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Theorem eldioph2 41071
Description: Construct a Diophantine set from a polynomial with witness variables drawn from any set whatsoever, via mzpcompact2 41061. (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 41061 . . 3 (𝑃 ∈ (mzPoly‘𝑆) → ∃𝑎 ∈ Fin ∃𝑏 ∈ (mzPoly‘𝑎)(𝑎𝑆𝑃 = (𝑒 ∈ (ℤ ↑m 𝑆) ↦ (𝑏‘(𝑒𝑎)))))
213ad2ant3 1135 . 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 6841 . . . . . . . . . 10 (𝑃 = (𝑒 ∈ (ℤ ↑m 𝑆) ↦ (𝑏‘(𝑒𝑎))) → (𝑃𝑢) = ((𝑒 ∈ (ℤ ↑m 𝑆) ↦ (𝑏‘(𝑒𝑎)))‘𝑢))
43eqeq1d 2738 . . . . . . . . 9 (𝑃 = (𝑒 ∈ (ℤ ↑m 𝑆) ↦ (𝑏‘(𝑒𝑎))) → ((𝑃𝑢) = 0 ↔ ((𝑒 ∈ (ℤ ↑m 𝑆) ↦ (𝑏‘(𝑒𝑎)))‘𝑢) = 0))
54anbi2d 629 . . . . . . . 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 3175 . . . . . . 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 727 . . . . 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 773 . . . . . . . . . . 11 ((((𝑁 ∈ ℕ0 ∧ (𝑆 ∈ V ∧ (1...𝑁) ⊆ 𝑆)) ∧ (𝑎 ∈ Fin ∧ 𝑏 ∈ (mzPoly‘𝑎))) ∧ 𝑎𝑆) → 𝑁 ∈ ℕ0)
10 simplrl 775 . . . . . . . . . . . 12 ((((𝑁 ∈ ℕ0 ∧ (𝑆 ∈ V ∧ (1...𝑁) ⊆ 𝑆)) ∧ (𝑎 ∈ Fin ∧ 𝑏 ∈ (mzPoly‘𝑎))) ∧ 𝑎𝑆) → 𝑎 ∈ Fin)
11 fzfi 13877 . . . . . . . . . . . 12 (1...𝑁) ∈ Fin
12 unfi 9116 . . . . . . . . . . . 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 4133 . . . . . . . . . . . 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 41069 . . . . . . . . . . 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 1371 . . . . . . . . . 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 6811 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑑:(1...𝑐)–1-1-onto→(𝑎 ∪ (1...𝑁)) → (𝑑𝑑) = ( I ↾ (𝑎 ∪ (1...𝑁))))
1918ad2antrl 726 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 5936 . . . . . . . . . . . . . . . . . . . . . . . . . . 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 4132 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 𝑎 ⊆ (𝑎 ∪ (1...𝑁))
22 resabs1 5967 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2793 . . . . . . . . . . . . . . . . . . . . . . . . . 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 6202 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑑𝑑) ↾ 𝑎) = (𝑑 ∘ (𝑑𝑎))
2624, 25eqtrdi 2792 . . . . . . . . . . . . . . . . . . . . . . . . 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 481 . . . . . . . . . . . . . . . . . . . . . . . 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 5818 . . . . . . . . . . . . . . . . . . . . . . 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 6216 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑒 ∘ ( I ↾ 𝑎)) = (𝑒𝑎)
30 coass 6217 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑒𝑑) ∘ (𝑑𝑎)) = (𝑒 ∘ (𝑑 ∘ (𝑑𝑎)))
3130eqcomi 2745 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑒 ∘ (𝑑 ∘ (𝑑𝑎))) = ((𝑒𝑑) ∘ (𝑑𝑎))
3228, 29, 313eqtr3g 2799 . . . . . . . . . . . . . . . . . . . . . 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 6846 . . . . . . . . . . . . . . . . . . . . 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 7392 . . . . . . . . . . . . . . . . . . . . . . 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 485 . . . . . . . . . . . . . . . . . . . . . . 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 6783 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑑:(1...𝑐)–1-1-onto→(𝑎 ∪ (1...𝑁)) → 𝑑:(1...𝑐)–1-1→(𝑎 ∪ (1...𝑁)))
3736ad2antrl 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)))) → 𝑑:(1...𝑐)–1-1→(𝑎 ∪ (1...𝑁)))
38 simpr 485 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((((𝑁 ∈ ℕ0 ∧ (𝑆 ∈ V ∧ (1...𝑁) ⊆ 𝑆)) ∧ (𝑎 ∈ Fin ∧ 𝑏 ∈ (mzPoly‘𝑎))) ∧ 𝑎𝑆) → 𝑎𝑆)
39 simprr 771 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝑁 ∈ ℕ0 ∧ (𝑆 ∈ V ∧ (1...𝑁) ⊆ 𝑆)) → (1...𝑁) ⊆ 𝑆)
4039ad2antrr 724 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((((𝑁 ∈ ℕ0 ∧ (𝑆 ∈ V ∧ (1...𝑁) ⊆ 𝑆)) ∧ (𝑎 ∈ Fin ∧ 𝑏 ∈ (mzPoly‘𝑎))) ∧ 𝑎𝑆) → (1...𝑁) ⊆ 𝑆)
4138, 40unssd 4146 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((((𝑁 ∈ ℕ0 ∧ (𝑆 ∈ V ∧ (1...𝑁) ⊆ 𝑆)) ∧ (𝑎 ∈ Fin ∧ 𝑏 ∈ (mzPoly‘𝑎))) ∧ 𝑎𝑆) → (𝑎 ∪ (1...𝑁)) ⊆ 𝑆)
4241ad2antrr 724 . . . . . . . . . . . . . . . . . . . . . . . . . 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 6744 . . . . . . . . . . . . . . . . . . . . . . . . . 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 6738 . . . . . . . . . . . . . . . . . . . . . . . . 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 481 . . . . . . . . . . . . . . . . . . . . . . 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 41023 . . . . . . . . . . . . . . . . . . . . . . 23 (((1...𝑐) ∈ V ∧ 𝑒 ∈ (ℤ ↑m 𝑆) ∧ 𝑑:(1...𝑐)⟶𝑆) → (𝑒𝑑) ∈ (ℤ ↑m (1...𝑐)))
