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Mathbox for Stefan O'Rear |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > eldiophss | Structured version Visualization version GIF version |
Description: Diophantine sets are sets of tuples of nonnegative integers. (Contributed by Stefan O'Rear, 10-Oct-2014.) (Revised by Stefan O'Rear, 6-May-2015.) |
Ref | Expression |
---|---|
eldiophss | ⊢ (𝐴 ∈ (Dioph‘𝐵) → 𝐴 ⊆ (ℕ0 ↑m (1...𝐵))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eldioph3b 41992 | . 2 ⊢ (𝐴 ∈ (Dioph‘𝐵) ↔ (𝐵 ∈ ℕ0 ∧ ∃𝑎 ∈ (mzPoly‘ℕ)𝐴 = {𝑏 ∣ ∃𝑐 ∈ (ℕ0 ↑m ℕ)(𝑏 = (𝑐 ↾ (1...𝐵)) ∧ (𝑎‘𝑐) = 0)})) | |
2 | simpr 484 | . . . 4 ⊢ (((𝐵 ∈ ℕ0 ∧ 𝑎 ∈ (mzPoly‘ℕ)) ∧ 𝐴 = {𝑏 ∣ ∃𝑐 ∈ (ℕ0 ↑m ℕ)(𝑏 = (𝑐 ↾ (1...𝐵)) ∧ (𝑎‘𝑐) = 0)}) → 𝐴 = {𝑏 ∣ ∃𝑐 ∈ (ℕ0 ↑m ℕ)(𝑏 = (𝑐 ↾ (1...𝐵)) ∧ (𝑎‘𝑐) = 0)}) | |
3 | vex 3470 | . . . . . . . 8 ⊢ 𝑑 ∈ V | |
4 | eqeq1 2728 | . . . . . . . . . 10 ⊢ (𝑏 = 𝑑 → (𝑏 = (𝑐 ↾ (1...𝐵)) ↔ 𝑑 = (𝑐 ↾ (1...𝐵)))) | |
5 | 4 | anbi1d 629 | . . . . . . . . 9 ⊢ (𝑏 = 𝑑 → ((𝑏 = (𝑐 ↾ (1...𝐵)) ∧ (𝑎‘𝑐) = 0) ↔ (𝑑 = (𝑐 ↾ (1...𝐵)) ∧ (𝑎‘𝑐) = 0))) |
6 | 5 | rexbidv 3170 | . . . . . . . 8 ⊢ (𝑏 = 𝑑 → (∃𝑐 ∈ (ℕ0 ↑m ℕ)(𝑏 = (𝑐 ↾ (1...𝐵)) ∧ (𝑎‘𝑐) = 0) ↔ ∃𝑐 ∈ (ℕ0 ↑m ℕ)(𝑑 = (𝑐 ↾ (1...𝐵)) ∧ (𝑎‘𝑐) = 0))) |
7 | 3, 6 | elab 3660 | . . . . . . 7 ⊢ (𝑑 ∈ {𝑏 ∣ ∃𝑐 ∈ (ℕ0 ↑m ℕ)(𝑏 = (𝑐 ↾ (1...𝐵)) ∧ (𝑎‘𝑐) = 0)} ↔ ∃𝑐 ∈ (ℕ0 ↑m ℕ)(𝑑 = (𝑐 ↾ (1...𝐵)) ∧ (𝑎‘𝑐) = 0)) |
8 | simpr 484 | . . . . . . . . . . 11 ⊢ ((((𝐵 ∈ ℕ0 ∧ 𝑎 ∈ (mzPoly‘ℕ)) ∧ 𝑐 ∈ (ℕ0 ↑m ℕ)) ∧ 𝑑 = (𝑐 ↾ (1...𝐵))) → 𝑑 = (𝑐 ↾ (1...𝐵))) | |
9 | elfznn 13527 | . . . . . . . . . . . . . 14 ⊢ (𝑎 ∈ (1...𝐵) → 𝑎 ∈ ℕ) | |
10 | 9 | ssriv 3978 | . . . . . . . . . . . . 13 ⊢ (1...𝐵) ⊆ ℕ |
