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Theorem eldioph4b 39270
 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 39223 . 2 (𝑆 ∈ (Dioph‘𝑁) → 𝑁 ∈ ℕ0)
2 eldioph4b.a . . . . . 6 𝑊 ∈ V
3 ovex 7184 . . . . . 6 (1...𝑁) ∈ V
42, 3unex 7461 . . . . 5 (𝑊 ∪ (1...𝑁)) ∈ V
54jctr 525 . . . 4 (𝑁 ∈ ℕ0 → (𝑁 ∈ ℕ0 ∧ (𝑊 ∪ (1...𝑁)) ∈ V))
6 eldioph4b.b . . . . . . 7 ¬ 𝑊 ∈ Fin
76intnanr 488 . . . . . 6 ¬ (𝑊 ∈ Fin ∧ (1...𝑁) ∈ Fin)
8 unfir 8778 . . . . . 6 ((𝑊 ∪ (1...𝑁)) ∈ Fin → (𝑊 ∈ Fin ∧ (1...𝑁) ∈ Fin))
97, 8mto 198 . . . . 5 ¬ (𝑊 ∪ (1...𝑁)) ∈ Fin
10 ssun2 4152 . . . . 5 (1...𝑁) ⊆ (𝑊 ∪ (1...𝑁))
119, 10pm3.2i 471 . . . 4 (¬ (𝑊 ∪ (1...𝑁)) ∈ Fin ∧ (1...𝑁) ⊆ (𝑊 ∪ (1...𝑁)))
12 eldioph2b 39222 . . . 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 586 . . 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 8424 . . . . . . . . . . . . . . 15 ((𝑢 ∈ (ℕ0m (𝑊 ∪ (1...𝑁))) ∧ (1...𝑁) ⊆ (𝑊 ∪ (1...𝑁))) → (𝑢 ↾ (1...𝑁)) ∈ (ℕ0m (1...𝑁)))
1510, 14mpan2 687 . . . . . . . . . . . . . 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 4151 . . . . . . . . . . . . . . . 16 𝑊 ⊆ (𝑊 ∪ (1...𝑁))
18 elmapssres 8424 . . . . . . . . . . . . . . . 16 ((𝑢 ∈ (ℕ0m (𝑊 ∪ (1...𝑁))) ∧ 𝑊 ⊆ (𝑊 ∪ (1...𝑁))) → (𝑢𝑊) ∈ (ℕ0m 𝑊))
1917, 18mpan2 687 . . . . . . . . . . . . . . 15 (𝑢 ∈ (ℕ0m (𝑊 ∪ (1...𝑁))) → (𝑢𝑊) ∈ (ℕ0m 𝑊))
2019adantr 481 . . . . . . . . . . . . . 14 ((𝑢 ∈ (ℕ0m (𝑊 ∪ (1...𝑁))) ∧ (𝑝𝑢) = 0) → (𝑢𝑊) ∈ (ℕ0m 𝑊))
21 uncom 4132 . . . . . . . . . . . . . . . . . 18 ((𝑢 ↾ (1...𝑁)) ∪ (𝑢𝑊)) = ((𝑢𝑊) ∪ (𝑢 ↾ (1...𝑁)))
22 resundi 5865 . . . . . . . . . . . . . . . . . 18 (𝑢 ↾ (𝑊 ∪ (1...𝑁))) = ((𝑢𝑊) ∪ (𝑢 ↾ (1...𝑁)))
2321, 22eqtr4i 2851 . . . . . . . . . . . . . . . . 17 ((𝑢 ↾ (1...𝑁)) ∪ (𝑢𝑊)) = (𝑢 ↾ (𝑊 ∪ (1...𝑁)))
24 elmapi 8421 . . . . . . . . . . . . . . . . . 18 (𝑢 ∈ (ℕ0m (𝑊 ∪ (1...𝑁))) → 𝑢:(𝑊 ∪ (1...𝑁))⟶ℕ0)
25 ffn 6510 . . . . . . . . . . . . . . . . . 18 (𝑢:(𝑊 ∪ (1...𝑁))⟶ℕ0𝑢 Fn (𝑊 ∪ (1...𝑁)))
