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Theorem eldioph4b 36393
 Description: Membership in Dioph expressed using a quantified union to add witness variables instead of a restriction to remove them. (Contributed by Stefan O'Rear, 16-Oct-2014.)
Hypotheses
Ref Expression
eldioph4b.a 𝑊 ∈ V
eldioph4b.b ¬ 𝑊 ∈ Fin
eldioph4b.c (𝑊 ∩ ℕ) = ∅
Assertion
Ref Expression
eldioph4b (𝑆 ∈ (Dioph‘𝑁) ↔ (𝑁 ∈ ℕ0 ∧ ∃𝑝 ∈ (mzPoly‘(𝑊 ∪ (1...𝑁)))𝑆 = {𝑡 ∈ (ℕ0𝑚 (1...𝑁)) ∣ ∃𝑤 ∈ (ℕ0𝑚 𝑊)(𝑝‘(𝑡𝑤)) = 0}))
Distinct variable groups:   𝑊,𝑝,𝑡,𝑤   𝑆,𝑝,𝑡,𝑤   𝑁,𝑝,𝑡,𝑤

Proof of Theorem eldioph4b
Dummy variable 𝑢 is distinct from all other variables.
StepHypRef Expression
1 eldiophelnn0 36345 . 2 (𝑆 ∈ (Dioph‘𝑁) → 𝑁 ∈ ℕ0)
2 eldioph4b.a . . . . . 6 𝑊 ∈ V
3 ovex 6577 . . . . . 6 (1...𝑁) ∈ V
42, 3unex 6854 . . . . 5 (𝑊 ∪ (1...𝑁)) ∈ V
54jctr 563 . . . 4 (𝑁 ∈ ℕ0 → (𝑁 ∈ ℕ0 ∧ (𝑊 ∪ (1...𝑁)) ∈ V))
6 eldioph4b.b . . . . . . 7 ¬ 𝑊 ∈ Fin
76intnanr 952 . . . . . 6 ¬ (𝑊 ∈ Fin ∧ (1...𝑁) ∈ Fin)
8 unfir 8113 . . . . . 6 ((𝑊 ∪ (1...𝑁)) ∈ Fin → (𝑊 ∈ Fin ∧ (1...𝑁) ∈ Fin))
97, 8mto 187 . . . . 5 ¬ (𝑊 ∪ (1...𝑁)) ∈ Fin
10 ssun2 3739 . . . . 5 (1...𝑁) ⊆ (𝑊 ∪ (1...𝑁))
119, 10pm3.2i 470 . . . 4 (¬ (𝑊 ∪ (1...𝑁)) ∈ Fin ∧ (1...𝑁) ⊆ (𝑊 ∪ (1...𝑁)))
12 eldioph2b 36344 . . . 4 (((𝑁 ∈ ℕ0 ∧ (𝑊 ∪ (1...𝑁)) ∈ V) ∧ (¬ (𝑊 ∪ (1...𝑁)) ∈ Fin ∧ (1...𝑁) ⊆ (𝑊 ∪ (1...𝑁)))) → (𝑆 ∈ (Dioph‘𝑁) ↔ ∃𝑝 ∈ (mzPoly‘(𝑊 ∪ (1...𝑁)))𝑆 = {𝑡 ∣ ∃𝑢 ∈ (ℕ0𝑚 (𝑊 ∪ (1...𝑁)))(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝𝑢) = 0)}))
135, 11, 12sylancl 693 . . 3 (𝑁 ∈ ℕ0 → (𝑆 ∈ (Dioph‘𝑁) ↔ ∃𝑝 ∈ (mzPoly‘(𝑊 ∪ (1...𝑁)))𝑆 = {𝑡 ∣ ∃𝑢 ∈ (ℕ0𝑚 (𝑊 ∪ (1...𝑁)))(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝𝑢) = 0)}))