4934, 35, 47, 48syl3anc 1371 . . . . . . . . . . . . . . . . . . . . . 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 5813 . . . . . . . . . . . . . . . . . . . . . . . 24 ( = (𝑒𝑑) → ( ∘ (𝑑𝑎)) = ((𝑒𝑑) ∘ (𝑑𝑎)))
5150fveq2d 6846 . . . . . . . . . . . . . . . . . . . . . . 23 ( = (𝑒𝑑) → (𝑏‘( ∘ (𝑑𝑎))) = (𝑏‘((𝑒𝑑) ∘ (𝑑𝑎))))
52 eqid 2736 . . . . . . . . . . . . . . . . . . . . . . 23 ( ∈ (ℤ ↑m (1...𝑐)) ↦ (𝑏‘( ∘ (𝑑𝑎)))) = ( ∈ (ℤ ↑m (1...𝑐)) ↦ (𝑏‘( ∘ (𝑑𝑎))))
53 fvex 6855 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑏‘((𝑒𝑑) ∘ (𝑑𝑎))) ∈ V
5451, 52, 53fvmpt 6948 . . . . . . . . . . . . . . . . . . . . . 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 2779 . . . . . . . . . . . . . . . . . . . 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 5205 . . . . . . . . . . . . . . . . . . 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 6844 . . . . . . . . . . . . . . . . . 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 2738 . . . . . . . . . . . . . . . . 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 629 . . . . . . . . . . . . . . . 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 3175 . . . . . . . . . . . . . . 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 775 . . . . . . . . . . . . . . . 16 (((𝑁 ∈ ℕ0 ∧ (𝑆 ∈ V ∧ (1...𝑁) ⊆ 𝑆)) ∧ (𝑎 ∈ Fin ∧ 𝑏 ∈ (mzPoly‘𝑎))) → 𝑆 ∈ V)
6463ad3antrrr 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)))) → 𝑆 ∈ V)
65 simprr 771 . . . . . . . . . . . . . . 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 41068 . . . . . . . . . . . . . . 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 1371 . . . . . . . . . . . . . 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 2776 . . . . . . . . . . . . 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 783 . . . . . . . . . . . . . 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 775 . . . . . . . . . . . . . 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 7392 . . . . . . . . . . . . . . 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 776 . . . . . . . . . . . . . . . 16 ((((𝑁 ∈ ℕ0 ∧ (𝑆 ∈ V ∧ (1...𝑁) ⊆ 𝑆)) ∧ (𝑎 ∈ Fin ∧ 𝑏 ∈ (mzPoly‘𝑎))) ∧ 𝑎𝑆) → 𝑏 ∈ (mzPoly‘𝑎))
7372ad2antrr 724 . . . . . . . . . . . . . . 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 6796 . . . . . . . . . . . . . . . . . 18 (𝑑:(1...𝑐)–1-1-onto→(𝑎 ∪ (1...𝑁)) → 𝑑:(𝑎 ∪ (1...𝑁))–1-1-onto→(1...𝑐))
75 f1of 6784 . . . . . . . . . . . . . . . . . 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 6708 . . . . . . . . . . . . . . . . 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 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)))) → (𝑑𝑎):𝑎⟶(1...𝑐))
80 mzprename 41058 . . . . . . . . . . . . . . 15 (((1...𝑐) ∈ V ∧ 𝑏 ∈ (mzPoly‘𝑎) ∧ (𝑑𝑎):𝑎⟶(1...𝑐)) → ( ∈ (ℤ ↑m (1...𝑐)) ↦ (𝑏‘( ∘ (𝑑𝑎)))) ∈ (mzPoly‘(1...𝑐)))
8171, 73, 79, 80syl3anc 1371 . . . . . . . . . . . . . 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 41067 . . . . . . . . . . . . . 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 1371 . . . . . . . . . . . . 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 413 . . . . . . . . . . 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 3205 . . . . . . . . . 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 420 . . . . . . . 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 1132 . . . . . . 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 418 . . . . . 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 715 . . . . 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 413 . . 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 3205 . 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 396  w3a 1087   = wceq 1541  wcel 2106  {cab 2713  wrex 3073  Vcvv 3445  cun 3908  wss 3910  cmpt 5188   I cid 5530  ccnv 5632  cres 5635  ccom 5637  wf 6492  1-1wf1 6493  1-1-ontowf1o 6495  cfv 6496  (class class class)co 7357  m cmap 8765  Fincfn 8883  0cc0 11051  1c1 11052  0cn0 12413  cz 12499  cuz 12763  ...cfz 13424  mzPolycmzp 41031  Diophcdioph 41064
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-of 7617  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-oadd 8416  df-er 8648  df-map 8767  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-dju 9837  df-card 9875  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-nn 12154  df-n0 12414  df-z 12500  df-uz 12764  df-fz 13425  df-hash 14231  df-mzpcl 41032  df-mzp 41033  df-dioph 41065
This theorem is referenced by:  eldioph2b  41072  diophin  41081  diophun  41082
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