Users' Mathboxes Mathbox for Stefan O'Rear < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  eldioph2 Structured version   Visualization version   GIF version

Theorem eldioph2 42749
Description: Construct a Diophantine set from a polynomial with witness variables drawn from any set whatsoever, via mzpcompact2 42739. (Contributed by Stefan O'Rear, 8-Oct-2014.) (Revised by Stefan O'Rear, 5-Jun-2015.)
Assertion
Ref Expression
eldioph2 ((𝑁 ∈ ℕ0 ∧ (𝑆 ∈ V ∧ (1...𝑁) ⊆ 𝑆) ∧ 𝑃 ∈ (mzPoly‘𝑆)) → {𝑡 ∣ ∃𝑢 ∈ (ℕ0m 𝑆)(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑃𝑢) = 0)} ∈ (Dioph‘𝑁))
Distinct variable groups:   𝑡,𝑃,𝑢   𝑡,𝑆,𝑢   𝑡,𝑁,𝑢

Proof of Theorem eldioph2
Dummy variables 𝑎 𝑏 𝑐 𝑒 𝑔 𝑑 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 mzpcompact2 42739 . . 3 (𝑃 ∈ (mzPoly‘𝑆) → ∃𝑎 ∈ Fin ∃𝑏 ∈ (mzPoly‘𝑎)(𝑎𝑆𝑃 = (𝑒 ∈ (ℤ ↑m 𝑆) ↦ (𝑏‘(𝑒𝑎)))))
213ad2ant3 1134 . 2 ((𝑁 ∈ ℕ0 ∧ (𝑆 ∈ V ∧ (1...𝑁) ⊆ 𝑆) ∧ 𝑃 ∈ (mzPoly‘𝑆)) → ∃𝑎 ∈ Fin ∃𝑏 ∈ (mzPoly‘𝑎)(𝑎𝑆𝑃 = (𝑒 ∈ (ℤ ↑m 𝑆) ↦ (𝑏‘(𝑒𝑎)))))
3 fveq1 6905 . . . . . . . . . 10 (𝑃 = (𝑒 ∈ (ℤ ↑m 𝑆) ↦ (𝑏‘(𝑒𝑎))) → (𝑃𝑢) = ((𝑒 ∈ (ℤ ↑m 𝑆) ↦ (𝑏‘(𝑒𝑎)))‘𝑢))
43eqeq1d 2736 . . . . . . . . 9 (𝑃 = (𝑒 ∈ (ℤ ↑m 𝑆) ↦ (𝑏‘(𝑒𝑎))) → ((𝑃𝑢) = 0 ↔ ((𝑒 ∈ (ℤ ↑m 𝑆) ↦ (𝑏‘(𝑒𝑎)))‘𝑢) = 0))
54anbi2d 630 . . . . . . . 8 (𝑃 = (𝑒 ∈ (ℤ ↑m 𝑆) ↦ (𝑏‘(𝑒𝑎))) → ((𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑃𝑢) = 0) ↔ (𝑡 = (𝑢 ↾ (1...𝑁)) ∧ ((𝑒 ∈ (ℤ ↑m 𝑆) ↦ (𝑏‘(𝑒𝑎)))‘𝑢) = 0)))
65rexbidv 3176 . . . . . . 7 (𝑃 = (𝑒 ∈ (ℤ ↑m 𝑆) ↦ (𝑏‘(𝑒𝑎))) → (∃𝑢 ∈ (ℕ0m 𝑆)(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑃𝑢) = 0) ↔ ∃𝑢 ∈ (ℕ0m 𝑆)(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ ((𝑒 ∈ (ℤ ↑m 𝑆) ↦ (𝑏‘(𝑒𝑎)))‘𝑢) = 0)))
76abbidv 2805 . . . . . 6 (𝑃 = (𝑒 ∈ (ℤ ↑m 𝑆) ↦ (𝑏‘(𝑒𝑎))) → {𝑡 ∣ ∃𝑢 ∈ (ℕ0m 𝑆)(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑃𝑢) = 0)} = {𝑡 ∣ ∃𝑢 ∈ (ℕ0m 𝑆)(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ ((𝑒 ∈ (ℤ ↑m 𝑆) ↦ (𝑏‘(𝑒𝑎)))‘𝑢) = 0)})
87ad2antll 729 . . . . 5 ((((𝑁 ∈ ℕ0 ∧ (𝑆 ∈ V ∧ (1...𝑁) ⊆ 𝑆) ∧ 𝑃 ∈ (mzPoly‘𝑆)) ∧ (𝑎 ∈ Fin ∧ 𝑏 ∈ (mzPoly‘𝑎))) ∧ (𝑎𝑆𝑃 = (𝑒 ∈ (ℤ ↑m 𝑆) ↦ (𝑏‘(𝑒𝑎))))) → {𝑡 ∣ ∃𝑢 ∈ (ℕ0m 𝑆)(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑃𝑢) = 0)} = {𝑡 ∣ ∃𝑢 ∈ (ℕ0m 𝑆)(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ ((𝑒 ∈ (ℤ ↑m 𝑆) ↦ (𝑏‘(𝑒𝑎)))‘𝑢) = 0)})
9 simplll 775 . . . . . . . . . . 11 ((((𝑁 ∈ ℕ0 ∧ (𝑆 ∈ V ∧ (1...𝑁) ⊆ 𝑆)) ∧ (𝑎 ∈ Fin ∧ 𝑏 ∈ (mzPoly‘𝑎))) ∧ 𝑎𝑆) → 𝑁 ∈ ℕ0)
10 simplrl 777 . . . . . . . . . . . 12 ((((𝑁 ∈ ℕ0 ∧ (𝑆 ∈ V ∧ (1...𝑁) ⊆ 𝑆)) ∧ (𝑎 ∈ Fin ∧ 𝑏 ∈ (mzPoly‘𝑎))) ∧ 𝑎𝑆) → 𝑎 ∈ Fin)
11 fzfi 14009 . . . . . . . . . . . 12 (1...𝑁) ∈ Fin
12 unfi 9209 . . . . . . . . . . . 12 ((𝑎 ∈ Fin ∧ (1...𝑁) ∈ Fin) → (𝑎 ∪ (1...𝑁)) ∈ Fin)
1310, 11, 12sylancl 586 . . . . . . . . . . 11 ((((𝑁 ∈ ℕ0 ∧ (𝑆 ∈ V ∧ (1...𝑁) ⊆ 𝑆)) ∧ (𝑎 ∈ Fin ∧ 𝑏 ∈ (mzPoly‘𝑎))) ∧ 𝑎𝑆) → (𝑎 ∪ (1...𝑁)) ∈ Fin)
14 ssun2 4188 . . . . . . . . . . . 12 (1...𝑁) ⊆ (𝑎 ∪ (1...𝑁))
