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Theorem diophren 36884
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 11337 . . . . . 6 ℤ ∈ V
2 difexg 4773 . . . . . 6 (ℤ ∈ V → (ℤ ∖ ℕ) ∈ V)
31, 2ax-mp 5 . . . . 5 (ℤ ∖ ℕ) ∈ V
4 ominf 8123 . . . . . 6 ¬ ω ∈ Fin
5 nnuz 11674 . . . . . . . . . 10 ℕ = (ℤ‘1)
6 0p1e1 11083 . . . . . . . . . . 11 (0 + 1) = 1
76fveq2i 6156 . . . . . . . . . 10 (ℤ‘(0 + 1)) = (ℤ‘1)
85, 7eqtr4i 2646 . . . . . . . . 9 ℕ = (ℤ‘(0 + 1))
98difeq2i 3708 . . . . . . . 8 (ℤ ∖ ℕ) = (ℤ ∖ (ℤ‘(0 + 1)))
10 0z 11339 . . . . . . . . 9 0 ∈ ℤ
11 lzenom 36840 . . . . . . . . 9 (0 ∈ ℤ → (ℤ ∖ (ℤ‘(0 + 1))) ≈ ω)
1210, 11ax-mp 5 . . . . . . . 8 (ℤ ∖ (ℤ‘(0 + 1))) ≈ ω
139, 12eqbrtri 4639 . . . . . . 7 (ℤ ∖ ℕ) ≈ ω
14 enfi 8127 . . . . . . 7 ((ℤ ∖ ℕ) ≈ ω → ((ℤ ∖ ℕ) ∈ Fin ↔ ω ∈ Fin))
1513, 14ax-mp 5 . . . . . 6 ((ℤ ∖ ℕ) ∈ Fin ↔ ω ∈ Fin)
164, 15mtbir 313 . . . . 5 ¬ (ℤ ∖ ℕ) ∈ Fin
17 incom 3788 . . . . . 6 ((ℤ ∖ ℕ) ∩ ℕ) = (ℕ ∩ (ℤ ∖ ℕ))
18 disjdif 4017 . . . . . 6 (ℕ ∩ (ℤ ∖ ℕ)) = ∅
1917, 18eqtri 2643 . . . . 5 ((ℤ ∖ ℕ) ∩ ℕ) = ∅
203, 16, 19eldioph4b 36882 . . . 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 806 . . . . . . . . . . . 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 6638 . . . . . . . . . . . . 13 (1...𝑁) ∈ V
2423mapco2 36785 . . . . . . . . . . . 12 ((𝑎 ∈ (ℕ0𝑚 (1...𝑀)) ∧ 𝐹:(1...𝑁)⟶(1...𝑀)) → (𝑎𝐹) ∈ (ℕ0𝑚 (1...𝑁)))
2521, 22, 24syl2anc 692 . . . . . . . . . . 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 3743 . . . . . . . . . . . . . . 15 (𝑐 = (𝑎𝐹) → (𝑐𝑑) = ((𝑎𝐹) ∪ 𝑑))
2726fveq2d 6157 . . . . . . . . . . . . . 14 (𝑐 = (𝑎𝐹) → (𝑏‘(𝑐𝑑)) = (𝑏‘((𝑎𝐹) ∪ 𝑑)))
2827eqeq1d 2623 . . . . . . . . . . . . 13 (𝑐 = (𝑎𝐹) → ((𝑏‘(𝑐𝑑)) = 0 ↔ (𝑏‘((𝑎𝐹) ∪ 𝑑)) = 0))
2928rexbidv 3046 . . . . . . . . . . . 12 (𝑐 = (𝑎𝐹) → (∃𝑑 ∈ (ℕ0𝑚 (ℤ ∖ ℕ))(𝑏‘(𝑐𝑑)) = 0 ↔ ∃𝑑 ∈ (ℕ0𝑚 (ℤ ∖ ℕ))(𝑏‘((𝑎𝐹) ∪ 𝑑)) = 0))
3029elrab3 3351 . . . . . . . . . . 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 808 . . . . . . . . . . . . . . 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 791 . . . . . . . . . . . . . . 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 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)))
