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Theorem diophren 38835
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 11800 . . . . . 6 ℤ ∈ V
2 difexg 5083 . . . . . 6 (ℤ ∈ V → (ℤ ∖ ℕ) ∈ V)
31, 2ax-mp 5 . . . . 5 (ℤ ∖ ℕ) ∈ V
4 ominf 8523 . . . . . 6 ¬ ω ∈ Fin
5 nnuz 12093 . . . . . . . . . 10 ℕ = (ℤ‘1)
6 0p1e1 11567 . . . . . . . . . . 11 (0 + 1) = 1
76fveq2i 6499 . . . . . . . . . 10 (ℤ‘(0 + 1)) = (ℤ‘1)
85, 7eqtr4i 2799 . . . . . . . . 9 ℕ = (ℤ‘(0 + 1))
98difeq2i 3980 . . . . . . . 8 (ℤ ∖ ℕ) = (ℤ ∖ (ℤ‘(0 + 1)))
10 0z 11802 . . . . . . . . 9 0 ∈ ℤ
11 lzenom 38791 . . . . . . . . 9 (0 ∈ ℤ → (ℤ ∖ (ℤ‘(0 + 1))) ≈ ω)
1210, 11ax-mp 5 . . . . . . . 8 (ℤ ∖ (ℤ‘(0 + 1))) ≈ ω
139, 12eqbrtri 4946 . . . . . . 7 (ℤ ∖ ℕ) ≈ ω
14 enfi 8527 . . . . . . 7 ((ℤ ∖ ℕ) ≈ ω → ((ℤ ∖ ℕ) ∈ Fin ↔ ω ∈ Fin))
1513, 14ax-mp 5 . . . . . 6 ((ℤ ∖ ℕ) ∈ Fin ↔ ω ∈ Fin)
164, 15mtbir 315 . . . . 5 ¬ (ℤ ∖ ℕ) ∈ Fin
17 incom 4060 . . . . . 6 ((ℤ ∖ ℕ) ∩ ℕ) = (ℕ ∩ (ℤ ∖ ℕ))
18 disjdif 4298 . . . . . 6 (ℕ ∩ (ℤ ∖ ℕ)) = ∅
1917, 18eqtri 2796 . . . . 5 ((ℤ ∖ ℕ) ∩ ℕ) = ∅
203, 16, 19eldioph4b 38833 . . . 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 477 . . . . . . . . . . . 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 771 . . . . . . . . . . . 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 7006 . . . . . . . . . . . . 13 (1...𝑁) ∈ V
2423mapco2 38736 . . . . . . . . . . . 12 ((𝑎 ∈ (ℕ0𝑚 (1...𝑀)) ∧ 𝐹:(1...𝑁)⟶(1...𝑀)) → (𝑎𝐹) ∈ (ℕ0𝑚 (1...𝑁)))
2521, 22, 24syl2anc 576 . . . . . . . . . . 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 4015 . . . . . . . . . . . . . 14 (𝑐 = (𝑎𝐹) → (𝑐𝑑) = ((𝑎𝐹) ∪ 𝑑))
2726fveqeq2d 6504 . . . . . . . . . . . . 13 (𝑐 = (𝑎𝐹) → ((𝑏‘(𝑐𝑑)) = 0 ↔ (𝑏‘((𝑎𝐹) ∪ 𝑑)) = 0))
2827rexbidv 3236 . . . . . . . . . . . 12 (𝑐 = (𝑎𝐹) → (∃𝑑 ∈ (ℕ0𝑚 (ℤ ∖ ℕ))(𝑏‘(𝑐𝑑)) = 0 ↔ ∃𝑑 ∈ (ℕ0𝑚 (ℤ ∖ ℕ))(𝑏‘((𝑎𝐹) ∪ 𝑑)) = 0))
2928elrab3 3591 . . . . . . . . . . 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))
