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Theorem rabdiophlem2 41843
Description: Lemma for arithmetic diophantine sets. Reuse a polynomial expression under a new quantifier. (Contributed by Stefan O'Rear, 10-Oct-2014.)
Hypothesis
Ref Expression
rabdiophlem2.1 𝑀 = (𝑁 + 1)
Assertion
Ref Expression
rabdiophlem2 ((𝑁 ∈ β„•0 ∧ (𝑒 ∈ (β„€ ↑m (1...𝑁)) ↦ 𝐴) ∈ (mzPolyβ€˜(1...𝑁))) β†’ (𝑑 ∈ (β„€ ↑m (1...𝑀)) ↦ ⦋(𝑑 β†Ύ (1...𝑁)) / π‘’β¦Œπ΄) ∈ (mzPolyβ€˜(1...𝑀)))
Distinct variable groups:   𝑒,𝑁,𝑑   𝑒,𝑀,𝑑   𝑑,𝐴
Allowed substitution hint:   𝐴(𝑒)
Proof of Theorem rabdiophlem2
Dummy variable π‘Ž is distinct from all other variables.
StepHypRef Expression
1 nfcv 2902 . . . . . 6 β„²π‘Žπ΄
2 nfcsb1v 3919 . . . . . 6 β„²π‘’β¦‹π‘Ž / π‘’β¦Œπ΄
3 csbeq1a 3908 . . . . . 6 (𝑒 = π‘Ž β†’ 𝐴 = β¦‹π‘Ž / π‘’β¦Œπ΄)
41, 2, 3cbvmpt 5260 . . . . 5 (𝑒 ∈ (β„€ ↑m (1...𝑁)) ↦ 𝐴) = (π‘Ž ∈ (β„€ ↑m (1...𝑁)) ↦ β¦‹π‘Ž / π‘’β¦Œπ΄)
54fveq1i 6893 . . . 4 ((𝑒 ∈ (β„€ ↑m (1...𝑁)) ↦ 𝐴)β€˜(𝑑 β†Ύ (1...𝑁))) = ((π‘Ž ∈ (β„€ ↑m (1...𝑁)) ↦ β¦‹π‘Ž / π‘’β¦Œπ΄)β€˜(𝑑 β†Ύ (1...𝑁)))
6 eqid 2731 . . . . 5 (π‘Ž ∈ (β„€ ↑m (1...𝑁)) ↦ β¦‹π‘Ž / π‘’β¦Œπ΄) = (π‘Ž ∈ (β„€ ↑m (1...𝑁)) ↦ β¦‹π‘Ž / π‘’β¦Œπ΄)
7 csbeq1 3897 . . . . 5 (π‘Ž = (𝑑 β†Ύ (1...𝑁)) β†’ β¦‹π‘Ž / π‘’β¦Œπ΄ = ⦋(𝑑 β†Ύ (1...𝑁)) / π‘’β¦Œπ΄)
8 rabdiophlem2.1 . . . . . . 7 𝑀 = (𝑁 + 1)
98mapfzcons1cl 41759 . . . . . 6 (𝑑 ∈ (β„€ ↑m (1...𝑀)) β†’ (𝑑 β†Ύ (1...𝑁)) ∈ (β„€ ↑m (1...𝑁)))
109adantl 481 . . . . 5 (((𝑁 ∈ β„•0 ∧ (𝑒 ∈ (β„€ ↑m (1...𝑁)) ↦ 𝐴) ∈ (mzPolyβ€˜(1...𝑁))) ∧ 𝑑 ∈ (β„€ ↑m (1...𝑀))) β†’ (𝑑 β†Ύ (1...𝑁)) ∈ (β„€ ↑m (1...𝑁)))
11 mzpf 41777 . . . . . . . 8 ((𝑒 ∈ (β„€ ↑m (1...𝑁)) ↦ 𝐴) ∈ (mzPolyβ€˜(1...𝑁)) β†’ (𝑒 ∈ (β„€ ↑m (1...𝑁)) ↦ 𝐴):(β„€ ↑m (1...𝑁))βŸΆβ„€)
12 eqid 2731 . . . . . . . . 9 (𝑒 ∈ (β„€ ↑m (1...𝑁)) ↦ 𝐴) = (𝑒 ∈ (β„€ ↑m (1...𝑁)) ↦ 𝐴)
1312fmpt 7112 . . . . . . . 8 (βˆ€π‘’ ∈ (β„€ ↑m (1...𝑁))𝐴 ∈ β„€ ↔ (𝑒 ∈ (β„€ ↑m (1...𝑁)) ↦ 𝐴):(β„€ ↑m (1...𝑁))βŸΆβ„€)
1411, 13sylibr 233 . . . . . . 7 ((𝑒 ∈ (β„€ ↑m (1...𝑁)) ↦ 𝐴) ∈ (mzPolyβ€˜(1...𝑁)) β†’ βˆ€π‘’ ∈ (β„€ ↑m (1...𝑁))𝐴 ∈ β„€)
