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Mathbox for Stefan O'Rear |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > eldioph3b | Structured version Visualization version GIF version |
Description: Define Diophantine sets in terms of polynomials with variables indexed by ℕ. This avoids a quantifier over the number of witness variables and will be easier to use than eldiophb 41058 in most cases. (Contributed by Stefan O'Rear, 10-Oct-2014.) |
Ref | Expression |
---|---|
eldioph3b | ⊢ (𝐴 ∈ (Dioph‘𝑁) ↔ (𝑁 ∈ ℕ0 ∧ ∃𝑝 ∈ (mzPoly‘ℕ)𝐴 = {𝑡 ∣ ∃𝑢 ∈ (ℕ0 ↑m ℕ)(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝‘𝑢) = 0)})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eldiophelnn0 41065 | . 2 ⊢ (𝐴 ∈ (Dioph‘𝑁) → 𝑁 ∈ ℕ0) | |
2 | nnex 12158 | . . 3 ⊢ ℕ ∈ V | |
3 | 1z 12532 | . . . . 5 ⊢ 1 ∈ ℤ | |
4 | nnuz 12805 | . . . . . 6 ⊢ ℕ = (ℤ≥‘1) | |
5 | 4 | uzinf 13869 | . . . . 5 ⊢ (1 ∈ ℤ → ¬ ℕ ∈ Fin) |
6 | 3, 5 | ax-mp 5 | . . . 4 ⊢ ¬ ℕ ∈ Fin |
7 | elfznn 13469 | . . . . 5 ⊢ (𝑝 ∈ (1...𝑁) → 𝑝 ∈ ℕ) | |
8 | 7 | ssriv 3948 | . . . 4 ⊢ (1...𝑁) ⊆ ℕ |
9 | eldioph2b 41064 | . . . 4 ⊢ (((𝑁 ∈ ℕ0 ∧ ℕ ∈ V) ∧ (¬ ℕ ∈ Fin ∧ (1...𝑁) ⊆ ℕ)) → (𝐴 ∈ (Dioph‘𝑁) ↔ ∃𝑝 ∈ (mzPoly‘ℕ)𝐴 = {𝑡 ∣ ∃𝑢 ∈ (ℕ0 ↑m ℕ)(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝‘𝑢) = 0)})) | |
10 | 6, 8, 9 | mpanr12 703 | . . 3 ⊢ ((𝑁 ∈ ℕ0 ∧ ℕ ∈ V) → (𝐴 ∈ (Dioph‘𝑁) ↔ ∃𝑝 ∈ (mzPoly‘ℕ)𝐴 = {𝑡 ∣ ∃𝑢 ∈ (ℕ0 ↑m ℕ)(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝‘𝑢) = 0)})) |
11 | 2, 10 | mpan2 689 | . 2 ⊢ (𝑁 ∈ ℕ0 → (𝐴 ∈ (Dioph‘𝑁) ↔ ∃𝑝 ∈ (mzPoly‘ℕ)𝐴 = {𝑡 ∣ ∃𝑢 ∈ (ℕ0 ↑m ℕ)(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝‘𝑢) = 0)})) |
12 | 1, 11 | biadanii 820 | 1 ⊢ (𝐴 ∈ (Dioph‘𝑁) ↔ (𝑁 ∈ ℕ0 ∧ ∃𝑝 ∈ (mzPoly‘ℕ)𝐴 = {𝑡 ∣ ∃𝑢 ∈ (ℕ0 ↑m ℕ)(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝‘𝑢) = 0)})) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 205 ∧ wa 396 = wceq 1541 ∈ wcel 2106 {cab 2713 ∃wrex 3073 Vcvv 3445 ⊆ wss 3910 ↾ cres 5635 ‘cfv 6496 (class class class)co 7356 ↑m cmap 8764 Fincfn 8882 0cc0 11050 1c1 11051 ℕcn 12152 ℕ0cn0 12412 ℤcz 12498 ...cfz 13423 mzPolycmzp 41023 Diophcdioph 41056 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-rep 5242 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7671 ax-inf2 9576 ax-cnex 11106 ax-resscn 11107 ax-1cn 11108 ax-icn 11109 ax-addcl 11110 ax-addrcl 11111 ax-mulcl 11112 ax-mulrcl 11113 ax-mulcom 11114 ax-addass 11115 ax-mulass 11116 ax-distr 11117 ax-i2m1 11118 ax-1ne0 11119 ax-1rid 11120 ax-rnegex 11121 ax-rrecex 11122 ax-cnre 11123 ax-pre-lttri 11124 ax-pre-lttrn 11125 ax-pre-ltadd 11126 ax-pre-mulgt0 11127 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3065 df-rex 3074 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-pss 3929 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-op 4593 df-uni 4866 df-int 4908 df-iun 4956 df-br 5106 df-opab 5168 df-mpt 5189 df-tr 5223 df-id 5531 df-eprel 5537 df-po 5545 df-so 5546 df-fr 5588 df-we 5590 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-pred 6253 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-riota 7312 df-ov 7359 df-oprab 7360 df-mpo 7361 df-of 7616 df-om 7802 df-1st 7920 df-2nd 7921 df-frecs 8211 df-wrecs 8242 df-recs 8316 df-rdg 8355 df-1o 8411 df-oadd 8415 df-er 8647 df-map 8766 df-en 8883 df-dom 8884 df-sdom 8885 df-fin 8886 df-dju 9836 df-card 9874 df-pnf 11190 df-mnf 11191 df-xr 11192 df-ltxr 11193 df-le 11194 df-sub 11386 df-neg 11387 df-nn 12153 df-n0 12413 df-z 12499 df-uz 12763 df-fz 13424 df-hash 14230 df-mzpcl 41024 df-mzp 41025 df-dioph 41057 |
This theorem is referenced by: eldioph3 41067 eldiophss 41075 diophrex 41076 |
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