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Theorem eldioph4b 37194
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 37146 . 2 (𝑆 ∈ (Dioph‘𝑁) → 𝑁 ∈ ℕ0)
2 eldioph4b.a . . . . . 6 𝑊 ∈ V
3 ovex 6663 . . . . . 6 (1...𝑁) ∈ V
42, 3unex 6941 . . . . 5 (𝑊 ∪ (1...𝑁)) ∈ V
54jctr 564 . . . 4 (𝑁 ∈ ℕ0 → (𝑁 ∈ ℕ0 ∧ (𝑊 ∪ (1...𝑁)) ∈ V))
6 eldioph4b.b . . . . . . 7 ¬ 𝑊 ∈ Fin
76intnanr 960 . . . . . 6 ¬ (𝑊 ∈ Fin ∧ (1...𝑁) ∈ Fin)
8 unfir 8213 . . . . . 6 ((𝑊 ∪ (1...𝑁)) ∈ Fin → (𝑊 ∈ Fin ∧ (1...𝑁) ∈ Fin))
97, 8mto 188 . . . . 5 ¬ (𝑊 ∪ (1...𝑁)) ∈ Fin
10 ssun2 3769 . . . . 5 (1...𝑁) ⊆ (𝑊 ∪ (1...𝑁))
119, 10pm3.2i 471 . . . 4 (¬ (𝑊 ∪ (1...𝑁)) ∈ Fin ∧ (1...𝑁) ⊆ (𝑊 ∪ (1...𝑁)))
12 eldioph2b 37145 . . . 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 7867 . . . . . . . . . . . . . . 15 ((𝑢 ∈ (ℕ0𝑚 (𝑊 ∪ (1...𝑁))) ∧ (1...𝑁) ⊆ (𝑊 ∪ (1...𝑁))) → (𝑢 ↾ (1...𝑁)) ∈ (ℕ0𝑚 (1...𝑁)))
1510, 14mpan2 706 . . . . . . . . . . . . . 14 (𝑢 ∈ (ℕ0𝑚 (𝑊 ∪ (1...𝑁))) → (𝑢 ↾ (1...𝑁)) ∈ (ℕ0𝑚 (1...𝑁)))
1615adantr 481 . . . . . . . . . . . . 13 ((𝑢 ∈ (ℕ0𝑚 (𝑊 ∪ (1...𝑁))) ∧ (𝑝𝑢) = 0) → (𝑢 ↾ (1...𝑁)) ∈ (ℕ0𝑚 (1...𝑁)))
17 ssun1 3768 . . . . . . . . . . . . . . . 16 𝑊 ⊆ (𝑊 ∪ (1...𝑁))
18 elmapssres 7867 . . . . . . . . . . . . . . . 16 ((𝑢 ∈ (ℕ0𝑚 (𝑊 ∪ (1...𝑁))) ∧ 𝑊 ⊆ (𝑊 ∪ (1...𝑁))) → (𝑢𝑊) ∈ (ℕ0𝑚 𝑊))
1917, 18mpan2 706 . . . . . . . . . . . . . . 15 (𝑢 ∈ (ℕ0𝑚 (𝑊 ∪ (1...𝑁))) → (𝑢𝑊) ∈ (ℕ0𝑚 𝑊))
2019adantr 481 . . . . . . . . . . . . . 14 ((𝑢 ∈ (ℕ0𝑚 (𝑊 ∪ (1...𝑁))) ∧ (𝑝𝑢) = 0) → (𝑢𝑊) ∈ (ℕ0𝑚 𝑊))
21 uncom 3749 . . . . . . . . . . . . . . . . . . 19 ((𝑢 ↾ (1...𝑁)) ∪ (𝑢𝑊)) = ((𝑢𝑊) ∪ (𝑢 ↾ (1...𝑁)))
22 resundi 5398 . . . . . . . . . . . . . . . . . . 19 (𝑢 ↾ (𝑊 ∪ (1...𝑁))) = ((𝑢𝑊) ∪ (𝑢 ↾ (1...𝑁)))
2321, 22eqtr4i 2645 . . . . . . . . . . . . . . . . . 18 ((𝑢 ↾ (1...𝑁)) ∪ (𝑢𝑊)) = (𝑢 ↾ (𝑊 ∪ (1...𝑁)))
