Theorem eldioph2 37325

Description: Construct a Diophantine set from a polynomial with witness variables drawn from any set whatsoever, via mzpcompact2 37315. (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 ∈ (ℕ0 ↑𝑚 𝑆)(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑃‘𝑢) = 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 ℎ 𝑑 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		mzpcompact2 37315	. . 3 ⊢ (𝑃 ∈ (mzPoly‘𝑆) → ∃𝑎 ∈ Fin ∃𝑏 ∈ (mzPoly‘𝑎)(𝑎 ⊆ 𝑆 ∧ 𝑃 = (𝑒 ∈ (ℤ ↑𝑚 𝑆) ↦ (𝑏‘(𝑒 ↾ 𝑎)))))
2	1	3ad2ant3 1084	. 2 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑆 ∈ V ∧ (1...𝑁) ⊆ 𝑆) ∧ 𝑃 ∈ (mzPoly‘𝑆)) → ∃𝑎 ∈ Fin ∃𝑏 ∈ (mzPoly‘𝑎)(𝑎 ⊆ 𝑆 ∧ 𝑃 = (𝑒 ∈ (ℤ ↑𝑚 𝑆) ↦ (𝑏‘(𝑒 ↾ 𝑎)))))
3		fveq1 6190	. . . . . . . . . 10 ⊢ (𝑃 = (𝑒 ∈ (ℤ ↑𝑚 𝑆) ↦ (𝑏‘(𝑒 ↾ 𝑎))) → (𝑃‘𝑢) = ((𝑒 ∈ (ℤ ↑𝑚 𝑆) ↦ (𝑏‘(𝑒 ↾ 𝑎)))‘𝑢))
4	3	eqeq1d 2624	. . . . . . . . 9 ⊢ (𝑃 = (𝑒 ∈ (ℤ ↑𝑚 𝑆) ↦ (𝑏‘(𝑒 ↾ 𝑎))) → ((𝑃‘𝑢) = 0 ↔ ((𝑒 ∈ (ℤ ↑𝑚 𝑆) ↦ (𝑏‘(𝑒 ↾ 𝑎)))‘𝑢) = 0))
5	4	anbi2d 740	. . . . . . . 8 ⊢ (𝑃 = (𝑒 ∈ (ℤ ↑𝑚 𝑆) ↦ (𝑏‘(𝑒 ↾ 𝑎))) → ((𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑃‘𝑢) = 0) ↔ (𝑡 = (𝑢 ↾ (1...𝑁)) ∧ ((𝑒 ∈ (ℤ ↑𝑚 𝑆) ↦ (𝑏‘(𝑒 ↾ 𝑎)))‘𝑢) = 0)))
6	5	rexbidv 3052	. . . . . . 7 ⊢ (𝑃 = (𝑒 ∈ (ℤ ↑𝑚 𝑆) ↦ (𝑏‘(𝑒 ↾ 𝑎))) → (∃𝑢 ∈ (ℕ0 ↑𝑚 𝑆)(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑃‘𝑢) = 0) ↔ ∃𝑢 ∈ (ℕ0 ↑𝑚 𝑆)(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ ((𝑒 ∈ (ℤ ↑𝑚 𝑆) ↦ (𝑏‘(𝑒 ↾ 𝑎)))‘𝑢) = 0)))
7	6	abbidv 2741	. . . . . 6 ⊢ (𝑃 = (𝑒 ∈ (ℤ ↑𝑚 𝑆) ↦ (𝑏‘(𝑒 ↾ 𝑎))) → {𝑡 ∣ ∃𝑢 ∈ (ℕ0 ↑𝑚 𝑆)(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑃‘𝑢) = 0)} = {𝑡 ∣ ∃𝑢 ∈ (ℕ0 ↑𝑚 𝑆)(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ ((𝑒 ∈ (ℤ ↑𝑚 𝑆) ↦ (𝑏‘(𝑒 ↾ 𝑎)))‘𝑢) = 0)})
8	7	ad2antll 765	. . . . 5 ⊢ ((((𝑁 ∈ ℕ0 ∧ (𝑆 ∈ V ∧ (1...𝑁) ⊆ 𝑆) ∧ 𝑃 ∈ (mzPoly‘𝑆)) ∧ (𝑎 ∈ Fin ∧ 𝑏 ∈ (mzPoly‘𝑎))) ∧ (𝑎 ⊆ 𝑆 ∧ 𝑃 = (𝑒 ∈ (ℤ ↑𝑚 𝑆) ↦ (𝑏‘(𝑒 ↾ 𝑎))))) → {𝑡 ∣ ∃𝑢 ∈ (ℕ0 ↑𝑚 𝑆)(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑃‘𝑢) = 0)} = {𝑡 ∣ ∃𝑢 ∈ (ℕ0 ↑𝑚 𝑆)(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ ((𝑒 ∈ (ℤ ↑𝑚 𝑆) ↦ (𝑏‘(𝑒 ↾ 𝑎)))‘𝑢) = 0)})
9		simplll 798	. . . . . . . . . . 11 ⊢ ((((𝑁 ∈ ℕ0 ∧ (𝑆 ∈ V ∧ (1...𝑁) ⊆ 𝑆)) ∧ (𝑎 ∈ Fin ∧ 𝑏 ∈ (mzPoly‘𝑎))) ∧ 𝑎 ⊆ 𝑆) → 𝑁 ∈ ℕ0)
10		simplrl 800	. . . . . . . . . . . 12 ⊢ ((((𝑁 ∈ ℕ0 ∧ (𝑆 ∈ V ∧ (1...𝑁) ⊆ 𝑆)) ∧ (𝑎 ∈ Fin ∧ 𝑏 ∈ (mzPoly‘𝑎))) ∧ 𝑎 ⊆ 𝑆) → 𝑎 ∈ Fin)
11		fzfi 12771	. . . . . . . . . . . 12 ⊢ (1...𝑁) ∈ Fin
12		unfi 8227	. . . . . . . . . . . 12 ⊢ ((𝑎 ∈ Fin ∧ (1...𝑁) ∈ Fin) → (𝑎 ∪ (1...𝑁)) ∈ Fin)
13	10, 11, 12	sylancl 694	. . . . . . . . . . 11 ⊢ ((((𝑁 ∈ ℕ0 ∧ (𝑆 ∈ V ∧ (1...𝑁) ⊆ 𝑆)) ∧ (𝑎 ∈ Fin ∧ 𝑏 ∈ (mzPoly‘𝑎))) ∧ 𝑎 ⊆ 𝑆) → (𝑎 ∪ (1...𝑁)) ∈ Fin)
14		ssun2 3777	. . . . . . . . . . . 12 ⊢ (1...𝑁) ⊆ (𝑎 ∪ (1...𝑁))
15	14	a1i 11	. . . . . . . . . . 11 ⊢ ((((𝑁 ∈ ℕ0 ∧ (𝑆 ∈ V ∧ (1...𝑁) ⊆ 𝑆)) ∧ (𝑎 ∈ Fin ∧ 𝑏 ∈ (mzPoly‘𝑎))) ∧ 𝑎 ⊆ 𝑆) → (1...𝑁) ⊆ (𝑎 ∪ (1...𝑁)))
