Theorem eldioph2 43705

Description: Construct a Diophantine set from a polynomial with witness variables drawn from any set whatsoever, via mzpcompact2 43695. (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 43695	. . 3 ⊢ (𝑃 ∈ (mzPoly‘𝑆) → ∃𝑎 ∈ Fin ∃𝑏 ∈ (mzPoly‘𝑎)(𝑎 ⊆ 𝑆 ∧ 𝑃 = (𝑒 ∈ (ℤ ↑m 𝑆) ↦ (𝑏‘(𝑒 ↾ 𝑎)))))
2	1	3ad2ant3 1153	. 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 6873	. . . . . . . . . 10 ⊢ (𝑃 = (𝑒 ∈ (ℤ ↑m 𝑆) ↦ (𝑏‘(𝑒 ↾ 𝑎))) → (𝑃‘𝑢) = ((𝑒 ∈ (ℤ ↑m 𝑆) ↦ (𝑏‘(𝑒 ↾ 𝑎)))‘𝑢))
4	3	eqeq1d 2762	. . . . . . . . 9 ⊢ (𝑃 = (𝑒 ∈ (ℤ ↑m 𝑆) ↦ (𝑏‘(𝑒 ↾ 𝑎))) → ((𝑃‘𝑢) = 0 ↔ ((𝑒 ∈ (ℤ ↑m 𝑆) ↦ (𝑏‘(𝑒 ↾ 𝑎)))‘𝑢) = 0))
5	4	anbi2d 642	. . . . . . . 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 3186	. . . . . . 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 2826	. . . . . 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 742	. . . . 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 787	. . . . . . . . . . 11 ⊢ ((((𝑁 ∈ ℕ0 ∧ (𝑆 ∈ V ∧ (1...𝑁) ⊆ 𝑆)) ∧ (𝑎 ∈ Fin ∧ 𝑏 ∈ (mzPoly‘𝑎))) ∧ 𝑎 ⊆ 𝑆) → 𝑁 ∈ ℕ0)
10		simplrl 789	. . . . . . . . . . . 12 ⊢ ((((𝑁 ∈ ℕ0 ∧ (𝑆 ∈ V ∧ (1...𝑁) ⊆ 𝑆)) ∧ (𝑎 ∈ Fin ∧ 𝑏 ∈ (mzPoly‘𝑎))) ∧ 𝑎 ⊆ 𝑆) → 𝑎 ∈ Fin)
11		fzfi 14069	. . . . . . . . . . . 12 ⊢ (1...𝑁) ∈ Fin
12		unfi 9165	. . . . . . . . . . . 12 ⊢ ((𝑎 ∈ Fin ∧ (1...𝑁) ∈ Fin) → (𝑎 ∪ (1...𝑁)) ∈ Fin)
13	10, 11, 12	sylancl 598	. . . . . . . . . . 11 ⊢ ((((𝑁 ∈ ℕ0 ∧ (𝑆 ∈ V ∧ (1...𝑁) ⊆ 𝑆)) ∧ (𝑎 ∈ Fin ∧ 𝑏 ∈ (mzPoly‘𝑎))) ∧ 𝑎 ⊆ 𝑆) → (𝑎 ∪ (1...𝑁)) ∈ Fin)
14		ssun2 4125	. . . . . . . . . . . 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 43703	. . . . . . . . . . 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 1398	. . . . . . . . . 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 6841	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑑:(1...𝑐)–1-1-onto→(𝑎 ∪ (1...𝑁)) → (𝑑 ∘ ◡𝑑) = ( I ↾ (𝑎 ∪ (1...𝑁))))
19	18	ad2antrl 741	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 5966	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 4124	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ 𝑎 ⊆ (𝑎 ∪ (1...𝑁))
22		resabs1 5994	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2812	. . . . . . . . . . . . . . . . . . . . . . . . . 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 6241	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑑 ∘ ◡𝑑) ↾ 𝑎) = (𝑑 ∘ (◡𝑑 ↾ 𝑎))
26	24, 25	eqtrdi 2811	. . . . . . . . . . . . . . . . . . . . . . . . 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 486	. . . . . . . . . . . . . . . . . . . . . . . 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 5837	. . . . . . . . . . . . . . . . . . . . . . 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 6256	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑒 ∘ ( I ↾ 𝑎)) = (𝑒 ↾ 𝑎)
30		coass 6257	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑒 ∘ 𝑑) ∘ (◡𝑑 ↾ 𝑎)) = (𝑒 ∘ (𝑑 ∘ (◡𝑑 ↾ 𝑎)))
