Theorem diophren 42774

Description: Change variables in a Diophantine set, using class notation. This allows already proved Diophantine sets to be reused in contexts with more variables. (Contributed by Stefan O'Rear, 16-Oct-2014.) (Revised by Stefan O'Rear, 5-Jun-2015.)

Assertion

Ref	Expression
diophren	⊢ ((𝑆 ∈ (Dioph‘𝑁) ∧ 𝑀 ∈ ℕ0 ∧ 𝐹:(1...𝑁)⟶(1...𝑀)) → {𝑎 ∈ (ℕ0 ↑m (1...𝑀)) ∣ (𝑎 ∘ 𝐹) ∈ 𝑆} ∈ (Dioph‘𝑀))

Distinct variable groups:   𝑆,𝑎   𝑀,𝑎   𝑁,𝑎   𝐹,𝑎

Proof of Theorem diophren

Dummy variables 𝑏 𝑐 𝑑 𝑒 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		zex 12514	. . . . . 6 ⊢ ℤ ∈ V
2		difexg 5279	. . . . . 6 ⊢ (ℤ ∈ V → (ℤ ∖ ℕ) ∈ V)
3	1, 2	ax-mp 5	. . . . 5 ⊢ (ℤ ∖ ℕ) ∈ V
4		ominf 9181	. . . . . 6 ⊢ ¬ ω ∈ Fin
5		nnuz 12812	. . . . . . . . . 10 ⊢ ℕ = (ℤ≥‘1)
6		0p1e1 12279	. . . . . . . . . . 11 ⊢ (0 + 1) = 1
7	6	fveq2i 6843	. . . . . . . . . 10 ⊢ (ℤ≥‘(0 + 1)) = (ℤ≥‘1)
8	5, 7	eqtr4i 2755	. . . . . . . . 9 ⊢ ℕ = (ℤ≥‘(0 + 1))
9	8	difeq2i 4082	. . . . . . . 8 ⊢ (ℤ ∖ ℕ) = (ℤ ∖ (ℤ≥‘(0 + 1)))
10		0z 12516	. . . . . . . . 9 ⊢ 0 ∈ ℤ
11		lzenom 42731	. . . . . . . . 9 ⊢ (0 ∈ ℤ → (ℤ ∖ (ℤ≥‘(0 + 1))) ≈ ω)
12	10, 11	ax-mp 5	. . . . . . . 8 ⊢ (ℤ ∖ (ℤ≥‘(0 + 1))) ≈ ω
13	9, 12	eqbrtri 5123	. . . . . . 7 ⊢ (ℤ ∖ ℕ) ≈ ω
14		enfi 9128	. . . . . . 7 ⊢ ((ℤ ∖ ℕ) ≈ ω → ((ℤ ∖ ℕ) ∈ Fin ↔ ω ∈ Fin))
15	13, 14	ax-mp 5	. . . . . 6 ⊢ ((ℤ ∖ ℕ) ∈ Fin ↔ ω ∈ Fin)
16	4, 15	mtbir 323	. . . . 5 ⊢ ¬ (ℤ ∖ ℕ) ∈ Fin
17		disjdifr 4432	. . . . 5 ⊢ ((ℤ ∖ ℕ) ∩ ℕ) = ∅
18	3, 16, 17	eldioph4b 42772	. . . 4 ⊢ (𝑆 ∈ (Dioph‘𝑁) ↔ (𝑁 ∈ ℕ0 ∧ ∃𝑏 ∈ (mzPoly‘((ℤ ∖ ℕ) ∪ (1...𝑁)))𝑆 = {𝑐 ∈ (ℕ0 ↑m (1...𝑁)) ∣ ∃𝑑 ∈ (ℕ0 ↑m (ℤ ∖ ℕ))(𝑏‘(𝑐 ∪ 𝑑)) = 0}))
19		simpr 484	. . . . . . . . . . . 12 ⊢ (((((𝑀 ∈ ℕ0 ∧ 𝐹:(1...𝑁)⟶(1...𝑀)) ∧ 𝑁 ∈ ℕ0) ∧ 𝑏 ∈ (mzPoly‘((ℤ ∖ ℕ) ∪ (1...𝑁)))) ∧ 𝑎 ∈ (ℕ0 ↑m (1...𝑀))) → 𝑎 ∈ (ℕ0 ↑m (1...𝑀)))
20		simp-4r 783	. . . . . . . . . . . 12 ⊢ (((((𝑀 ∈ ℕ0 ∧ 𝐹:(1...𝑁)⟶(1...𝑀)) ∧ 𝑁 ∈ ℕ0) ∧ 𝑏 ∈ (mzPoly‘((ℤ ∖ ℕ) ∪ (1...𝑁)))) ∧ 𝑎 ∈ (ℕ0 ↑m (1...𝑀))) → 𝐹:(1...𝑁)⟶(1...𝑀))
21		ovex 7402	. . . . . . . . . . . . 13 ⊢ (1...𝑁) ∈ V
22	21	mapco2 42676	. . . . . . . . . . . 12 ⊢ ((𝑎 ∈ (ℕ0 ↑m (1...𝑀)) ∧ 𝐹:(1...𝑁)⟶(1...𝑀)) → (𝑎 ∘ 𝐹) ∈ (ℕ0 ↑m (1...𝑁)))
23	19, 20, 22	syl2anc 584	. . . . . . . . . . 11 ⊢ (((((𝑀 ∈ ℕ0 ∧ 𝐹:(1...𝑁)⟶(1...𝑀)) ∧ 𝑁 ∈ ℕ0) ∧ 𝑏 ∈ (mzPoly‘((ℤ ∖ ℕ) ∪ (1...𝑁)))) ∧ 𝑎 ∈ (ℕ0 ↑m (1...𝑀))) → (𝑎 ∘ 𝐹) ∈ (ℕ0 ↑m (1...𝑁)))
24		uneq1 4120	. . . . . . . . . . . . . 14 ⊢ (𝑐 = (𝑎 ∘ 𝐹) → (𝑐 ∪ 𝑑) = ((𝑎 ∘ 𝐹) ∪ 𝑑))
25	24	fveqeq2d 6848	. . . . . . . . . . . . 13 ⊢ (𝑐 = (𝑎 ∘ 𝐹) → ((𝑏‘(𝑐 ∪ 𝑑)) = 0 ↔ (𝑏‘((𝑎 ∘ 𝐹) ∪ 𝑑)) = 0))
26	25	rexbidv 3157	. . . . . . . . . . . 12 ⊢ (𝑐 = (𝑎 ∘ 𝐹) → (∃𝑑 ∈ (ℕ0 ↑m (ℤ ∖ ℕ))(𝑏‘(𝑐 ∪ 𝑑)) = 0 ↔ ∃𝑑 ∈ (ℕ0 ↑m (ℤ ∖ ℕ))(𝑏‘((𝑎 ∘ 𝐹) ∪ 𝑑)) = 0))