16		eldioph2lem1 37323	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑎 ∪ (1...𝑁)) ∈ Fin ∧ (1...𝑁) ⊆ (𝑎 ∪ (1...𝑁))) → ∃𝑐 ∈ (ℤ≥‘𝑁)∃𝑑 ∈ V (𝑑:(1...𝑐)–1-1-onto→(𝑎 ∪ (1...𝑁)) ∧ (𝑑 ↾ (1...𝑁)) = ( I ↾ (1...𝑁))))
17	9, 13, 15, 16	syl3anc 1326	. . . . . . . . . 10 ⊢ ((((𝑁 ∈ ℕ0 ∧ (𝑆 ∈ V ∧ (1...𝑁) ⊆ 𝑆)) ∧ (𝑎 ∈ Fin ∧ 𝑏 ∈ (mzPoly‘𝑎))) ∧ 𝑎 ⊆ 𝑆) → ∃𝑐 ∈ (ℤ≥‘𝑁)∃𝑑 ∈ V (𝑑:(1...𝑐)–1-1-onto→(𝑎 ∪ (1...𝑁)) ∧ (𝑑 ↾ (1...𝑁)) = ( I ↾ (1...𝑁))))
18		f1ococnv2 6163	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑑:(1...𝑐)–1-1-onto→(𝑎 ∪ (1...𝑁)) → (𝑑 ∘ ◡𝑑) = ( I ↾ (𝑎 ∪ (1...𝑁))))
19	18	ad2antrl 764	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((((((𝑁 ∈ ℕ0 ∧ (𝑆 ∈ V ∧ (1...𝑁) ⊆ 𝑆)) ∧ (𝑎 ∈ Fin ∧ 𝑏 ∈ (mzPoly‘𝑎))) ∧ 𝑎 ⊆ 𝑆) ∧ (𝑐 ∈ (ℤ≥‘𝑁) ∧ 𝑑 ∈ V)) ∧ (𝑑:(1...𝑐)–1-1-onto→(𝑎 ∪ (1...𝑁)) ∧ (𝑑 ↾ (1...𝑁)) = ( I ↾ (1...𝑁)))) → (𝑑 ∘ ◡𝑑) = ( I ↾ (𝑎 ∪ (1...𝑁))))
20	19	reseq1d 5395	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((((((𝑁 ∈ ℕ0 ∧ (𝑆 ∈ V ∧ (1...𝑁) ⊆ 𝑆)) ∧ (𝑎 ∈ Fin ∧ 𝑏 ∈ (mzPoly‘𝑎))) ∧ 𝑎 ⊆ 𝑆) ∧ (𝑐 ∈ (ℤ≥‘𝑁) ∧ 𝑑 ∈ V)) ∧ (𝑑:(1...𝑐)–1-1-onto→(𝑎 ∪ (1...𝑁)) ∧ (𝑑 ↾ (1...𝑁)) = ( I ↾ (1...𝑁)))) → ((𝑑 ∘ ◡𝑑) ↾ 𝑎) = (( I ↾ (𝑎 ∪ (1...𝑁))) ↾ 𝑎))
21		ssun1 3776	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ 𝑎 ⊆ (𝑎 ∪ (1...𝑁))
22		resabs1 5427	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑎 ⊆ (𝑎 ∪ (1...𝑁)) → (( I ↾ (𝑎 ∪ (1...𝑁))) ↾ 𝑎) = ( I ↾ 𝑎))
23	21, 22	ax-mp 5	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (( I ↾ (𝑎 ∪ (1...𝑁))) ↾ 𝑎) = ( I ↾ 𝑎)
24	20, 23	syl6req 2673	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((((((𝑁 ∈ ℕ0 ∧ (𝑆 ∈ V ∧ (1...𝑁) ⊆ 𝑆)) ∧ (𝑎 ∈ Fin ∧ 𝑏 ∈ (mzPoly‘𝑎))) ∧ 𝑎 ⊆ 𝑆) ∧ (𝑐 ∈ (ℤ≥‘𝑁) ∧ 𝑑 ∈ V)) ∧ (𝑑:(1...𝑐)–1-1-onto→(𝑎 ∪ (1...𝑁)) ∧ (𝑑 ↾ (1...𝑁)) = ( I ↾ (1...𝑁)))) → ( I ↾ 𝑎) = ((𝑑 ∘ ◡𝑑) ↾ 𝑎))
25		resco 5639	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑑 ∘ ◡𝑑) ↾ 𝑎) = (𝑑 ∘ (◡𝑑 ↾ 𝑎))
26	24, 25	syl6eq 2672	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((((𝑁 ∈ ℕ0 ∧ (𝑆 ∈ V ∧ (1...𝑁) ⊆ 𝑆)) ∧ (𝑎 ∈ Fin ∧ 𝑏 ∈ (mzPoly‘𝑎))) ∧ 𝑎 ⊆ 𝑆) ∧ (𝑐 ∈ (ℤ≥‘𝑁) ∧ 𝑑 ∈ V)) ∧ (𝑑:(1...𝑐)–1-1-onto→(𝑎 ∪ (1...𝑁)) ∧ (𝑑 ↾ (1...𝑁)) = ( I ↾ (1...𝑁)))) → ( I ↾ 𝑎) = (𝑑 ∘ (◡𝑑 ↾ 𝑎)))
27	26	adantr 481	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((((((𝑁 ∈ ℕ0 ∧ (𝑆 ∈ V ∧ (1...𝑁) ⊆ 𝑆)) ∧ (𝑎 ∈ Fin ∧ 𝑏 ∈ (mzPoly‘𝑎))) ∧ 𝑎 ⊆ 𝑆) ∧ (𝑐 ∈ (ℤ≥‘𝑁) ∧ 𝑑 ∈ V)) ∧ (𝑑:(1...𝑐)–1-1-onto→(𝑎 ∪ (1...𝑁)) ∧ (𝑑 ↾ (1...𝑁)) = ( I ↾ (1...𝑁)))) ∧ 𝑒 ∈ (ℤ ↑𝑚 𝑆)) → ( I ↾ 𝑎) = (𝑑 ∘ (◡𝑑 ↾ 𝑎)))
28	27	coeq2d 5284	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((((((𝑁 ∈ ℕ0 ∧ (𝑆 ∈ V ∧ (1...𝑁) ⊆ 𝑆)) ∧ (𝑎 ∈ Fin ∧ 𝑏 ∈ (mzPoly‘𝑎))) ∧ 𝑎 ⊆ 𝑆) ∧ (𝑐 ∈ (ℤ≥‘𝑁) ∧ 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 ∘ (◡𝑑 ↾ 𝑎))))
29		coires1 5653	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑒 ∘ ( I ↾ 𝑎)) = (𝑒 ↾ 𝑎)
