Theorem eldioph2 43041

Description: Construct a Diophantine set from a polynomial with witness variables drawn from any set whatsoever, via mzpcompact2 43031. (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 43031	. . 3 ⊢ (𝑃 ∈ (mzPoly‘𝑆) → ∃𝑎 ∈ Fin ∃𝑏 ∈ (mzPoly‘𝑎)(𝑎 ⊆ 𝑆 ∧ 𝑃 = (𝑒 ∈ (ℤ ↑m 𝑆) ↦ (𝑏‘(𝑒 ↾ 𝑎)))))
2	1	3ad2ant3 1136	. 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 6832	. . . . . . . . . 10 ⊢ (𝑃 = (𝑒 ∈ (ℤ ↑m 𝑆) ↦ (𝑏‘(𝑒 ↾ 𝑎))) → (𝑃‘𝑢) = ((𝑒 ∈ (ℤ ↑m 𝑆) ↦ (𝑏‘(𝑒 ↾ 𝑎)))‘𝑢))
4	3	eqeq1d 2737	. . . . . . . . 9 ⊢ (𝑃 = (𝑒 ∈ (ℤ ↑m 𝑆) ↦ (𝑏‘(𝑒 ↾ 𝑎))) → ((𝑃‘𝑢) = 0 ↔ ((𝑒 ∈ (ℤ ↑m 𝑆) ↦ (𝑏‘(𝑒 ↾ 𝑎)))‘𝑢) = 0))
5	4	anbi2d 631	. . . . . . . 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 3159	. . . . . . 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 2801	. . . . . 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 730	. . . . 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 775	. . . . . . . . . . 11 ⊢ ((((𝑁 ∈ ℕ0 ∧ (𝑆 ∈ V ∧ (1...𝑁) ⊆ 𝑆)) ∧ (𝑎 ∈ Fin ∧ 𝑏 ∈ (mzPoly‘𝑎))) ∧ 𝑎 ⊆ 𝑆) → 𝑁 ∈ ℕ0)
10		simplrl 777	. . . . . . . . . . . 12 ⊢ ((((𝑁 ∈ ℕ0 ∧ (𝑆 ∈ V ∧ (1...𝑁) ⊆ 𝑆)) ∧ (𝑎 ∈ Fin ∧ 𝑏 ∈ (mzPoly‘𝑎))) ∧ 𝑎 ⊆ 𝑆) → 𝑎 ∈ Fin)
11		fzfi 13897	. . . . . . . . . . . 12 ⊢ (1...𝑁) ∈ Fin
12		unfi 9097	. . . . . . . . . . . 12 ⊢ ((𝑎 ∈ Fin ∧ (1...𝑁) ∈ Fin) → (𝑎 ∪ (1...𝑁)) ∈ Fin)
13	10, 11, 12	sylancl 587	. . . . . . . . . . 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 43039	. . . . . . . . . . 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 1374	. . . . . . . . . 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 6800	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑑:(1...𝑐)–1-1-onto→(𝑎 ∪ (1...𝑁)) → (𝑑 ∘ ◡𝑑) = ( I ↾ (𝑎 ∪ (1...𝑁))))
19	18	ad2antrl 729	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 5936	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 5964	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2787	. . . . . . . . . . . . . . . . . . . . . . . . . 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 6207	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑑 ∘ ◡𝑑) ↾ 𝑎) = (𝑑 ∘ (◡𝑑 ↾ 𝑎))
26	24, 25	eqtrdi 2786	. . . . . . . . . . . . . . . . . . . . . . . . 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 5810	. . . . . . . . . . . . . . . . . . . . . . 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 6222	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑒 ∘ ( I ↾ 𝑎)) = (𝑒 ↾ 𝑎)
30		coass 6223	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑒 ∘ 𝑑) ∘ (◡𝑑 ↾ 𝑎)) = (𝑒 ∘ (𝑑 ∘ (◡𝑑 ↾ 𝑎)))
31	30	eqcomi 2744	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑒 ∘ (𝑑 ∘ (◡𝑑 ↾ 𝑎))) = ((𝑒 ∘ 𝑑) ∘ (◡𝑑 ↾ 𝑎))
32	28, 29, 31	3eqtr3g 2793	. . . . . . . . . . . . . . . . . . . . . 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 6837	. . . . . . . . . . . . . . . . . . . . 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 7393	. . . . . . . . . . . . . . . . . . . . . . 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 6772	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑑:(1...𝑐)–1-1-onto→(𝑎 ∪ (1...𝑁)) → 𝑑:(1...𝑐)–1-1→(𝑎 ∪ (1...𝑁)))
37	36	ad2antrl 729	. . . . . . . . . . . . . . . . . . . . . . . . . 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 773	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑆 ∈ V ∧ (1...𝑁) ⊆ 𝑆)) → (1...𝑁) ⊆ 𝑆)
