 ∈ (ℤ ↑𝑚 𝑆) ↦ (𝑏‘(𝑒𝑐))) → (∃𝑢 ∈ (ℕ0𝑚 𝑆)(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝𝑢) = 0) ↔ ∃𝑢 ∈ (ℕ0𝑚 𝑆)(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ ((𝑒 ∈ (ℤ ↑𝑚 𝑆) ↦ (𝑏‘(𝑒𝑐)))‘𝑢) = 0)))
1817abbidv 2879 . . . . . . . . . 10 (𝑝 = (𝑒 ∈ (ℤ ↑𝑚 𝑆) ↦ (𝑏‘(𝑒𝑐))) → {𝑡 ∣ ∃𝑢 ∈ (ℕ0𝑚 𝑆)(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝𝑢) = 0)} = {𝑡 ∣ ∃𝑢 ∈ (ℕ0𝑚 𝑆)(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ ((𝑒 ∈ (ℤ ↑𝑚 𝑆) ↦ (𝑏‘(𝑒𝑐)))‘𝑢) = 0)})
1918eqeq2d 2770 . . . . . . . . 9 (𝑝 = (𝑒 ∈ (ℤ ↑𝑚 𝑆) ↦ (𝑏‘(𝑒𝑐))) → ({𝑡 ∣ ∃𝑑 ∈ (ℕ0𝑚 (1...𝑎))(𝑡 = (𝑑 ↾ (1...𝑁)) ∧ (𝑏𝑑) = 0)} = {𝑡 ∣ ∃𝑢 ∈ (ℕ0𝑚 𝑆)(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝𝑢) = 0)} ↔ {𝑡 ∣ ∃𝑑 ∈ (ℕ0𝑚 (1...𝑎))(𝑡 = (𝑑 ↾ (1...𝑁)) ∧ (𝑏𝑑) = 0)} = {𝑡 ∣ ∃𝑢 ∈ (ℕ0𝑚 𝑆)(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ ((𝑒 ∈ (ℤ ↑𝑚 𝑆) ↦ (𝑏‘(𝑒𝑐)))‘𝑢) = 0)}))
2019rspcev 3449 . . . . . . . 8 (((𝑒 ∈ (ℤ ↑𝑚 𝑆) ↦ (𝑏‘(𝑒𝑐))) ∈ (mzPoly‘𝑆) ∧ {𝑡 ∣ ∃𝑑 ∈ (ℕ0𝑚 (1...𝑎))(𝑡 = (𝑑 ↾ (1...𝑁)) ∧ (𝑏𝑑) = 0)} = {𝑡 ∣ ∃𝑢 ∈ (ℕ0𝑚 𝑆)(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ ((𝑒 ∈ (ℤ ↑𝑚 𝑆) ↦ (𝑏‘(𝑒𝑐)))‘𝑢) = 0)}) → ∃𝑝 ∈ (mzPoly‘𝑆){𝑡 ∣ ∃𝑑 ∈ (ℕ0𝑚 (1...𝑎))(𝑡 = (𝑑 ↾ (1...𝑁)) ∧ (𝑏𝑑) = 0)} = {𝑡 ∣ ∃𝑢 ∈ (ℕ0𝑚 𝑆)(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝𝑢) = 0)})
219, 13, 20syl2anc 696 . . . . . . 7 ((((((𝑁 ∈ ℕ0𝑆 ∈ V) ∧ (¬ 𝑆 ∈ Fin ∧ (1...𝑁) ⊆ 𝑆)) ∧ (𝑎 ∈ (ℤ𝑁) ∧ 𝑏 ∈ (mzPoly‘(1...𝑎)))) ∧ 𝑐 ∈ V) ∧ (𝑐:(1...𝑎)–1-1𝑆 ∧ (𝑐 ↾ (1...𝑁)) = ( I ↾ (1...𝑁)))) → ∃𝑝 ∈ (mzPoly‘𝑆){𝑡 ∣ ∃𝑑 ∈ (ℕ0𝑚 (1...𝑎))(𝑡 = (𝑑 ↾ (1...𝑁)) ∧ (𝑏𝑑) = 0)} = {𝑡 ∣ ∃𝑢 ∈ (ℕ0𝑚 𝑆)(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝𝑢) = 0)})
22 simplll 815 . . . . . . . . 9 ((((𝑁 ∈ ℕ0𝑆 ∈ V) ∧ (¬ 𝑆 ∈ Fin ∧ (1...𝑁) ⊆ 𝑆)) ∧ (𝑎 ∈ (ℤ𝑁) ∧ 𝑏 ∈ (mzPoly‘(1...𝑎)))) → 𝑁 ∈ ℕ0)
