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Theorem diophren 37896
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 ∈ ℕ0𝐹:(1...𝑁)⟶(1...𝑀)) → {𝑎 ∈ (ℕ0𝑚 (1...𝑀)) ∣ (𝑎𝐹) ∈ 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.
StepHypRef Expression
1 zex 11586 . . . . . 6 ℤ ∈ V
2 difexg 4942 . . . . . 6 (ℤ ∈ V → (ℤ ∖ ℕ) ∈ V)
31, 2ax-mp 5 . . . . 5 (ℤ ∖ ℕ) ∈ V
4 ominf 8326 . . . . . 6 ¬ ω ∈ Fin
5 nnuz 11923 . . . . . . . . . 10 ℕ = (ℤ‘1)
6 0p1e1 11332 . . . . . . . . . . 11 (0 + 1) = 1
76fveq2i 6333 . . . . . . . . . 10 (ℤ‘(0 + 1)) = (ℤ‘1)
85, 7eqtr4i 2796 . . . . . . . . 9 ℕ = (ℤ‘(0 + 1))
98difeq2i 3876 . . . . . . . 8 (ℤ ∖ ℕ) = (ℤ ∖ (ℤ‘(0 + 1)))
10 0z 11588 . . . . . . . . 9 0 ∈ ℤ
11 lzenom 37852 . . . . . . . . 9 (0 ∈ ℤ → (ℤ ∖ (ℤ‘(0 + 1))) ≈ ω)
1210, 11ax-mp 5 . . . . . . . 8 (ℤ ∖ (ℤ‘(0 + 1))) ≈ ω
139, 12eqbrtri 4807 . . . . . . 7 (ℤ ∖ ℕ) ≈ ω
14 enfi 8330 . . . . . . 7 ((ℤ ∖ ℕ) ≈ ω → ((ℤ ∖ ℕ) ∈ Fin ↔ ω ∈ Fin))
1513, 14ax-mp 5 . . . . . 6 ((ℤ ∖ ℕ) ∈ Fin ↔ ω ∈ Fin)
164, 15mtbir 312 . . . . 5 ¬ (ℤ ∖ ℕ) ∈ Fin
17 incom 3956 . . . . . 6 ((ℤ ∖ ℕ) ∩ ℕ) = (ℕ ∩ (ℤ ∖ ℕ))
18 disjdif 4182 . . . . . 6 (ℕ ∩ (ℤ ∖ ℕ)) = ∅
1917, 18eqtri 2793 . . . . 5 ((ℤ ∖ ℕ) ∩ ℕ) = ∅
203, 16, 19eldioph4b 37894 . . . 4 (𝑆 ∈ (Dioph‘𝑁) ↔ (𝑁 ∈ ℕ0 ∧ ∃𝑏 ∈ (mzPoly‘((ℤ ∖ ℕ) ∪ (1...𝑁)))𝑆 = {𝑐 ∈ (ℕ0𝑚 (1...𝑁)) ∣ ∃𝑑 ∈ (ℕ0𝑚 (ℤ ∖ ℕ))(𝑏‘(𝑐𝑑)) = 0}))
21 simpr 471 . . . . . . . . . . . 12 (((((𝑀 ∈ ℕ0𝐹:(1...𝑁)⟶(1...𝑀)) ∧ 𝑁 ∈ ℕ0) ∧ 𝑏 ∈ (mzPoly‘((ℤ ∖ ℕ) ∪ (1...𝑁)))) ∧ 𝑎 ∈ (ℕ0𝑚 (1...𝑀))) → 𝑎 ∈ (ℕ0𝑚 (1...𝑀)))
22 simp-4r 770 . . . . . . . . . . . 12 (((((𝑀 ∈ ℕ0𝐹:(1...𝑁)⟶(1...𝑀)) ∧ 𝑁 ∈ ℕ0) ∧ 𝑏 ∈ (mzPoly‘((ℤ ∖ ℕ) ∪ (1...𝑁)))) ∧ 𝑎 ∈ (ℕ0𝑚 (1...𝑀))) → 𝐹:(1...𝑁)⟶(1...𝑀))
23 ovex 6821 . . . . . . . . . . . . 13 (1...𝑁) ∈ V
2423mapco2 37797 . . . . . . . . . . . 12 ((𝑎 ∈ (ℕ0𝑚 (1...𝑀)) ∧ 𝐹:(1...𝑁)⟶(1...𝑀)) → (𝑎𝐹) ∈ (ℕ0𝑚 (1...𝑁)))
2521, 22, 24syl2anc 573 . . . . . . . . . . 11 (((((𝑀 ∈ ℕ0𝐹:(1...𝑁)⟶(1...𝑀)) ∧ 𝑁 ∈ ℕ0) ∧ 𝑏 ∈ (mzPoly‘((ℤ ∖ ℕ) ∪ (1...𝑁)))) ∧ 𝑎 ∈ (ℕ0𝑚 (1...𝑀))) → (𝑎𝐹) ∈ (ℕ0𝑚 (1...𝑁)))
26 uneq1 3911 . . . . . . . . . . . . . . 15 (𝑐 = (𝑎𝐹) → (𝑐𝑑) = ((𝑎𝐹) ∪ 𝑑))
2726fveq2d 6334 . . . . . . . . . . . . . 14 (𝑐 = (𝑎𝐹) → (𝑏‘(𝑐𝑑)) = (𝑏‘((𝑎𝐹) ∪ 𝑑)))
2827eqeq1d 2773 . . . . . . . . . . . . 13 (𝑐 = (𝑎𝐹) → ((𝑏‘(𝑐𝑑)) = 0 ↔ (𝑏‘((𝑎𝐹) ∪ 𝑑)) = 0))
2928rexbidv 3200 . . . . . . . . . . . 12 (𝑐 = (𝑎𝐹) → (∃𝑑 ∈ (ℕ0𝑚 (ℤ ∖ ℕ))(𝑏‘(𝑐𝑑)) = 0 ↔ ∃𝑑 ∈ (ℕ0𝑚 (ℤ ∖ ℕ))(𝑏‘((𝑎𝐹) ∪ 𝑑)) = 0))
