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Theorem eldioph4b 38218
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 38170 . 2 (𝑆 ∈ (Dioph‘𝑁) → 𝑁 ∈ ℕ0)
2 eldioph4b.a . . . . . 6 𝑊 ∈ V
3 ovex 6936 . . . . . 6 (1...𝑁) ∈ V
42, 3unex 7215 . . . . 5 (𝑊 ∪ (1...𝑁)) ∈ V
54jctr 522 . . . 4 (𝑁 ∈ ℕ0 → (𝑁 ∈ ℕ0 ∧ (𝑊 ∪ (1...𝑁)) ∈ V))
6 eldioph4b.b . . . . . . 7 ¬ 𝑊 ∈ Fin
76intnanr 483 . . . . . 6 ¬ (𝑊 ∈ Fin ∧ (1...𝑁) ∈ Fin)
8 unfir 8496 . . . . . 6 ((𝑊 ∪ (1...𝑁)) ∈ Fin → (𝑊 ∈ Fin ∧ (1...𝑁) ∈ Fin))
97, 8mto 189 . . . . 5 ¬ (𝑊 ∪ (1...𝑁)) ∈ Fin
10 ssun2 4003 . . . . 5 (1...𝑁) ⊆ (𝑊 ∪ (1...𝑁))
119, 10pm3.2i 464 . . . 4 (¬ (𝑊 ∪ (1...𝑁)) ∈ Fin ∧ (1...𝑁) ⊆ (𝑊 ∪ (1...𝑁)))
12 eldioph2b 38169 . . . 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 582 . . 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 8146 . . . . . . . . . . . . . . 15 ((𝑢 ∈ (ℕ0𝑚 (𝑊 ∪ (1...𝑁))) ∧ (1...𝑁) ⊆ (𝑊 ∪ (1...𝑁))) → (𝑢 ↾ (1...𝑁)) ∈ (ℕ0𝑚 (1...𝑁)))
1510, 14mpan2 684 . . . . . . . . . . . . . 14 (𝑢 ∈ (ℕ0𝑚 (𝑊 ∪ (1...𝑁))) → (𝑢 ↾ (1...𝑁)) ∈ (ℕ0𝑚 (1...𝑁)))
1615adantr 474 . . . . . . . . . . . . 13 ((𝑢 ∈ (ℕ0𝑚 (𝑊 ∪ (1...𝑁))) ∧ (𝑝𝑢) = 0) → (𝑢 ↾ (1...𝑁)) ∈ (ℕ0𝑚 (1...𝑁)))
17 ssun1 4002 . . . . . . . . . . . . . . . 16 𝑊 ⊆ (𝑊 ∪ (1...𝑁))
18 elmapssres 8146 . . . . . . . . . . . . . . . 16 ((𝑢 ∈ (ℕ0𝑚 (𝑊 ∪ (1...𝑁))) ∧ 𝑊 ⊆ (𝑊 ∪ (1...𝑁))) → (𝑢𝑊) ∈ (ℕ0𝑚 𝑊))
1917, 18mpan2 684 . . . . . . . . . . . . . . 15 (𝑢 ∈ (ℕ0𝑚 (𝑊 ∪ (1...𝑁))) → (𝑢𝑊) ∈ (ℕ0𝑚 𝑊))
2019adantr 474 . . . . . . . . . . . . . 14 ((𝑢 ∈ (ℕ0𝑚 (𝑊 ∪ (1...𝑁))) ∧ (𝑝𝑢) = 0) → (𝑢𝑊) ∈ (ℕ0𝑚 𝑊))
21 uncom 3983 . . . . . . . . . . . . . . . . . 18 ((𝑢 ↾ (1...𝑁)) ∪ (𝑢𝑊)) = ((𝑢𝑊) ∪ (𝑢 ↾ (1...𝑁)))
22 resundi 5646 . . . . . . . . . . . . . . . . . 18 (𝑢 ↾ (𝑊 ∪ (1...𝑁))) = ((𝑢𝑊) ∪ (𝑢 ↾ (1...𝑁)))
2321, 22eqtr4i 2851 . . . . . . . . . . . . . . . . 17 ((𝑢 ↾ (1...𝑁)) ∪ (𝑢𝑊)) = (𝑢 ↾ (𝑊 ∪ (1...𝑁)))
24 elmapi 8143 . . . . . . . . . . . . . . . . . 18 (𝑢 ∈ (ℕ0𝑚 (𝑊 ∪ (1...𝑁))) → 𝑢:(𝑊 ∪ (1...𝑁))⟶ℕ0)
25 ffn 6277 . . . . . . . . . . . . . . . . . 18 (𝑢:(𝑊 ∪ (1...𝑁))⟶ℕ0𝑢 Fn (𝑊 ∪ (1...𝑁)))