3025, 29syl 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))
31 simp-5r 773 . . . . . . . . . . . . . . 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))
32 simplr 756 . . . . . . . . . . . . . . 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)))
33 simpr 477 . . . . . . . . . . . . . . 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)))
34 coundi 5936 . . . . . . . . . . . . . . . 16 ((𝑎𝑑) ∘ (𝐹 ∪ ( I ↾ (ℤ ∖ ℕ)))) = (((𝑎𝑑) ∘ 𝐹) ∪ ((𝑎𝑑) ∘ ( I ↾ (ℤ ∖ ℕ))))
35 coundir 5937 . . . . . . . . . . . . . . . . . . 19 ((𝑎𝑑) ∘ 𝐹) = ((𝑎𝐹) ∪ (𝑑𝐹))
36 elmapi 8226 . . . . . . . . . . . . . . . . . . . . . 22 (𝑑 ∈ (ℕ0𝑚 (ℤ ∖ ℕ)) → 𝑑:(ℤ ∖ ℕ)⟶ℕ0)
37363ad2ant3 1115 . . . . . . . . . . . . . . . . . . . . 21 ((𝐹:(1...𝑁)⟶(1...𝑀) ∧ 𝑎 ∈ (ℕ0𝑚 (1...𝑀)) ∧ 𝑑 ∈ (ℕ0𝑚 (ℤ ∖ ℕ))) → 𝑑:(ℤ ∖ ℕ)⟶ℕ0)
38 simp1 1116 . . . . . . . . . . . . . . . . . . . . 21 ((𝐹:(1...𝑁)⟶(1...𝑀) ∧ 𝑎 ∈ (ℕ0𝑚 (1...𝑀)) ∧ 𝑑 ∈ (ℕ0𝑚 (ℤ ∖ ℕ))) → 𝐹:(1...𝑁)⟶(1...𝑀))
39 incom 4060 . . . . . . . . . . . . . . . . . . . . . . 23 ((ℤ ∖ ℕ) ∩ (1...𝑀)) = ((1...𝑀) ∩ (ℤ ∖ ℕ))
40 fz1ssnn 12752 . . . . . . . . . . . . . . . . . . . . . . . 24 (1...𝑀) ⊆ ℕ
41 ssdisj 4286 . . . . . . . . . . . . . . . . . . . . . . . 24 (((1...𝑀) ⊆ ℕ ∧ (ℕ ∩ (ℤ ∖ ℕ)) = ∅) → ((1...𝑀) ∩ (ℤ ∖ ℕ)) = ∅)
4240, 18, 41mp2an 679 . . . . . . . . . . . . . . . . . . . . . . 23 ((1...𝑀) ∩ (ℤ ∖ ℕ)) = ∅
4339, 42eqtri 2796 . . . . . . . . . . . . . . . . . . . . . 22 ((ℤ ∖ ℕ) ∩ (1...𝑀)) = ∅
4443a1i 11 . . . . . . . . . . . . . . . . . . . . 21 ((𝐹:(1...𝑁)⟶(1...𝑀) ∧ 𝑎 ∈ (ℕ0𝑚 (1...𝑀)) ∧ 𝑑 ∈ (ℕ0𝑚 (ℤ ∖ ℕ))) → ((ℤ ∖ ℕ) ∩ (1...𝑀)) = ∅)
45 coeq0i 38774 . . . . . . . . . . . . . . . . . . . . 21 ((𝑑:(ℤ ∖ ℕ)⟶ℕ0𝐹:(1...𝑁)⟶(1...𝑀) ∧ ((ℤ ∖ ℕ) ∩ (1...𝑀)) = ∅) → (𝑑𝐹) = ∅)
4637, 38, 44, 45syl3anc 1351 . . . . . . . . . . . . . . . . . . . 20 ((𝐹:(1...𝑁)⟶(1...𝑀) ∧ 𝑎 ∈ (ℕ0𝑚 (1...𝑀)) ∧ 𝑑 ∈ (ℕ0𝑚 (ℤ ∖ ℕ))) → (𝑑𝐹) = ∅)
4746uneq2d 4022 . . . . . . . . . . . . . . . . . . 19 ((𝐹:(1...𝑁)⟶(1...𝑀) ∧ 𝑎 ∈ (ℕ0𝑚 (1...𝑀)) ∧ 𝑑 ∈ (ℕ0𝑚 (ℤ ∖ ℕ))) → ((𝑎𝐹) ∪ (𝑑𝐹)) = ((𝑎𝐹) ∪ ∅))
