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Theorem eldiophss 42090
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.)
Assertion
Ref Expression
eldiophss (𝐴 ∈ (Diophβ€˜π΅) β†’ 𝐴 βŠ† (β„•0 ↑m (1...𝐡)))
Proof of Theorem eldiophss
Dummy variables π‘Ž 𝑏 𝑐 𝑑 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eldioph3b 42081 . 2 (𝐴 ∈ (Diophβ€˜π΅) ↔ (𝐡 ∈ β„•0 ∧ βˆƒπ‘Ž ∈ (mzPolyβ€˜β„•)𝐴 = {𝑏 ∣ βˆƒπ‘ ∈ (β„•0 ↑m β„•)(𝑏 = (𝑐 β†Ύ (1...𝐡)) ∧ (π‘Žβ€˜π‘) = 0)}))
2 simpr 484 . . . 4 (((𝐡 ∈ β„•0 ∧ π‘Ž ∈ (mzPolyβ€˜β„•)) ∧ 𝐴 = {𝑏 ∣ βˆƒπ‘ ∈ (β„•0 ↑m β„•)(𝑏 = (𝑐 β†Ύ (1...𝐡)) ∧ (π‘Žβ€˜π‘) = 0)}) β†’ 𝐴 = {𝑏 ∣ βˆƒπ‘ ∈ (β„•0 ↑m β„•)(𝑏 = (𝑐 β†Ύ (1...𝐡)) ∧ (π‘Žβ€˜π‘) = 0)})
3 vex 3472 . . . . . . . 8 𝑑 ∈ V
4 eqeq1 2730 . . . . . . . . . 10 (𝑏 = 𝑑 β†’ (𝑏 = (𝑐 β†Ύ (1...𝐡)) ↔ 𝑑 = (𝑐 β†Ύ (1...𝐡))))
54anbi1d 629 . . . . . . . . 9 (𝑏 = 𝑑 β†’ ((𝑏 = (𝑐 β†Ύ (1...𝐡)) ∧ (π‘Žβ€˜π‘) = 0) ↔ (𝑑 = (𝑐 β†Ύ (1...𝐡)) ∧ (π‘Žβ€˜π‘) = 0)))
65rexbidv 3172 . . . . . . . 8 (𝑏 = 𝑑 β†’ (βˆƒπ‘ ∈ (β„•0 ↑m β„•)(𝑏 = (𝑐 β†Ύ (1...𝐡)) ∧ (π‘Žβ€˜π‘) = 0) ↔ βˆƒπ‘ ∈ (β„•0 ↑m β„•)(𝑑 = (𝑐 β†Ύ (1...𝐡)) ∧ (π‘Žβ€˜π‘) = 0)))
73, 6elab 3663 . . . . . . 7 (𝑑 ∈ {𝑏 ∣ βˆƒπ‘ ∈ (β„•0 ↑m β„•)(𝑏 = (𝑐 β†Ύ (1...𝐡)) ∧ (π‘Žβ€˜π‘) = 0)} ↔ βˆƒπ‘ ∈ (β„•0 ↑m β„•)(𝑑 = (𝑐 β†Ύ (1...𝐡)) ∧ (π‘Žβ€˜π‘) = 0))
8 simpr 484 . . . . . . . . . . 11 ((((𝐡 ∈ β„•0 ∧ π‘Ž ∈ (mzPolyβ€˜β„•)) ∧ 𝑐 ∈ (β„•0 ↑m β„•)) ∧ 𝑑 = (𝑐 β†Ύ (1...𝐡))) β†’ 𝑑 = (𝑐 β†Ύ (1...𝐡)))
9 elfznn 13536 . . . . . . . . . . . . . 14 (π‘Ž ∈ (1...𝐡) β†’ π‘Ž ∈ β„•)
109ssriv 3981 . . . . . . . . . . . . 13 (1...𝐡) βŠ† β„•
11 elmapssres 8863 . . . . . . . . . . . . 13 ((𝑐 ∈ (β„•0 ↑m β„•) ∧ (1...𝐡) βŠ† β„•) β†’ (𝑐 β†Ύ (1...𝐡)) ∈ (β„•0 ↑m (1...𝐡)))
1210, 11mpan2 688 . . . . . . . . . . . 12 (𝑐 ∈ (β„•0 ↑m β„•) β†’ (𝑐 β†Ύ (1...𝐡)) ∈ (β„•0 ↑m (1...𝐡)))
1312ad2antlr 724 . . . . . . . . . . 11 ((((𝐡 ∈ β„•0 ∧ π‘Ž ∈ (mzPolyβ€˜β„•)) ∧ 𝑐 ∈ (β„•0 ↑m β„•)) ∧ 𝑑 = (𝑐 β†Ύ (1...𝐡))) β†’ (𝑐 β†Ύ (1...𝐡)) ∈ (β„•0 ↑m (1...𝐡)))
148, 13eqeltrd 2827 . . . . . . . . . 10 ((((𝐡 ∈ β„•0 ∧ π‘Ž ∈ (mzPolyβ€˜β„•)) ∧ 𝑐 ∈ (β„•0 ↑m β„•)) ∧ 𝑑 = (𝑐 β†Ύ (1...𝐡))) β†’ 𝑑 ∈ (β„•0 ↑m (1...𝐡)))
1514ex 412 . . . . . . . . 9 (((𝐡 ∈ β„•0 ∧ π‘Ž ∈ (mzPolyβ€˜β„•)) ∧ 𝑐 ∈ (β„•0 ↑m β„•)) β†’ (𝑑 = (𝑐 β†Ύ (1...𝐡)) β†’ 𝑑 ∈ (β„•0 ↑m (1...𝐡))))
1615adantrd 491 . . . . . . . 8 (((𝐡 ∈ β„•0 ∧ π‘Ž ∈ (mzPolyβ€˜β„•)) ∧ 𝑐 ∈ (β„•0 ↑m β„•)) β†’ ((𝑑 = (𝑐 β†Ύ (1...𝐡)) ∧ (π‘Žβ€˜π‘) = 0) β†’ 𝑑 ∈ (β„•0 ↑m (1...𝐡))))
1716rexlimdva 3149 . . . . . . 7 ((𝐡 ∈ β„•0 ∧ π‘Ž ∈ (mzPolyβ€˜β„•)) β†’ (βˆƒπ‘ ∈ (β„•0 ↑m β„•)(𝑑 = (𝑐 β†Ύ (1...𝐡)) ∧ (π‘Žβ€˜π‘) = 0) β†’ 𝑑 ∈ (β„•0 ↑m (1...𝐡))))
187, 17biimtrid 241 . . . . . 6 ((𝐡 ∈ β„•0 ∧ π‘Ž ∈ (mzPolyβ€˜β„•)) β†’ (𝑑 ∈ {𝑏 ∣ βˆƒπ‘ ∈ (β„•0 ↑m β„•)(𝑏 = (𝑐 β†Ύ (1...𝐡)) ∧ (π‘Žβ€˜π‘) = 0)} β†’ 𝑑 ∈ (β„•0 ↑m (1...𝐡))))