35 coundi 5600 . . . . . . . . . . . . . . . 16 ((𝑎𝑑) ∘ (𝐹 ∪ ( I ↾ (ℤ ∖ ℕ)))) = (((𝑎𝑑) ∘ 𝐹) ∪ ((𝑎𝑑) ∘ ( I ↾ (ℤ ∖ ℕ))))
36 coundir 5601 . . . . . . . . . . . . . . . . . . 19 ((𝑎𝑑) ∘ 𝐹) = ((𝑎𝐹) ∪ (𝑑𝐹))
37 elmapi 7830 . . . . . . . . . . . . . . . . . . . . . 22 (𝑑 ∈ (ℕ0𝑚 (ℤ ∖ ℕ)) → 𝑑:(ℤ ∖ ℕ)⟶ℕ0)
38373ad2ant3 1082 . . . . . . . . . . . . . . . . . . . . 21 ((𝐹:(1...𝑁)⟶(1...𝑀) ∧ 𝑎 ∈ (ℕ0𝑚 (1...𝑀)) ∧ 𝑑 ∈ (ℕ0𝑚 (ℤ ∖ ℕ))) → 𝑑:(ℤ ∖ ℕ)⟶ℕ0)
39 simp1 1059 . . . . . . . . . . . . . . . . . . . . 21 ((𝐹:(1...𝑁)⟶(1...𝑀) ∧ 𝑎 ∈ (ℕ0𝑚 (1...𝑀)) ∧ 𝑑 ∈ (ℕ0𝑚 (ℤ ∖ ℕ))) → 𝐹:(1...𝑁)⟶(1...𝑀))
40 incom 3788 . . . . . . . . . . . . . . . . . . . . . . 23 ((ℤ ∖ ℕ) ∩ (1...𝑀)) = ((1...𝑀) ∩ (ℤ ∖ ℕ))
41 fz1ssnn 12321 . . . . . . . . . . . . . . . . . . . . . . . 24 (1...𝑀) ⊆ ℕ
42 ssdisj 4003 . . . . . . . . . . . . . . . . . . . . . . . 24 (((1...𝑀) ⊆ ℕ ∧ (ℕ ∩ (ℤ ∖ ℕ)) = ∅) → ((1...𝑀) ∩ (ℤ ∖ ℕ)) = ∅)
4341, 18, 42mp2an 707 . . . . . . . . . . . . . . . . . . . . . . 23 ((1...𝑀) ∩ (ℤ ∖ ℕ)) = ∅
4440, 43eqtri 2643 . . . . . . . . . . . . . . . . . . . . . 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 36823 . . . . . . . . . . . . . . . . . . . . 21 ((𝑑:(ℤ ∖ ℕ)⟶ℕ0𝐹:(1...𝑁)⟶(1...𝑀) ∧ ((ℤ ∖ ℕ) ∩ (1...𝑀)) = ∅) → (𝑑𝐹) = ∅)
4738, 39, 45, 46syl3anc 1323 . . . . . . . . . . . . . . . . . . . 20 ((𝐹:(1...𝑁)⟶(1...𝑀) ∧ 𝑎 ∈ (ℕ0𝑚 (1...𝑀)) ∧ 𝑑 ∈ (ℕ0𝑚 (ℤ ∖ ℕ))) → (𝑑𝐹) = ∅)
4847uneq2d 3750 . . . . . . . . . . . . . . . . . . 19 ((𝐹:(1...𝑁)⟶(1...𝑀) ∧ 𝑎 ∈ (ℕ0𝑚 (1...𝑀)) ∧ 𝑑 ∈ (ℕ0𝑚 (ℤ ∖ ℕ))) → ((𝑎𝐹) ∪ (𝑑𝐹)) = ((𝑎𝐹) ∪ ∅))
4936, 48syl5eq 2667 . . . . . . . . . . . . . . . . . 18 ((𝐹:(1...𝑁)⟶(1...𝑀) ∧ 𝑎 ∈ (ℕ0𝑚 (1...𝑀)) ∧ 𝑑 ∈ (ℕ0𝑚 (ℤ ∖ ℕ))) → ((𝑎𝑑) ∘ 𝐹) = ((𝑎𝐹) ∪ ∅))
50 un0 3944 . . . . . . . . . . . . . . . . . 18 ((𝑎𝐹) ∪ ∅) = (𝑎𝐹)
5149, 50syl6eq 2671 . . . . . . . . . . . . . . . . 17 ((𝐹:(1...𝑁)⟶(1...𝑀) ∧ 𝑎 ∈ (ℕ0𝑚 (1...𝑀)) ∧ 𝑑 ∈ (ℕ0𝑚 (ℤ ∖ ℕ))) → ((𝑎𝑑) ∘ 𝐹) = (𝑎𝐹))
52 coundir 5601 . . . . . . . . . . . . . . . . . . 19 ((𝑎𝑑) ∘ ( I ↾ (ℤ ∖ ℕ))) = ((𝑎 ∘ ( I ↾ (ℤ ∖ ℕ))) ∪ (𝑑 ∘ ( I ↾ (ℤ ∖ ℕ))))
53 elmapi 7830 . . . . . . . . . . . . . . . . . . . . . 22 (𝑎 ∈ (ℕ0𝑚 (1...𝑀)) → 𝑎:(1...𝑀)⟶ℕ0)
54533ad2ant2 1081 . . . . . . . . . . . . . . . . . . . . 21 ((𝐹:(1...𝑁)⟶(1...𝑀) ∧ 𝑎 ∈ (ℕ0𝑚 (1...𝑀)) ∧ 𝑑 ∈ (ℕ0𝑚 (ℤ ∖ ℕ))) → 𝑎:(1...𝑀)⟶ℕ0)