4835, 47syl5eq 2820 . . . . . . . . . . . . . . . . . 18 ((𝐹:(1...𝑁)⟶(1...𝑀) ∧ 𝑎 ∈ (ℕ0𝑚 (1...𝑀)) ∧ 𝑑 ∈ (ℕ0𝑚 (ℤ ∖ ℕ))) → ((𝑎𝑑) ∘ 𝐹) = ((𝑎𝐹) ∪ ∅))
49 un0 4224 . . . . . . . . . . . . . . . . . 18 ((𝑎𝐹) ∪ ∅) = (𝑎𝐹)
5048, 49syl6eq 2824 . . . . . . . . . . . . . . . . 17 ((𝐹:(1...𝑁)⟶(1...𝑀) ∧ 𝑎 ∈ (ℕ0𝑚 (1...𝑀)) ∧ 𝑑 ∈ (ℕ0𝑚 (ℤ ∖ ℕ))) → ((𝑎𝑑) ∘ 𝐹) = (𝑎𝐹))
51 coundir 5937 . . . . . . . . . . . . . . . . . . 19 ((𝑎𝑑) ∘ ( I ↾ (ℤ ∖ ℕ))) = ((𝑎 ∘ ( I ↾ (ℤ ∖ ℕ))) ∪ (𝑑 ∘ ( I ↾ (ℤ ∖ ℕ))))
52 elmapi 8226 . . . . . . . . . . . . . . . . . . . . . 22 (𝑎 ∈ (ℕ0𝑚 (1...𝑀)) → 𝑎:(1...𝑀)⟶ℕ0)
53523ad2ant2 1114 . . . . . . . . . . . . . . . . . . . . 21 ((𝐹:(1...𝑁)⟶(1...𝑀) ∧ 𝑎 ∈ (ℕ0𝑚 (1...𝑀)) ∧ 𝑑 ∈ (ℕ0𝑚 (ℤ ∖ ℕ))) → 𝑎:(1...𝑀)⟶ℕ0)
54 f1oi 6478 . . . . . . . . . . . . . . . . . . . . . . 23 ( I ↾ (ℤ ∖ ℕ)):(ℤ ∖ ℕ)–1-1-onto→(ℤ ∖ ℕ)
55 f1of 6441 . . . . . . . . . . . . . . . . . . . . . . 23 (( I ↾ (ℤ ∖ ℕ)):(ℤ ∖ ℕ)–1-1-onto→(ℤ ∖ ℕ) → ( I ↾ (ℤ ∖ ℕ)):(ℤ ∖ ℕ)⟶(ℤ ∖ ℕ))
5654, 55ax-mp 5 . . . . . . . . . . . . . . . . . . . . . 22 ( I ↾ (ℤ ∖ ℕ)):(ℤ ∖ ℕ)⟶(ℤ ∖ ℕ)
57 coeq0i 38774 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑎:(1...𝑀)⟶ℕ0 ∧ ( I ↾ (ℤ ∖ ℕ)):(ℤ ∖ ℕ)⟶(ℤ ∖ ℕ) ∧ ((1...𝑀) ∩ (ℤ ∖ ℕ)) = ∅) → (𝑎 ∘ ( I ↾ (ℤ ∖ ℕ))) = ∅)
5856, 42, 57mp3an23 1432 . . . . . . . . . . . . . . . . . . . . 21 (𝑎:(1...𝑀)⟶ℕ0 → (𝑎 ∘ ( I ↾ (ℤ ∖ ℕ))) = ∅)
5953, 58syl 17 . . . . . . . . . . . . . . . . . . . 20 ((𝐹:(1...𝑁)⟶(1...𝑀) ∧ 𝑎 ∈ (ℕ0𝑚 (1...𝑀)) ∧ 𝑑 ∈ (ℕ0𝑚 (ℤ ∖ ℕ))) → (𝑎 ∘ ( I ↾ (ℤ ∖ ℕ))) = ∅)
60 coires1 5953 . . . . . . . . . . . . . . . . . . . . . 22 (𝑑 ∘ ( I ↾ (ℤ ∖ ℕ))) = (𝑑 ↾ (ℤ ∖ ℕ))
61 ffn 6341 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑑:(ℤ ∖ ℕ)⟶ℕ0𝑑 Fn (ℤ ∖ ℕ))
62 fnresdm 6296 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑑 Fn (ℤ ∖ ℕ) → (𝑑 ↾ (ℤ ∖ ℕ)) = 𝑑)
6336, 61, 623syl 18 . . . . . . . . . . . . . . . . . . . . . 22 (𝑑 ∈ (ℕ0𝑚 (ℤ ∖ ℕ)) → (𝑑 ↾ (ℤ ∖ ℕ)) = 𝑑)
6460, 63syl5eq 2820 . . . . . . . . . . . . . . . . . . . . 21 (𝑑 ∈ (ℕ0𝑚 (ℤ ∖ ℕ)) → (𝑑 ∘ ( I ↾ (ℤ ∖ ℕ))) = 𝑑)