40	39	ad2antrr 727	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 727	. . . . . . . . . . . . . . . . . . . . . . . . . 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 6734	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑑:(1...𝑐)–1-1→(𝑎 ∪ (1...𝑁)) ∧ (𝑎 ∪ (1...𝑁)) ⊆ 𝑆) → 𝑑:(1...𝑐)–1-1→𝑆)
44	37, 42, 43	syl2anc 585	. . . . . . . . . . . . . . . . . . . . . . . . 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 6729	. . . . . . . . . . . . . . . . . . . . . . . . 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 42993	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((1...𝑐) ∈ V ∧ 𝑒 ∈ (ℤ ↑m 𝑆) ∧ 𝑑:(1...𝑐)⟶𝑆) → (𝑒 ∘ 𝑑) ∈ (ℤ ↑m (1...𝑐)))
49	34, 35, 47, 48	syl3anc 1374	. . . . . . . . . . . . . . . . . . . . . 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 5805	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (ℎ = (𝑒 ∘ 𝑑) → (ℎ ∘ (◡𝑑 ↾ 𝑎)) = ((𝑒 ∘ 𝑑) ∘ (◡𝑑 ↾ 𝑎)))
51	50	fveq2d 6837	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (ℎ = (𝑒 ∘ 𝑑) → (𝑏‘(ℎ ∘ (◡𝑑 ↾ 𝑎))) = (𝑏‘((𝑒 ∘ 𝑑) ∘ (◡𝑑 ↾ 𝑎))))
52		eqid 2735	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (ℎ ∈ (ℤ ↑m (1...𝑐)) ↦ (𝑏‘(ℎ ∘ (◡𝑑 ↾ 𝑎)))) = (ℎ ∈ (ℤ ↑m (1...𝑐)) ↦ (𝑏‘(ℎ ∘ (◡𝑑 ↾ 𝑎))))
53		fvex 6846	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑏‘((𝑒 ∘ 𝑑) ∘ (◡𝑑 ↾ 𝑎))) ∈ V
54	51, 52, 53	fvmpt 6940	. . . . . . . . . . . . . . . . . . . . . 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 2773	. . . . . . . . . . . . . . . . . . . 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 5190	. . . . . . . . . . . . . . . . . . 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 6835	. . . . . . . . . . . . . . . . . 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 2737	. . . . . . . . . . . . . . . . 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 631	. . . . . . . . . . . . . . . 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 3159	. . . . . . . . . . . . . . 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 2801	. . . . . . . . . . . . . 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 777	. . . . . . . . . . . . . . . 16 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑆 ∈ V ∧ (1...𝑁) ⊆ 𝑆)) ∧ (𝑎 ∈ Fin ∧ 𝑏 ∈ (mzPoly‘𝑎))) → 𝑆 ∈ V)
64	63	ad3antrrr 731	. . . . . . . . . . . . . . 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 773	. . . . . . . . . . . . . . 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 43038	. . . . . . . . . . . . . . 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 1374	. . . . . . . . . . . . . 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 2770	. . . . . . . . . . . . 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 785	. . . . . . . . . . . . . 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 777	. . . . . . . . . . . . . 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 7393	. . . . . . . . . . . . . . 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 778	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑁 ∈ ℕ0 ∧ (𝑆 ∈ V ∧ (1...𝑁) ⊆ 𝑆)) ∧ (𝑎 ∈ Fin ∧ 𝑏 ∈ (mzPoly‘𝑎))) ∧ 𝑎 ⊆ 𝑆) → 𝑏 ∈ (mzPoly‘𝑎))
73	72	ad2antrr 727	. . . . . . . . . . . . . . 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 6785	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑑:(1...𝑐)–1-1-onto→(𝑎 ∪ (1...𝑁)) → ◡𝑑:(𝑎 ∪ (1...𝑁))–1-1-onto→(1...𝑐))
75		f1of 6773	. . . . . . . . . . . . . . . . . 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 6699	. . . . . . . . . . . . . . . . 17 ⊢ ((◡𝑑:(𝑎 ∪ (1...𝑁))⟶(1...𝑐) ∧ 𝑎 ⊆ (𝑎 ∪ (1...𝑁))) → (◡𝑑 ↾ 𝑎):𝑎⟶(1...𝑐))
