Theorem mzpcl34 37294

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 1062	. 2 ⊢ ((𝑃 ∈ (mzPolyCld‘𝑉) ∧ 𝐹 ∈ 𝑃 ∧ 𝐺 ∈ 𝑃) → 𝐹 ∈ 𝑃)
2		simp3 1063	. 2 ⊢ ((𝑃 ∈ (mzPolyCld‘𝑉) ∧ 𝐹 ∈ 𝑃 ∧ 𝐺 ∈ 𝑃) → 𝐺 ∈ 𝑃)
3		simp1 1061	. . . 4 ⊢ ((𝑃 ∈ (mzPolyCld‘𝑉) ∧ 𝐹 ∈ 𝑃 ∧ 𝐺 ∈ 𝑃) → 𝑃 ∈ (mzPolyCld‘𝑉))
4	3	elfvexd 6222	. . . . 5 ⊢ ((𝑃 ∈ (mzPolyCld‘𝑉) ∧ 𝐹 ∈ 𝑃 ∧ 𝐺 ∈ 𝑃) → 𝑉 ∈ V)
5		elmzpcl 37289	. . . . 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 222	. . 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 797	. 2 ⊢ ((𝑃 ∈ (mzPolyCld‘𝑉) ∧ 𝐹 ∈ 𝑃 ∧ 𝐺 ∈ 𝑃) → ∀𝑓 ∈ 𝑃 ∀𝑔 ∈ 𝑃 ((𝑓 ∘𝑓 + 𝑔) ∈ 𝑃 ∧ (𝑓 ∘𝑓 · 𝑔) ∈ 𝑃))
9		oveq1 6657	. . . . 5 ⊢ (𝑓 = 𝐹 → (𝑓 ∘𝑓 + 𝑔) = (𝐹 ∘𝑓 + 𝑔))
10	9	eleq1d 2686	. . . 4 ⊢ (𝑓 = 𝐹 → ((𝑓 ∘𝑓 + 𝑔) ∈ 𝑃 ↔ (𝐹 ∘𝑓 + 𝑔) ∈ 𝑃))
11		oveq1 6657	. . . . 5 ⊢ (𝑓 = 𝐹 → (𝑓 ∘𝑓 · 𝑔) = (𝐹 ∘𝑓 · 𝑔))
12	11	eleq1d 2686	. . . 4 ⊢ (𝑓 = 𝐹 → ((𝑓 ∘𝑓 · 𝑔) ∈ 𝑃 ↔ (𝐹 ∘𝑓 · 𝑔) ∈ 𝑃))
13	10, 12	anbi12d 747	. . 3 ⊢ (𝑓 = 𝐹 → (((𝑓 ∘𝑓 + 𝑔) ∈ 𝑃 ∧ (𝑓 ∘𝑓 · 𝑔) ∈ 𝑃) ↔ ((𝐹 ∘𝑓 + 𝑔) ∈ 𝑃 ∧ (𝐹 ∘𝑓 · 𝑔) ∈ 𝑃)))
14		oveq2 6658	. . . . 5 ⊢ (𝑔 = 𝐺 → (𝐹 ∘𝑓 + 𝑔) = (𝐹 ∘𝑓 + 𝐺))
15	14	eleq1d 2686	. . . 4 ⊢ (𝑔 = 𝐺 → ((𝐹 ∘𝑓 + 𝑔) ∈ 𝑃 ↔ (𝐹 ∘𝑓 + 𝐺) ∈ 𝑃))
16		oveq2 6658	. . . . 5 ⊢ (𝑔 = 𝐺 → (𝐹 ∘𝑓 · 𝑔) = (𝐹 ∘𝑓 · 𝐺))
17	16	eleq1d 2686	. . . 4 ⊢ (𝑔 = 𝐺 → ((𝐹 ∘𝑓 · 𝑔) ∈ 𝑃 ↔ (𝐹 ∘𝑓 · 𝐺) ∈ 𝑃))
18	15, 17	anbi12d 747	. . 3 ⊢ (𝑔 = 𝐺 → (((𝐹 ∘𝑓 + 𝑔) ∈ 𝑃 ∧ (𝐹 ∘𝑓 · 𝑔) ∈ 𝑃) ↔ ((𝐹 ∘𝑓 + 𝐺) ∈ 𝑃 ∧ (𝐹 ∘𝑓 · 𝐺) ∈ 𝑃)))
19	13, 18	rspc2va 3323	. 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 1325	1 ⊢ ((𝑃 ∈ (mzPolyCld‘𝑉) ∧ 𝐹 ∈ 𝑃 ∧ 𝐺 ∈ 𝑃) → ((𝐹 ∘𝑓 + 𝐺) ∈ 𝑃 ∧ (𝐹 ∘𝑓 · 𝐺) ∈ 𝑃))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990  ∀wral 2912  Vcvv 3200   ⊆ wss 3574  {csn 4177   ↦ cmpt 4729   × cxp 5112  ‘cfv 5888  (class class class)co 6650   ∘_𝑓 cof 6895   ↑_𝑚 cmap 7857   + caddc 9939   · cmul 9941  ℤcz 11377  mzPolyCldcmzpcl 37284

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fv 5896  df-ov 6653  df-mzpcl 37286

This theorem is referenced by:  mzpincl  37297  mzpadd  37301  mzpmul  37302