23 simplrl 819 . . . . . . . . 9 ((((𝑁 ∈ ℕ0𝑆 ∈ V) ∧ (¬ 𝑆 ∈ Fin ∧ (1...𝑁) ⊆ 𝑆)) ∧ (𝑎 ∈ (ℤ𝑁) ∧ 𝑏 ∈ (mzPoly‘(1...𝑎)))) → ¬ 𝑆 ∈ Fin)
24 simplrr 820 . . . . . . . . 9 ((((𝑁 ∈ ℕ0𝑆 ∈ V) ∧ (¬ 𝑆 ∈ Fin ∧ (1...𝑁) ⊆ 𝑆)) ∧ (𝑎 ∈ (ℤ𝑁) ∧ 𝑏 ∈ (mzPoly‘(1...𝑎)))) → (1...𝑁) ⊆ 𝑆)
25 simprl 811 . . . . . . . . 9 ((((𝑁 ∈ ℕ0𝑆 ∈ V) ∧ (¬ 𝑆 ∈ Fin ∧ (1...𝑁) ⊆ 𝑆)) ∧ (𝑎 ∈ (ℤ𝑁) ∧ 𝑏 ∈ (mzPoly‘(1...𝑎)))) → 𝑎 ∈ (ℤ𝑁))
26 eldioph2lem2 37826 . . . . . . . . 9 (((𝑁 ∈ ℕ0 ∧ ¬ 𝑆 ∈ Fin) ∧ ((1...𝑁) ⊆ 𝑆𝑎 ∈ (ℤ𝑁))) → ∃𝑐(𝑐:(1...𝑎)–1-1𝑆 ∧ (𝑐 ↾ (1...𝑁)) = ( I ↾ (1...𝑁))))
2722, 23, 24, 25, 26syl22anc 1478 . . . . . . . 8 ((((𝑁 ∈ ℕ0𝑆 ∈ V) ∧ (¬ 𝑆 ∈ Fin ∧ (1...𝑁) ⊆ 𝑆)) ∧ (𝑎 ∈ (ℤ𝑁) ∧ 𝑏 ∈ (mzPoly‘(1...𝑎)))) → ∃𝑐(𝑐:(1...𝑎)–1-1𝑆 ∧ (𝑐 ↾ (1...𝑁)) = ( I ↾ (1...𝑁))))
28 rexv 3360 . . . . . . . 8 (∃𝑐 ∈ V (𝑐:(1...𝑎)–1-1𝑆 ∧ (𝑐 ↾ (1...𝑁)) = ( I ↾ (1...𝑁))) ↔ ∃𝑐(𝑐:(1...𝑎)–1-1𝑆 ∧ (𝑐 ↾ (1...𝑁)) = ( I ↾ (1...𝑁))))
2927, 28sylibr 224 . . . . . . 7 ((((𝑁 ∈ ℕ0𝑆 ∈ V) ∧ (¬ 𝑆 ∈ Fin ∧ (1...𝑁) ⊆ 𝑆)) ∧ (𝑎 ∈ (ℤ𝑁) ∧ 𝑏 ∈ (mzPoly‘(1...𝑎)))) → ∃𝑐 ∈ V (𝑐:(1...𝑎)–1-1𝑆 ∧ (𝑐 ↾ (1...𝑁)) = ( I ↾ (1...𝑁))))
3021, 29r19.29a 3216 . . . . . 6 ((((𝑁 ∈ ℕ0𝑆 ∈ V) ∧ (¬ 𝑆 ∈ Fin ∧ (1...𝑁) ⊆ 𝑆)) ∧ (𝑎 ∈ (ℤ𝑁) ∧ 𝑏 ∈ (mzPoly‘(1...𝑎)))) → ∃𝑝 ∈ (mzPoly‘𝑆){𝑡 ∣ ∃𝑑 ∈ (ℕ0𝑚 (1...𝑎))(𝑡 = (𝑑 ↾ (1...𝑁)) ∧ (𝑏𝑑) = 0)} = {𝑡 ∣ ∃𝑢 ∈ (ℕ0𝑚 𝑆)(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝𝑢) = 0)})
31 eqeq1 2764 . . . . . . 7 (𝐴 = {𝑡 ∣ ∃𝑑 ∈ (ℕ0𝑚 (1...𝑎))(𝑡 = (𝑑 ↾ (1...𝑁)) ∧ (𝑏𝑑) = 0)} → (𝐴 = {𝑡 ∣ ∃𝑢 ∈ (ℕ0𝑚 𝑆)(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝𝑢) = 0)} ↔ {𝑡 ∣ ∃𝑑 ∈ (ℕ0𝑚 (1...𝑎))(𝑡 = (𝑑 ↾ (1...𝑁)) ∧ (𝑏𝑑) = 0)} = {𝑡 ∣ ∃𝑢 ∈ (ℕ0𝑚 𝑆)(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝𝑢) = 0)}))