14 elmapssres 7768 . . . . . . . . . . . . . . 15 ((𝑢 ∈ (ℕ0𝑚 (𝑊 ∪ (1...𝑁))) ∧ (1...𝑁) ⊆ (𝑊 ∪ (1...𝑁))) → (𝑢 ↾ (1...𝑁)) ∈ (ℕ0𝑚 (1...𝑁)))
1510, 14mpan2 703 . . . . . . . . . . . . . 14 (𝑢 ∈ (ℕ0𝑚 (𝑊 ∪ (1...𝑁))) → (𝑢 ↾ (1...𝑁)) ∈ (ℕ0𝑚 (1...𝑁)))
1615adantr 480 . . . . . . . . . . . . 13 ((𝑢 ∈ (ℕ0𝑚 (𝑊 ∪ (1...𝑁))) ∧ (𝑝𝑢) = 0) → (𝑢 ↾ (1...𝑁)) ∈ (ℕ0𝑚 (1...𝑁)))
17 ssun1 3738 . . . . . . . . . . . . . . . 16 𝑊 ⊆ (𝑊 ∪ (1...𝑁))
18 elmapssres 7768 . . . . . . . . . . . . . . . 16 ((𝑢 ∈ (ℕ0𝑚 (𝑊 ∪ (1...𝑁))) ∧ 𝑊 ⊆ (𝑊 ∪ (1...𝑁))) → (𝑢𝑊) ∈ (ℕ0𝑚 𝑊))
1917, 18mpan2 703 . . . . . . . . . . . . . . 15 (𝑢 ∈ (ℕ0𝑚 (𝑊 ∪ (1...𝑁))) → (𝑢𝑊) ∈ (ℕ0𝑚 𝑊))
2019adantr 480 . . . . . . . . . . . . . 14 ((𝑢 ∈ (ℕ0𝑚 (𝑊 ∪ (1...𝑁))) ∧ (𝑝𝑢) = 0) → (𝑢𝑊) ∈ (ℕ0𝑚 𝑊))
21 uncom 3719 . . . . . . . . . . . . . . . . . . 19 ((𝑢 ↾ (1...𝑁)) ∪ (𝑢𝑊)) = ((𝑢𝑊) ∪ (𝑢 ↾ (1...𝑁)))
22 resundi 5330 . . . . . . . . . . . . . . . . . . 19 (𝑢 ↾ (𝑊 ∪ (1...𝑁))) = ((𝑢𝑊) ∪ (𝑢 ↾ (1...𝑁)))
2321, 22eqtr4i 2635 . . . . . . . . . . . . . . . . . 18 ((𝑢 ↾ (1...𝑁)) ∪ (𝑢𝑊)) = (𝑢 ↾ (𝑊 ∪ (1...𝑁)))
24 elmapi 7765 . . . . . . . . . . . . . . . . . . 19 (𝑢 ∈ (ℕ0𝑚 (𝑊 ∪ (1...𝑁))) → 𝑢:(𝑊 ∪ (1...𝑁))⟶ℕ0)
25 ffn 5958 . . . . . . . . . . . . . . . . . . 19 (𝑢:(𝑊 ∪ (1...𝑁))⟶ℕ0𝑢 Fn (𝑊 ∪ (1...𝑁)))
26 fnresdm 5914 . . . . . . . . . . . . . . . . . . 19 (𝑢 Fn (𝑊 ∪ (1...𝑁)) → (𝑢 ↾ (𝑊 ∪ (1...𝑁))) = 𝑢)
2724, 25, 263syl 18 . . . . . . . . . . . . . . . . . 18 (𝑢 ∈ (ℕ0𝑚 (𝑊 ∪ (1...𝑁))) → (𝑢 ↾ (𝑊 ∪ (1...𝑁))) = 𝑢)
2823, 27syl5eq 2656 . . . . . . . . . . . . . . . . 17 (𝑢 ∈ (ℕ0𝑚 (𝑊 ∪ (1...𝑁))) → ((𝑢 ↾ (1...𝑁)) ∪ (𝑢𝑊)) = 𝑢)