31	30	eqcomi 2769	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑒 ∘ (𝑑 ∘ (◡𝑑 ↾ 𝑎))) = ((𝑒 ∘ 𝑑) ∘ (◡𝑑 ↾ 𝑎))
32	28, 29, 31	3eqtr3g 2818	. . . . . . . . . . . . . . . . . . . . . 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 6878	. . . . . . . . . . . . . . . . . . . . 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 7444	. . . . . . . . . . . . . . . . . . . . . . 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 490	. . . . . . . . . . . . . . . . . . . . . . 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 6812	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑑:(1...𝑐)–1-1-onto→(𝑎 ∪ (1...𝑁)) → 𝑑:(1...𝑐)–1-1→(𝑎 ∪ (1...𝑁)))
37	36	ad2antrl 741	. . . . . . . . . . . . . . . . . . . . . . . . . 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 490	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((((𝑁 ∈ ℕ0 ∧ (𝑆 ∈ V ∧ (1...𝑁) ⊆ 𝑆)) ∧ (𝑎 ∈ Fin ∧ 𝑏 ∈ (mzPoly‘𝑎))) ∧ 𝑎 ⊆ 𝑆) → 𝑎 ⊆ 𝑆)
39		simprr 785	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑆 ∈ V ∧ (1...𝑁) ⊆ 𝑆)) → (1...𝑁) ⊆ 𝑆)
40	39	ad2antrr 739	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((((𝑁 ∈ ℕ0 ∧ (𝑆 ∈ V ∧ (1...𝑁) ⊆ 𝑆)) ∧ (𝑎 ∈ Fin ∧ 𝑏 ∈ (mzPoly‘𝑎))) ∧ 𝑎 ⊆ 𝑆) → (1...𝑁) ⊆ 𝑆)
41	38, 40	unssd 4138	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((((𝑁 ∈ ℕ0 ∧ (𝑆 ∈ V ∧ (1...𝑁) ⊆ 𝑆)) ∧ (𝑎 ∈ Fin ∧ 𝑏 ∈ (mzPoly‘𝑎))) ∧ 𝑎 ⊆ 𝑆) → (𝑎 ∪ (1...𝑁)) ⊆ 𝑆)
42	41	ad2antrr 739	. . . . . . . . . . . . . . . . . . . . . . . . . 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 6774	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑑:(1...𝑐)–1-1→(𝑎 ∪ (1...𝑁)) ∧ (𝑎 ∪ (1...𝑁)) ⊆ 𝑆) → 𝑑:(1...𝑐)–1-1→𝑆)
44	37, 42, 43	syl2anc 596	. . . . . . . . . . . . . . . . . . . . . . . . 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 6767	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑑:(1...𝑐)–1-1→𝑆 → 𝑑:(1...𝑐)⟶𝑆)
46	44, 45	syl 18	. . . . . . . . . . . . . . . . . . . . . . . 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 486	. . . . . . . . . . . . . . . . . . . . . . 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 43657	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((1...𝑐) ∈ V ∧ 𝑒 ∈ (ℤ ↑m 𝑆) ∧ 𝑑:(1...𝑐)⟶𝑆) → (𝑒 ∘ 𝑑) ∈ (ℤ ↑m (1...𝑐)))
49	34, 35, 47, 48	syl3anc 1398	. . . . . . . . . . . . . . . . . . . . . 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 5832	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (ℎ = (𝑒 ∘ 𝑑) → (ℎ ∘ (◡𝑑 ↾ 𝑎)) = ((𝑒 ∘ 𝑑) ∘ (◡𝑑 ↾ 𝑎)))
51	50	fveq2d 6878	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (ℎ = (𝑒 ∘ 𝑑) → (𝑏‘(ℎ ∘ (◡𝑑 ↾ 𝑎))) = (𝑏‘((𝑒 ∘ 𝑑) ∘ (◡𝑑 ↾ 𝑎))))
52		eqid 2760	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (ℎ ∈ (ℤ ↑m (1...𝑐)) ↦ (𝑏‘(ℎ ∘ (◡𝑑 ↾ 𝑎)))) = (ℎ ∈ (ℤ ↑m (1...𝑐)) ↦ (𝑏‘(ℎ ∘ (◡𝑑 ↾ 𝑎))))
53		fvex 6887	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑏‘((𝑒 ∘ 𝑑) ∘ (◡𝑑 ↾ 𝑎))) ∈ V