3231rexbidv 3190 . . . . . 6 (𝐴 = {𝑡 ∣ ∃𝑑 ∈ (ℕ0𝑚 (1...𝑎))(𝑡 = (𝑑 ↾ (1...𝑁)) ∧ (𝑏𝑑) = 0)} → (∃𝑝 ∈ (mzPoly‘𝑆)𝐴 = {𝑡 ∣ ∃𝑢 ∈ (ℕ0𝑚 𝑆)(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝𝑢) = 0)} ↔ ∃𝑝 ∈ (mzPoly‘𝑆){𝑡 ∣ ∃𝑑 ∈ (ℕ0𝑚 (1...𝑎))(𝑡 = (𝑑 ↾ (1...𝑁)) ∧ (𝑏𝑑) = 0)} = {𝑡 ∣ ∃𝑢 ∈ (ℕ0𝑚 𝑆)(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝𝑢) = 0)}))
3330, 32syl5ibrcom 237 . . . . 5 ((((𝑁 ∈ ℕ0𝑆 ∈ V) ∧ (¬ 𝑆 ∈ Fin ∧ (1...𝑁) ⊆ 𝑆)) ∧ (𝑎 ∈ (ℤ𝑁) ∧ 𝑏 ∈ (mzPoly‘(1...𝑎)))) → (𝐴 = {𝑡 ∣ ∃𝑑 ∈ (ℕ0𝑚 (1...𝑎))(𝑡 = (𝑑 ↾ (1...𝑁)) ∧ (𝑏𝑑) = 0)} → ∃𝑝 ∈ (mzPoly‘𝑆)𝐴 = {𝑡 ∣ ∃𝑢 ∈ (ℕ0𝑚 𝑆)(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝𝑢) = 0)}))
3433rexlimdvva 3176 . . . 4 (((𝑁 ∈ ℕ0𝑆 ∈ V) ∧ (¬ 𝑆 ∈ Fin ∧ (1...𝑁) ⊆ 𝑆)) → (∃𝑎 ∈ (ℤ𝑁)∃𝑏 ∈ (mzPoly‘(1...𝑎))𝐴 = {𝑡 ∣ ∃𝑑 ∈ (ℕ0𝑚 (1...𝑎))(𝑡 = (𝑑 ↾ (1...𝑁)) ∧ (𝑏𝑑) = 0)} → ∃𝑝 ∈ (mzPoly‘𝑆)𝐴 = {𝑡 ∣ ∃𝑢 ∈ (ℕ0𝑚 𝑆)(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝𝑢) = 0)}))
3534adantld 484 . . 3 (((𝑁 ∈ ℕ0𝑆 ∈ V) ∧ (¬ 𝑆 ∈ Fin ∧ (1...𝑁) ⊆ 𝑆)) → ((𝑁 ∈ ℕ0 ∧ ∃𝑎 ∈ (ℤ𝑁)∃𝑏 ∈ (mzPoly‘(1...𝑎))𝐴 = {𝑡 ∣ ∃𝑑 ∈ (ℕ0𝑚 (1...𝑎))(𝑡 = (𝑑 ↾ (1...𝑁)) ∧ (𝑏𝑑) = 0)}) → ∃𝑝 ∈ (mzPoly‘𝑆)𝐴 = {𝑡 ∣ ∃𝑢 ∈ (ℕ0𝑚 𝑆)(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝𝑢) = 0)}))
361, 35syl5bi 232 . 2 (((𝑁 ∈ ℕ0𝑆 ∈ V) ∧ (¬ 𝑆 ∈ Fin ∧ (1...𝑁) ⊆ 𝑆)) → (𝐴 ∈ (Dioph‘𝑁) → ∃𝑝 ∈ (mzPoly‘𝑆)𝐴 = {𝑡 ∣ ∃𝑢 ∈ (ℕ0𝑚 𝑆)(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝𝑢) = 0)}))
37 simpr 479 . . . . 5 (((((𝑁 ∈ ℕ0𝑆 ∈ V) ∧ (¬ 𝑆 ∈ Fin ∧ (1...𝑁) ⊆ 𝑆)) ∧ 𝑝 ∈ (mzPoly‘𝑆)) ∧ 𝐴 = {𝑡 ∣ ∃𝑢 ∈ (ℕ0𝑚 𝑆)(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝𝑢) = 0)}) → 𝐴 = {𝑡 ∣ ∃𝑢 ∈ (ℕ0𝑚 𝑆)(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝𝑢) = 0)})
38 simplll 815 . . . . . . 7 ((((𝑁 ∈ ℕ0𝑆 ∈ V) ∧ (¬ 𝑆 ∈ Fin ∧ (1...𝑁) ⊆ 𝑆)) ∧ 𝑝 ∈ (mzPoly‘𝑆)) → 𝑁 ∈ ℕ0)