2928fveq2d 6107 . . . . . . . . . . . . . . . 16 (𝑢 ∈ (ℕ0𝑚 (𝑊 ∪ (1...𝑁))) → (𝑝‘((𝑢 ↾ (1...𝑁)) ∪ (𝑢𝑊))) = (𝑝𝑢))
3029eqeq1d 2612 . . . . . . . . . . . . . . 15 (𝑢 ∈ (ℕ0𝑚 (𝑊 ∪ (1...𝑁))) → ((𝑝‘((𝑢 ↾ (1...𝑁)) ∪ (𝑢𝑊))) = 0 ↔ (𝑝𝑢) = 0))
3130biimpar 501 . . . . . . . . . . . . . 14 ((𝑢 ∈ (ℕ0𝑚 (𝑊 ∪ (1...𝑁))) ∧ (𝑝𝑢) = 0) → (𝑝‘((𝑢 ↾ (1...𝑁)) ∪ (𝑢𝑊))) = 0)
32 uneq2 3723 . . . . . . . . . . . . . . . . 17 (𝑤 = (𝑢𝑊) → ((𝑢 ↾ (1...𝑁)) ∪ 𝑤) = ((𝑢 ↾ (1...𝑁)) ∪ (𝑢𝑊)))
3332fveq2d 6107 . . . . . . . . . . . . . . . 16 (𝑤 = (𝑢𝑊) → (𝑝‘((𝑢 ↾ (1...𝑁)) ∪ 𝑤)) = (𝑝‘((𝑢 ↾ (1...𝑁)) ∪ (𝑢𝑊))))
3433eqeq1d 2612 . . . . . . . . . . . . . . 15 (𝑤 = (𝑢𝑊) → ((𝑝‘((𝑢 ↾ (1...𝑁)) ∪ 𝑤)) = 0 ↔ (𝑝‘((𝑢 ↾ (1...𝑁)) ∪ (𝑢𝑊))) = 0))
3534rspcev 3282 . . . . . . . . . . . . . 14 (((𝑢𝑊) ∈ (ℕ0𝑚 𝑊) ∧ (𝑝‘((𝑢 ↾ (1...𝑁)) ∪ (𝑢𝑊))) = 0) → ∃𝑤 ∈ (ℕ0𝑚 𝑊)(𝑝‘((𝑢 ↾ (1...𝑁)) ∪ 𝑤)) = 0)
3620, 31, 35syl2anc 691 . . . . . . . . . . . . 13 ((𝑢 ∈ (ℕ0𝑚 (𝑊 ∪ (1...𝑁))) ∧ (𝑝𝑢) = 0) → ∃𝑤 ∈ (ℕ0𝑚 𝑊)(𝑝‘((𝑢 ↾ (1...𝑁)) ∪ 𝑤)) = 0)
3716, 36jca 553 . . . . . . . . . . . 12 ((𝑢 ∈ (ℕ0𝑚 (𝑊 ∪ (1...𝑁))) ∧ (𝑝𝑢) = 0) → ((𝑢 ↾ (1...𝑁)) ∈ (ℕ0𝑚 (1...𝑁)) ∧ ∃𝑤 ∈ (ℕ0𝑚 𝑊)(𝑝‘((𝑢 ↾ (1...𝑁)) ∪ 𝑤)) = 0))
38 eleq1 2676 . . . . . . . . . . . . 13 (𝑡 = (𝑢 ↾ (1...𝑁)) → (𝑡 ∈ (ℕ0𝑚 (1...𝑁)) ↔ (𝑢 ↾ (1...𝑁)) ∈ (ℕ0𝑚 (1...𝑁))))
39 uneq1 3722 . . . . . . . . . . . . . . . 16 (𝑡 = (𝑢 ↾ (1...𝑁)) → (𝑡𝑤) = ((𝑢 ↾ (1...𝑁)) ∪ 𝑤))
4039fveq2d 6107 . . . . . . . . . . . . . . 15 (𝑡 = (𝑢 ↾ (1...𝑁)) → (𝑝‘(𝑡𝑤)) = (𝑝‘((𝑢 ↾ (1...𝑁)) ∪ 𝑤)))
4140eqeq1d 2612 . . . . . . . . . . . . . 14 (𝑡 = (𝑢 ↾ (1...𝑁)) → ((𝑝‘(𝑡𝑤)) = 0 ↔ (𝑝‘((𝑢 ↾ (1...𝑁)) ∪ 𝑤)) = 0))