24 elmapi 7864 . . . . . . . . . . . . . . . . . . 19 (𝑢 ∈ (ℕ0𝑚 (𝑊 ∪ (1...𝑁))) → 𝑢:(𝑊 ∪ (1...𝑁))⟶ℕ0)
25 ffn 6032 . . . . . . . . . . . . . . . . . . 19 (𝑢:(𝑊 ∪ (1...𝑁))⟶ℕ0𝑢 Fn (𝑊 ∪ (1...𝑁)))
26 fnresdm 5988 . . . . . . . . . . . . . . . . . . 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 2666 . . . . . . . . . . . . . . . . 17 (𝑢 ∈ (ℕ0𝑚 (𝑊 ∪ (1...𝑁))) → ((𝑢 ↾ (1...𝑁)) ∪ (𝑢𝑊)) = 𝑢)
2928fveq2d 6182 . . . . . . . . . . . . . . . 16 (𝑢 ∈ (ℕ0𝑚 (𝑊 ∪ (1...𝑁))) → (𝑝‘((𝑢 ↾ (1...𝑁)) ∪ (𝑢𝑊))) = (𝑝𝑢))
3029eqeq1d 2622 . . . . . . . . . . . . . . 15 (𝑢 ∈ (ℕ0𝑚 (𝑊 ∪ (1...𝑁))) → ((𝑝‘((𝑢 ↾ (1...𝑁)) ∪ (𝑢𝑊))) = 0 ↔ (𝑝𝑢) = 0))
3130biimpar 502 . . . . . . . . . . . . . 14 ((𝑢 ∈ (ℕ0𝑚 (𝑊 ∪ (1...𝑁))) ∧ (𝑝𝑢) = 0) → (𝑝‘((𝑢 ↾ (1...𝑁)) ∪ (𝑢𝑊))) = 0)
32 uneq2 3753 . . . . . . . . . . . . . . . . 17 (𝑤 = (𝑢𝑊) → ((𝑢 ↾ (1...𝑁)) ∪ 𝑤) = ((𝑢 ↾ (1...𝑁)) ∪ (𝑢𝑊)))
3332fveq2d 6182 . . . . . . . . . . . . . . . 16 (𝑤 = (𝑢𝑊) → (𝑝‘((𝑢 ↾ (1...𝑁)) ∪ 𝑤)) = (𝑝‘((𝑢 ↾ (1...𝑁)) ∪ (𝑢𝑊))))
3433eqeq1d 2622 . . . . . . . . . . . . . . 15 (𝑤 = (𝑢𝑊) → ((𝑝‘((𝑢 ↾ (1...𝑁)) ∪ 𝑤)) = 0 ↔ (𝑝‘((𝑢 ↾ (1...𝑁)) ∪ (𝑢𝑊))) = 0))
3534rspcev 3304 . . . . . . . . . . . . . 14 (((𝑢𝑊) ∈ (ℕ0𝑚 𝑊) ∧ (𝑝‘((𝑢 ↾ (1...𝑁)) ∪ (𝑢𝑊))) = 0) → ∃𝑤 ∈ (ℕ0𝑚 𝑊)(𝑝‘((𝑢 ↾ (1...𝑁)) ∪ 𝑤)) = 0)
3620, 31, 35syl2anc 692 . . . . . . . . . . . . 13 ((𝑢 ∈ (ℕ0𝑚 (𝑊 ∪ (1...𝑁))) ∧ (𝑝𝑢) = 0) → ∃𝑤 ∈ (ℕ0𝑚 𝑊)(𝑝‘((𝑢 ↾ (1...𝑁)) ∪ 𝑤)) = 0)
3716, 36jca 554 . . . . . . . . . . . 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 2687 . . . . . . . . . . . . 13 (𝑡 = (𝑢 ↾ (1...𝑁)) → (𝑡 ∈ (ℕ0𝑚 (1...𝑁)) ↔ (𝑢 ↾ (1...𝑁)) ∈ (ℕ0𝑚 (1...𝑁))))
39 uneq1 3752 . . . . . . . . . . . . . . . 16 (𝑡 = (𝑢 ↾ (1...𝑁)) → (𝑡𝑤) = ((𝑢 ↾ (1...𝑁)) ∪ 𝑤))
4039fveq2d 6182 . . . . . . . . . . . . . . 15 (𝑡 = (𝑢 ↾ (1...𝑁)) → (𝑝‘(𝑡𝑤)) = (𝑝‘((𝑢 ↾ (1...𝑁)) ∪ 𝑤)))