55 f1oi 6136 . . . . . . . . . . . . . . . . . . . . . . 23 ( I ↾ (ℤ ∖ ℕ)):(ℤ ∖ ℕ)–1-1-onto→(ℤ ∖ ℕ)
56 f1of 6099 . . . . . . . . . . . . . . . . . . . . . . 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 36823 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑎:(1...𝑀)⟶ℕ0 ∧ ( I ↾ (ℤ ∖ ℕ)):(ℤ ∖ ℕ)⟶(ℤ ∖ ℕ) ∧ ((1...𝑀) ∩ (ℤ ∖ ℕ)) = ∅) → (𝑎 ∘ ( I ↾ (ℤ ∖ ℕ))) = ∅)
5957, 43, 58mp3an23 1413 . . . . . . . . . . . . . . . . . . . . 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 5617 . . . . . . . . . . . . . . . . . . . . . 22 (𝑑 ∘ ( I ↾ (ℤ ∖ ℕ))) = (𝑑 ↾ (ℤ ∖ ℕ))
62 ffn 6007 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑑:(ℤ ∖ ℕ)⟶ℕ0𝑑 Fn (ℤ ∖ ℕ))
63 fnresdm 5963 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑑 Fn (ℤ ∖ ℕ) → (𝑑 ↾ (ℤ ∖ ℕ)) = 𝑑)
6437, 62, 633syl 18 . . . . . . . . . . . . . . . . . . . . . 22 (𝑑 ∈ (ℕ0𝑚 (ℤ ∖ ℕ)) → (𝑑 ↾ (ℤ ∖ ℕ)) = 𝑑)
6561, 64syl5eq 2667 . . . . . . . . . . . . . . . . . . . . 21 (𝑑 ∈ (ℕ0𝑚 (ℤ ∖ ℕ)) → (𝑑 ∘ ( I ↾ (ℤ ∖ ℕ))) = 𝑑)
66653ad2ant3 1082 . . . . . . . . . . . . . . . . . . . 20 ((𝐹:(1...𝑁)⟶(1...𝑀) ∧ 𝑎 ∈ (ℕ0𝑚 (1...𝑀)) ∧ 𝑑 ∈ (ℕ0𝑚 (ℤ ∖ ℕ))) → (𝑑 ∘ ( I ↾ (ℤ ∖ ℕ))) = 𝑑)
6760, 66uneq12d 3751 . . . . . . . . . . . . . . . . . . 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 2667 . . . . . . . . . . . . . . . . . 18 ((𝐹:(1...𝑁)⟶(1...𝑀) ∧ 𝑎 ∈ (ℕ0𝑚 (1...𝑀)) ∧ 𝑑 ∈ (ℕ0𝑚 (ℤ ∖ ℕ))) → ((𝑎𝑑) ∘ ( I ↾ (ℤ ∖ ℕ))) = (∅ ∪ 𝑑))
69 uncom 3740 . . . . . . . . . . . . . . . . . . 19 (∅ ∪ 𝑑) = (𝑑 ∪ ∅)
70 un0 3944 . . . . . . . . . . . . . . . . . . 19 (𝑑 ∪ ∅) = 𝑑
7169, 70eqtri 2643 . . . . . . . . . . . . . . . . . 18 (∅ ∪ 𝑑) = 𝑑
7268, 71syl6eq 2671 . . . . . . . . . . . . . . . . 17 ((𝐹:(1...𝑁)⟶(1...𝑀) ∧ 𝑎 ∈ (ℕ0𝑚 (1...𝑀)) ∧ 𝑑 ∈ (ℕ0𝑚 (ℤ ∖ ℕ))) → ((𝑎𝑑) ∘ ( I ↾ (ℤ ∖ ℕ))) = 𝑑)
7351, 72uneq12d 3751 . . . . . . . . . . . . . . . 16 ((𝐹:(1...𝑁)⟶(1...𝑀) ∧ 𝑎 ∈ (ℕ0𝑚 (1...𝑀)) ∧ 𝑑 ∈ (ℕ0𝑚 (ℤ ∖ ℕ))) → (((𝑎𝑑) ∘ 𝐹) ∪ ((𝑎𝑑) ∘ ( I ↾ (ℤ ∖ ℕ)))) = ((𝑎𝐹) ∪ 𝑑))
7435, 73syl5req 2668 . . . . . . . . . . . . . . 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 1323 . . . . . . . . . . . . . 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 6157 . . . . . . . . . . . . 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 11349 . . . . . . . . . . . . . . . . 17 0 ⊆ ℤ
