Theorem mzpcl34 42687

Description: Defining properties 3 and 4 of a polynomially closed function set 𝑃: it is closed under pointwise addition and multiplication. (Contributed by Stefan O'Rear, 4-Oct-2014.)

Assertion

Ref	Expression
mzpcl34	⊢ ((𝑃 ∈ (mzPolyCld‘𝑉) ∧ 𝐹 ∈ 𝑃 ∧ 𝐺 ∈ 𝑃) → ((𝐹 ∘f + 𝐺) ∈ 𝑃 ∧ (𝐹 ∘f · 𝐺) ∈ 𝑃))

Proof of Theorem mzpcl34

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

Step	Hyp	Ref	Expression
1		simp2 1137	. 2 ⊢ ((𝑃 ∈ (mzPolyCld‘𝑉) ∧ 𝐹 ∈ 𝑃 ∧ 𝐺 ∈ 𝑃) → 𝐹 ∈ 𝑃)
2		simp3 1138	. 2 ⊢ ((𝑃 ∈ (mzPolyCld‘𝑉) ∧ 𝐹 ∈ 𝑃 ∧ 𝐺 ∈ 𝑃) → 𝐺 ∈ 𝑃)
3		simp1 1136	. . . 4 ⊢ ((𝑃 ∈ (mzPolyCld‘𝑉) ∧ 𝐹 ∈ 𝑃 ∧ 𝐺 ∈ 𝑃) → 𝑃 ∈ (mzPolyCld‘𝑉))
4	3	elfvexd 6959	. . . . 5 ⊢ ((𝑃 ∈ (mzPolyCld‘𝑉) ∧ 𝐹 ∈ 𝑃 ∧ 𝐺 ∈ 𝑃) → 𝑉 ∈ V)
5		elmzpcl 42682	. . . . 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 ∈ 𝑃 ∧ 𝐺 ∈ 𝑃) → (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	3, 6	mpbid 232	. . 3 ⊢ ((𝑃 ∈ (mzPolyCld‘𝑉) ∧ 𝐹 ∈ 𝑃 ∧ 𝐺 ∈ 𝑃) → (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	7	simprrd 773	. 2 ⊢ ((𝑃 ∈ (mzPolyCld‘𝑉) ∧ 𝐹 ∈ 𝑃 ∧ 𝐺 ∈ 𝑃) → ∀𝑓 ∈ 𝑃 ∀𝑔 ∈ 𝑃 ((𝑓 ∘f + 𝑔) ∈ 𝑃 ∧ (𝑓 ∘f · 𝑔) ∈ 𝑃))
9		oveq1 7455	. . . . 5 ⊢ (𝑓 = 𝐹 → (𝑓 ∘f + 𝑔) = (𝐹 ∘f + 𝑔))
10	9	eleq1d 2829	. . . 4 ⊢ (𝑓 = 𝐹 → ((𝑓 ∘f + 𝑔) ∈ 𝑃 ↔ (𝐹 ∘f + 𝑔) ∈ 𝑃))
11		oveq1 7455	. . . . 5 ⊢ (𝑓 = 𝐹 → (𝑓 ∘f · 𝑔) = (𝐹 ∘f · 𝑔))
12	11	eleq1d 2829	. . . 4 ⊢ (𝑓 = 𝐹 → ((𝑓 ∘f · 𝑔) ∈ 𝑃 ↔ (𝐹 ∘f · 𝑔) ∈ 𝑃))
13	10, 12	anbi12d 631	. . 3 ⊢ (𝑓 = 𝐹 → (((𝑓 ∘f + 𝑔) ∈ 𝑃 ∧ (𝑓 ∘f · 𝑔) ∈ 𝑃) ↔ ((𝐹 ∘f + 𝑔) ∈ 𝑃 ∧ (𝐹 ∘f · 𝑔) ∈ 𝑃)))
14		oveq2 7456	. . . . 5 ⊢ (𝑔 = 𝐺 → (𝐹 ∘f + 𝑔) = (𝐹 ∘f + 𝐺))
15	14	eleq1d 2829	. . . 4 ⊢ (𝑔 = 𝐺 → ((𝐹 ∘f + 𝑔) ∈ 𝑃 ↔ (𝐹 ∘f + 𝐺) ∈ 𝑃))
16		oveq2 7456	. . . . 5 ⊢ (𝑔 = 𝐺 → (𝐹 ∘f · 𝑔) = (𝐹 ∘f · 𝐺))
17	16	eleq1d 2829	. . . 4 ⊢ (𝑔 = 𝐺 → ((𝐹 ∘f · 𝑔) ∈ 𝑃 ↔ (𝐹 ∘f · 𝐺) ∈ 𝑃))
18	15, 17	anbi12d 631	. . 3 ⊢ (𝑔 = 𝐺 → (((𝐹 ∘f + 𝑔) ∈ 𝑃 ∧ (𝐹 ∘f · 𝑔) ∈ 𝑃) ↔ ((𝐹 ∘f + 𝐺) ∈ 𝑃 ∧ (𝐹 ∘f · 𝐺) ∈ 𝑃)))
19	13, 18	rspc2va 3647	. 2 ⊢ (((𝐹 ∈ 𝑃 ∧ 𝐺 ∈ 𝑃) ∧ ∀𝑓 ∈ 𝑃 ∀𝑔 ∈ 𝑃 ((𝑓 ∘f + 𝑔) ∈ 𝑃 ∧ (𝑓 ∘f · 𝑔) ∈ 𝑃)) → ((𝐹 ∘f + 𝐺) ∈ 𝑃 ∧ (𝐹 ∘f · 𝐺) ∈ 𝑃))
20	1, 2, 8, 19	syl21anc 837	1 ⊢ ((𝑃 ∈ (mzPolyCld‘𝑉) ∧ 𝐹 ∈ 𝑃 ∧ 𝐺 ∈ 𝑃) → ((𝐹 ∘f + 𝐺) ∈ 𝑃 ∧ (𝐹 ∘f · 𝐺) ∈ 𝑃))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1537   ∈ wcel 2108  ∀wral 3067  Vcvv 3488   ⊆ wss 3976  {csn 4648   ↦ cmpt 5249   × cxp 5698  ‘cfv 6573  (class class class)co 7448   ∘_f cof 7712   ↑_m cmap 8884   + caddc 11187   · cmul 11189  ℤcz 12639  mzPolyCldcmzpcl 42677

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-iota 6525  df-fun 6575  df-fv 6581  df-ov 7451  df-mzpcl 42679

This theorem is referenced by:  mzpincl  42690  mzpadd  42694  mzpmul  42695