39 simpllr 817 . . . . . . 7 ((((𝑁 ∈ ℕ0𝑆 ∈ V) ∧ (¬ 𝑆 ∈ Fin ∧ (1...𝑁) ⊆ 𝑆)) ∧ 𝑝 ∈ (mzPoly‘𝑆)) → 𝑆 ∈ V)
40 simplrr 820 . . . . . . 7 ((((𝑁 ∈ ℕ0𝑆 ∈ V) ∧ (¬ 𝑆 ∈ Fin ∧ (1...𝑁) ⊆ 𝑆)) ∧ 𝑝 ∈ (mzPoly‘𝑆)) → (1...𝑁) ⊆ 𝑆)
41 simpr 479 . . . . . . 7 ((((𝑁 ∈ ℕ0𝑆 ∈ V) ∧ (¬ 𝑆 ∈ Fin ∧ (1...𝑁) ⊆ 𝑆)) ∧ 𝑝 ∈ (mzPoly‘𝑆)) → 𝑝 ∈ (mzPoly‘𝑆))
42 eldioph2 37827 . . . . . . 7 ((𝑁 ∈ ℕ0 ∧ (𝑆 ∈ V ∧ (1...𝑁) ⊆ 𝑆) ∧ 𝑝 ∈ (mzPoly‘𝑆)) → {𝑡 ∣ ∃𝑢 ∈ (ℕ0𝑚 𝑆)(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝𝑢) = 0)} ∈ (Dioph‘𝑁))
4338, 39, 40, 41, 42syl121anc 1482 . . . . . 6 ((((𝑁 ∈ ℕ0𝑆 ∈ V) ∧ (¬ 𝑆 ∈ Fin ∧ (1...𝑁) ⊆ 𝑆)) ∧ 𝑝 ∈ (mzPoly‘𝑆)) → {𝑡 ∣ ∃𝑢 ∈ (ℕ0𝑚 𝑆)(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝𝑢) = 0)} ∈ (Dioph‘𝑁))
4443adantr 472 . . . . 5 (((((𝑁 ∈ ℕ0𝑆 ∈ V) ∧ (¬ 𝑆 ∈ Fin ∧ (1...𝑁) ⊆ 𝑆)) ∧ 𝑝 ∈ (mzPoly‘𝑆)) ∧ 𝐴 = {𝑡 ∣ ∃𝑢 ∈ (ℕ0𝑚 𝑆)(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝𝑢) = 0)}) → {𝑡 ∣ ∃𝑢 ∈ (ℕ0𝑚 𝑆)(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝𝑢) = 0)} ∈ (Dioph‘𝑁))
4537, 44eqeltrd 2839 . . . 4 (((((𝑁 ∈ ℕ0𝑆 ∈ V) ∧ (¬ 𝑆 ∈ Fin ∧ (1...𝑁) ⊆ 𝑆)) ∧ 𝑝 ∈ (mzPoly‘𝑆)) ∧ 𝐴 = {𝑡 ∣ ∃𝑢 ∈ (ℕ0𝑚 𝑆)(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝𝑢) = 0)}) → 𝐴 ∈ (Dioph‘𝑁))
4645ex 449 . . 3 ((((𝑁 ∈ ℕ0𝑆 ∈ V) ∧ (¬ 𝑆 ∈ Fin ∧ (1...𝑁) ⊆ 𝑆)) ∧ 𝑝 ∈ (mzPoly‘𝑆)) → (𝐴 = {𝑡 ∣ ∃𝑢 ∈ (ℕ0𝑚 𝑆)(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝𝑢) = 0)} → 𝐴 ∈ (Dioph‘𝑁)))
4746rexlimdva 3169 . 2 (((𝑁 ∈ ℕ0𝑆 ∈ V) ∧ (¬ 𝑆 ∈ Fin ∧ (1...𝑁) ⊆ 𝑆)) → (∃𝑝 ∈ (mzPoly‘𝑆)𝐴 = {𝑡 ∣ ∃𝑢 ∈ (ℕ0𝑚 𝑆)(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝𝑢) = 0)} → 𝐴 ∈ (Dioph‘𝑁)))