11 | elmapssres 8857 | . . . . . . . . . . . . 13 ⊢ ((𝑐 ∈ (ℕ0 ↑m ℕ) ∧ (1...𝐵) ⊆ ℕ) → (𝑐 ↾ (1...𝐵)) ∈ (ℕ0 ↑m (1...𝐵))) | |
12 | 10, 11 | mpan2 688 | . . . . . . . . . . . 12 ⊢ (𝑐 ∈ (ℕ0 ↑m ℕ) → (𝑐 ↾ (1...𝐵)) ∈ (ℕ0 ↑m (1...𝐵))) |
13 | 12 | ad2antlr 724 | . . . . . . . . . . 11 ⊢ ((((𝐵 ∈ ℕ0 ∧ 𝑎 ∈ (mzPoly‘ℕ)) ∧ 𝑐 ∈ (ℕ0 ↑m ℕ)) ∧ 𝑑 = (𝑐 ↾ (1...𝐵))) → (𝑐 ↾ (1...𝐵)) ∈ (ℕ0 ↑m (1...𝐵))) |
14 | 8, 13 | eqeltrd 2825 | . . . . . . . . . 10 ⊢ ((((𝐵 ∈ ℕ0 ∧ 𝑎 ∈ (mzPoly‘ℕ)) ∧ 𝑐 ∈ (ℕ0 ↑m ℕ)) ∧ 𝑑 = (𝑐 ↾ (1...𝐵))) → 𝑑 ∈ (ℕ0 ↑m (1...𝐵))) |
15 | 14 | ex 412 | . . . . . . . . 9 ⊢ (((𝐵 ∈ ℕ0 ∧ 𝑎 ∈ (mzPoly‘ℕ)) ∧ 𝑐 ∈ (ℕ0 ↑m ℕ)) → (𝑑 = (𝑐 ↾ (1...𝐵)) → 𝑑 ∈ (ℕ0 ↑m (1...𝐵)))) |
16 | 15 | adantrd 491 | . . . . . . . 8 ⊢ (((𝐵 ∈ ℕ0 ∧ 𝑎 ∈ (mzPoly‘ℕ)) ∧ 𝑐 ∈ (ℕ0 ↑m ℕ)) → ((𝑑 = (𝑐 ↾ (1...𝐵)) ∧ (𝑎‘𝑐) = 0) → 𝑑 ∈ (ℕ0 ↑m (1...𝐵)))) |
17 | 16 | rexlimdva 3147 | . . . . . . 7 ⊢ ((𝐵 ∈ ℕ0 ∧ 𝑎 ∈ (mzPoly‘ℕ)) → (∃𝑐 ∈ (ℕ0 ↑m ℕ)(𝑑 = (𝑐 ↾ (1...𝐵)) ∧ (𝑎‘𝑐) = 0) → 𝑑 ∈ (ℕ0 ↑m (1...𝐵)))) |
18 | 7, 17 | biimtrid 241 | . . . . . 6 ⊢ ((𝐵 ∈ ℕ0 ∧ 𝑎 ∈ (mzPoly‘ℕ)) → (𝑑 ∈ {𝑏 ∣ ∃𝑐 ∈ (ℕ0 ↑m ℕ)(𝑏 = (𝑐 ↾ (1...𝐵)) ∧ (𝑎‘𝑐) = 0)} → 𝑑 ∈ (ℕ0 ↑m (1...𝐵)))) |
19 | 18 | ssrdv 3980 | . . . . 5 ⊢ ((𝐵 ∈ ℕ0 ∧ 𝑎 ∈ (mzPoly‘ℕ)) → {𝑏 ∣ ∃𝑐 ∈ (ℕ0 ↑m ℕ)(𝑏 = (𝑐 ↾ (1...𝐵)) ∧ (𝑎‘𝑐) = 0)} ⊆ (ℕ0 ↑m (1...𝐵))) |
20 | 19 | adantr 480 | . . . 4 ⊢ (((𝐵 ∈ ℕ0 ∧ 𝑎 ∈ (mzPoly‘ℕ)) ∧ 𝐴 = {𝑏 ∣ ∃𝑐 ∈ (ℕ0 ↑m ℕ)(𝑏 = (𝑐 ↾ (1...𝐵)) ∧ (𝑎‘𝑐) = 0)}) → {𝑏 ∣ ∃𝑐 ∈ (ℕ0 ↑m ℕ)(𝑏 = (𝑐 ↾ (1...𝐵)) ∧ (𝑎‘𝑐) = 0)} ⊆ (ℕ0 ↑m (1...𝐵))) |
21 | 2, 20 | eqsstrd 4012 | . . 3 ⊢ (((𝐵 ∈ ℕ0 ∧ 𝑎 ∈ (mzPoly‘ℕ)) ∧ 𝐴 = {𝑏 ∣ ∃𝑐 ∈ (ℕ0 ↑m ℕ)(𝑏 = (𝑐 ↾ (1...𝐵)) ∧ (𝑎‘𝑐) = 0)}) → 𝐴 ⊆ (ℕ0 ↑m (1...𝐵))) |
