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Theorem eldioph4b 36275
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 36227 . 2 (𝑆 ∈ (Dioph‘𝑁) → 𝑁 ∈ ℕ0)
2 eldioph4b.a . . . . . 6 𝑊 ∈ V
3 ovex 6453 . . . . . 6 (1...𝑁) ∈ V
42, 3unex 6728 . . . . 5 (𝑊 ∪ (1...𝑁)) ∈ V
54jctr 562 . . . 4 (𝑁 ∈ ℕ0 → (𝑁 ∈ ℕ0 ∧ (𝑊 ∪ (1...𝑁)) ∈ V))
6 eldioph4b.b . . . . . . 7 ¬ 𝑊 ∈ Fin
76intnanr 951 . . . . . 6 ¬ (𝑊 ∈ Fin ∧ (1...𝑁) ∈ Fin)
8 unfir 7987 . . . . . 6 ((𝑊 ∪ (1...𝑁)) ∈ Fin → (𝑊 ∈ Fin ∧ (1...𝑁) ∈ Fin))
97, 8mto 186 . . . . 5 ¬ (𝑊 ∪ (1...𝑁)) ∈ Fin
10 ssun2 3643 . . . . 5 (1...𝑁) ⊆ (𝑊 ∪ (1...𝑁))
119, 10pm3.2i 469 . . . 4 (¬ (𝑊 ∪ (1...𝑁)) ∈ Fin ∧ (1...𝑁) ⊆ (𝑊 ∪ (1...𝑁)))
12 eldioph2b 36226 . . . 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 692 . . 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 7642 . . . . . . . . . . . . . . 15 ((𝑢 ∈ (ℕ0𝑚 (𝑊 ∪ (1...𝑁))) ∧ (1...𝑁) ⊆ (𝑊 ∪ (1...𝑁))) → (𝑢 ↾ (1...𝑁)) ∈ (ℕ0𝑚 (1...𝑁)))
1510, 14mpan2 702 . . . . . . . . . . . . . 14 (𝑢 ∈ (ℕ0𝑚 (𝑊 ∪ (1...𝑁))) → (𝑢 ↾ (1...𝑁)) ∈ (ℕ0𝑚 (1...𝑁)))
1615adantr 479 . . . . . . . . . . . . 13 ((𝑢 ∈ (ℕ0𝑚 (𝑊 ∪ (1...𝑁))) ∧ (𝑝𝑢) = 0) → (𝑢 ↾ (1...𝑁)) ∈ (ℕ0𝑚 (1...𝑁)))
17 ssun1 3642 . . . . . . . . . . . . . . . 16 𝑊 ⊆ (𝑊 ∪ (1...𝑁))
18 elmapssres 7642 . . . . . . . . . . . . . . . 16 ((𝑢 ∈ (ℕ0𝑚 (𝑊 ∪ (1...𝑁))) ∧ 𝑊 ⊆ (𝑊 ∪ (1...𝑁))) → (𝑢𝑊) ∈ (ℕ0𝑚 𝑊))
1917, 18mpan2 702 . . . . . . . . . . . . . . 15 (𝑢 ∈ (ℕ0𝑚 (𝑊 ∪ (1...𝑁))) → (𝑢𝑊) ∈ (ℕ0𝑚 𝑊))
2019adantr 479 . . . . . . . . . . . . . 14 ((𝑢 ∈ (ℕ0𝑚 (𝑊 ∪ (1...𝑁))) ∧ (𝑝𝑢) = 0) → (𝑢𝑊) ∈ (ℕ0𝑚 𝑊))
21 uncom 3623 . . . . . . . . . . . . . . . . . . 19 ((𝑢 ↾ (1...𝑁)) ∪ (𝑢𝑊)) = ((𝑢𝑊) ∪ (𝑢 ↾ (1...𝑁)))
22 resundi 5221 . . . . . . . . . . . . . . . . . . 19 (𝑢 ↾ (𝑊 ∪ (1...𝑁))) = ((𝑢𝑊) ∪ (𝑢 ↾ (1...𝑁)))
2321, 22eqtr4i 2539 . . . . . . . . . . . . . . . . . 18 ((𝑢 ↾ (1...𝑁)) ∪ (𝑢𝑊)) = (𝑢 ↾ (𝑊 ∪ (1...𝑁)))
