Theorem rabdiophlem2 40624

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)) / 𝑢⦌𝐴) ∈ (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.

Step	Hyp	Ref	Expression
1		nfcv 2907	. . . . . 6 ⊢ Ⅎ𝑎𝐴
2		nfcsb1v 3857	. . . . . 6 ⊢ Ⅎ𝑢⦋𝑎 / 𝑢⦌𝐴
3		csbeq1a 3846	. . . . . 6 ⊢ (𝑢 = 𝑎 → 𝐴 = ⦋𝑎 / 𝑢⦌𝐴)
4	1, 2, 3	cbvmpt 5185	. . . . 5 ⊢ (𝑢 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴) = (𝑎 ∈ (ℤ ↑m (1...𝑁)) ↦ ⦋𝑎 / 𝑢⦌𝐴)
5	4	fveq1i 6775	. . . 4 ⊢ ((𝑢 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴)‘(𝑡 ↾ (1...𝑁))) = ((𝑎 ∈ (ℤ ↑m (1...𝑁)) ↦ ⦋𝑎 / 𝑢⦌𝐴)‘(𝑡 ↾ (1...𝑁)))
6		eqid 2738	. . . . 5 ⊢ (𝑎 ∈ (ℤ ↑m (1...𝑁)) ↦ ⦋𝑎 / 𝑢⦌𝐴) = (𝑎 ∈ (ℤ ↑m (1...𝑁)) ↦ ⦋𝑎 / 𝑢⦌𝐴)
7		csbeq1 3835	. . . . 5 ⊢ (𝑎 = (𝑡 ↾ (1...𝑁)) → ⦋𝑎 / 𝑢⦌𝐴 = ⦋(𝑡 ↾ (1...𝑁)) / 𝑢⦌𝐴)
8		rabdiophlem2.1	. . . . . . 7 ⊢ 𝑀 = (𝑁 + 1)
9	8	mapfzcons1cl 40540	. . . . . 6 ⊢ (𝑡 ∈ (ℤ ↑m (1...𝑀)) → (𝑡 ↾ (1...𝑁)) ∈ (ℤ ↑m (1...𝑁)))
10	9	adantl 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 40558	. . . . . . . 8 ⊢ ((𝑢 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁)) → (𝑢 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴):(ℤ ↑m (1...𝑁))⟶ℤ)
12		eqid 2738	. . . . . . . . 9 ⊢ (𝑢 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴) = (𝑢 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴)
13	12	fmpt 6984	. . . . . . . 8 ⊢ (∀𝑢 ∈ (ℤ ↑m (1...𝑁))𝐴 ∈ ℤ ↔ (𝑢 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴):(ℤ ↑m (1...𝑁))⟶ℤ)
14	11, 13	sylibr 233	. . . . . . 7 ⊢ ((𝑢 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁)) → ∀𝑢 ∈ (ℤ ↑m (1...𝑁))𝐴 ∈ ℤ)
15	14	ad2antlr 724	. . . . . 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 3857	. . . . . . . 8 ⊢ Ⅎ𝑢⦋(𝑡 ↾ (1...𝑁)) / 𝑢⦌𝐴
17	16	nfel1 2923	. . . . . . 7 ⊢ Ⅎ𝑢⦋(𝑡 ↾ (1...𝑁)) / 𝑢⦌𝐴 ∈ ℤ
18		csbeq1a 3846	. . . . . . . 8 ⊢ (𝑢 = (𝑡 ↾ (1...𝑁)) → 𝐴 = ⦋(𝑡 ↾ (1...𝑁)) / 𝑢⦌𝐴)
19	18	eleq1d 2823	. . . . . . 7 ⊢ (𝑢 = (𝑡 ↾ (1...𝑁)) → (𝐴 ∈ ℤ ↔ ⦋(𝑡 ↾ (1...𝑁)) / 𝑢⦌𝐴 ∈ ℤ))
20	17, 19	rspc 3549	. . . . . 6 ⊢ ((𝑡 ↾ (1...𝑁)) ∈ (ℤ ↑m (1...𝑁)) → (∀𝑢 ∈ (ℤ ↑m (1...𝑁))𝐴 ∈ ℤ → ⦋(𝑡 ↾ (1...𝑁)) / 𝑢⦌𝐴 ∈ ℤ))
21	10, 15, 20	sylc 65	. . . . 5 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑢 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁))) ∧ 𝑡 ∈ (ℤ ↑m (1...𝑀))) → ⦋(𝑡 ↾ (1...𝑁)) / 𝑢⦌𝐴 ∈ ℤ)
22	6, 7, 10, 21	fvmptd3 6898	. . . 4 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑢 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁))) ∧ 𝑡 ∈ (ℤ ↑m (1...𝑀))) → ((𝑎 ∈ (ℤ ↑m (1...𝑁)) ↦ ⦋𝑎 / 𝑢⦌𝐴)‘(𝑡 ↾ (1...𝑁))) = ⦋(𝑡 ↾ (1...𝑁)) / 𝑢⦌𝐴)
23	5, 22	eqtr2id 2791	. . 3 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑢 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁))) ∧ 𝑡 ∈ (ℤ ↑m (1...𝑀))) → ⦋(𝑡 ↾ (1...𝑁)) / 𝑢⦌𝐴 = ((𝑢 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴)‘(𝑡 ↾ (1...𝑁))))
24	23	mpteq2dva 5174	. 2 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑢 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁))) → (𝑡 ∈ (ℤ ↑m (1...𝑀)) ↦ ⦋(𝑡 ↾ (1...𝑁)) / 𝑢⦌𝐴) = (𝑡 ∈ (ℤ ↑m (1...𝑀)) ↦ ((𝑢 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴)‘(𝑡 ↾ (1...𝑁)))))
25		ovexd 7310	. . 3 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑢 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁))) → (1...𝑀) ∈ V)
26		fzssp1 13299	. . . . 5 ⊢ (1...𝑁) ⊆ (1...(𝑁 + 1))
27	8	oveq2i 7286	. . . . 5 ⊢ (1...𝑀) = (1...(𝑁 + 1))
28	26, 27	sseqtrri 3958	. . . 4 ⊢ (1...𝑁) ⊆ (1...𝑀)
29	28	a1i 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 40572	. . 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)))
32	25, 29, 30, 31	syl3anc 1370	. 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)))
33	24, 32	eqeltrd 2839	1 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑢 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁))) → (𝑡 ∈ (ℤ ↑m (1...𝑀)) ↦ ⦋(𝑡 ↾ (1...𝑁)) / 𝑢⦌𝐴) ∈ (mzPoly‘(1...𝑀)))

