Theorem mzpclall 36308

Description: The set of all functions with the signature of a polynomial is a polynomially closed set. This is a lemma to show that the intersection in df-mzp 36305 is well-defined. (Contributed by Stefan O'Rear, 4-Oct-2014.)

Assertion

Ref	Expression
mzpclall	⊢ (𝑉 ∈ V → (ℤ ↑𝑚 (ℤ ↑𝑚 𝑉)) ∈ (mzPolyCld‘𝑉))

Proof of Theorem mzpclall

Dummy variables 𝑣 𝑓 𝑔 𝑎 𝑏 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		oveq2 6557	. . . 4 ⊢ (𝑣 = 𝑉 → (ℤ ↑𝑚 𝑣) = (ℤ ↑𝑚 𝑉))
2	1	oveq2d 6565	. . 3 ⊢ (𝑣 = 𝑉 → (ℤ ↑𝑚 (ℤ ↑𝑚 𝑣)) = (ℤ ↑𝑚 (ℤ ↑𝑚 𝑉)))
3		fveq2 6103	. . 3 ⊢ (𝑣 = 𝑉 → (mzPolyCld‘𝑣) = (mzPolyCld‘𝑉))
4	2, 3	eleq12d 2682	. 2 ⊢ (𝑣 = 𝑉 → ((ℤ ↑𝑚 (ℤ ↑𝑚 𝑣)) ∈ (mzPolyCld‘𝑣) ↔ (ℤ ↑𝑚 (ℤ ↑𝑚 𝑉)) ∈ (mzPolyCld‘𝑉)))
5		ssid 3587	. . 3 ⊢ (ℤ ↑𝑚 (ℤ ↑𝑚 𝑣)) ⊆ (ℤ ↑𝑚 (ℤ ↑𝑚 𝑣))
6		ovex 6577	. . . . . . 7 ⊢ (ℤ ↑𝑚 𝑣) ∈ V
7		zex 11263	. . . . . . 7 ⊢ ℤ ∈ V
8	6, 7	constmap 36294	. . . . . 6 ⊢ (𝑓 ∈ ℤ → ((ℤ ↑𝑚 𝑣) × {𝑓}) ∈ (ℤ ↑𝑚 (ℤ ↑𝑚 𝑣)))
9	8	rgen 2906	. . . . 5 ⊢ ∀𝑓 ∈ ℤ ((ℤ ↑𝑚 𝑣) × {𝑓}) ∈ (ℤ ↑𝑚 (ℤ ↑𝑚 𝑣))
10		vex 3176	. . . . . . . . . . 11 ⊢ 𝑣 ∈ V
11	7, 10	elmap 7772	. . . . . . . . . 10 ⊢ (𝑔 ∈ (ℤ ↑𝑚 𝑣) ↔ 𝑔:𝑣⟶ℤ)
12		ffvelrn 6265	. . . . . . . . . 10 ⊢ ((𝑔:𝑣⟶ℤ ∧ 𝑓 ∈ 𝑣) → (𝑔‘𝑓) ∈ ℤ)
13	11, 12	sylanb 488	. . . . . . . . 9 ⊢ ((𝑔 ∈ (ℤ ↑𝑚 𝑣) ∧ 𝑓 ∈ 𝑣) → (𝑔‘𝑓) ∈ ℤ)
14	13	ancoms 468	. . . . . . . 8 ⊢ ((𝑓 ∈ 𝑣 ∧ 𝑔 ∈ (ℤ ↑𝑚 𝑣)) → (𝑔‘𝑓) ∈ ℤ)