3029elrab3 3516 . . . . . . . . . . 11 ((𝑎𝐹) ∈ (ℕ0𝑚 (1...𝑁)) → ((𝑎𝐹) ∈ {𝑐 ∈ (ℕ0𝑚 (1...𝑁)) ∣ ∃𝑑 ∈ (ℕ0𝑚 (ℤ ∖ ℕ))(𝑏‘(𝑐𝑑)) = 0} ↔ ∃𝑑 ∈ (ℕ0𝑚 (ℤ ∖ ℕ))(𝑏‘((𝑎𝐹) ∪ 𝑑)) = 0))
3125, 30syl 17 . . . . . . . . . 10 (((((𝑀 ∈ ℕ0𝐹:(1...𝑁)⟶(1...𝑀)) ∧ 𝑁 ∈ ℕ0) ∧ 𝑏 ∈ (mzPoly‘((ℤ ∖ ℕ) ∪ (1...𝑁)))) ∧ 𝑎 ∈ (ℕ0𝑚 (1...𝑀))) → ((𝑎𝐹) ∈ {𝑐 ∈ (ℕ0𝑚 (1...𝑁)) ∣ ∃𝑑 ∈ (ℕ0𝑚 (ℤ ∖ ℕ))(𝑏‘(𝑐𝑑)) = 0} ↔ ∃𝑑 ∈ (ℕ0𝑚 (ℤ ∖ ℕ))(𝑏‘((𝑎𝐹) ∪ 𝑑)) = 0))
32 simp-5r 774 . . . . . . . . . . . . . . 15 ((((((𝑀 ∈ ℕ0𝐹:(1...𝑁)⟶(1...𝑀)) ∧ 𝑁 ∈ ℕ0) ∧ 𝑏 ∈ (mzPoly‘((ℤ ∖ ℕ) ∪ (1...𝑁)))) ∧ 𝑎 ∈ (ℕ0𝑚 (1...𝑀))) ∧ 𝑑 ∈ (ℕ0𝑚 (ℤ ∖ ℕ))) → 𝐹:(1...𝑁)⟶(1...𝑀))
33 simplr 752 . . . . . . . . . . . . . . 15 ((((((𝑀 ∈ ℕ0𝐹:(1...𝑁)⟶(1...𝑀)) ∧ 𝑁 ∈ ℕ0) ∧ 𝑏 ∈ (mzPoly‘((ℤ ∖ ℕ) ∪ (1...𝑁)))) ∧ 𝑎 ∈ (ℕ0𝑚 (1...𝑀))) ∧ 𝑑 ∈ (ℕ0𝑚 (ℤ ∖ ℕ))) → 𝑎 ∈ (ℕ0𝑚 (1...𝑀)))
34 simpr 471 . . . . . . . . . . . . . . 15 ((((((𝑀 ∈ ℕ0𝐹:(1...𝑁)⟶(1...𝑀)) ∧ 𝑁 ∈ ℕ0) ∧ 𝑏 ∈ (mzPoly‘((ℤ ∖ ℕ) ∪ (1...𝑁)))) ∧ 𝑎 ∈ (ℕ0𝑚 (1...𝑀))) ∧ 𝑑 ∈ (ℕ0𝑚 (ℤ ∖ ℕ))) → 𝑑 ∈ (ℕ0𝑚 (ℤ ∖ ℕ)))
35 coundi 5778 . . . . . . . . . . . . . . . 16 ((𝑎𝑑) ∘ (𝐹 ∪ ( I ↾ (ℤ ∖ ℕ)))) = (((𝑎𝑑) ∘ 𝐹) ∪ ((𝑎𝑑) ∘ ( I ↾ (ℤ ∖ ℕ))))
36 coundir 5779 . . . . . . . . . . . . . . . . . . 19 ((𝑎𝑑) ∘ 𝐹) = ((𝑎𝐹) ∪ (𝑑𝐹))
37 elmapi 8029 . . . . . . . . . . . . . . . . . . . . . 22 (𝑑 ∈ (ℕ0𝑚 (ℤ ∖ ℕ)) → 𝑑:(ℤ ∖ ℕ)⟶ℕ0)
38373ad2ant3 1129 . . . . . . . . . . . . . . . . . . . . 21 ((𝐹:(1...𝑁)⟶(1...𝑀) ∧ 𝑎 ∈ (ℕ0𝑚 (1...𝑀)) ∧ 𝑑 ∈ (ℕ0𝑚 (ℤ ∖ ℕ))) → 𝑑:(ℤ ∖ ℕ)⟶ℕ0)
39 simp1 1130 . . . . . . . . . . . . . . . . . . . . 21 ((𝐹:(1...𝑁)⟶(1...𝑀) ∧ 𝑎 ∈ (ℕ0𝑚 (1...𝑀)) ∧ 𝑑 ∈ (ℕ0𝑚 (ℤ ∖ ℕ))) → 𝐹:(1...𝑁)⟶(1...𝑀))
40 incom 3956 . . . . . . . . . . . . . . . . . . . . . . 23 ((ℤ ∖ ℕ) ∩ (1...𝑀)) = ((1...𝑀) ∩ (ℤ ∖ ℕ))
41 fz1ssnn 12572 . . . . . . . . . . . . . . . . . . . . . . . 24 (1...𝑀) ⊆ ℕ
42 ssdisj 4170 . . . . . . . . . . . . . . . . . . . . . . . 24 (((1...𝑀) ⊆ ℕ ∧ (ℕ ∩ (ℤ ∖ ℕ)) = ∅) → ((1...𝑀) ∩ (ℤ ∖ ℕ)) = ∅)
4341, 18, 42mp2an 672 . . . . . . . . . . . . . . . . . . . . . . 23 ((1...𝑀) ∩ (ℤ ∖ ℕ)) = ∅
4440, 43eqtri 2793 . . . . . . . . . . . . . . . . . . . . . 22 ((ℤ ∖ ℕ) ∩ (1...𝑀)) = ∅
4544a1i 11 . . . . . . . . . . . . . . . . . . . . 21 ((𝐹:(1...𝑁)⟶(1...𝑀) ∧ 𝑎 ∈ (ℕ0𝑚 (1...𝑀)) ∧ 𝑑 ∈ (ℕ0𝑚 (ℤ ∖ ℕ))) → ((ℤ ∖ ℕ) ∩ (1...𝑀)) = ∅)
46 coeq0i 37835 . . . . . . . . . . . . . . . . . . . . 21 ((𝑑:(ℤ ∖ ℕ)⟶ℕ0𝐹:(1...𝑁)⟶(1...𝑀) ∧ ((ℤ ∖ ℕ) ∩ (1...𝑀)) = ∅) → (𝑑𝐹) = ∅)
4738, 39, 45, 46syl3anc 1476 . . . . . . . . . . . . . . . . . . . 20 ((𝐹:(1...𝑁)⟶(1...𝑀) ∧ 𝑎 ∈ (ℕ0𝑚 (1...𝑀)) ∧ 𝑑 ∈ (ℕ0𝑚 (ℤ ∖ ℕ))) → (𝑑𝐹) = ∅)
