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Theorem rabdiophlem2 36843
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 ∈ ℕ0 ∧ (𝑢 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁))) → (𝑡 ∈ (ℤ ↑𝑚 (1...𝑀)) ↦ (𝑡 ↾ (1...𝑁)) / 𝑢𝐴) ∈ (mzPoly‘(1...𝑀)))
Distinct variable groups:   𝑢,𝑁,𝑡   𝑢,𝑀,𝑡   𝑡,𝐴
Allowed substitution hint:   𝐴(𝑢)

Proof of Theorem rabdiophlem2
Dummy variable 𝑎 is distinct from all other variables.
StepHypRef Expression
1 nfcv 2761 . . . . . 6 𝑎𝐴
2 nfcsb1v 3530 . . . . . 6 𝑢𝑎 / 𝑢𝐴
3 csbeq1a 3523 . . . . . 6 (𝑢 = 𝑎𝐴 = 𝑎 / 𝑢𝐴)
41, 2, 3cbvmpt 4709 . . . . 5 (𝑢 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝐴) = (𝑎 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝑎 / 𝑢𝐴)
54fveq1i 6149 . . . 4 ((𝑢 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝐴)‘(𝑡 ↾ (1...𝑁))) = ((𝑎 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝑎 / 𝑢𝐴)‘(𝑡 ↾ (1...𝑁)))
6 rabdiophlem2.1 . . . . . . 7 𝑀 = (𝑁 + 1)
76mapfzcons1cl 36758 . . . . . 6 (𝑡 ∈ (ℤ ↑𝑚 (1...𝑀)) → (𝑡 ↾ (1...𝑁)) ∈ (ℤ ↑𝑚 (1...𝑁)))
87adantl 482 . . . . 5 (((𝑁 ∈ ℕ0 ∧ (𝑢 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁))) ∧ 𝑡 ∈ (ℤ ↑𝑚 (1...𝑀))) → (𝑡 ↾ (1...𝑁)) ∈ (ℤ ↑𝑚 (1...𝑁)))
9 mzpf 36776 . . . . . . . 8 ((𝑢 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁)) → (𝑢 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝐴):(ℤ ↑𝑚 (1...𝑁))⟶ℤ)
10 eqid 2621 . . . . . . . . 9 (𝑢 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝐴) = (𝑢 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝐴)
1110fmpt 6337 . . . . . . . 8 (∀𝑢 ∈ (ℤ ↑𝑚 (1...𝑁))𝐴 ∈ ℤ ↔ (𝑢 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝐴):(ℤ ↑𝑚 (1...𝑁))⟶ℤ)
129, 11sylibr 224 . . . . . . 7 ((𝑢 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁)) → ∀𝑢 ∈ (ℤ ↑𝑚 (1...𝑁))𝐴 ∈ ℤ)
1312ad2antlr 762 . . . . . 6 (((𝑁 ∈ ℕ0 ∧ (𝑢 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁))) ∧ 𝑡 ∈ (ℤ ↑𝑚 (1...𝑀))) → ∀𝑢 ∈ (ℤ ↑𝑚 (1...𝑁))𝐴 ∈ ℤ)
14 nfcsb1v 3530 . . . . . . . 8 𝑢(𝑡 ↾ (1...𝑁)) / 𝑢𝐴
1514nfel1 2775 . . . . . . 7 𝑢(𝑡 ↾ (1...𝑁)) / 𝑢𝐴 ∈ ℤ
16 csbeq1a 3523 . . . . . . . 8 (𝑢 = (𝑡 ↾ (1...𝑁)) → 𝐴 = (𝑡 ↾ (1...𝑁)) / 𝑢𝐴)
1716eleq1d 2683 . . . . . . 7 (𝑢 = (𝑡 ↾ (1...𝑁)) → (𝐴 ∈ ℤ ↔ (𝑡 ↾ (1...𝑁)) / 𝑢𝐴 ∈ ℤ))
1815, 17rspc 3289 . . . . . 6 ((𝑡 ↾ (1...𝑁)) ∈ (ℤ ↑𝑚 (1...𝑁)) → (∀𝑢 ∈ (ℤ ↑𝑚 (1...𝑁))𝐴 ∈ ℤ → (𝑡 ↾ (1...𝑁)) / 𝑢𝐴 ∈ ℤ))
198, 13, 18sylc 65 . . . . 5 (((𝑁 ∈ ℕ0 ∧ (𝑢 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁))) ∧ 𝑡 ∈ (ℤ ↑𝑚 (1...𝑀))) → (𝑡 ↾ (1...𝑁)) / 𝑢𝐴 ∈ ℤ)
20 csbeq1 3517 . . . . . 6 (𝑎 = (𝑡 ↾ (1...𝑁)) → 𝑎 / 𝑢𝐴 = (𝑡 ↾ (1...𝑁)) / 𝑢𝐴)
21 eqid 2621 . . . . . 6 (𝑎 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝑎 / 𝑢𝐴) = (𝑎 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝑎 / 𝑢𝐴)
2220, 21fvmptg 6237 . . . . 5 (((𝑡 ↾ (1...𝑁)) ∈ (ℤ ↑𝑚 (1...𝑁)) ∧ (𝑡 ↾ (1...𝑁)) / 𝑢𝐴 ∈ ℤ) → ((𝑎 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝑎 / 𝑢𝐴)‘(𝑡 ↾ (1...𝑁))) = (𝑡 ↾ (1...𝑁)) / 𝑢𝐴)
238, 19, 22syl2anc 692 . . . 4 (((𝑁 ∈ ℕ0 ∧ (𝑢 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁))) ∧ 𝑡 ∈ (ℤ ↑𝑚 (1...𝑀))) → ((𝑎 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝑎 / 𝑢𝐴)‘(𝑡 ↾ (1...𝑁))) = (𝑡 ↾ (1...𝑁)) / 𝑢𝐴)