24 elmapi 7639 . . . . . . . . . . . . . . . . . . 19 (𝑢 ∈ (ℕ0𝑚 (𝑊 ∪ (1...𝑁))) → 𝑢:(𝑊 ∪ (1...𝑁))⟶ℕ0)
25 ffn 5843 . . . . . . . . . . . . . . . . . . 19 (𝑢:(𝑊 ∪ (1...𝑁))⟶ℕ0𝑢 Fn (𝑊 ∪ (1...𝑁)))
26 fnresdm 5799 . . . . . . . . . . . . . . . . . . 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 2560 . . . . . . . . . . . . . . . . 17 (𝑢 ∈ (ℕ0𝑚 (𝑊 ∪ (1...𝑁))) → ((𝑢 ↾ (1...𝑁)) ∪ (𝑢𝑊)) = 𝑢)
2928fveq2d 5990 . . . . . . . . . . . . . . . 16 (𝑢 ∈ (ℕ0𝑚 (𝑊 ∪ (1...𝑁))) → (𝑝‘((𝑢 ↾ (1...𝑁)) ∪ (𝑢𝑊))) = (𝑝𝑢))
3029eqeq1d 2516 . . . . . . . . . . . . . . 15 (𝑢 ∈ (ℕ0𝑚 (𝑊 ∪ (1...𝑁))) → ((𝑝‘((𝑢 ↾ (1...𝑁)) ∪ (𝑢𝑊))) = 0 ↔ (𝑝𝑢) = 0))
3130biimpar 500 . . . . . . . . . . . . . 14 ((𝑢 ∈ (ℕ0𝑚 (𝑊 ∪ (1...𝑁))) ∧ (𝑝𝑢) = 0) → (𝑝‘((𝑢 ↾ (1...𝑁)) ∪ (𝑢𝑊))) = 0)
32 uneq2 3627 . . . . . . . . . . . . . . . . 17 (𝑤 = (𝑢𝑊) → ((𝑢 ↾ (1...𝑁)) ∪ 𝑤) = ((𝑢 ↾ (1...𝑁)) ∪ (𝑢𝑊)))
3332fveq2d 5990 . . . . . . . . . . . . . . . 16 (𝑤 = (𝑢𝑊) → (𝑝‘((𝑢 ↾ (1...𝑁)) ∪ 𝑤)) = (𝑝‘((𝑢 ↾ (1...𝑁)) ∪ (𝑢𝑊))))
3433eqeq1d 2516 . . . . . . . . . . . . . . 15 (𝑤 = (𝑢𝑊) → ((𝑝‘((𝑢 ↾ (1...𝑁)) ∪ 𝑤)) = 0 ↔ (𝑝‘((𝑢 ↾ (1...𝑁)) ∪ (𝑢𝑊))) = 0))
3534rspcev 3186 . . . . . . . . . . . . . 14 (((𝑢𝑊) ∈ (ℕ0𝑚 𝑊) ∧ (𝑝‘((𝑢 ↾ (1...𝑁)) ∪ (𝑢𝑊))) = 0) → ∃𝑤 ∈ (ℕ0𝑚 𝑊)(𝑝‘((𝑢 ↾ (1...𝑁)) ∪ 𝑤)) = 0)
3620, 31, 35syl2anc 690 . . . . . . . . . . . . 13 ((𝑢 ∈ (ℕ0𝑚 (𝑊 ∪ (1...𝑁))) ∧ (𝑝𝑢) = 0) → ∃𝑤 ∈ (ℕ0𝑚 𝑊)(𝑝‘((𝑢 ↾ (1...𝑁)) ∪ 𝑤)) = 0)
3716, 36jca 552 . . . . . . . . . . . 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 2580 . . . . . . . . . . . . 13 (𝑡 = (𝑢 ↾ (1...𝑁)) → (𝑡 ∈ (ℕ0𝑚 (1...𝑁)) ↔ (𝑢 ↾ (1...𝑁)) ∈ (ℕ0𝑚 (1...𝑁))))
39 uneq1 3626 . . . . . . . . . . . . . . . 16 (𝑡 = (𝑢 ↾ (1...𝑁)) → (𝑡𝑤) = ((𝑢 ↾ (1...𝑁)) ∪ 𝑤))
4039fveq2d 5990 . . . . . . . . . . . . . . 15 (𝑡 = (𝑢 ↾ (1...𝑁)) → (𝑝‘(𝑡𝑤)) = (𝑝‘((𝑢 ↾ (1...𝑁)) ∪ 𝑤)))
4140eqeq1d 2516 . . . . . . . . . . . . . 14 (𝑡 = (𝑢 ↾ (1...𝑁)) → ((𝑝‘(𝑡𝑤)) = 0 ↔ (𝑝‘((𝑢 ↾ (1...𝑁)) ∪ 𝑤)) = 0))