15		eqid 2610	. . . . . . . 8 ⊢ (𝑔 ∈ (ℤ ↑𝑚 𝑣) ↦ (𝑔‘𝑓)) = (𝑔 ∈ (ℤ ↑𝑚 𝑣) ↦ (𝑔‘𝑓))
16	14, 15	fmptd 6292	. . . . . . 7 ⊢ (𝑓 ∈ 𝑣 → (𝑔 ∈ (ℤ ↑𝑚 𝑣) ↦ (𝑔‘𝑓)):(ℤ ↑𝑚 𝑣)⟶ℤ)
17	7, 6	elmap 7772	. . . . . . 7 ⊢ ((𝑔 ∈ (ℤ ↑𝑚 𝑣) ↦ (𝑔‘𝑓)) ∈ (ℤ ↑𝑚 (ℤ ↑𝑚 𝑣)) ↔ (𝑔 ∈ (ℤ ↑𝑚 𝑣) ↦ (𝑔‘𝑓)):(ℤ ↑𝑚 𝑣)⟶ℤ)
18	16, 17	sylibr 223	. . . . . 6 ⊢ (𝑓 ∈ 𝑣 → (𝑔 ∈ (ℤ ↑𝑚 𝑣) ↦ (𝑔‘𝑓)) ∈ (ℤ ↑𝑚 (ℤ ↑𝑚 𝑣)))
19	18	rgen 2906	. . . . 5 ⊢ ∀𝑓 ∈ 𝑣 (𝑔 ∈ (ℤ ↑𝑚 𝑣) ↦ (𝑔‘𝑓)) ∈ (ℤ ↑𝑚 (ℤ ↑𝑚 𝑣))
20	9, 19	pm3.2i 470	. . . 4 ⊢ (∀𝑓 ∈ ℤ ((ℤ ↑𝑚 𝑣) × {𝑓}) ∈ (ℤ ↑𝑚 (ℤ ↑𝑚 𝑣)) ∧ ∀𝑓 ∈ 𝑣 (𝑔 ∈ (ℤ ↑𝑚 𝑣) ↦ (𝑔‘𝑓)) ∈ (ℤ ↑𝑚 (ℤ ↑𝑚 𝑣)))
21		zaddcl 11294	. . . . . . . . 9 ⊢ ((𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ) → (𝑎 + 𝑏) ∈ ℤ)
22	21	adantl 481	. . . . . . . 8 ⊢ (((𝑓:(ℤ ↑𝑚 𝑣)⟶ℤ ∧ 𝑔:(ℤ ↑𝑚 𝑣)⟶ℤ) ∧ (𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ)) → (𝑎 + 𝑏) ∈ ℤ)
23		simpl 472	. . . . . . . 8 ⊢ ((𝑓:(ℤ ↑𝑚 𝑣)⟶ℤ ∧ 𝑔:(ℤ ↑𝑚 𝑣)⟶ℤ) → 𝑓:(ℤ ↑𝑚 𝑣)⟶ℤ)
24		simpr 476	. . . . . . . 8 ⊢ ((𝑓:(ℤ ↑𝑚 𝑣)⟶ℤ ∧ 𝑔:(ℤ ↑𝑚 𝑣)⟶ℤ) → 𝑔:(ℤ ↑𝑚 𝑣)⟶ℤ)
25	6	a1i 11	. . . . . . . 8 ⊢ ((𝑓:(ℤ ↑𝑚 𝑣)⟶ℤ ∧ 𝑔:(ℤ ↑𝑚 𝑣)⟶ℤ) → (ℤ ↑𝑚 𝑣) ∈ V)
26		inidm 3784	. . . . . . . 8 ⊢ ((ℤ ↑𝑚 𝑣) ∩ (ℤ ↑𝑚 𝑣)) = (ℤ ↑𝑚 𝑣)
27	22, 23, 24, 25, 25, 26	off 6810	. . . . . . 7 ⊢ ((𝑓:(ℤ ↑𝑚 𝑣)⟶ℤ ∧ 𝑔:(ℤ ↑𝑚 𝑣)⟶ℤ) → (𝑓 ∘𝑓 + 𝑔):(ℤ ↑𝑚 𝑣)⟶ℤ)
28		zmulcl 11303	. . . . . . . . 9 ⊢ ((𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ) → (𝑎 · 𝑏) ∈ ℤ)
29	28	adantl 481	. . . . . . . 8 ⊢ (((𝑓:(ℤ ↑𝑚 𝑣)⟶ℤ ∧ 𝑔:(ℤ ↑𝑚 𝑣)⟶ℤ) ∧ (𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ)) → (𝑎 · 𝑏) ∈ ℤ)
