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Theorem eldioph3 41806
Description: Inference version of eldioph3b 41805 with quantifier expanded. (Contributed by Stefan O'Rear, 10-Oct-2014.)
Assertion
Ref Expression
eldioph3 ((𝑁 ∈ β„•0 ∧ 𝑃 ∈ (mzPolyβ€˜β„•)) β†’ {𝑑 ∣ βˆƒπ‘’ ∈ (β„•0 ↑m β„•)(𝑑 = (𝑒 β†Ύ (1...𝑁)) ∧ (π‘ƒβ€˜π‘’) = 0)} ∈ (Diophβ€˜π‘))
Distinct variable groups:   𝑑,𝑁,𝑒   𝑑,𝑃,𝑒
Proof of Theorem eldioph3
Dummy variables π‘Ž 𝑏 𝑝 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpl 481 . 2 ((𝑁 ∈ β„•0 ∧ 𝑃 ∈ (mzPolyβ€˜β„•)) β†’ 𝑁 ∈ β„•0)
2 simpr 483 . . 3 ((𝑁 ∈ β„•0 ∧ 𝑃 ∈ (mzPolyβ€˜β„•)) β†’ 𝑃 ∈ (mzPolyβ€˜β„•))
3 eqidd 2731 . . 3 ((𝑁 ∈ β„•0 ∧ 𝑃 ∈ (mzPolyβ€˜β„•)) β†’ {𝑑 ∣ βˆƒπ‘’ ∈ (β„•0 ↑m β„•)(𝑑 = (𝑒 β†Ύ (1...𝑁)) ∧ (π‘ƒβ€˜π‘’) = 0)} = {𝑑 ∣ βˆƒπ‘’ ∈ (β„•0 ↑m β„•)(𝑑 = (𝑒 β†Ύ (1...𝑁)) ∧ (π‘ƒβ€˜π‘’) = 0)})
4 fveq1 6889 . . . . . . . . 9 (𝑝 = 𝑃 β†’ (π‘β€˜π‘) = (π‘ƒβ€˜π‘))
54eqeq1d 2732 . . . . . . . 8 (𝑝 = 𝑃 β†’ ((π‘β€˜π‘) = 0 ↔ (π‘ƒβ€˜π‘) = 0))
65anbi2d 627 . . . . . . 7 (𝑝 = 𝑃 β†’ ((π‘Ž = (𝑏 β†Ύ (1...𝑁)) ∧ (π‘β€˜π‘) = 0) ↔ (π‘Ž = (𝑏 β†Ύ (1...𝑁)) ∧ (π‘ƒβ€˜π‘) = 0)))
76rexbidv 3176 . . . . . 6 (𝑝 = 𝑃 β†’ (βˆƒπ‘ ∈ (β„•0 ↑m β„•)(π‘Ž = (𝑏 β†Ύ (1...𝑁)) ∧ (π‘β€˜π‘) = 0) ↔ βˆƒπ‘ ∈ (β„•0 ↑m β„•)(π‘Ž = (𝑏 β†Ύ (1...𝑁)) ∧ (π‘ƒβ€˜π‘) = 0)))
87abbidv 2799 . . . . 5 (𝑝 = 𝑃 β†’ {π‘Ž ∣ βˆƒπ‘ ∈ (β„•0 ↑m β„•)(π‘Ž = (𝑏 β†Ύ (1...𝑁)) ∧ (π‘β€˜π‘) = 0)} = {π‘Ž ∣ βˆƒπ‘ ∈ (β„•0 ↑m β„•)(π‘Ž = (𝑏 β†Ύ (1...𝑁)) ∧ (π‘ƒβ€˜π‘) = 0)})
9 eqeq1 2734 . . . . . . . . 9 (π‘Ž = 𝑑 β†’ (π‘Ž = (𝑏 β†Ύ (1...𝑁)) ↔ 𝑑 = (𝑏 β†Ύ (1...𝑁))))
109anbi1d 628 . . . . . . . 8 (π‘Ž = 𝑑 β†’ ((π‘Ž = (𝑏 β†Ύ (1...𝑁)) ∧ (π‘ƒβ€˜π‘) = 0) ↔ (𝑑 = (𝑏 β†Ύ (1...𝑁)) ∧ (π‘ƒβ€˜π‘) = 0)))