22 | 21 | r19.29an 3150 | . 2 ⊢ ((𝐵 ∈ ℕ0 ∧ ∃𝑎 ∈ (mzPoly‘ℕ)𝐴 = {𝑏 ∣ ∃𝑐 ∈ (ℕ0 ↑m ℕ)(𝑏 = (𝑐 ↾ (1...𝐵)) ∧ (𝑎‘𝑐) = 0)}) → 𝐴 ⊆ (ℕ0 ↑m (1...𝐵))) |
23 | 1, 22 | sylbi 216 | 1 ⊢ (𝐴 ∈ (Dioph‘𝐵) → 𝐴 ⊆ (ℕ0 ↑m (1...𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1533 ∈ wcel 2098 {cab 2701 ∃wrex 3062 ⊆ wss 3940 ↾ cres 5668 ‘cfv 6533 (class class class)co 7401 ↑m cmap 8816 0cc0 11106 1c1 11107 ℕcn 12209 ℕ0cn0 12469 ...cfz 13481 mzPolycmzp 41949 Diophcdioph 41982 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-rep 5275 ax-sep 5289 ax-nul 5296 ax-pow 5353 ax-pr 5417 ax-un 7718 ax-inf2 9632 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3959 df-nul 4315 df-if 4521 df-pw 4596 df-sn 4621 df-pr 4623 df-op 4627 df-uni 4900 df-int 4941 df-iun 4989 df-br 5139 df-opab 5201 df-mpt 5222 df-tr 5256 df-id 5564 df-eprel 5570 df-po 5578 df-so 5579 df-fr 5621 df-we 5623 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-pred 6290 df-ord 6357 df-on 6358 df-lim 6359 df-suc 6360 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-riota 7357 df-ov 7404 df-oprab 7405 df-mpo 7406 df-of 7663 df-om 7849 df-1st 7968 df-2nd 7969 df-frecs 8261 df-wrecs 8292 df-recs 8366 df-rdg 8405 df-1o 8461 df-oadd 8465 df-er 8699 df-map 8818 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-dju 9892 df-card 9930 df-pnf 11247 df-mnf 11248 df-xr 11249 df-ltxr 11250 df-le 11251 df-sub 11443 df-neg 11444 df-nn 12210 df-n0 12470 df-z 12556 df-uz 12820 df-fz 13482 df-hash 14288 df-mzpcl 41950 df-mzp 41951 df-dioph 41983 |
This theorem is referenced by: (None) |
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