27	26	elrab3 3657	. . . . . . . . . . 11 ⊢ ((𝑎 ∘ 𝐹) ∈ (ℕ0 ↑m (1...𝑁)) → ((𝑎 ∘ 𝐹) ∈ {𝑐 ∈ (ℕ0 ↑m (1...𝑁)) ∣ ∃𝑑 ∈ (ℕ0 ↑m (ℤ ∖ ℕ))(𝑏‘(𝑐 ∪ 𝑑)) = 0} ↔ ∃𝑑 ∈ (ℕ0 ↑m (ℤ ∖ ℕ))(𝑏‘((𝑎 ∘ 𝐹) ∪ 𝑑)) = 0))
28	23, 27	syl 17	. . . . . . . . . 10 ⊢ (((((𝑀 ∈ ℕ0 ∧ 𝐹:(1...𝑁)⟶(1...𝑀)) ∧ 𝑁 ∈ ℕ0) ∧ 𝑏 ∈ (mzPoly‘((ℤ ∖ ℕ) ∪ (1...𝑁)))) ∧ 𝑎 ∈ (ℕ0 ↑m (1...𝑀))) → ((𝑎 ∘ 𝐹) ∈ {𝑐 ∈ (ℕ0 ↑m (1...𝑁)) ∣ ∃𝑑 ∈ (ℕ0 ↑m (ℤ ∖ ℕ))(𝑏‘(𝑐 ∪ 𝑑)) = 0} ↔ ∃𝑑 ∈ (ℕ0 ↑m (ℤ ∖ ℕ))(𝑏‘((𝑎 ∘ 𝐹) ∪ 𝑑)) = 0))
29		simp-5r 785	. . . . . . . . . . . . . . 15 ⊢ ((((((𝑀 ∈ ℕ0 ∧ 𝐹:(1...𝑁)⟶(1...𝑀)) ∧ 𝑁 ∈ ℕ0) ∧ 𝑏 ∈ (mzPoly‘((ℤ ∖ ℕ) ∪ (1...𝑁)))) ∧ 𝑎 ∈ (ℕ0 ↑m (1...𝑀))) ∧ 𝑑 ∈ (ℕ0 ↑m (ℤ ∖ ℕ))) → 𝐹:(1...𝑁)⟶(1...𝑀))
30		simplr 768	. . . . . . . . . . . . . . 15 ⊢ ((((((𝑀 ∈ ℕ0 ∧ 𝐹:(1...𝑁)⟶(1...𝑀)) ∧ 𝑁 ∈ ℕ0) ∧ 𝑏 ∈ (mzPoly‘((ℤ ∖ ℕ) ∪ (1...𝑁)))) ∧ 𝑎 ∈ (ℕ0 ↑m (1...𝑀))) ∧ 𝑑 ∈ (ℕ0 ↑m (ℤ ∖ ℕ))) → 𝑎 ∈ (ℕ0 ↑m (1...𝑀)))
31		simpr 484	. . . . . . . . . . . . . . 15 ⊢ ((((((𝑀 ∈ ℕ0 ∧ 𝐹:(1...𝑁)⟶(1...𝑀)) ∧ 𝑁 ∈ ℕ0) ∧ 𝑏 ∈ (mzPoly‘((ℤ ∖ ℕ) ∪ (1...𝑁)))) ∧ 𝑎 ∈ (ℕ0 ↑m (1...𝑀))) ∧ 𝑑 ∈ (ℕ0 ↑m (ℤ ∖ ℕ))) → 𝑑 ∈ (ℕ0 ↑m (ℤ ∖ ℕ)))
32		coundi 6208	. . . . . . . . . . . . . . . 16 ⊢ ((𝑎 ∪ 𝑑) ∘ (𝐹 ∪ ( I ↾ (ℤ ∖ ℕ)))) = (((𝑎 ∪ 𝑑) ∘ 𝐹) ∪ ((𝑎 ∪ 𝑑) ∘ ( I ↾ (ℤ ∖ ℕ))))
33		coundir 6209	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑎 ∪ 𝑑) ∘ 𝐹) = ((𝑎 ∘ 𝐹) ∪ (𝑑 ∘ 𝐹))
34		elmapi 8799	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑑 ∈ (ℕ0 ↑m (ℤ ∖ ℕ)) → 𝑑:(ℤ ∖ ℕ)⟶ℕ0)
35	34	3ad2ant3 1135	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐹:(1...𝑁)⟶(1...𝑀) ∧ 𝑎 ∈ (ℕ0 ↑m (1...𝑀)) ∧ 𝑑 ∈ (ℕ0 ↑m (ℤ ∖ ℕ))) → 𝑑:(ℤ ∖ ℕ)⟶ℕ0)
36		simp1 1136	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐹:(1...𝑁)⟶(1...𝑀) ∧ 𝑎 ∈ (ℕ0 ↑m (1...𝑀)) ∧ 𝑑 ∈ (ℕ0 ↑m (ℤ ∖ ℕ))) → 𝐹:(1...𝑁)⟶(1...𝑀))
37		incom 4168	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((ℤ ∖ ℕ) ∩ (1...𝑀)) = ((1...𝑀) ∩ (ℤ ∖ ℕ))
38		fz1ssnn 13492	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (1...𝑀) ⊆ ℕ
39		disjdif 4431	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (ℕ ∩ (ℤ ∖ ℕ)) = ∅
40		ssdisj 4419	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((1...𝑀) ⊆ ℕ ∧ (ℕ ∩ (ℤ ∖ ℕ)) = ∅) → ((1...𝑀) ∩ (ℤ ∖ ℕ)) = ∅)
41	38, 39, 40	mp2an 692	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((1...𝑀) ∩ (ℤ ∖ ℕ)) = ∅
42	37, 41	eqtri 2752	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((ℤ ∖ ℕ) ∩ (1...𝑀)) = ∅
43	42	a1i 11	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐹:(1...𝑁)⟶(1...𝑀) ∧ 𝑎 ∈ (ℕ0 ↑m (1...𝑀)) ∧ 𝑑 ∈ (ℕ0 ↑m (ℤ ∖ ℕ))) → ((ℤ ∖ ℕ) ∩ (1...𝑀)) = ∅)
44		coeq0i 42714	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑑:(ℤ ∖ ℕ)⟶ℕ0 ∧ 𝐹:(1...𝑁)⟶(1...𝑀) ∧ ((ℤ ∖ ℕ) ∩ (1...𝑀)) = ∅) → (𝑑 ∘ 𝐹) = ∅)
45	35, 36, 43, 44	syl3anc 1373	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐹:(1...𝑁)⟶(1...𝑀) ∧ 𝑎 ∈ (ℕ0 ↑m (1...𝑀)) ∧ 𝑑 ∈ (ℕ0 ↑m (ℤ ∖ ℕ))) → (𝑑 ∘ 𝐹) = ∅)
