Theorem mzpcl2 41401

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 486	. 2 ⊢ ((𝑃 ∈ (mzPolyCld‘𝑉) ∧ 𝐹 ∈ 𝑉) → 𝐹 ∈ 𝑉)
2		simpl 484	. . . 4 ⊢ ((𝑃 ∈ (mzPolyCld‘𝑉) ∧ 𝐹 ∈ 𝑉) → 𝑃 ∈ (mzPolyCld‘𝑉))
3		elfvex 6926	. . . . . 6 ⊢ (𝑃 ∈ (mzPolyCld‘𝑉) → 𝑉 ∈ V)
4	3	adantr 482	. . . . 5 ⊢ ((𝑃 ∈ (mzPolyCld‘𝑉) ∧ 𝐹 ∈ 𝑉) → 𝑉 ∈ V)
5		elmzpcl 41397	. . . . 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 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 6888	. . . . 5 ⊢ (𝑓 = 𝐹 → (𝑔‘𝑓) = (𝑔‘𝐹))
11	10	mpteq2dv 5249	. . . 4 ⊢ (𝑓 = 𝐹 → (𝑔 ∈ (ℤ ↑m 𝑉) ↦ (𝑔‘𝑓)) = (𝑔 ∈ (ℤ ↑m 𝑉) ↦ (𝑔‘𝐹)))
12	11	eleq1d 2819	. . 3 ⊢ (𝑓 = 𝐹 → ((𝑔 ∈ (ℤ ↑m 𝑉) ↦ (𝑔‘𝑓)) ∈ 𝑃 ↔ (𝑔 ∈ (ℤ ↑m 𝑉) ↦ (𝑔‘𝐹)) ∈ 𝑃))
13	12	rspcva 3610	. 2 ⊢ ((𝐹 ∈ 𝑉 ∧ ∀𝑓 ∈ 𝑉 (𝑔 ∈ (ℤ ↑m 𝑉) ↦ (𝑔‘𝑓)) ∈ 𝑃) → (𝑔 ∈ (ℤ ↑m 𝑉) ↦ (𝑔‘𝐹)) ∈ 𝑃)
14	1, 9, 13	syl2anc 585	1 ⊢ ((𝑃 ∈ (mzPolyCld‘𝑉) ∧ 𝐹 ∈ 𝑉) → (𝑔 ∈ (ℤ ↑m 𝑉) ↦ (𝑔‘𝐹)) ∈ 𝑃)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 397   = wceq 1542   ∈ wcel 2107  ∀wral 3062  Vcvv 3475   ⊆ wss 3947  {csn 4627   ↦ cmpt 5230   × cxp 5673  ‘cfv 6540  (class class class)co 7404   ∘_f cof 7663   ↑_m cmap 8816   + caddc 11109   · cmul 11111  ℤcz 12554  mzPolyCldcmzpcl 41392

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-iota 6492  df-fun 6542  df-fv 6548  df-ov 7407  df-mzpcl 41394

This theorem is referenced by:  mzpincl  41405  mzpproj  41408