4140eqeq1d 2622 . . . . . . . . . . . . . 14 (𝑡 = (𝑢 ↾ (1...𝑁)) → ((𝑝‘(𝑡𝑤)) = 0 ↔ (𝑝‘((𝑢 ↾ (1...𝑁)) ∪ 𝑤)) = 0))
4241rexbidv 3048 . . . . . . . . . . . . 13 (𝑡 = (𝑢 ↾ (1...𝑁)) → (∃𝑤 ∈ (ℕ0𝑚 𝑊)(𝑝‘(𝑡𝑤)) = 0 ↔ ∃𝑤 ∈ (ℕ0𝑚 𝑊)(𝑝‘((𝑢 ↾ (1...𝑁)) ∪ 𝑤)) = 0))
4338, 42anbi12d 746 . . . . . . . . . . . 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 237 . . . . . . . . . . 11 ((𝑢 ∈ (ℕ0𝑚 (𝑊 ∪ (1...𝑁))) ∧ (𝑝𝑢) = 0) → (𝑡 = (𝑢 ↾ (1...𝑁)) → (𝑡 ∈ (ℕ0𝑚 (1...𝑁)) ∧ ∃𝑤 ∈ (ℕ0𝑚 𝑊)(𝑝‘(𝑡𝑤)) = 0)))
4544expimpd 628 . . . . . . . . . 10 (𝑢 ∈ (ℕ0𝑚 (𝑊 ∪ (1...𝑁))) → (((𝑝𝑢) = 0 ∧ 𝑡 = (𝑢 ↾ (1...𝑁))) → (𝑡 ∈ (ℕ0𝑚 (1...𝑁)) ∧ ∃𝑤 ∈ (ℕ0𝑚 𝑊)(𝑝‘(𝑡𝑤)) = 0)))
4645ancomsd 470 . . . . . . . . 9 (𝑢 ∈ (ℕ0𝑚 (𝑊 ∪ (1...𝑁))) → ((𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝𝑢) = 0) → (𝑡 ∈ (ℕ0𝑚 (1...𝑁)) ∧ ∃𝑤 ∈ (ℕ0𝑚 𝑊)(𝑝‘(𝑡𝑤)) = 0)))
4746rexlimiv 3023 . . . . . . . 8 (∃𝑢 ∈ (ℕ0𝑚 (𝑊 ∪ (1...𝑁)))(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝𝑢) = 0) → (𝑡 ∈ (ℕ0𝑚 (1...𝑁)) ∧ ∃𝑤 ∈ (ℕ0𝑚 𝑊)(𝑝‘(𝑡𝑤)) = 0))
48 uncom 3749 . . . . . . . . . . . 12 (𝑡𝑤) = (𝑤𝑡)
49 fz1ssnn 12357 . . . . . . . . . . . . . . . . . . . 20 (1...𝑁) ⊆ ℕ
50 sslin 3831 . . . . . . . . . . . . . . . . . . . 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 3629 . . . . . . . . . . . . . . . . . 18 (𝑊 ∩ (1...𝑁)) ⊆ ∅
54 ss0 3965 . . . . . . . . . . . . . . . . . 18 ((𝑊 ∩ (1...𝑁)) ⊆ ∅ → (𝑊 ∩ (1...𝑁)) = ∅)
5553, 54ax-mp 5 . . . . . . . . . . . . . . . . 17 (𝑊 ∩ (1...𝑁)) = ∅
5655reseq2i 5382 . . . . . . . . . . . . . . . 16 (𝑤 ↾ (𝑊 ∩ (1...𝑁))) = (𝑤 ↾ ∅)
57 res0 5389 . . . . . . . . . . . . . . . 16 (𝑤 ↾ ∅) = ∅
5856, 57eqtri 2642 . . . . . . . . . . . . . . 15 (𝑤 ↾ (𝑊 ∩ (1...𝑁))) = ∅
5955reseq2i 5382 . . . . . . . . . . . . . . . 16 (𝑡 ↾ (𝑊 ∩ (1...𝑁))) = (𝑡 ↾ ∅)
60 res0 5389 . . . . . . . . . . . . . . . 16 (𝑡 ↾ ∅) = ∅