46	45	uneq2d 4127	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐹:(1...𝑁)⟶(1...𝑀) ∧ 𝑎 ∈ (ℕ0 ↑m (1...𝑀)) ∧ 𝑑 ∈ (ℕ0 ↑m (ℤ ∖ ℕ))) → ((𝑎 ∘ 𝐹) ∪ (𝑑 ∘ 𝐹)) = ((𝑎 ∘ 𝐹) ∪ ∅))
47	33, 46	eqtrid 2776	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐹:(1...𝑁)⟶(1...𝑀) ∧ 𝑎 ∈ (ℕ0 ↑m (1...𝑀)) ∧ 𝑑 ∈ (ℕ0 ↑m (ℤ ∖ ℕ))) → ((𝑎 ∪ 𝑑) ∘ 𝐹) = ((𝑎 ∘ 𝐹) ∪ ∅))
48		un0 4353	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑎 ∘ 𝐹) ∪ ∅) = (𝑎 ∘ 𝐹)
49	47, 48	eqtrdi 2780	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐹:(1...𝑁)⟶(1...𝑀) ∧ 𝑎 ∈ (ℕ0 ↑m (1...𝑀)) ∧ 𝑑 ∈ (ℕ0 ↑m (ℤ ∖ ℕ))) → ((𝑎 ∪ 𝑑) ∘ 𝐹) = (𝑎 ∘ 𝐹))
50		coundir 6209	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑎 ∪ 𝑑) ∘ ( I ↾ (ℤ ∖ ℕ))) = ((𝑎 ∘ ( I ↾ (ℤ ∖ ℕ))) ∪ (𝑑 ∘ ( I ↾ (ℤ ∖ ℕ))))
51		elmapi 8799	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑎 ∈ (ℕ0 ↑m (1...𝑀)) → 𝑎:(1...𝑀)⟶ℕ0)
52	51	3ad2ant2 1134	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐹:(1...𝑁)⟶(1...𝑀) ∧ 𝑎 ∈ (ℕ0 ↑m (1...𝑀)) ∧ 𝑑 ∈ (ℕ0 ↑m (ℤ ∖ ℕ))) → 𝑎:(1...𝑀)⟶ℕ0)
53		f1oi 6820	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ( I ↾ (ℤ ∖ ℕ)):(ℤ ∖ ℕ)–1-1-onto→(ℤ ∖ ℕ)
54		f1of 6782	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (( I ↾ (ℤ ∖ ℕ)):(ℤ ∖ ℕ)–1-1-onto→(ℤ ∖ ℕ) → ( I ↾ (ℤ ∖ ℕ)):(ℤ ∖ ℕ)⟶(ℤ ∖ ℕ))
55	53, 54	ax-mp 5	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ( I ↾ (ℤ ∖ ℕ)):(ℤ ∖ ℕ)⟶(ℤ ∖ ℕ)
56		coeq0i 42714	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑎:(1...𝑀)⟶ℕ0 ∧ ( I ↾ (ℤ ∖ ℕ)):(ℤ ∖ ℕ)⟶(ℤ ∖ ℕ) ∧ ((1...𝑀) ∩ (ℤ ∖ ℕ)) = ∅) → (𝑎 ∘ ( I ↾ (ℤ ∖ ℕ))) = ∅)
57	55, 41, 56	mp3an23 1455	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑎:(1...𝑀)⟶ℕ0 → (𝑎 ∘ ( I ↾ (ℤ ∖ ℕ))) = ∅)
58	52, 57	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐹:(1...𝑁)⟶(1...𝑀) ∧ 𝑎 ∈ (ℕ0 ↑m (1...𝑀)) ∧ 𝑑 ∈ (ℕ0 ↑m (ℤ ∖ ℕ))) → (𝑎 ∘ ( I ↾ (ℤ ∖ ℕ))) = ∅)
59		coires1 6225	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑑 ∘ ( I ↾ (ℤ ∖ ℕ))) = (𝑑 ↾ (ℤ ∖ ℕ))
60		ffn 6670	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑑:(ℤ ∖ ℕ)⟶ℕ0 → 𝑑 Fn (ℤ ∖ ℕ))
61		fnresdm 6619	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑑 Fn (ℤ ∖ ℕ) → (𝑑 ↾ (ℤ ∖ ℕ)) = 𝑑)
62	34, 60, 61	3syl 18	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑑 ∈ (ℕ0 ↑m (ℤ ∖ ℕ)) → (𝑑 ↾ (ℤ ∖ ℕ)) = 𝑑)
63	59, 62	eqtrid 2776	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑑 ∈ (ℕ0 ↑m (ℤ ∖ ℕ)) → (𝑑 ∘ ( I ↾ (ℤ ∖ ℕ))) = 𝑑)
64	63	3ad2ant3 1135	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐹:(1...𝑁)⟶(1...𝑀) ∧ 𝑎 ∈ (ℕ0 ↑m (1...𝑀)) ∧ 𝑑 ∈ (ℕ0 ↑m (ℤ ∖ ℕ))) → (𝑑 ∘ ( I ↾ (ℤ ∖ ℕ))) = 𝑑)
65	58, 64	uneq12d 4128	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐹:(1...𝑁)⟶(1...𝑀) ∧ 𝑎 ∈ (ℕ0 ↑m (1...𝑀)) ∧ 𝑑 ∈ (ℕ0 ↑m (ℤ ∖ ℕ))) → ((𝑎 ∘ ( I ↾ (ℤ ∖ ℕ))) ∪ (𝑑 ∘ ( I ↾ (ℤ ∖ ℕ)))) = (∅ ∪ 𝑑))