1514a1i 11 . . . . . . . . . . 11 ((((𝑁 ∈ ℕ0 ∧ (𝑆 ∈ V ∧ (1...𝑁) ⊆ 𝑆)) ∧ (𝑎 ∈ Fin ∧ 𝑏 ∈ (mzPoly‘𝑎))) ∧ 𝑎𝑆) → (1...𝑁) ⊆ (𝑎 ∪ (1...𝑁)))
16 eldioph2lem1 42747 . . . . . . . . . . 11 ((𝑁 ∈ ℕ0 ∧ (𝑎 ∪ (1...𝑁)) ∈ Fin ∧ (1...𝑁) ⊆ (𝑎 ∪ (1...𝑁))) → ∃𝑐 ∈ (ℤ𝑁)∃𝑑 ∈ V (𝑑:(1...𝑐)–1-1-onto→(𝑎 ∪ (1...𝑁)) ∧ (𝑑 ↾ (1...𝑁)) = ( I ↾ (1...𝑁))))
179, 13, 15, 16syl3anc 1370 . . . . . . . . . 10 ((((𝑁 ∈ ℕ0 ∧ (𝑆 ∈ V ∧ (1...𝑁) ⊆ 𝑆)) ∧ (𝑎 ∈ Fin ∧ 𝑏 ∈ (mzPoly‘𝑎))) ∧ 𝑎𝑆) → ∃𝑐 ∈ (ℤ𝑁)∃𝑑 ∈ V (𝑑:(1...𝑐)–1-1-onto→(𝑎 ∪ (1...𝑁)) ∧ (𝑑 ↾ (1...𝑁)) = ( I ↾ (1...𝑁))))
18 f1ococnv2 6875 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑑:(1...𝑐)–1-1-onto→(𝑎 ∪ (1...𝑁)) → (𝑑𝑑) = ( I ↾ (𝑎 ∪ (1...𝑁))))
1918ad2antrl 728 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((((((𝑁 ∈ ℕ0 ∧ (𝑆 ∈ V ∧ (1...𝑁) ⊆ 𝑆)) ∧ (𝑎 ∈ Fin ∧ 𝑏 ∈ (mzPoly‘𝑎))) ∧ 𝑎𝑆) ∧ (𝑐 ∈ (ℤ𝑁) ∧ 𝑑 ∈ V)) ∧ (𝑑:(1...𝑐)–1-1-onto→(𝑎 ∪ (1...𝑁)) ∧ (𝑑 ↾ (1...𝑁)) = ( I ↾ (1...𝑁)))) → (𝑑𝑑) = ( I ↾ (𝑎 ∪ (1...𝑁))))
2019reseq1d 5998 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((((((𝑁 ∈ ℕ0 ∧ (𝑆 ∈ V ∧ (1...𝑁) ⊆ 𝑆)) ∧ (𝑎 ∈ Fin ∧ 𝑏 ∈ (mzPoly‘𝑎))) ∧ 𝑎𝑆) ∧ (𝑐 ∈ (ℤ𝑁) ∧ 𝑑 ∈ V)) ∧ (𝑑:(1...𝑐)–1-1-onto→(𝑎 ∪ (1...𝑁)) ∧ (𝑑 ↾ (1...𝑁)) = ( I ↾ (1...𝑁)))) → ((𝑑𝑑) ↾ 𝑎) = (( I ↾ (𝑎 ∪ (1...𝑁))) ↾ 𝑎))
21 ssun1 4187 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 𝑎 ⊆ (𝑎 ∪ (1...𝑁))
22 resabs1 6026 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑎 ⊆ (𝑎 ∪ (1...𝑁)) → (( I ↾ (𝑎 ∪ (1...𝑁))) ↾ 𝑎) = ( I ↾ 𝑎))
2321, 22ax-mp 5 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (( I ↾ (𝑎 ∪ (1...𝑁))) ↾ 𝑎) = ( I ↾ 𝑎)
2420, 23eqtr2di 2791 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((((((𝑁 ∈ ℕ0 ∧ (𝑆 ∈ V ∧ (1...𝑁) ⊆ 𝑆)) ∧ (𝑎 ∈ Fin ∧ 𝑏 ∈ (mzPoly‘𝑎))) ∧ 𝑎𝑆) ∧ (𝑐 ∈ (ℤ𝑁) ∧ 𝑑 ∈ V)) ∧ (𝑑:(1...𝑐)–1-1-onto→(𝑎 ∪ (1...𝑁)) ∧ (𝑑 ↾ (1...𝑁)) = ( I ↾ (1...𝑁)))) → ( I ↾ 𝑎) = ((𝑑𝑑) ↾ 𝑎))
25 resco 6271 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑑𝑑) ↾ 𝑎) = (𝑑 ∘ (𝑑𝑎))
2624, 25eqtrdi 2790 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((((𝑁 ∈ ℕ0 ∧ (𝑆 ∈ V ∧ (1...𝑁) ⊆ 𝑆)) ∧ (𝑎 ∈ Fin ∧ 𝑏 ∈ (mzPoly‘𝑎))) ∧ 𝑎𝑆) ∧ (𝑐 ∈ (ℤ𝑁) ∧ 𝑑 ∈ V)) ∧ (𝑑:(1...𝑐)–1-1-onto→(𝑎 ∪ (1...𝑁)) ∧ (𝑑 ↾ (1...𝑁)) = ( I ↾ (1...𝑁)))) → ( I ↾ 𝑎) = (𝑑 ∘ (𝑑𝑎)))
2726adantr 480 . . . . . . . . . . . . . . . . . . . . . . . 24 (((((((𝑁 ∈ ℕ0 ∧ (𝑆 ∈ V ∧ (1...𝑁) ⊆ 𝑆)) ∧ (𝑎 ∈ Fin ∧ 𝑏 ∈ (mzPoly‘𝑎))) ∧ 𝑎𝑆) ∧ (𝑐 ∈ (ℤ𝑁) ∧ 𝑑 ∈ V)) ∧ (𝑑:(1...𝑐)–1-1-onto→(𝑎 ∪ (1...𝑁)) ∧ (𝑑 ↾ (1...𝑁)) = ( I ↾ (1...𝑁)))) ∧ 𝑒 ∈ (ℤ ↑m 𝑆)) → ( I ↾ 𝑎) = (𝑑 ∘ (𝑑𝑎)))
2827coeq2d 5875 . . . . . . . . . . . . . . . . . . . . . . 23 (((((((𝑁 ∈ ℕ0 ∧ (𝑆 ∈ V ∧ (1...𝑁) ⊆ 𝑆)) ∧ (𝑎 ∈ Fin ∧ 𝑏 ∈ (mzPoly‘𝑎))) ∧ 𝑎𝑆) ∧ (𝑐 ∈ (ℤ𝑁) ∧ 𝑑 ∈ V)) ∧ (𝑑:(1...𝑐)–1-1-onto→(𝑎 ∪ (1...𝑁)) ∧ (𝑑 ↾ (1...𝑁)) = ( I ↾ (1...𝑁)))) ∧ 𝑒 ∈ (ℤ ↑m 𝑆)) → (𝑒 ∘ ( I ↾ 𝑎)) = (𝑒 ∘ (𝑑 ∘ (𝑑𝑎))))
29 coires1 6285 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑒 ∘ ( I ↾ 𝑎)) = (𝑒𝑎)
