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Theorem eldioph2 37935
Description: Construct a Diophantine set from a polynomial with witness variables drawn from any set whatsoever, via mzpcompact2 37925. (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 37925 . . 3 (𝑃 ∈ (mzPoly‘𝑆) → ∃𝑎 ∈ Fin ∃𝑏 ∈ (mzPoly‘𝑎)(𝑎𝑆𝑃 = (𝑒 ∈ (ℤ ↑𝑚 𝑆) ↦ (𝑏‘(𝑒𝑎)))))
213ad2ant3 1165 . 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 6373 . . . . . . . . . 10 (𝑃 = (𝑒 ∈ (ℤ ↑𝑚 𝑆) ↦ (𝑏‘(𝑒𝑎))) → (𝑃𝑢) = ((𝑒 ∈ (ℤ ↑𝑚 𝑆) ↦ (𝑏‘(𝑒𝑎)))‘𝑢))
43eqeq1d 2766 . . . . . . . . 9 (𝑃 = (𝑒 ∈ (ℤ ↑𝑚 𝑆) ↦ (𝑏‘(𝑒𝑎))) → ((𝑃𝑢) = 0 ↔ ((𝑒 ∈ (ℤ ↑𝑚 𝑆) ↦ (𝑏‘(𝑒𝑎)))‘𝑢) = 0))
54anbi2d 622 . . . . . . . 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 3198 . . . . . . 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 2883 . . . . . 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 720 . . . . 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 791 . . . . . . . . . . 11 ((((𝑁 ∈ ℕ0 ∧ (𝑆 ∈ V ∧ (1...𝑁) ⊆ 𝑆)) ∧ (𝑎 ∈ Fin ∧ 𝑏 ∈ (mzPoly‘𝑎))) ∧ 𝑎𝑆) → 𝑁 ∈ ℕ0)
10 simplrl 795 . . . . . . . . . . . 12 ((((𝑁 ∈ ℕ0 ∧ (𝑆 ∈ V ∧ (1...𝑁) ⊆ 𝑆)) ∧ (𝑎 ∈ Fin ∧ 𝑏 ∈ (mzPoly‘𝑎))) ∧ 𝑎𝑆) → 𝑎 ∈ Fin)
11 fzfi 12978 . . . . . . . . . . . 12 (1...𝑁) ∈ Fin
12 unfi 8433 . . . . . . . . . . . 12 ((𝑎 ∈ Fin ∧ (1...𝑁) ∈ Fin) → (𝑎 ∪ (1...𝑁)) ∈ Fin)
1310, 11, 12sylancl 580 . . . . . . . . . . 11 ((((𝑁 ∈ ℕ0 ∧ (𝑆 ∈ V ∧ (1...𝑁) ⊆ 𝑆)) ∧ (𝑎 ∈ Fin ∧ 𝑏 ∈ (mzPoly‘𝑎))) ∧ 𝑎𝑆) → (𝑎 ∪ (1...𝑁)) ∈ Fin)
14 ssun2 3938 . . . . . . . . . . . 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 37933 . . . . . . . . . . 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 1490 . . . . . . . . . 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 6345 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑑:(1...𝑐)–1-1-onto→(𝑎 ∪ (1...𝑁)) → (𝑑𝑑) = ( I ↾ (𝑎 ∪ (1...𝑁))))
1918ad2antrl 719 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 5563 . . . . . . . . . . . . . . . . . . . . . . . . . . 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 3937 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 𝑎 ⊆ (𝑎 ∪ (1...𝑁))
22 resabs1 5601 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2815 . . . . . . . . . . . . . . . . . . . . . . . . . 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 5824 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑑𝑑) ↾ 𝑎) = (𝑑 ∘ (𝑑𝑎))
2624, 25syl6eq 2814 . . . . . . . . . . . . . . . . . . . . . . . . 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 472 . . . . . . . . . . . . . . . . . . . . . . . 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 5452 . . . . . . . . . . . . . . . . . . . . . . 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 5838 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑒 ∘ ( I ↾ 𝑎)) = (𝑒𝑎)
30 coass 5839 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑒𝑑) ∘ (𝑑𝑎)) = (𝑒 ∘ (𝑑 ∘ (𝑑𝑎)))
3130eqcomi 2773 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑒 ∘ (𝑑 ∘ (𝑑𝑎))) = ((𝑒𝑑) ∘ (𝑑𝑎))
3228, 29, 313eqtr3g 2821 . . . . . . . . . . . . . . . . . . . . . 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 6378 . . . . . . . . . . . . . . . . . . . . 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 6875 . . . . . . . . . . . . . . . . . . . . . . 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 6318 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑑:(1...𝑐)–1-1-onto→(𝑎 ∪ (1...𝑁)) → 𝑑:(1...𝑐)–1-1→(𝑎 ∪ (1...𝑁)))
3736ad2antrl 719 . . . . . . . . . . . . . . . . . . . . . . . . . 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 789 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝑁 ∈ ℕ0 ∧ (𝑆 ∈ V ∧ (1...𝑁) ⊆ 𝑆)) → (1...𝑁) ⊆ 𝑆)