245, 23syl5req 2668 . . 3 (((𝑁 ∈ ℕ0 ∧ (𝑢 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁))) ∧ 𝑡 ∈ (ℤ ↑𝑚 (1...𝑀))) → (𝑡 ↾ (1...𝑁)) / 𝑢𝐴 = ((𝑢 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝐴)‘(𝑡 ↾ (1...𝑁))))
2524mpteq2dva 4704 . 2 ((𝑁 ∈ ℕ0 ∧ (𝑢 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁))) → (𝑡 ∈ (ℤ ↑𝑚 (1...𝑀)) ↦ (𝑡 ↾ (1...𝑁)) / 𝑢𝐴) = (𝑡 ∈ (ℤ ↑𝑚 (1...𝑀)) ↦ ((𝑢 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝐴)‘(𝑡 ↾ (1...𝑁)))))
26 ovex 6632 . . . 4 (1...𝑀) ∈ V
2726a1i 11 . . 3 ((𝑁 ∈ ℕ0 ∧ (𝑢 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁))) → (1...𝑀) ∈ V)
28 fzssp1 12326 . . . . 5 (1...𝑁) ⊆ (1...(𝑁 + 1))
296oveq2i 6615 . . . . 5 (1...𝑀) = (1...(𝑁 + 1))
3028, 29sseqtr4i 3617 . . . 4 (1...𝑁) ⊆ (1...𝑀)
3130a1i 11 . . 3 ((𝑁 ∈ ℕ0 ∧ (𝑢 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁))) → (1...𝑁) ⊆ (1...𝑀))
32 simpr 477 . . 3 ((𝑁 ∈ ℕ0 ∧ (𝑢 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁))) → (𝑢 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁)))
33 mzpresrename 36790 . . 3 (((1...𝑀) ∈ V ∧ (1...𝑁) ⊆ (1...𝑀) ∧ (𝑢 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁))) → (𝑡 ∈ (ℤ ↑𝑚 (1...𝑀)) ↦ ((𝑢 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝐴)‘(𝑡 ↾ (1...𝑁)))) ∈ (mzPoly‘(1...𝑀)))
3427, 31, 32, 33syl3anc 1323 . 2 ((𝑁 ∈ ℕ0 ∧ (𝑢 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁))) → (𝑡 ∈ (ℤ ↑𝑚 (1...𝑀)) ↦ ((𝑢 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝐴)‘(𝑡 ↾ (1...𝑁)))) ∈ (mzPoly‘(1...𝑀)))
3525, 34eqeltrd 2698 1 ((𝑁 ∈ ℕ0 ∧ (𝑢 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁))) → (𝑡 ∈ (ℤ ↑𝑚 (1...𝑀)) ↦ (𝑡 ↾ (1...𝑁)) / 𝑢𝐴) ∈ (mzPoly‘(1...𝑀)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wcel 1987  wral 2907  Vcvv 3186  csb 3514  wss 3555  cmpt 4673  cres 5076  wf 5843  cfv 5847  (class class class)co 6604  𝑚 cmap 7802  1c1 9881   + caddc 9883  0cn0 11236  cz 11321  ...cfz 12268  mzPolycmzp 36762
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902  ax-cnex 9936  ax-resscn 9937  ax-1cn 9938  ax-icn 9939  ax-addcl 9940  ax-addrcl 9941  ax-mulcl 9942  ax-mulrcl 9943  ax-mulcom 9944  ax-addass 9945  ax-mulass 9946  ax-distr 9947  ax-i2m1 9948  ax-1ne0 9949  ax-1rid 9950  ax-rnegex 9951  ax-rrecex 9952  ax-cnre 9953  ax-pre-lttri 9954  ax-pre-lttrn 9955  ax-pre-ltadd 9956  ax-pre-mulgt0 9957
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-int 4441  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-riota 6565  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-of 6850  df-om 7013  df-1st 7113  df-2nd 7114  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-er 7687  df-map 7804  df-en 7900  df-dom 7901  df-sdom 7902  df-pnf 10020  df-mnf 10021  df-xr 10022  df-ltxr 10023  df-le 10024  df-sub 10212  df-neg 10213  df-nn 10965  df-n0 11237  df-z 11322  df-uz 11632  df-fz 12269  df-mzpcl 36763  df-mzp 36764
This theorem is referenced by:  elnn0rabdioph  36844  dvdsrabdioph  36851
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