54	51, 52, 53	fvmpt 6982	. . . . . . . . . . . . . . . . . . . . . 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 18	. . . . . . . . . . . . . . . . . . . . 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 2798	. . . . . . . . . . . . . . . . . . . 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 5198	. . . . . . . . . . . . . . . . . . 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 6876	. . . . . . . . . . . . . . . . . 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 2762	. . . . . . . . . . . . . . . . 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 642	. . . . . . . . . . . . . . . 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 3186	. . . . . . . . . . . . . . 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 2826	. . . . . . . . . . . . . 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 789	. . . . . . . . . . . . . . . 16 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑆 ∈ V ∧ (1...𝑁) ⊆ 𝑆)) ∧ (𝑎 ∈ Fin ∧ 𝑏 ∈ (mzPoly‘𝑎))) → 𝑆 ∈ V)
64	63	ad3antrrr 743	. . . . . . . . . . . . . . 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 785	. . . . . . . . . . . . . . 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 43702	. . . . . . . . . . . . . . 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 1398	. . . . . . . . . . . . . 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 2795	. . . . . . . . . . . . 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 797	. . . . . . . . . . . . . 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 789	. . . . . . . . . . . . . 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 7444	. . . . . . . . . . . . . . 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 790	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑁 ∈ ℕ0 ∧ (𝑆 ∈ V ∧ (1...𝑁) ⊆ 𝑆)) ∧ (𝑎 ∈ Fin ∧ 𝑏 ∈ (mzPoly‘𝑎))) ∧ 𝑎 ⊆ 𝑆) → 𝑏 ∈ (mzPoly‘𝑎))
73	72	ad2antrr 739	. . . . . . . . . . . . . . 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 6826	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑑:(1...𝑐)–1-1-onto→(𝑎 ∪ (1...𝑁)) → ◡𝑑:(𝑎 ∪ (1...𝑁))–1-1-onto→(1...𝑐))
75		f1of 6813	. . . . . . . . . . . . . . . . . 18 ⊢ (◡𝑑:(𝑎 ∪ (1...𝑁))–1-1-onto→(1...𝑐) → ◡𝑑:(𝑎 ∪ (1...𝑁))⟶(1...𝑐))
76	74, 75	syl 18	. . . . . . . . . . . . . . . . 17 ⊢ (𝑑:(1...𝑐)–1-1-onto→(𝑎 ∪ (1...𝑁)) → ◡𝑑:(𝑎 ∪ (1...𝑁))⟶(1...𝑐))
77		fssres 6737	. . . . . . . . . . . . . . . . 17 ⊢ ((◡𝑑:(𝑎 ∪ (1...𝑁))⟶(1...𝑐) ∧ 𝑎 ⊆ (𝑎 ∪ (1...𝑁))) → (◡𝑑 ↾ 𝑎):𝑎⟶(1...𝑐))
78	76, 21, 77	sylancl 598	. . . . . . . . . . . . . . . 16 ⊢ (𝑑:(1...𝑐)–1-1-onto→(𝑎 ∪ (1...𝑁)) → (◡𝑑 ↾ 𝑎):𝑎⟶(1...𝑐))