30		coass 5654	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑒 ∘ 𝑑) ∘ (◡𝑑 ↾ 𝑎)) = (𝑒 ∘ (𝑑 ∘ (◡𝑑 ↾ 𝑎)))
31	30	eqcomi 2631	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑒 ∘ (𝑑 ∘ (◡𝑑 ↾ 𝑎))) = ((𝑒 ∘ 𝑑) ∘ (◡𝑑 ↾ 𝑎))
32	28, 29, 31	3eqtr3g 2679	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((((((𝑁 ∈ ℕ0 ∧ (𝑆 ∈ V ∧ (1...𝑁) ⊆ 𝑆)) ∧ (𝑎 ∈ Fin ∧ 𝑏 ∈ (mzPoly‘𝑎))) ∧ 𝑎 ⊆ 𝑆) ∧ (𝑐 ∈ (ℤ≥‘𝑁) ∧ 𝑑 ∈ V)) ∧ (𝑑:(1...𝑐)–1-1-onto→(𝑎 ∪ (1...𝑁)) ∧ (𝑑 ↾ (1...𝑁)) = ( I ↾ (1...𝑁)))) ∧ 𝑒 ∈ (ℤ ↑𝑚 𝑆)) → (𝑒 ↾ 𝑎) = ((𝑒 ∘ 𝑑) ∘ (◡𝑑 ↾ 𝑎)))
33	32	fveq2d 6195	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((((((𝑁 ∈ ℕ0 ∧ (𝑆 ∈ V ∧ (1...𝑁) ⊆ 𝑆)) ∧ (𝑎 ∈ Fin ∧ 𝑏 ∈ (mzPoly‘𝑎))) ∧ 𝑎 ⊆ 𝑆) ∧ (𝑐 ∈ (ℤ≥‘𝑁) ∧ 𝑑 ∈ V)) ∧ (𝑑:(1...𝑐)–1-1-onto→(𝑎 ∪ (1...𝑁)) ∧ (𝑑 ↾ (1...𝑁)) = ( I ↾ (1...𝑁)))) ∧ 𝑒 ∈ (ℤ ↑𝑚 𝑆)) → (𝑏‘(𝑒 ↾ 𝑎)) = (𝑏‘((𝑒 ∘ 𝑑) ∘ (◡𝑑 ↾ 𝑎))))
34		ovexd 6680	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((((((𝑁 ∈ ℕ0 ∧ (𝑆 ∈ V ∧ (1...𝑁) ⊆ 𝑆)) ∧ (𝑎 ∈ Fin ∧ 𝑏 ∈ (mzPoly‘𝑎))) ∧ 𝑎 ⊆ 𝑆) ∧ (𝑐 ∈ (ℤ≥‘𝑁) ∧ 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))) ∧ 𝑎 ⊆ 𝑆) ∧ (𝑐 ∈ (ℤ≥‘𝑁) ∧ 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 6136	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑑:(1...𝑐)–1-1-onto→(𝑎 ∪ (1...𝑁)) → 𝑑:(1...𝑐)–1-1→(𝑎 ∪ (1...𝑁)))
37	36	ad2antrl 764	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((((((𝑁 ∈ ℕ0 ∧ (𝑆 ∈ V ∧ (1...𝑁) ⊆ 𝑆)) ∧ (𝑎 ∈ Fin ∧ 𝑏 ∈ (mzPoly‘𝑎))) ∧ 𝑎 ⊆ 𝑆) ∧ (𝑐 ∈ (ℤ≥‘𝑁) ∧ 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))) ∧ 𝑎 ⊆ 𝑆) → 𝑎 ⊆ 𝑆)
39		simprr 796	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑆 ∈ V ∧ (1...𝑁) ⊆ 𝑆)) → (1...𝑁) ⊆ 𝑆)
40	39	ad2antrr 762	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((((𝑁 ∈ ℕ0 ∧ (𝑆 ∈ V ∧ (1...𝑁) ⊆ 𝑆)) ∧ (𝑎 ∈ Fin ∧ 𝑏 ∈ (mzPoly‘𝑎))) ∧ 𝑎 ⊆ 𝑆) → (1...𝑁) ⊆ 𝑆)
41	38, 40	unssd 3789	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((((𝑁 ∈ ℕ0 ∧ (𝑆 ∈ V ∧ (1...𝑁) ⊆ 𝑆)) ∧ (𝑎 ∈ Fin ∧ 𝑏 ∈ (mzPoly‘𝑎))) ∧ 𝑎 ⊆ 𝑆) → (𝑎 ∪ (1...𝑁)) ⊆ 𝑆)
42	41	ad2antrr 762	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((((((𝑁 ∈ ℕ0 ∧ (𝑆 ∈ V ∧ (1...𝑁) ⊆ 𝑆)) ∧ (𝑎 ∈ Fin ∧ 𝑏 ∈ (mzPoly‘𝑎))) ∧ 𝑎 ⊆ 𝑆) ∧ (𝑐 ∈ (ℤ≥‘𝑁) ∧ 𝑑 ∈ V)) ∧ (𝑑:(1...𝑐)–1-1-onto→(𝑎 ∪ (1...𝑁)) ∧ (𝑑 ↾ (1...𝑁)) = ( I ↾ (1...𝑁)))) → (𝑎 ∪ (1...𝑁)) ⊆ 𝑆)
43		f1ss 6106	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑑:(1...𝑐)–1-1→(𝑎 ∪ (1...𝑁)) ∧ (𝑎 ∪ (1...𝑁)) ⊆ 𝑆) → 𝑑:(1...𝑐)–1-1→𝑆)
44	37, 42, 43	syl2anc 693	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((((𝑁 ∈ ℕ0 ∧ (𝑆 ∈ V ∧ (1...𝑁) ⊆ 𝑆)) ∧ (𝑎 ∈ Fin ∧ 𝑏 ∈ (mzPoly‘𝑎))) ∧ 𝑎 ⊆ 𝑆) ∧ (𝑐 ∈ (ℤ≥‘𝑁) ∧ 𝑑 ∈ V)) ∧ (𝑑:(1...𝑐)–1-1-onto→(𝑎 ∪ (1...𝑁)) ∧ (𝑑 ↾ (1...𝑁)) = ( I ↾ (1...𝑁)))) → 𝑑:(1...𝑐)–1-1→𝑆)
45		f1f 6101	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑑:(1...𝑐)–1-1→𝑆 → 𝑑:(1...𝑐)⟶𝑆)