26 fnresdm 6232 . . . . . . . . . . . . . . . . . 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 6440 . . . . . . . . . . . . . . 15 (𝑢 ∈ (ℕ0𝑚 (𝑊 ∪ (1...𝑁))) → ((𝑝‘((𝑢 ↾ (1...𝑁)) ∪ (𝑢𝑊))) = 0 ↔ (𝑝𝑢) = 0))
3029biimpar 471 . . . . . . . . . . . . . 14 ((𝑢 ∈ (ℕ0𝑚 (𝑊 ∪ (1...𝑁))) ∧ (𝑝𝑢) = 0) → (𝑝‘((𝑢 ↾ (1...𝑁)) ∪ (𝑢𝑊))) = 0)
31 uneq2 3987 . . . . . . . . . . . . . . . 16 (𝑤 = (𝑢𝑊) → ((𝑢 ↾ (1...𝑁)) ∪ 𝑤) = ((𝑢 ↾ (1...𝑁)) ∪ (𝑢𝑊)))
3231fveqeq2d 6440 . . . . . . . . . . . . . . 15 (𝑤 = (𝑢𝑊) → ((𝑝‘((𝑢 ↾ (1...𝑁)) ∪ 𝑤)) = 0 ↔ (𝑝‘((𝑢 ↾ (1...𝑁)) ∪ (𝑢𝑊))) = 0))
3332rspcev 3525 . . . . . . . . . . . . . 14 (((𝑢𝑊) ∈ (ℕ0𝑚 𝑊) ∧ (𝑝‘((𝑢 ↾ (1...𝑁)) ∪ (𝑢𝑊))) = 0) → ∃𝑤 ∈ (ℕ0𝑚 𝑊)(𝑝‘((𝑢 ↾ (1...𝑁)) ∪ 𝑤)) = 0)
3420, 30, 33syl2anc 581 . . . . . . . . . . . . 13 ((𝑢 ∈ (ℕ0𝑚 (𝑊 ∪ (1...𝑁))) ∧ (𝑝𝑢) = 0) → ∃𝑤 ∈ (ℕ0𝑚 𝑊)(𝑝‘((𝑢 ↾ (1...𝑁)) ∪ 𝑤)) = 0)
3516, 34jca 509 . . . . . . . . . . . 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 2893 . . . . . . . . . . . . 13 (𝑡 = (𝑢 ↾ (1...𝑁)) → (𝑡 ∈ (ℕ0𝑚 (1...𝑁)) ↔ (𝑢 ↾ (1...𝑁)) ∈ (ℕ0𝑚 (1...𝑁))))
37 uneq1 3986 . . . . . . . . . . . . . . 15 (𝑡 = (𝑢 ↾ (1...𝑁)) → (𝑡𝑤) = ((𝑢 ↾ (1...𝑁)) ∪ 𝑤))
3837fveqeq2d 6440 . . . . . . . . . . . . . 14 (𝑡 = (𝑢 ↾ (1...𝑁)) → ((𝑝‘(𝑡𝑤)) = 0 ↔ (𝑝‘((𝑢 ↾ (1...𝑁)) ∪ 𝑤)) = 0))
3938rexbidv 3261 . . . . . . . . . . . . 13 (𝑡 = (𝑢 ↾ (1...𝑁)) → (∃𝑤 ∈ (ℕ0𝑚 𝑊)(𝑝‘(𝑡𝑤)) = 0 ↔ ∃𝑤 ∈ (ℕ0𝑚 𝑊)(𝑝‘((𝑢 ↾ (1...𝑁)) ∪ 𝑤)) = 0))
4036, 39anbi12d 626 . . . . . . . . . . . 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 239 . . . . . . . . . . 11 ((𝑢 ∈ (ℕ0𝑚 (𝑊 ∪ (1...𝑁))) ∧ (𝑝𝑢) = 0) → (𝑡 = (𝑢 ↾ (1...𝑁)) → (𝑡 ∈ (ℕ0𝑚 (1...𝑁)) ∧ ∃𝑤 ∈ (ℕ0𝑚 𝑊)(𝑝‘(𝑡𝑤)) = 0)))
4241expimpd 447 . . . . . . . . . 10 (𝑢 ∈ (ℕ0𝑚 (𝑊 ∪ (1...𝑁))) → (((𝑝𝑢) = 0 ∧ 𝑡 = (𝑢 ↾ (1...𝑁))) → (𝑡 ∈ (ℕ0𝑚 (1...𝑁)) ∧ ∃𝑤 ∈ (ℕ0𝑚 𝑊)(𝑝‘(𝑡𝑤)) = 0)))
4342ancomsd 459 . . . . . . . . 9 (𝑢 ∈ (ℕ0𝑚 (𝑊 ∪ (1...𝑁))) → ((𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝𝑢) = 0) → (𝑡 ∈ (ℕ0𝑚 (1...𝑁)) ∧ ∃𝑤 ∈ (ℕ0𝑚 𝑊)(𝑝‘(𝑡𝑤)) = 0)))
4443rexlimiv 3235 . . . . . . . 8 (∃𝑢 ∈ (ℕ0𝑚 (𝑊 ∪ (1...𝑁)))(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝𝑢) = 0) → (𝑡 ∈ (ℕ0𝑚 (1...𝑁)) ∧ ∃𝑤 ∈ (ℕ0𝑚 𝑊)(𝑝‘(𝑡𝑤)) = 0))