66	50, 65	eqtrid 2776	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐹:(1...𝑁)⟶(1...𝑀) ∧ 𝑎 ∈ (ℕ0 ↑m (1...𝑀)) ∧ 𝑑 ∈ (ℕ0 ↑m (ℤ ∖ ℕ))) → ((𝑎 ∪ 𝑑) ∘ ( I ↾ (ℤ ∖ ℕ))) = (∅ ∪ 𝑑))
67		uncom 4117	. . . . . . . . . . . . . . . . . . 19 ⊢ (∅ ∪ 𝑑) = (𝑑 ∪ ∅)
68		un0 4353	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑑 ∪ ∅) = 𝑑
69	67, 68	eqtri 2752	. . . . . . . . . . . . . . . . . 18 ⊢ (∅ ∪ 𝑑) = 𝑑
70	66, 69	eqtrdi 2780	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐹:(1...𝑁)⟶(1...𝑀) ∧ 𝑎 ∈ (ℕ0 ↑m (1...𝑀)) ∧ 𝑑 ∈ (ℕ0 ↑m (ℤ ∖ ℕ))) → ((𝑎 ∪ 𝑑) ∘ ( I ↾ (ℤ ∖ ℕ))) = 𝑑)
71	49, 70	uneq12d 4128	. . . . . . . . . . . . . . . 16 ⊢ ((𝐹:(1...𝑁)⟶(1...𝑀) ∧ 𝑎 ∈ (ℕ0 ↑m (1...𝑀)) ∧ 𝑑 ∈ (ℕ0 ↑m (ℤ ∖ ℕ))) → (((𝑎 ∪ 𝑑) ∘ 𝐹) ∪ ((𝑎 ∪ 𝑑) ∘ ( I ↾ (ℤ ∖ ℕ)))) = ((𝑎 ∘ 𝐹) ∪ 𝑑))
72	32, 71	eqtr2id 2777	. . . . . . . . . . . . . . 15 ⊢ ((𝐹:(1...𝑁)⟶(1...𝑀) ∧ 𝑎 ∈ (ℕ0 ↑m (1...𝑀)) ∧ 𝑑 ∈ (ℕ0 ↑m (ℤ ∖ ℕ))) → ((𝑎 ∘ 𝐹) ∪ 𝑑) = ((𝑎 ∪ 𝑑) ∘ (𝐹 ∪ ( I ↾ (ℤ ∖ ℕ)))))
73	29, 30, 31, 72	syl3anc 1373	. . . . . . . . . . . . . 14 ⊢ ((((((𝑀 ∈ ℕ0 ∧ 𝐹:(1...𝑁)⟶(1...𝑀)) ∧ 𝑁 ∈ ℕ0) ∧ 𝑏 ∈ (mzPoly‘((ℤ ∖ ℕ) ∪ (1...𝑁)))) ∧ 𝑎 ∈ (ℕ0 ↑m (1...𝑀))) ∧ 𝑑 ∈ (ℕ0 ↑m (ℤ ∖ ℕ))) → ((𝑎 ∘ 𝐹) ∪ 𝑑) = ((𝑎 ∪ 𝑑) ∘ (𝐹 ∪ ( I ↾ (ℤ ∖ ℕ)))))
74	73	fveq2d 6844	. . . . . . . . . . . . 13 ⊢ ((((((𝑀 ∈ ℕ0 ∧ 𝐹:(1...𝑁)⟶(1...𝑀)) ∧ 𝑁 ∈ ℕ0) ∧ 𝑏 ∈ (mzPoly‘((ℤ ∖ ℕ) ∪ (1...𝑁)))) ∧ 𝑎 ∈ (ℕ0 ↑m (1...𝑀))) ∧ 𝑑 ∈ (ℕ0 ↑m (ℤ ∖ ℕ))) → (𝑏‘((𝑎 ∘ 𝐹) ∪ 𝑑)) = (𝑏‘((𝑎 ∪ 𝑑) ∘ (𝐹 ∪ ( I ↾ (ℤ ∖ ℕ))))))
75		nn0ssz 12528	. . . . . . . . . . . . . . . . 17 ⊢ ℕ0 ⊆ ℤ
76		mapss 8839	. . . . . . . . . . . . . . . . 17 ⊢ ((ℤ ∈ V ∧ ℕ0 ⊆ ℤ) → (ℕ0 ↑m ((ℤ ∖ ℕ) ∪ (1...𝑀))) ⊆ (ℤ ↑m ((ℤ ∖ ℕ) ∪ (1...𝑀))))
77	1, 75, 76	mp2an 692	. . . . . . . . . . . . . . . 16 ⊢ (ℕ0 ↑m ((ℤ ∖ ℕ) ∪ (1...𝑀))) ⊆ (ℤ ↑m ((ℤ ∖ ℕ) ∪ (1...𝑀)))
78	41	reseq2i 5936	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑎 ↾ ((1...𝑀) ∩ (ℤ ∖ ℕ))) = (𝑎 ↾ ∅)
79		res0 5943	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑎 ↾ ∅) = ∅
80	78, 79	eqtri 2752	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑎 ↾ ((1...𝑀) ∩ (ℤ ∖ ℕ))) = ∅
81	41	reseq2i 5936	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑑 ↾ ((1...𝑀) ∩ (ℤ ∖ ℕ))) = (𝑑 ↾ ∅)
82		res0 5943	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑑 ↾ ∅) = ∅
83	81, 82	eqtri 2752	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑑 ↾ ((1...𝑀) ∩ (ℤ ∖ ℕ))) = ∅
84	80, 83	eqtr4i 2755	. . . . . . . . . . . . . . . . 17 ⊢ (𝑎 ↾ ((1...𝑀) ∩ (ℤ ∖ ℕ))) = (𝑑 ↾ ((1...𝑀) ∩ (ℤ ∖ ℕ)))
85		elmapresaun 8830	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑎 ∈ (ℕ0 ↑m (1...𝑀)) ∧ 𝑑 ∈ (ℕ0 ↑m (ℤ ∖ ℕ)) ∧ (𝑎 ↾ ((1...𝑀) ∩ (ℤ ∖ ℕ))) = (𝑑 ↾ ((1...𝑀) ∩ (ℤ ∖ ℕ)))) → (𝑎 ∪ 𝑑) ∈ (ℕ0 ↑m ((1...𝑀) ∪ (ℤ ∖ ℕ))))