26 fnresdm 6462 . . . . . . . . . . . . . . . . . 18 (𝑢 Fn (𝑊 ∪ (1...𝑁)) → (𝑢 ↾ (𝑊 ∪ (1...𝑁))) = 𝑢)
2724, 25, 263syl 18 . . . . . . . . . . . . . . . . 17 (𝑢 ∈ (ℕ0m (𝑊 ∪ (1...𝑁))) → (𝑢 ↾ (𝑊 ∪ (1...𝑁))) = 𝑢)
2823, 27syl5eq 2872 . . . . . . . . . . . . . . . 16 (𝑢 ∈ (ℕ0m (𝑊 ∪ (1...𝑁))) → ((𝑢 ↾ (1...𝑁)) ∪ (𝑢𝑊)) = 𝑢)
2928fveqeq2d 6674 . . . . . . . . . . . . . . 15 (𝑢 ∈ (ℕ0m (𝑊 ∪ (1...𝑁))) → ((𝑝‘((𝑢 ↾ (1...𝑁)) ∪ (𝑢𝑊))) = 0 ↔ (𝑝𝑢) = 0))
3029biimpar 478 . . . . . . . . . . . . . 14 ((𝑢 ∈ (ℕ0m (𝑊 ∪ (1...𝑁))) ∧ (𝑝𝑢) = 0) → (𝑝‘((𝑢 ↾ (1...𝑁)) ∪ (𝑢𝑊))) = 0)
31 uneq2 4136 . . . . . . . . . . . . . . . 16 (𝑤 = (𝑢𝑊) → ((𝑢 ↾ (1...𝑁)) ∪ 𝑤) = ((𝑢 ↾ (1...𝑁)) ∪ (𝑢𝑊)))
3231fveqeq2d 6674 . . . . . . . . . . . . . . 15 (𝑤 = (𝑢𝑊) → ((𝑝‘((𝑢 ↾ (1...𝑁)) ∪ 𝑤)) = 0 ↔ (𝑝‘((𝑢 ↾ (1...𝑁)) ∪ (𝑢𝑊))) = 0))
3332rspcev 3626 . . . . . . . . . . . . . 14 (((𝑢𝑊) ∈ (ℕ0m 𝑊) ∧ (𝑝‘((𝑢 ↾ (1...𝑁)) ∪ (𝑢𝑊))) = 0) → ∃𝑤 ∈ (ℕ0m 𝑊)(𝑝‘((𝑢 ↾ (1...𝑁)) ∪ 𝑤)) = 0)
3420, 30, 33syl2anc 584 . . . . . . . . . . . . 13 ((𝑢 ∈ (ℕ0m (𝑊 ∪ (1...𝑁))) ∧ (𝑝𝑢) = 0) → ∃𝑤 ∈ (ℕ0m 𝑊)(𝑝‘((𝑢 ↾ (1...𝑁)) ∪ 𝑤)) = 0)
3516, 34jca 512 . . . . . . . . . . . 12 ((𝑢 ∈ (ℕ0m (𝑊 ∪ (1...𝑁))) ∧ (𝑝𝑢) = 0) → ((𝑢 ↾ (1...𝑁)) ∈ (ℕ0m (1...𝑁)) ∧ ∃𝑤 ∈ (ℕ0m 𝑊)(𝑝‘((𝑢 ↾ (1...𝑁)) ∪ 𝑤)) = 0))
36 eleq1 2904 . . . . . . . . . . . . 13 (𝑡 = (𝑢 ↾ (1...𝑁)) → (𝑡 ∈ (ℕ0m (1...𝑁)) ↔ (𝑢 ↾ (1...𝑁)) ∈ (ℕ0m (1...𝑁))))
37 uneq1 4135 . . . . . . . . . . . . . . 15 (𝑡 = (𝑢 ↾ (1...𝑁)) → (𝑡𝑤) = ((𝑢 ↾ (1...𝑁)) ∪ 𝑤))
3837fveqeq2d 6674 . . . . . . . . . . . . . 14 (𝑡 = (𝑢 ↾ (1...𝑁)) → ((𝑝‘(𝑡𝑤)) = 0 ↔ (𝑝‘((𝑢 ↾ (1...𝑁)) ∪ 𝑤)) = 0))
3938rexbidv 3301 . . . . . . . . . . . . 13 (𝑡 = (𝑢 ↾ (1...𝑁)) → (∃𝑤 ∈ (ℕ0m 𝑊)(𝑝‘(𝑡𝑤)) = 0 ↔ ∃𝑤 ∈ (ℕ0m 𝑊)(𝑝‘((𝑢 ↾ (1...𝑁)) ∪ 𝑤)) = 0))
4036, 39anbi12d 630 . . . . . . . . . . . 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)))