4847uneq2d 3918 . . . . . . . . . . . . . . . . . . 19 ((𝐹:(1...𝑁)⟶(1...𝑀) ∧ 𝑎 ∈ (ℕ0𝑚 (1...𝑀)) ∧ 𝑑 ∈ (ℕ0𝑚 (ℤ ∖ ℕ))) → ((𝑎𝐹) ∪ (𝑑𝐹)) = ((𝑎𝐹) ∪ ∅))
4936, 48syl5eq 2817 . . . . . . . . . . . . . . . . . 18 ((𝐹:(1...𝑁)⟶(1...𝑀) ∧ 𝑎 ∈ (ℕ0𝑚 (1...𝑀)) ∧ 𝑑 ∈ (ℕ0𝑚 (ℤ ∖ ℕ))) → ((𝑎𝑑) ∘ 𝐹) = ((𝑎𝐹) ∪ ∅))
50 un0 4111 . . . . . . . . . . . . . . . . . 18 ((𝑎𝐹) ∪ ∅) = (𝑎𝐹)
5149, 50syl6eq 2821 . . . . . . . . . . . . . . . . 17 ((𝐹:(1...𝑁)⟶(1...𝑀) ∧ 𝑎 ∈ (ℕ0𝑚 (1...𝑀)) ∧ 𝑑 ∈ (ℕ0𝑚 (ℤ ∖ ℕ))) → ((𝑎𝑑) ∘ 𝐹) = (𝑎𝐹))
52 coundir 5779 . . . . . . . . . . . . . . . . . . 19 ((𝑎𝑑) ∘ ( I ↾ (ℤ ∖ ℕ))) = ((𝑎 ∘ ( I ↾ (ℤ ∖ ℕ))) ∪ (𝑑 ∘ ( I ↾ (ℤ ∖ ℕ))))
53 elmapi 8029 . . . . . . . . . . . . . . . . . . . . . 22 (𝑎 ∈ (ℕ0𝑚 (1...𝑀)) → 𝑎:(1...𝑀)⟶ℕ0)
54533ad2ant2 1128 . . . . . . . . . . . . . . . . . . . . 21 ((𝐹:(1...𝑁)⟶(1...𝑀) ∧ 𝑎 ∈ (ℕ0𝑚 (1...𝑀)) ∧ 𝑑 ∈ (ℕ0𝑚 (ℤ ∖ ℕ))) → 𝑎:(1...𝑀)⟶ℕ0)
55 f1oi 6313 . . . . . . . . . . . . . . . . . . . . . . 23 ( I ↾ (ℤ ∖ ℕ)):(ℤ ∖ ℕ)–1-1-onto→(ℤ ∖ ℕ)
56 f1of 6276 . . . . . . . . . . . . . . . . . . . . . . 23 (( I ↾ (ℤ ∖ ℕ)):(ℤ ∖ ℕ)–1-1-onto→(ℤ ∖ ℕ) → ( I ↾ (ℤ ∖ ℕ)):(ℤ ∖ ℕ)⟶(ℤ ∖ ℕ))
5755, 56ax-mp 5 . . . . . . . . . . . . . . . . . . . . . 22 ( I ↾ (ℤ ∖ ℕ)):(ℤ ∖ ℕ)⟶(ℤ ∖ ℕ)
58 coeq0i 37835 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑎:(1...𝑀)⟶ℕ0 ∧ ( I ↾ (ℤ ∖ ℕ)):(ℤ ∖ ℕ)⟶(ℤ ∖ ℕ) ∧ ((1...𝑀) ∩ (ℤ ∖ ℕ)) = ∅) → (𝑎 ∘ ( I ↾ (ℤ ∖ ℕ))) = ∅)
5957, 43, 58mp3an23 1564 . . . . . . . . . . . . . . . . . . . . 21 (𝑎:(1...𝑀)⟶ℕ0 → (𝑎 ∘ ( I ↾ (ℤ ∖ ℕ))) = ∅)
6054, 59syl 17 . . . . . . . . . . . . . . . . . . . 20 ((𝐹:(1...𝑁)⟶(1...𝑀) ∧ 𝑎 ∈ (ℕ0𝑚 (1...𝑀)) ∧ 𝑑 ∈ (ℕ0𝑚 (ℤ ∖ ℕ))) → (𝑎 ∘ ( I ↾ (ℤ ∖ ℕ))) = ∅)
61 coires1 5795 . . . . . . . . . . . . . . . . . . . . . 22 (𝑑 ∘ ( I ↾ (ℤ ∖ ℕ))) = (𝑑 ↾ (ℤ ∖ ℕ))
62 ffn 6183 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑑:(ℤ ∖ ℕ)⟶ℕ0𝑑 Fn (ℤ ∖ ℕ))
63 fnresdm 6138 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑑 Fn (ℤ ∖ ℕ) → (𝑑 ↾ (ℤ ∖ ℕ)) = 𝑑)
6437, 62, 633syl 18 . . . . . . . . . . . . . . . . . . . . . 22 (𝑑 ∈ (ℕ0𝑚 (ℤ ∖ ℕ)) → (𝑑 ↾ (ℤ ∖ ℕ)) = 𝑑)
6561, 64syl5eq 2817 . . . . . . . . . . . . . . . . . . . . 21 (𝑑 ∈ (ℕ0𝑚 (ℤ ∖ ℕ)) → (𝑑 ∘ ( I ↾ (ℤ ∖ ℕ))) = 𝑑)
66653ad2ant3 1129 . . . . . . . . . . . . . . . . . . . 20 ((𝐹:(1...𝑁)⟶(1...𝑀) ∧ 𝑎 ∈ (ℕ0𝑚 (1...𝑀)) ∧ 𝑑 ∈ (ℕ0𝑚 (ℤ ∖ ℕ))) → (𝑑 ∘ ( I ↾ (ℤ ∖ ℕ))) = 𝑑)
6760, 66uneq12d 3919 . . . . . . . . . . . . . . . . . . 19 ((𝐹:(1...𝑁)⟶(1...𝑀) ∧ 𝑎 ∈ (ℕ0𝑚 (1...𝑀)) ∧ 𝑑 ∈ (ℕ0𝑚 (ℤ ∖ ℕ))) → ((𝑎 ∘ ( I ↾ (ℤ ∖ ℕ))) ∪ (𝑑 ∘ ( I ↾ (ℤ ∖ ℕ)))) = (∅ ∪ 𝑑))