4241rexbidv 2938 . . . . . . . . . . . . 13 (𝑡 = (𝑢 ↾ (1...𝑁)) → (∃𝑤 ∈ (ℕ0𝑚 𝑊)(𝑝‘(𝑡𝑤)) = 0 ↔ ∃𝑤 ∈ (ℕ0𝑚 𝑊)(𝑝‘((𝑢 ↾ (1...𝑁)) ∪ 𝑤)) = 0))
4338, 42anbi12d 742 . . . . . . . . . . . 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 235 . . . . . . . . . . 11 ((𝑢 ∈ (ℕ0𝑚 (𝑊 ∪ (1...𝑁))) ∧ (𝑝𝑢) = 0) → (𝑡 = (𝑢 ↾ (1...𝑁)) → (𝑡 ∈ (ℕ0𝑚 (1...𝑁)) ∧ ∃𝑤 ∈ (ℕ0𝑚 𝑊)(𝑝‘(𝑡𝑤)) = 0)))
4544expimpd 626 . . . . . . . . . 10 (𝑢 ∈ (ℕ0𝑚 (𝑊 ∪ (1...𝑁))) → (((𝑝𝑢) = 0 ∧ 𝑡 = (𝑢 ↾ (1...𝑁))) → (𝑡 ∈ (ℕ0𝑚 (1...𝑁)) ∧ ∃𝑤 ∈ (ℕ0𝑚 𝑊)(𝑝‘(𝑡𝑤)) = 0)))
4645ancomsd 468 . . . . . . . . 9 (𝑢 ∈ (ℕ0𝑚 (𝑊 ∪ (1...𝑁))) → ((𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝𝑢) = 0) → (𝑡 ∈ (ℕ0𝑚 (1...𝑁)) ∧ ∃𝑤 ∈ (ℕ0𝑚 𝑊)(𝑝‘(𝑡𝑤)) = 0)))
4746rexlimiv 2913 . . . . . . . 8 (∃𝑢 ∈ (ℕ0𝑚 (𝑊 ∪ (1...𝑁)))(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝𝑢) = 0) → (𝑡 ∈ (ℕ0𝑚 (1...𝑁)) ∧ ∃𝑤 ∈ (ℕ0𝑚 𝑊)(𝑝‘(𝑡𝑤)) = 0))
48 uncom 3623 . . . . . . . . . . . 12 (𝑡𝑤) = (𝑤𝑡)
49 fz1ssnn 12108 . . . . . . . . . . . . . . . . . . . 20 (1...𝑁) ⊆ ℕ
50 sslin 3704 . . . . . . . . . . . . . . . . . . . 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 3504 . . . . . . . . . . . . . . . . . 18 (𝑊 ∩ (1...𝑁)) ⊆ ∅
54 ss0 3829 . . . . . . . . . . . . . . . . . 18 ((𝑊 ∩ (1...𝑁)) ⊆ ∅ → (𝑊 ∩ (1...𝑁)) = ∅)
5553, 54ax-mp 5 . . . . . . . . . . . . . . . . 17 (𝑊 ∩ (1...𝑁)) = ∅
5655reseq2i 5205 . . . . . . . . . . . . . . . 16 (𝑤 ↾ (𝑊 ∩ (1...𝑁))) = (𝑤 ↾ ∅)
57 res0 5212 . . . . . . . . . . . . . . . 16 (𝑤 ↾ ∅) = ∅
5856, 57eqtri 2536 . . . . . . . . . . . . . . 15 (𝑤 ↾ (𝑊 ∩ (1...𝑁))) = ∅
5955reseq2i 5205 . . . . . . . . . . . . . . . 16 (𝑡 ↾ (𝑊 ∩ (1...𝑁))) = (𝑡 ↾ ∅)
60 res0 5212 . . . . . . . . . . . . . . . 16 (𝑡 ↾ ∅) = ∅
6159, 60eqtri 2536 . . . . . . . . . . . . . . 15 (𝑡 ↾ (𝑊 ∩ (1...𝑁))) = ∅
6258, 61eqtr4i 2539 . . . . . . . . . . . . . 14 (𝑤 ↾ (𝑊 ∩ (1...𝑁))) = (𝑡 ↾ (𝑊 ∩ (1...𝑁)))
