Theorem rabdiophlem2 42790

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 2903	. . . . . 6 ⊢ Ⅎ𝑎𝐴
2		nfcsb1v 3933	. . . . . 6 ⊢ Ⅎ𝑢⦋𝑎 / 𝑢⦌𝐴
3		csbeq1a 3922	. . . . . 6 ⊢ (𝑢 = 𝑎 → 𝐴 = ⦋𝑎 / 𝑢⦌𝐴)
4	1, 2, 3	cbvmpt 5259	. . . . 5 ⊢ (𝑢 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴) = (𝑎 ∈ (ℤ ↑m (1...𝑁)) ↦ ⦋𝑎 / 𝑢⦌𝐴)
5	4	fveq1i 6908	. . . 4 ⊢ ((𝑢 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴)‘(𝑡 ↾ (1...𝑁))) = ((𝑎 ∈ (ℤ ↑m (1...𝑁)) ↦ ⦋𝑎 / 𝑢⦌𝐴)‘(𝑡 ↾ (1...𝑁)))
6		eqid 2735	. . . . 5 ⊢ (𝑎 ∈ (ℤ ↑m (1...𝑁)) ↦ ⦋𝑎 / 𝑢⦌𝐴) = (𝑎 ∈ (ℤ ↑m (1...𝑁)) ↦ ⦋𝑎 / 𝑢⦌𝐴)
7		csbeq1 3911	. . . . 5 ⊢ (𝑎 = (𝑡 ↾ (1...𝑁)) → ⦋𝑎 / 𝑢⦌𝐴 = ⦋(𝑡 ↾ (1...𝑁)) / 𝑢⦌𝐴)
8		rabdiophlem2.1	. . . . . . 7 ⊢ 𝑀 = (𝑁 + 1)
9	8	mapfzcons1cl 42706	. . . . . 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 42724	. . . . . . . 8 ⊢ ((𝑢 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁)) → (𝑢 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴):(ℤ ↑m (1...𝑁))⟶ℤ)
12		eqid 2735	. . . . . . . . 9 ⊢ (𝑢 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴) = (𝑢 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴)
13	12	fmpt 7130	. . . . . . . 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 3933	. . . . . . . 8 ⊢ Ⅎ𝑢⦋(𝑡 ↾ (1...𝑁)) / 𝑢⦌𝐴
17	16	nfel1 2920	. . . . . . 7 ⊢ Ⅎ𝑢⦋(𝑡 ↾ (1...𝑁)) / 𝑢⦌𝐴 ∈ ℤ
18		csbeq1a 3922	. . . . . . . 8 ⊢ (𝑢 = (𝑡 ↾ (1...𝑁)) → 𝐴 = ⦋(𝑡 ↾ (1...𝑁)) / 𝑢⦌𝐴)
19	18	eleq1d 2824	. . . . . . 7 ⊢ (𝑢 = (𝑡 ↾ (1...𝑁)) → (𝐴 ∈ ℤ ↔ ⦋(𝑡 ↾ (1...𝑁)) / 𝑢⦌𝐴 ∈ ℤ))
20	17, 19	rspc 3610	. . . . . 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 7039	. . . 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 2788	. . 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 5248	. 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 7466	. . 3 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑢 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁))) → (1...𝑀) ∈ V)
26		fzssp1 13604	. . . . 5 ⊢ (1...𝑁) ⊆ (1...(𝑁 + 1))
27	8	oveq2i 7442	. . . . 5 ⊢ (1...𝑀) = (1...(𝑁 + 1))
28	26, 27	sseqtrri 4033	. . . 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 42738	. . 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 395   = wceq 1537   ∈ wcel 2106  ∀wral 3059  Vcvv 3478  ⦋csb 3908   ⊆ wss 3963   ↦ cmpt 5231   ↾ cres 5691  ⟶wf 6559  ‘cfv 6563  (class class class)co 7431   ↑_m cmap 8865  1c1 11154   + caddc 11156  ℕ₀cn0 12524  ℤcz 12611  ...cfz 13544  mzPolycmzp 42710

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-of 7697  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-er 8744  df-map 8867  df-en 8985  df-dom 8986  df-sdom 8987  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-nn 12265  df-n0 12525  df-z 12612  df-uz 12877  df-fz 13545  df-mzpcl 42711  df-mzp 42712

This theorem is referenced by:  elnn0rabdioph  42791  dvdsrabdioph  42798