30 coass 6286 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑒𝑑) ∘ (𝑑𝑎)) = (𝑒 ∘ (𝑑 ∘ (𝑑𝑎)))
3130eqcomi 2743 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑒 ∘ (𝑑 ∘ (𝑑𝑎))) = ((𝑒𝑑) ∘ (𝑑𝑎))
3228, 29, 313eqtr3g 2797 . . . . . . . . . . . . . . . . . . . . . 22 (((((((𝑁 ∈ ℕ0 ∧ (𝑆 ∈ V ∧ (1...𝑁) ⊆ 𝑆)) ∧ (𝑎 ∈ Fin ∧ 𝑏 ∈ (mzPoly‘𝑎))) ∧ 𝑎𝑆) ∧ (𝑐 ∈ (ℤ𝑁) ∧ 𝑑 ∈ V)) ∧ (𝑑:(1...𝑐)–1-1-onto→(𝑎 ∪ (1...𝑁)) ∧ (𝑑 ↾ (1...𝑁)) = ( I ↾ (1...𝑁)))) ∧ 𝑒 ∈ (ℤ ↑m 𝑆)) → (𝑒𝑎) = ((𝑒𝑑) ∘ (𝑑𝑎)))
3332fveq2d 6910 . . . . . . . . . . . . . . . . . . . . 21 (((((((𝑁 ∈ ℕ0 ∧ (𝑆 ∈ V ∧ (1...𝑁) ⊆ 𝑆)) ∧ (𝑎 ∈ Fin ∧ 𝑏 ∈ (mzPoly‘𝑎))) ∧ 𝑎𝑆) ∧ (𝑐 ∈ (ℤ𝑁) ∧ 𝑑 ∈ V)) ∧ (𝑑:(1...𝑐)–1-1-onto→(𝑎 ∪ (1...𝑁)) ∧ (𝑑 ↾ (1...𝑁)) = ( I ↾ (1...𝑁)))) ∧ 𝑒 ∈ (ℤ ↑m 𝑆)) → (𝑏‘(𝑒𝑎)) = (𝑏‘((𝑒𝑑) ∘ (𝑑𝑎))))
34 ovexd 7465 . . . . . . . . . . . . . . . . . . . . . . 23 (((((((𝑁 ∈ ℕ0 ∧ (𝑆 ∈ V ∧ (1...𝑁) ⊆ 𝑆)) ∧ (𝑎 ∈ Fin ∧ 𝑏 ∈ (mzPoly‘𝑎))) ∧ 𝑎𝑆) ∧ (𝑐 ∈ (ℤ𝑁) ∧ 𝑑 ∈ V)) ∧ (𝑑:(1...𝑐)–1-1-onto→(𝑎 ∪ (1...𝑁)) ∧ (𝑑 ↾ (1...𝑁)) = ( I ↾ (1...𝑁)))) ∧ 𝑒 ∈ (ℤ ↑m 𝑆)) → (1...𝑐) ∈ V)
35 simpr 484 . . . . . . . . . . . . . . . . . . . . . . 23 (((((((𝑁 ∈ ℕ0 ∧ (𝑆 ∈ V ∧ (1...𝑁) ⊆ 𝑆)) ∧ (𝑎 ∈ Fin ∧ 𝑏 ∈ (mzPoly‘𝑎))) ∧ 𝑎𝑆) ∧ (𝑐 ∈ (ℤ𝑁) ∧ 𝑑 ∈ V)) ∧ (𝑑:(1...𝑐)–1-1-onto→(𝑎 ∪ (1...𝑁)) ∧ (𝑑 ↾ (1...𝑁)) = ( I ↾ (1...𝑁)))) ∧ 𝑒 ∈ (ℤ ↑m 𝑆)) → 𝑒 ∈ (ℤ ↑m 𝑆))
36 f1of1 6847 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑑:(1...𝑐)–1-1-onto→(𝑎 ∪ (1...𝑁)) → 𝑑:(1...𝑐)–1-1→(𝑎 ∪ (1...𝑁)))
3736ad2antrl 728 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((((((𝑁 ∈ ℕ0 ∧ (𝑆 ∈ V ∧ (1...𝑁) ⊆ 𝑆)) ∧ (𝑎 ∈ Fin ∧ 𝑏 ∈ (mzPoly‘𝑎))) ∧ 𝑎𝑆) ∧ (𝑐 ∈ (ℤ𝑁) ∧ 𝑑 ∈ V)) ∧ (𝑑:(1...𝑐)–1-1-onto→(𝑎 ∪ (1...𝑁)) ∧ (𝑑 ↾ (1...𝑁)) = ( I ↾ (1...𝑁)))) → 𝑑:(1...𝑐)–1-1→(𝑎 ∪ (1...𝑁)))
38 simpr 484 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((((𝑁 ∈ ℕ0 ∧ (𝑆 ∈ V ∧ (1...𝑁) ⊆ 𝑆)) ∧ (𝑎 ∈ Fin ∧ 𝑏 ∈ (mzPoly‘𝑎))) ∧ 𝑎𝑆) → 𝑎𝑆)
39 simprr 773 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝑁 ∈ ℕ0 ∧ (𝑆 ∈ V ∧ (1...𝑁) ⊆ 𝑆)) → (1...𝑁) ⊆ 𝑆)
4039ad2antrr 726 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((((𝑁 ∈ ℕ0 ∧ (𝑆 ∈ V ∧ (1...𝑁) ⊆ 𝑆)) ∧ (𝑎 ∈ Fin ∧ 𝑏 ∈ (mzPoly‘𝑎))) ∧ 𝑎𝑆) → (1...𝑁) ⊆ 𝑆)
4138, 40unssd 4201 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((((𝑁 ∈ ℕ0 ∧ (𝑆 ∈ V ∧ (1...𝑁) ⊆ 𝑆)) ∧ (𝑎 ∈ Fin ∧ 𝑏 ∈ (mzPoly‘𝑎))) ∧ 𝑎𝑆) → (𝑎 ∪ (1...𝑁)) ⊆ 𝑆)
4241ad2antrr 726 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((((((𝑁 ∈ ℕ0 ∧ (𝑆 ∈ V ∧ (1...𝑁) ⊆ 𝑆)) ∧ (𝑎 ∈ Fin ∧ 𝑏 ∈ (mzPoly‘𝑎))) ∧ 𝑎𝑆) ∧ (𝑐 ∈ (ℤ𝑁) ∧ 𝑑 ∈ V)) ∧ (𝑑:(1...𝑐)–1-1-onto→(𝑎 ∪ (1...𝑁)) ∧ (𝑑 ↾ (1...𝑁)) = ( I ↾ (1...𝑁)))) → (𝑎 ∪ (1...𝑁)) ⊆ 𝑆)
43 f1ss 6809 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑑:(1...𝑐)–1-1→(𝑎 ∪ (1...𝑁)) ∧ (𝑎 ∪ (1...𝑁)) ⊆ 𝑆) → 𝑑:(1...𝑐)–1-1𝑆)