46	44, 45	syl 17	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((((𝑁 ∈ ℕ0 ∧ (𝑆 ∈ V ∧ (1...𝑁) ⊆ 𝑆)) ∧ (𝑎 ∈ Fin ∧ 𝑏 ∈ (mzPoly‘𝑎))) ∧ 𝑎 ⊆ 𝑆) ∧ (𝑐 ∈ (ℤ≥‘𝑁) ∧ 𝑑 ∈ V)) ∧ (𝑑:(1...𝑐)–1-1-onto→(𝑎 ∪ (1...𝑁)) ∧ (𝑑 ↾ (1...𝑁)) = ( I ↾ (1...𝑁)))) → 𝑑:(1...𝑐)⟶𝑆)
47	46	adantr 481	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((((((𝑁 ∈ ℕ0 ∧ (𝑆 ∈ V ∧ (1...𝑁) ⊆ 𝑆)) ∧ (𝑎 ∈ Fin ∧ 𝑏 ∈ (mzPoly‘𝑎))) ∧ 𝑎 ⊆ 𝑆) ∧ (𝑐 ∈ (ℤ≥‘𝑁) ∧ 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 37277	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((1...𝑐) ∈ V ∧ 𝑒 ∈ (ℤ ↑𝑚 𝑆) ∧ 𝑑:(1...𝑐)⟶𝑆) → (𝑒 ∘ 𝑑) ∈ (ℤ ↑𝑚 (1...𝑐)))
49	34, 35, 47, 48	syl3anc 1326	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((((((𝑁 ∈ ℕ0 ∧ (𝑆 ∈ V ∧ (1...𝑁) ⊆ 𝑆)) ∧ (𝑎 ∈ Fin ∧ 𝑏 ∈ (mzPoly‘𝑎))) ∧ 𝑎 ⊆ 𝑆) ∧ (𝑐 ∈ (ℤ≥‘𝑁) ∧ 𝑑 ∈ V)) ∧ (𝑑:(1...𝑐)–1-1-onto→(𝑎 ∪ (1...𝑁)) ∧ (𝑑 ↾ (1...𝑁)) = ( I ↾ (1...𝑁)))) ∧ 𝑒 ∈ (ℤ ↑𝑚 𝑆)) → (𝑒 ∘ 𝑑) ∈ (ℤ ↑𝑚 (1...𝑐)))
50		coeq1 5279	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (ℎ = (𝑒 ∘ 𝑑) → (ℎ ∘ (◡𝑑 ↾ 𝑎)) = ((𝑒 ∘ 𝑑) ∘ (◡𝑑 ↾ 𝑎)))
51	50	fveq2d 6195	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (ℎ = (𝑒 ∘ 𝑑) → (𝑏‘(ℎ ∘ (◡𝑑 ↾ 𝑎))) = (𝑏‘((𝑒 ∘ 𝑑) ∘ (◡𝑑 ↾ 𝑎))))
52		eqid 2622	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (ℎ ∈ (ℤ ↑𝑚 (1...𝑐)) ↦ (𝑏‘(ℎ ∘ (◡𝑑 ↾ 𝑎)))) = (ℎ ∈ (ℤ ↑𝑚 (1...𝑐)) ↦ (𝑏‘(ℎ ∘ (◡𝑑 ↾ 𝑎))))
53		fvex 6201	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑏‘((𝑒 ∘ 𝑑) ∘ (◡𝑑 ↾ 𝑎))) ∈ V
54	51, 52, 53	fvmpt 6282	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑒 ∘ 𝑑) ∈ (ℤ ↑𝑚 (1...𝑐)) → ((ℎ ∈ (ℤ ↑𝑚 (1...𝑐)) ↦ (𝑏‘(ℎ ∘ (◡𝑑 ↾ 𝑎))))‘(𝑒 ∘ 𝑑)) = (𝑏‘((𝑒 ∘ 𝑑) ∘ (◡𝑑 ↾ 𝑎))))
55	49, 54	syl 17	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((((((𝑁 ∈ ℕ0 ∧ (𝑆 ∈ V ∧ (1...𝑁) ⊆ 𝑆)) ∧ (𝑎 ∈ Fin ∧ 𝑏 ∈ (mzPoly‘𝑎))) ∧ 𝑎 ⊆ 𝑆) ∧ (𝑐 ∈ (ℤ≥‘𝑁) ∧ 𝑑 ∈ V)) ∧ (𝑑:(1...𝑐)–1-1-onto→(𝑎 ∪ (1...𝑁)) ∧ (𝑑 ↾ (1...𝑁)) = ( I ↾ (1...𝑁)))) ∧ 𝑒 ∈ (ℤ ↑𝑚 𝑆)) → ((ℎ ∈ (ℤ ↑𝑚 (1...𝑐)) ↦ (𝑏‘(ℎ ∘ (◡𝑑 ↾ 𝑎))))‘(𝑒 ∘ 𝑑)) = (𝑏‘((𝑒 ∘ 𝑑) ∘ (◡𝑑 ↾ 𝑎))))
56	33, 55	eqtr4d 2659	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((((𝑁 ∈ ℕ0 ∧ (𝑆 ∈ V ∧ (1...𝑁) ⊆ 𝑆)) ∧ (𝑎 ∈ Fin ∧ 𝑏 ∈ (mzPoly‘𝑎))) ∧ 𝑎 ⊆ 𝑆) ∧ (𝑐 ∈ (ℤ≥‘𝑁) ∧ 𝑑 ∈ V)) ∧ (𝑑:(1...𝑐)–1-1-onto→(𝑎 ∪ (1...𝑁)) ∧ (𝑑 ↾ (1...𝑁)) = ( I ↾ (1...𝑁)))) ∧ 𝑒 ∈ (ℤ ↑𝑚 𝑆)) → (𝑏‘(𝑒 ↾ 𝑎)) = ((ℎ ∈ (ℤ ↑𝑚 (1...𝑐)) ↦ (𝑏‘(ℎ ∘ (◡𝑑 ↾ 𝑎))))‘(𝑒 ∘ 𝑑)))
57	56	mpteq2dva 4744	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((((𝑁 ∈ ℕ0 ∧ (𝑆 ∈ V ∧ (1...𝑁) ⊆ 𝑆)) ∧ (𝑎 ∈ Fin ∧ 𝑏 ∈ (mzPoly‘𝑎))) ∧ 𝑎 ⊆ 𝑆) ∧ (𝑐 ∈ (ℤ≥‘𝑁) ∧ 𝑑 ∈ V)) ∧ (𝑑:(1...𝑐)–1-1-onto→(𝑎 ∪ (1...𝑁)) ∧ (𝑑 ↾ (1...𝑁)) = ( I ↾ (1...𝑁)))) → (𝑒 ∈ (ℤ ↑𝑚 𝑆) ↦ (𝑏‘(𝑒 ↾ 𝑎))) = (𝑒 ∈ (ℤ ↑𝑚 𝑆) ↦ ((ℎ ∈ (ℤ ↑𝑚 (1...𝑐)) ↦ (𝑏‘(ℎ ∘ (◡𝑑 ↾ 𝑎))))‘(𝑒 ∘ 𝑑))))
58	57	fveq1d 6193	. . . . . . . . . . . . . . . . . 18 ⊢ ((((((𝑁 ∈ ℕ0 ∧ (𝑆 ∈ V ∧ (1...𝑁) ⊆ 𝑆)) ∧ (𝑎 ∈ Fin ∧ 𝑏 ∈ (mzPoly‘𝑎))) ∧ 𝑎 ⊆ 𝑆) ∧ (𝑐 ∈ (ℤ≥‘𝑁) ∧ 𝑑 ∈ V)) ∧ (𝑑:(1...𝑐)–1-1-onto→(𝑎 ∪ (1...𝑁)) ∧ (𝑑 ↾ (1...𝑁)) = ( I ↾ (1...𝑁)))) → ((𝑒 ∈ (ℤ ↑𝑚 𝑆) ↦ (𝑏‘(𝑒 ↾ 𝑎)))‘𝑢) = ((𝑒 ∈ (ℤ ↑𝑚 𝑆) ↦ ((ℎ ∈ (ℤ ↑𝑚 (1...𝑐)) ↦ (𝑏‘(ℎ ∘ (◡𝑑 ↾ 𝑎))))‘(𝑒 ∘ 𝑑)))‘𝑢))
