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Theorem rabdiophlem2 42813
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 2905 . . . . . 6 𝑎𝐴
2 nfcsb1v 3923 . . . . . 6 𝑢𝑎 / 𝑢𝐴
3 csbeq1a 3913 . . . . . 6 (𝑢 = 𝑎𝐴 = 𝑎 / 𝑢𝐴)
41, 2, 3cbvmpt 5253 . . . . 5 (𝑢 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴) = (𝑎 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝑎 / 𝑢𝐴)
54fveq1i 6907 . . . 4 ((𝑢 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴)‘(𝑡 ↾ (1...𝑁))) = ((𝑎 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝑎 / 𝑢𝐴)‘(𝑡 ↾ (1...𝑁)))
6 eqid 2737 . . . . 5 (𝑎 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝑎 / 𝑢𝐴) = (𝑎 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝑎 / 𝑢𝐴)
7 csbeq1 3902 . . . . 5 (𝑎 = (𝑡 ↾ (1...𝑁)) → 𝑎 / 𝑢𝐴 = (𝑡 ↾ (1...𝑁)) / 𝑢𝐴)
8 rabdiophlem2.1 . . . . . . 7 𝑀 = (𝑁 + 1)
98mapfzcons1cl 42729 . . . . . 6 (𝑡 ∈ (ℤ ↑m (1...𝑀)) → (𝑡 ↾ (1...𝑁)) ∈ (ℤ ↑m (1...𝑁)))
109adantl 481 . . . . 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 42747 . . . . . . . 8 ((𝑢 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁)) → (𝑢 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴):(ℤ ↑m (1...𝑁))⟶ℤ)
12 eqid 2737 . . . . . . . . 9 (𝑢 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴) = (𝑢 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴)
1312fmpt 7130 . . . . . . . 8 (∀𝑢 ∈ (ℤ ↑m (1...𝑁))𝐴 ∈ ℤ ↔ (𝑢 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴):(ℤ ↑m (1...𝑁))⟶ℤ)
1411, 13sylibr 234 . . . . . . 7 ((𝑢 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁)) → ∀𝑢 ∈ (ℤ ↑m (1...𝑁))𝐴 ∈ ℤ)
1514ad2antlr 727 . . . . . 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 3923 . . . . . . . 8 𝑢(𝑡 ↾ (1...𝑁)) / 𝑢𝐴
1716nfel1 2922 . . . . . . 7 𝑢(𝑡 ↾ (1...𝑁)) / 𝑢𝐴 ∈ ℤ
18 csbeq1a 3913 . . . . . . . 8 (𝑢 = (𝑡 ↾ (1...𝑁)) → 𝐴 = (𝑡 ↾ (1...𝑁)) / 𝑢𝐴)
1918eleq1d 2826 . . . . . . 7 (𝑢 = (𝑡 ↾ (1...𝑁)) → (𝐴 ∈ ℤ ↔ (𝑡 ↾ (1...𝑁)) / 𝑢𝐴 ∈ ℤ))
2017, 19rspc 3610 . . . . . 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 7039 . . . 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, 22eqtr2id 2790 . . 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 5242 . 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 7466 . . 3 ((𝑁 ∈ ℕ0 ∧ (𝑢 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁))) → (1...𝑀) ∈ V)
26 fzssp1 13607 . . . . 5 (1...𝑁) ⊆ (1...(𝑁 + 1))
278oveq2i 7442 . . . . 5 (1...𝑀) = (1...(𝑁 + 1))
2826, 27sseqtrri 4033 . . . 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 484 . . 3 ((𝑁 ∈ ℕ0 ∧ (𝑢 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁))) → (𝑢 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁)))
31 mzpresrename 42761 . . 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 1373 . 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 2841 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 395   = wceq 1540  wcel 2108  wral 3061  Vcvv 3480  csb 3899  wss 3951  cmpt 5225  cres 5687  wf 6557  cfv 6561  (class class class)co 7431  m cmap 8866  1c1 11156   + caddc 11158  0cn0 12526  cz 12613  ...cfz 13547  mzPolycmzp 42733
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 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-of 7697  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-er 8745  df-map 8868  df-en 8986  df-dom 8987  df-sdom 8988  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-nn 12267  df-n0 12527  df-z 12614  df-uz 12879  df-fz 13548  df-mzpcl 42734  df-mzp 42735
This theorem is referenced by:  elnn0rabdioph  42814  dvdsrabdioph  42821
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