4836, 47impbid 202 1 (((𝑁 ∈ ℕ0𝑆 ∈ V) ∧ (¬ 𝑆 ∈ Fin ∧ (1...𝑁) ⊆ 𝑆)) → (𝐴 ∈ (Dioph‘𝑁) ↔ ∃𝑝 ∈ (mzPoly‘𝑆)𝐴 = {𝑡 ∣ ∃𝑢 ∈ (ℕ0𝑚 𝑆)(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝𝑢) = 0)}))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 383  w3a 1072   = wceq 1632  wex 1853  wcel 2139  {cab 2746  wrex 3051  Vcvv 3340  wss 3715  cmpt 4881   I cid 5173  cres 5268  ccom 5270  wf 6045  1-1wf1 6046  cfv 6049  (class class class)co 6813  𝑚 cmap 8023  Fincfn 8121  0cc0 10128  1c1 10129  0cn0 11484  cz 11569  cuz 11879  ...cfz 12519  mzPolycmzp 37787  Diophcdioph 37820
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114  ax-cnex 10184  ax-resscn 10185  ax-1cn 10186  ax-icn 10187  ax-addcl 10188  ax-addrcl 10189  ax-mulcl 10190  ax-mulrcl 10191  ax-mulcom 10192  ax-addass 10193  ax-mulass 10194  ax-distr 10195  ax-i2m1 10196  ax-1ne0 10197  ax-1rid 10198  ax-rnegex 10199  ax-rrecex 10200  ax-cnre 10201  ax-pre-lttri 10202  ax-pre-lttrn 10203  ax-pre-ltadd 10204  ax-pre-mulgt0 10205
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-nel 3036  df-ral 3055  df-rex 3056  df-reu 3057  df-rmo 3058  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-int 4628  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-riota 6774  df-ov 6816  df-oprab 6817  df-mpt2 6818  df-of 7062  df-om 7231  df-1st 7333  df-2nd 7334  df-wrecs 7576  df-recs 7637  df-rdg 7675  df-1o 7729  df-oadd 7733  df-er 7911  df-map 8025  df-en 8122  df-dom 8123  df-sdom 8124  df-fin 8125  df-card 8955  df-cda 9182  df-pnf 10268  df-mnf 10269  df-xr 10270  df-ltxr 10271  df-le 10272  df-sub 10460  df-neg 10461  df-nn 11213  df-n0 11485  df-z 11570  df-uz 11880  df-fz 12520  df-hash 13312  df-mzpcl 37788  df-mzp 37789  df-dioph 37821
This theorem is referenced by:  eldioph3b  37830  diophin  37838  diophun  37839  eldioph4b  37877
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