78 mapss 7851 . . . . . . . . . . . . . . . . 17 ((ℤ ∈ V ∧ ℕ0 ⊆ ℤ) → (ℕ0𝑚 ((ℤ ∖ ℕ) ∪ (1...𝑀))) ⊆ (ℤ ↑𝑚 ((ℤ ∖ ℕ) ∪ (1...𝑀))))
791, 77, 78mp2an 707 . . . . . . . . . . . . . . . 16 (ℕ0𝑚 ((ℤ ∖ ℕ) ∪ (1...𝑀))) ⊆ (ℤ ↑𝑚 ((ℤ ∖ ℕ) ∪ (1...𝑀)))
8043reseq2i 5358 . . . . . . . . . . . . . . . . . . 19 (𝑎 ↾ ((1...𝑀) ∩ (ℤ ∖ ℕ))) = (𝑎 ↾ ∅)
81 res0 5365 . . . . . . . . . . . . . . . . . . 19 (𝑎 ↾ ∅) = ∅
8280, 81eqtri 2643 . . . . . . . . . . . . . . . . . 18 (𝑎 ↾ ((1...𝑀) ∩ (ℤ ∖ ℕ))) = ∅
8343reseq2i 5358 . . . . . . . . . . . . . . . . . . 19 (𝑑 ↾ ((1...𝑀) ∩ (ℤ ∖ ℕ))) = (𝑑 ↾ ∅)
84 res0 5365 . . . . . . . . . . . . . . . . . . 19 (𝑑 ↾ ∅) = ∅
8583, 84eqtri 2643 . . . . . . . . . . . . . . . . . 18 (𝑑 ↾ ((1...𝑀) ∩ (ℤ ∖ ℕ))) = ∅
8682, 85eqtr4i 2646 . . . . . . . . . . . . . . . . 17 (𝑎 ↾ ((1...𝑀) ∩ (ℤ ∖ ℕ))) = (𝑑 ↾ ((1...𝑀) ∩ (ℤ ∖ ℕ)))
87 elmapresaun 36841 . . . . . . . . . . . . . . . . . 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 3740 . . . . . . . . . . . . . . . . . . 19 ((1...𝑀) ∪ (ℤ ∖ ℕ)) = ((ℤ ∖ ℕ) ∪ (1...𝑀))
8988oveq2i 6621 . . . . . . . . . . . . . . . . . 18 (ℕ0𝑚 ((1...𝑀) ∪ (ℤ ∖ ℕ))) = (ℕ0𝑚 ((ℤ ∖ ℕ) ∪ (1...𝑀)))
9087, 89syl6eleq 2708 . . . . . . . . . . . . . . . . 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 1410 . . . . . . . . . . . . . . . 16 ((𝑎 ∈ (ℕ0𝑚 (1...𝑀)) ∧ 𝑑 ∈ (ℕ0𝑚 (ℤ ∖ ℕ))) → (𝑎𝑑) ∈ (ℕ0𝑚 ((ℤ ∖ ℕ) ∪ (1...𝑀))))
9279, 91sseldi 3585 . . . . . . . . . . . . . . 15 ((𝑎 ∈ (ℕ0𝑚 (1...𝑀)) ∧ 𝑑 ∈ (ℕ0𝑚 (ℤ ∖ ℕ))) → (𝑎𝑑) ∈ (ℤ ↑𝑚 ((ℤ ∖ ℕ) ∪ (1...𝑀))))
9392adantll 749 . . . . . . . . . . . . . 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 5244 . . . . . . . . . . . . . . . 16 (𝑒 = (𝑎𝑑) → (𝑒 ∘ (𝐹 ∪ ( I ↾ (ℤ ∖ ℕ)))) = ((𝑎𝑑) ∘ (𝐹 ∪ ( I ↾ (ℤ ∖ ℕ)))))
9594fveq2d 6157 . . . . . . . . . . . . . . 15 (𝑒 = (𝑎𝑑) → (𝑏‘(𝑒 ∘ (𝐹 ∪ ( I ↾ (ℤ ∖ ℕ))))) = (𝑏‘((𝑎𝑑) ∘ (𝐹 ∪ ( I ↾ (ℤ ∖ ℕ))))))
96 eqid 2621 . . . . . . . . . . . . . . 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 6163 . . . . . . . . . . . . . . 15 (𝑏‘((𝑎𝑑) ∘ (𝐹 ∪ ( I ↾ (ℤ ∖ ℕ))))) ∈ V
