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Theorem eldioph2 36840
 Description: Construct a Diophantine set from a polynomial with witness variables drawn from any set whatsoever, via mzpcompact2 36830. (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 36830 . . 3 (𝑃 ∈ (mzPoly‘𝑆) → ∃𝑎 ∈ Fin ∃𝑏 ∈ (mzPoly‘𝑎)(𝑎𝑆𝑃 = (𝑒 ∈ (ℤ ↑𝑚 𝑆) ↦ (𝑏‘(𝑒𝑎)))))
213ad2ant3 1082 . 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 6152 . . . . . . . . . 10 (𝑃 = (𝑒 ∈ (ℤ ↑𝑚 𝑆) ↦ (𝑏‘(𝑒𝑎))) → (𝑃𝑢) = ((𝑒 ∈ (ℤ ↑𝑚 𝑆) ↦ (𝑏‘(𝑒𝑎)))‘𝑢))
43eqeq1d 2623 . . . . . . . . 9 (𝑃 = (𝑒 ∈ (ℤ ↑𝑚 𝑆) ↦ (𝑏‘(𝑒𝑎))) → ((𝑃𝑢) = 0 ↔ ((𝑒 ∈ (ℤ ↑𝑚 𝑆) ↦ (𝑏‘(𝑒𝑎)))‘𝑢) = 0))
54anbi2d 739 . . . . . . . 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 3046 . . . . . . 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 2738 . . . . . 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 764 . . . . 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 797 . . . . . . . . . . 11 ((((𝑁 ∈ ℕ0 ∧ (𝑆 ∈ V ∧ (1...𝑁) ⊆ 𝑆)) ∧ (𝑎 ∈ Fin ∧ 𝑏 ∈ (mzPoly‘𝑎))) ∧ 𝑎𝑆) → 𝑁 ∈ ℕ0)
10 simplrl 799 . . . . . . . . . . . 12 ((((𝑁 ∈ ℕ0 ∧ (𝑆 ∈ V ∧ (1...𝑁) ⊆ 𝑆)) ∧ (𝑎 ∈ Fin ∧ 𝑏 ∈ (mzPoly‘𝑎))) ∧ 𝑎𝑆) → 𝑎 ∈ Fin)
11 fzfi 12719 . . . . . . . . . . . 12 (1...𝑁) ∈ Fin
12 unfi 8179 . . . . . . . . . . . 12 ((𝑎 ∈ Fin ∧ (1...𝑁) ∈ Fin) → (𝑎 ∪ (1...𝑁)) ∈ Fin)
1310, 11, 12sylancl 693 . . . . . . . . . . 11 ((((𝑁 ∈ ℕ0 ∧ (𝑆 ∈ V ∧ (1...𝑁) ⊆ 𝑆)) ∧ (𝑎 ∈ Fin ∧ 𝑏 ∈ (mzPoly‘𝑎))) ∧ 𝑎𝑆) → (𝑎 ∪ (1...𝑁)) ∈ Fin)
14 ssun2 3760 . . . . . . . . . . . 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 36838 . . . . . . . . . . 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 1323 . . . . . . . . . 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 6125 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑑:(1...𝑐)–1-1-onto→(𝑎 ∪ (1...𝑁)) → (𝑑𝑑) = ( I ↾ (𝑎 ∪ (1...𝑁))))
1918ad2antrl 763 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 5360 . . . . . . . . . . . . . . . . . . . . . . . . . . 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 3759 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 𝑎 ⊆ (𝑎 ∪ (1...𝑁))
22 resabs1 5391 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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, 23syl6req 2672 . . . . . . . . . . . . . . . . . . . . . . . . . 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 5603 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑑𝑑) ↾ 𝑎) = (𝑑 ∘ (𝑑𝑎))
2624, 25syl6eq 2671 . . . . . . . . . . . . . . . . . . . . . . . . 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 5249 . . . . . . . . . . . . . . . . . . . . . . 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 5617 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑒 ∘ ( I ↾ 𝑎)) = (𝑒𝑎)
30 coass 5618 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑒𝑑) ∘ (𝑑𝑎)) = (𝑒 ∘ (𝑑 ∘ (𝑑𝑎)))
3130eqcomi 2630 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑒 ∘ (𝑑 ∘ (𝑑𝑎))) = ((𝑒𝑑) ∘ (𝑑𝑎))
3228, 29, 313eqtr3g 2678 . . . . . . . . . . . . . . . . . . . . . 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 6157 . . . . . . . . . . . . . . . . . . . . 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 6640 . . . . . . . . . . . . . . . . . . . . . . 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 477 . . . . . . . . . . . . . . . . . . . . . . 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 6098 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑑:(1...𝑐)–1-1-onto→(𝑎 ∪ (1...𝑁)) → 𝑑:(1...𝑐)–1-1→(𝑎 ∪ (1...𝑁)))
