Theorem mzpcl34 43503

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 1155	. 2 ⊢ ((𝑃 ∈ (mzPolyCld‘𝑉) ∧ 𝐹 ∈ 𝑃 ∧ 𝐺 ∈ 𝑃) → 𝐹 ∈ 𝑃)
2		simp3 1156	. 2 ⊢ ((𝑃 ∈ (mzPolyCld‘𝑉) ∧ 𝐹 ∈ 𝑃 ∧ 𝐺 ∈ 𝑃) → 𝐺 ∈ 𝑃)
3		simp1 1154	. . . 4 ⊢ ((𝑃 ∈ (mzPolyCld‘𝑉) ∧ 𝐹 ∈ 𝑃 ∧ 𝐺 ∈ 𝑃) → 𝑃 ∈ (mzPolyCld‘𝑉))
4	3	elfvexd 6924	. . . . 5 ⊢ ((𝑃 ∈ (mzPolyCld‘𝑉) ∧ 𝐹 ∈ 𝑃 ∧ 𝐺 ∈ 𝑃) → 𝑉 ∈ V)
5		elmzpcl 43498	. . . . 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 18	. . . 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 235	. . 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 786	. 2 ⊢ ((𝑃 ∈ (mzPolyCld‘𝑉) ∧ 𝐹 ∈ 𝑃 ∧ 𝐺 ∈ 𝑃) → ∀𝑓 ∈ 𝑃 ∀𝑔 ∈ 𝑃 ((𝑓 ∘f + 𝑔) ∈ 𝑃 ∧ (𝑓 ∘f · 𝑔) ∈ 𝑃))
9		oveq1 7430	. . . . 5 ⊢ (𝑓 = 𝐹 → (𝑓 ∘f + 𝑔) = (𝐹 ∘f + 𝑔))
10	9	eleq1d 2851	. . . 4 ⊢ (𝑓 = 𝐹 → ((𝑓 ∘f + 𝑔) ∈ 𝑃 ↔ (𝐹 ∘f + 𝑔) ∈ 𝑃))
11		oveq1 7430	. . . . 5 ⊢ (𝑓 = 𝐹 → (𝑓 ∘f · 𝑔) = (𝐹 ∘f · 𝑔))
12	11	eleq1d 2851	. . . 4 ⊢ (𝑓 = 𝐹 → ((𝑓 ∘f · 𝑔) ∈ 𝑃 ↔ (𝐹 ∘f · 𝑔) ∈ 𝑃))
13	10, 12	anbi12d 644	. . 3 ⊢ (𝑓 = 𝐹 → (((𝑓 ∘f + 𝑔) ∈ 𝑃 ∧ (𝑓 ∘f · 𝑔) ∈ 𝑃) ↔ ((𝐹 ∘f + 𝑔) ∈ 𝑃 ∧ (𝐹 ∘f · 𝑔) ∈ 𝑃)))
14		oveq2 7431	. . . . 5 ⊢ (𝑔 = 𝐺 → (𝐹 ∘f + 𝑔) = (𝐹 ∘f + 𝐺))
15	14	eleq1d 2851	. . . 4 ⊢ (𝑔 = 𝐺 → ((𝐹 ∘f + 𝑔) ∈ 𝑃 ↔ (𝐹 ∘f + 𝐺) ∈ 𝑃))
16		oveq2 7431	. . . . 5 ⊢ (𝑔 = 𝐺 → (𝐹 ∘f · 𝑔) = (𝐹 ∘f · 𝐺))
17	16	eleq1d 2851	. . . 4 ⊢ (𝑔 = 𝐺 → ((𝐹 ∘f · 𝑔) ∈ 𝑃 ↔ (𝐹 ∘f · 𝐺) ∈ 𝑃))
18	15, 17	anbi12d 644	. . 3 ⊢ (𝑔 = 𝐺 → (((𝐹 ∘f + 𝑔) ∈ 𝑃 ∧ (𝐹 ∘f · 𝑔) ∈ 𝑃) ↔ ((𝐹 ∘f + 𝐺) ∈ 𝑃 ∧ (𝐹 ∘f · 𝐺) ∈ 𝑃)))
19	13, 18	rspc2va 3596	. 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 851	1 ⊢ ((𝑃 ∈ (mzPolyCld‘𝑉) ∧ 𝐹 ∈ 𝑃 ∧ 𝐺 ∈ 𝑃) → ((𝐹 ∘f + 𝐺) ∈ 𝑃 ∧ (𝐹 ∘f · 𝐺) ∈ 𝑃))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2146  ∀wral 3082  Vcvv 3458   ⊆ wss 3908  {csn 4594   ↦ cmpt 5197   × cxp 5664  ‘cfv 6543  (class class class)co 7423   ∘_f cof 7685   ↑_m cmap 8833   + caddc 11121   · cmul 11123  ℤcz 12609  mzPolyCldcmzpcl 43493

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2738  ax-sep 5262  ax-nul 5274  ax-pow 5341  ax-pr 5409

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2570  df-eu 2600  df-clab 2745  df-cleq 2758  df-clel 2841  df-nfc 2915  df-ne 2962  df-ral 3083  df-rex 3093  df-rab 3420  df-v 3460  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4290  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4878  df-br 5115  df-opab 5179  df-mpt 5198  df-id 5561  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-iota 6499  df-fun 6545  df-fv 6551  df-ov 7426  df-mzpcl 43495

This theorem is used by:  mzpincl  43506  mzpadd  43510  mzpmul  43511