4241rexbidv 3034 . . . . . . . . . . . . 13 (𝑡 = (𝑢 ↾ (1...𝑁)) → (∃𝑤 ∈ (ℕ0𝑚 𝑊)(𝑝‘(𝑡𝑤)) = 0 ↔ ∃𝑤 ∈ (ℕ0𝑚 𝑊)(𝑝‘((𝑢 ↾ (1...𝑁)) ∪ 𝑤)) = 0))
4338, 42anbi12d 743 . . . . . . . . . . . 12 (𝑡 = (𝑢 ↾ (1...𝑁)) → ((𝑡 ∈ (ℕ0𝑚 (1...𝑁)) ∧ ∃𝑤 ∈ (ℕ0𝑚 𝑊)(𝑝‘(𝑡𝑤)) = 0) ↔ ((𝑢 ↾ (1...𝑁)) ∈ (ℕ0𝑚 (1...𝑁)) ∧ ∃𝑤 ∈ (ℕ0𝑚 𝑊)(𝑝‘((𝑢 ↾ (1...𝑁)) ∪ 𝑤)) = 0)))
4437, 43syl5ibrcom 236 . . . . . . . . . . 11 ((𝑢 ∈ (ℕ0𝑚 (𝑊 ∪ (1...𝑁))) ∧ (𝑝𝑢) = 0) → (𝑡 = (𝑢 ↾ (1...𝑁)) → (𝑡 ∈ (ℕ0𝑚 (1...𝑁)) ∧ ∃𝑤 ∈ (ℕ0𝑚 𝑊)(𝑝‘(𝑡𝑤)) = 0)))
4544expimpd 627 . . . . . . . . . 10 (𝑢 ∈ (ℕ0𝑚 (𝑊 ∪ (1...𝑁))) → (((𝑝𝑢) = 0 ∧ 𝑡 = (𝑢 ↾ (1...𝑁))) → (𝑡 ∈ (ℕ0𝑚 (1...𝑁)) ∧ ∃𝑤 ∈ (ℕ0𝑚 𝑊)(𝑝‘(𝑡𝑤)) = 0)))
4645ancomsd 469 . . . . . . . . 9 (𝑢 ∈ (ℕ0𝑚 (𝑊 ∪ (1...𝑁))) → ((𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝𝑢) = 0) → (𝑡 ∈ (ℕ0𝑚 (1...𝑁)) ∧ ∃𝑤 ∈ (ℕ0𝑚 𝑊)(𝑝‘(𝑡𝑤)) = 0)))
4746rexlimiv 3009 . . . . . . . 8 (∃𝑢 ∈ (ℕ0𝑚 (𝑊 ∪ (1...𝑁)))(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝𝑢) = 0) → (𝑡 ∈ (ℕ0𝑚 (1...𝑁)) ∧ ∃𝑤 ∈ (ℕ0𝑚 𝑊)(𝑝‘(𝑡𝑤)) = 0))
48 uncom 3719 . . . . . . . . . . . 12 (𝑡𝑤) = (𝑤𝑡)
49 fz1ssnn 12243 . . . . . . . . . . . . . . . . . . . 20 (1...𝑁) ⊆ ℕ
50 sslin 3801 . . . . . . . . . . . . . . . . . . . 20 ((1...𝑁) ⊆ ℕ → (𝑊 ∩ (1...𝑁)) ⊆ (𝑊 ∩ ℕ))
5149, 50ax-mp 5 . . . . . . . . . . . . . . . . . . 19 (𝑊 ∩ (1...𝑁)) ⊆ (𝑊 ∩ ℕ)
52 eldioph4b.c . . . . . . . . . . . . . . . . . . 19 (𝑊 ∩ ℕ) = ∅
5351, 52sseqtri 3600 . . . . . . . . . . . . . . . . . 18 (𝑊 ∩ (1...𝑁)) ⊆ ∅
54 ss0 3926 . . . . . . . . . . . . . . . . . 18 ((𝑊 ∩ (1...𝑁)) ⊆ ∅ → (𝑊 ∩ (1...𝑁)) = ∅)
5553, 54ax-mp 5 . . . . . . . . . . . . . . . . 17 (𝑊 ∩ (1...𝑁)) = ∅