4039ad2antrr 717 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((((𝑁 ∈ ℕ0 ∧ (𝑆 ∈ V ∧ (1...𝑁) ⊆ 𝑆)) ∧ (𝑎 ∈ Fin ∧ 𝑏 ∈ (mzPoly‘𝑎))) ∧ 𝑎𝑆) → (1...𝑁) ⊆ 𝑆)
4138, 40unssd 3950 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((((𝑁 ∈ ℕ0 ∧ (𝑆 ∈ V ∧ (1...𝑁) ⊆ 𝑆)) ∧ (𝑎 ∈ Fin ∧ 𝑏 ∈ (mzPoly‘𝑎))) ∧ 𝑎𝑆) → (𝑎 ∪ (1...𝑁)) ⊆ 𝑆)
4241ad2antrr 717 . . . . . . . . . . . . . . . . . . . . . . . . . 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 6287 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑑:(1...𝑐)–1-1→(𝑎 ∪ (1...𝑁)) ∧ (𝑎 ∪ (1...𝑁)) ⊆ 𝑆) → 𝑑:(1...𝑐)–1-1𝑆)
4437, 42, 43syl2anc 579 . . . . . . . . . . . . . . . . . . . . . . . . 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 6282 . . . . . . . . . . . . . . . . . . . . . . . . 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 472 . . . . . . . . . . . . . . . . . . . . . . 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 37887 . . . . . . . . . . . . . . . . . . . . . . 23 (((1...𝑐) ∈ V ∧ 𝑒 ∈ (ℤ ↑𝑚 𝑆) ∧ 𝑑:(1...𝑐)⟶𝑆) → (𝑒𝑑) ∈ (ℤ ↑𝑚 (1...𝑐)))
4934, 35, 47, 48syl3anc 1490 . . . . . . . . . . . . . . . . . . . . . 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 5447 . . . . . . . . . . . . . . . . . . . . . . . 24 ( = (𝑒𝑑) → ( ∘ (𝑑𝑎)) = ((𝑒𝑑) ∘ (𝑑𝑎)))
5150fveq2d 6378 . . . . . . . . . . . . . . . . . . . . . . 23 ( = (𝑒𝑑) → (𝑏‘( ∘ (𝑑𝑎))) = (𝑏‘((𝑒𝑑) ∘ (𝑑𝑎))))
52 eqid 2764 . . . . . . . . . . . . . . . . . . . . . . 23 ( ∈ (ℤ ↑𝑚 (1...𝑐)) ↦ (𝑏‘( ∘ (𝑑𝑎)))) = ( ∈ (ℤ ↑𝑚 (1...𝑐)) ↦ (𝑏‘( ∘ (𝑑𝑎))))
53 fvex 6387 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑏‘((𝑒𝑑) ∘ (𝑑𝑎))) ∈ V
5451, 52, 53fvmpt 6470 . . . . . . . . . . . . . . . . . . . . . 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 2801 . . . . . . . . . . . . . . . . . . . 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 4902 . . . . . . . . . . . . . . . . . . 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 6376 . . . . . . . . . . . . . . . . . 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 2766 . . . . . . . . . . . . . . . . 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 622 . . . . . . . . . . . . . . . 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 3198 . . . . . . . . . . . . . . 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 2883 . . . . . . . . . . . . . 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 795 . . . . . . . . . . . . . . . 16 (((𝑁 ∈ ℕ0 ∧ (𝑆 ∈ V ∧ (1...𝑁) ⊆ 𝑆)) ∧ (𝑎 ∈ Fin ∧ 𝑏 ∈ (mzPoly‘𝑎))) → 𝑆 ∈ V)
6463ad3antrrr 721 . . . . . . . . . . . . . . 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 789 . . . . . . . . . . . . . . 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 37932 . . . . . . . . . . . . . . 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 1490 . . . . . . . . . . . . . 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 2798 . . . . . . . . . . . . 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 805 . . . . . . . . . . . . . 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 795 . . . . . . . . . . . . . 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 6875 . . . . . . . . . . . . . . 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 796 . . . . . . . . . . . . . . . 16 ((((𝑁 ∈ ℕ0 ∧ (𝑆 ∈ V ∧ (1...𝑁) ⊆ 𝑆)) ∧ (𝑎 ∈ Fin ∧ 𝑏 ∈ (mzPoly‘𝑎))) ∧ 𝑎𝑆) → 𝑏 ∈ (mzPoly‘𝑎))
7372ad2antrr 717 . . . . . . . . . . . . . . 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 6331 . . . . . . . . . . . . . . . . . 18 (𝑑:(1...𝑐)–1-1-onto→(𝑎 ∪ (1...𝑁)) → 𝑑:(𝑎 ∪ (1...𝑁))–1-1-onto→(1...𝑐))
75 f1of 6319 . . . . . . . . . . . . . . . . . 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 6251 . . . . . . . . . . . . . . . . 17 ((𝑑:(𝑎 ∪ (1...𝑁))⟶(1...𝑐) ∧ 𝑎 ⊆ (𝑎 ∪ (1...𝑁))) → (𝑑𝑎):𝑎⟶(1...𝑐))
7876, 21, 77sylancl 580 . . . . . . . . . . . . . . . 16 (𝑑:(1...𝑐)–1-1-onto→(𝑎 ∪ (1...𝑁)) → (𝑑𝑎):𝑎⟶(1...𝑐))
