Theorem mzpcl2 42020

Description: Defining property 2 of a polynomially closed function set 𝑃: it contains all projections. (Contributed by Stefan O'Rear, 4-Oct-2014.)

Assertion

Ref	Expression
mzpcl2	⊢ ((𝑃 ∈ (mzPolyCld‘𝑉) ∧ 𝐹 ∈ 𝑉) → (𝑔 ∈ (ℤ ↑m 𝑉) ↦ (𝑔‘𝐹)) ∈ 𝑃)

Distinct variable groups:   𝑔,𝑉   𝑃,𝑔   𝑔,𝐹

Proof of Theorem mzpcl2

Dummy variable 𝑓 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpr 484	. 2 ⊢ ((𝑃 ∈ (mzPolyCld‘𝑉) ∧ 𝐹 ∈ 𝑉) → 𝐹 ∈ 𝑉)
2		simpl 482	. . . 4 ⊢ ((𝑃 ∈ (mzPolyCld‘𝑉) ∧ 𝐹 ∈ 𝑉) → 𝑃 ∈ (mzPolyCld‘𝑉))
3		elfvex 6920	. . . . . 6 ⊢ (𝑃 ∈ (mzPolyCld‘𝑉) → 𝑉 ∈ V)
4	3	adantr 480	. . . . 5 ⊢ ((𝑃 ∈ (mzPolyCld‘𝑉) ∧ 𝐹 ∈ 𝑉) → 𝑉 ∈ V)
5		elmzpcl 42016	. . . . 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 ∈ 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)))))
7	2, 6	mpbid 231	. . 3 ⊢ ((𝑃 ∈ (mzPolyCld‘𝑉) ∧ 𝐹 ∈ 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))))
8		simprlr 777	. . 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 ∈ 𝑉 (𝑔 ∈ (ℤ ↑m 𝑉) ↦ (𝑔‘𝑓)) ∈ 𝑃)
9	7, 8	syl 17	. 2 ⊢ ((𝑃 ∈ (mzPolyCld‘𝑉) ∧ 𝐹 ∈ 𝑉) → ∀𝑓 ∈ 𝑉 (𝑔 ∈ (ℤ ↑m 𝑉) ↦ (𝑔‘𝑓)) ∈ 𝑃)
10		fveq2 6882	. . . . 5 ⊢ (𝑓 = 𝐹 → (𝑔‘𝑓) = (𝑔‘𝐹))
11	10	mpteq2dv 5241	. . . 4 ⊢ (𝑓 = 𝐹 → (𝑔 ∈ (ℤ ↑m 𝑉) ↦ (𝑔‘𝑓)) = (𝑔 ∈ (ℤ ↑m 𝑉) ↦ (𝑔‘𝐹)))
12	11	eleq1d 2810	. . 3 ⊢ (𝑓 = 𝐹 → ((𝑔 ∈ (ℤ ↑m 𝑉) ↦ (𝑔‘𝑓)) ∈ 𝑃 ↔ (𝑔 ∈ (ℤ ↑m 𝑉) ↦ (𝑔‘𝐹)) ∈ 𝑃))
13	12	rspcva 3602	. 2 ⊢ ((𝐹 ∈ 𝑉 ∧ ∀𝑓 ∈ 𝑉 (𝑔 ∈ (ℤ ↑m 𝑉) ↦ (𝑔‘𝑓)) ∈ 𝑃) → (𝑔 ∈ (ℤ ↑m 𝑉) ↦ (𝑔‘𝐹)) ∈ 𝑃)
14	1, 9, 13	syl2anc 583	1 ⊢ ((𝑃 ∈ (mzPolyCld‘𝑉) ∧ 𝐹 ∈ 𝑉) → (𝑔 ∈ (ℤ ↑m 𝑉) ↦ (𝑔‘𝐹)) ∈ 𝑃)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1533   ∈ wcel 2098  ∀wral 3053  Vcvv 3466   ⊆ wss 3941  {csn 4621   ↦ cmpt 5222   × cxp 5665  ‘cfv 6534  (class class class)co 7402   ∘_f cof 7662   ↑_m cmap 8817   + caddc 11110   · cmul 11112  ℤcz 12557  mzPolyCldcmzpcl 42011

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-sep 5290  ax-nul 5297  ax-pow 5354  ax-pr 5418

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-pw 4597  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-br 5140  df-opab 5202  df-mpt 5223  df-id 5565  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-iota 6486  df-fun 6536  df-fv 6542  df-ov 7405  df-mzpcl 42013

This theorem is referenced by:  mzpincl  42024  mzpproj  42027