Theorem rabdiophlem2 42986

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 2896	. . . . . 6 ⊢ Ⅎ𝑎𝐴
2		nfcsb1v 3871	. . . . . 6 ⊢ Ⅎ𝑢⦋𝑎 / 𝑢⦌𝐴
3		csbeq1a 3861	. . . . . 6 ⊢ (𝑢 = 𝑎 → 𝐴 = ⦋𝑎 / 𝑢⦌𝐴)
4	1, 2, 3	cbvmpt 5198	. . . . 5 ⊢ (𝑢 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴) = (𝑎 ∈ (ℤ ↑m (1...𝑁)) ↦ ⦋𝑎 / 𝑢⦌𝐴)
5	4	fveq1i 6833	. . . 4 ⊢ ((𝑢 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴)‘(𝑡 ↾ (1...𝑁))) = ((𝑎 ∈ (ℤ ↑m (1...𝑁)) ↦ ⦋𝑎 / 𝑢⦌𝐴)‘(𝑡 ↾ (1...𝑁)))
6		eqid 2734	. . . . 5 ⊢ (𝑎 ∈ (ℤ ↑m (1...𝑁)) ↦ ⦋𝑎 / 𝑢⦌𝐴) = (𝑎 ∈ (ℤ ↑m (1...𝑁)) ↦ ⦋𝑎 / 𝑢⦌𝐴)
7		csbeq1 3850	. . . . 5 ⊢ (𝑎 = (𝑡 ↾ (1...𝑁)) → ⦋𝑎 / 𝑢⦌𝐴 = ⦋(𝑡 ↾ (1...𝑁)) / 𝑢⦌𝐴)
8		rabdiophlem2.1	. . . . . . 7 ⊢ 𝑀 = (𝑁 + 1)
9	8	mapfzcons1cl 42902	. . . . . 6 ⊢ (𝑡 ∈ (ℤ ↑m (1...𝑀)) → (𝑡 ↾ (1...𝑁)) ∈ (ℤ ↑m (1...𝑁)))
10	9	adantl 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 42920	. . . . . . . 8 ⊢ ((𝑢 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁)) → (𝑢 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴):(ℤ ↑m (1...𝑁))⟶ℤ)
12		eqid 2734	. . . . . . . . 9 ⊢ (𝑢 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴) = (𝑢 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴)
13	12	fmpt 7053	. . . . . . . 8 ⊢ (∀𝑢 ∈ (ℤ ↑m (1...𝑁))𝐴 ∈ ℤ ↔ (𝑢 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴):(ℤ ↑m (1...𝑁))⟶ℤ)
14	11, 13	sylibr 234	. . . . . . 7 ⊢ ((𝑢 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁)) → ∀𝑢 ∈ (ℤ ↑m (1...𝑁))𝐴 ∈ ℤ)
15	14	ad2antlr 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 3871	. . . . . . . 8 ⊢ Ⅎ𝑢⦋(𝑡 ↾ (1...𝑁)) / 𝑢⦌𝐴
17	16	nfel1 2913	. . . . . . 7 ⊢ Ⅎ𝑢⦋(𝑡 ↾ (1...𝑁)) / 𝑢⦌𝐴 ∈ ℤ
18		csbeq1a 3861	. . . . . . . 8 ⊢ (𝑢 = (𝑡 ↾ (1...𝑁)) → 𝐴 = ⦋(𝑡 ↾ (1...𝑁)) / 𝑢⦌𝐴)
19	18	eleq1d 2819	. . . . . . 7 ⊢ (𝑢 = (𝑡 ↾ (1...𝑁)) → (𝐴 ∈ ℤ ↔ ⦋(𝑡 ↾ (1...𝑁)) / 𝑢⦌𝐴 ∈ ℤ))
20	17, 19	rspc 3562	. . . . . 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 6962	. . . 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 2782	. . 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 5189	. 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 7391	. . 3 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑢 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁))) → (1...𝑀) ∈ V)
26		fzssp1 13481	. . . . 5 ⊢ (1...𝑁) ⊆ (1...(𝑁 + 1))
27	8	oveq2i 7367	. . . . 5 ⊢ (1...𝑀) = (1...(𝑁 + 1))
28	26, 27	sseqtrri 3981	. . . 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 484	. . 3 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑢 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁))) → (𝑢 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁)))
31		mzpresrename 42934	. . 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 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)))
33	24, 32	eqeltrd 2834	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 395   = wceq 1541   ∈ wcel 2113  ∀wral 3049  Vcvv 3438  ⦋csb 3847   ⊆ wss 3899   ↦ cmpt 5177   ↾ cres 5624  ⟶wf 6486  ‘cfv 6490  (class class class)co 7356   ↑_m cmap 8761  1c1 11025   + caddc 11027  ℕ₀cn0 12399  ℤcz 12486  ...cfz 13421  mzPolycmzp 42906

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2706  ax-rep 5222  ax-sep 5239  ax-nul 5249  ax-pow 5308  ax-pr 5375  ax-un 7678  ax-cnex 11080  ax-resscn 11081  ax-1cn 11082  ax-icn 11083  ax-addcl 11084  ax-addrcl 11085  ax-mulcl 11086  ax-mulrcl 11087  ax-mulcom 11088  ax-addass 11089  ax-mulass 11090  ax-distr 11091  ax-i2m1 11092  ax-1ne0 11093  ax-1rid 11094  ax-rnegex 11095  ax-rrecex 11096  ax-cnre 11097  ax-pre-lttri 11098  ax-pre-lttrn 11099  ax-pre-ltadd 11100  ax-pre-mulgt0 11101

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2809  df-nfc 2883  df-ne 2931  df-nel 3035  df-ral 3050  df-rex 3059  df-reu 3349  df-rab 3398  df-v 3440  df-sbc 3739  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4579  df-pr 4581  df-op 4585  df-uni 4862  df-int 4901  df-iun 4946  df-br 5097  df-opab 5159  df-mpt 5178  df-tr 5204  df-id 5517  df-eprel 5522  df-po 5530  df-so 5531  df-fr 5575  df-we 5577  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635  df-pred 6257  df-ord 6318  df-on 6319  df-lim 6320  df-suc 6321  df-iota 6446  df-fun 6492  df-fn 6493  df-f 6494  df-f1 6495  df-fo 6496  df-f1o 6497  df-fv 6498  df-riota 7313  df-ov 7359  df-oprab 7360  df-mpo 7361  df-of 7620  df-om 7807  df-1st 7931  df-2nd 7932  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-rdg 8339  df-er 8633  df-map 8763  df-en 8882  df-dom 8883  df-sdom 8884  df-pnf 11166  df-mnf 11167  df-xr 11168  df-ltxr 11169  df-le 11170  df-sub 11364  df-neg 11365  df-nn 12144  df-n0 12400  df-z 12487  df-uz 12750  df-fz 13422  df-mzpcl 42907  df-mzp 42908

This theorem is referenced by:  elnn0rabdioph  42987  dvdsrabdioph  42994