6852, 67syl5eq 2817 . . . . . . . . . . . . . . . . . 18 ((𝐹:(1...𝑁)⟶(1...𝑀) ∧ 𝑎 ∈ (ℕ0𝑚 (1...𝑀)) ∧ 𝑑 ∈ (ℕ0𝑚 (ℤ ∖ ℕ))) → ((𝑎𝑑) ∘ ( I ↾ (ℤ ∖ ℕ))) = (∅ ∪ 𝑑))
69 uncom 3908 . . . . . . . . . . . . . . . . . . 19 (∅ ∪ 𝑑) = (𝑑 ∪ ∅)
70 un0 4111 . . . . . . . . . . . . . . . . . . 19 (𝑑 ∪ ∅) = 𝑑
7169, 70eqtri 2793 . . . . . . . . . . . . . . . . . 18 (∅ ∪ 𝑑) = 𝑑
7268, 71syl6eq 2821 . . . . . . . . . . . . . . . . 17 ((𝐹:(1...𝑁)⟶(1...𝑀) ∧ 𝑎 ∈ (ℕ0𝑚 (1...𝑀)) ∧ 𝑑 ∈ (ℕ0𝑚 (ℤ ∖ ℕ))) → ((𝑎𝑑) ∘ ( I ↾ (ℤ ∖ ℕ))) = 𝑑)
7351, 72uneq12d 3919 . . . . . . . . . . . . . . . 16 ((𝐹:(1...𝑁)⟶(1...𝑀) ∧ 𝑎 ∈ (ℕ0𝑚 (1...𝑀)) ∧ 𝑑 ∈ (ℕ0𝑚 (ℤ ∖ ℕ))) → (((𝑎𝑑) ∘ 𝐹) ∪ ((𝑎𝑑) ∘ ( I ↾ (ℤ ∖ ℕ)))) = ((𝑎𝐹) ∪ 𝑑))
7435, 73syl5req 2818 . . . . . . . . . . . . . . 15 ((𝐹:(1...𝑁)⟶(1...𝑀) ∧ 𝑎 ∈ (ℕ0𝑚 (1...𝑀)) ∧ 𝑑 ∈ (ℕ0𝑚 (ℤ ∖ ℕ))) → ((𝑎𝐹) ∪ 𝑑) = ((𝑎𝑑) ∘ (𝐹 ∪ ( I ↾ (ℤ ∖ ℕ)))))
7532, 33, 34, 74syl3anc 1476 . . . . . . . . . . . . . 14 ((((((𝑀 ∈ ℕ0𝐹:(1...𝑁)⟶(1...𝑀)) ∧ 𝑁 ∈ ℕ0) ∧ 𝑏 ∈ (mzPoly‘((ℤ ∖ ℕ) ∪ (1...𝑁)))) ∧ 𝑎 ∈ (ℕ0𝑚 (1...𝑀))) ∧ 𝑑 ∈ (ℕ0𝑚 (ℤ ∖ ℕ))) → ((𝑎𝐹) ∪ 𝑑) = ((𝑎𝑑) ∘ (𝐹 ∪ ( I ↾ (ℤ ∖ ℕ)))))
7675fveq2d 6334 . . . . . . . . . . . . 13 ((((((𝑀 ∈ ℕ0𝐹:(1...𝑁)⟶(1...𝑀)) ∧ 𝑁 ∈ ℕ0) ∧ 𝑏 ∈ (mzPoly‘((ℤ ∖ ℕ) ∪ (1...𝑁)))) ∧ 𝑎 ∈ (ℕ0𝑚 (1...𝑀))) ∧ 𝑑 ∈ (ℕ0𝑚 (ℤ ∖ ℕ))) → (𝑏‘((𝑎𝐹) ∪ 𝑑)) = (𝑏‘((𝑎𝑑) ∘ (𝐹 ∪ ( I ↾ (ℤ ∖ ℕ))))))
77 nn0ssz 11598 . . . . . . . . . . . . . . . . 17 0 ⊆ ℤ
78 mapss 8052 . . . . . . . . . . . . . . . . 17 ((ℤ ∈ V ∧ ℕ0 ⊆ ℤ) → (ℕ0𝑚 ((ℤ ∖ ℕ) ∪ (1...𝑀))) ⊆ (ℤ ↑𝑚 ((ℤ ∖ ℕ) ∪ (1...𝑀))))
791, 77, 78mp2an 672 . . . . . . . . . . . . . . . 16 (ℕ0𝑚 ((ℤ ∖ ℕ) ∪ (1...𝑀))) ⊆ (ℤ ↑𝑚 ((ℤ ∖ ℕ) ∪ (1...𝑀)))
8043reseq2i 5529 . . . . . . . . . . . . . . . . . . 19 (𝑎 ↾ ((1...𝑀) ∩ (ℤ ∖ ℕ))) = (𝑎 ↾ ∅)
81 res0 5536 . . . . . . . . . . . . . . . . . . 19 (𝑎 ↾ ∅) = ∅
8280, 81eqtri 2793 . . . . . . . . . . . . . . . . . 18 (𝑎 ↾ ((1...𝑀) ∩ (ℤ ∖ ℕ))) = ∅
8343reseq2i 5529 . . . . . . . . . . . . . . . . . . 19 (𝑑 ↾ ((1...𝑀) ∩ (ℤ ∖ ℕ))) = (𝑑 ↾ ∅)
84 res0 5536 . . . . . . . . . . . . . . . . . . 19 (𝑑 ↾ ∅) = ∅
8583, 84eqtri 2793 . . . . . . . . . . . . . . . . . 18 (𝑑 ↾ ((1...𝑀) ∩ (ℤ ∖ ℕ))) = ∅
8682, 85eqtr4i 2796 . . . . . . . . . . . . . . . . 17 (𝑎 ↾ ((1...𝑀) ∩ (ℤ ∖ ℕ))) = (𝑑 ↾ ((1...𝑀) ∩ (ℤ ∖ ℕ)))
87 elmapresaun 37853 . . . . . . . . . . . . . . . . . 18 ((𝑎 ∈ (ℕ0𝑚 (1...𝑀)) ∧ 𝑑 ∈ (ℕ0𝑚 (ℤ ∖ ℕ)) ∧ (𝑎 ↾ ((1...𝑀) ∩ (ℤ ∖ ℕ))) = (𝑑 ↾ ((1...𝑀) ∩ (ℤ ∖ ℕ)))) → (𝑎𝑑) ∈ (ℕ0𝑚 ((1...𝑀) ∪ (ℤ ∖ ℕ))))