4135, 40syl5ibrcom 248 . . . . . . . . . . 11 ((𝑢 ∈ (ℕ0m (𝑊 ∪ (1...𝑁))) ∧ (𝑝𝑢) = 0) → (𝑡 = (𝑢 ↾ (1...𝑁)) → (𝑡 ∈ (ℕ0m (1...𝑁)) ∧ ∃𝑤 ∈ (ℕ0m 𝑊)(𝑝‘(𝑡𝑤)) = 0)))
4241expimpd 454 . . . . . . . . . 10 (𝑢 ∈ (ℕ0m (𝑊 ∪ (1...𝑁))) → (((𝑝𝑢) = 0 ∧ 𝑡 = (𝑢 ↾ (1...𝑁))) → (𝑡 ∈ (ℕ0m (1...𝑁)) ∧ ∃𝑤 ∈ (ℕ0m 𝑊)(𝑝‘(𝑡𝑤)) = 0)))
4342ancomsd 466 . . . . . . . . 9 (𝑢 ∈ (ℕ0m (𝑊 ∪ (1...𝑁))) → ((𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝𝑢) = 0) → (𝑡 ∈ (ℕ0m (1...𝑁)) ∧ ∃𝑤 ∈ (ℕ0m 𝑊)(𝑝‘(𝑡𝑤)) = 0)))
4443rexlimiv 3284 . . . . . . . 8 (∃𝑢 ∈ (ℕ0m (𝑊 ∪ (1...𝑁)))(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝𝑢) = 0) → (𝑡 ∈ (ℕ0m (1...𝑁)) ∧ ∃𝑤 ∈ (ℕ0m 𝑊)(𝑝‘(𝑡𝑤)) = 0))
45 uncom 4132 . . . . . . . . . . . 12 (𝑡𝑤) = (𝑤𝑡)
46 fz1ssnn 12931 . . . . . . . . . . . . . . . . . . . 20 (1...𝑁) ⊆ ℕ
47 sslin 4214 . . . . . . . . . . . . . . . . . . . 20 ((1...𝑁) ⊆ ℕ → (𝑊 ∩ (1...𝑁)) ⊆ (𝑊 ∩ ℕ))
4846, 47ax-mp 5 . . . . . . . . . . . . . . . . . . 19 (𝑊 ∩ (1...𝑁)) ⊆ (𝑊 ∩ ℕ)
49 eldioph4b.c . . . . . . . . . . . . . . . . . . 19 (𝑊 ∩ ℕ) = ∅
5048, 49sseqtri 4006 . . . . . . . . . . . . . . . . . 18 (𝑊 ∩ (1...𝑁)) ⊆ ∅
51 ss0 4355 . . . . . . . . . . . . . . . . . 18 ((𝑊 ∩ (1...𝑁)) ⊆ ∅ → (𝑊 ∩ (1...𝑁)) = ∅)
5250, 51ax-mp 5 . . . . . . . . . . . . . . . . 17 (𝑊 ∩ (1...𝑁)) = ∅
5352reseq2i 5848 . . . . . . . . . . . . . . . 16 (𝑤 ↾ (𝑊 ∩ (1...𝑁))) = (𝑤 ↾ ∅)
54 res0 5855 . . . . . . . . . . . . . . . 16 (𝑤 ↾ ∅) = ∅
5553, 54eqtri 2848 . . . . . . . . . . . . . . 15 (𝑤 ↾ (𝑊 ∩ (1...𝑁))) = ∅
5652reseq2i 5848 . . . . . . . . . . . . . . . 16 (𝑡 ↾ (𝑊 ∩ (1...𝑁))) = (𝑡 ↾ ∅)
57 res0 5855 . . . . . . . . . . . . . . . 16 (𝑡 ↾ ∅) = ∅
5856, 57eqtri 2848 . . . . . . . . . . . . . . 15 (𝑡 ↾ (𝑊 ∩ (1...𝑁))) = ∅
5955, 58eqtr4i 2851 . . . . . . . . . . . . . 14 (𝑤 ↾ (𝑊 ∩ (1...𝑁))) = (𝑡 ↾ (𝑊 ∩ (1...𝑁)))
60 elmapresaun 8437 . . . . . . . . . . . . . 14 ((𝑤 ∈ (ℕ0m 𝑊) ∧ 𝑡 ∈ (ℕ0m (1...𝑁)) ∧ (𝑤 ↾ (𝑊 ∩ (1...𝑁))) = (𝑡 ↾ (𝑊 ∩ (1...𝑁)))) → (𝑤𝑡) ∈ (ℕ0m (𝑊 ∪ (1...𝑁))))