30	29, 23, 24, 25, 25, 26	off 6810	. . . . . . 7 ⊢ ((𝑓:(ℤ ↑𝑚 𝑣)⟶ℤ ∧ 𝑔:(ℤ ↑𝑚 𝑣)⟶ℤ) → (𝑓 ∘𝑓 · 𝑔):(ℤ ↑𝑚 𝑣)⟶ℤ)
31	27, 30	jca 553	. . . . . 6 ⊢ ((𝑓:(ℤ ↑𝑚 𝑣)⟶ℤ ∧ 𝑔:(ℤ ↑𝑚 𝑣)⟶ℤ) → ((𝑓 ∘𝑓 + 𝑔):(ℤ ↑𝑚 𝑣)⟶ℤ ∧ (𝑓 ∘𝑓 · 𝑔):(ℤ ↑𝑚 𝑣)⟶ℤ))
32	7, 6	elmap 7772	. . . . . . 7 ⊢ (𝑓 ∈ (ℤ ↑𝑚 (ℤ ↑𝑚 𝑣)) ↔ 𝑓:(ℤ ↑𝑚 𝑣)⟶ℤ)
33	7, 6	elmap 7772	. . . . . . 7 ⊢ (𝑔 ∈ (ℤ ↑𝑚 (ℤ ↑𝑚 𝑣)) ↔ 𝑔:(ℤ ↑𝑚 𝑣)⟶ℤ)
34	32, 33	anbi12i 729	. . . . . 6 ⊢ ((𝑓 ∈ (ℤ ↑𝑚 (ℤ ↑𝑚 𝑣)) ∧ 𝑔 ∈ (ℤ ↑𝑚 (ℤ ↑𝑚 𝑣))) ↔ (𝑓:(ℤ ↑𝑚 𝑣)⟶ℤ ∧ 𝑔:(ℤ ↑𝑚 𝑣)⟶ℤ))
35	7, 6	elmap 7772	. . . . . . 7 ⊢ ((𝑓 ∘𝑓 + 𝑔) ∈ (ℤ ↑𝑚 (ℤ ↑𝑚 𝑣)) ↔ (𝑓 ∘𝑓 + 𝑔):(ℤ ↑𝑚 𝑣)⟶ℤ)
36	7, 6	elmap 7772	. . . . . . 7 ⊢ ((𝑓 ∘𝑓 · 𝑔) ∈ (ℤ ↑𝑚 (ℤ ↑𝑚 𝑣)) ↔ (𝑓 ∘𝑓 · 𝑔):(ℤ ↑𝑚 𝑣)⟶ℤ)
37	35, 36	anbi12i 729	. . . . . 6 ⊢ (((𝑓 ∘𝑓 + 𝑔) ∈ (ℤ ↑𝑚 (ℤ ↑𝑚 𝑣)) ∧ (𝑓 ∘𝑓 · 𝑔) ∈ (ℤ ↑𝑚 (ℤ ↑𝑚 𝑣))) ↔ ((𝑓 ∘𝑓 + 𝑔):(ℤ ↑𝑚 𝑣)⟶ℤ ∧ (𝑓 ∘𝑓 · 𝑔):(ℤ ↑𝑚 𝑣)⟶ℤ))
38	31, 34, 37	3imtr4i 280	. . . . 5 ⊢ ((𝑓 ∈ (ℤ ↑𝑚 (ℤ ↑𝑚 𝑣)) ∧ 𝑔 ∈ (ℤ ↑𝑚 (ℤ ↑𝑚 𝑣))) → ((𝑓 ∘𝑓 + 𝑔) ∈ (ℤ ↑𝑚 (ℤ ↑𝑚 𝑣)) ∧ (𝑓 ∘𝑓 · 𝑔) ∈ (ℤ ↑𝑚 (ℤ ↑𝑚 𝑣))))
39	38	rgen2a 2960	. . . 4 ⊢ ∀𝑓 ∈ (ℤ ↑𝑚 (ℤ ↑𝑚 𝑣))∀𝑔 ∈ (ℤ ↑𝑚 (ℤ ↑𝑚 𝑣))((𝑓 ∘𝑓 + 𝑔) ∈ (ℤ ↑𝑚 (ℤ ↑𝑚 𝑣)) ∧ (𝑓 ∘𝑓 · 𝑔) ∈ (ℤ ↑𝑚 (ℤ ↑𝑚 𝑣)))
40	20, 39	pm3.2i 470	. . 3 ⊢ ((∀𝑓 ∈ ℤ ((ℤ ↑𝑚 𝑣) × {𝑓}) ∈ (ℤ ↑𝑚 (ℤ ↑𝑚 𝑣)) ∧ ∀𝑓 ∈ 𝑣 (𝑔 ∈ (ℤ ↑𝑚 𝑣) ↦ (𝑔‘𝑓)) ∈ (ℤ ↑𝑚 (ℤ ↑𝑚 𝑣))) ∧ ∀𝑓 ∈ (ℤ ↑𝑚 (ℤ ↑𝑚 𝑣))∀𝑔 ∈ (ℤ ↑𝑚 (ℤ ↑𝑚 𝑣))((𝑓 ∘𝑓 + 𝑔) ∈ (ℤ ↑𝑚 (ℤ ↑𝑚 𝑣)) ∧ (𝑓 ∘𝑓 · 𝑔) ∈ (ℤ ↑𝑚 (ℤ ↑𝑚 𝑣))))