3736ad2antrl 763 . . . . . . . . . . . . . . . . . . . . . . . . . 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 477 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((((𝑁 ∈ ℕ0 ∧ (𝑆 ∈ V ∧ (1...𝑁) ⊆ 𝑆)) ∧ (𝑎 ∈ Fin ∧ 𝑏 ∈ (mzPoly‘𝑎))) ∧ 𝑎𝑆) → 𝑎𝑆)
39 simprr 795 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝑁 ∈ ℕ0 ∧ (𝑆 ∈ V ∧ (1...𝑁) ⊆ 𝑆)) → (1...𝑁) ⊆ 𝑆)
4039ad2antrr 761 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((((𝑁 ∈ ℕ0 ∧ (𝑆 ∈ V ∧ (1...𝑁) ⊆ 𝑆)) ∧ (𝑎 ∈ Fin ∧ 𝑏 ∈ (mzPoly‘𝑎))) ∧ 𝑎𝑆) → (1...𝑁) ⊆ 𝑆)
4138, 40unssd 3772 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((((𝑁 ∈ ℕ0 ∧ (𝑆 ∈ V ∧ (1...𝑁) ⊆ 𝑆)) ∧ (𝑎 ∈ Fin ∧ 𝑏 ∈ (mzPoly‘𝑎))) ∧ 𝑎𝑆) → (𝑎 ∪ (1...𝑁)) ⊆ 𝑆)
4241ad2antrr 761 . . . . . . . . . . . . . . . . . . . . . . . . . 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 6068 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑑:(1...𝑐)–1-1→(𝑎 ∪ (1...𝑁)) ∧ (𝑎 ∪ (1...𝑁)) ⊆ 𝑆) → 𝑑:(1...𝑐)–1-1𝑆)
4437, 42, 43syl2anc 692 . . . . . . . . . . . . . . . . . . . . . . . . 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 6063 . . . . . . . . . . . . . . . . . . . . . . . . 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 36792 . . . . . . . . . . . . . . . . . . . . . . 23 (((1...𝑐) ∈ V ∧ 𝑒 ∈ (ℤ ↑𝑚 𝑆) ∧ 𝑑:(1...𝑐)⟶𝑆) → (𝑒𝑑) ∈ (ℤ ↑𝑚 (1...𝑐)))
4934, 35, 47, 48syl3anc 1323 . . . . . . . . . . . . . . . . . . . . . 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 5244 . . . . . . . . . . . . . . . . . . . . . . . 24 ( = (𝑒𝑑) → ( ∘ (𝑑𝑎)) = ((𝑒𝑑) ∘ (𝑑𝑎)))
5150fveq2d 6157 . . . . . . . . . . . . . . . . . . . . . . 23 ( = (𝑒𝑑) → (𝑏‘( ∘ (𝑑𝑎))) = (𝑏‘((𝑒𝑑) ∘ (𝑑𝑎))))
52 eqid 2621 . . . . . . . . . . . . . . . . . . . . . . 23 ( ∈ (ℤ ↑𝑚 (1...𝑐)) ↦ (𝑏‘( ∘ (𝑑𝑎)))) = ( ∈ (ℤ ↑𝑚 (1...𝑐)) ↦ (𝑏‘( ∘ (𝑑𝑎))))
53 fvex 6163 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑏‘((𝑒𝑑) ∘ (𝑑𝑎))) ∈ V
5451, 52, 53fvmpt 6244 . . . . . . . . . . . . . . . . . . . . . 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 2658 . . . . . . . . . . . . . . . . . . . 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 4709 . . . . . . . . . . . . . . . . . . 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 6155 . . . . . . . . . . . . . . . . . 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 2623 . . . . . . . . . . . . . . . . 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 739 . . . . . . . . . . . . . . . 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 3046 . . . . . . . . . . . . . . 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 2738 . . . . . . . . . . . . . 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 799 . . . . . . . . . . . . . . . 16 (((𝑁 ∈ ℕ0 ∧ (𝑆 ∈ V ∧ (1...𝑁) ⊆ 𝑆)) ∧ (𝑎 ∈ Fin ∧ 𝑏 ∈ (mzPoly‘𝑎))) → 𝑆 ∈ V)
6463ad3antrrr 765 . . . . . . . . . . . . . . 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 795 . . . . . . . . . . . . . . 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 36837 . . . . . . . . . . . . . . 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 1323 . . . . . . . . . . . . . 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 2655 . . . . . . . . . . . . 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 807 . . . . . . . . . . . . . 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 799 . . . . . . . . . . . . . 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 6640 . . . . . . . . . . . . . . 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 800 . . . . . . . . . . . . . . . 16 ((((𝑁 ∈ ℕ0 ∧ (𝑆 ∈ V ∧ (1...𝑁) ⊆ 𝑆)) ∧ (𝑎 ∈ Fin ∧ 𝑏 ∈ (mzPoly‘𝑎))) ∧ 𝑎𝑆) → 𝑏 ∈ (mzPoly‘𝑎))