63 elmapresaun 36234 . . . . . . . . . . . . . 14 ((𝑤 ∈ (ℕ0𝑚 𝑊) ∧ 𝑡 ∈ (ℕ0𝑚 (1...𝑁)) ∧ (𝑤 ↾ (𝑊 ∩ (1...𝑁))) = (𝑡 ↾ (𝑊 ∩ (1...𝑁)))) → (𝑤𝑡) ∈ (ℕ0𝑚 (𝑊 ∪ (1...𝑁))))
6462, 63mp3an3 1404 . . . . . . . . . . . . 13 ((𝑤 ∈ (ℕ0𝑚 𝑊) ∧ 𝑡 ∈ (ℕ0𝑚 (1...𝑁))) → (𝑤𝑡) ∈ (ℕ0𝑚 (𝑊 ∪ (1...𝑁))))
6564ancoms 467 . . . . . . . . . . . 12 ((𝑡 ∈ (ℕ0𝑚 (1...𝑁)) ∧ 𝑤 ∈ (ℕ0𝑚 𝑊)) → (𝑤𝑡) ∈ (ℕ0𝑚 (𝑊 ∪ (1...𝑁))))
6648, 65syl5eqel 2596 . . . . . . . . . . 11 ((𝑡 ∈ (ℕ0𝑚 (1...𝑁)) ∧ 𝑤 ∈ (ℕ0𝑚 𝑊)) → (𝑡𝑤) ∈ (ℕ0𝑚 (𝑊 ∪ (1...𝑁))))
6766adantr 479 . . . . . . . . . 10 (((𝑡 ∈ (ℕ0𝑚 (1...𝑁)) ∧ 𝑤 ∈ (ℕ0𝑚 𝑊)) ∧ (𝑝‘(𝑡𝑤)) = 0) → (𝑡𝑤) ∈ (ℕ0𝑚 (𝑊 ∪ (1...𝑁))))
6848reseq1i 5204 . . . . . . . . . . . 12 ((𝑡𝑤) ↾ (1...𝑁)) = ((𝑤𝑡) ↾ (1...𝑁))
69 elmapresaunres2 36235 . . . . . . . . . . . . . 14 ((𝑤 ∈ (ℕ0𝑚 𝑊) ∧ 𝑡 ∈ (ℕ0𝑚 (1...𝑁)) ∧ (𝑤 ↾ (𝑊 ∩ (1...𝑁))) = (𝑡 ↾ (𝑊 ∩ (1...𝑁)))) → ((𝑤𝑡) ↾ (1...𝑁)) = 𝑡)
7062, 69mp3an3 1404 . . . . . . . . . . . . 13 ((𝑤 ∈ (ℕ0𝑚 𝑊) ∧ 𝑡 ∈ (ℕ0𝑚 (1...𝑁))) → ((𝑤𝑡) ↾ (1...𝑁)) = 𝑡)
7170ancoms 467 . . . . . . . . . . . 12 ((𝑡 ∈ (ℕ0𝑚 (1...𝑁)) ∧ 𝑤 ∈ (ℕ0𝑚 𝑊)) → ((𝑤𝑡) ↾ (1...𝑁)) = 𝑡)
7268, 71syl5req 2561 . . . . . . . . . . 11 ((𝑡 ∈ (ℕ0𝑚 (1...𝑁)) ∧ 𝑤 ∈ (ℕ0𝑚 𝑊)) → 𝑡 = ((𝑡𝑤) ↾ (1...𝑁)))
7372adantr 479 . . . . . . . . . 10 (((𝑡 ∈ (ℕ0𝑚 (1...𝑁)) ∧ 𝑤 ∈ (ℕ0𝑚 𝑊)) ∧ (𝑝‘(𝑡𝑤)) = 0) → 𝑡 = ((𝑡𝑤) ↾ (1...𝑁)))
74 simpr 475 . . . . . . . . . 10 (((𝑡 ∈ (ℕ0𝑚 (1...𝑁)) ∧ 𝑤 ∈ (ℕ0𝑚 𝑊)) ∧ (𝑝‘(𝑡𝑤)) = 0) → (𝑝‘(𝑡𝑤)) = 0)
75 reseq1 5202 . . . . . . . . . . . . 13 (𝑢 = (𝑡𝑤) → (𝑢 ↾ (1...𝑁)) = ((𝑡𝑤) ↾ (1...𝑁)))
7675eqeq2d 2524 . . . . . . . . . . . 12 (𝑢 = (𝑡𝑤) → (𝑡 = (𝑢 ↾ (1...𝑁)) ↔ 𝑡 = ((𝑡𝑤) ↾ (1...𝑁))))
77 fveq2 5986 . . . . . . . . . . . . 13 (𝑢 = (𝑡𝑤) → (𝑝𝑢) = (𝑝‘(𝑡𝑤)))
7877eqeq1d 2516 . . . . . . . . . . . 12 (𝑢 = (𝑡𝑤) → ((𝑝𝑢) = 0 ↔ (𝑝‘(𝑡𝑤)) = 0))
7976, 78anbi12d 742 . . . . . . . . . . 11 (𝑢 = (𝑡𝑤) → ((𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝𝑢) = 0) ↔ (𝑡 = ((𝑡𝑤) ↾ (1...𝑁)) ∧ (𝑝‘(𝑡𝑤)) = 0)))
