Theorem mzpcl34 43187

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 1143	. 2 ⊢ ((𝑃 ∈ (mzPolyCld‘𝑉) ∧ 𝐹 ∈ 𝑃 ∧ 𝐺 ∈ 𝑃) → 𝐹 ∈ 𝑃)
2		simp3 1144	. 2 ⊢ ((𝑃 ∈ (mzPolyCld‘𝑉) ∧ 𝐹 ∈ 𝑃 ∧ 𝐺 ∈ 𝑃) → 𝐺 ∈ 𝑃)
3		simp1 1142	. . . 4 ⊢ ((𝑃 ∈ (mzPolyCld‘𝑉) ∧ 𝐹 ∈ 𝑃 ∧ 𝐺 ∈ 𝑃) → 𝑃 ∈ (mzPolyCld‘𝑉))
4	3	elfvexd 6870	. . . . 5 ⊢ ((𝑃 ∈ (mzPolyCld‘𝑉) ∧ 𝐹 ∈ 𝑃 ∧ 𝐺 ∈ 𝑃) → 𝑉 ∈ V)
5		elmzpcl 43182	. . . . 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 233	. . 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 779	. 2 ⊢ ((𝑃 ∈ (mzPolyCld‘𝑉) ∧ 𝐹 ∈ 𝑃 ∧ 𝐺 ∈ 𝑃) → ∀𝑓 ∈ 𝑃 ∀𝑔 ∈ 𝑃 ((𝑓 ∘f + 𝑔) ∈ 𝑃 ∧ (𝑓 ∘f · 𝑔) ∈ 𝑃))
9		oveq1 7370	. . . . 5 ⊢ (𝑓 = 𝐹 → (𝑓 ∘f + 𝑔) = (𝐹 ∘f + 𝑔))
10	9	eleq1d 2825	. . . 4 ⊢ (𝑓 = 𝐹 → ((𝑓 ∘f + 𝑔) ∈ 𝑃 ↔ (𝐹 ∘f + 𝑔) ∈ 𝑃))
11		oveq1 7370	. . . . 5 ⊢ (𝑓 = 𝐹 → (𝑓 ∘f · 𝑔) = (𝐹 ∘f · 𝑔))
12	11	eleq1d 2825	. . . 4 ⊢ (𝑓 = 𝐹 → ((𝑓 ∘f · 𝑔) ∈ 𝑃 ↔ (𝐹 ∘f · 𝑔) ∈ 𝑃))
13	10, 12	anbi12d 638	. . 3 ⊢ (𝑓 = 𝐹 → (((𝑓 ∘f + 𝑔) ∈ 𝑃 ∧ (𝑓 ∘f · 𝑔) ∈ 𝑃) ↔ ((𝐹 ∘f + 𝑔) ∈ 𝑃 ∧ (𝐹 ∘f · 𝑔) ∈ 𝑃)))
14		oveq2 7371	. . . . 5 ⊢ (𝑔 = 𝐺 → (𝐹 ∘f + 𝑔) = (𝐹 ∘f + 𝐺))
15	14	eleq1d 2825	. . . 4 ⊢ (𝑔 = 𝐺 → ((𝐹 ∘f + 𝑔) ∈ 𝑃 ↔ (𝐹 ∘f + 𝐺) ∈ 𝑃))
16		oveq2 7371	. . . . 5 ⊢ (𝑔 = 𝐺 → (𝐹 ∘f · 𝑔) = (𝐹 ∘f · 𝐺))
17	16	eleq1d 2825	. . . 4 ⊢ (𝑔 = 𝐺 → ((𝐹 ∘f · 𝑔) ∈ 𝑃 ↔ (𝐹 ∘f · 𝐺) ∈ 𝑃))
18	15, 17	anbi12d 638	. . 3 ⊢ (𝑔 = 𝐺 → (((𝐹 ∘f + 𝑔) ∈ 𝑃 ∧ (𝐹 ∘f · 𝑔) ∈ 𝑃) ↔ ((𝐹 ∘f + 𝐺) ∈ 𝑃 ∧ (𝐹 ∘f · 𝐺) ∈ 𝑃)))
19	13, 18	rspc2va 3579	. 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 843	1 ⊢ ((𝑃 ∈ (mzPolyCld‘𝑉) ∧ 𝐹 ∈ 𝑃 ∧ 𝐺 ∈ 𝑃) → ((𝐹 ∘f + 𝐺) ∈ 𝑃 ∧ (𝐹 ∘f · 𝐺) ∈ 𝑃))

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   ∧ w3a 1092   = wceq 1547   ∈ wcel 2119  ∀wral 3054  Vcvv 3432   ⊆ wss 3890  {csn 4562   ↦ cmpt 5160   × cxp 5623  ‘cfv 6492  (class class class)co 7363   ∘_f cof 7625   ↑_m cmap 8770   + caddc 11039   · cmul 11041  ℤcz 12522  mzPolyCldcmzpcl 43177

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2712  ax-sep 5225  ax-nul 5235  ax-pow 5301  ax-pr 5369

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2719  df-cleq 2732  df-clel 2815  df-nfc 2889  df-ne 2936  df-ral 3055  df-rex 3065  df-rab 3393  df-v 3434  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4269  df-if 4462  df-pw 4538  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4846  df-br 5080  df-opab 5142  df-mpt 5161  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-iota 6448  df-fun 6494  df-fv 6500  df-ov 7366  df-mzpcl 43179

This theorem is referenced by:  mzpincl  43190  mzpadd  43194  mzpmul  43195