7978ad2antrl 719 . . . . . . . . . . . . . . 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 37922 . . . . . . . . . . . . . . 15 (((1...𝑐) ∈ V ∧ 𝑏 ∈ (mzPoly‘𝑎) ∧ (𝑑𝑎):𝑎⟶(1...𝑐)) → ( ∈ (ℤ ↑𝑚 (1...𝑐)) ↦ (𝑏‘( ∘ (𝑑𝑎)))) ∈ (mzPoly‘(1...𝑐)))
8171, 73, 79, 80syl3anc 1490 . . . . . . . . . . . . . 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 37931 . . . . . . . . . . . . . 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 1490 . . . . . . . . . . . . 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 2843 . . . . . . . . . . . 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 401 . . . . . . . . . . 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 3184 . . . . . . . . . 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 410 . . . . . . . 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 1162 . . . . . . 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 408 . . . . . 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 708 . . . . 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 2843 . . . 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 401 . . 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 3184 . 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 1107   = wceq 1652  wcel 2155  {cab 2750  wrex 3055  Vcvv 3349  cun 3729  wss 3731  cmpt 4887   I cid 5183  ccnv 5275  cres 5278  ccom 5280  wf 6063  1-1wf1 6064  1-1-ontowf1o 6066  cfv 6067  (class class class)co 6841  𝑚 cmap 8059  Fincfn 8159  0cc0 10188  1c1 10189  0cn0 11537  cz 11623  cuz 11885  ...cfz 12532  mzPolycmzp 37895  Diophcdioph 37928
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2069  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2349  ax-ext 2742  ax-rep 4929  ax-sep 4940  ax-nul 4948  ax-pow 5000  ax-pr 5061  ax-un 7146  ax-cnex 10244  ax-resscn 10245  ax-1cn 10246  ax-icn 10247  ax-addcl 10248  ax-addrcl 10249  ax-mulcl 10250  ax-mulrcl 10251  ax-mulcom 10252  ax-addass 10253  ax-mulass 10254  ax-distr 10255  ax-i2m1 10256  ax-1ne0 10257  ax-1rid 10258  ax-rnegex 10259  ax-rrecex 10260  ax-cnre 10261  ax-pre-lttri 10262  ax-pre-lttrn 10263  ax-pre-ltadd 10264  ax-pre-mulgt0 10265
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2062  df-mo 2564  df-eu 2581  df-clab 2751  df-cleq 2757  df-clel 2760  df-nfc 2895  df-ne 2937  df-nel 3040  df-ral 3059  df-rex 3060  df-reu 3061  df-rmo 3062  df-rab 3063  df-v 3351  df-sbc 3596  df-csb 3691  df-dif 3734  df-un 3736  df-in 3738  df-ss 3745  df-pss 3747  df-nul 4079  df-if 4243  df-pw 4316  df-sn 4334  df-pr 4336  df-tp 4338  df-op 4340  df-uni 4594  df-int 4633  df-iun 4677  df-br 4809  df-opab 4871  df-mpt 4888  df-tr 4911  df-id 5184  df-eprel 5189  df-po 5197  df-so 5198  df-fr 5235  df-we 5237  df-xp 5282  df-rel 5283  df-cnv 5284  df-co 5285  df-dm 5286  df-rn 5287  df-res 5288  df-ima 5289  df-pred 5864  df-ord 5910  df-on 5911  df-lim 5912  df-suc 5913  df-iota 6030  df-fun 6069  df-fn 6070  df-f 6071  df-f1 6072  df-fo 6073  df-f1o 6074  df-fv 6075  df-riota 6802  df-ov 6844  df-oprab 6845  df-mpt2 6846  df-of 7094  df-om 7263  df-1st 7365  df-2nd 7366  df-wrecs 7609  df-recs 7671  df-rdg 7709  df-1o 7763  df-oadd 7767  df-er 7946  df-map 8061  df-en 8160  df-dom 8161  df-sdom 8162  df-fin 8163  df-card 9015  df-cda 9242  df-pnf 10329  df-mnf 10330  df-xr 10331  df-ltxr 10332  df-le 10333  df-sub 10521  df-neg 10522  df-nn 11274  df-n0 11538  df-z 11624  df-uz 11886  df-fz 12533  df-hash 13321  df-mzpcl 37896  df-mzp 37897  df-dioph 37929
This theorem is referenced by:  eldioph2b  37936  diophin  37946  diophun  37947
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