6159, 60mp3an3 1443 . . . . . . . . . . . . 13 ((𝑤 ∈ (ℕ0m 𝑊) ∧ 𝑡 ∈ (ℕ0m (1...𝑁))) → (𝑤𝑡) ∈ (ℕ0m (𝑊 ∪ (1...𝑁))))
6261ancoms 459 . . . . . . . . . . . 12 ((𝑡 ∈ (ℕ0m (1...𝑁)) ∧ 𝑤 ∈ (ℕ0m 𝑊)) → (𝑤𝑡) ∈ (ℕ0m (𝑊 ∪ (1...𝑁))))
6345, 62eqeltrid 2921 . . . . . . . . . . 11 ((𝑡 ∈ (ℕ0m (1...𝑁)) ∧ 𝑤 ∈ (ℕ0m 𝑊)) → (𝑡𝑤) ∈ (ℕ0m (𝑊 ∪ (1...𝑁))))
6463adantr 481 . . . . . . . . . 10 (((𝑡 ∈ (ℕ0m (1...𝑁)) ∧ 𝑤 ∈ (ℕ0m 𝑊)) ∧ (𝑝‘(𝑡𝑤)) = 0) → (𝑡𝑤) ∈ (ℕ0m (𝑊 ∪ (1...𝑁))))
6545reseq1i 5847 . . . . . . . . . . . 12 ((𝑡𝑤) ↾ (1...𝑁)) = ((𝑤𝑡) ↾ (1...𝑁))
66 elmapresaunres2 39230 . . . . . . . . . . . . . 14 ((𝑤 ∈ (ℕ0m 𝑊) ∧ 𝑡 ∈ (ℕ0m (1...𝑁)) ∧ (𝑤 ↾ (𝑊 ∩ (1...𝑁))) = (𝑡 ↾ (𝑊 ∩ (1...𝑁)))) → ((𝑤𝑡) ↾ (1...𝑁)) = 𝑡)
6759, 66mp3an3 1443 . . . . . . . . . . . . 13 ((𝑤 ∈ (ℕ0m 𝑊) ∧ 𝑡 ∈ (ℕ0m (1...𝑁))) → ((𝑤𝑡) ↾ (1...𝑁)) = 𝑡)
6867ancoms 459 . . . . . . . . . . . 12 ((𝑡 ∈ (ℕ0m (1...𝑁)) ∧ 𝑤 ∈ (ℕ0m 𝑊)) → ((𝑤𝑡) ↾ (1...𝑁)) = 𝑡)
6965, 68syl5req 2873 . . . . . . . . . . 11 ((𝑡 ∈ (ℕ0m (1...𝑁)) ∧ 𝑤 ∈ (ℕ0m 𝑊)) → 𝑡 = ((𝑡𝑤) ↾ (1...𝑁)))
7069adantr 481 . . . . . . . . . 10 (((𝑡 ∈ (ℕ0m (1...𝑁)) ∧ 𝑤 ∈ (ℕ0m 𝑊)) ∧ (𝑝‘(𝑡𝑤)) = 0) → 𝑡 = ((𝑡𝑤) ↾ (1...𝑁)))
71 simpr 485 . . . . . . . . . 10 (((𝑡 ∈ (ℕ0m (1...𝑁)) ∧ 𝑤 ∈ (ℕ0m 𝑊)) ∧ (𝑝‘(𝑡𝑤)) = 0) → (𝑝‘(𝑡𝑤)) = 0)
72 reseq1 5845 . . . . . . . . . . . . 13 (𝑢 = (𝑡𝑤) → (𝑢 ↾ (1...𝑁)) = ((𝑡𝑤) ↾ (1...𝑁)))
7372eqeq2d 2835 . . . . . . . . . . . 12 (𝑢 = (𝑡𝑤) → (𝑡 = (𝑢 ↾ (1...𝑁)) ↔ 𝑡 = ((𝑡𝑤) ↾ (1...𝑁))))
74 fveqeq2 6675 . . . . . . . . . . . 12 (𝑢 = (𝑡𝑤) → ((𝑝𝑢) = 0 ↔ (𝑝‘(𝑡𝑤)) = 0))
7573, 74anbi12d 630 . . . . . . . . . . 11 (𝑢 = (𝑡𝑤) → ((𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝𝑢) = 0) ↔ (𝑡 = ((𝑡𝑤) ↾ (1...𝑁)) ∧ (𝑝‘(𝑡𝑤)) = 0)))
7675rspcev 3626 . . . . . . . . . 10 (((𝑡𝑤) ∈ (ℕ0m (𝑊 ∪ (1...𝑁))) ∧ (𝑡 = ((𝑡𝑤) ↾ (1...𝑁)) ∧ (𝑝‘(𝑡𝑤)) = 0)) → ∃𝑢 ∈ (ℕ0m (𝑊 ∪ (1...𝑁)))(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝𝑢) = 0))