65643ad2ant3 1115 . . . . . . . . . . . . . . . . . . . 20 ((𝐹:(1...𝑁)⟶(1...𝑀) ∧ 𝑎 ∈ (ℕ0𝑚 (1...𝑀)) ∧ 𝑑 ∈ (ℕ0𝑚 (ℤ ∖ ℕ))) → (𝑑 ∘ ( I ↾ (ℤ ∖ ℕ))) = 𝑑)
6659, 65uneq12d 4023 . . . . . . . . . . . . . . . . . . 19 ((𝐹:(1...𝑁)⟶(1...𝑀) ∧ 𝑎 ∈ (ℕ0𝑚 (1...𝑀)) ∧ 𝑑 ∈ (ℕ0𝑚 (ℤ ∖ ℕ))) → ((𝑎 ∘ ( I ↾ (ℤ ∖ ℕ))) ∪ (𝑑 ∘ ( I ↾ (ℤ ∖ ℕ)))) = (∅ ∪ 𝑑))
6751, 66syl5eq 2820 . . . . . . . . . . . . . . . . . 18 ((𝐹:(1...𝑁)⟶(1...𝑀) ∧ 𝑎 ∈ (ℕ0𝑚 (1...𝑀)) ∧ 𝑑 ∈ (ℕ0𝑚 (ℤ ∖ ℕ))) → ((𝑎𝑑) ∘ ( I ↾ (ℤ ∖ ℕ))) = (∅ ∪ 𝑑))
68 uncom 4012 . . . . . . . . . . . . . . . . . . 19 (∅ ∪ 𝑑) = (𝑑 ∪ ∅)
69 un0 4224 . . . . . . . . . . . . . . . . . . 19 (𝑑 ∪ ∅) = 𝑑
7068, 69eqtri 2796 . . . . . . . . . . . . . . . . . 18 (∅ ∪ 𝑑) = 𝑑
7167, 70syl6eq 2824 . . . . . . . . . . . . . . . . 17 ((𝐹:(1...𝑁)⟶(1...𝑀) ∧ 𝑎 ∈ (ℕ0𝑚 (1...𝑀)) ∧ 𝑑 ∈ (ℕ0𝑚 (ℤ ∖ ℕ))) → ((𝑎𝑑) ∘ ( I ↾ (ℤ ∖ ℕ))) = 𝑑)
7250, 71uneq12d 4023 . . . . . . . . . . . . . . . 16 ((𝐹:(1...𝑁)⟶(1...𝑀) ∧ 𝑎 ∈ (ℕ0𝑚 (1...𝑀)) ∧ 𝑑 ∈ (ℕ0𝑚 (ℤ ∖ ℕ))) → (((𝑎𝑑) ∘ 𝐹) ∪ ((𝑎𝑑) ∘ ( I ↾ (ℤ ∖ ℕ)))) = ((𝑎𝐹) ∪ 𝑑))
7334, 72syl5req 2821 . . . . . . . . . . . . . . 15 ((𝐹:(1...𝑁)⟶(1...𝑀) ∧ 𝑎 ∈ (ℕ0𝑚 (1...𝑀)) ∧ 𝑑 ∈ (ℕ0𝑚 (ℤ ∖ ℕ))) → ((𝑎𝐹) ∪ 𝑑) = ((𝑎𝑑) ∘ (𝐹 ∪ ( I ↾ (ℤ ∖ ℕ)))))
7431, 32, 33, 73syl3anc 1351 . . . . . . . . . . . . . 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)))))
7574fveq2d 6500 . . . . . . . . . . . . 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))))))
76 nn0ssz 11814 . . . . . . . . . . . . . . . . 17 0 ⊆ ℤ
77 mapss 8249 . . . . . . . . . . . . . . . . 17 ((ℤ ∈ V ∧ ℕ0 ⊆ ℤ) → (ℕ0𝑚 ((ℤ ∖ ℕ) ∪ (1...𝑀))) ⊆ (ℤ ↑𝑚 ((ℤ ∖ ℕ) ∪ (1...𝑀))))
781, 76, 77mp2an 679 . . . . . . . . . . . . . . . 16 (ℕ0𝑚 ((ℤ ∖ ℕ) ∪ (1...𝑀))) ⊆ (ℤ ↑𝑚 ((ℤ ∖ ℕ) ∪ (1...𝑀)))
7942reseq2i 5689 . . . . . . . . . . . . . . . . . . 19 (𝑎 ↾ ((1...𝑀) ∩ (ℤ ∖ ℕ))) = (𝑎 ↾ ∅)
80 res0 5696 . . . . . . . . . . . . . . . . . . 19 (𝑎 ↾ ∅) = ∅
8179, 80eqtri 2796 . . . . . . . . . . . . . . . . . 18 (𝑎 ↾ ((1...𝑀) ∩ (ℤ ∖ ℕ))) = ∅