1110rexbidv 3176 . . . . . . 7 (π‘Ž = 𝑑 β†’ (βˆƒπ‘ ∈ (β„•0 ↑m β„•)(π‘Ž = (𝑏 β†Ύ (1...𝑁)) ∧ (π‘ƒβ€˜π‘) = 0) ↔ βˆƒπ‘ ∈ (β„•0 ↑m β„•)(𝑑 = (𝑏 β†Ύ (1...𝑁)) ∧ (π‘ƒβ€˜π‘) = 0)))
12 reseq1 5974 . . . . . . . . . 10 (𝑏 = 𝑒 β†’ (𝑏 β†Ύ (1...𝑁)) = (𝑒 β†Ύ (1...𝑁)))
1312eqeq2d 2741 . . . . . . . . 9 (𝑏 = 𝑒 β†’ (𝑑 = (𝑏 β†Ύ (1...𝑁)) ↔ 𝑑 = (𝑒 β†Ύ (1...𝑁))))
14 fveqeq2 6899 . . . . . . . . 9 (𝑏 = 𝑒 β†’ ((π‘ƒβ€˜π‘) = 0 ↔ (π‘ƒβ€˜π‘’) = 0))
1513, 14anbi12d 629 . . . . . . . 8 (𝑏 = 𝑒 β†’ ((𝑑 = (𝑏 β†Ύ (1...𝑁)) ∧ (π‘ƒβ€˜π‘) = 0) ↔ (𝑑 = (𝑒 β†Ύ (1...𝑁)) ∧ (π‘ƒβ€˜π‘’) = 0)))
1615cbvrexvw 3233 . . . . . . 7 (βˆƒπ‘ ∈ (β„•0 ↑m β„•)(𝑑 = (𝑏 β†Ύ (1...𝑁)) ∧ (π‘ƒβ€˜π‘) = 0) ↔ βˆƒπ‘’ ∈ (β„•0 ↑m β„•)(𝑑 = (𝑒 β†Ύ (1...𝑁)) ∧ (π‘ƒβ€˜π‘’) = 0))
1711, 16bitrdi 286 . . . . . 6 (π‘Ž = 𝑑 β†’ (βˆƒπ‘ ∈ (β„•0 ↑m β„•)(π‘Ž = (𝑏 β†Ύ (1...𝑁)) ∧ (π‘ƒβ€˜π‘) = 0) ↔ βˆƒπ‘’ ∈ (β„•0 ↑m β„•)(𝑑 = (𝑒 β†Ύ (1...𝑁)) ∧ (π‘ƒβ€˜π‘’) = 0)))
1817cbvabv 2803 . . . . 5 {π‘Ž ∣ βˆƒπ‘ ∈ (β„•0 ↑m β„•)(π‘Ž = (𝑏 β†Ύ (1...𝑁)) ∧ (π‘ƒβ€˜π‘) = 0)} = {𝑑 ∣ βˆƒπ‘’ ∈ (β„•0 ↑m β„•)(𝑑 = (𝑒 β†Ύ (1...𝑁)) ∧ (π‘ƒβ€˜π‘’) = 0)}
198, 18eqtrdi 2786 . . . 4 (𝑝 = 𝑃 β†’ {π‘Ž ∣ βˆƒπ‘ ∈ (β„•0 ↑m β„•)(π‘Ž = (𝑏 β†Ύ (1...𝑁)) ∧ (π‘β€˜π‘) = 0)} = {𝑑 ∣ βˆƒπ‘’ ∈ (β„•0 ↑m β„•)(𝑑 = (𝑒 β†Ύ (1...𝑁)) ∧ (π‘ƒβ€˜π‘’) = 0)})
2019rspceeqv 3632 . . 3 ((𝑃 ∈ (mzPolyβ€˜β„•) ∧ {𝑑 ∣ βˆƒπ‘’ ∈ (β„•0 ↑m β„•)(𝑑 = (𝑒 β†Ύ (1...𝑁)) ∧ (π‘ƒβ€˜π‘’) = 0)} = {𝑑 ∣ βˆƒπ‘’ ∈ (β„•0 ↑m β„•)(𝑑 = (𝑒 β†Ύ (1...𝑁)) ∧ (π‘ƒβ€˜π‘’) = 0)}) β†’ βˆƒπ‘ ∈ (mzPolyβ€˜β„•){𝑑 ∣ βˆƒπ‘’ ∈ (β„•0 ↑m β„•)(𝑑 = (𝑒 β†Ύ (1...𝑁)) ∧ (π‘ƒβ€˜π‘’) = 0)} = {π‘Ž ∣ βˆƒπ‘ ∈ (β„•0 ↑m β„•)(π‘Ž = (𝑏 β†Ύ (1...𝑁)) ∧ (π‘β€˜π‘) = 0)})