1514ad2antlr 724 . . . . . 6 (((𝑁 ∈ β„•0 ∧ (𝑒 ∈ (β„€ ↑m (1...𝑁)) ↦ 𝐴) ∈ (mzPolyβ€˜(1...𝑁))) ∧ 𝑑 ∈ (β„€ ↑m (1...𝑀))) β†’ βˆ€π‘’ ∈ (β„€ ↑m (1...𝑁))𝐴 ∈ β„€)
16 nfcsb1v 3919 . . . . . . . 8 Ⅎ𝑒⦋(𝑑 β†Ύ (1...𝑁)) / π‘’β¦Œπ΄
1716nfel1 2918 . . . . . . 7 Ⅎ𝑒⦋(𝑑 β†Ύ (1...𝑁)) / π‘’β¦Œπ΄ ∈ β„€
18 csbeq1a 3908 . . . . . . . 8 (𝑒 = (𝑑 β†Ύ (1...𝑁)) β†’ 𝐴 = ⦋(𝑑 β†Ύ (1...𝑁)) / π‘’β¦Œπ΄)
1918eleq1d 2817 . . . . . . 7 (𝑒 = (𝑑 β†Ύ (1...𝑁)) β†’ (𝐴 ∈ β„€ ↔ ⦋(𝑑 β†Ύ (1...𝑁)) / π‘’β¦Œπ΄ ∈ β„€))
2017, 19rspc 3601 . . . . . 6 ((𝑑 β†Ύ (1...𝑁)) ∈ (β„€ ↑m (1...𝑁)) β†’ (βˆ€π‘’ ∈ (β„€ ↑m (1...𝑁))𝐴 ∈ β„€ β†’ ⦋(𝑑 β†Ύ (1...𝑁)) / π‘’β¦Œπ΄ ∈ β„€))
2110, 15, 20sylc 65 . . . . 5 (((𝑁 ∈ β„•0 ∧ (𝑒 ∈ (β„€ ↑m (1...𝑁)) ↦ 𝐴) ∈ (mzPolyβ€˜(1...𝑁))) ∧ 𝑑 ∈ (β„€ ↑m (1...𝑀))) β†’ ⦋(𝑑 β†Ύ (1...𝑁)) / π‘’β¦Œπ΄ ∈ β„€)
226, 7, 10, 21fvmptd3 7022 . . . 4 (((𝑁 ∈ β„•0 ∧ (𝑒 ∈ (β„€ ↑m (1...𝑁)) ↦ 𝐴) ∈ (mzPolyβ€˜(1...𝑁))) ∧ 𝑑 ∈ (β„€ ↑m (1...𝑀))) β†’ ((π‘Ž ∈ (β„€ ↑m (1...𝑁)) ↦ β¦‹π‘Ž / π‘’β¦Œπ΄)β€˜(𝑑 β†Ύ (1...𝑁))) = ⦋(𝑑 β†Ύ (1...𝑁)) / π‘’β¦Œπ΄)
235, 22eqtr2id 2784 . . 3 (((𝑁 ∈ β„•0 ∧ (𝑒 ∈ (β„€ ↑m (1...𝑁)) ↦ 𝐴) ∈ (mzPolyβ€˜(1...𝑁))) ∧ 𝑑 ∈ (β„€ ↑m (1...𝑀))) β†’ ⦋(𝑑 β†Ύ (1...𝑁)) / π‘’β¦Œπ΄ = ((𝑒 ∈ (β„€ ↑m (1...𝑁)) ↦ 𝐴)β€˜(𝑑 β†Ύ (1...𝑁))))
2423mpteq2dva 5249 . 2 ((𝑁 ∈ β„•0 ∧ (𝑒 ∈ (β„€ ↑m (1...𝑁)) ↦ 𝐴) ∈ (mzPolyβ€˜(1...𝑁))) β†’ (𝑑 ∈ (β„€ ↑m (1...𝑀)) ↦ ⦋(𝑑 β†Ύ (1...𝑁)) / π‘’β¦Œπ΄) = (𝑑 ∈ (β„€ ↑m (1...𝑀)) ↦ ((𝑒 ∈ (β„€ ↑m (1...𝑁)) ↦ 𝐴)β€˜(𝑑 β†Ύ (1...𝑁)))))
25 ovexd 7447 . . 3 ((𝑁 ∈ β„•0 ∧ (𝑒 ∈ (β„€ ↑m (1...𝑁)) ↦ 𝐴) ∈ (mzPolyβ€˜(1...𝑁))) β†’ (1...𝑀) ∈ V)
26 fzssp1 13549 . . . . 5 (1...𝑁) βŠ† (1...(𝑁 + 1))
278oveq2i 7423 . . . . 5 (1...𝑀) = (1...(𝑁 + 1))
2826, 27sseqtrri 4020 . . . 4 (1...𝑁) βŠ† (1...𝑀)
2928a1i 11 . . 3 ((𝑁 ∈ β„•0 ∧ (𝑒 ∈ (β„€ ↑m (1...𝑁)) ↦ 𝐴) ∈ (mzPolyβ€˜(1...𝑁))) β†’ (1...𝑁) βŠ† (1...𝑀))
30 simpr 484 . . 3 ((𝑁 ∈ β„•0 ∧ (𝑒 ∈ (β„€ ↑m (1...𝑁)) ↦ 𝐴) ∈ (mzPolyβ€˜(1...𝑁))) β†’ (𝑒 ∈ (β„€ ↑m (1...𝑁)) ↦ 𝐴) ∈ (mzPolyβ€˜(1...𝑁)))