8242reseq2i 5689 . . . . . . . . . . . . . . . . . . 19 (𝑑 ↾ ((1...𝑀) ∩ (ℤ ∖ ℕ))) = (𝑑 ↾ ∅)
83 res0 5696 . . . . . . . . . . . . . . . . . . 19 (𝑑 ↾ ∅) = ∅
8482, 83eqtri 2796 . . . . . . . . . . . . . . . . . 18 (𝑑 ↾ ((1...𝑀) ∩ (ℤ ∖ ℕ))) = ∅
8581, 84eqtr4i 2799 . . . . . . . . . . . . . . . . 17 (𝑎 ↾ ((1...𝑀) ∩ (ℤ ∖ ℕ))) = (𝑑 ↾ ((1...𝑀) ∩ (ℤ ∖ ℕ)))
86 elmapresaun 38792 . . . . . . . . . . . . . . . . . 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))))
87 uncom 4012 . . . . . . . . . . . . . . . . . . 19 ((1...𝑀) ∪ (ℤ ∖ ℕ)) = ((ℤ ∖ ℕ) ∪ (1...𝑀))
8887oveq2i 6985 . . . . . . . . . . . . . . . . . 18 (ℕ0𝑚 ((1...𝑀) ∪ (ℤ ∖ ℕ))) = (ℕ0𝑚 ((ℤ ∖ ℕ) ∪ (1...𝑀)))
8986, 88syl6eleq 2870 . . . . . . . . . . . . . . . . 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))))
9085, 89mp3an3 1429 . . . . . . . . . . . . . . . 16 ((𝑎 ∈ (ℕ0𝑚 (1...𝑀)) ∧ 𝑑 ∈ (ℕ0𝑚 (ℤ ∖ ℕ))) → (𝑎𝑑) ∈ (ℕ0𝑚 ((ℤ ∖ ℕ) ∪ (1...𝑀))))
9178, 90sseldi 3850 . . . . . . . . . . . . . . 15 ((𝑎 ∈ (ℕ0𝑚 (1...𝑀)) ∧ 𝑑 ∈ (ℕ0𝑚 (ℤ ∖ ℕ))) → (𝑎𝑑) ∈ (ℤ ↑𝑚 ((ℤ ∖ ℕ) ∪ (1...𝑀))))
9291adantll 701 . . . . . . . . . . . . . 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))))
93 coeq1 5574 . . . . . . . . . . . . . . . 16 (𝑒 = (𝑎𝑑) → (𝑒 ∘ (𝐹 ∪ ( I ↾ (ℤ ∖ ℕ)))) = ((𝑎𝑑) ∘ (𝐹 ∪ ( I ↾ (ℤ ∖ ℕ)))))
9493fveq2d 6500 . . . . . . . . . . . . . . 15 (𝑒 = (𝑎𝑑) → (𝑏‘(𝑒 ∘ (𝐹 ∪ ( I ↾ (ℤ ∖ ℕ))))) = (𝑏‘((𝑎𝑑) ∘ (𝐹 ∪ ( I ↾ (ℤ ∖ ℕ))))))
95 eqid 2772 . . . . . . . . . . . . . . 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))))))
96 fvex 6509 . . . . . . . . . . . . . . 15 (𝑏‘((𝑎𝑑) ∘ (𝐹 ∪ ( I ↾ (ℤ ∖ ℕ))))) ∈ V
9794, 95, 96fvmpt 6593 . . . . . . . . . . . . . 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))))))
9892, 97syl 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))))))
9975, 98eqtr4d 2811 . . . . . . . . . . . 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)))
10099eqeq1d 2774 . . . . . . . . . . 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))