59	58	eqeq1d 2624	. . . . . . . . . . . . . . . . 17 ⊢ ((((((𝑁 ∈ ℕ0 ∧ (𝑆 ∈ V ∧ (1...𝑁) ⊆ 𝑆)) ∧ (𝑎 ∈ Fin ∧ 𝑏 ∈ (mzPoly‘𝑎))) ∧ 𝑎 ⊆ 𝑆) ∧ (𝑐 ∈ (ℤ≥‘𝑁) ∧ 𝑑 ∈ V)) ∧ (𝑑:(1...𝑐)–1-1-onto→(𝑎 ∪ (1...𝑁)) ∧ (𝑑 ↾ (1...𝑁)) = ( I ↾ (1...𝑁)))) → (((𝑒 ∈ (ℤ ↑𝑚 𝑆) ↦ (𝑏‘(𝑒 ↾ 𝑎)))‘𝑢) = 0 ↔ ((𝑒 ∈ (ℤ ↑𝑚 𝑆) ↦ ((ℎ ∈ (ℤ ↑𝑚 (1...𝑐)) ↦ (𝑏‘(ℎ ∘ (◡𝑑 ↾ 𝑎))))‘(𝑒 ∘ 𝑑)))‘𝑢) = 0))
60	59	anbi2d 740	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝑁 ∈ ℕ0 ∧ (𝑆 ∈ V ∧ (1...𝑁) ⊆ 𝑆)) ∧ (𝑎 ∈ Fin ∧ 𝑏 ∈ (mzPoly‘𝑎))) ∧ 𝑎 ⊆ 𝑆) ∧ (𝑐 ∈ (ℤ≥‘𝑁) ∧ 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‘(𝑒 ↾ 𝑎)))‘𝑢) = 0) ↔ (𝑡 = (𝑢 ↾ (1...𝑁)) ∧ ((𝑒 ∈ (ℤ ↑𝑚 𝑆) ↦ ((ℎ ∈ (ℤ ↑𝑚 (1...𝑐)) ↦ (𝑏‘(ℎ ∘ (◡𝑑 ↾ 𝑎))))‘(𝑒 ∘ 𝑑)))‘𝑢) = 0)))
61	60	rexbidv 3052	. . . . . . . . . . . . . . 15 ⊢ ((((((𝑁 ∈ ℕ0 ∧ (𝑆 ∈ V ∧ (1...𝑁) ⊆ 𝑆)) ∧ (𝑎 ∈ Fin ∧ 𝑏 ∈ (mzPoly‘𝑎))) ∧ 𝑎 ⊆ 𝑆) ∧ (𝑐 ∈ (ℤ≥‘𝑁) ∧ 𝑑 ∈ V)) ∧ (𝑑:(1...𝑐)–1-1-onto→(𝑎 ∪ (1...𝑁)) ∧ (𝑑 ↾ (1...𝑁)) = ( I ↾ (1...𝑁)))) → (∃𝑢 ∈ (ℕ0 ↑𝑚 𝑆)(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ ((𝑒 ∈ (ℤ ↑𝑚 𝑆) ↦ (𝑏‘(𝑒 ↾ 𝑎)))‘𝑢) = 0) ↔ ∃𝑢 ∈ (ℕ0 ↑𝑚 𝑆)(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ ((𝑒 ∈ (ℤ ↑𝑚 𝑆) ↦ ((ℎ ∈ (ℤ ↑𝑚 (1...𝑐)) ↦ (𝑏‘(ℎ ∘ (◡𝑑 ↾ 𝑎))))‘(𝑒 ∘ 𝑑)))‘𝑢) = 0)))
62	61	abbidv 2741	. . . . . . . . . . . . . 14 ⊢ ((((((𝑁 ∈ ℕ0 ∧ (𝑆 ∈ V ∧ (1...𝑁) ⊆ 𝑆)) ∧ (𝑎 ∈ Fin ∧ 𝑏 ∈ (mzPoly‘𝑎))) ∧ 𝑎 ⊆ 𝑆) ∧ (𝑐 ∈ (ℤ≥‘𝑁) ∧ 𝑑 ∈ V)) ∧ (𝑑:(1...𝑐)–1-1-onto→(𝑎 ∪ (1...𝑁)) ∧ (𝑑 ↾ (1...𝑁)) = ( I ↾ (1...𝑁)))) → {𝑡 ∣ ∃𝑢 ∈ (ℕ0 ↑𝑚 𝑆)(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ ((𝑒 ∈ (ℤ ↑𝑚 𝑆) ↦ (𝑏‘(𝑒 ↾ 𝑎)))‘𝑢) = 0)} = {𝑡 ∣ ∃𝑢 ∈ (ℕ0 ↑𝑚 𝑆)(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ ((𝑒 ∈ (ℤ ↑𝑚 𝑆) ↦ ((ℎ ∈ (ℤ ↑𝑚 (1...𝑐)) ↦ (𝑏‘(ℎ ∘ (◡𝑑 ↾ 𝑎))))‘(𝑒 ∘ 𝑑)))‘𝑢) = 0)})
63		simplrl 800	. . . . . . . . . . . . . . . 16 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑆 ∈ V ∧ (1...𝑁) ⊆ 𝑆)) ∧ (𝑎 ∈ Fin ∧ 𝑏 ∈ (mzPoly‘𝑎))) → 𝑆 ∈ V)
64	63	ad3antrrr 766	. . . . . . . . . . . . . . 15 ⊢ ((((((𝑁 ∈ ℕ0 ∧ (𝑆 ∈ V ∧ (1...𝑁) ⊆ 𝑆)) ∧ (𝑎 ∈ Fin ∧ 𝑏 ∈ (mzPoly‘𝑎))) ∧ 𝑎 ⊆ 𝑆) ∧ (𝑐 ∈ (ℤ≥‘𝑁) ∧ 𝑑 ∈ V)) ∧ (𝑑:(1...𝑐)–1-1-onto→(𝑎 ∪ (1...𝑁)) ∧ (𝑑 ↾ (1...𝑁)) = ( I ↾ (1...𝑁)))) → 𝑆 ∈ V)
65		simprr 796	. . . . . . . . . . . . . . 15 ⊢ ((((((𝑁 ∈ ℕ0 ∧ (𝑆 ∈ V ∧ (1...𝑁) ⊆ 𝑆)) ∧ (𝑎 ∈ Fin ∧ 𝑏 ∈ (mzPoly‘𝑎))) ∧ 𝑎 ⊆ 𝑆) ∧ (𝑐 ∈ (ℤ≥‘𝑁) ∧ 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 37322	. . . . . . . . . . . . . . 15 ⊢ ((𝑆 ∈ V ∧ 𝑑:(1...𝑐)–1-1→𝑆 ∧ (𝑑 ↾ (1...𝑁)) = ( I ↾ (1...𝑁))) → {𝑡 ∣ ∃𝑢 ∈ (ℕ0 ↑𝑚 𝑆)(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ ((𝑒 ∈ (ℤ ↑𝑚 𝑆) ↦ ((ℎ ∈ (ℤ ↑𝑚 (1...𝑐)) ↦ (𝑏‘(ℎ ∘ (◡𝑑 ↾ 𝑎))))‘(𝑒 ∘ 𝑑)))‘𝑢) = 0)} = {𝑡 ∣ ∃𝑔 ∈ (ℕ0 ↑𝑚 (1...𝑐))(𝑡 = (𝑔 ↾ (1...𝑁)) ∧ ((ℎ ∈ (ℤ ↑𝑚 (1...𝑐)) ↦ (𝑏‘(ℎ ∘ (◡𝑑 ↾ 𝑎))))‘𝑔) = 0)})