78	76, 21, 77	sylancl 587	. . . . . . . . . . . . . . . 16 ⊢ (𝑑:(1...𝑐)–1-1-onto→(𝑎 ∪ (1...𝑁)) → (◡𝑑 ↾ 𝑎):𝑎⟶(1...𝑐))
79	78	ad2antrl 729	. . . . . . . . . . . . . . 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 43028	. . . . . . . . . . . . . . 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 1374	. . . . . . . . . . . . . 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 43037	. . . . . . . . . . . . . 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 1374	. . . . . . . . . . . . 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 2835	. . . . . . . . . . . 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 3192	. . . . . . . . . 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 1133	. . . . . . 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 718	. . . . 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 2835	. . . 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 3192	. 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 1087   = wceq 1542   ∈ wcel 2114  {cab 2713  ∃wrex 3059  Vcvv 3439   ∪ cun 3898   ⊆ wss 3900   ↦ cmpt 5178   I cid 5517  ^◡ccnv 5622   ↾ cres 5625   ∘ ccom 5627  ⟶wf 6487  –1-1→wf1 6488  –1-1-onto→wf1o 6490  ‘cfv 6491  (class class class)co 7358   ↑_m cmap 8765  Fincfn 8885  0cc0 11028  1c1 11029  ℕ₀cn0 12403  ℤcz 12490  ℤ_≥cuz 12753  ...cfz 13425  mzPolycmzp 43001  Diophcdioph 43034

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2183  ax-ext 2707  ax-rep 5223  ax-sep 5240  ax-nul 5250  ax-pow 5309  ax-pr 5376  ax-un 7680  ax-cnex 11084  ax-resscn 11085  ax-1cn 11086  ax-icn 11087  ax-addcl 11088  ax-addrcl 11089  ax-mulcl 11090  ax-mulrcl 11091  ax-mulcom 11092  ax-addass 11093  ax-mulass 11094  ax-distr 11095  ax-i2m1 11096  ax-1ne0 11097  ax-1rid 11098  ax-rnegex 11099  ax-rrecex 11100  ax-cnre 11101  ax-pre-lttri 11102  ax-pre-lttrn 11103  ax-pre-ltadd 11104  ax-pre-mulgt0 11105

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2538  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2810  df-nfc 2884  df-ne 2932  df-nel 3036  df-ral 3051  df-rex 3060  df-reu 3350  df-rab 3399  df-v 3441  df-sbc 3740  df-csb 3849  df-dif 3903  df-un 3905  df-in 3907  df-ss 3917  df-pss 3920  df-nul 4285  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4863  df-int 4902  df-iun 4947  df-br 5098  df-opab 5160  df-mpt 5179  df-tr 5205  df-id 5518  df-eprel 5523  df-po 5531  df-so 5532  df-fr 5576  df-we 5578  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-pred 6258  df-ord 6319  df-on 6320  df-lim 6321  df-suc 6322  df-iota 6447  df-fun 6493  df-fn 6494  df-f 6495  df-f1 6496  df-fo 6497  df-f1o 6498  df-fv 6499  df-riota 7315  df-ov 7361  df-oprab 7362  df-mpo 7363  df-of 7622  df-om 7809  df-1st 7933  df-2nd 7934  df-frecs 8223  df-wrecs 8254  df-recs 8303  df-rdg 8341  df-1o 8397  df-oadd 8401  df-er 8635  df-map 8767  df-en 8886  df-dom 8887  df-sdom 8888  df-fin 8889  df-dju 9815  df-card 9853  df-pnf 11170  df-mnf 11171  df-xr 11172  df-ltxr 11173  df-le 11174  df-sub 11368  df-neg 11369  df-nn 12148  df-n0 12404  df-z 12491  df-uz 12754  df-fz 13426  df-hash 14256  df-mzpcl 43002  df-mzp 43003  df-dioph 43035

This theorem is referenced by:  eldioph2b  43042  diophin  43051  diophun  43052