Users' Mathboxes Mathbox for Stefan O'Rear < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  rabdiophlem2 Structured version   Visualization version   GIF version

Theorem rabdiophlem2 38044
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 2907 . . . . . 6 𝑎𝐴
2 nfcsb1v 3707 . . . . . 6 𝑢𝑎 / 𝑢𝐴
3 csbeq1a 3700 . . . . . 6 (𝑢 = 𝑎𝐴 = 𝑎 / 𝑢𝐴)
41, 2, 3cbvmpt 4908 . . . . 5 (𝑢 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝐴) = (𝑎 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝑎 / 𝑢𝐴)
54fveq1i 6376 . . . 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 37959 . . . . . 6 (𝑡 ∈ (ℤ ↑𝑚 (1...𝑀)) → (𝑡 ↾ (1...𝑁)) ∈ (ℤ ↑𝑚 (1...𝑁)))
87adantl 473 . . . . 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 37977 . . . . . . . 8 ((𝑢 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁)) → (𝑢 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝐴):(ℤ ↑𝑚 (1...𝑁))⟶ℤ)
10 eqid 2765 . . . . . . . . 9 (𝑢 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝐴) = (𝑢 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝐴)
1110fmpt 6570 . . . . . . . 8 (∀𝑢 ∈ (ℤ ↑𝑚 (1...𝑁))𝐴 ∈ ℤ ↔ (𝑢 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝐴):(ℤ ↑𝑚 (1...𝑁))⟶ℤ)
129, 11sylibr 225 . . . . . . 7 ((𝑢 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁)) → ∀𝑢 ∈ (ℤ ↑𝑚 (1...𝑁))𝐴 ∈ ℤ)
1312ad2antlr 718 . . . . . 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 3707 . . . . . . . 8 𝑢(𝑡 ↾ (1...𝑁)) / 𝑢𝐴
1514nfel1 2922 . . . . . . 7 𝑢(𝑡 ↾ (1...𝑁)) / 𝑢𝐴 ∈ ℤ
16 csbeq1a 3700 . . . . . . . 8 (𝑢 = (𝑡 ↾ (1...𝑁)) → 𝐴 = (𝑡 ↾ (1...𝑁)) / 𝑢𝐴)
1716eleq1d 2829 . . . . . . 7 (𝑢 = (𝑡 ↾ (1...𝑁)) → (𝐴 ∈ ℤ ↔ (𝑡 ↾ (1...𝑁)) / 𝑢𝐴 ∈ ℤ))
1815, 17rspc 3455 . . . . . 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 3694 . . . . . 6 (𝑎 = (𝑡 ↾ (1...𝑁)) → 𝑎 / 𝑢𝐴 = (𝑡 ↾ (1...𝑁)) / 𝑢𝐴)
21 eqid 2765 . . . . . 6 (𝑎 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝑎 / 𝑢𝐴) = (𝑎 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝑎 / 𝑢𝐴)
2220, 21fvmptg 6469 . . . . 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 579 . . . 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 2812 . . 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 4903 . 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 ovexd 6876 . . 3 ((𝑁 ∈ ℕ0 ∧ (𝑢 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁))) → (1...𝑀) ∈ V)
27 fzssp1 12591 . . . . 5 (1...𝑁) ⊆ (1...(𝑁 + 1))
286oveq2i 6853 . . . . 5 (1...𝑀) = (1...(𝑁 + 1))
2927, 28sseqtr4i 3798 . . . 4 (1...𝑁) ⊆ (1...𝑀)
3029a1i 11 . . 3 ((𝑁 ∈ ℕ0 ∧ (𝑢 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁))) → (1...𝑁) ⊆ (1...𝑀))
31 simpr 477 . . 3 ((𝑁 ∈ ℕ0 ∧ (𝑢 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁))) → (𝑢 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁)))
32 mzpresrename 37991 . . 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)))
3326, 30, 31, 32syl3anc 1490 . 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)))
3425, 33eqeltrd 2844 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 1652  wcel 2155  wral 3055  Vcvv 3350  csb 3691  wss 3732  cmpt 4888  cres 5279  wf 6064  cfv 6068  (class class class)co 6842  𝑚 cmap 8060  1c1 10190   + caddc 10192  0cn0 11538  cz 11624  ...cfz 12533  mzPolycmzp 37963
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4930  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147  ax-cnex 10245  ax-resscn 10246  ax-1cn 10247  ax-icn 10248  ax-addcl 10249  ax-addrcl 10250  ax-mulcl 10251  ax-mulrcl 10252  ax-mulcom 10253  ax-addass 10254  ax-mulass 10255  ax-distr 10256  ax-i2m1 10257  ax-1ne0 10258  ax-1rid 10259  ax-rnegex 10260  ax-rrecex 10261  ax-cnre 10262  ax-pre-lttri 10263  ax-pre-lttrn 10264  ax-pre-ltadd 10265  ax-pre-mulgt0 10266
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-nel 3041  df-ral 3060  df-rex 3061  df-reu 3062  df-rab 3064  df-v 3352  df-sbc 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-pss 3748  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-tp 4339  df-op 4341  df-uni 4595  df-int 4634  df-iun 4678  df-br 4810  df-opab 4872  df-mpt 4889  df-tr 4912  df-id 5185  df-eprel 5190  df-po 5198  df-so 5199  df-fr 5236  df-we 5238  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-pred 5865  df-ord 5911  df-on 5912  df-lim 5913  df-suc 5914  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-f1 6073  df-fo 6074  df-f1o 6075  df-fv 6076  df-riota 6803  df-ov 6845  df-oprab 6846  df-mpt2 6847  df-of 7095  df-om 7264  df-1st 7366  df-2nd 7367  df-wrecs 7610  df-recs 7672  df-rdg 7710  df-er 7947  df-map 8062  df-en 8161  df-dom 8162  df-sdom 8163  df-pnf 10330  df-mnf 10331  df-xr 10332  df-ltxr 10333  df-le 10334  df-sub 10522  df-neg 10523  df-nn 11275  df-n0 11539  df-z 11625  df-uz 11887  df-fz 12534  df-mzpcl 37964  df-mzp 37965
This theorem is referenced by:  elnn0rabdioph  38045  dvdsrabdioph  38052
  Copyright terms: Public domain W3C validator