45 uncom 3983 . . . . . . . . . . . 12 (𝑡𝑤) = (𝑤𝑡)
46 fz1ssnn 12664 . . . . . . . . . . . . . . . . . . . 20 (1...𝑁) ⊆ ℕ
47 sslin 4062 . . . . . . . . . . . . . . . . . . . 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 3861 . . . . . . . . . . . . . . . . . 18 (𝑊 ∩ (1...𝑁)) ⊆ ∅
51 ss0 4198 . . . . . . . . . . . . . . . . . 18 ((𝑊 ∩ (1...𝑁)) ⊆ ∅ → (𝑊 ∩ (1...𝑁)) = ∅)
5250, 51ax-mp 5 . . . . . . . . . . . . . . . . 17 (𝑊 ∩ (1...𝑁)) = ∅
5352reseq2i 5625 . . . . . . . . . . . . . . . 16 (𝑤 ↾ (𝑊 ∩ (1...𝑁))) = (𝑤 ↾ ∅)
54 res0 5632 . . . . . . . . . . . . . . . 16 (𝑤 ↾ ∅) = ∅
5553, 54eqtri 2848 . . . . . . . . . . . . . . 15 (𝑤 ↾ (𝑊 ∩ (1...𝑁))) = ∅
5652reseq2i 5625 . . . . . . . . . . . . . . . 16 (𝑡 ↾ (𝑊 ∩ (1...𝑁))) = (𝑡 ↾ ∅)
57 res0 5632 . . . . . . . . . . . . . . . 16 (𝑡 ↾ ∅) = ∅
5856, 57eqtri 2848 . . . . . . . . . . . . . . 15 (𝑡 ↾ (𝑊 ∩ (1...𝑁))) = ∅
5955, 58eqtr4i 2851 . . . . . . . . . . . . . 14 (𝑤 ↾ (𝑊 ∩ (1...𝑁))) = (𝑡 ↾ (𝑊 ∩ (1...𝑁)))
60 elmapresaun 38177 . . . . . . . . . . . . . 14 ((𝑤 ∈ (ℕ0𝑚 𝑊) ∧ 𝑡 ∈ (ℕ0𝑚 (1...𝑁)) ∧ (𝑤 ↾ (𝑊 ∩ (1...𝑁))) = (𝑡 ↾ (𝑊 ∩ (1...𝑁)))) → (𝑤𝑡) ∈ (ℕ0𝑚 (𝑊 ∪ (1...𝑁))))
6159, 60mp3an3 1580 . . . . . . . . . . . . 13 ((𝑤 ∈ (ℕ0𝑚 𝑊) ∧ 𝑡 ∈ (ℕ0𝑚 (1...𝑁))) → (𝑤𝑡) ∈ (ℕ0𝑚 (𝑊 ∪ (1...𝑁))))
6261ancoms 452 . . . . . . . . . . . 12 ((𝑡 ∈ (ℕ0𝑚 (1...𝑁)) ∧ 𝑤 ∈ (ℕ0𝑚 𝑊)) → (𝑤𝑡) ∈ (ℕ0𝑚 (𝑊 ∪ (1...𝑁))))
6345, 62syl5eqel 2909 . . . . . . . . . . 11 ((𝑡 ∈ (ℕ0𝑚 (1...𝑁)) ∧ 𝑤 ∈ (ℕ0𝑚 𝑊)) → (𝑡𝑤) ∈ (ℕ0𝑚 (𝑊 ∪ (1...𝑁))))
6463adantr 474 . . . . . . . . . 10 (((𝑡 ∈ (ℕ0𝑚 (1...𝑁)) ∧ 𝑤 ∈ (ℕ0𝑚 𝑊)) ∧ (𝑝‘(𝑡𝑤)) = 0) → (𝑡𝑤) ∈ (ℕ0𝑚 (𝑊 ∪ (1...𝑁))))
6545reseq1i 5624 . . . . . . . . . . . 12 ((𝑡𝑤) ↾ (1...𝑁)) = ((𝑤𝑡) ↾ (1...𝑁))
66 elmapresaunres2 38178 . . . . . . . . . . . . . 14 ((𝑤 ∈ (ℕ0𝑚 𝑊) ∧ 𝑡 ∈ (ℕ0𝑚 (1...𝑁)) ∧ (𝑤 ↾ (𝑊 ∩ (1...𝑁))) = (𝑡 ↾ (𝑊 ∩ (1...𝑁)))) → ((𝑤𝑡) ↾ (1...𝑁)) = 𝑡)
6759, 66mp3an3 1580 . . . . . . . . . . . . 13 ((𝑤 ∈ (ℕ0𝑚 𝑊) ∧ 𝑡 ∈ (ℕ0𝑚 (1...𝑁))) → ((𝑤𝑡) ↾ (1...𝑁)) = 𝑡)
6867ancoms 452 . . . . . . . . . . . 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 474 . . . . . . . . . 10 (((𝑡 ∈ (ℕ0𝑚 (1...𝑁)) ∧ 𝑤 ∈ (ℕ0𝑚 𝑊)) ∧ (𝑝‘(𝑡𝑤)) = 0) → 𝑡 = ((𝑡𝑤) ↾ (1...𝑁)))