6159, 60eqtri 2642 . . . . . . . . . . . . . . 15 (𝑡 ↾ (𝑊 ∩ (1...𝑁))) = ∅
6258, 61eqtr4i 2645 . . . . . . . . . . . . . 14 (𝑤 ↾ (𝑊 ∩ (1...𝑁))) = (𝑡 ↾ (𝑊 ∩ (1...𝑁)))
63 elmapresaun 37153 . . . . . . . . . . . . . 14 ((𝑤 ∈ (ℕ0𝑚 𝑊) ∧ 𝑡 ∈ (ℕ0𝑚 (1...𝑁)) ∧ (𝑤 ↾ (𝑊 ∩ (1...𝑁))) = (𝑡 ↾ (𝑊 ∩ (1...𝑁)))) → (𝑤𝑡) ∈ (ℕ0𝑚 (𝑊 ∪ (1...𝑁))))
6462, 63mp3an3 1411 . . . . . . . . . . . . 13 ((𝑤 ∈ (ℕ0𝑚 𝑊) ∧ 𝑡 ∈ (ℕ0𝑚 (1...𝑁))) → (𝑤𝑡) ∈ (ℕ0𝑚 (𝑊 ∪ (1...𝑁))))
6564ancoms 469 . . . . . . . . . . . 12 ((𝑡 ∈ (ℕ0𝑚 (1...𝑁)) ∧ 𝑤 ∈ (ℕ0𝑚 𝑊)) → (𝑤𝑡) ∈ (ℕ0𝑚 (𝑊 ∪ (1...𝑁))))
6648, 65syl5eqel 2703 . . . . . . . . . . 11 ((𝑡 ∈ (ℕ0𝑚 (1...𝑁)) ∧ 𝑤 ∈ (ℕ0𝑚 𝑊)) → (𝑡𝑤) ∈ (ℕ0𝑚 (𝑊 ∪ (1...𝑁))))
6766adantr 481 . . . . . . . . . 10 (((𝑡 ∈ (ℕ0𝑚 (1...𝑁)) ∧ 𝑤 ∈ (ℕ0𝑚 𝑊)) ∧ (𝑝‘(𝑡𝑤)) = 0) → (𝑡𝑤) ∈ (ℕ0𝑚 (𝑊 ∪ (1...𝑁))))
6848reseq1i 5381 . . . . . . . . . . . 12 ((𝑡𝑤) ↾ (1...𝑁)) = ((𝑤𝑡) ↾ (1...𝑁))
69 elmapresaunres2 37154 . . . . . . . . . . . . . 14 ((𝑤 ∈ (ℕ0𝑚 𝑊) ∧ 𝑡 ∈ (ℕ0𝑚 (1...𝑁)) ∧ (𝑤 ↾ (𝑊 ∩ (1...𝑁))) = (𝑡 ↾ (𝑊 ∩ (1...𝑁)))) → ((𝑤𝑡) ↾ (1...𝑁)) = 𝑡)
7062, 69mp3an3 1411 . . . . . . . . . . . . 13 ((𝑤 ∈ (ℕ0𝑚 𝑊) ∧ 𝑡 ∈ (ℕ0𝑚 (1...𝑁))) → ((𝑤𝑡) ↾ (1...𝑁)) = 𝑡)
7170ancoms 469 . . . . . . . . . . . 12 ((𝑡 ∈ (ℕ0𝑚 (1...𝑁)) ∧ 𝑤 ∈ (ℕ0𝑚 𝑊)) → ((𝑤𝑡) ↾ (1...𝑁)) = 𝑡)
7268, 71syl5req 2667 . . . . . . . . . . 11 ((𝑡 ∈ (ℕ0𝑚 (1...𝑁)) ∧ 𝑤 ∈ (ℕ0𝑚 𝑊)) → 𝑡 = ((𝑡𝑤) ↾ (1...𝑁)))
7372adantr 481 . . . . . . . . . 10 (((𝑡 ∈ (ℕ0𝑚 (1...𝑁)) ∧ 𝑤 ∈ (ℕ0𝑚 𝑊)) ∧ (𝑝‘(𝑡𝑤)) = 0) → 𝑡 = ((𝑡𝑤) ↾ (1...𝑁)))
74 simpr 477 . . . . . . . . . 10 (((𝑡 ∈ (ℕ0𝑚 (1...𝑁)) ∧ 𝑤 ∈ (ℕ0𝑚 𝑊)) ∧ (𝑝‘(𝑡𝑤)) = 0) → (𝑝‘(𝑡𝑤)) = 0)
75 reseq1 5379 . . . . . . . . . . . . 13 (𝑢 = (𝑡𝑤) → (𝑢 ↾ (1...𝑁)) = ((𝑡𝑤) ↾ (1...𝑁)))
7675eqeq2d 2630 . . . . . . . . . . . 12 (𝑢 = (𝑡𝑤) → (𝑡 = (𝑢 ↾ (1...𝑁)) ↔ 𝑡 = ((𝑡𝑤) ↾ (1...𝑁))))
77 fveq2 6178 . . . . . . . . . . . . 13 (𝑢 = (𝑡𝑤) → (𝑝𝑢) = (𝑝‘(𝑡𝑤)))