8079rspcev 3186 . . . . . . . . . 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 1315 . . . . . . . . 9 (((𝑡 ∈ (ℕ0𝑚 (1...𝑁)) ∧ 𝑤 ∈ (ℕ0𝑚 𝑊)) ∧ (𝑝‘(𝑡𝑤)) = 0) → ∃𝑢 ∈ (ℕ0𝑚 (𝑊 ∪ (1...𝑁)))(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝𝑢) = 0))
8281r19.29an 2963 . . . . . . . 8 ((𝑡 ∈ (ℕ0𝑚 (1...𝑁)) ∧ ∃𝑤 ∈ (ℕ0𝑚 𝑊)(𝑝‘(𝑡𝑤)) = 0) → ∃𝑢 ∈ (ℕ0𝑚 (𝑊 ∪ (1...𝑁)))(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝𝑢) = 0))
8347, 82impbii 197 . . . . . . 7 (∃𝑢 ∈ (ℕ0𝑚 (𝑊 ∪ (1...𝑁)))(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝𝑢) = 0) ↔ (𝑡 ∈ (ℕ0𝑚 (1...𝑁)) ∧ ∃𝑤 ∈ (ℕ0𝑚 𝑊)(𝑝‘(𝑡𝑤)) = 0))
8483abbii 2630 . . . . . 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 2809 . . . . . 6 {𝑡 ∈ (ℕ0𝑚 (1...𝑁)) ∣ ∃𝑤 ∈ (ℕ0𝑚 𝑊)(𝑝‘(𝑡𝑤)) = 0} = {𝑡 ∣ (𝑡 ∈ (ℕ0𝑚 (1...𝑁)) ∧ ∃𝑤 ∈ (ℕ0𝑚 𝑊)(𝑝‘(𝑡𝑤)) = 0)}
8684, 85eqtr4i 2539 . . . . 5 {𝑡 ∣ ∃𝑢 ∈ (ℕ0𝑚 (𝑊 ∪ (1...𝑁)))(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝𝑢) = 0)} = {𝑡 ∈ (ℕ0𝑚 (1...𝑁)) ∣ ∃𝑤 ∈ (ℕ0𝑚 𝑊)(𝑝‘(𝑡𝑤)) = 0}
8786eqeq2i 2526 . . . 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 2927 . . 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 274 . 2 (𝑁 ∈ ℕ0 → (𝑆 ∈ (Dioph‘𝑁) ↔ ∃𝑝 ∈ (mzPoly‘(𝑊 ∪ (1...𝑁)))𝑆 = {𝑡 ∈ (ℕ0𝑚 (1...𝑁)) ∣ ∃𝑤 ∈ (ℕ0𝑚 𝑊)(𝑝‘(𝑡𝑤)) = 0}))
901, 89biadan2 671 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 194  wa 382   = wceq 1474  wcel 1938  {cab 2500  wrex 2801  {crab 2804  Vcvv 3077  cun 3442  cin 3443  wss 3444  c0 3777  cres 4934   Fn wfn 5684  wf 5685  cfv 5689  (class class class)co 6425  𝑚 cmap 7618  Fincfn 7715  0cc0 9689  1c1 9690  cn 10773  0cn0 11045  ...cfz 12062  mzPolycmzp 36185  Diophcdioph 36218