79	78	ad2antrl 741	. . . . . . . . . . . . . . 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 43692	. . . . . . . . . . . . . . 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 1398	. . . . . . . . . . . . . 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 43701	. . . . . . . . . . . . . 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 1398	. . . . . . . . . . . . 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 2860	. . . . . . . . . . . 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 418	. . . . . . . . . . 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 3219	. . . . . . . . . 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 16	. . . . . . . . 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 425	. . . . . . . 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 1150	. . . . . . 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 423	. . . . . 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 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)) ∧ ((𝑒 ∈ (ℤ ↑m 𝑆) ↦ (𝑏‘(𝑒 ↾ 𝑎)))‘𝑢) = 0)} ∈ (Dioph‘𝑁))
92	8, 91	eqeltrd 2860	. . . 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 418	. . 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 3219	. 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 16	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))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145  {cab 2738  ∃wrex 3086  Vcvv 3450   ∪ cun 3897   ⊆ wss 3899   ↦ cmpt 5186   I cid 5542  ^◡ccnv 5647   ↾ cres 5650   ∘ ccom 5652  ⟶wf 6524  –1-1→wf1 6525  –1-1-onto→wf1o 6527  ‘cfv 6528  (class class class)co 7409   ↑_m cmap 8826  Fincfn 8952  0cc0 11157  1c1 11158  ℕ₀cn0 12561  ℤcz 12648  ℤ_≥cuz 12920  ...cfz 13594  mzPolycmzp 43665  Diophcdioph 43698

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7735  ax-cnex 11213  ax-resscn 11214  ax-1cn 11215  ax-icn 11216  ax-addcl 11217  ax-addrcl 11218  ax-mulcl 11219  ax-mulrcl 11220  ax-mulcom 11221  ax-addass 11222  ax-mulass 11223  ax-distr 11224  ax-i2m1 11225  ax-1ne0 11226  ax-1rid 11227  ax-rnegex 11228  ax-rrecex 11229  ax-cnre 11230  ax-pre-lttri 11231  ax-pre-lttrn 11232  ax-pre-ltadd 11233  ax-pre-mulgt0 11234

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-nel 3062  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-int 4908  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5543  df-eprel 5548  df-po 5556  df-so 5557  df-fr 5601  df-we 5603  df-xp 5654  df-rel 5655  df-cnv 5656  df-co 5657  df-dm 5658  df-rn 5659  df-res 5660  df-ima 5661  df-pred 6294  df-ord 6355  df-on 6356  df-lim 6357  df-suc 6358  df-iota 6484  df-fun 6530  df-fn 6531  df-f 6532  df-f1 6533  df-fo 6534  df-f1o 6535  df-fv 6536  df-riota 7366  df-ov 7412  df-oprab 7413  df-mpo 7414  df-of 7677  df-om 7862  df-1st 7985  df-2nd 7986  df-frecs 8278  df-wrecs 8309  df-recs 8358  df-rdg 8397  df-1o 8455  df-oadd 8459  df-er 8696  df-map 8828  df-en 8953  df-dom 8954  df-sdom 8955  df-fin 8956  df-dju 9939  df-card 9977  df-pnf 11302  df-mnf 11303  df-xr 11304  df-ltxr 11305  df-le 11306  df-sub 11500  df-neg 11501  df-nn 12291  df-n0 12562  df-z 12649  df-uz 12921  df-fz 13595  df-hash 14428  df-mzpcl 43666  df-mzp 43667  df-dioph 43699

This theorem is used by:  eldioph2b  43706  diophin  43715  diophun  43716