41		elmzpcl 36307	. . . 4 ⊢ (𝑣 ∈ V → ((ℤ ↑𝑚 (ℤ ↑𝑚 𝑣)) ∈ (mzPolyCld‘𝑣) ↔ ((ℤ ↑𝑚 (ℤ ↑𝑚 𝑣)) ⊆ (ℤ ↑𝑚 (ℤ ↑𝑚 𝑣)) ∧ ((∀𝑓 ∈ ℤ ((ℤ ↑𝑚 𝑣) × {𝑓}) ∈ (ℤ ↑𝑚 (ℤ ↑𝑚 𝑣)) ∧ ∀𝑓 ∈ 𝑣 (𝑔 ∈ (ℤ ↑𝑚 𝑣) ↦ (𝑔‘𝑓)) ∈ (ℤ ↑𝑚 (ℤ ↑𝑚 𝑣))) ∧ ∀𝑓 ∈ (ℤ ↑𝑚 (ℤ ↑𝑚 𝑣))∀𝑔 ∈ (ℤ ↑𝑚 (ℤ ↑𝑚 𝑣))((𝑓 ∘𝑓 + 𝑔) ∈ (ℤ ↑𝑚 (ℤ ↑𝑚 𝑣)) ∧ (𝑓 ∘𝑓 · 𝑔) ∈ (ℤ ↑𝑚 (ℤ ↑𝑚 𝑣)))))))
42	10, 41	ax-mp 5	. . 3 ⊢ ((ℤ ↑𝑚 (ℤ ↑𝑚 𝑣)) ∈ (mzPolyCld‘𝑣) ↔ ((ℤ ↑𝑚 (ℤ ↑𝑚 𝑣)) ⊆ (ℤ ↑𝑚 (ℤ ↑𝑚 𝑣)) ∧ ((∀𝑓 ∈ ℤ ((ℤ ↑𝑚 𝑣) × {𝑓}) ∈ (ℤ ↑𝑚 (ℤ ↑𝑚 𝑣)) ∧ ∀𝑓 ∈ 𝑣 (𝑔 ∈ (ℤ ↑𝑚 𝑣) ↦ (𝑔‘𝑓)) ∈ (ℤ ↑𝑚 (ℤ ↑𝑚 𝑣))) ∧ ∀𝑓 ∈ (ℤ ↑𝑚 (ℤ ↑𝑚 𝑣))∀𝑔 ∈ (ℤ ↑𝑚 (ℤ ↑𝑚 𝑣))((𝑓 ∘𝑓 + 𝑔) ∈ (ℤ ↑𝑚 (ℤ ↑𝑚 𝑣)) ∧ (𝑓 ∘𝑓 · 𝑔) ∈ (ℤ ↑𝑚 (ℤ ↑𝑚 𝑣))))))
43	5, 40, 42	mpbir2an 957	. 2 ⊢ (ℤ ↑𝑚 (ℤ ↑𝑚 𝑣)) ∈ (mzPolyCld‘𝑣)
44	4, 43	vtoclg 3239	1 ⊢ (𝑉 ∈ V → (ℤ ↑𝑚 (ℤ ↑𝑚 𝑉)) ∈ (mzPolyCld‘𝑉))

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896  Vcvv 3173   ⊆ wss 3540  {csn 4125   ↦ cmpt 4643   × cxp 5036  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549   ∘_𝑓 cof 6793   ↑_𝑚 cmap 7744   + caddc 9818   · cmul 9820  ℤcz 11254  mzPolyCldcmzpcl 36302

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-of 6795  df-om 6958  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-er 7629  df-map 7746  df-en 7842  df-dom 7843  df-sdom 7844  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-n0 11170  df-z 11255  df-mzpcl 36304

This theorem is referenced by:  mzpcln0  36309  mzpincl  36315  mzpf  36317