4437, 42, 43syl2anc 584 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((((𝑁 ∈ ℕ0 ∧ (𝑆 ∈ V ∧ (1...𝑁) ⊆ 𝑆)) ∧ (𝑎 ∈ Fin ∧ 𝑏 ∈ (mzPoly‘𝑎))) ∧ 𝑎𝑆) ∧ (𝑐 ∈ (ℤ𝑁) ∧ 𝑑 ∈ V)) ∧ (𝑑:(1...𝑐)–1-1-onto→(𝑎 ∪ (1...𝑁)) ∧ (𝑑 ↾ (1...𝑁)) = ( I ↾ (1...𝑁)))) → 𝑑:(1...𝑐)–1-1𝑆)
45 f1f 6804 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑑:(1...𝑐)–1-1𝑆𝑑:(1...𝑐)⟶𝑆)
4644, 45syl 17 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((((𝑁 ∈ ℕ0 ∧ (𝑆 ∈ V ∧ (1...𝑁) ⊆ 𝑆)) ∧ (𝑎 ∈ Fin ∧ 𝑏 ∈ (mzPoly‘𝑎))) ∧ 𝑎𝑆) ∧ (𝑐 ∈ (ℤ𝑁) ∧ 𝑑 ∈ V)) ∧ (𝑑:(1...𝑐)–1-1-onto→(𝑎 ∪ (1...𝑁)) ∧ (𝑑 ↾ (1...𝑁)) = ( I ↾ (1...𝑁)))) → 𝑑:(1...𝑐)⟶𝑆)
4746adantr 480 . . . . . . . . . . . . . . . . . . . . . . 23 (((((((𝑁 ∈ ℕ0 ∧ (𝑆 ∈ V ∧ (1...𝑁) ⊆ 𝑆)) ∧ (𝑎 ∈ Fin ∧ 𝑏 ∈ (mzPoly‘𝑎))) ∧ 𝑎𝑆) ∧ (𝑐 ∈ (ℤ𝑁) ∧ 𝑑 ∈ V)) ∧ (𝑑:(1...𝑐)–1-1-onto→(𝑎 ∪ (1...𝑁)) ∧ (𝑑 ↾ (1...𝑁)) = ( I ↾ (1...𝑁)))) ∧ 𝑒 ∈ (ℤ ↑m 𝑆)) → 𝑑:(1...𝑐)⟶𝑆)
48 mapco2g 42701 . . . . . . . . . . . . . . . . . . . . . . 23 (((1...𝑐) ∈ V ∧ 𝑒 ∈ (ℤ ↑m 𝑆) ∧ 𝑑:(1...𝑐)⟶𝑆) → (𝑒𝑑) ∈ (ℤ ↑m (1...𝑐)))
4934, 35, 47, 48syl3anc 1370 . . . . . . . . . . . . . . . . . . . . . 22 (((((((𝑁 ∈ ℕ0 ∧ (𝑆 ∈ V ∧ (1...𝑁) ⊆ 𝑆)) ∧ (𝑎 ∈ Fin ∧ 𝑏 ∈ (mzPoly‘𝑎))) ∧ 𝑎𝑆) ∧ (𝑐 ∈ (ℤ𝑁) ∧ 𝑑 ∈ V)) ∧ (𝑑:(1...𝑐)–1-1-onto→(𝑎 ∪ (1...𝑁)) ∧ (𝑑 ↾ (1...𝑁)) = ( I ↾ (1...𝑁)))) ∧ 𝑒 ∈ (ℤ ↑m 𝑆)) → (𝑒𝑑) ∈ (ℤ ↑m (1...𝑐)))
50 coeq1 5870 . . . . . . . . . . . . . . . . . . . . . . . 24 ( = (𝑒𝑑) → ( ∘ (𝑑𝑎)) = ((𝑒𝑑) ∘ (𝑑𝑎)))
5150fveq2d 6910 . . . . . . . . . . . . . . . . . . . . . . 23 ( = (𝑒𝑑) → (𝑏‘( ∘ (𝑑𝑎))) = (𝑏‘((𝑒𝑑) ∘ (𝑑𝑎))))
52 eqid 2734 . . . . . . . . . . . . . . . . . . . . . . 23 ( ∈ (ℤ ↑m (1...𝑐)) ↦ (𝑏‘( ∘ (𝑑𝑎)))) = ( ∈ (ℤ ↑m (1...𝑐)) ↦ (𝑏‘( ∘ (𝑑𝑎))))
53 fvex 6919 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑏‘((𝑒𝑑) ∘ (𝑑𝑎))) ∈ V
5451, 52, 53fvmpt 7015 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑒𝑑) ∈ (ℤ ↑m (1...𝑐)) → (( ∈ (ℤ ↑m (1...𝑐)) ↦ (𝑏‘( ∘ (𝑑𝑎))))‘(𝑒𝑑)) = (𝑏‘((𝑒𝑑) ∘ (𝑑𝑎))))
5549, 54syl 17 . . . . . . . . . . . . . . . . . . . . 21 (((((((𝑁 ∈ ℕ0 ∧ (𝑆 ∈ V ∧ (1...𝑁) ⊆ 𝑆)) ∧ (𝑎 ∈ Fin ∧ 𝑏 ∈ (mzPoly‘𝑎))) ∧ 𝑎𝑆) ∧ (𝑐 ∈ (ℤ𝑁) ∧ 𝑑 ∈ V)) ∧ (𝑑:(1...𝑐)–1-1-onto→(𝑎 ∪ (1...𝑁)) ∧ (𝑑 ↾ (1...𝑁)) = ( I ↾ (1...𝑁)))) ∧ 𝑒 ∈ (ℤ ↑m 𝑆)) → (( ∈ (ℤ ↑m (1...𝑐)) ↦ (𝑏‘( ∘ (𝑑𝑎))))‘(𝑒𝑑)) = (𝑏‘((𝑒𝑑) ∘ (𝑑𝑎))))
5633, 55eqtr4d 2777 . . . . . . . . . . . . . . . . . . . 20 (((((((𝑁 ∈ ℕ0 ∧ (𝑆 ∈ V ∧ (1...𝑁) ⊆ 𝑆)) ∧ (𝑎 ∈ Fin ∧ 𝑏 ∈ (mzPoly‘𝑎))) ∧ 𝑎𝑆) ∧ (𝑐 ∈ (ℤ𝑁) ∧ 𝑑 ∈ V)) ∧ (𝑑:(1...𝑐)–1-1-onto→(𝑎 ∪ (1...𝑁)) ∧ (𝑑 ↾ (1...𝑁)) = ( I ↾ (1...𝑁)))) ∧ 𝑒 ∈ (ℤ ↑m 𝑆)) → (𝑏‘(𝑒𝑎)) = (( ∈ (ℤ ↑m (1...𝑐)) ↦ (𝑏‘( ∘ (𝑑𝑎))))‘(𝑒𝑑)))
5756mpteq2dva 5247 . . . . . . . . . . . . . . . . . . 19 ((((((𝑁 ∈ ℕ0 ∧ (𝑆 ∈ V ∧ (1...𝑁) ⊆ 𝑆)) ∧ (𝑎 ∈ Fin ∧ 𝑏 ∈ (mzPoly‘𝑎))) ∧ 𝑎𝑆) ∧ (𝑐 ∈ (ℤ𝑁) ∧ 𝑑 ∈ V)) ∧ (𝑑:(1...𝑐)–1-1-onto→(𝑎 ∪ (1...𝑁)) ∧ (𝑑 ↾ (1...𝑁)) = ( I ↾ (1...𝑁)))) → (𝑒 ∈ (ℤ ↑m 𝑆) ↦ (𝑏‘(𝑒𝑎))) = (𝑒 ∈ (ℤ ↑m 𝑆) ↦ (( ∈ (ℤ ↑m (1...𝑐)) ↦ (𝑏‘( ∘ (𝑑𝑎))))‘(𝑒𝑑))))