9895, 96, 97fvmpt 6244 . . . . . . . . . . . . . 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 2658 . . . . . . . . . . . 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 2623 . . . . . . . . . . 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 3043 . . . . . . . . . 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 3179 . . . . . . . 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 797 . . . . . . . . 9 ((((𝑀 ∈ ℕ0𝐹:(1...𝑁)⟶(1...𝑀)) ∧ 𝑁 ∈ ℕ0) ∧ 𝑏 ∈ (mzPoly‘((ℤ ∖ ℕ) ∪ (1...𝑁)))) → 𝑀 ∈ ℕ0)
106 ovex 6638 . . . . . . . . . . . 12 (1...𝑀) ∈ V
1073, 106unex 6916 . . . . . . . . . . 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 477 . . . . . . . . . 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 3788 . . . . . . . . . . . . . . 15 ((ℤ ∖ ℕ) ∩ (1...𝑁)) = ((1...𝑁) ∩ (ℤ ∖ ℕ))
113 fz1ssnn 12321 . . . . . . . . . . . . . . . 16 (1...𝑁) ⊆ ℕ
114 ssdisj 4003 . . . . . . . . . . . . . . . 16 (((1...𝑁) ⊆ ℕ ∧ (ℕ ∩ (ℤ ∖ ℕ)) = ∅) → ((1...𝑁) ∩ (ℤ ∖ ℕ)) = ∅)
115113, 18, 114mp2an 707 . . . . . . . . . . . . . . 15 ((1...𝑁) ∩ (ℤ ∖ ℕ)) = ∅
116112, 115eqtri 2643 . . . . . . . . . . . . . 14 ((ℤ ∖ ℕ) ∩ (1...𝑁)) = ∅
117116a1i 11 . . . . . . . . . . . . 13 (𝐹:(1...𝑁)⟶(1...𝑀) → ((ℤ ∖ ℕ) ∩ (1...𝑁)) = ∅)
118 fun 6028 . . . . . . . . . . . . 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 1322 . . . . . . . . . . . 12 (𝐹:(1...𝑁)⟶(1...𝑀) → (( I ↾ (ℤ ∖ ℕ)) ∪ 𝐹):((ℤ ∖ ℕ) ∪ (1...𝑁))⟶((ℤ ∖ ℕ) ∪ (1...𝑀)))
120 uncom 3740 . . . . . . . . . . . . 13 (( I ↾ (ℤ ∖ ℕ)) ∪ 𝐹) = (𝐹 ∪ ( I ↾ (ℤ ∖ ℕ)))
121120feq1i 5998 . . . . . . . . . . . 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 766 . . . . . . . . . 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 36819 . . . . . . . . . 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 1323 . . . . . . . . 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 36883 . . . . . . . . 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 692 . . . . . . . 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 2698 . . . . . . 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 2687 . . . . . . . . 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 3180 . . . . . . . 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 2683 . . . . . . 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 3025 . . . . 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 628 . . . 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 446 . 2 ((𝑆 ∈ (Dioph‘𝑁) ∧ (𝑀 ∈ ℕ0𝐹:(1...𝑁)⟶(1...𝑀))) → {𝑎 ∈ (ℕ0𝑚 (1...𝑀)) ∣ (𝑎𝐹) ∈ 𝑆} ∈ (Dioph‘𝑀))