86		uncom 4117	. . . . . . . . . . . . . . . . . . 19 ⊢ ((1...𝑀) ∪ (ℤ ∖ ℕ)) = ((ℤ ∖ ℕ) ∪ (1...𝑀))
87	86	oveq2i 7380	. . . . . . . . . . . . . . . . . 18 ⊢ (ℕ0 ↑m ((1...𝑀) ∪ (ℤ ∖ ℕ))) = (ℕ0 ↑m ((ℤ ∖ ℕ) ∪ (1...𝑀)))
88	85, 87	eleqtrdi 2838	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑎 ∈ (ℕ0 ↑m (1...𝑀)) ∧ 𝑑 ∈ (ℕ0 ↑m (ℤ ∖ ℕ)) ∧ (𝑎 ↾ ((1...𝑀) ∩ (ℤ ∖ ℕ))) = (𝑑 ↾ ((1...𝑀) ∩ (ℤ ∖ ℕ)))) → (𝑎 ∪ 𝑑) ∈ (ℕ0 ↑m ((ℤ ∖ ℕ) ∪ (1...𝑀))))
89	84, 88	mp3an3 1452	. . . . . . . . . . . . . . . 16 ⊢ ((𝑎 ∈ (ℕ0 ↑m (1...𝑀)) ∧ 𝑑 ∈ (ℕ0 ↑m (ℤ ∖ ℕ))) → (𝑎 ∪ 𝑑) ∈ (ℕ0 ↑m ((ℤ ∖ ℕ) ∪ (1...𝑀))))
90	77, 89	sselid 3941	. . . . . . . . . . . . . . 15 ⊢ ((𝑎 ∈ (ℕ0 ↑m (1...𝑀)) ∧ 𝑑 ∈ (ℕ0 ↑m (ℤ ∖ ℕ))) → (𝑎 ∪ 𝑑) ∈ (ℤ ↑m ((ℤ ∖ ℕ) ∪ (1...𝑀))))
91	90	adantll 714	. . . . . . . . . . . . . 14 ⊢ ((((((𝑀 ∈ ℕ0 ∧ 𝐹:(1...𝑁)⟶(1...𝑀)) ∧ 𝑁 ∈ ℕ0) ∧ 𝑏 ∈ (mzPoly‘((ℤ ∖ ℕ) ∪ (1...𝑁)))) ∧ 𝑎 ∈ (ℕ0 ↑m (1...𝑀))) ∧ 𝑑 ∈ (ℕ0 ↑m (ℤ ∖ ℕ))) → (𝑎 ∪ 𝑑) ∈ (ℤ ↑m ((ℤ ∖ ℕ) ∪ (1...𝑀))))
92		coeq1 5811	. . . . . . . . . . . . . . . 16 ⊢ (𝑒 = (𝑎 ∪ 𝑑) → (𝑒 ∘ (𝐹 ∪ ( I ↾ (ℤ ∖ ℕ)))) = ((𝑎 ∪ 𝑑) ∘ (𝐹 ∪ ( I ↾ (ℤ ∖ ℕ)))))
93	92	fveq2d 6844	. . . . . . . . . . . . . . 15 ⊢ (𝑒 = (𝑎 ∪ 𝑑) → (𝑏‘(𝑒 ∘ (𝐹 ∪ ( I ↾ (ℤ ∖ ℕ))))) = (𝑏‘((𝑎 ∪ 𝑑) ∘ (𝐹 ∪ ( I ↾ (ℤ ∖ ℕ))))))
94		eqid 2729	. . . . . . . . . . . . . . 15 ⊢ (𝑒 ∈ (ℤ ↑m ((ℤ ∖ ℕ) ∪ (1...𝑀))) ↦ (𝑏‘(𝑒 ∘ (𝐹 ∪ ( I ↾ (ℤ ∖ ℕ)))))) = (𝑒 ∈ (ℤ ↑m ((ℤ ∖ ℕ) ∪ (1...𝑀))) ↦ (𝑏‘(𝑒 ∘ (𝐹 ∪ ( I ↾ (ℤ ∖ ℕ))))))
95		fvex 6853	. . . . . . . . . . . . . . 15 ⊢ (𝑏‘((𝑎 ∪ 𝑑) ∘ (𝐹 ∪ ( I ↾ (ℤ ∖ ℕ))))) ∈ V
96	93, 94, 95	fvmpt 6950	. . . . . . . . . . . . . 14 ⊢ ((𝑎 ∪ 𝑑) ∈ (ℤ ↑m ((ℤ ∖ ℕ) ∪ (1...𝑀))) → ((𝑒 ∈ (ℤ ↑m ((ℤ ∖ ℕ) ∪ (1...𝑀))) ↦ (𝑏‘(𝑒 ∘ (𝐹 ∪ ( I ↾ (ℤ ∖ ℕ))))))‘(𝑎 ∪ 𝑑)) = (𝑏‘((𝑎 ∪ 𝑑) ∘ (𝐹 ∪ ( I ↾ (ℤ ∖ ℕ))))))
97	91, 96	syl 17	. . . . . . . . . . . . 13 ⊢ ((((((𝑀 ∈ ℕ0 ∧ 𝐹:(1...𝑁)⟶(1...𝑀)) ∧ 𝑁 ∈ ℕ0) ∧ 𝑏 ∈ (mzPoly‘((ℤ ∖ ℕ) ∪ (1...𝑁)))) ∧ 𝑎 ∈ (ℕ0 ↑m (1...𝑀))) ∧ 𝑑 ∈ (ℕ0 ↑m (ℤ ∖ ℕ))) → ((𝑒 ∈ (ℤ ↑m ((ℤ ∖ ℕ) ∪ (1...𝑀))) ↦ (𝑏‘(𝑒 ∘ (𝐹 ∪ ( I ↾ (ℤ ∖ ℕ))))))‘(𝑎 ∪ 𝑑)) = (𝑏‘((𝑎 ∪ 𝑑) ∘ (𝐹 ∪ ( I ↾ (ℤ ∖ ℕ))))))
98	74, 97	eqtr4d 2767	. . . . . . . . . . . 12 ⊢ ((((((𝑀 ∈ ℕ0 ∧ 𝐹:(1...𝑁)⟶(1...𝑀)) ∧ 𝑁 ∈ ℕ0) ∧ 𝑏 ∈ (mzPoly‘((ℤ ∖ ℕ) ∪ (1...𝑁)))) ∧ 𝑎 ∈ (ℕ0 ↑m (1...𝑀))) ∧ 𝑑 ∈ (ℕ0 ↑m (ℤ ∖ ℕ))) → (𝑏‘((𝑎 ∘ 𝐹) ∪ 𝑑)) = ((𝑒 ∈ (ℤ ↑m ((ℤ ∖ ℕ) ∪ (1...𝑀))) ↦ (𝑏‘(𝑒 ∘ (𝐹 ∪ ( I ↾ (ℤ ∖ ℕ))))))‘(𝑎 ∪ 𝑑)))
99	98	eqeq1d 2731	. . . . . . . . . . 11 ⊢ ((((((𝑀 ∈ ℕ0 ∧ 𝐹:(1...𝑁)⟶(1...𝑀)) ∧ 𝑁 ∈ ℕ0) ∧ 𝑏 ∈ (mzPoly‘((ℤ ∖ ℕ) ∪ (1...𝑁)))) ∧ 𝑎 ∈ (ℕ0 ↑m (1...𝑀))) ∧ 𝑑 ∈ (ℕ0 ↑m (ℤ ∖ ℕ))) → ((𝑏‘((𝑎 ∘ 𝐹) ∪ 𝑑)) = 0 ↔ ((𝑒 ∈ (ℤ ↑m ((ℤ ∖ ℕ) ∪ (1...𝑀))) ↦ (𝑏‘(𝑒 ∘ (𝐹 ∪ ( I ↾ (ℤ ∖ ℕ))))))‘(𝑎 ∪ 𝑑)) = 0))
100	99	rexbidva 3155	. . . . . . . . . 10 ⊢ (((((𝑀 ∈ ℕ0 ∧ 𝐹:(1...𝑁)⟶(1...𝑀)) ∧ 𝑁 ∈ ℕ0) ∧ 𝑏 ∈ (mzPoly‘((ℤ ∖ ℕ) ∪ (1...𝑁)))) ∧ 𝑎 ∈ (ℕ0 ↑m (1...𝑀))) → (∃𝑑 ∈ (ℕ0 ↑m (ℤ ∖ ℕ))(𝑏‘((𝑎 ∘ 𝐹) ∪ 𝑑)) = 0 ↔ ∃𝑑 ∈ (ℕ0 ↑m (ℤ ∖ ℕ))((𝑒 ∈ (ℤ ↑m ((ℤ ∖ ℕ) ∪ (1...𝑀))) ↦ (𝑏‘(𝑒 ∘ (𝐹 ∪ ( I ↾ (ℤ ∖ ℕ))))))‘(𝑎 ∪ 𝑑)) = 0))