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1539   ∈ wcel 2106  ∀wral 3064  Vcvv 3432  ⦋csb 3832   ⊆ wss 3887   ↦ cmpt 5157   ↾ cres 5591  ⟶wf 6429  ‘cfv 6433  (class class class)co 7275   ↑_m cmap 8615  1c1 10872   + caddc 10874  ℕ₀cn0 12233  ℤcz 12319  ...cfz 13239  mzPolycmzp 40544

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588  ax-cnex 10927  ax-resscn 10928  ax-1cn 10929  ax-icn 10930  ax-addcl 10931  ax-addrcl 10932  ax-mulcl 10933  ax-mulrcl 10934  ax-mulcom 10935  ax-addass 10936  ax-mulass 10937  ax-distr 10938  ax-i2m1 10939  ax-1ne0 10940  ax-1rid 10941  ax-rnegex 10942  ax-rrecex 10943  ax-cnre 10944  ax-pre-lttri 10945  ax-pre-lttrn 10946  ax-pre-ltadd 10947  ax-pre-mulgt0 10948

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-int 4880  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-riota 7232  df-ov 7278  df-oprab 7279  df-mpo 7280  df-of 7533  df-om 7713  df-1st 7831  df-2nd 7832  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-rdg 8241  df-er 8498  df-map 8617  df-en 8734  df-dom 8735  df-sdom 8736  df-pnf 11011  df-mnf 11012  df-xr 11013  df-ltxr 11014  df-le 11015  df-sub 11207  df-neg 11208  df-nn 11974  df-n0 12234  df-z 12320  df-uz 12583  df-fz 13240  df-mzpcl 40545  df-mzp 40546

This theorem is referenced by:  elnn0rabdioph  40625  dvdsrabdioph  40632