67	64, 44, 65, 66	syl3anc 1326	. . . . . . . . . . . . . 14 ⊢ ((((((𝑁 ∈ ℕ0 ∧ (𝑆 ∈ V ∧ (1...𝑁) ⊆ 𝑆)) ∧ (𝑎 ∈ Fin ∧ 𝑏 ∈ (mzPoly‘𝑎))) ∧ 𝑎 ⊆ 𝑆) ∧ (𝑐 ∈ (ℤ≥‘𝑁) ∧ 𝑑 ∈ V)) ∧ (𝑑:(1...𝑐)–1-1-onto→(𝑎 ∪ (1...𝑁)) ∧ (𝑑 ↾ (1...𝑁)) = ( I ↾ (1...𝑁)))) → {𝑡 ∣ ∃𝑢 ∈ (ℕ0 ↑𝑚 𝑆)(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ ((𝑒 ∈ (ℤ ↑𝑚 𝑆) ↦ ((ℎ ∈ (ℤ ↑𝑚 (1...𝑐)) ↦ (𝑏‘(ℎ ∘ (◡𝑑 ↾ 𝑎))))‘(𝑒 ∘ 𝑑)))‘𝑢) = 0)} = {𝑡 ∣ ∃𝑔 ∈ (ℕ0 ↑𝑚 (1...𝑐))(𝑡 = (𝑔 ↾ (1...𝑁)) ∧ ((ℎ ∈ (ℤ ↑𝑚 (1...𝑐)) ↦ (𝑏‘(ℎ ∘ (◡𝑑 ↾ 𝑎))))‘𝑔) = 0)})
68	62, 67	eqtrd 2656	. . . . . . . . . . . . 13 ⊢ ((((((𝑁 ∈ ℕ0 ∧ (𝑆 ∈ V ∧ (1...𝑁) ⊆ 𝑆)) ∧ (𝑎 ∈ Fin ∧ 𝑏 ∈ (mzPoly‘𝑎))) ∧ 𝑎 ⊆ 𝑆) ∧ (𝑐 ∈ (ℤ≥‘𝑁) ∧ 𝑑 ∈ V)) ∧ (𝑑:(1...𝑐)–1-1-onto→(𝑎 ∪ (1...𝑁)) ∧ (𝑑 ↾ (1...𝑁)) = ( I ↾ (1...𝑁)))) → {𝑡 ∣ ∃𝑢 ∈ (ℕ0 ↑𝑚 𝑆)(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ ((𝑒 ∈ (ℤ ↑𝑚 𝑆) ↦ (𝑏‘(𝑒 ↾ 𝑎)))‘𝑢) = 0)} = {𝑡 ∣ ∃𝑔 ∈ (ℕ0 ↑𝑚 (1...𝑐))(𝑡 = (𝑔 ↾ (1...𝑁)) ∧ ((ℎ ∈ (ℤ ↑𝑚 (1...𝑐)) ↦ (𝑏‘(ℎ ∘ (◡𝑑 ↾ 𝑎))))‘𝑔) = 0)})
69		simp-5l 808	. . . . . . . . . . . . . 14 ⊢ ((((((𝑁 ∈ ℕ0 ∧ (𝑆 ∈ V ∧ (1...𝑁) ⊆ 𝑆)) ∧ (𝑎 ∈ Fin ∧ 𝑏 ∈ (mzPoly‘𝑎))) ∧ 𝑎 ⊆ 𝑆) ∧ (𝑐 ∈ (ℤ≥‘𝑁) ∧ 𝑑 ∈ V)) ∧ (𝑑:(1...𝑐)–1-1-onto→(𝑎 ∪ (1...𝑁)) ∧ (𝑑 ↾ (1...𝑁)) = ( I ↾ (1...𝑁)))) → 𝑁 ∈ ℕ0)
70		simplrl 800	. . . . . . . . . . . . . 14 ⊢ ((((((𝑁 ∈ ℕ0 ∧ (𝑆 ∈ V ∧ (1...𝑁) ⊆ 𝑆)) ∧ (𝑎 ∈ Fin ∧ 𝑏 ∈ (mzPoly‘𝑎))) ∧ 𝑎 ⊆ 𝑆) ∧ (𝑐 ∈ (ℤ≥‘𝑁) ∧ 𝑑 ∈ V)) ∧ (𝑑:(1...𝑐)–1-1-onto→(𝑎 ∪ (1...𝑁)) ∧ (𝑑 ↾ (1...𝑁)) = ( I ↾ (1...𝑁)))) → 𝑐 ∈ (ℤ≥‘𝑁))
71		ovexd 6680	. . . . . . . . . . . . . . 15 ⊢ ((((((𝑁 ∈ ℕ0 ∧ (𝑆 ∈ V ∧ (1...𝑁) ⊆ 𝑆)) ∧ (𝑎 ∈ Fin ∧ 𝑏 ∈ (mzPoly‘𝑎))) ∧ 𝑎 ⊆ 𝑆) ∧ (𝑐 ∈ (ℤ≥‘𝑁) ∧ 𝑑 ∈ V)) ∧ (𝑑:(1...𝑐)–1-1-onto→(𝑎 ∪ (1...𝑁)) ∧ (𝑑 ↾ (1...𝑁)) = ( I ↾ (1...𝑁)))) → (1...𝑐) ∈ V)
72		simplrr 801	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑁 ∈ ℕ0 ∧ (𝑆 ∈ V ∧ (1...𝑁) ⊆ 𝑆)) ∧ (𝑎 ∈ Fin ∧ 𝑏 ∈ (mzPoly‘𝑎))) ∧ 𝑎 ⊆ 𝑆) → 𝑏 ∈ (mzPoly‘𝑎))
73	72	ad2antrr 762	. . . . . . . . . . . . . . 15 ⊢ ((((((𝑁 ∈ ℕ0 ∧ (𝑆 ∈ V ∧ (1...𝑁) ⊆ 𝑆)) ∧ (𝑎 ∈ Fin ∧ 𝑏 ∈ (mzPoly‘𝑎))) ∧ 𝑎 ⊆ 𝑆) ∧ (𝑐 ∈ (ℤ≥‘𝑁) ∧ 𝑑 ∈ V)) ∧ (𝑑:(1...𝑐)–1-1-onto→(𝑎 ∪ (1...𝑁)) ∧ (𝑑 ↾ (1...𝑁)) = ( I ↾ (1...𝑁)))) → 𝑏 ∈ (mzPoly‘𝑎))
74		f1ocnv 6149	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑑:(1...𝑐)–1-1-onto→(𝑎 ∪ (1...𝑁)) → ◡𝑑:(𝑎 ∪ (1...𝑁))–1-1-onto→(1...𝑐))
75		f1of 6137	. . . . . . . . . . . . . . . . . 18 ⊢ (◡𝑑:(𝑎 ∪ (1...𝑁))–1-1-onto→(1...𝑐) → ◡𝑑:(𝑎 ∪ (1...𝑁))⟶(1...𝑐))
76	74, 75	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (𝑑:(1...𝑐)–1-1-onto→(𝑎 ∪ (1...𝑁)) → ◡𝑑:(𝑎 ∪ (1...𝑁))⟶(1...𝑐))
77		fssres 6070	. . . . . . . . . . . . . . . . 17 ⊢ ((◡𝑑:(𝑎 ∪ (1...𝑁))⟶(1...𝑐) ∧ 𝑎 ⊆ (𝑎 ∪ (1...𝑁))) → (◡𝑑 ↾ 𝑎):𝑎⟶(1...𝑐))
78	76, 21, 77	sylancl 694	. . . . . . . . . . . . . . . 16 ⊢ (𝑑:(1...𝑐)–1-1-onto→(𝑎 ∪ (1...𝑁)) → (◡𝑑 ↾ 𝑎):𝑎⟶(1...𝑐))