5857fveq1d 6908 . . . . . . . . . . . . . . . . . 18 ((((((𝑁 ∈ ℕ0 ∧ (𝑆 ∈ V ∧ (1...𝑁) ⊆ 𝑆)) ∧ (𝑎 ∈ Fin ∧ 𝑏 ∈ (mzPoly‘𝑎))) ∧ 𝑎𝑆) ∧ (𝑐 ∈ (ℤ𝑁) ∧ 𝑑 ∈ V)) ∧ (𝑑:(1...𝑐)–1-1-onto→(𝑎 ∪ (1...𝑁)) ∧ (𝑑 ↾ (1...𝑁)) = ( I ↾ (1...𝑁)))) → ((𝑒 ∈ (ℤ ↑m 𝑆) ↦ (𝑏‘(𝑒𝑎)))‘𝑢) = ((𝑒 ∈ (ℤ ↑m 𝑆) ↦ (( ∈ (ℤ ↑m (1...𝑐)) ↦ (𝑏‘( ∘ (𝑑𝑎))))‘(𝑒𝑑)))‘𝑢))
5958eqeq1d 2736 . . . . . . . . . . . . . . . . 17 ((((((𝑁 ∈ ℕ0 ∧ (𝑆 ∈ V ∧ (1...𝑁) ⊆ 𝑆)) ∧ (𝑎 ∈ Fin ∧ 𝑏 ∈ (mzPoly‘𝑎))) ∧ 𝑎𝑆) ∧ (𝑐 ∈ (ℤ𝑁) ∧ 𝑑 ∈ V)) ∧ (𝑑:(1...𝑐)–1-1-onto→(𝑎 ∪ (1...𝑁)) ∧ (𝑑 ↾ (1...𝑁)) = ( I ↾ (1...𝑁)))) → (((𝑒 ∈ (ℤ ↑m 𝑆) ↦ (𝑏‘(𝑒𝑎)))‘𝑢) = 0 ↔ ((𝑒 ∈ (ℤ ↑m 𝑆) ↦ (( ∈ (ℤ ↑m (1...𝑐)) ↦ (𝑏‘( ∘ (𝑑𝑎))))‘(𝑒𝑑)))‘𝑢) = 0))
6059anbi2d 630 . . . . . . . . . . . . . . . 16 ((((((𝑁 ∈ ℕ0 ∧ (𝑆 ∈ V ∧ (1...𝑁) ⊆ 𝑆)) ∧ (𝑎 ∈ Fin ∧ 𝑏 ∈ (mzPoly‘𝑎))) ∧ 𝑎𝑆) ∧ (𝑐 ∈ (ℤ𝑁) ∧ 𝑑 ∈ V)) ∧ (𝑑:(1...𝑐)–1-1-onto→(𝑎 ∪ (1...𝑁)) ∧ (𝑑 ↾ (1...𝑁)) = ( I ↾ (1...𝑁)))) → ((𝑡 = (𝑢 ↾ (1...𝑁)) ∧ ((𝑒 ∈ (ℤ ↑m 𝑆) ↦ (𝑏‘(𝑒𝑎)))‘𝑢) = 0) ↔ (𝑡 = (𝑢 ↾ (1...𝑁)) ∧ ((𝑒 ∈ (ℤ ↑m 𝑆) ↦ (( ∈ (ℤ ↑m (1...𝑐)) ↦ (𝑏‘( ∘ (𝑑𝑎))))‘(𝑒𝑑)))‘𝑢) = 0)))
6160rexbidv 3176 . . . . . . . . . . . . . . 15 ((((((𝑁 ∈ ℕ0 ∧ (𝑆 ∈ V ∧ (1...𝑁) ⊆ 𝑆)) ∧ (𝑎 ∈ Fin ∧ 𝑏 ∈ (mzPoly‘𝑎))) ∧ 𝑎𝑆) ∧ (𝑐 ∈ (ℤ𝑁) ∧ 𝑑 ∈ V)) ∧ (𝑑:(1...𝑐)–1-1-onto→(𝑎 ∪ (1...𝑁)) ∧ (𝑑 ↾ (1...𝑁)) = ( I ↾ (1...𝑁)))) → (∃𝑢 ∈ (ℕ0m 𝑆)(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ ((𝑒 ∈ (ℤ ↑m 𝑆) ↦ (𝑏‘(𝑒𝑎)))‘𝑢) = 0) ↔ ∃𝑢 ∈ (ℕ0m 𝑆)(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ ((𝑒 ∈ (ℤ ↑m 𝑆) ↦ (( ∈ (ℤ ↑m (1...𝑐)) ↦ (𝑏‘( ∘ (𝑑𝑎))))‘(𝑒𝑑)))‘𝑢) = 0)))
6261abbidv 2805 . . . . . . . . . . . . . 14 ((((((𝑁 ∈ ℕ0 ∧ (𝑆 ∈ V ∧ (1...𝑁) ⊆ 𝑆)) ∧ (𝑎 ∈ Fin ∧ 𝑏 ∈ (mzPoly‘𝑎))) ∧ 𝑎𝑆) ∧ (𝑐 ∈ (ℤ𝑁) ∧ 𝑑 ∈ V)) ∧ (𝑑:(1...𝑐)–1-1-onto→(𝑎 ∪ (1...𝑁)) ∧ (𝑑 ↾ (1...𝑁)) = ( I ↾ (1...𝑁)))) → {𝑡 ∣ ∃𝑢 ∈ (ℕ0m 𝑆)(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ ((𝑒 ∈ (ℤ ↑m 𝑆) ↦ (𝑏‘(𝑒𝑎)))‘𝑢) = 0)} = {𝑡 ∣ ∃𝑢 ∈ (ℕ0m 𝑆)(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ ((𝑒 ∈ (ℤ ↑m 𝑆) ↦ (( ∈ (ℤ ↑m (1...𝑐)) ↦ (𝑏‘( ∘ (𝑑𝑎))))‘(𝑒𝑑)))‘𝑢) = 0)})
63 simplrl 777 . . . . . . . . . . . . . . . 16 (((𝑁 ∈ ℕ0 ∧ (𝑆 ∈ V ∧ (1...𝑁) ⊆ 𝑆)) ∧ (𝑎 ∈ Fin ∧ 𝑏 ∈ (mzPoly‘𝑎))) → 𝑆 ∈ V)
6463ad3antrrr 730 . . . . . . . . . . . . . . 15 ((((((𝑁 ∈ ℕ0 ∧ (𝑆 ∈ V ∧ (1...𝑁) ⊆ 𝑆)) ∧ (𝑎 ∈ Fin ∧ 𝑏 ∈ (mzPoly‘𝑎))) ∧ 𝑎𝑆) ∧ (𝑐 ∈ (ℤ𝑁) ∧ 𝑑 ∈ V)) ∧ (𝑑:(1...𝑐)–1-1-onto→(𝑎 ∪ (1...𝑁)) ∧ (𝑑 ↾ (1...𝑁)) = ( I ↾ (1...𝑁)))) → 𝑆 ∈ V)
65 simprr 773 . . . . . . . . . . . . . . 15 ((((((𝑁 ∈ ℕ0 ∧ (𝑆 ∈ V ∧ (1...𝑁) ⊆ 𝑆)) ∧ (𝑎 ∈ Fin ∧ 𝑏 ∈ (mzPoly‘𝑎))) ∧ 𝑎𝑆) ∧ (𝑐 ∈ (ℤ𝑁) ∧ 𝑑 ∈ V)) ∧ (𝑑:(1...𝑐)–1-1-onto→(𝑎 ∪ (1...𝑁)) ∧ (𝑑 ↾ (1...𝑁)) = ( I ↾ (1...𝑁)))) → (𝑑 ↾ (1...𝑁)) = ( I ↾ (1...𝑁)))
66 diophrw 42746 . . . . . . . . . . . . . . 15 ((𝑆 ∈ V ∧ 𝑑:(1...𝑐)–1-1𝑆 ∧ (𝑑 ↾ (1...𝑁)) = ( I ↾ (1...𝑁))) → {𝑡 ∣ ∃𝑢 ∈ (ℕ0m 𝑆)(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ ((𝑒 ∈ (ℤ ↑m 𝑆) ↦ (( ∈ (ℤ ↑m (1...𝑐)) ↦ (𝑏‘( ∘ (𝑑𝑎))))‘(𝑒𝑑)))‘𝑢) = 0)} = {𝑡 ∣ ∃𝑔 ∈ (ℕ0m (1...𝑐))(𝑡 = (𝑔 ↾ (1...𝑁)) ∧ (( ∈ (ℤ ↑m (1...𝑐)) ↦ (𝑏‘( ∘ (𝑑𝑎))))‘𝑔) = 0)})
