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Theorem eldiophss 42259
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 42250 . 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 483 . . . 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 3467 . . . . . . . 8 𝑑 ∈ V
4 eqeq1 2729 . . . . . . . . . 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 3169 . . . . . . . 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 3659 . . . . . . 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 483 . . . . . . . . . . 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 13562 . . . . . . . . . . . . . 14 (π‘Ž ∈ (1...𝐡) β†’ π‘Ž ∈ β„•)
109ssriv 3976 . . . . . . . . . . . . 13 (1...𝐡) βŠ† β„•
11 elmapssres 8884 . . . . . . . . . . . . 13 ((𝑐 ∈ (β„•0 ↑m β„•) ∧ (1...𝐡) βŠ† β„•) β†’ (𝑐 β†Ύ (1...𝐡)) ∈ (β„•0 ↑m (1...𝐡)))
1210, 11mpan2 689 . . . . . . . . . . . 12 (𝑐 ∈ (β„•0 ↑m β„•) β†’ (𝑐 β†Ύ (1...𝐡)) ∈ (β„•0 ↑m (1...𝐡)))
1312ad2antlr 725 . . . . . . . . . . 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 2825 . . . . . . . . . 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 411 . . . . . . . . 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 490 . . . . . . . 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 3145 . . . . . . 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 3978 . . . . 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 479 . . . 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 4011 . . 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 3148 . 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 394   = wceq 1533   ∈ wcel 2098  {cab 2702  βˆƒwrex 3060   βŠ† wss 3939   β†Ύ cres 5674  β€˜cfv 6543  (class class class)co 7416   ↑m cmap 8843  0cc0 11138  1c1 11139  β„•cn 12242  β„•0cn0 12502  ...cfz 13516  mzPolycmzp 42207  Diophcdioph 42240
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 2166  ax-ext 2696  ax-rep 5280  ax-sep 5294  ax-nul 5301  ax-pow 5359  ax-pr 5423  ax-un 7738  ax-inf2 9664  ax-cnex 11194  ax-resscn 11195  ax-1cn 11196  ax-icn 11197  ax-addcl 11198  ax-addrcl 11199  ax-mulcl 11200  ax-mulrcl 11201  ax-mulcom 11202  ax-addass 11203  ax-mulass 11204  ax-distr 11205  ax-i2m1 11206  ax-1ne0 11207  ax-1rid 11208  ax-rnegex 11209  ax-rrecex 11210  ax-cnre 11211  ax-pre-lttri 11212  ax-pre-lttrn 11213  ax-pre-ltadd 11214  ax-pre-mulgt0 11215
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  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 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-nel 3037  df-ral 3052  df-rex 3061  df-reu 3365  df-rab 3420  df-v 3465  df-sbc 3769  df-csb 3885  df-dif 3942  df-un 3944  df-in 3946  df-ss 3956  df-pss 3959  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-int 4945  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5227  df-tr 5261  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7372  df-ov 7419  df-oprab 7420  df-mpo 7421  df-of 7682  df-om 7869  df-1st 7991  df-2nd 7992  df-frecs 8285  df-wrecs 8316  df-recs 8390  df-rdg 8429  df-1o 8485  df-oadd 8489  df-er 8723  df-map 8845  df-en 8963  df-dom 8964  df-sdom 8965  df-fin 8966  df-dju 9924  df-card 9962  df-pnf 11280  df-mnf 11281  df-xr 11282  df-ltxr 11283  df-le 11284  df-sub 11476  df-neg 11477  df-nn 12243  df-n0 12503  df-z 12589  df-uz 12853  df-fz 13517  df-hash 14322  df-mzpcl 42208  df-mzp 42209  df-dioph 42241
This theorem is referenced by: (None)
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