1371363impb 1257 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 384  w3a 1036   = wceq 1480  wcel 1987  wrex 2908  {crab 2911  Vcvv 3189  cdif 3556  cun 3557  cin 3558  wss 3559  c0 3896   class class class wbr 4618  cmpt 4678   I cid 4989  cres 5081  ccom 5083   Fn wfn 5847  wf 5848  1-1-ontowf1o 5851  cfv 5852  (class class class)co 6610  ωcom 7019  𝑚 cmap 7809  cen 7903  Fincfn 7906  0cc0 9887  1c1 9888   + caddc 9890  cn 10971  0cn0 11243  cz 11328  cuz 11638  ...cfz 12275  mzPolycmzp 36792  Diophcdioph 36825
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909  ax-inf2 8489  ax-cnex 9943  ax-resscn 9944  ax-1cn 9945  ax-icn 9946  ax-addcl 9947  ax-addrcl 9948  ax-mulcl 9949  ax-mulrcl 9950  ax-mulcom 9951  ax-addass 9952  ax-mulass 9953  ax-distr 9954  ax-i2m1 9955  ax-1ne0 9956  ax-1rid 9957  ax-rnegex 9958  ax-rrecex 9959  ax-cnre 9960  ax-pre-lttri 9961  ax-pre-lttrn 9962  ax-pre-ltadd 9963  ax-pre-mulgt0 9964
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-pss 3575  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-int 4446  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5644  df-ord 5690  df-on 5691  df-lim 5692  df-suc 5693  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-riota 6571  df-ov 6613  df-oprab 6614  df-mpt2 6615  df-of 6857  df-om 7020  df-1st 7120  df-2nd 7121  df-wrecs 7359  df-recs 7420  df-rdg 7458  df-1o 7512  df-oadd 7516  df-er 7694  df-map 7811  df-en 7907  df-dom 7908  df-sdom 7909  df-fin 7910  df-card 8716  df-cda 8941  df-pnf 10027  df-mnf 10028  df-xr 10029  df-ltxr 10030  df-le 10031  df-sub 10219  df-neg 10220  df-nn 10972  df-n0 11244  df-z 11329  df-uz 11639  df-fz 12276  df-hash 13065  df-mzpcl 36793  df-mzp 36794  df-dioph 36826
This theorem is referenced by:  rabrenfdioph  36885
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