Theorem eldioph2 42755

Description: Construct a Diophantine set from a polynomial with witness variables drawn from any set whatsoever, via mzpcompact2 42745. (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 42745	. . 3 ⊢ (𝑃 ∈ (mzPoly‘𝑆) → ∃𝑎 ∈ Fin ∃𝑏 ∈ (mzPoly‘𝑎)(𝑎 ⊆ 𝑆 ∧ 𝑃 = (𝑒 ∈ (ℤ ↑m 𝑆) ↦ (𝑏‘(𝑒 ↾ 𝑎)))))
2	1	3ad2ant3 1135	. 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 6821	. . . . . . . . . 10 ⊢ (𝑃 = (𝑒 ∈ (ℤ ↑m 𝑆) ↦ (𝑏‘(𝑒 ↾ 𝑎))) → (𝑃‘𝑢) = ((𝑒 ∈ (ℤ ↑m 𝑆) ↦ (𝑏‘(𝑒 ↾ 𝑎)))‘𝑢))
4	3	eqeq1d 2731	. . . . . . . . 9 ⊢ (𝑃 = (𝑒 ∈ (ℤ ↑m 𝑆) ↦ (𝑏‘(𝑒 ↾ 𝑎))) → ((𝑃‘𝑢) = 0 ↔ ((𝑒 ∈ (ℤ ↑m 𝑆) ↦ (𝑏‘(𝑒 ↾ 𝑎)))‘𝑢) = 0))
5	4	anbi2d 630	. . . . . . . 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 3153	. . . . . . 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 2795	. . . . . 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 729	. . . . 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 774	. . . . . . . . . . 11 ⊢ ((((𝑁 ∈ ℕ0 ∧ (𝑆 ∈ V ∧ (1...𝑁) ⊆ 𝑆)) ∧ (𝑎 ∈ Fin ∧ 𝑏 ∈ (mzPoly‘𝑎))) ∧ 𝑎 ⊆ 𝑆) → 𝑁 ∈ ℕ0)
10		simplrl 776	. . . . . . . . . . . 12 ⊢ ((((𝑁 ∈ ℕ0 ∧ (𝑆 ∈ V ∧ (1...𝑁) ⊆ 𝑆)) ∧ (𝑎 ∈ Fin ∧ 𝑏 ∈ (mzPoly‘𝑎))) ∧ 𝑎 ⊆ 𝑆) → 𝑎 ∈ Fin)
11		fzfi 13879	. . . . . . . . . . . 12 ⊢ (1...𝑁) ∈ Fin
12		unfi 9085	. . . . . . . . . . . 12 ⊢ ((𝑎 ∈ Fin ∧ (1...𝑁) ∈ Fin) → (𝑎 ∪ (1...𝑁)) ∈ Fin)
13	10, 11, 12	sylancl 586	. . . . . . . . . . 11 ⊢ ((((𝑁 ∈ ℕ0 ∧ (𝑆 ∈ V ∧ (1...𝑁) ⊆ 𝑆)) ∧ (𝑎 ∈ Fin ∧ 𝑏 ∈ (mzPoly‘𝑎))) ∧ 𝑎 ⊆ 𝑆) → (𝑎 ∪ (1...𝑁)) ∈ Fin)
14		ssun2 4130	. . . . . . . . . . . 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 42753	. . . . . . . . . . 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 1373	. . . . . . . . . 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 6791	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑑:(1...𝑐)–1-1-onto→(𝑎 ∪ (1...𝑁)) → (𝑑 ∘ ◡𝑑) = ( I ↾ (𝑎 ∪ (1...𝑁))))
19	18	ad2antrl 728	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 5929	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 4129	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ 𝑎 ⊆ (𝑎 ∪ (1...𝑁))
22		resabs1 5957	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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	eqtr2di 2781	. . . . . . . . . . . . . . . . . . . . . . . . . 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 6199	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑑 ∘ ◡𝑑) ↾ 𝑎) = (𝑑 ∘ (◡𝑑 ↾ 𝑎))
26	24, 25	eqtrdi 2780	. . . . . . . . . . . . . . . . . . . . . . . . 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 480	. . . . . . . . . . . . . . . . . . . . . . . 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 5805	. . . . . . . . . . . . . . . . . . . . . . 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 6213	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑒 ∘ ( I ↾ 𝑎)) = (𝑒 ↾ 𝑎)
30		coass 6214	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑒 ∘ 𝑑) ∘ (◡𝑑 ↾ 𝑎)) = (𝑒 ∘ (𝑑 ∘ (◡𝑑 ↾ 𝑎)))
31	30	eqcomi 2738	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑒 ∘ (𝑑 ∘ (◡𝑑 ↾ 𝑎))) = ((𝑒 ∘ 𝑑) ∘ (◡𝑑 ↾ 𝑎))
32	28, 29, 31	3eqtr3g 2787	. . . . . . . . . . . . . . . . . . . . . 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 6826	. . . . . . . . . . . . . . . . . . . . 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 7384	. . . . . . . . . . . . . . . . . . . . . . 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 484	. . . . . . . . . . . . . . . . . . . . . . 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 6763	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑑:(1...𝑐)–1-1-onto→(𝑎 ∪ (1...𝑁)) → 𝑑:(1...𝑐)–1-1→(𝑎 ∪ (1...𝑁)))
