Theorem rabdiophlem2 43230

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 2898	. . . . . 6 ⊢ Ⅎ𝑎𝐴
2		nfcsb1v 3861	. . . . . 6 ⊢ Ⅎ𝑢⦋𝑎 / 𝑢⦌𝐴
3		csbeq1a 3851	. . . . . 6 ⊢ (𝑢 = 𝑎 → 𝐴 = ⦋𝑎 / 𝑢⦌𝐴)
4	1, 2, 3	cbvmpt 5187	. . . . 5 ⊢ (𝑢 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴) = (𝑎 ∈ (ℤ ↑m (1...𝑁)) ↦ ⦋𝑎 / 𝑢⦌𝐴)
5	4	fveq1i 6841	. . . 4 ⊢ ((𝑢 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴)‘(𝑡 ↾ (1...𝑁))) = ((𝑎 ∈ (ℤ ↑m (1...𝑁)) ↦ ⦋𝑎 / 𝑢⦌𝐴)‘(𝑡 ↾ (1...𝑁)))
6		eqid 2736	. . . . 5 ⊢ (𝑎 ∈ (ℤ ↑m (1...𝑁)) ↦ ⦋𝑎 / 𝑢⦌𝐴) = (𝑎 ∈ (ℤ ↑m (1...𝑁)) ↦ ⦋𝑎 / 𝑢⦌𝐴)
7		csbeq1 3840	. . . . 5 ⊢ (𝑎 = (𝑡 ↾ (1...𝑁)) → ⦋𝑎 / 𝑢⦌𝐴 = ⦋(𝑡 ↾ (1...𝑁)) / 𝑢⦌𝐴)
8		rabdiophlem2.1	. . . . . . 7 ⊢ 𝑀 = (𝑁 + 1)
9	8	mapfzcons1cl 43150	. . . . . 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 43168	. . . . . . . 8 ⊢ ((𝑢 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁)) → (𝑢 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴):(ℤ ↑m (1...𝑁))⟶ℤ)
12		eqid 2736	. . . . . . . . 9 ⊢ (𝑢 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴) = (𝑢 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴)
13	12	fmpt 7062	. . . . . . . 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 728	. . . . . 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 3861	. . . . . . . 8 ⊢ Ⅎ𝑢⦋(𝑡 ↾ (1...𝑁)) / 𝑢⦌𝐴
17	16	nfel1 2915	. . . . . . 7 ⊢ Ⅎ𝑢⦋(𝑡 ↾ (1...𝑁)) / 𝑢⦌𝐴 ∈ ℤ
18		csbeq1a 3851	. . . . . . . 8 ⊢ (𝑢 = (𝑡 ↾ (1...𝑁)) → 𝐴 = ⦋(𝑡 ↾ (1...𝑁)) / 𝑢⦌𝐴)
19	18	eleq1d 2821	. . . . . . 7 ⊢ (𝑢 = (𝑡 ↾ (1...𝑁)) → (𝐴 ∈ ℤ ↔ ⦋(𝑡 ↾ (1...𝑁)) / 𝑢⦌𝐴 ∈ ℤ))
20	17, 19	rspc 3552	. . . . . 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 6971	. . . 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 2784	. . 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 5178	. 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 7402	. . 3 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑢 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁))) → (1...𝑀) ∈ V)
26		fzssp1 13521	. . . . 5 ⊢ (1...𝑁) ⊆ (1...(𝑁 + 1))
27	8	oveq2i 7378	. . . . 5 ⊢ (1...𝑀) = (1...(𝑁 + 1))
28	26, 27	sseqtrri 3971	. . . 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 43182	. . 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 1374	. 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 2836	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 1542   ∈ wcel 2114  ∀wral 3051  Vcvv 3429  ⦋csb 3837   ⊆ wss 3889   ↦ cmpt 5166   ↾ cres 5633  ⟶wf 6494  ‘cfv 6498  (class class class)co 7367   ↑_m cmap 8773  1c1 11039   + caddc 11041  ℕ₀cn0 12437  ℤcz 12524  ...cfz 13461  mzPolycmzp 43154

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689  ax-cnex 11094  ax-resscn 11095  ax-1cn 11096  ax-icn 11097  ax-addcl 11098  ax-addrcl 11099  ax-mulcl 11100  ax-mulrcl 11101  ax-mulcom 11102  ax-addass 11103  ax-mulass 11104  ax-distr 11105  ax-i2m1 11106  ax-1ne0 11107  ax-1rid 11108  ax-rnegex 11109  ax-rrecex 11110  ax-cnre 11111  ax-pre-lttri 11112  ax-pre-lttrn 11113  ax-pre-ltadd 11114  ax-pre-mulgt0 11115

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3062  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-pss 3909  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-int 4890  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-tr 5193  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6265  df-ord 6326  df-on 6327  df-lim 6328  df-suc 6329  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-riota 7324  df-ov 7370  df-oprab 7371  df-mpo 7372  df-of 7631  df-om 7818  df-1st 7942  df-2nd 7943  df-frecs 8231  df-wrecs 8262  df-recs 8311  df-rdg 8349  df-er 8643  df-map 8775  df-en 8894  df-dom 8895  df-sdom 8896  df-pnf 11181  df-mnf 11182  df-xr 11183  df-ltxr 11184  df-le 11185  df-sub 11379  df-neg 11380  df-nn 12175  df-n0 12438  df-z 12525  df-uz 12789  df-fz 13462  df-mzpcl 43155  df-mzp 43156

This theorem is referenced by:  elnn0rabdioph  43231  dvdsrabdioph  43238