7877eqeq1d 2622 . . . . . . . . . . . 12 (𝑢 = (𝑡𝑤) → ((𝑝𝑢) = 0 ↔ (𝑝‘(𝑡𝑤)) = 0))
7976, 78anbi12d 746 . . . . . . . . . . 11 (𝑢 = (𝑡𝑤) → ((𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝𝑢) = 0) ↔ (𝑡 = ((𝑡𝑤) ↾ (1...𝑁)) ∧ (𝑝‘(𝑡𝑤)) = 0)))
8079rspcev 3304 . . . . . . . . . 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 1322 . . . . . . . . 9 (((𝑡 ∈ (ℕ0𝑚 (1...𝑁)) ∧ 𝑤 ∈ (ℕ0𝑚 𝑊)) ∧ (𝑝‘(𝑡𝑤)) = 0) → ∃𝑢 ∈ (ℕ0𝑚 (𝑊 ∪ (1...𝑁)))(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝𝑢) = 0))
8281r19.29an 3073 . . . . . . . 8 ((𝑡 ∈ (ℕ0𝑚 (1...𝑁)) ∧ ∃𝑤 ∈ (ℕ0𝑚 𝑊)(𝑝‘(𝑡𝑤)) = 0) → ∃𝑢 ∈ (ℕ0𝑚 (𝑊 ∪ (1...𝑁)))(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝𝑢) = 0))
8347, 82impbii 199 . . . . . . 7 (∃𝑢 ∈ (ℕ0𝑚 (𝑊 ∪ (1...𝑁)))(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝𝑢) = 0) ↔ (𝑡 ∈ (ℕ0𝑚 (1...𝑁)) ∧ ∃𝑤 ∈ (ℕ0𝑚 𝑊)(𝑝‘(𝑡𝑤)) = 0))
8483abbii 2737 . . . . . 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 2918 . . . . . 6 {𝑡 ∈ (ℕ0𝑚 (1...𝑁)) ∣ ∃𝑤 ∈ (ℕ0𝑚 𝑊)(𝑝‘(𝑡𝑤)) = 0} = {𝑡 ∣ (𝑡 ∈ (ℕ0𝑚 (1...𝑁)) ∧ ∃𝑤 ∈ (ℕ0𝑚 𝑊)(𝑝‘(𝑡𝑤)) = 0)}
8684, 85eqtr4i 2645 . . . . 5 {𝑡 ∣ ∃𝑢 ∈ (ℕ0𝑚 (𝑊 ∪ (1...𝑁)))(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝𝑢) = 0)} = {𝑡 ∈ (ℕ0𝑚 (1...𝑁)) ∣ ∃𝑤 ∈ (ℕ0𝑚 𝑊)(𝑝‘(𝑡𝑤)) = 0}
8786eqeq2i 2632 . . . 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 3037 . . 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 276 . 2 (𝑁 ∈ ℕ0 → (𝑆 ∈ (Dioph‘𝑁) ↔ ∃𝑝 ∈ (mzPoly‘(𝑊 ∪ (1...𝑁)))𝑆 = {𝑡 ∈ (ℕ0𝑚 (1...𝑁)) ∣ ∃𝑤 ∈ (ℕ0𝑚 𝑊)(𝑝‘(𝑡𝑤)) = 0}))
901, 89biadan2 673 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 196  wa 384   = wceq 1481  wcel 1988  {cab 2606  wrex 2910  {crab 2913  Vcvv 3195  cun 3565  cin 3566  wss 3567  c0 3907  cres 5106   Fn wfn 5871  wf 5872  cfv 5876  (class class class)co 6635  𝑚 cmap 7842  Fincfn 7940  0cc0 9921  1c1 9922  cn 11005  0cn0 11277  ...cfz 12311  mzPolycmzp 37104  Diophcdioph 37137