31 mzpresrename 41791 . . 3 (((1...𝑀) ∈ V ∧ (1...𝑁) βŠ† (1...𝑀) ∧ (𝑒 ∈ (β„€ ↑m (1...𝑁)) ↦ 𝐴) ∈ (mzPolyβ€˜(1...𝑁))) β†’ (𝑑 ∈ (β„€ ↑m (1...𝑀)) ↦ ((𝑒 ∈ (β„€ ↑m (1...𝑁)) ↦ 𝐴)β€˜(𝑑 β†Ύ (1...𝑁)))) ∈ (mzPolyβ€˜(1...𝑀)))
3225, 29, 30, 31syl3anc 1370 . 2 ((𝑁 ∈ β„•0 ∧ (𝑒 ∈ (β„€ ↑m (1...𝑁)) ↦ 𝐴) ∈ (mzPolyβ€˜(1...𝑁))) β†’ (𝑑 ∈ (β„€ ↑m (1...𝑀)) ↦ ((𝑒 ∈ (β„€ ↑m (1...𝑁)) ↦ 𝐴)β€˜(𝑑 β†Ύ (1...𝑁)))) ∈ (mzPolyβ€˜(1...𝑀)))
3324, 32eqeltrd 2832 1 ((𝑁 ∈ β„•0 ∧ (𝑒 ∈ (β„€ ↑m (1...𝑁)) ↦ 𝐴) ∈ (mzPolyβ€˜(1...𝑁))) β†’ (𝑑 ∈ (β„€ ↑m (1...𝑀)) ↦ ⦋(𝑑 β†Ύ (1...𝑁)) / π‘’β¦Œπ΄) ∈ (mzPolyβ€˜(1...𝑀)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2105  βˆ€wral 3060  Vcvv 3473  β¦‹csb 3894   βŠ† wss 3949   ↦ cmpt 5232   β†Ύ cres 5679  βŸΆwf 6540  β€˜cfv 6544  (class class class)co 7412   ↑m cmap 8823  1c1 11114   + caddc 11116  β„•0cn0 12477  β„€cz 12563  ...cfz 13489  mzPolycmzp 41763
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7728  ax-cnex 11169  ax-resscn 11170  ax-1cn 11171  ax-icn 11172  ax-addcl 11173  ax-addrcl 11174  ax-mulcl 11175  ax-mulrcl 11176  ax-mulcom 11177  ax-addass 11178  ax-mulass 11179  ax-distr 11180  ax-i2m1 11181  ax-1ne0 11182  ax-1rid 11183  ax-rnegex 11184  ax-rrecex 11185  ax-cnre 11186  ax-pre-lttri 11187  ax-pre-lttrn 11188  ax-pre-ltadd 11189  ax-pre-mulgt0 11190
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-reu 3376  df-rab 3432  df-v 3475  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 7368  df-ov 7415  df-oprab 7416  df-mpo 7417  df-of 7673  df-om 7859  df-1st 7978  df-2nd 7979  df-frecs 8269  df-wrecs 8300  df-recs 8374  df-rdg 8413  df-er 8706  df-map 8825  df-en 8943  df-dom 8944  df-sdom 8945  df-pnf 11255  df-mnf 11256  df-xr 11257  df-ltxr 11258  df-le 11259  df-sub 11451  df-neg 11452  df-nn 12218  df-n0 12478  df-z 12564  df-uz 12828  df-fz 13490  df-mzpcl 41764  df-mzp 41765
This theorem is referenced by:  elnn0rabdioph  41844  dvdsrabdioph  41851
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