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Theorem eldiophss 41512
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 41503 . 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 486 . . . 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 3479 . . . . . . . 8 𝑑 ∈ V
4 eqeq1 2737 . . . . . . . . . 10 (𝑏 = 𝑑 β†’ (𝑏 = (𝑐 β†Ύ (1...𝐡)) ↔ 𝑑 = (𝑐 β†Ύ (1...𝐡))))
54anbi1d 631 . . . . . . . . 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 3179 . . . . . . . 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 3669 . . . . . . 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 486 . . . . . . . . . . 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 13530 . . . . . . . . . . . . . 14 (π‘Ž ∈ (1...𝐡) β†’ π‘Ž ∈ β„•)
109ssriv 3987 . . . . . . . . . . . . 13 (1...𝐡) βŠ† β„•
11 elmapssres 8861 . . . . . . . . . . . . 13 ((𝑐 ∈ (β„•0 ↑m β„•) ∧ (1...𝐡) βŠ† β„•) β†’ (𝑐 β†Ύ (1...𝐡)) ∈ (β„•0 ↑m (1...𝐡)))
1210, 11mpan2 690 . . . . . . . . . . . 12 (𝑐 ∈ (β„•0 ↑m β„•) β†’ (𝑐 β†Ύ (1...𝐡)) ∈ (β„•0 ↑m (1...𝐡)))
1312ad2antlr 726 . . . . . . . . . . 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 2834 . . . . . . . . . 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 414 . . . . . . . . 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 493 . . . . . . . 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 3156 . . . . . . 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 3989 . . . . 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 482 . . . 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 4021 . . 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 3159 . 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 397   = wceq 1542   ∈ wcel 2107  {cab 2710  βˆƒwrex 3071   βŠ† wss 3949   β†Ύ cres 5679  β€˜cfv 6544  (class class class)co 7409   ↑m cmap 8820  0cc0 11110  1c1 11111  β„•cn 12212  β„•0cn0 12472  ...cfz 13484  mzPolycmzp 41460  Diophcdioph 41493
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725  ax-inf2 9636  ax-cnex 11166  ax-resscn 11167  ax-1cn 11168  ax-icn 11169  ax-addcl 11170  ax-addrcl 11171  ax-mulcl 11172  ax-mulrcl 11173  ax-mulcom 11174  ax-addass 11175  ax-mulass 11176  ax-distr 11177  ax-i2m1 11178  ax-1ne0 11179  ax-1rid 11180  ax-rnegex 11181  ax-rrecex 11182  ax-cnre 11183  ax-pre-lttri 11184  ax-pre-lttrn 11185  ax-pre-ltadd 11186  ax-pre-mulgt0 11187
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-int 4952  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-riota 7365  df-ov 7412  df-oprab 7413  df-mpo 7414  df-of 7670  df-om 7856  df-1st 7975  df-2nd 7976  df-frecs 8266  df-wrecs 8297  df-recs 8371  df-rdg 8410  df-1o 8466  df-oadd 8470  df-er 8703  df-map 8822  df-en 8940  df-dom 8941  df-sdom 8942  df-fin 8943  df-dju 9896  df-card 9934  df-pnf 11250  df-mnf 11251  df-xr 11252  df-ltxr 11253  df-le 11254  df-sub 11446  df-neg 11447  df-nn 12213  df-n0 12473  df-z 12559  df-uz 12823  df-fz 13485  df-hash 14291  df-mzpcl 41461  df-mzp 41462  df-dioph 41494
This theorem is referenced by: (None)
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