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Theorem rabdiophlem2 39277
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 2974 . . . . . 6 𝑎𝐴
2 nfcsb1v 3904 . . . . . 6 𝑢𝑎 / 𝑢𝐴
3 csbeq1a 3894 . . . . . 6 (𝑢 = 𝑎𝐴 = 𝑎 / 𝑢𝐴)
41, 2, 3cbvmpt 5158 . . . . 5 (𝑢 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴) = (𝑎 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝑎 / 𝑢𝐴)
54fveq1i 6664 . . . 4 ((𝑢 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴)‘(𝑡 ↾ (1...𝑁))) = ((𝑎 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝑎 / 𝑢𝐴)‘(𝑡 ↾ (1...𝑁)))
6 eqid 2818 . . . . 5 (𝑎 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝑎 / 𝑢𝐴) = (𝑎 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝑎 / 𝑢𝐴)
7 csbeq1 3883 . . . . 5 (𝑎 = (𝑡 ↾ (1...𝑁)) → 𝑎 / 𝑢𝐴 = (𝑡 ↾ (1...𝑁)) / 𝑢𝐴)
8 rabdiophlem2.1 . . . . . . 7 𝑀 = (𝑁 + 1)
98mapfzcons1cl 39193 . . . . . 6 (𝑡 ∈ (ℤ ↑m (1...𝑀)) → (𝑡 ↾ (1...𝑁)) ∈ (ℤ ↑m (1...𝑁)))
109adantl 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)))
11 mzpf 39211 . . . . . . . 8 ((𝑢 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁)) → (𝑢 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴):(ℤ ↑m (1...𝑁))⟶ℤ)
12 eqid 2818 . . . . . . . . 9 (𝑢 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴) = (𝑢 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴)
1312fmpt 6866 . . . . . . . 8 (∀𝑢 ∈ (ℤ ↑m (1...𝑁))𝐴 ∈ ℤ ↔ (𝑢 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴):(ℤ ↑m (1...𝑁))⟶ℤ)
1411, 13sylibr 235 . . . . . . 7 ((𝑢 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁)) → ∀𝑢 ∈ (ℤ ↑m (1...𝑁))𝐴 ∈ ℤ)
1514ad2antlr 723 . . . . . 6 (((𝑁 ∈ ℕ0 ∧ (𝑢 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁))) ∧ 𝑡 ∈ (ℤ ↑m (1...𝑀))) → ∀𝑢 ∈ (ℤ ↑m (1...𝑁))𝐴 ∈ ℤ)
16 nfcsb1v 3904 . . . . . . . 8 𝑢(𝑡 ↾ (1...𝑁)) / 𝑢𝐴
1716nfel1 2991 . . . . . . 7 𝑢(𝑡 ↾ (1...𝑁)) / 𝑢𝐴 ∈ ℤ
18 csbeq1a 3894 . . . . . . . 8 (𝑢 = (𝑡 ↾ (1...𝑁)) → 𝐴 = (𝑡 ↾ (1...𝑁)) / 𝑢𝐴)
1918eleq1d 2894 . . . . . . 7 (𝑢 = (𝑡 ↾ (1...𝑁)) → (𝐴 ∈ ℤ ↔ (𝑡 ↾ (1...𝑁)) / 𝑢𝐴 ∈ ℤ))
2017, 19rspc 3608 . . . . . 6 ((𝑡 ↾ (1...𝑁)) ∈ (ℤ ↑m (1...𝑁)) → (∀𝑢 ∈ (ℤ ↑m (1...𝑁))𝐴 ∈ ℤ → (𝑡 ↾ (1...𝑁)) / 𝑢𝐴 ∈ ℤ))
2110, 15, 20sylc 65 . . . . 5 (((𝑁 ∈ ℕ0 ∧ (𝑢 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁))) ∧ 𝑡 ∈ (ℤ ↑m (1...𝑀))) → (𝑡 ↾ (1...𝑁)) / 𝑢𝐴 ∈ ℤ)
226, 7, 10, 21fvmptd3 6783 . . . 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)
235, 22syl5req 2866 . . 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))))
2423mpteq2dva 5152 . 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)))))
25 ovexd 7180 . . 3 ((𝑁 ∈ ℕ0 ∧ (𝑢 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁))) → (1...𝑀) ∈ V)
26 fzssp1 12938 . . . . 5 (1...𝑁) ⊆ (1...(𝑁 + 1))
278oveq2i 7156 . . . . 5 (1...𝑀) = (1...(𝑁 + 1))
2826, 27sseqtrri 4001 . . . 4 (1...𝑁) ⊆ (1...𝑀)
2928a1i 11 . . 3 ((𝑁 ∈ ℕ0 ∧ (𝑢 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁))) → (1...𝑁) ⊆ (1...𝑀))
30 simpr 485 . . 3 ((𝑁 ∈ ℕ0 ∧ (𝑢 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁))) → (𝑢 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁)))
31 mzpresrename 39225 . . 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)))
3225, 29, 30, 31syl3anc 1363 . 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)))
3324, 32eqeltrd 2910 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 396   = wceq 1528  wcel 2105  wral 3135  Vcvv 3492  csb 3880  wss 3933  cmpt 5137  cres 5550  wf 6344  cfv 6348  (class class class)co 7145  m cmap 8395  1c1 10526   + caddc 10528  0cn0 11885  cz 11969  ...cfz 12880  mzPolycmzp 39197
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450  ax-cnex 10581  ax-resscn 10582  ax-1cn 10583  ax-icn 10584  ax-addcl 10585  ax-addrcl 10586  ax-mulcl 10587  ax-mulrcl 10588  ax-mulcom 10589  ax-addass 10590  ax-mulass 10591  ax-distr 10592  ax-i2m1 10593  ax-1ne0 10594  ax-1rid 10595  ax-rnegex 10596  ax-rrecex 10597  ax-cnre 10598  ax-pre-lttri 10599  ax-pre-lttrn 10600  ax-pre-ltadd 10601  ax-pre-mulgt0 10602
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-nel 3121  df-ral 3140  df-rex 3141  df-reu 3142  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-int 4868  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-of 7398  df-om 7570  df-1st 7678  df-2nd 7679  df-wrecs 7936  df-recs 7997  df-rdg 8035  df-er 8278  df-map 8397  df-en 8498  df-dom 8499  df-sdom 8500  df-pnf 10665  df-mnf 10666  df-xr 10667  df-ltxr 10668  df-le 10669  df-sub 10860  df-neg 10861  df-nn 11627  df-n0 11886  df-z 11970  df-uz 12232  df-fz 12881  df-mzpcl 39198  df-mzp 39199
This theorem is referenced by:  elnn0rabdioph  39278  dvdsrabdioph  39285
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