5655reseq2i 5314 . . . . . . . . . . . . . . . 16 (𝑤 ↾ (𝑊 ∩ (1...𝑁))) = (𝑤 ↾ ∅)
57 res0 5321 . . . . . . . . . . . . . . . 16 (𝑤 ↾ ∅) = ∅
5856, 57eqtri 2632 . . . . . . . . . . . . . . 15 (𝑤 ↾ (𝑊 ∩ (1...𝑁))) = ∅
5955reseq2i 5314 . . . . . . . . . . . . . . . 16 (𝑡 ↾ (𝑊 ∩ (1...𝑁))) = (𝑡 ↾ ∅)
60 res0 5321 . . . . . . . . . . . . . . . 16 (𝑡 ↾ ∅) = ∅
6159, 60eqtri 2632 . . . . . . . . . . . . . . 15 (𝑡 ↾ (𝑊 ∩ (1...𝑁))) = ∅
6258, 61eqtr4i 2635 . . . . . . . . . . . . . 14 (𝑤 ↾ (𝑊 ∩ (1...𝑁))) = (𝑡 ↾ (𝑊 ∩ (1...𝑁)))
63 elmapresaun 36352 . . . . . . . . . . . . . 14 ((𝑤 ∈ (ℕ0𝑚 𝑊) ∧ 𝑡 ∈ (ℕ0𝑚 (1...𝑁)) ∧ (𝑤 ↾ (𝑊 ∩ (1...𝑁))) = (𝑡 ↾ (𝑊 ∩ (1...𝑁)))) → (𝑤𝑡) ∈ (ℕ0𝑚 (𝑊 ∪ (1...𝑁))))
6462, 63mp3an3 1405 . . . . . . . . . . . . 13 ((𝑤 ∈ (ℕ0𝑚 𝑊) ∧ 𝑡 ∈ (ℕ0𝑚 (1...𝑁))) → (𝑤𝑡) ∈ (ℕ0𝑚 (𝑊 ∪ (1...𝑁))))
6564ancoms 468 . . . . . . . . . . . 12 ((𝑡 ∈ (ℕ0𝑚 (1...𝑁)) ∧ 𝑤 ∈ (ℕ0𝑚 𝑊)) → (𝑤𝑡) ∈ (ℕ0𝑚 (𝑊 ∪ (1...𝑁))))
6648, 65syl5eqel 2692 . . . . . . . . . . 11 ((𝑡 ∈ (ℕ0𝑚 (1...𝑁)) ∧ 𝑤 ∈ (ℕ0𝑚 𝑊)) → (𝑡𝑤) ∈ (ℕ0𝑚 (𝑊 ∪ (1...𝑁))))
6766adantr 480 . . . . . . . . . 10 (((𝑡 ∈ (ℕ0𝑚 (1...𝑁)) ∧ 𝑤 ∈ (ℕ0𝑚 𝑊)) ∧ (𝑝‘(𝑡𝑤)) = 0) → (𝑡𝑤) ∈ (ℕ0𝑚 (𝑊 ∪ (1...𝑁))))
6848reseq1i 5313 . . . . . . . . . . . 12 ((𝑡𝑤) ↾ (1...𝑁)) = ((𝑤𝑡) ↾ (1...𝑁))
69 elmapresaunres2 36353 . . . . . . . . . . . . . 14 ((𝑤 ∈ (ℕ0𝑚 𝑊) ∧ 𝑡 ∈ (ℕ0𝑚 (1...𝑁)) ∧ (𝑤 ↾ (𝑊 ∩ (1...𝑁))) = (𝑡 ↾ (𝑊 ∩ (1...𝑁)))) → ((𝑤𝑡) ↾ (1...𝑁)) = 𝑡)
7062, 69mp3an3 1405 . . . . . . . . . . . . 13 ((𝑤 ∈ (ℕ0𝑚 𝑊) ∧ 𝑡 ∈ (ℕ0𝑚 (1...𝑁))) → ((𝑤𝑡) ↾ (1...𝑁)) = 𝑡)
7170ancoms 468 . . . . . . . . . . . 12 ((𝑡 ∈ (ℕ0𝑚 (1...𝑁)) ∧ 𝑤 ∈ (ℕ0𝑚 𝑊)) → ((𝑤𝑡) ↾ (1...𝑁)) = 𝑡)