101	28, 100	bitrd 279	. . . . . . . . 9 ⊢ (((((𝑀 ∈ ℕ0 ∧ 𝐹:(1...𝑁)⟶(1...𝑀)) ∧ 𝑁 ∈ ℕ0) ∧ 𝑏 ∈ (mzPoly‘((ℤ ∖ ℕ) ∪ (1...𝑁)))) ∧ 𝑎 ∈ (ℕ0 ↑m (1...𝑀))) → ((𝑎 ∘ 𝐹) ∈ {𝑐 ∈ (ℕ0 ↑m (1...𝑁)) ∣ ∃𝑑 ∈ (ℕ0 ↑m (ℤ ∖ ℕ))(𝑏‘(𝑐 ∪ 𝑑)) = 0} ↔ ∃𝑑 ∈ (ℕ0 ↑m (ℤ ∖ ℕ))((𝑒 ∈ (ℤ ↑m ((ℤ ∖ ℕ) ∪ (1...𝑀))) ↦ (𝑏‘(𝑒 ∘ (𝐹 ∪ ( I ↾ (ℤ ∖ ℕ))))))‘(𝑎 ∪ 𝑑)) = 0))
102	101	rabbidva 3409	. . . . . . . 8 ⊢ ((((𝑀 ∈ ℕ0 ∧ 𝐹:(1...𝑁)⟶(1...𝑀)) ∧ 𝑁 ∈ ℕ0) ∧ 𝑏 ∈ (mzPoly‘((ℤ ∖ ℕ) ∪ (1...𝑁)))) → {𝑎 ∈ (ℕ0 ↑m (1...𝑀)) ∣ (𝑎 ∘ 𝐹) ∈ {𝑐 ∈ (ℕ0 ↑m (1...𝑁)) ∣ ∃𝑑 ∈ (ℕ0 ↑m (ℤ ∖ ℕ))(𝑏‘(𝑐 ∪ 𝑑)) = 0}} = {𝑎 ∈ (ℕ0 ↑m (1...𝑀)) ∣ ∃𝑑 ∈ (ℕ0 ↑m (ℤ ∖ ℕ))((𝑒 ∈ (ℤ ↑m ((ℤ ∖ ℕ) ∪ (1...𝑀))) ↦ (𝑏‘(𝑒 ∘ (𝐹 ∪ ( I ↾ (ℤ ∖ ℕ))))))‘(𝑎 ∪ 𝑑)) = 0})
103		simplll 774	. . . . . . . . 9 ⊢ ((((𝑀 ∈ ℕ0 ∧ 𝐹:(1...𝑁)⟶(1...𝑀)) ∧ 𝑁 ∈ ℕ0) ∧ 𝑏 ∈ (mzPoly‘((ℤ ∖ ℕ) ∪ (1...𝑁)))) → 𝑀 ∈ ℕ0)
104		ovex 7402	. . . . . . . . . . . 12 ⊢ (1...𝑀) ∈ V
105	3, 104	unex 7700	. . . . . . . . . . 11 ⊢ ((ℤ ∖ ℕ) ∪ (1...𝑀)) ∈ V
106	105	a1i 11	. . . . . . . . . 10 ⊢ ((((𝑀 ∈ ℕ0 ∧ 𝐹:(1...𝑁)⟶(1...𝑀)) ∧ 𝑁 ∈ ℕ0) ∧ 𝑏 ∈ (mzPoly‘((ℤ ∖ ℕ) ∪ (1...𝑁)))) → ((ℤ ∖ ℕ) ∪ (1...𝑀)) ∈ V)
107		simpr 484	. . . . . . . . . 10 ⊢ ((((𝑀 ∈ ℕ0 ∧ 𝐹:(1...𝑁)⟶(1...𝑀)) ∧ 𝑁 ∈ ℕ0) ∧ 𝑏 ∈ (mzPoly‘((ℤ ∖ ℕ) ∪ (1...𝑁)))) → 𝑏 ∈ (mzPoly‘((ℤ ∖ ℕ) ∪ (1...𝑁))))
108	55	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝐹:(1...𝑁)⟶(1...𝑀) → ( I ↾ (ℤ ∖ ℕ)):(ℤ ∖ ℕ)⟶(ℤ ∖ ℕ))
109		id 22	. . . . . . . . . . . . 13 ⊢ (𝐹:(1...𝑁)⟶(1...𝑀) → 𝐹:(1...𝑁)⟶(1...𝑀))
110		incom 4168	. . . . . . . . . . . . . . 15 ⊢ ((ℤ ∖ ℕ) ∩ (1...𝑁)) = ((1...𝑁) ∩ (ℤ ∖ ℕ))
111		fz1ssnn 13492	. . . . . . . . . . . . . . . 16 ⊢ (1...𝑁) ⊆ ℕ
112		ssdisj 4419	. . . . . . . . . . . . . . . 16 ⊢ (((1...𝑁) ⊆ ℕ ∧ (ℕ ∩ (ℤ ∖ ℕ)) = ∅) → ((1...𝑁) ∩ (ℤ ∖ ℕ)) = ∅)
113	111, 39, 112	mp2an 692	. . . . . . . . . . . . . . 15 ⊢ ((1...𝑁) ∩ (ℤ ∖ ℕ)) = ∅
114	110, 113	eqtri 2752	. . . . . . . . . . . . . 14 ⊢ ((ℤ ∖ ℕ) ∩ (1...𝑁)) = ∅
115	114	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝐹:(1...𝑁)⟶(1...𝑀) → ((ℤ ∖ ℕ) ∩ (1...𝑁)) = ∅)
116		fun 6704	. . . . . . . . . . . . 13 ⊢ (((( I ↾ (ℤ ∖ ℕ)):(ℤ ∖ ℕ)⟶(ℤ ∖ ℕ) ∧ 𝐹:(1...𝑁)⟶(1...𝑀)) ∧ ((ℤ ∖ ℕ) ∩ (1...𝑁)) = ∅) → (( I ↾ (ℤ ∖ ℕ)) ∪ 𝐹):((ℤ ∖ ℕ) ∪ (1...𝑁))⟶((ℤ ∖ ℕ) ∪ (1...𝑀)))
117	108, 109, 115, 116	syl21anc 837	. . . . . . . . . . . 12 ⊢ (𝐹:(1...𝑁)⟶(1...𝑀) → (( I ↾ (ℤ ∖ ℕ)) ∪ 𝐹):((ℤ ∖ ℕ) ∪ (1...𝑁))⟶((ℤ ∖ ℕ) ∪ (1...𝑀)))
118		uncom 4117	. . . . . . . . . . . . 13 ⊢ (( I ↾ (ℤ ∖ ℕ)) ∪ 𝐹) = (𝐹 ∪ ( I ↾ (ℤ ∖ ℕ)))
119	118	feq1i 6661	. . . . . . . . . . . 12 ⊢ ((( I ↾ (ℤ ∖ ℕ)) ∪ 𝐹):((ℤ ∖ ℕ) ∪ (1...𝑁))⟶((ℤ ∖ ℕ) ∪ (1...𝑀)) ↔ (𝐹 ∪ ( I ↾ (ℤ ∖ ℕ))):((ℤ ∖ ℕ) ∪ (1...𝑁))⟶((ℤ ∖ ℕ) ∪ (1...𝑀)))
120	117, 119	sylib 218	. . . . . . . . . . 11 ⊢ (𝐹:(1...𝑁)⟶(1...𝑀) → (𝐹 ∪ ( I ↾ (ℤ ∖ ℕ))):((ℤ ∖ ℕ) ∪ (1...𝑁))⟶((ℤ ∖ ℕ) ∪ (1...𝑀)))