88 uncom 3908 . . . . . . . . . . . . . . . . . . 19 ((1...𝑀) ∪ (ℤ ∖ ℕ)) = ((ℤ ∖ ℕ) ∪ (1...𝑀))
8988oveq2i 6802 . . . . . . . . . . . . . . . . . 18 (ℕ0𝑚 ((1...𝑀) ∪ (ℤ ∖ ℕ))) = (ℕ0𝑚 ((ℤ ∖ ℕ) ∪ (1...𝑀)))
9087, 89syl6eleq 2860 . . . . . . . . . . . . . . . . 17 ((𝑎 ∈ (ℕ0𝑚 (1...𝑀)) ∧ 𝑑 ∈ (ℕ0𝑚 (ℤ ∖ ℕ)) ∧ (𝑎 ↾ ((1...𝑀) ∩ (ℤ ∖ ℕ))) = (𝑑 ↾ ((1...𝑀) ∩ (ℤ ∖ ℕ)))) → (𝑎𝑑) ∈ (ℕ0𝑚 ((ℤ ∖ ℕ) ∪ (1...𝑀))))
9186, 90mp3an3 1561 . . . . . . . . . . . . . . . 16 ((𝑎 ∈ (ℕ0𝑚 (1...𝑀)) ∧ 𝑑 ∈ (ℕ0𝑚 (ℤ ∖ ℕ))) → (𝑎𝑑) ∈ (ℕ0𝑚 ((ℤ ∖ ℕ) ∪ (1...𝑀))))
9279, 91sseldi 3750 . . . . . . . . . . . . . . 15 ((𝑎 ∈ (ℕ0𝑚 (1...𝑀)) ∧ 𝑑 ∈ (ℕ0𝑚 (ℤ ∖ ℕ))) → (𝑎𝑑) ∈ (ℤ ↑𝑚 ((ℤ ∖ ℕ) ∪ (1...𝑀))))
9392adantll 693 . . . . . . . . . . . . . 14 ((((((𝑀 ∈ ℕ0𝐹:(1...𝑁)⟶(1...𝑀)) ∧ 𝑁 ∈ ℕ0) ∧ 𝑏 ∈ (mzPoly‘((ℤ ∖ ℕ) ∪ (1...𝑁)))) ∧ 𝑎 ∈ (ℕ0𝑚 (1...𝑀))) ∧ 𝑑 ∈ (ℕ0𝑚 (ℤ ∖ ℕ))) → (𝑎𝑑) ∈ (ℤ ↑𝑚 ((ℤ ∖ ℕ) ∪ (1...𝑀))))
94 coeq1 5416 . . . . . . . . . . . . . . . 16 (𝑒 = (𝑎𝑑) → (𝑒 ∘ (𝐹 ∪ ( I ↾ (ℤ ∖ ℕ)))) = ((𝑎𝑑) ∘ (𝐹 ∪ ( I ↾ (ℤ ∖ ℕ)))))
9594fveq2d 6334 . . . . . . . . . . . . . . 15 (𝑒 = (𝑎𝑑) → (𝑏‘(𝑒 ∘ (𝐹 ∪ ( I ↾ (ℤ ∖ ℕ))))) = (𝑏‘((𝑎𝑑) ∘ (𝐹 ∪ ( I ↾ (ℤ ∖ ℕ))))))
96 eqid 2771 . . . . . . . . . . . . . . 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))))))
97 fvex 6340 . . . . . . . . . . . . . . 15 (𝑏‘((𝑎𝑑) ∘ (𝐹 ∪ ( I ↾ (ℤ ∖ ℕ))))) ∈ V
9895, 96, 97fvmpt 6422 . . . . . . . . . . . . . 14 ((𝑎𝑑) ∈ (ℤ ↑𝑚 ((ℤ ∖ ℕ) ∪ (1...𝑀))) → ((𝑒 ∈ (ℤ ↑𝑚 ((ℤ ∖ ℕ) ∪ (1...𝑀))) ↦ (𝑏‘(𝑒 ∘ (𝐹 ∪ ( I ↾ (ℤ ∖ ℕ))))))‘(𝑎𝑑)) = (𝑏‘((𝑎𝑑) ∘ (𝐹 ∪ ( I ↾ (ℤ ∖ ℕ))))))
9993, 98syl 17 . . . . . . . . . . . . 13 ((((((𝑀 ∈ ℕ0𝐹:(1...𝑁)⟶(1...𝑀)) ∧ 𝑁 ∈ ℕ0) ∧ 𝑏 ∈ (mzPoly‘((ℤ ∖ ℕ) ∪ (1...𝑁)))) ∧ 𝑎 ∈ (ℕ0𝑚 (1...𝑀))) ∧ 𝑑 ∈ (ℕ0𝑚 (ℤ ∖ ℕ))) → ((𝑒 ∈ (ℤ ↑𝑚 ((ℤ ∖ ℕ) ∪ (1...𝑀))) ↦ (𝑏‘(𝑒 ∘ (𝐹 ∪ ( I ↾ (ℤ ∖ ℕ))))))‘(𝑎𝑑)) = (𝑏‘((𝑎𝑑) ∘ (𝐹 ∪ ( I ↾ (ℤ ∖ ℕ))))))
10076, 99eqtr4d 2808 . . . . . . . . . . . 12 ((((((𝑀 ∈ ℕ0𝐹:(1...𝑁)⟶(1...𝑀)) ∧ 𝑁 ∈ ℕ0) ∧ 𝑏 ∈ (mzPoly‘((ℤ ∖ ℕ) ∪ (1...𝑁)))) ∧ 𝑎 ∈ (ℕ0𝑚 (1...𝑀))) ∧ 𝑑 ∈ (ℕ0𝑚 (ℤ ∖ ℕ))) → (𝑏‘((𝑎𝐹) ∪ 𝑑)) = ((𝑒 ∈ (ℤ ↑𝑚 ((ℤ ∖ ℕ) ∪ (1...𝑀))) ↦ (𝑏‘(𝑒 ∘ (𝐹 ∪ ( I ↾ (ℤ ∖ ℕ))))))‘(𝑎𝑑)))
101100eqeq1d 2773 . . . . . . . . . . 11 ((((((𝑀 ∈ ℕ0𝐹:(1...𝑁)⟶(1...𝑀)) ∧ 𝑁 ∈ ℕ0) ∧ 𝑏 ∈ (mzPoly‘((ℤ ∖ ℕ) ∪ (1...𝑁)))) ∧ 𝑎 ∈ (ℕ0𝑚 (1...𝑀))) ∧ 𝑑 ∈ (ℕ0𝑚 (ℤ ∖ ℕ))) → ((𝑏‘((𝑎𝐹) ∪ 𝑑)) = 0 ↔ ((𝑒 ∈ (ℤ ↑𝑚 ((ℤ ∖ ℕ) ∪ (1...𝑀))) ↦ (𝑏‘(𝑒 ∘ (𝐹 ∪ ( I ↾ (ℤ ∖ ℕ))))))‘(𝑎𝑑)) = 0))