71 simpr 479 . . . . . . . . . 10 (((𝑡 ∈ (ℕ0𝑚 (1...𝑁)) ∧ 𝑤 ∈ (ℕ0𝑚 𝑊)) ∧ (𝑝‘(𝑡𝑤)) = 0) → (𝑝‘(𝑡𝑤)) = 0)
72 reseq1 5622 . . . . . . . . . . . . 13 (𝑢 = (𝑡𝑤) → (𝑢 ↾ (1...𝑁)) = ((𝑡𝑤) ↾ (1...𝑁)))
7372eqeq2d 2834 . . . . . . . . . . . 12 (𝑢 = (𝑡𝑤) → (𝑡 = (𝑢 ↾ (1...𝑁)) ↔ 𝑡 = ((𝑡𝑤) ↾ (1...𝑁))))
74 fveqeq2 6441 . . . . . . . . . . . 12 (𝑢 = (𝑡𝑤) → ((𝑝𝑢) = 0 ↔ (𝑝‘(𝑡𝑤)) = 0))
7573, 74anbi12d 626 . . . . . . . . . . 11 (𝑢 = (𝑡𝑤) → ((𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝𝑢) = 0) ↔ (𝑡 = ((𝑡𝑤) ↾ (1...𝑁)) ∧ (𝑝‘(𝑡𝑤)) = 0)))
7675rspcev 3525 . . . . . . . . . 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 872 . . . . . . . . 9 (((𝑡 ∈ (ℕ0𝑚 (1...𝑁)) ∧ 𝑤 ∈ (ℕ0𝑚 𝑊)) ∧ (𝑝‘(𝑡𝑤)) = 0) → ∃𝑢 ∈ (ℕ0𝑚 (𝑊 ∪ (1...𝑁)))(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝𝑢) = 0))
7877r19.29an 3286 . . . . . . . 8 ((𝑡 ∈ (ℕ0𝑚 (1...𝑁)) ∧ ∃𝑤 ∈ (ℕ0𝑚 𝑊)(𝑝‘(𝑡𝑤)) = 0) → ∃𝑢 ∈ (ℕ0𝑚 (𝑊 ∪ (1...𝑁)))(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝𝑢) = 0))
7944, 78impbii 201 . . . . . . 7 (∃𝑢 ∈ (ℕ0𝑚 (𝑊 ∪ (1...𝑁)))(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝𝑢) = 0) ↔ (𝑡 ∈ (ℕ0𝑚 (1...𝑁)) ∧ ∃𝑤 ∈ (ℕ0𝑚 𝑊)(𝑝‘(𝑡𝑤)) = 0))
8079abbii 2943 . . . . . 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 3125 . . . . . 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 2836 . . . 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 3250 . . 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 279 . 2 (𝑁 ∈ ℕ0 → (𝑆 ∈ (Dioph‘𝑁) ↔ ∃𝑝 ∈ (mzPoly‘(𝑊 ∪ (1...𝑁)))𝑆 = {𝑡 ∈ (ℕ0𝑚 (1...𝑁)) ∣ ∃𝑤 ∈ (ℕ0𝑚 𝑊)(𝑝‘(𝑡𝑤)) = 0}))
861, 85biadanii 859 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 198  wa 386   = wceq 1658  wcel 2166  {cab 2810  wrex 3117  {crab 3120  Vcvv 3413  cun 3795  cin 3796  wss 3797  c0 4143  cres 5343   Fn wfn 6117  wf 6118  cfv 6122  (class class class)co 6904  𝑚 cmap 8121  Fincfn 8221  0cc0 10251  1c1 10252  cn 11349  0cn0 11617  ...cfz 12618  mzPolycmzp 38128  Diophcdioph 38161