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-rep 4762  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934  ax-cnex 9977  ax-resscn 9978  ax-1cn 9979  ax-icn 9980  ax-addcl 9981  ax-addrcl 9982  ax-mulcl 9983  ax-mulrcl 9984  ax-mulcom 9985  ax-addass 9986  ax-mulass 9987  ax-distr 9988  ax-i2m1 9989  ax-1ne0 9990  ax-1rid 9991  ax-rnegex 9992  ax-rrecex 9993  ax-cnre 9994  ax-pre-lttri 9995  ax-pre-lttrn 9996  ax-pre-ltadd 9997  ax-pre-mulgt0 9998
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-nel 2895  df-ral 2914  df-rex 2915  df-reu 2916  df-rmo 2917  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-pss 3583  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-tp 4173  df-op 4175  df-uni 4428  df-int 4467  df-iun 4513  df-br 4645  df-opab 4704  df-mpt 4721  df-tr 4744  df-id 5014  df-eprel 5019  df-po 5025  df-so 5026  df-fr 5063  df-we 5065  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-pred 5668  df-ord 5714  df-on 5715  df-lim 5716  df-suc 5717  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-fv 5884  df-riota 6596  df-ov 6638  df-oprab 6639  df-mpt2 6640  df-of 6882  df-om 7051  df-1st 7153  df-2nd 7154  df-wrecs 7392  df-recs 7453  df-rdg 7491  df-1o 7545  df-oadd 7549  df-er 7727  df-map 7844  df-en 7941  df-dom 7942  df-sdom 7943  df-fin 7944  df-card 8750  df-cda 8975  df-pnf 10061  df-mnf 10062  df-xr 10063  df-ltxr 10064  df-le 10065  df-sub 10253  df-neg 10254  df-nn 11006  df-n0 11278  df-z 11363  df-uz 11673  df-fz 12312  df-hash 13101  df-mzpcl 37105  df-mzp 37106  df-dioph 37138
This theorem is referenced by:  eldioph4i  37195  diophren  37196
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