7372ad2antrr 761 . . . . . . . . . . . . . . 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 6111 . . . . . . . . . . . . . . . . . 18 (𝑑:(1...𝑐)–1-1-onto→(𝑎 ∪ (1...𝑁)) → 𝑑:(𝑎 ∪ (1...𝑁))–1-1-onto→(1...𝑐))
75 f1of 6099 . . . . . . . . . . . . . . . . . 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 6032 . . . . . . . . . . . . . . . . 17 ((𝑑:(𝑎 ∪ (1...𝑁))⟶(1...𝑐) ∧ 𝑎 ⊆ (𝑎 ∪ (1...𝑁))) → (𝑑𝑎):𝑎⟶(1...𝑐))
7876, 21, 77sylancl 693 . . . . . . . . . . . . . . . 16 (𝑑:(1...𝑐)–1-1-onto→(𝑎 ∪ (1...𝑁)) → (𝑑𝑎):𝑎⟶(1...𝑐))
7978ad2antrl 763 . . . . . . . . . . . . . . 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 36827 . . . . . . . . . . . . . . 15 (((1...𝑐) ∈ V ∧ 𝑏 ∈ (mzPoly‘𝑎) ∧ (𝑑𝑎):𝑎⟶(1...𝑐)) → ( ∈ (ℤ ↑𝑚 (1...𝑐)) ↦ (𝑏‘( ∘ (𝑑𝑎)))) ∈ (mzPoly‘(1...𝑐)))
8171, 73, 79, 80syl3anc 1323 . . . . . . . . . . . . . 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 36836 . . . . . . . . . . . . . 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 1323 . . . . . . . . . . . . 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 2698 . . . . . . . . . . . 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 450 . . . . . . . . . . 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 3032 . . . . . . . . . 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 629 . . . . . . . 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 1079 . . . . . . 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 448 . . . . . 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 752 . . . . 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 2698 . . . 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 450 . . 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 3032 . 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 384   ∧ w3a 1036   = wceq 1480   ∈ wcel 1987  {cab 2607  ∃wrex 2908  Vcvv 3189   ∪ cun 3557   ⊆ wss 3559   ↦ cmpt 4678   I cid 4989  ◡ccnv 5078   ↾ cres 5081   ∘ ccom 5083  ⟶wf 5848  –1-1→wf1 5849  –1-1-onto→wf1o 5851  ‘cfv 5852  (class class class)co 6610   ↑𝑚 cmap 7809  Fincfn 7907  0cc0 9888  1c1 9889  ℕ0cn0 11244  ℤcz 11329  ℤ≥cuz 11639  ...cfz 12276  mzPolycmzp 36800  Diophcdioph 36833 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909  ax-cnex 9944  ax-resscn 9945  ax-1cn 9946  ax-icn 9947  ax-addcl 9948  ax-addrcl 9949  ax-mulcl 9950  ax-mulrcl 9951  ax-mulcom 9952  ax-addass 9953  ax-mulass 9954  ax-distr 9955  ax-i2m1 9956  ax-1ne0 9957  ax-1rid 9958  ax-rnegex 9959  ax-rrecex 9960  ax-cnre 9961  ax-pre-lttri 9962  ax-pre-lttrn 9963  ax-pre-ltadd 9964  ax-pre-mulgt0 9965 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-pss 3575  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-int 4446  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5644  df-ord 5690  df-on 5691  df-lim 5692  df-suc 5693  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-riota 6571  df-ov 6613  df-oprab 6614  df-mpt2 6615  df-of 6857  df-om 7020  df-1st 7120  df-2nd 7121  df-wrecs 7359  df-recs 7420  df-rdg 7458  df-1o 7512  df-oadd 7516  df-er 7694  df-map 7811  df-en 7908  df-dom 7909  df-sdom 7910  df-fin 7911  df-card 8717  df-cda 8942  df-pnf 10028  df-mnf 10029  df-xr 10030  df-ltxr 10031  df-le 10032  df-sub 10220  df-neg 10221  df-nn 10973  df-n0 11245  df-z 11330  df-uz 11640  df-fz 12277  df-hash 13066  df-mzpcl 36801  df-mzp 36802  df-dioph 36834 This theorem is referenced by:  eldioph2b  36841  diophin  36851  diophun  36852
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