6764, 44, 65, 66syl3anc 1370 . . . . . . . . . . . . . 14 ((((((𝑁 ∈ ℕ0 ∧ (𝑆 ∈ V ∧ (1...𝑁) ⊆ 𝑆)) ∧ (𝑎 ∈ Fin ∧ 𝑏 ∈ (mzPoly‘𝑎))) ∧ 𝑎𝑆) ∧ (𝑐 ∈ (ℤ𝑁) ∧ 𝑑 ∈ V)) ∧ (𝑑:(1...𝑐)–1-1-onto→(𝑎 ∪ (1...𝑁)) ∧ (𝑑 ↾ (1...𝑁)) = ( I ↾ (1...𝑁)))) → {𝑡 ∣ ∃𝑢 ∈ (ℕ0m 𝑆)(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ ((𝑒 ∈ (ℤ ↑m 𝑆) ↦ (( ∈ (ℤ ↑m (1...𝑐)) ↦ (𝑏‘( ∘ (𝑑𝑎))))‘(𝑒𝑑)))‘𝑢) = 0)} = {𝑡 ∣ ∃𝑔 ∈ (ℕ0m (1...𝑐))(𝑡 = (𝑔 ↾ (1...𝑁)) ∧ (( ∈ (ℤ ↑m (1...𝑐)) ↦ (𝑏‘( ∘ (𝑑𝑎))))‘𝑔) = 0)})
6862, 67eqtrd 2774 . . . . . . . . . . . . 13 ((((((𝑁 ∈ ℕ0 ∧ (𝑆 ∈ V ∧ (1...𝑁) ⊆ 𝑆)) ∧ (𝑎 ∈ Fin ∧ 𝑏 ∈ (mzPoly‘𝑎))) ∧ 𝑎𝑆) ∧ (𝑐 ∈ (ℤ𝑁) ∧ 𝑑 ∈ V)) ∧ (𝑑:(1...𝑐)–1-1-onto→(𝑎 ∪ (1...𝑁)) ∧ (𝑑 ↾ (1...𝑁)) = ( I ↾ (1...𝑁)))) → {𝑡 ∣ ∃𝑢 ∈ (ℕ0m 𝑆)(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ ((𝑒 ∈ (ℤ ↑m 𝑆) ↦ (𝑏‘(𝑒𝑎)))‘𝑢) = 0)} = {𝑡 ∣ ∃𝑔 ∈ (ℕ0m (1...𝑐))(𝑡 = (𝑔 ↾ (1...𝑁)) ∧ (( ∈ (ℤ ↑m (1...𝑐)) ↦ (𝑏‘( ∘ (𝑑𝑎))))‘𝑔) = 0)})
69 simp-5l 785 . . . . . . . . . . . . . 14 ((((((𝑁 ∈ ℕ0 ∧ (𝑆 ∈ V ∧ (1...𝑁) ⊆ 𝑆)) ∧ (𝑎 ∈ Fin ∧ 𝑏 ∈ (mzPoly‘𝑎))) ∧ 𝑎𝑆) ∧ (𝑐 ∈ (ℤ𝑁) ∧ 𝑑 ∈ V)) ∧ (𝑑:(1...𝑐)–1-1-onto→(𝑎 ∪ (1...𝑁)) ∧ (𝑑 ↾ (1...𝑁)) = ( I ↾ (1...𝑁)))) → 𝑁 ∈ ℕ0)
70 simplrl 777 . . . . . . . . . . . . . 14 ((((((𝑁 ∈ ℕ0 ∧ (𝑆 ∈ V ∧ (1...𝑁) ⊆ 𝑆)) ∧ (𝑎 ∈ Fin ∧ 𝑏 ∈ (mzPoly‘𝑎))) ∧ 𝑎𝑆) ∧ (𝑐 ∈ (ℤ𝑁) ∧ 𝑑 ∈ V)) ∧ (𝑑:(1...𝑐)–1-1-onto→(𝑎 ∪ (1...𝑁)) ∧ (𝑑 ↾ (1...𝑁)) = ( I ↾ (1...𝑁)))) → 𝑐 ∈ (ℤ𝑁))
71 ovexd 7465 . . . . . . . . . . . . . . 15 ((((((𝑁 ∈ ℕ0 ∧ (𝑆 ∈ V ∧ (1...𝑁) ⊆ 𝑆)) ∧ (𝑎 ∈ Fin ∧ 𝑏 ∈ (mzPoly‘𝑎))) ∧ 𝑎𝑆) ∧ (𝑐 ∈ (ℤ𝑁) ∧ 𝑑 ∈ V)) ∧ (𝑑:(1...𝑐)–1-1-onto→(𝑎 ∪ (1...𝑁)) ∧ (𝑑 ↾ (1...𝑁)) = ( I ↾ (1...𝑁)))) → (1...𝑐) ∈ V)
72 simplrr 778 . . . . . . . . . . . . . . . 16 ((((𝑁 ∈ ℕ0 ∧ (𝑆 ∈ V ∧ (1...𝑁) ⊆ 𝑆)) ∧ (𝑎 ∈ Fin ∧ 𝑏 ∈ (mzPoly‘𝑎))) ∧ 𝑎𝑆) → 𝑏 ∈ (mzPoly‘𝑎))
7372ad2antrr 726 . . . . . . . . . . . . . . 15 ((((((𝑁 ∈ ℕ0 ∧ (𝑆 ∈ V ∧ (1...𝑁) ⊆ 𝑆)) ∧ (𝑎 ∈ Fin ∧ 𝑏 ∈ (mzPoly‘𝑎))) ∧ 𝑎𝑆) ∧ (𝑐 ∈ (ℤ𝑁) ∧ 𝑑 ∈ V)) ∧ (𝑑:(1...𝑐)–1-1-onto→(𝑎 ∪ (1...𝑁)) ∧ (𝑑 ↾ (1...𝑁)) = ( I ↾ (1...𝑁)))) → 𝑏 ∈ (mzPoly‘𝑎))
74 f1ocnv 6860 . . . . . . . . . . . . . . . . . 18 (𝑑:(1...𝑐)–1-1-onto→(𝑎 ∪ (1...𝑁)) → 𝑑:(𝑎 ∪ (1...𝑁))–1-1-onto→(1...𝑐))
75 f1of 6848 . . . . . . . . . . . . . . . . . 18 (𝑑:(𝑎 ∪ (1...𝑁))–1-1-onto→(1...𝑐) → 𝑑:(𝑎 ∪ (1...𝑁))⟶(1...𝑐))
7674, 75syl 17 . . . . . . . . . . . . . . . . 17 (𝑑:(1...𝑐)–1-1-onto→(𝑎 ∪ (1...𝑁)) → 𝑑:(𝑎 ∪ (1...𝑁))⟶(1...𝑐))
77 fssres 6774 . . . . . . . . . . . . . . . . 17 ((𝑑:(𝑎 ∪ (1...𝑁))⟶(1...𝑐) ∧ 𝑎 ⊆ (𝑎 ∪ (1...𝑁))) → (𝑑𝑎):𝑎⟶(1...𝑐))
7876, 21, 77sylancl 586 . . . . . . . . . . . . . . . 16 (𝑑:(1...𝑐)–1-1-onto→(𝑎 ∪ (1...𝑁)) → (𝑑𝑎):𝑎⟶(1...𝑐))
7978ad2antrl 728 . . . . . . . . . . . . . . 15 ((((((𝑁 ∈ ℕ0 ∧ (𝑆 ∈ V ∧ (1...𝑁) ⊆ 𝑆)) ∧ (𝑎 ∈ Fin ∧ 𝑏 ∈ (mzPoly‘𝑎))) ∧ 𝑎𝑆) ∧ (𝑐 ∈ (ℤ𝑁) ∧ 𝑑 ∈ V)) ∧ (𝑑:(1...𝑐)–1-1-onto→(𝑎 ∪ (1...𝑁)) ∧ (𝑑 ↾ (1...𝑁)) = ( I ↾ (1...𝑁)))) → (𝑑𝑎):𝑎⟶(1...𝑐))
80 mzprename 42736 . . . . . . . . . . . . . . 15 (((1...𝑐) ∈ V ∧ 𝑏 ∈ (mzPoly‘𝑎) ∧ (𝑑𝑎):𝑎⟶(1...𝑐)) → ( ∈ (ℤ ↑m (1...𝑐)) ↦ (𝑏‘( ∘ (𝑑𝑎)))) ∈ (mzPoly‘(1...𝑐)))