102101rexbidva 3197 . . . . . . . . . 10 (((((𝑀 ∈ ℕ0𝐹:(1...𝑁)⟶(1...𝑀)) ∧ 𝑁 ∈ ℕ0) ∧ 𝑏 ∈ (mzPoly‘((ℤ ∖ ℕ) ∪ (1...𝑁)))) ∧ 𝑎 ∈ (ℕ0𝑚 (1...𝑀))) → (∃𝑑 ∈ (ℕ0𝑚 (ℤ ∖ ℕ))(𝑏‘((𝑎𝐹) ∪ 𝑑)) = 0 ↔ ∃𝑑 ∈ (ℕ0𝑚 (ℤ ∖ ℕ))((𝑒 ∈ (ℤ ↑𝑚 ((ℤ ∖ ℕ) ∪ (1...𝑀))) ↦ (𝑏‘(𝑒 ∘ (𝐹 ∪ ( I ↾ (ℤ ∖ ℕ))))))‘(𝑎𝑑)) = 0))
10331, 102bitrd 268 . . . . . . . . 9 (((((𝑀 ∈ ℕ0𝐹:(1...𝑁)⟶(1...𝑀)) ∧ 𝑁 ∈ ℕ0) ∧ 𝑏 ∈ (mzPoly‘((ℤ ∖ ℕ) ∪ (1...𝑁)))) ∧ 𝑎 ∈ (ℕ0𝑚 (1...𝑀))) → ((𝑎𝐹) ∈ {𝑐 ∈ (ℕ0𝑚 (1...𝑁)) ∣ ∃𝑑 ∈ (ℕ0𝑚 (ℤ ∖ ℕ))(𝑏‘(𝑐𝑑)) = 0} ↔ ∃𝑑 ∈ (ℕ0𝑚 (ℤ ∖ ℕ))((𝑒 ∈ (ℤ ↑𝑚 ((ℤ ∖ ℕ) ∪ (1...𝑀))) ↦ (𝑏‘(𝑒 ∘ (𝐹 ∪ ( I ↾ (ℤ ∖ ℕ))))))‘(𝑎𝑑)) = 0))
104103rabbidva 3338 . . . . . . . 8 ((((𝑀 ∈ ℕ0𝐹:(1...𝑁)⟶(1...𝑀)) ∧ 𝑁 ∈ ℕ0) ∧ 𝑏 ∈ (mzPoly‘((ℤ ∖ ℕ) ∪ (1...𝑁)))) → {𝑎 ∈ (ℕ0𝑚 (1...𝑀)) ∣ (𝑎𝐹) ∈ {𝑐 ∈ (ℕ0𝑚 (1...𝑁)) ∣ ∃𝑑 ∈ (ℕ0𝑚 (ℤ ∖ ℕ))(𝑏‘(𝑐𝑑)) = 0}} = {𝑎 ∈ (ℕ0𝑚 (1...𝑀)) ∣ ∃𝑑 ∈ (ℕ0𝑚 (ℤ ∖ ℕ))((𝑒 ∈ (ℤ ↑𝑚 ((ℤ ∖ ℕ) ∪ (1...𝑀))) ↦ (𝑏‘(𝑒 ∘ (𝐹 ∪ ( I ↾ (ℤ ∖ ℕ))))))‘(𝑎𝑑)) = 0})
105 simplll 758 . . . . . . . . 9 ((((𝑀 ∈ ℕ0𝐹:(1...𝑁)⟶(1...𝑀)) ∧ 𝑁 ∈ ℕ0) ∧ 𝑏 ∈ (mzPoly‘((ℤ ∖ ℕ) ∪ (1...𝑁)))) → 𝑀 ∈ ℕ0)
106 ovex 6821 . . . . . . . . . . . 12 (1...𝑀) ∈ V
1073, 106unex 7101 . . . . . . . . . . 11 ((ℤ ∖ ℕ) ∪ (1...𝑀)) ∈ V
108107a1i 11 . . . . . . . . . 10 ((((𝑀 ∈ ℕ0𝐹:(1...𝑁)⟶(1...𝑀)) ∧ 𝑁 ∈ ℕ0) ∧ 𝑏 ∈ (mzPoly‘((ℤ ∖ ℕ) ∪ (1...𝑁)))) → ((ℤ ∖ ℕ) ∪ (1...𝑀)) ∈ V)
109 simpr 471 . . . . . . . . . 10 ((((𝑀 ∈ ℕ0𝐹:(1...𝑁)⟶(1...𝑀)) ∧ 𝑁 ∈ ℕ0) ∧ 𝑏 ∈ (mzPoly‘((ℤ ∖ ℕ) ∪ (1...𝑁)))) → 𝑏 ∈ (mzPoly‘((ℤ ∖ ℕ) ∪ (1...𝑁))))
11057a1i 11 . . . . . . . . . . . . 13 (𝐹:(1...𝑁)⟶(1...𝑀) → ( I ↾ (ℤ ∖ ℕ)):(ℤ ∖ ℕ)⟶(ℤ ∖ ℕ))
111 id 22 . . . . . . . . . . . . 13 (𝐹:(1...𝑁)⟶(1...𝑀) → 𝐹:(1...𝑁)⟶(1...𝑀))
112 incom 3956 . . . . . . . . . . . . . . 15 ((ℤ ∖ ℕ) ∩ (1...𝑁)) = ((1...𝑁) ∩ (ℤ ∖ ℕ))
113 fz1ssnn 12572 . . . . . . . . . . . . . . . 16 (1...𝑁) ⊆ ℕ
114 ssdisj 4170 . . . . . . . . . . . . . . . 16 (((1...𝑁) ⊆ ℕ ∧ (ℕ ∩ (ℤ ∖ ℕ)) = ∅) → ((1...𝑁) ∩ (ℤ ∖ ℕ)) = ∅)
115113, 18, 114mp2an 672 . . . . . . . . . . . . . . 15 ((1...𝑁) ∩ (ℤ ∖ ℕ)) = ∅
116112, 115eqtri 2793 . . . . . . . . . . . . . 14 ((ℤ ∖ ℕ) ∩ (1...𝑁)) = ∅
117116a1i 11 . . . . . . . . . . . . 13 (𝐹:(1...𝑁)⟶(1...𝑀) → ((ℤ ∖ ℕ) ∩ (1...𝑁)) = ∅)
118 fun 6204 . . . . . . . . . . . . 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)))
119110, 111, 117, 118syl21anc 1475 . . . . . . . . . . . 12 (𝐹:(1...𝑁)⟶(1...𝑀) → (( I ↾ (ℤ ∖ ℕ)) ∪ 𝐹):((ℤ ∖ ℕ) ∪ (1...𝑁))⟶((ℤ ∖ ℕ) ∪ (1...𝑀)))
120 uncom 3908 . . . . . . . . . . . . 13 (( I ↾ (ℤ ∖ ℕ)) ∪ 𝐹) = (𝐹 ∪ ( I ↾ (ℤ ∖ ℕ)))
121120feq1i 6174 . . . . . . . . . . . 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)))