101100rexbidva 3235 . . . . . . . . . 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))
10230, 101bitrd 271 . . . . . . . . 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))
103102rabbidva 3396 . . . . . . . 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})
104 simplll 762 . . . . . . . . 9 ((((𝑀 ∈ ℕ0𝐹:(1...𝑁)⟶(1...𝑀)) ∧ 𝑁 ∈ ℕ0) ∧ 𝑏 ∈ (mzPoly‘((ℤ ∖ ℕ) ∪ (1...𝑁)))) → 𝑀 ∈ ℕ0)
105 ovex 7006 . . . . . . . . . . . 12 (1...𝑀) ∈ V
1063, 105unex 7284 . . . . . . . . . . 11 ((ℤ ∖ ℕ) ∪ (1...𝑀)) ∈ V
107106a1i 11 . . . . . . . . . 10 ((((𝑀 ∈ ℕ0𝐹:(1...𝑁)⟶(1...𝑀)) ∧ 𝑁 ∈ ℕ0) ∧ 𝑏 ∈ (mzPoly‘((ℤ ∖ ℕ) ∪ (1...𝑁)))) → ((ℤ ∖ ℕ) ∪ (1...𝑀)) ∈ V)
108 simpr 477 . . . . . . . . . 10 ((((𝑀 ∈ ℕ0𝐹:(1...𝑁)⟶(1...𝑀)) ∧ 𝑁 ∈ ℕ0) ∧ 𝑏 ∈ (mzPoly‘((ℤ ∖ ℕ) ∪ (1...𝑁)))) → 𝑏 ∈ (mzPoly‘((ℤ ∖ ℕ) ∪ (1...𝑁))))
10956a1i 11 . . . . . . . . . . . . 13 (𝐹:(1...𝑁)⟶(1...𝑀) → ( I ↾ (ℤ ∖ ℕ)):(ℤ ∖ ℕ)⟶(ℤ ∖ ℕ))
110 id 22 . . . . . . . . . . . . 13 (𝐹:(1...𝑁)⟶(1...𝑀) → 𝐹:(1...𝑁)⟶(1...𝑀))
111 incom 4060 . . . . . . . . . . . . . . 15 ((ℤ ∖ ℕ) ∩ (1...𝑁)) = ((1...𝑁) ∩ (ℤ ∖ ℕ))
112 fz1ssnn 12752 . . . . . . . . . . . . . . . 16 (1...𝑁) ⊆ ℕ
113 ssdisj 4286 . . . . . . . . . . . . . . . 16 (((1...𝑁) ⊆ ℕ ∧ (ℕ ∩ (ℤ ∖ ℕ)) = ∅) → ((1...𝑁) ∩ (ℤ ∖ ℕ)) = ∅)
114112, 18, 113mp2an 679 . . . . . . . . . . . . . . 15 ((1...𝑁) ∩ (ℤ ∖ ℕ)) = ∅
115111, 114eqtri 2796 . . . . . . . . . . . . . 14 ((ℤ ∖ ℕ) ∩ (1...𝑁)) = ∅
116115a1i 11 . . . . . . . . . . . . 13 (𝐹:(1...𝑁)⟶(1...𝑀) → ((ℤ ∖ ℕ) ∩ (1...𝑁)) = ∅)
117 fun 6366 . . . . . . . . . . . . 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)))
118109, 110, 116, 117syl21anc 825 . . . . . . . . . . . 12 (𝐹:(1...𝑁)⟶(1...𝑀) → (( I ↾ (ℤ ∖ ℕ)) ∪ 𝐹):((ℤ ∖ ℕ) ∪ (1...𝑁))⟶((ℤ ∖ ℕ) ∪ (1...𝑀)))
119 uncom 4012 . . . . . . . . . . . . 13 (( I ↾ (ℤ ∖ ℕ)) ∪ 𝐹) = (𝐹 ∪ ( I ↾ (ℤ ∖ ℕ)))
120119feq1i 6332 . . . . . . . . . . . 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)))
121118, 120sylib 210 . . . . . . . . . . 11 (𝐹:(1...𝑁)⟶(1...𝑀) → (𝐹 ∪ ( I ↾ (ℤ ∖ ℕ))):((ℤ ∖ ℕ) ∪ (1...𝑁))⟶((ℤ ∖ ℕ) ∪ (1...𝑀)))