121	120	ad3antlr 731	. . . . . . . . . 10 ⊢ ((((𝑀 ∈ ℕ0 ∧ 𝐹:(1...𝑁)⟶(1...𝑀)) ∧ 𝑁 ∈ ℕ0) ∧ 𝑏 ∈ (mzPoly‘((ℤ ∖ ℕ) ∪ (1...𝑁)))) → (𝐹 ∪ ( I ↾ (ℤ ∖ ℕ))):((ℤ ∖ ℕ) ∪ (1...𝑁))⟶((ℤ ∖ ℕ) ∪ (1...𝑀)))
122		mzprename 42710	. . . . . . . . . 10 ⊢ ((((ℤ ∖ ℕ) ∪ (1...𝑀)) ∈ V ∧ 𝑏 ∈ (mzPoly‘((ℤ ∖ ℕ) ∪ (1...𝑁))) ∧ (𝐹 ∪ ( I ↾ (ℤ ∖ ℕ))):((ℤ ∖ ℕ) ∪ (1...𝑁))⟶((ℤ ∖ ℕ) ∪ (1...𝑀))) → (𝑒 ∈ (ℤ ↑m ((ℤ ∖ ℕ) ∪ (1...𝑀))) ↦ (𝑏‘(𝑒 ∘ (𝐹 ∪ ( I ↾ (ℤ ∖ ℕ)))))) ∈ (mzPoly‘((ℤ ∖ ℕ) ∪ (1...𝑀))))
123	106, 107, 121, 122	syl3anc 1373	. . . . . . . . 9 ⊢ ((((𝑀 ∈ ℕ0 ∧ 𝐹:(1...𝑁)⟶(1...𝑀)) ∧ 𝑁 ∈ ℕ0) ∧ 𝑏 ∈ (mzPoly‘((ℤ ∖ ℕ) ∪ (1...𝑁)))) → (𝑒 ∈ (ℤ ↑m ((ℤ ∖ ℕ) ∪ (1...𝑀))) ↦ (𝑏‘(𝑒 ∘ (𝐹 ∪ ( I ↾ (ℤ ∖ ℕ)))))) ∈ (mzPoly‘((ℤ ∖ ℕ) ∪ (1...𝑀))))
124	3, 16, 17	eldioph4i 42773	. . . . . . . . 9 ⊢ ((𝑀 ∈ ℕ0 ∧ (𝑒 ∈ (ℤ ↑m ((ℤ ∖ ℕ) ∪ (1...𝑀))) ↦ (𝑏‘(𝑒 ∘ (𝐹 ∪ ( I ↾ (ℤ ∖ ℕ)))))) ∈ (mzPoly‘((ℤ ∖ ℕ) ∪ (1...𝑀)))) → {𝑎 ∈ (ℕ0 ↑m (1...𝑀)) ∣ ∃𝑑 ∈ (ℕ0 ↑m (ℤ ∖ ℕ))((𝑒 ∈ (ℤ ↑m ((ℤ ∖ ℕ) ∪ (1...𝑀))) ↦ (𝑏‘(𝑒 ∘ (𝐹 ∪ ( I ↾ (ℤ ∖ ℕ))))))‘(𝑎 ∪ 𝑑)) = 0} ∈ (Dioph‘𝑀))
125	103, 123, 124	syl2anc 584	. . . . . . . 8 ⊢ ((((𝑀 ∈ ℕ0 ∧ 𝐹:(1...𝑁)⟶(1...𝑀)) ∧ 𝑁 ∈ ℕ0) ∧ 𝑏 ∈ (mzPoly‘((ℤ ∖ ℕ) ∪ (1...𝑁)))) → {𝑎 ∈ (ℕ0 ↑m (1...𝑀)) ∣ ∃𝑑 ∈ (ℕ0 ↑m (ℤ ∖ ℕ))((𝑒 ∈ (ℤ ↑m ((ℤ ∖ ℕ) ∪ (1...𝑀))) ↦ (𝑏‘(𝑒 ∘ (𝐹 ∪ ( I ↾ (ℤ ∖ ℕ))))))‘(𝑎 ∪ 𝑑)) = 0} ∈ (Dioph‘𝑀))
126	102, 125	eqeltrd 2828	. . . . . . 7 ⊢ ((((𝑀 ∈ ℕ0 ∧ 𝐹:(1...𝑁)⟶(1...𝑀)) ∧ 𝑁 ∈ ℕ0) ∧ 𝑏 ∈ (mzPoly‘((ℤ ∖ ℕ) ∪ (1...𝑁)))) → {𝑎 ∈ (ℕ0 ↑m (1...𝑀)) ∣ (𝑎 ∘ 𝐹) ∈ {𝑐 ∈ (ℕ0 ↑m (1...𝑁)) ∣ ∃𝑑 ∈ (ℕ0 ↑m (ℤ ∖ ℕ))(𝑏‘(𝑐 ∪ 𝑑)) = 0}} ∈ (Dioph‘𝑀))
127		eleq2 2817	. . . . . . . . 9 ⊢ (𝑆 = {𝑐 ∈ (ℕ0 ↑m (1...𝑁)) ∣ ∃𝑑 ∈ (ℕ0 ↑m (ℤ ∖ ℕ))(𝑏‘(𝑐 ∪ 𝑑)) = 0} → ((𝑎 ∘ 𝐹) ∈ 𝑆 ↔ (𝑎 ∘ 𝐹) ∈ {𝑐 ∈ (ℕ0 ↑m (1...𝑁)) ∣ ∃𝑑 ∈ (ℕ0 ↑m (ℤ ∖ ℕ))(𝑏‘(𝑐 ∪ 𝑑)) = 0}))
128	127	rabbidv 3410	. . . . . . . 8 ⊢ (𝑆 = {𝑐 ∈ (ℕ0 ↑m (1...𝑁)) ∣ ∃𝑑 ∈ (ℕ0 ↑m (ℤ ∖ ℕ))(𝑏‘(𝑐 ∪ 𝑑)) = 0} → {𝑎 ∈ (ℕ0 ↑m (1...𝑀)) ∣ (𝑎 ∘ 𝐹) ∈ 𝑆} = {𝑎 ∈ (ℕ0 ↑m (1...𝑀)) ∣ (𝑎 ∘ 𝐹) ∈ {𝑐 ∈ (ℕ0 ↑m (1...𝑁)) ∣ ∃𝑑 ∈ (ℕ0 ↑m (ℤ ∖ ℕ))(𝑏‘(𝑐 ∪ 𝑑)) = 0}})
129	128	eleq1d 2813	. . . . . . 7 ⊢ (𝑆 = {𝑐 ∈ (ℕ0 ↑m (1...𝑁)) ∣ ∃𝑑 ∈ (ℕ0 ↑m (ℤ ∖ ℕ))(𝑏‘(𝑐 ∪ 𝑑)) = 0} → ({𝑎 ∈ (ℕ0 ↑m (1...𝑀)) ∣ (𝑎 ∘ 𝐹) ∈ 𝑆} ∈ (Dioph‘𝑀) ↔ {𝑎 ∈ (ℕ0 ↑m (1...𝑀)) ∣ (𝑎 ∘ 𝐹) ∈ {𝑐 ∈ (ℕ0 ↑m (1...𝑁)) ∣ ∃𝑑 ∈ (ℕ0 ↑m (ℤ ∖ ℕ))(𝑏‘(𝑐 ∪ 𝑑)) = 0}} ∈ (Dioph‘𝑀)))
130	126, 129	syl5ibrcom 247	. . . . . 6 ⊢ ((((𝑀 ∈ ℕ0 ∧ 𝐹:(1...𝑁)⟶(1...𝑀)) ∧ 𝑁 ∈ ℕ0) ∧ 𝑏 ∈ (mzPoly‘((ℤ ∖ ℕ) ∪ (1...𝑁)))) → (𝑆 = {𝑐 ∈ (ℕ0 ↑m (1...𝑁)) ∣ ∃𝑑 ∈ (ℕ0 ↑m (ℤ ∖ ℕ))(𝑏‘(𝑐 ∪ 𝑑)) = 0} → {𝑎 ∈ (ℕ0 ↑m (1...𝑀)) ∣ (𝑎 ∘ 𝐹) ∈ 𝑆} ∈ (Dioph‘𝑀)))
131	130	rexlimdva 3134	. . . . 5 ⊢ (((𝑀 ∈ ℕ0 ∧ 𝐹:(1...𝑁)⟶(1...𝑀)) ∧ 𝑁 ∈ ℕ0) → (∃𝑏 ∈ (mzPoly‘((ℤ ∖ ℕ) ∪ (1...𝑁)))𝑆 = {𝑐 ∈ (ℕ0 ↑m (1...𝑁)) ∣ ∃𝑑 ∈ (ℕ0 ↑m (ℤ ∖ ℕ))(𝑏‘(𝑐 ∪ 𝑑)) = 0} → {𝑎 ∈ (ℕ0 ↑m (1...𝑀)) ∣ (𝑎 ∘ 𝐹) ∈ 𝑆} ∈ (Dioph‘𝑀)))