122119, 121sylib 208 . . . . . . . . . . 11 (𝐹:(1...𝑁)⟶(1...𝑀) → (𝐹 ∪ ( I ↾ (ℤ ∖ ℕ))):((ℤ ∖ ℕ) ∪ (1...𝑁))⟶((ℤ ∖ ℕ) ∪ (1...𝑀)))
123122ad3antlr 710 . . . . . . . . . 10 ((((𝑀 ∈ ℕ0𝐹:(1...𝑁)⟶(1...𝑀)) ∧ 𝑁 ∈ ℕ0) ∧ 𝑏 ∈ (mzPoly‘((ℤ ∖ ℕ) ∪ (1...𝑁)))) → (𝐹 ∪ ( I ↾ (ℤ ∖ ℕ))):((ℤ ∖ ℕ) ∪ (1...𝑁))⟶((ℤ ∖ ℕ) ∪ (1...𝑀)))
124 mzprename 37831 . . . . . . . . . 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))))
125108, 109, 123, 124syl3anc 1476 . . . . . . . . 9 ((((𝑀 ∈ ℕ0𝐹:(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))))
1263, 16, 19eldioph4i 37895 . . . . . . . . 9 ((𝑀 ∈ ℕ0 ∧ (𝑒 ∈ (ℤ ↑𝑚 ((ℤ ∖ ℕ) ∪ (1...𝑀))) ↦ (𝑏‘(𝑒 ∘ (𝐹 ∪ ( I ↾ (ℤ ∖ ℕ)))))) ∈ (mzPoly‘((ℤ ∖ ℕ) ∪ (1...𝑀)))) → {𝑎 ∈ (ℕ0𝑚 (1...𝑀)) ∣ ∃𝑑 ∈ (ℕ0𝑚 (ℤ ∖ ℕ))((𝑒 ∈ (ℤ ↑𝑚 ((ℤ ∖ ℕ) ∪ (1...𝑀))) ↦ (𝑏‘(𝑒 ∘ (𝐹 ∪ ( I ↾ (ℤ ∖ ℕ))))))‘(𝑎𝑑)) = 0} ∈ (Dioph‘𝑀))
127105, 125, 126syl2anc 573 . . . . . . . 8 ((((𝑀 ∈ ℕ0𝐹:(1...𝑁)⟶(1...𝑀)) ∧ 𝑁 ∈ ℕ0) ∧ 𝑏 ∈ (mzPoly‘((ℤ ∖ ℕ) ∪ (1...𝑁)))) → {𝑎 ∈ (ℕ0𝑚 (1...𝑀)) ∣ ∃𝑑 ∈ (ℕ0𝑚 (ℤ ∖ ℕ))((𝑒 ∈ (ℤ ↑𝑚 ((ℤ ∖ ℕ) ∪ (1...𝑀))) ↦ (𝑏‘(𝑒 ∘ (𝐹 ∪ ( I ↾ (ℤ ∖ ℕ))))))‘(𝑎𝑑)) = 0} ∈ (Dioph‘𝑀))
128104, 127eqeltrd 2850 . . . . . . 7 ((((𝑀 ∈ ℕ0𝐹:(1...𝑁)⟶(1...𝑀)) ∧ 𝑁 ∈ ℕ0) ∧ 𝑏 ∈ (mzPoly‘((ℤ ∖ ℕ) ∪ (1...𝑁)))) → {𝑎 ∈ (ℕ0𝑚 (1...𝑀)) ∣ (𝑎𝐹) ∈ {𝑐 ∈ (ℕ0𝑚 (1...𝑁)) ∣ ∃𝑑 ∈ (ℕ0𝑚 (ℤ ∖ ℕ))(𝑏‘(𝑐𝑑)) = 0}} ∈ (Dioph‘𝑀))
129 eleq2 2839 . . . . . . . . 9 (𝑆 = {𝑐 ∈ (ℕ0𝑚 (1...𝑁)) ∣ ∃𝑑 ∈ (ℕ0𝑚 (ℤ ∖ ℕ))(𝑏‘(𝑐𝑑)) = 0} → ((𝑎𝐹) ∈ 𝑆 ↔ (𝑎𝐹) ∈ {𝑐 ∈ (ℕ0𝑚 (1...𝑁)) ∣ ∃𝑑 ∈ (ℕ0𝑚 (ℤ ∖ ℕ))(𝑏‘(𝑐𝑑)) = 0}))
130129rabbidv 3339 . . . . . . . 8 (𝑆 = {𝑐 ∈ (ℕ0𝑚 (1...𝑁)) ∣ ∃𝑑 ∈ (ℕ0𝑚 (ℤ ∖ ℕ))(𝑏‘(𝑐𝑑)) = 0} → {𝑎 ∈ (ℕ0𝑚 (1...𝑀)) ∣ (𝑎𝐹) ∈ 𝑆} = {𝑎 ∈ (ℕ0𝑚 (1...𝑀)) ∣ (𝑎𝐹) ∈ {𝑐 ∈ (ℕ0𝑚 (1...𝑁)) ∣ ∃𝑑 ∈ (ℕ0𝑚 (ℤ ∖ ℕ))(𝑏‘(𝑐𝑑)) = 0}})
131130eleq1d 2835 . . . . . . 7 (𝑆 = {𝑐 ∈ (ℕ0𝑚 (1...𝑁)) ∣ ∃𝑑 ∈ (ℕ0𝑚 (ℤ ∖ ℕ))(𝑏‘(𝑐𝑑)) = 0} → ({𝑎 ∈ (ℕ0𝑚 (1...𝑀)) ∣ (𝑎𝐹) ∈ 𝑆} ∈ (Dioph‘𝑀) ↔ {𝑎 ∈ (ℕ0𝑚 (1...𝑀)) ∣ (𝑎𝐹) ∈ {𝑐 ∈ (ℕ0𝑚 (1...𝑁)) ∣ ∃𝑑 ∈ (ℕ0𝑚 (ℤ ∖ ℕ))(𝑏‘(𝑐𝑑)) = 0}} ∈ (Dioph‘𝑀)))
132128, 131syl5ibrcom 237 . . . . . 6 ((((𝑀 ∈ ℕ0𝐹:(1...𝑁)⟶(1...𝑀)) ∧ 𝑁 ∈ ℕ0) ∧ 𝑏 ∈ (mzPoly‘((ℤ ∖ ℕ) ∪ (1...𝑁)))) → (𝑆 = {𝑐 ∈ (ℕ0𝑚 (1...𝑁)) ∣ ∃𝑑 ∈ (ℕ0𝑚 (ℤ ∖ ℕ))(𝑏‘(𝑐𝑑)) = 0} → {𝑎 ∈ (ℕ0𝑚 (1...𝑀)) ∣ (𝑎𝐹) ∈ 𝑆} ∈ (Dioph‘𝑀)))
133132rexlimdva 3179 . . . . 5 (((𝑀 ∈ ℕ0𝐹:(1...𝑁)⟶(1...𝑀)) ∧ 𝑁 ∈ ℕ0) → (∃𝑏 ∈ (mzPoly‘((ℤ ∖ ℕ) ∪ (1...𝑁)))𝑆 = {𝑐 ∈ (ℕ0𝑚 (1...𝑁)) ∣ ∃𝑑 ∈ (ℕ0𝑚 (ℤ ∖ ℕ))(𝑏‘(𝑐𝑑)) = 0} → {𝑎 ∈ (ℕ0𝑚 (1...𝑀)) ∣ (𝑎𝐹) ∈ 𝑆} ∈ (Dioph‘𝑀)))