122121ad3antlr 718 . . . . . . . . . 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)))
123 mzprename 38770 . . . . . . . . . 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))))
124107, 108, 122, 123syl3anc 1351 . . . . . . . . 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))))
1253, 16, 19eldioph4i 38834 . . . . . . . . 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))
126104, 124, 125syl2anc 576 . . . . . . . 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))
127103, 126eqeltrd 2860 . . . . . . 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))
128 eleq2 2848 . . . . . . . . 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}))
129128rabbidv 3397 . . . . . . . 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}})
130129eleq1d 2844 . . . . . . 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)))
131127, 130syl5ibrcom 239 . . . . . 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)))
132131rexlimdva 3223 . . . . 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)))
133132expimpd 446 . . . 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)))
13420, 133syl5bi 234 . . 3 ((𝑀 ∈ ℕ0𝐹:(1...𝑁)⟶(1...𝑀)) → (𝑆 ∈ (Dioph‘𝑁) → {𝑎 ∈ (ℕ0𝑚 (1...𝑀)) ∣ (𝑎𝐹) ∈ 𝑆} ∈ (Dioph‘𝑀)))
135134impcom 399 . 2 ((𝑆 ∈ (Dioph‘𝑁) ∧ (𝑀 ∈ ℕ0𝐹:(1...𝑁)⟶(1...𝑀))) → {𝑎 ∈ (ℕ0𝑚 (1...𝑀)) ∣ (𝑎𝐹) ∈ 𝑆} ∈ (Dioph‘𝑀))
1361353impb 1095 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 198  wa 387  w3a 1068   = wceq 1507  wcel 2050  wrex 3083  {crab 3086  Vcvv 3409  cdif 3820  cun 3821  cin 3822  wss 3823  c0 4172   class class class wbr 4925  cmpt 5004   I cid 5307  cres 5405  ccom 5407   Fn wfn 6180  wf 6181  1-1-ontowf1o 6184  cfv 6185  (class class class)co 6974  ωcom 7394  𝑚 cmap 8204  cen 8301  Fincfn 8304  0cc0 10333  1c1 10334   + caddc 10336  cn 11437  0cn0 11705  cz 11791  cuz 12056  ...cfz 12706  mzPolycmzp 38743  Diophcdioph 38776