132	131	expimpd 453	. . . 4 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝐹:(1...𝑁)⟶(1...𝑀)) → ((𝑁 ∈ ℕ0 ∧ ∃𝑏 ∈ (mzPoly‘((ℤ ∖ ℕ) ∪ (1...𝑁)))𝑆 = {𝑐 ∈ (ℕ0 ↑m (1...𝑁)) ∣ ∃𝑑 ∈ (ℕ0 ↑m (ℤ ∖ ℕ))(𝑏‘(𝑐 ∪ 𝑑)) = 0}) → {𝑎 ∈ (ℕ0 ↑m (1...𝑀)) ∣ (𝑎 ∘ 𝐹) ∈ 𝑆} ∈ (Dioph‘𝑀)))
133	18, 132	biimtrid 242	. . 3 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝐹:(1...𝑁)⟶(1...𝑀)) → (𝑆 ∈ (Dioph‘𝑁) → {𝑎 ∈ (ℕ0 ↑m (1...𝑀)) ∣ (𝑎 ∘ 𝐹) ∈ 𝑆} ∈ (Dioph‘𝑀)))
134	133	impcom 407	. 2 ⊢ ((𝑆 ∈ (Dioph‘𝑁) ∧ (𝑀 ∈ ℕ0 ∧ 𝐹:(1...𝑁)⟶(1...𝑀))) → {𝑎 ∈ (ℕ0 ↑m (1...𝑀)) ∣ (𝑎 ∘ 𝐹) ∈ 𝑆} ∈ (Dioph‘𝑀))
135	134	3impb 1114	1 ⊢ ((𝑆 ∈ (Dioph‘𝑁) ∧ 𝑀 ∈ ℕ0 ∧ 𝐹:(1...𝑁)⟶(1...𝑀)) → {𝑎 ∈ (ℕ0 ↑m (1...𝑀)) ∣ (𝑎 ∘ 𝐹) ∈ 𝑆} ∈ (Dioph‘𝑀))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109  ∃wrex 3053  {crab 3402  Vcvv 3444   ∖ cdif 3908   ∪ cun 3909   ∩ cin 3910   ⊆ wss 3911  ∅c0 4292   class class class wbr 5102   ↦ cmpt 5183   I cid 5525   ↾ cres 5633   ∘ ccom 5635   Fn wfn 6494  ⟶wf 6495  –1-1-onto→wf1o 6498  ‘cfv 6499  (class class class)co 7369  ωcom 7822   ↑_m cmap 8776   ≈ cen 8892  Fincfn 8895  0cc0 11044  1c1 11045   + caddc 11047  ℕcn 12162  ℕ₀cn0 12418  ℤcz 12505  ℤ_≥cuz 12769  ...cfz 13444  mzPolycmzp 42683  Diophcdioph 42716

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 5229  ax-sep 5246  ax-nul 5256  ax-pow 5315  ax-pr 5382  ax-un 7691  ax-inf2 9570  ax-cnex 11100  ax-resscn 11101  ax-1cn 11102  ax-icn 11103  ax-addcl 11104  ax-addrcl 11105  ax-mulcl 11106  ax-mulrcl 11107  ax-mulcom 11108  ax-addass 11109  ax-mulass 11110  ax-distr 11111  ax-i2m1 11112  ax-1ne0 11113  ax-1rid 11114  ax-rnegex 11115  ax-rrecex 11116  ax-cnre 11117  ax-pre-lttri 11118  ax-pre-lttrn 11119  ax-pre-ltadd 11120  ax-pre-mulgt0 11121

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 3352  df-rab 3403  df-v 3446  df-sbc 3751  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3931  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-int 4907  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-tr 5210  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6262  df-ord 6323  df-on 6324  df-lim 6325  df-suc 6326  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-riota 7326  df-ov 7372  df-oprab 7373  df-mpo 7374  df-of 7633  df-om 7823  df-1st 7947  df-2nd 7948  df-frecs 8237  df-wrecs 8268  df-recs 8317  df-rdg 8355  df-1o 8411  df-oadd 8415  df-er 8648  df-map 8778  df-en 8896  df-dom 8897  df-sdom 8898  df-fin 8899  df-dju 9830  df-card 9868  df-pnf 11186  df-mnf 11187  df-xr 11188  df-ltxr 11189  df-le 11190  df-sub 11383  df-neg 11384  df-nn 12163  df-n0 12419  df-z 12506  df-uz 12770  df-fz 13445  df-hash 14272  df-mzpcl 42684  df-mzp 42685  df-dioph 42717

This theorem is referenced by:  rabrenfdioph  42775