37	36	ad2antrl 728	. . . . . . . . . . . . . . . . . . . . . . . . . 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 484	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((((𝑁 ∈ ℕ0 ∧ (𝑆 ∈ V ∧ (1...𝑁) ⊆ 𝑆)) ∧ (𝑎 ∈ Fin ∧ 𝑏 ∈ (mzPoly‘𝑎))) ∧ 𝑎 ⊆ 𝑆) → 𝑎 ⊆ 𝑆)
39		simprr 772	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑆 ∈ V ∧ (1...𝑁) ⊆ 𝑆)) → (1...𝑁) ⊆ 𝑆)
40	39	ad2antrr 726	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((((𝑁 ∈ ℕ0 ∧ (𝑆 ∈ V ∧ (1...𝑁) ⊆ 𝑆)) ∧ (𝑎 ∈ Fin ∧ 𝑏 ∈ (mzPoly‘𝑎))) ∧ 𝑎 ⊆ 𝑆) → (1...𝑁) ⊆ 𝑆)
41	38, 40	unssd 4143	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((((𝑁 ∈ ℕ0 ∧ (𝑆 ∈ V ∧ (1...𝑁) ⊆ 𝑆)) ∧ (𝑎 ∈ Fin ∧ 𝑏 ∈ (mzPoly‘𝑎))) ∧ 𝑎 ⊆ 𝑆) → (𝑎 ∪ (1...𝑁)) ⊆ 𝑆)
42	41	ad2antrr 726	. . . . . . . . . . . . . . . . . . . . . . . . . 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 6725	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑑:(1...𝑐)–1-1→(𝑎 ∪ (1...𝑁)) ∧ (𝑎 ∪ (1...𝑁)) ⊆ 𝑆) → 𝑑:(1...𝑐)–1-1→𝑆)
44	37, 42, 43	syl2anc 584	. . . . . . . . . . . . . . . . . . . . . . . . 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 6720	. . . . . . . . . . . . . . . . . . . . . . . . 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 480	. . . . . . . . . . . . . . . . . . . . . . 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 42707	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((1...𝑐) ∈ V ∧ 𝑒 ∈ (ℤ ↑m 𝑆) ∧ 𝑑:(1...𝑐)⟶𝑆) → (𝑒 ∘ 𝑑) ∈ (ℤ ↑m (1...𝑐)))
49	34, 35, 47, 48	syl3anc 1373	. . . . . . . . . . . . . . . . . . . . . 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 5800	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (ℎ = (𝑒 ∘ 𝑑) → (ℎ ∘ (◡𝑑 ↾ 𝑎)) = ((𝑒 ∘ 𝑑) ∘ (◡𝑑 ↾ 𝑎)))
51	50	fveq2d 6826	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (ℎ = (𝑒 ∘ 𝑑) → (𝑏‘(ℎ ∘ (◡𝑑 ↾ 𝑎))) = (𝑏‘((𝑒 ∘ 𝑑) ∘ (◡𝑑 ↾ 𝑎))))
52		eqid 2729	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (ℎ ∈ (ℤ ↑m (1...𝑐)) ↦ (𝑏‘(ℎ ∘ (◡𝑑 ↾ 𝑎)))) = (ℎ ∈ (ℤ ↑m (1...𝑐)) ↦ (𝑏‘(ℎ ∘ (◡𝑑 ↾ 𝑎))))
53		fvex 6835	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑏‘((𝑒 ∘ 𝑑) ∘ (◡𝑑 ↾ 𝑎))) ∈ V
54	51, 52, 53	fvmpt 6930	. . . . . . . . . . . . . . . . . . . . . 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 2767	. . . . . . . . . . . . . . . . . . . 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 5185	. . . . . . . . . . . . . . . . . . 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 6824	. . . . . . . . . . . . . . . . . 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 2731	. . . . . . . . . . . . . . . . 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 630	. . . . . . . . . . . . . . . 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 3153	. . . . . . . . . . . . . . 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 2795	. . . . . . . . . . . . . 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 776	. . . . . . . . . . . . . . . 16 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑆 ∈ V ∧ (1...𝑁) ⊆ 𝑆)) ∧ (𝑎 ∈ Fin ∧ 𝑏 ∈ (mzPoly‘𝑎))) → 𝑆 ∈ V)
64	63	ad3antrrr 730	. . . . . . . . . . . . . . 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 772	. . . . . . . . . . . . . . 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 42752	. . . . . . . . . . . . . . 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 1373	. . . . . . . . . . . . . 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 2764	. . . . . . . . . . . . 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 784	. . . . . . . . . . . . . 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 776	. . . . . . . . . . . . . 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 7384	. . . . . . . . . . . . . . 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 777	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑁 ∈ ℕ0 ∧ (𝑆 ∈ V ∧ (1...𝑁) ⊆ 𝑆)) ∧ (𝑎 ∈ Fin ∧ 𝑏 ∈ (mzPoly‘𝑎))) ∧ 𝑎 ⊆ 𝑆) → 𝑏 ∈ (mzPoly‘𝑎))