212, 3, 20syl2anc 582 . 2 ((𝑁 ∈ β„•0 ∧ 𝑃 ∈ (mzPolyβ€˜β„•)) β†’ βˆƒπ‘ ∈ (mzPolyβ€˜β„•){𝑑 ∣ βˆƒπ‘’ ∈ (β„•0 ↑m β„•)(𝑑 = (𝑒 β†Ύ (1...𝑁)) ∧ (π‘ƒβ€˜π‘’) = 0)} = {π‘Ž ∣ βˆƒπ‘ ∈ (β„•0 ↑m β„•)(π‘Ž = (𝑏 β†Ύ (1...𝑁)) ∧ (π‘β€˜π‘) = 0)})
22 eldioph3b 41805 . 2 ({𝑑 ∣ βˆƒπ‘’ ∈ (β„•0 ↑m β„•)(𝑑 = (𝑒 β†Ύ (1...𝑁)) ∧ (π‘ƒβ€˜π‘’) = 0)} ∈ (Diophβ€˜π‘) ↔ (𝑁 ∈ β„•0 ∧ βˆƒπ‘ ∈ (mzPolyβ€˜β„•){𝑑 ∣ βˆƒπ‘’ ∈ (β„•0 ↑m β„•)(𝑑 = (𝑒 β†Ύ (1...𝑁)) ∧ (π‘ƒβ€˜π‘’) = 0)} = {π‘Ž ∣ βˆƒπ‘ ∈ (β„•0 ↑m β„•)(π‘Ž = (𝑏 β†Ύ (1...𝑁)) ∧ (π‘β€˜π‘) = 0)}))
231, 21, 22sylanbrc 581 1 ((𝑁 ∈ β„•0 ∧ 𝑃 ∈ (mzPolyβ€˜β„•)) β†’ {𝑑 ∣ βˆƒπ‘’ ∈ (β„•0 ↑m β„•)(𝑑 = (𝑒 β†Ύ (1...𝑁)) ∧ (π‘ƒβ€˜π‘’) = 0)} ∈ (Diophβ€˜π‘))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 394   = wceq 1539   ∈ wcel 2104  {cab 2707  βˆƒwrex 3068   β†Ύ cres 5677  β€˜cfv 6542  (class class class)co 7411   ↑m cmap 8822  0cc0 11112  1c1 11113  β„•cn 12216  β„•0cn0 12476  ...cfz 13488  mzPolycmzp 41762  Diophcdioph 41795
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727  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 395  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-reu 3375  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-int 4950  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-riota 7367  df-ov 7414  df-oprab 7415  df-mpo 7416  df-of 7672  df-om 7858  df-1st 7977  df-2nd 7978  df-frecs 8268  df-wrecs 8299  df-recs 8373  df-rdg 8412  df-1o 8468  df-oadd 8472  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 13489  df-hash 14295  df-mzpcl 41763  df-mzp 41764  df-dioph 41796
This theorem is referenced by:  diophrex  41815
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