8171, 73, 79, 80syl3anc 1370 . . . . . . . . . . . . . 14 ((((((𝑁 ∈ ℕ0 ∧ (𝑆 ∈ V ∧ (1...𝑁) ⊆ 𝑆)) ∧ (𝑎 ∈ Fin ∧ 𝑏 ∈ (mzPoly‘𝑎))) ∧ 𝑎𝑆) ∧ (𝑐 ∈ (ℤ𝑁) ∧ 𝑑 ∈ V)) ∧ (𝑑:(1...𝑐)–1-1-onto→(𝑎 ∪ (1...𝑁)) ∧ (𝑑 ↾ (1...𝑁)) = ( I ↾ (1...𝑁)))) → ( ∈ (ℤ ↑m (1...𝑐)) ↦ (𝑏‘( ∘ (𝑑𝑎)))) ∈ (mzPoly‘(1...𝑐)))
82 eldioph 42745 . . . . . . . . . . . . . 14 ((𝑁 ∈ ℕ0𝑐 ∈ (ℤ𝑁) ∧ ( ∈ (ℤ ↑m (1...𝑐)) ↦ (𝑏‘( ∘ (𝑑𝑎)))) ∈ (mzPoly‘(1...𝑐))) → {𝑡 ∣ ∃𝑔 ∈ (ℕ0m (1...𝑐))(𝑡 = (𝑔 ↾ (1...𝑁)) ∧ (( ∈ (ℤ ↑m (1...𝑐)) ↦ (𝑏‘( ∘ (𝑑𝑎))))‘𝑔) = 0)} ∈ (Dioph‘𝑁))
8369, 70, 81, 82syl3anc 1370 . . . . . . . . . . . . 13 ((((((𝑁 ∈ ℕ0 ∧ (𝑆 ∈ V ∧ (1...𝑁) ⊆ 𝑆)) ∧ (𝑎 ∈ Fin ∧ 𝑏 ∈ (mzPoly‘𝑎))) ∧ 𝑎𝑆) ∧ (𝑐 ∈ (ℤ𝑁) ∧ 𝑑 ∈ V)) ∧ (𝑑:(1...𝑐)–1-1-onto→(𝑎 ∪ (1...𝑁)) ∧ (𝑑 ↾ (1...𝑁)) = ( I ↾ (1...𝑁)))) → {𝑡 ∣ ∃𝑔 ∈ (ℕ0m (1...𝑐))(𝑡 = (𝑔 ↾ (1...𝑁)) ∧ (( ∈ (ℤ ↑m (1...𝑐)) ↦ (𝑏‘( ∘ (𝑑𝑎))))‘𝑔) = 0)} ∈ (Dioph‘𝑁))
8468, 83eqeltrd 2838 . . . . . . . . . . . 12 ((((((𝑁 ∈ ℕ0 ∧ (𝑆 ∈ V ∧ (1...𝑁) ⊆ 𝑆)) ∧ (𝑎 ∈ Fin ∧ 𝑏 ∈ (mzPoly‘𝑎))) ∧ 𝑎𝑆) ∧ (𝑐 ∈ (ℤ𝑁) ∧ 𝑑 ∈ V)) ∧ (𝑑:(1...𝑐)–1-1-onto→(𝑎 ∪ (1...𝑁)) ∧ (𝑑 ↾ (1...𝑁)) = ( I ↾ (1...𝑁)))) → {𝑡 ∣ ∃𝑢 ∈ (ℕ0m 𝑆)(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ ((𝑒 ∈ (ℤ ↑m 𝑆) ↦ (𝑏‘(𝑒𝑎)))‘𝑢) = 0)} ∈ (Dioph‘𝑁))
8584ex 412 . . . . . . . . . . 11 (((((𝑁 ∈ ℕ0 ∧ (𝑆 ∈ V ∧ (1...𝑁) ⊆ 𝑆)) ∧ (𝑎 ∈ Fin ∧ 𝑏 ∈ (mzPoly‘𝑎))) ∧ 𝑎𝑆) ∧ (𝑐 ∈ (ℤ𝑁) ∧ 𝑑 ∈ V)) → ((𝑑:(1...𝑐)–1-1-onto→(𝑎 ∪ (1...𝑁)) ∧ (𝑑 ↾ (1...𝑁)) = ( I ↾ (1...𝑁))) → {𝑡 ∣ ∃𝑢 ∈ (ℕ0m 𝑆)(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ ((𝑒 ∈ (ℤ ↑m 𝑆) ↦ (𝑏‘(𝑒𝑎)))‘𝑢) = 0)} ∈ (Dioph‘𝑁)))
8685rexlimdvva 3210 . . . . . . . . . 10 ((((𝑁 ∈ ℕ0 ∧ (𝑆 ∈ V ∧ (1...𝑁) ⊆ 𝑆)) ∧ (𝑎 ∈ Fin ∧ 𝑏 ∈ (mzPoly‘𝑎))) ∧ 𝑎𝑆) → (∃𝑐 ∈ (ℤ𝑁)∃𝑑 ∈ V (𝑑:(1...𝑐)–1-1-onto→(𝑎 ∪ (1...𝑁)) ∧ (𝑑 ↾ (1...𝑁)) = ( I ↾ (1...𝑁))) → {𝑡 ∣ ∃𝑢 ∈ (ℕ0m 𝑆)(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ ((𝑒 ∈ (ℤ ↑m 𝑆) ↦ (𝑏‘(𝑒𝑎)))‘𝑢) = 0)} ∈ (Dioph‘𝑁)))
8717, 86mpd 15 . . . . . . . . 9 ((((𝑁 ∈ ℕ0 ∧ (𝑆 ∈ V ∧ (1...𝑁) ⊆ 𝑆)) ∧ (𝑎 ∈ Fin ∧ 𝑏 ∈ (mzPoly‘𝑎))) ∧ 𝑎𝑆) → {𝑡 ∣ ∃𝑢 ∈ (ℕ0m 𝑆)(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ ((𝑒 ∈ (ℤ ↑m 𝑆) ↦ (𝑏‘(𝑒𝑎)))‘𝑢) = 0)} ∈ (Dioph‘𝑁))
8887exp31 419 . . . . . . . 8 ((𝑁 ∈ ℕ0 ∧ (𝑆 ∈ V ∧ (1...𝑁) ⊆ 𝑆)) → ((𝑎 ∈ Fin ∧ 𝑏 ∈ (mzPoly‘𝑎)) → (𝑎𝑆 → {𝑡 ∣ ∃𝑢 ∈ (ℕ0m 𝑆)(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ ((𝑒 ∈ (ℤ ↑m 𝑆) ↦ (𝑏‘(𝑒𝑎)))‘𝑢) = 0)} ∈ (Dioph‘𝑁))))
89883adant3 1131 . . . . . . 7 ((𝑁 ∈ ℕ0 ∧ (𝑆 ∈ V ∧ (1...𝑁) ⊆ 𝑆) ∧ 𝑃 ∈ (mzPoly‘𝑆)) → ((𝑎 ∈ Fin ∧ 𝑏 ∈ (mzPoly‘𝑎)) → (𝑎𝑆 → {𝑡 ∣ ∃𝑢 ∈ (ℕ0m 𝑆)(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ ((𝑒 ∈ (ℤ ↑m 𝑆) ↦ (𝑏‘(𝑒𝑎)))‘𝑢) = 0)} ∈ (Dioph‘𝑁))))
9089imp31 417 . . . . . 6 ((((𝑁 ∈ ℕ0 ∧ (𝑆 ∈ V ∧ (1...𝑁) ⊆ 𝑆) ∧ 𝑃 ∈ (mzPoly‘𝑆)) ∧ (𝑎 ∈ Fin ∧ 𝑏 ∈ (mzPoly‘𝑎))) ∧ 𝑎𝑆) → {𝑡 ∣ ∃𝑢 ∈ (ℕ0m 𝑆)(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ ((𝑒 ∈ (ℤ ↑m 𝑆) ↦ (𝑏‘(𝑒𝑎)))‘𝑢) = 0)} ∈ (Dioph‘𝑁))