73	72	ad2antrr 726	. . . . . . . . . . . . . . 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 6776	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑑:(1...𝑐)–1-1-onto→(𝑎 ∪ (1...𝑁)) → ◡𝑑:(𝑎 ∪ (1...𝑁))–1-1-onto→(1...𝑐))
75		f1of 6764	. . . . . . . . . . . . . . . . . 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 6690	. . . . . . . . . . . . . . . . 17 ⊢ ((◡𝑑:(𝑎 ∪ (1...𝑁))⟶(1...𝑐) ∧ 𝑎 ⊆ (𝑎 ∪ (1...𝑁))) → (◡𝑑 ↾ 𝑎):𝑎⟶(1...𝑐))
78	76, 21, 77	sylancl 586	. . . . . . . . . . . . . . . 16 ⊢ (𝑑:(1...𝑐)–1-1-onto→(𝑎 ∪ (1...𝑁)) → (◡𝑑 ↾ 𝑎):𝑎⟶(1...𝑐))
79	78	ad2antrl 728	. . . . . . . . . . . . . . 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 42742	. . . . . . . . . . . . . . 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 1373	. . . . . . . . . . . . . 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 42751	. . . . . . . . . . . . . 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 1373	. . . . . . . . . . . . 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 2828	. . . . . . . . . . . 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 412	. . . . . . . . . . 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 3186	. . . . . . . . . 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 419	. . . . . . . 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 1132	. . . . . . 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 417	. . . . . 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 717	. . . . 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 2828	. . . 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 412	. . 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 3186	. 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 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109  {cab 2707  ∃wrex 3053  Vcvv 3436   ∪ cun 3901   ⊆ wss 3903   ↦ cmpt 5173   I cid 5513  ^◡ccnv 5618   ↾ cres 5621   ∘ ccom 5623  ⟶wf 6478  –1-1→wf1 6479  –1-1-onto→wf1o 6481  ‘cfv 6482  (class class class)co 7349   ↑_m cmap 8753  Fincfn 8872  0cc0 11009  1c1 11010  ℕ₀cn0 12384  ℤcz 12471  ℤ_≥cuz 12735  ...cfz 13410  mzPolycmzp 42715  Diophcdioph 42748

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5218  ax-sep 5235  ax-nul 5245  ax-pow 5304  ax-pr 5371  ax-un 7671  ax-cnex 11065  ax-resscn 11066  ax-1cn 11067  ax-icn 11068  ax-addcl 11069  ax-addrcl 11070  ax-mulcl 11071  ax-mulrcl 11072  ax-mulcom 11073  ax-addass 11074  ax-mulass 11075  ax-distr 11076  ax-i2m1 11077  ax-1ne0 11078  ax-1rid 11079  ax-rnegex 11080  ax-rrecex 11081  ax-cnre 11082  ax-pre-lttri 11083  ax-pre-lttrn 11084  ax-pre-ltadd 11085  ax-pre-mulgt0 11086

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-reu 3344  df-rab 3395  df-v 3438  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-int 4897  df-iun 4943  df-br 5093  df-opab 5155  df-mpt 5174  df-tr 5200  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-pred 6249  df-ord 6310  df-on 6311  df-lim 6312  df-suc 6313  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-f1 6487  df-fo 6488  df-f1o 6489  df-fv 6490  df-riota 7306  df-ov 7352  df-oprab 7353  df-mpo 7354  df-of 7613  df-om 7800  df-1st 7924  df-2nd 7925  df-frecs 8214  df-wrecs 8245  df-recs 8294  df-rdg 8332  df-1o 8388  df-oadd 8392  df-er 8625  df-map 8755  df-en 8873  df-dom 8874  df-sdom 8875  df-fin 8876  df-dju 9797  df-card 9835  df-pnf 11151  df-mnf 11152  df-xr 11153  df-ltxr 11154  df-le 11155  df-sub 11349  df-neg 11350  df-nn 12129  df-n0 12385  df-z 12472  df-uz 12736  df-fz 13411  df-hash 14238  df-mzpcl 42716  df-mzp 42717  df-dioph 42749

This theorem is referenced by:  eldioph2b  42756  diophin  42765  diophun  42766