7268, 71syl5req 2657 . . . . . . . . . . 11 ((𝑡 ∈ (ℕ0𝑚 (1...𝑁)) ∧ 𝑤 ∈ (ℕ0𝑚 𝑊)) → 𝑡 = ((𝑡𝑤) ↾ (1...𝑁)))
7372adantr 480 . . . . . . . . . 10 (((𝑡 ∈ (ℕ0𝑚 (1...𝑁)) ∧ 𝑤 ∈ (ℕ0𝑚 𝑊)) ∧ (𝑝‘(𝑡𝑤)) = 0) → 𝑡 = ((𝑡𝑤) ↾ (1...𝑁)))
74 simpr 476 . . . . . . . . . 10 (((𝑡 ∈ (ℕ0𝑚 (1...𝑁)) ∧ 𝑤 ∈ (ℕ0𝑚 𝑊)) ∧ (𝑝‘(𝑡𝑤)) = 0) → (𝑝‘(𝑡𝑤)) = 0)
75 reseq1 5311 . . . . . . . . . . . . 13 (𝑢 = (𝑡𝑤) → (𝑢 ↾ (1...𝑁)) = ((𝑡𝑤) ↾ (1...𝑁)))
7675eqeq2d 2620 . . . . . . . . . . . 12 (𝑢 = (𝑡𝑤) → (𝑡 = (𝑢 ↾ (1...𝑁)) ↔ 𝑡 = ((𝑡𝑤) ↾ (1...𝑁))))
77 fveq2 6103 . . . . . . . . . . . . 13 (𝑢 = (𝑡𝑤) → (𝑝𝑢) = (𝑝‘(𝑡𝑤)))
7877eqeq1d 2612 . . . . . . . . . . . 12 (𝑢 = (𝑡𝑤) → ((𝑝𝑢) = 0 ↔ (𝑝‘(𝑡𝑤)) = 0))
7976, 78anbi12d 743 . . . . . . . . . . 11 (𝑢 = (𝑡𝑤) → ((𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝𝑢) = 0) ↔ (𝑡 = ((𝑡𝑤) ↾ (1...𝑁)) ∧ (𝑝‘(𝑡𝑤)) = 0)))
8079rspcev 3282 . . . . . . . . . 10 (((𝑡𝑤) ∈ (ℕ0𝑚 (𝑊 ∪ (1...𝑁))) ∧ (𝑡 = ((𝑡𝑤) ↾ (1...𝑁)) ∧ (𝑝‘(𝑡𝑤)) = 0)) → ∃𝑢 ∈ (ℕ0𝑚 (𝑊 ∪ (1...𝑁)))(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝𝑢) = 0))
8167, 73, 74, 80syl12anc 1316 . . . . . . . . 9 (((𝑡 ∈ (ℕ0𝑚 (1...𝑁)) ∧ 𝑤 ∈ (ℕ0𝑚 𝑊)) ∧ (𝑝‘(𝑡𝑤)) = 0) → ∃𝑢 ∈ (ℕ0𝑚 (𝑊 ∪ (1...𝑁)))(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝𝑢) = 0))
8281r19.29an 3059 . . . . . . . 8 ((𝑡 ∈ (ℕ0𝑚 (1...𝑁)) ∧ ∃𝑤 ∈ (ℕ0𝑚 𝑊)(𝑝‘(𝑡𝑤)) = 0) → ∃𝑢 ∈ (ℕ0𝑚 (𝑊 ∪ (1...𝑁)))(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝𝑢) = 0))
8347, 82impbii 198 . . . . . . 7 (∃𝑢 ∈ (ℕ0𝑚 (𝑊 ∪ (1...𝑁)))(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝𝑢) = 0) ↔ (𝑡 ∈ (ℕ0𝑚 (1...𝑁)) ∧ ∃𝑤 ∈ (ℕ0𝑚 𝑊)(𝑝‘(𝑡𝑤)) = 0))