79	78	ad2antrl 764	. . . . . . . . . . . . . . 15 ⊢ ((((((𝑁 ∈ ℕ0 ∧ (𝑆 ∈ V ∧ (1...𝑁) ⊆ 𝑆)) ∧ (𝑎 ∈ Fin ∧ 𝑏 ∈ (mzPoly‘𝑎))) ∧ 𝑎 ⊆ 𝑆) ∧ (𝑐 ∈ (ℤ≥‘𝑁) ∧ 𝑑 ∈ V)) ∧ (𝑑:(1...𝑐)–1-1-onto→(𝑎 ∪ (1...𝑁)) ∧ (𝑑 ↾ (1...𝑁)) = ( I ↾ (1...𝑁)))) → (◡𝑑 ↾ 𝑎):𝑎⟶(1...𝑐))
80		mzprename 37312	. . . . . . . . . . . . . . 15 ⊢ (((1...𝑐) ∈ V ∧ 𝑏 ∈ (mzPoly‘𝑎) ∧ (◡𝑑 ↾ 𝑎):𝑎⟶(1...𝑐)) → (ℎ ∈ (ℤ ↑𝑚 (1...𝑐)) ↦ (𝑏‘(ℎ ∘ (◡𝑑 ↾ 𝑎)))) ∈ (mzPoly‘(1...𝑐)))
81	71, 73, 79, 80	syl3anc 1326	. . . . . . . . . . . . . 14 ⊢ ((((((𝑁 ∈ ℕ0 ∧ (𝑆 ∈ V ∧ (1...𝑁) ⊆ 𝑆)) ∧ (𝑎 ∈ Fin ∧ 𝑏 ∈ (mzPoly‘𝑎))) ∧ 𝑎 ⊆ 𝑆) ∧ (𝑐 ∈ (ℤ≥‘𝑁) ∧ 𝑑 ∈ V)) ∧ (𝑑:(1...𝑐)–1-1-onto→(𝑎 ∪ (1...𝑁)) ∧ (𝑑 ↾ (1...𝑁)) = ( I ↾ (1...𝑁)))) → (ℎ ∈ (ℤ ↑𝑚 (1...𝑐)) ↦ (𝑏‘(ℎ ∘ (◡𝑑 ↾ 𝑎)))) ∈ (mzPoly‘(1...𝑐)))
82		eldioph 37321	. . . . . . . . . . . . . 14 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑐 ∈ (ℤ≥‘𝑁) ∧ (ℎ ∈ (ℤ ↑𝑚 (1...𝑐)) ↦ (𝑏‘(ℎ ∘ (◡𝑑 ↾ 𝑎)))) ∈ (mzPoly‘(1...𝑐))) → {𝑡 ∣ ∃𝑔 ∈ (ℕ0 ↑𝑚 (1...𝑐))(𝑡 = (𝑔 ↾ (1...𝑁)) ∧ ((ℎ ∈ (ℤ ↑𝑚 (1...𝑐)) ↦ (𝑏‘(ℎ ∘ (◡𝑑 ↾ 𝑎))))‘𝑔) = 0)} ∈ (Dioph‘𝑁))
83	69, 70, 81, 82	syl3anc 1326	. . . . . . . . . . . . 13 ⊢ ((((((𝑁 ∈ ℕ0 ∧ (𝑆 ∈ V ∧ (1...𝑁) ⊆ 𝑆)) ∧ (𝑎 ∈ Fin ∧ 𝑏 ∈ (mzPoly‘𝑎))) ∧ 𝑎 ⊆ 𝑆) ∧ (𝑐 ∈ (ℤ≥‘𝑁) ∧ 𝑑 ∈ V)) ∧ (𝑑:(1...𝑐)–1-1-onto→(𝑎 ∪ (1...𝑁)) ∧ (𝑑 ↾ (1...𝑁)) = ( I ↾ (1...𝑁)))) → {𝑡 ∣ ∃𝑔 ∈ (ℕ0 ↑𝑚 (1...𝑐))(𝑡 = (𝑔 ↾ (1...𝑁)) ∧ ((ℎ ∈ (ℤ ↑𝑚 (1...𝑐)) ↦ (𝑏‘(ℎ ∘ (◡𝑑 ↾ 𝑎))))‘𝑔) = 0)} ∈ (Dioph‘𝑁))
84	68, 83	eqeltrd 2701	. . . . . . . . . . . 12 ⊢ ((((((𝑁 ∈ ℕ0 ∧ (𝑆 ∈ V ∧ (1...𝑁) ⊆ 𝑆)) ∧ (𝑎 ∈ Fin ∧ 𝑏 ∈ (mzPoly‘𝑎))) ∧ 𝑎 ⊆ 𝑆) ∧ (𝑐 ∈ (ℤ≥‘𝑁) ∧ 𝑑 ∈ V)) ∧ (𝑑:(1...𝑐)–1-1-onto→(𝑎 ∪ (1...𝑁)) ∧ (𝑑 ↾ (1...𝑁)) = ( I ↾ (1...𝑁)))) → {𝑡 ∣ ∃𝑢 ∈ (ℕ0 ↑𝑚 𝑆)(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ ((𝑒 ∈ (ℤ ↑𝑚 𝑆) ↦ (𝑏‘(𝑒 ↾ 𝑎)))‘𝑢) = 0)} ∈ (Dioph‘𝑁))
85	84	ex 450	. . . . . . . . . . 11 ⊢ (((((𝑁 ∈ ℕ0 ∧ (𝑆 ∈ V ∧ (1...𝑁) ⊆ 𝑆)) ∧ (𝑎 ∈ Fin ∧ 𝑏 ∈ (mzPoly‘𝑎))) ∧ 𝑎 ⊆ 𝑆) ∧ (𝑐 ∈ (ℤ≥‘𝑁) ∧ 𝑑 ∈ V)) → ((𝑑:(1...𝑐)–1-1-onto→(𝑎 ∪ (1...𝑁)) ∧ (𝑑 ↾ (1...𝑁)) = ( I ↾ (1...𝑁))) → {𝑡 ∣ ∃𝑢 ∈ (ℕ0 ↑𝑚 𝑆)(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ ((𝑒 ∈ (ℤ ↑𝑚 𝑆) ↦ (𝑏‘(𝑒 ↾ 𝑎)))‘𝑢) = 0)} ∈ (Dioph‘𝑁)))
86	85	rexlimdvva 3038	. . . . . . . . . 10 ⊢ ((((𝑁 ∈ ℕ0 ∧ (𝑆 ∈ V ∧ (1...𝑁) ⊆ 𝑆)) ∧ (𝑎 ∈ Fin ∧ 𝑏 ∈ (mzPoly‘𝑎))) ∧ 𝑎 ⊆ 𝑆) → (∃𝑐 ∈ (ℤ≥‘𝑁)∃𝑑 ∈ V (𝑑:(1...𝑐)–1-1-onto→(𝑎 ∪ (1...𝑁)) ∧ (𝑑 ↾ (1...𝑁)) = ( I ↾ (1...𝑁))) → {𝑡 ∣ ∃𝑢 ∈ (ℕ0 ↑𝑚 𝑆)(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ ((𝑒 ∈ (ℤ ↑𝑚 𝑆) ↦ (𝑏‘(𝑒 ↾ 𝑎)))‘𝑢) = 0)} ∈ (Dioph‘𝑁)))
87	17, 86	mpd 15	. . . . . . . . 9 ⊢ ((((𝑁 ∈ ℕ0 ∧ (𝑆 ∈ V ∧ (1...𝑁) ⊆ 𝑆)) ∧ (𝑎 ∈ Fin ∧ 𝑏 ∈ (mzPoly‘𝑎))) ∧ 𝑎 ⊆ 𝑆) → {𝑡 ∣ ∃𝑢 ∈ (ℕ0 ↑𝑚 𝑆)(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ ((𝑒 ∈ (ℤ ↑𝑚 𝑆) ↦ (𝑏‘(𝑒 ↾ 𝑎)))‘𝑢) = 0)} ∈ (Dioph‘𝑁))
88	87	exp31 630	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑆 ∈ V ∧ (1...𝑁) ⊆ 𝑆)) → ((𝑎 ∈ Fin ∧ 𝑏 ∈ (mzPoly‘𝑎)) → (𝑎 ⊆ 𝑆 → {𝑡 ∣ ∃𝑢 ∈ (ℕ0 ↑𝑚 𝑆)(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ ((𝑒 ∈ (ℤ ↑𝑚 𝑆) ↦ (𝑏‘(𝑒 ↾ 𝑎)))‘𝑢) = 0)} ∈ (Dioph‘𝑁))))
