Theorem mzpcl34 42154

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 1134	. 2 ⊢ ((𝑃 ∈ (mzPolyCld‘𝑉) ∧ 𝐹 ∈ 𝑃 ∧ 𝐺 ∈ 𝑃) → 𝐹 ∈ 𝑃)
2		simp3 1135	. 2 ⊢ ((𝑃 ∈ (mzPolyCld‘𝑉) ∧ 𝐹 ∈ 𝑃 ∧ 𝐺 ∈ 𝑃) → 𝐺 ∈ 𝑃)
3		simp1 1133	. . . 4 ⊢ ((𝑃 ∈ (mzPolyCld‘𝑉) ∧ 𝐹 ∈ 𝑃 ∧ 𝐺 ∈ 𝑃) → 𝑃 ∈ (mzPolyCld‘𝑉))
4	3	elfvexd 6939	. . . . 5 ⊢ ((𝑃 ∈ (mzPolyCld‘𝑉) ∧ 𝐹 ∈ 𝑃 ∧ 𝐺 ∈ 𝑃) → 𝑉 ∈ V)
5		elmzpcl 42149	. . . . 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 231	. . 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 772	. 2 ⊢ ((𝑃 ∈ (mzPolyCld‘𝑉) ∧ 𝐹 ∈ 𝑃 ∧ 𝐺 ∈ 𝑃) → ∀𝑓 ∈ 𝑃 ∀𝑔 ∈ 𝑃 ((𝑓 ∘f + 𝑔) ∈ 𝑃 ∧ (𝑓 ∘f · 𝑔) ∈ 𝑃))
9		oveq1 7431	. . . . 5 ⊢ (𝑓 = 𝐹 → (𝑓 ∘f + 𝑔) = (𝐹 ∘f + 𝑔))
10	9	eleq1d 2813	. . . 4 ⊢ (𝑓 = 𝐹 → ((𝑓 ∘f + 𝑔) ∈ 𝑃 ↔ (𝐹 ∘f + 𝑔) ∈ 𝑃))
11		oveq1 7431	. . . . 5 ⊢ (𝑓 = 𝐹 → (𝑓 ∘f · 𝑔) = (𝐹 ∘f · 𝑔))
12	11	eleq1d 2813	. . . 4 ⊢ (𝑓 = 𝐹 → ((𝑓 ∘f · 𝑔) ∈ 𝑃 ↔ (𝐹 ∘f · 𝑔) ∈ 𝑃))
13	10, 12	anbi12d 630	. . 3 ⊢ (𝑓 = 𝐹 → (((𝑓 ∘f + 𝑔) ∈ 𝑃 ∧ (𝑓 ∘f · 𝑔) ∈ 𝑃) ↔ ((𝐹 ∘f + 𝑔) ∈ 𝑃 ∧ (𝐹 ∘f · 𝑔) ∈ 𝑃)))
14		oveq2 7432	. . . . 5 ⊢ (𝑔 = 𝐺 → (𝐹 ∘f + 𝑔) = (𝐹 ∘f + 𝐺))
15	14	eleq1d 2813	. . . 4 ⊢ (𝑔 = 𝐺 → ((𝐹 ∘f + 𝑔) ∈ 𝑃 ↔ (𝐹 ∘f + 𝐺) ∈ 𝑃))
16		oveq2 7432	. . . . 5 ⊢ (𝑔 = 𝐺 → (𝐹 ∘f · 𝑔) = (𝐹 ∘f · 𝐺))
17	16	eleq1d 2813	. . . 4 ⊢ (𝑔 = 𝐺 → ((𝐹 ∘f · 𝑔) ∈ 𝑃 ↔ (𝐹 ∘f · 𝐺) ∈ 𝑃))
18	15, 17	anbi12d 630	. . 3 ⊢ (𝑔 = 𝐺 → (((𝐹 ∘f + 𝑔) ∈ 𝑃 ∧ (𝐹 ∘f · 𝑔) ∈ 𝑃) ↔ ((𝐹 ∘f + 𝐺) ∈ 𝑃 ∧ (𝐹 ∘f · 𝐺) ∈ 𝑃)))
19	13, 18	rspc2va 3621	. 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 836	1 ⊢ ((𝑃 ∈ (mzPolyCld‘𝑉) ∧ 𝐹 ∈ 𝑃 ∧ 𝐺 ∈ 𝑃) → ((𝐹 ∘f + 𝐺) ∈ 𝑃 ∧ (𝐹 ∘f · 𝐺) ∈ 𝑃))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 394   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098  ∀wral 3057  Vcvv 3471   ⊆ wss 3947  {csn 4630   ↦ cmpt 5233   × cxp 5678  ‘cfv 6551  (class class class)co 7424   ∘_f cof 7687   ↑_m cmap 8849   + caddc 11147   · cmul 11149  ℤcz 12594  mzPolyCldcmzpcl 42144

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 2166  ax-ext 2698  ax-sep 5301  ax-nul 5308  ax-pow 5367  ax-pr 5431

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2937  df-ral 3058  df-rex 3067  df-rab 3429  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4325  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4911  df-br 5151  df-opab 5213  df-mpt 5234  df-id 5578  df-xp 5686  df-rel 5687  df-cnv 5688  df-co 5689  df-dm 5690  df-iota 6503  df-fun 6553  df-fv 6559  df-ov 7427  df-mzpcl 42146

This theorem is referenced by:  mzpincl  42157  mzpadd  42161  mzpmul  42162