7764, 70, 71, 76syl12anc 834 . . . . . . . . 9 (((𝑡 ∈ (ℕ0m (1...𝑁)) ∧ 𝑤 ∈ (ℕ0m 𝑊)) ∧ (𝑝‘(𝑡𝑤)) = 0) → ∃𝑢 ∈ (ℕ0m (𝑊 ∪ (1...𝑁)))(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝𝑢) = 0))
7877r19.29an 3292 . . . . . . . 8 ((𝑡 ∈ (ℕ0m (1...𝑁)) ∧ ∃𝑤 ∈ (ℕ0m 𝑊)(𝑝‘(𝑡𝑤)) = 0) → ∃𝑢 ∈ (ℕ0m (𝑊 ∪ (1...𝑁)))(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝𝑢) = 0))
7944, 78impbii 210 . . . . . . 7 (∃𝑢 ∈ (ℕ0m (𝑊 ∪ (1...𝑁)))(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝𝑢) = 0) ↔ (𝑡 ∈ (ℕ0m (1...𝑁)) ∧ ∃𝑤 ∈ (ℕ0m 𝑊)(𝑝‘(𝑡𝑤)) = 0))
8079abbii 2890 . . . . . 6 {𝑡 ∣ ∃𝑢 ∈ (ℕ0m (𝑊 ∪ (1...𝑁)))(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝𝑢) = 0)} = {𝑡 ∣ (𝑡 ∈ (ℕ0m (1...𝑁)) ∧ ∃𝑤 ∈ (ℕ0m 𝑊)(𝑝‘(𝑡𝑤)) = 0)}
81 df-rab 3151 . . . . . 6 {𝑡 ∈ (ℕ0m (1...𝑁)) ∣ ∃𝑤 ∈ (ℕ0m 𝑊)(𝑝‘(𝑡𝑤)) = 0} = {𝑡 ∣ (𝑡 ∈ (ℕ0m (1...𝑁)) ∧ ∃𝑤 ∈ (ℕ0m 𝑊)(𝑝‘(𝑡𝑤)) = 0)}
8280, 81eqtr4i 2851 . . . . 5 {𝑡 ∣ ∃𝑢 ∈ (ℕ0m (𝑊 ∪ (1...𝑁)))(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝𝑢) = 0)} = {𝑡 ∈ (ℕ0m (1...𝑁)) ∣ ∃𝑤 ∈ (ℕ0m 𝑊)(𝑝‘(𝑡𝑤)) = 0}
8382eqeq2i 2837 . . . 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})
8483rexbii 3251 . . 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})
8513, 84syl6bb 288 . 2 (𝑁 ∈ ℕ0 → (𝑆 ∈ (Dioph‘𝑁) ↔ ∃𝑝 ∈ (mzPoly‘(𝑊 ∪ (1...𝑁)))𝑆 = {𝑡 ∈ (ℕ0m (1...𝑁)) ∣ ∃𝑤 ∈ (ℕ0m 𝑊)(𝑝‘(𝑡𝑤)) = 0}))
861, 85biadanii 819 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 207   ∧ wa 396   = wceq 1530   ∈ wcel 2106  {cab 2802  ∃wrex 3143  {crab 3146  Vcvv 3499   ∪ cun 3937   ∩ cin 3938   ⊆ wss 3939  ∅c0 4294   ↾ cres 5555   Fn wfn 6346  ⟶wf 6347  ‘cfv 6351  (class class class)co 7151   ↑m cmap 8399  Fincfn 8501  0cc0 10529  1c1 10530  ℕcn 11630  ℕ0cn0 11889  ...cfz 12885  mzPolycmzp 39181  Diophcdioph 39214 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 1904  ax-6 1963  ax-7 2008  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2152  ax-12 2167  ax-13 2385  ax-ext 2796  ax-rep 5186  ax-sep 5199  ax-nul 