89	88	3adant3 1081	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑆 ∈ V ∧ (1...𝑁) ⊆ 𝑆) ∧ 𝑃 ∈ (mzPoly‘𝑆)) → ((𝑎 ∈ Fin ∧ 𝑏 ∈ (mzPoly‘𝑎)) → (𝑎 ⊆ 𝑆 → {𝑡 ∣ ∃𝑢 ∈ (ℕ0 ↑𝑚 𝑆)(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ ((𝑒 ∈ (ℤ ↑𝑚 𝑆) ↦ (𝑏‘(𝑒 ↾ 𝑎)))‘𝑢) = 0)} ∈ (Dioph‘𝑁))))
90	89	imp31 448	. . . . . 6 ⊢ ((((𝑁 ∈ ℕ0 ∧ (𝑆 ∈ V ∧ (1...𝑁) ⊆ 𝑆) ∧ 𝑃 ∈ (mzPoly‘𝑆)) ∧ (𝑎 ∈ Fin ∧ 𝑏 ∈ (mzPoly‘𝑎))) ∧ 𝑎 ⊆ 𝑆) → {𝑡 ∣ ∃𝑢 ∈ (ℕ0 ↑𝑚 𝑆)(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ ((𝑒 ∈ (ℤ ↑𝑚 𝑆) ↦ (𝑏‘(𝑒 ↾ 𝑎)))‘𝑢) = 0)} ∈ (Dioph‘𝑁))
91	90	adantrr 753	. . . . 5 ⊢ ((((𝑁 ∈ ℕ0 ∧ (𝑆 ∈ V ∧ (1...𝑁) ⊆ 𝑆) ∧ 𝑃 ∈ (mzPoly‘𝑆)) ∧ (𝑎 ∈ Fin ∧ 𝑏 ∈ (mzPoly‘𝑎))) ∧ (𝑎 ⊆ 𝑆 ∧ 𝑃 = (𝑒 ∈ (ℤ ↑𝑚 𝑆) ↦ (𝑏‘(𝑒 ↾ 𝑎))))) → {𝑡 ∣ ∃𝑢 ∈ (ℕ0 ↑𝑚 𝑆)(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ ((𝑒 ∈ (ℤ ↑𝑚 𝑆) ↦ (𝑏‘(𝑒 ↾ 𝑎)))‘𝑢) = 0)} ∈ (Dioph‘𝑁))
92	8, 91	eqeltrd 2701	. . . 4 ⊢ ((((𝑁 ∈ ℕ0 ∧ (𝑆 ∈ V ∧ (1...𝑁) ⊆ 𝑆) ∧ 𝑃 ∈ (mzPoly‘𝑆)) ∧ (𝑎 ∈ Fin ∧ 𝑏 ∈ (mzPoly‘𝑎))) ∧ (𝑎 ⊆ 𝑆 ∧ 𝑃 = (𝑒 ∈ (ℤ ↑𝑚 𝑆) ↦ (𝑏‘(𝑒 ↾ 𝑎))))) → {𝑡 ∣ ∃𝑢 ∈ (ℕ0 ↑𝑚 𝑆)(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑃‘𝑢) = 0)} ∈ (Dioph‘𝑁))
93	92	ex 450	. . 3 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑆 ∈ V ∧ (1...𝑁) ⊆ 𝑆) ∧ 𝑃 ∈ (mzPoly‘𝑆)) ∧ (𝑎 ∈ Fin ∧ 𝑏 ∈ (mzPoly‘𝑎))) → ((𝑎 ⊆ 𝑆 ∧ 𝑃 = (𝑒 ∈ (ℤ ↑𝑚 𝑆) ↦ (𝑏‘(𝑒 ↾ 𝑎)))) → {𝑡 ∣ ∃𝑢 ∈ (ℕ0 ↑𝑚 𝑆)(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑃‘𝑢) = 0)} ∈ (Dioph‘𝑁)))
94	93	rexlimdvva 3038	. 2 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑆 ∈ V ∧ (1...𝑁) ⊆ 𝑆) ∧ 𝑃 ∈ (mzPoly‘𝑆)) → (∃𝑎 ∈ Fin ∃𝑏 ∈ (mzPoly‘𝑎)(𝑎 ⊆ 𝑆 ∧ 𝑃 = (𝑒 ∈ (ℤ ↑𝑚 𝑆) ↦ (𝑏‘(𝑒 ↾ 𝑎)))) → {𝑡 ∣ ∃𝑢 ∈ (ℕ0 ↑𝑚 𝑆)(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑃‘𝑢) = 0)} ∈ (Dioph‘𝑁)))
95	2, 94	mpd 15	1 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑆 ∈ V ∧ (1...𝑁) ⊆ 𝑆) ∧ 𝑃 ∈ (mzPoly‘𝑆)) → {𝑡 ∣ ∃𝑢 ∈ (ℕ0 ↑𝑚 𝑆)(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑃‘𝑢) = 0)} ∈ (Dioph‘𝑁))

Syntax hints:   → wi 4   ∧ wa 384   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990  {cab 2608  ∃wrex 2913  Vcvv 3200   ∪ cun 3572   ⊆ wss 3574   ↦ cmpt 4729   I cid 5023  ^◡ccnv 5113   ↾ cres 5116   ∘ ccom 5118  ⟶wf 5884  –1-1→wf1 5885  –1-1-onto→wf1o 5887  ‘cfv 5888  (class class class)co 6650   ↑_𝑚 cmap 7857  Fincfn 7955  0cc0 9936  1c1 9937  ℕ₀cn0 11292  ℤcz 11377  ℤ_≥cuz 11687  ...cfz 12326  mzPolycmzp 37285  Diophcdioph 37318

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-of 6897  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-card 8765  df-cda 8990  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-n0 11293  df-z 11378  df-uz 11688  df-fz 12327  df-hash 13118  df-mzpcl 37286  df-mzp 37287  df-dioph 37319

This theorem is referenced by:  eldioph2b  37326  diophin  37336  diophun  37337