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-8 2168  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2390  ax-ext 2802  ax-rep 4993  ax-sep 5004  ax-nul 5012  ax-pow 5064  ax-pr 5126  ax-un 7208  ax-cnex 10307  ax-resscn 10308  ax-1cn 10309  ax-icn 10310  ax-addcl 10311  ax-addrcl 10312  ax-mulcl 10313  ax-mulrcl 10314  ax-mulcom 10315  ax-addass 10316  ax-mulass 10317  ax-distr 10318  ax-i2m1 10319  ax-1ne0 10320  ax-1rid 10321  ax-rnegex 10322  ax-rrecex 10323  ax-cnre 10324  ax-pre-lttri 10325  ax-pre-lttrn 10326  ax-pre-ltadd 10327  ax-pre-mulgt0 10328
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3or 1114  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2604  df-eu 2639  df-clab 2811  df-cleq 2817  df-clel 2820  df-nfc 2957  df-ne 2999  df-nel 3102  df-ral 3121  df-rex 3122  df-reu 3123  df-rmo 3124  df-rab 3125  df-v 3415  df-sbc 3662  df-csb 3757  df-dif 3800  df-un 3802  df-in 3804  df-ss 3811  df-pss 3813  df-nul 4144  df-if 4306  df-pw 4379  df-sn 4397  df-pr 4399  df-tp 4401  df-op 4403  df-uni 4658  df-int 4697  df-iun 4741  df-br 4873  df-opab 4935  df-mpt 4952  df-tr 4975  df-id 5249  df-eprel 5254  df-po 5262  df-so 5263  df-fr 5300  df-we 5302  df-xp 5347  df-rel 5348  df-cnv 5349  df-co 5350  df-dm 5351  df-rn 5352  df-res 5353  df-ima 5354  df-pred 5919  df-ord 5965  df-on 5966  df-lim 5967  df-suc 5968  df-iota 6085  df-fun 6124  df-fn 6125  df-f 6126  df-f1 6127  df-fo 6128  df-f1o 6129  df-fv 6130  df-riota 6865  df-ov 6907  df-oprab 6908  df-mpt2 6909  df-of 7156  df-om 7326  df-1st 7427  df-2nd 7428  df-wrecs 7671  df-recs 7733  df-rdg 7771  df-1o 7825  df-oadd 7829  df-er 8008  df-map 8123  df-en 8222  df-dom 8223  df-sdom 8224  df-fin 8225  df-card 9077  df-cda 9304  df-pnf 10392  df-mnf 10393  df-xr 10394  df-ltxr 10395  df-le 10396  df-sub 10586  df-neg 10587  df-nn 11350  df-n0 11618  df-z 11704  df-uz 11968  df-fz 12619  df-hash 13410  df-mzpcl 38129  df-mzp 38130  df-dioph 38162
This theorem is referenced by:  eldioph4i  38219  diophren  38220
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