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2744  ax-rep 5045  ax-sep 5056  ax-nul 5063  ax-pow 5115  ax-pr 5182  ax-un 7277  ax-inf2 8896  ax-cnex 10389  ax-resscn 10390  ax-1cn 10391  ax-icn 10392  ax-addcl 10393  ax-addrcl 10394  ax-mulcl 10395  ax-mulrcl 10396  ax-mulcom 10397  ax-addass 10398  ax-mulass 10399  ax-distr 10400  ax-i2m1 10401  ax-1ne0 10402  ax-1rid 10403  ax-rnegex 10404  ax-rrecex 10405  ax-cnre 10406  ax-pre-lttri 10407  ax-pre-lttrn 10408  ax-pre-ltadd 10409  ax-pre-mulgt0 10410
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3or 1069  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2584  df-clab 2753  df-cleq 2765  df-clel 2840  df-nfc 2912  df-ne 2962  df-nel 3068  df-ral 3087  df-rex 3088  df-reu 3089  df-rmo 3090  df-rab 3091  df-v 3411  df-sbc 3676  df-csb 3781  df-dif 3826  df-un 3828  df-in 3830  df-ss 3837  df-pss 3839  df-nul 4173  df-if 4345  df-pw 4418  df-sn 4436  df-pr 4438  df-tp 4440  df-op 4442  df-uni 4709  df-int 4746  df-iun 4790  df-br 4926  df-opab 4988  df-mpt 5005  df-tr 5027  df-id 5308  df-eprel 5313  df-po 5322  df-so 5323  df-fr 5362  df-we 5364  df-xp 5409  df-rel 5410  df-cnv 5411  df-co 5412  df-dm 5413  df-rn 5414  df-res 5415  df-ima 5416  df-pred 5983  df-ord 6029  df-on 6030  df-lim 6031  df-suc 6032  df-iota 6149  df-fun 6187  df-fn 6188  df-f 6189  df-f1 6190  df-fo 6191  df-f1o 6192  df-fv 6193  df-riota 6935  df-ov 6977  df-oprab 6978  df-mpo 6979  df-of 7225  df-om 7395  df-1st 7499  df-2nd 7500  df-wrecs 7748  df-recs 7810  df-rdg 7848  df-1o 7903  df-oadd 7907  df-er 8087  df-map 8206  df-en 8305  df-dom 8306  df-sdom 8307  df-fin 8308  df-dju 9122  df-card 9160  df-pnf 10474  df-mnf 10475  df-xr 10476  df-ltxr 10477  df-le 10478  df-sub 10670  df-neg 10671  df-nn 11438  df-n0 11706  df-z 11792  df-uz 12057  df-fz 12707  df-hash 13504  df-mzpcl 38744  df-mzp 38745  df-dioph 38777
This theorem is referenced by:  rabrenfdioph  38836
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