8483abbii 2726 . . . . . 6 {𝑡 ∣ ∃𝑢 ∈ (ℕ0𝑚 (𝑊 ∪ (1...𝑁)))(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝𝑢) = 0)} = {𝑡 ∣ (𝑡 ∈ (ℕ0𝑚 (1...𝑁)) ∧ ∃𝑤 ∈ (ℕ0𝑚 𝑊)(𝑝‘(𝑡𝑤)) = 0)}
85 df-rab 2905 . . . . . 6 {𝑡 ∈ (ℕ0𝑚 (1...𝑁)) ∣ ∃𝑤 ∈ (ℕ0𝑚 𝑊)(𝑝‘(𝑡𝑤)) = 0} = {𝑡 ∣ (𝑡 ∈ (ℕ0𝑚 (1...𝑁)) ∧ ∃𝑤 ∈ (ℕ0𝑚 𝑊)(𝑝‘(𝑡𝑤)) = 0)}
8684, 85eqtr4i 2635 . . . . 5 {𝑡 ∣ ∃𝑢 ∈ (ℕ0𝑚 (𝑊 ∪ (1...𝑁)))(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝𝑢) = 0)} = {𝑡 ∈ (ℕ0𝑚 (1...𝑁)) ∣ ∃𝑤 ∈ (ℕ0𝑚 𝑊)(𝑝‘(𝑡𝑤)) = 0}
8786eqeq2i 2622 . . . 4 (𝑆 = {𝑡 ∣ ∃𝑢 ∈ (ℕ0𝑚 (𝑊 ∪ (1...𝑁)))(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝𝑢) = 0)} ↔ 𝑆 = {𝑡 ∈ (ℕ0𝑚 (1...𝑁)) ∣ ∃𝑤 ∈ (ℕ0𝑚 𝑊)(𝑝‘(𝑡𝑤)) = 0})
8887rexbii 3023 . . 3 (∃𝑝 ∈ (mzPoly‘(𝑊 ∪ (1...𝑁)))𝑆 = {𝑡 ∣ ∃𝑢 ∈ (ℕ0𝑚 (𝑊 ∪ (1...𝑁)))(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝𝑢) = 0)} ↔ ∃𝑝 ∈ (mzPoly‘(𝑊 ∪ (1...𝑁)))𝑆 = {𝑡 ∈ (ℕ0𝑚 (1...𝑁)) ∣ ∃𝑤 ∈ (ℕ0𝑚 𝑊)(𝑝‘(𝑡𝑤)) = 0})
8913, 88syl6bb 275 . 2 (𝑁 ∈ ℕ0 → (𝑆 ∈ (Dioph‘𝑁) ↔ ∃𝑝 ∈ (mzPoly‘(𝑊 ∪ (1...𝑁)))𝑆 = {𝑡 ∈ (ℕ0𝑚 (1...𝑁)) ∣ ∃𝑤 ∈ (ℕ0𝑚 𝑊)(𝑝‘(𝑡𝑤)) = 0}))
901, 89biadan2 672 1 (𝑆 ∈ (Dioph‘𝑁) ↔ (𝑁 ∈ ℕ0 ∧ ∃𝑝 ∈ (mzPoly‘(𝑊 ∪ (1...𝑁)))𝑆 = {𝑡 ∈ (ℕ0𝑚 (1...𝑁)) ∣ ∃𝑤 ∈ (ℕ0𝑚 𝑊)(𝑝‘(𝑡𝑤)) = 0}))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  {cab 2596  ∃wrex 2897  {crab 2900  Vcvv 3173   ∪ cun 3538   ∩ cin 3539   ⊆ wss 3540  ∅c0 3874   ↾ cres 5040   Fn wfn 5799  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549   ↑𝑚 cmap 7744  Fincfn 7841  0cc0 9815  1c1 9816  ℕcn 10897  ℕ0cn0 11169  ...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-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:  eldioph4i  36394  diophren  36395
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