Theorem mzpcl1 42886

Description: Defining property 1 of a polynomially closed function set 𝑃: it contains all constant functions. (Contributed by Stefan O'Rear, 4-Oct-2014.)

Assertion

Ref	Expression
mzpcl1	⊢ ((𝑃 ∈ (mzPolyCld‘𝑉) ∧ 𝐹 ∈ ℤ) → ((ℤ ↑m 𝑉) × {𝐹}) ∈ 𝑃)

Proof of Theorem mzpcl1

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

Step	Hyp	Ref	Expression
1		simpr 484	. 2 ⊢ ((𝑃 ∈ (mzPolyCld‘𝑉) ∧ 𝐹 ∈ ℤ) → 𝐹 ∈ ℤ)
2		simpl 482	. . . 4 ⊢ ((𝑃 ∈ (mzPolyCld‘𝑉) ∧ 𝐹 ∈ ℤ) → 𝑃 ∈ (mzPolyCld‘𝑉))
3		elfvex 6866	. . . . . 6 ⊢ (𝑃 ∈ (mzPolyCld‘𝑉) → 𝑉 ∈ V)
4	3	adantr 480	. . . . 5 ⊢ ((𝑃 ∈ (mzPolyCld‘𝑉) ∧ 𝐹 ∈ ℤ) → 𝑉 ∈ V)
5		elmzpcl 42883	. . . . 5 ⊢ (𝑉 ∈ V → (𝑃 ∈ (mzPolyCld‘𝑉) ↔ (𝑃 ⊆ (ℤ ↑m (ℤ ↑m 𝑉)) ∧ ((∀𝑓 ∈ ℤ ((ℤ ↑m 𝑉) × {𝑓}) ∈ 𝑃 ∧ ∀𝑓 ∈ 𝑉 (𝑔 ∈ (ℤ ↑m 𝑉) ↦ (𝑔‘𝑓)) ∈ 𝑃) ∧ ∀𝑓 ∈ 𝑃 ∀𝑔 ∈ 𝑃 ((𝑓 ∘f + 𝑔) ∈ 𝑃 ∧ (𝑓 ∘f · 𝑔) ∈ 𝑃)))))
6	4, 5	syl 17	. . . 4 ⊢ ((𝑃 ∈ (mzPolyCld‘𝑉) ∧ 𝐹 ∈ ℤ) → (𝑃 ∈ (mzPolyCld‘𝑉) ↔ (𝑃 ⊆ (ℤ ↑m (ℤ ↑m 𝑉)) ∧ ((∀𝑓 ∈ ℤ ((ℤ ↑m 𝑉) × {𝑓}) ∈ 𝑃 ∧ ∀𝑓 ∈ 𝑉 (𝑔 ∈ (ℤ ↑m 𝑉) ↦ (𝑔‘𝑓)) ∈ 𝑃) ∧ ∀𝑓 ∈ 𝑃 ∀𝑔 ∈ 𝑃 ((𝑓 ∘f + 𝑔) ∈ 𝑃 ∧ (𝑓 ∘f · 𝑔) ∈ 𝑃)))))
7	2, 6	mpbid 232	. . 3 ⊢ ((𝑃 ∈ (mzPolyCld‘𝑉) ∧ 𝐹 ∈ ℤ) → (𝑃 ⊆ (ℤ ↑m (ℤ ↑m 𝑉)) ∧ ((∀𝑓 ∈ ℤ ((ℤ ↑m 𝑉) × {𝑓}) ∈ 𝑃 ∧ ∀𝑓 ∈ 𝑉 (𝑔 ∈ (ℤ ↑m 𝑉) ↦ (𝑔‘𝑓)) ∈ 𝑃) ∧ ∀𝑓 ∈ 𝑃 ∀𝑔 ∈ 𝑃 ((𝑓 ∘f + 𝑔) ∈ 𝑃 ∧ (𝑓 ∘f · 𝑔) ∈ 𝑃))))
8		simprll 778	. . 3 ⊢ ((𝑃 ⊆ (ℤ ↑m (ℤ ↑m 𝑉)) ∧ ((∀𝑓 ∈ ℤ ((ℤ ↑m 𝑉) × {𝑓}) ∈ 𝑃 ∧ ∀𝑓 ∈ 𝑉 (𝑔 ∈ (ℤ ↑m 𝑉) ↦ (𝑔‘𝑓)) ∈ 𝑃) ∧ ∀𝑓 ∈ 𝑃 ∀𝑔 ∈ 𝑃 ((𝑓 ∘f + 𝑔) ∈ 𝑃 ∧ (𝑓 ∘f · 𝑔) ∈ 𝑃))) → ∀𝑓 ∈ ℤ ((ℤ ↑m 𝑉) × {𝑓}) ∈ 𝑃)
9	7, 8	syl 17	. 2 ⊢ ((𝑃 ∈ (mzPolyCld‘𝑉) ∧ 𝐹 ∈ ℤ) → ∀𝑓 ∈ ℤ ((ℤ ↑m 𝑉) × {𝑓}) ∈ 𝑃)
10		sneq 4587	. . . . 5 ⊢ (𝑓 = 𝐹 → {𝑓} = {𝐹})
11	10	xpeq2d 5651	. . . 4 ⊢ (𝑓 = 𝐹 → ((ℤ ↑m 𝑉) × {𝑓}) = ((ℤ ↑m 𝑉) × {𝐹}))
12	11	eleq1d 2818	. . 3 ⊢ (𝑓 = 𝐹 → (((ℤ ↑m 𝑉) × {𝑓}) ∈ 𝑃 ↔ ((ℤ ↑m 𝑉) × {𝐹}) ∈ 𝑃))
13	12	rspcva 3571	. 2 ⊢ ((𝐹 ∈ ℤ ∧ ∀𝑓 ∈ ℤ ((ℤ ↑m 𝑉) × {𝑓}) ∈ 𝑃) → ((ℤ ↑m 𝑉) × {𝐹}) ∈ 𝑃)
14	1, 9, 13	syl2anc 584	1 ⊢ ((𝑃 ∈ (mzPolyCld‘𝑉) ∧ 𝐹 ∈ ℤ) → ((ℤ ↑m 𝑉) × {𝐹}) ∈ 𝑃)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1541   ∈ wcel 2113  ∀wral 3048  Vcvv 3437   ⊆ wss 3898  {csn 4577   ↦ cmpt 5176   × cxp 5619  ‘cfv 6489  (class class class)co 7355   ∘_f cof 7617   ↑_m cmap 8759   + caddc 11020   · cmul 11022  ℤcz 12479  mzPolyCldcmzpcl 42878

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 2705  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-ral 3049  df-rex 3058  df-rab 3397  df-v 3439  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4283  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4861  df-br 5096  df-opab 5158  df-mpt 5177  df-id 5516  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-iota 6445  df-fun 6491  df-fv 6497  df-ov 7358  df-mzpcl 42880

This theorem is referenced by:  mzpincl  42891  mzpconst  42892