9190adantrr 717 . . . . 5 ((((𝑁 ∈ ℕ0 ∧ (𝑆 ∈ V ∧ (1...𝑁) ⊆ 𝑆) ∧ 𝑃 ∈ (mzPoly‘𝑆)) ∧ (𝑎 ∈ Fin ∧ 𝑏 ∈ (mzPoly‘𝑎))) ∧ (𝑎𝑆𝑃 = (𝑒 ∈ (ℤ ↑m 𝑆) ↦ (𝑏‘(𝑒𝑎))))) → {𝑡 ∣ ∃𝑢 ∈ (ℕ0m 𝑆)(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ ((𝑒 ∈ (ℤ ↑m 𝑆) ↦ (𝑏‘(𝑒𝑎)))‘𝑢) = 0)} ∈ (Dioph‘𝑁))
928, 91eqeltrd 2838 . . . 4 ((((𝑁 ∈ ℕ0 ∧ (𝑆 ∈ V ∧ (1...𝑁) ⊆ 𝑆) ∧ 𝑃 ∈ (mzPoly‘𝑆)) ∧ (𝑎 ∈ Fin ∧ 𝑏 ∈ (mzPoly‘𝑎))) ∧ (𝑎𝑆𝑃 = (𝑒 ∈ (ℤ ↑m 𝑆) ↦ (𝑏‘(𝑒𝑎))))) → {𝑡 ∣ ∃𝑢 ∈ (ℕ0m 𝑆)(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑃𝑢) = 0)} ∈ (Dioph‘𝑁))
9392ex 412 . . 3 (((𝑁 ∈ ℕ0 ∧ (𝑆 ∈ V ∧ (1...𝑁) ⊆ 𝑆) ∧ 𝑃 ∈ (mzPoly‘𝑆)) ∧ (𝑎 ∈ Fin ∧ 𝑏 ∈ (mzPoly‘𝑎))) → ((𝑎𝑆𝑃 = (𝑒 ∈ (ℤ ↑m 𝑆) ↦ (𝑏‘(𝑒𝑎)))) → {𝑡 ∣ ∃𝑢 ∈ (ℕ0m 𝑆)(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑃𝑢) = 0)} ∈ (Dioph‘𝑁)))
9493rexlimdvva 3210 . 2 ((𝑁 ∈ ℕ0 ∧ (𝑆 ∈ V ∧ (1...𝑁) ⊆ 𝑆) ∧ 𝑃 ∈ (mzPoly‘𝑆)) → (∃𝑎 ∈ Fin ∃𝑏 ∈ (mzPoly‘𝑎)(𝑎𝑆𝑃 = (𝑒 ∈ (ℤ ↑m 𝑆) ↦ (𝑏‘(𝑒𝑎)))) → {𝑡 ∣ ∃𝑢 ∈ (ℕ0m 𝑆)(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑃𝑢) = 0)} ∈ (Dioph‘𝑁)))
952, 94mpd 15 1 ((𝑁 ∈ ℕ0 ∧ (𝑆 ∈ V ∧ (1...𝑁) ⊆ 𝑆) ∧ 𝑃 ∈ (mzPoly‘𝑆)) → {𝑡 ∣ ∃𝑢 ∈ (ℕ0m 𝑆)(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑃𝑢) = 0)} ∈ (Dioph‘𝑁))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1086   = wceq 1536  wcel 2105  {cab 2711  wrex 3067  Vcvv 3477  cun 3960  wss 3962  cmpt 5230   I cid 5581  ccnv 5687  cres 5690  ccom 5692  wf 6558  1-1wf1 6559  1-1-ontowf1o 6561  cfv 6562  (class class class)co 7430  m cmap 8864  Fincfn 8983  0cc0 11152  1c1 11153  0cn0 12523  cz 12610  cuz 12875  ...cfz 13543  mzPolycmzp 42709  Diophcdioph 42742
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753  ax-cnex 11208  ax-resscn 11209  ax-1cn 11210  ax-icn 11211  ax-addcl 11212  ax-addrcl 11213  ax-mulcl 11214  ax-mulrcl 11215  ax-mulcom 11216  ax-addass 11217  ax-mulass 11218  ax-distr 11219  ax-i2m1 11220  ax-1ne0 11221  ax-1rid 11222  ax-rnegex 11223  ax-rrecex 11224  ax-cnre 11225  ax-pre-lttri 11226  ax-pre-lttrn 11227  ax-pre-ltadd 11228  ax-pre-mulgt0 11229
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-int 4951  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-pred 6322  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-riota 7387  df-ov 7433  df-oprab 7434  df-mpo 7435  df-of 7696  df-om 7887  df-1st 8012  df-2nd 8013  df-frecs 8304  df-wrecs 8335  df-recs 8409  df-rdg 8448  df-1o 8504  df-oadd 8508  df-er 8743  df-map 8866  df-en 8984  df-dom 8985  df-sdom 8986  df-fin 8987  df-dju 9938  df-card 9976  df-pnf 11294  df-mnf 11295  df-xr 11296  df-ltxr 11297  df-le 11298  df-sub 11491  df-neg 11492  df-nn 12264  df-n0 12524  df-z 12611  df-uz 12876  df-fz 13544  df-hash 14366  df-mzpcl 42710  df-mzp 42711  df-dioph 42743
This theorem is referenced by:  eldioph2b  42750  diophin  42759  diophun  42760
  Copyright terms: Public domain W3C validator