1918ssrdv 3983 . . . . 5 ((𝐡 ∈ β„•0 ∧ π‘Ž ∈ (mzPolyβ€˜β„•)) β†’ {𝑏 ∣ βˆƒπ‘ ∈ (β„•0 ↑m β„•)(𝑏 = (𝑐 β†Ύ (1...𝐡)) ∧ (π‘Žβ€˜π‘) = 0)} βŠ† (β„•0 ↑m (1...𝐡)))
2019adantr 480 . . . 4 (((𝐡 ∈ β„•0 ∧ π‘Ž ∈ (mzPolyβ€˜β„•)) ∧ 𝐴 = {𝑏 ∣ βˆƒπ‘ ∈ (β„•0 ↑m β„•)(𝑏 = (𝑐 β†Ύ (1...𝐡)) ∧ (π‘Žβ€˜π‘) = 0)}) β†’ {𝑏 ∣ βˆƒπ‘ ∈ (β„•0 ↑m β„•)(𝑏 = (𝑐 β†Ύ (1...𝐡)) ∧ (π‘Žβ€˜π‘) = 0)} βŠ† (β„•0 ↑m (1...𝐡)))
212, 20eqsstrd 4015 . . 3 (((𝐡 ∈ β„•0 ∧ π‘Ž ∈ (mzPolyβ€˜β„•)) ∧ 𝐴 = {𝑏 ∣ βˆƒπ‘ ∈ (β„•0 ↑m β„•)(𝑏 = (𝑐 β†Ύ (1...𝐡)) ∧ (π‘Žβ€˜π‘) = 0)}) β†’ 𝐴 βŠ† (β„•0 ↑m (1...𝐡)))
2221r19.29an 3152 . 2 ((𝐡 ∈ β„•0 ∧ βˆƒπ‘Ž ∈ (mzPolyβ€˜β„•)𝐴 = {𝑏 ∣ βˆƒπ‘ ∈ (β„•0 ↑m β„•)(𝑏 = (𝑐 β†Ύ (1...𝐡)) ∧ (π‘Žβ€˜π‘) = 0)}) β†’ 𝐴 βŠ† (β„•0 ↑m (1...𝐡)))
231, 22sylbi 216 1 (𝐴 ∈ (Diophβ€˜π΅) β†’ 𝐴 βŠ† (β„•0 ↑m (1...𝐡)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 395   = wceq 1533   ∈ wcel 2098  {cab 2703  βˆƒwrex 3064   βŠ† wss 3943   β†Ύ cres 5671  β€˜cfv 6537  (class class class)co 7405   ↑m cmap 8822  0cc0 11112  1c1 11113  β„•cn 12216  β„•0cn0 12476  ...cfz 13490  mzPolycmzp 42038  Diophcdioph 42071
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 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7722  ax-inf2 9638  ax-cnex 11168  ax-resscn 11169  ax-1cn 11170  ax-icn 11171  ax-addcl 11172  ax-addrcl 11173  ax-mulcl 11174  ax-mulrcl 11175  ax-mulcom 11176  ax-addass 11177  ax-mulass 11178  ax-distr 11179  ax-i2m1 11180  ax-1ne0 11181  ax-1rid 11182  ax-rnegex 11183  ax-rrecex 11184  ax-cnre 11185  ax-pre-lttri 11186  ax-pre-lttrn 11187  ax-pre-ltadd 11188  ax-pre-mulgt0 11189
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 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-nel 3041  df-ral 3056  df-rex 3065  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-pss 3962  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-int 4944  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-pred 6294  df-ord 6361  df-on 6362  df-lim 6363  df-suc 6364  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7361  df-ov 7408  df-oprab 7409  df-mpo 7410  df-of 7667  df-om 7853  df-1st 7974  df-2nd 7975  df-frecs 8267  df-wrecs 8298  df-recs 8372  df-rdg 8411  df-1o 8467  df-oadd 8471  df-er 8705  df-map 8824  df-en 8942  df-dom 8943  df-sdom 8944  df-fin 8945  df-dju 9898  df-card 9936  df-pnf 11254  df-mnf 11255  df-xr 11256  df-ltxr 11257  df-le 11258  df-sub 11450  df-neg 11451  df-nn 12217  df-n0 12477  df-z 12563  df-uz 12827  df-fz 13491  df-hash 14296  df-mzpcl 42039  df-mzp 42040  df-dioph 42072
This theorem is referenced by: (None)
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