5206  ax-pow 5262  ax-pr 5325  ax-un 7454  ax-cnex 10585  ax-resscn 10586  ax-1cn 10587  ax-icn 10588  ax-addcl 10589  ax-addrcl 10590  ax-mulcl 10591  ax-mulrcl 10592  ax-mulcom 10593  ax-addass 10594  ax-mulass 10595  ax-distr 10596  ax-i2m1 10597  ax-1ne0 10598  ax-1rid 10599  ax-rnegex 10600  ax-rrecex 10601  ax-cnre 10602  ax-pre-lttri 10603  ax-pre-lttrn 10604  ax-pre-ltadd 10605  ax-pre-mulgt0 10606 This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2615  df-eu 2649  df-clab 2803  df-cleq 2817  df-clel 2897  df-nfc 2967  df-ne 3021  df-nel 3128  df-ral 3147  df-rex 3148  df-reu 3149  df-rmo 3150  df-rab 3151  df-v 3501  df-sbc 3776  df-csb 3887  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-pss 3957  df-nul 4295  df-if 4470  df-pw 4543  df-sn 4564  df-pr 4566  df-tp 4568  df-op 4570  df-uni 4837  df-int 4874  df-iun 4918  df-br 5063  df-opab 5125  df-mpt 5143  df-tr 5169  df-id 5458  df-eprel 5463  df-po 5472  df-so 5473  df-fr 5512  df-we 5514  df-xp 5559  df-rel 5560  df-cnv 5561  df-co 5562  df-dm 5563  df-rn 5564  df-res 5565  df-ima 5566  df-pred 6145  df-ord 6191  df-on 6192  df-lim 6193  df-suc 6194  df-iota 6311  df-fun 6353  df-fn 6354  df-f 6355  df-f1 6356  df-fo 6357  df-f1o 6358  df-fv 6359  df-riota 7109  df-ov 7154  df-oprab 7155  df-mpo 7156  df-of 7402  df-om 7572  df-1st 7683  df-2nd 7684  df-wrecs 7941  df-recs 8002  df-rdg 8040  df-1o 8096  df-oadd 8100  df-er 8282  df-map 8401  df-en 8502  df-dom 8503  df-sdom 8504  df-fin 8505  df-dju 9322  df-card 9360  df-pnf 10669  df-mnf 10670  df-xr 10671  df-ltxr 10672  df-le 10673  df-sub 10864  df-neg 10865  df-nn 11631  df-n0 11890  df-z 11974  df-uz 12236  df-fz 12886  df-hash 13684  df-mzpcl 39182  df-mzp 39183  df-dioph 39215 This theorem is referenced by:  eldioph4i  39271  diophren  39272
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