Theorem mzpcl2 42742

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 6852	. . . . . 6 ⊢ (𝑃 ∈ (mzPolyCld‘𝑉) → 𝑉 ∈ V)
4	3	adantr 480	. . . . 5 ⊢ ((𝑃 ∈ (mzPolyCld‘𝑉) ∧ 𝐹 ∈ 𝑉) → 𝑉 ∈ V)
5		elmzpcl 42738	. . . . 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 232	. . 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 779	. . 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 6817	. . . . 5 ⊢ (𝑓 = 𝐹 → (𝑔‘𝑓) = (𝑔‘𝐹))
11	10	mpteq2dv 5183	. . . 4 ⊢ (𝑓 = 𝐹 → (𝑔 ∈ (ℤ ↑m 𝑉) ↦ (𝑔‘𝑓)) = (𝑔 ∈ (ℤ ↑m 𝑉) ↦ (𝑔‘𝐹)))
12	11	eleq1d 2814	. . 3 ⊢ (𝑓 = 𝐹 → ((𝑔 ∈ (ℤ ↑m 𝑉) ↦ (𝑔‘𝑓)) ∈ 𝑃 ↔ (𝑔 ∈ (ℤ ↑m 𝑉) ↦ (𝑔‘𝐹)) ∈ 𝑃))
13	12	rspcva 3573	. 2 ⊢ ((𝐹 ∈ 𝑉 ∧ ∀𝑓 ∈ 𝑉 (𝑔 ∈ (ℤ ↑m 𝑉) ↦ (𝑔‘𝑓)) ∈ 𝑃) → (𝑔 ∈ (ℤ ↑m 𝑉) ↦ (𝑔‘𝐹)) ∈ 𝑃)
14	1, 9, 13	syl2anc 584	1 ⊢ ((𝑃 ∈ (mzPolyCld‘𝑉) ∧ 𝐹 ∈ 𝑉) → (𝑔 ∈ (ℤ ↑m 𝑉) ↦ (𝑔‘𝐹)) ∈ 𝑃)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1541   ∈ wcel 2110  ∀wral 3045  Vcvv 3434   ⊆ wss 3900  {csn 4574   ↦ cmpt 5170   × cxp 5612  ‘cfv 6477  (class class class)co 7341   ∘_f cof 7603   ↑_m cmap 8745   + caddc 11001   · cmul 11003  ℤcz 12460  mzPolyCldcmzpcl 42733

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 2112  ax-9 2120  ax-10 2143  ax-11 2159  ax-12 2179  ax-ext 2702  ax-sep 5232  ax-nul 5242  ax-pow 5301  ax-pr 5368

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 2067  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rab 3394  df-v 3436  df-dif 3903  df-un 3905  df-in 3907  df-ss 3917  df-nul 4282  df-if 4474  df-pw 4550  df-sn 4575  df-pr 4577  df-op 4581  df-uni 4858  df-br 5090  df-opab 5152  df-mpt 5171  df-id 5509  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-iota 6433  df-fun 6479  df-fv 6485  df-ov 7344  df-mzpcl 42735

This theorem is referenced by:  mzpincl  42746  mzpproj  42749