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1700  ax-4 1713  ax-5 1793  ax-6 1838  ax-7 1885  ax-8 1940  ax-9 1947  ax-10 1966  ax-11 1971  ax-12 1983  ax-13 2137  ax-ext 2494  ax-rep 4597  ax-sep 4607  ax-nul 4616  ax-pow 4668  ax-pr 4732  ax-un 6721  ax-cnex 9745  ax-resscn 9746  ax-1cn 9747  ax-icn 9748  ax-addcl 9749  ax-addrcl 9750  ax-mulcl 9751  ax-mulrcl 9752  ax-mulcom 9753  ax-addass 9754  ax-mulass 9755  ax-distr 9756  ax-i2m1 9757  ax-1ne0 9758  ax-1rid 9759  ax-rnegex 9760  ax-rrecex 9761  ax-cnre 9762  ax-pre-lttri 9763  ax-pre-lttrn 9764  ax-pre-ltadd 9765  ax-pre-mulgt0 9766
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1699  df-sb 1831  df-eu 2366  df-mo 2367  df-clab 2501  df-cleq 2507  df-clel 2510  df-nfc 2644  df-ne 2686  df-nel 2687  df-ral 2805  df-rex 2806  df-reu 2807  df-rmo 2808  df-rab 2809  df-v 3079  df-sbc 3307  df-csb 3404  df-dif 3447  df-un 3449  df-in 3451  df-ss 3458  df-pss 3460  df-nul 3778  df-if 3940  df-pw 4013  df-sn 4029  df-pr 4031  df-tp 4033  df-op 4035  df-uni 4271  df-int 4309  df-iun 4355  df-br 4482  df-opab 4542  df-mpt 4543  df-tr 4579  df-eprel 4843  df-id 4847  df-po 4853  df-so 4854  df-fr 4891  df-we 4893  df-xp 4938  df-rel 4939  df-cnv 4940  df-co 4941  df-dm 4942  df-rn 4943  df-res 4944  df-ima 4945  df-pred 5487  df-ord 5533  df-on 5534  df-lim 5535  df-suc 5536  df-iota 5653  df-fun 5691  df-fn 5692  df-f 5693  df-f1 5694  df-fo 5695  df-f1o 5696  df-fv 5697  df-riota 6387  df-ov 6428  df-oprab 6429  df-mpt2 6430  df-of 6669  df-om 6832  df-1st 6932  df-2nd 6933  df-wrecs 7167  df-recs 7229  df-rdg 7267  df-1o 7321  df-oadd 7325  df-er 7503  df-map 7620  df-en 7716  df-dom 7717  df-sdom 7718  df-fin 7719  df-card 8522  df-cda 8747  df-pnf 9829  df-mnf 9830  df-xr 9831  df-ltxr 9832  df-le 9833  df-sub 10017  df-neg 10018  df-nn 10774  df-n0 11046  df-z 11117  df-uz 11424  df-fz 12063  df-hash 12845  df-mzpcl 36186  df-mzp 36187  df-dioph 36219
This theorem is referenced by:  eldioph4i  36276  diophren  36277
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