134133expimpd 441 . . . 4 ((𝑀 ∈ ℕ0𝐹:(1...𝑁)⟶(1...𝑀)) → ((𝑁 ∈ ℕ0 ∧ ∃𝑏 ∈ (mzPoly‘((ℤ ∖ ℕ) ∪ (1...𝑁)))𝑆 = {𝑐 ∈ (ℕ0𝑚 (1...𝑁)) ∣ ∃𝑑 ∈ (ℕ0𝑚 (ℤ ∖ ℕ))(𝑏‘(𝑐𝑑)) = 0}) → {𝑎 ∈ (ℕ0𝑚 (1...𝑀)) ∣ (𝑎𝐹) ∈ 𝑆} ∈ (Dioph‘𝑀)))
13520, 134syl5bi 232 . . 3 ((𝑀 ∈ ℕ0𝐹:(1...𝑁)⟶(1...𝑀)) → (𝑆 ∈ (Dioph‘𝑁) → {𝑎 ∈ (ℕ0𝑚 (1...𝑀)) ∣ (𝑎𝐹) ∈ 𝑆} ∈ (Dioph‘𝑀)))
136135impcom 394 . 2 ((𝑆 ∈ (Dioph‘𝑁) ∧ (𝑀 ∈ ℕ0𝐹:(1...𝑁)⟶(1...𝑀))) → {𝑎 ∈ (ℕ0𝑚 (1...𝑀)) ∣ (𝑎𝐹) ∈ 𝑆} ∈ (Dioph‘𝑀))
1371363impb 1107 1 ((𝑆 ∈ (Dioph‘𝑁) ∧ 𝑀 ∈ ℕ0𝐹:(1...𝑁)⟶(1...𝑀)) → {𝑎 ∈ (ℕ0𝑚 (1...𝑀)) ∣ (𝑎𝐹) ∈ 𝑆} ∈ (Dioph‘𝑀))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382  w3a 1071   = wceq 1631  wcel 2145  wrex 3062  {crab 3065  Vcvv 3351  cdif 3720  cun 3721  cin 3722  wss 3723  c0 4063   class class class wbr 4786  cmpt 4863   I cid 5156  cres 5251  ccom 5253   Fn wfn 6024  wf 6025  1-1-ontowf1o 6028  cfv 6029  (class class class)co 6791  ωcom 7210  𝑚 cmap 8007  cen 8104  Fincfn 8107  0cc0 10136  1c1 10137   + caddc 10139  cn 11220  0cn0 11492  cz 11577  cuz 11886  ...cfz 12526  mzPolycmzp 37804  Diophcdioph 37837
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7094  ax-inf2 8700  ax-cnex 10192  ax-resscn 10193  ax-1cn 10194  ax-icn 10195  ax-addcl 10196  ax-addrcl 10197  ax-mulcl 10198  ax-mulrcl 10199  ax-mulcom 10200  ax-addass 10201  ax-mulass 10202  ax-distr 10203  ax-i2m1 10204  ax-1ne0 10205  ax-1rid 10206  ax-rnegex 10207  ax-rrecex 10208  ax-cnre 10209  ax-pre-lttri 10210  ax-pre-lttrn 10211  ax-pre-ltadd 10212  ax-pre-mulgt0 10213
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-tp 4321  df-op 4323  df-uni 4575  df-int 4612  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-tr 4887  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5821  df-ord 5867  df-on 5868  df-lim 5869  df-suc 5870  df-iota 5992  df-fun 6031  df-fn 6032  df-f 6033  df-f1 6034  df-fo 6035  df-f1o 6036  df-fv 6037  df-riota 6752  df-ov 6794  df-oprab 6795  df-mpt2 6796  df-of 7042  df-om 7211  df-1st 7313  df-2nd 7314  df-wrecs 7557  df-recs 7619  df-rdg 7657  df-1o 7711  df-oadd 7715  df-er 7894  df-map 8009  df-en 8108  df-dom 8109  df-sdom 8110  df-fin 8111  df-card 8963  df-cda 9190  df-pnf 10276  df-mnf 10277  df-xr 10278  df-ltxr 10279  df-le 10280  df-sub 10468  df-neg 10469  df-nn 11221  df-n0 11493  df-z 11578  df-uz 11887  df-fz 12527  df-hash 13315  df-mzpcl 37805  df-mzp 37806  df-dioph 37838
This theorem is referenced by:  rabrenfdioph  37897
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