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Theorem mzpcl34 39461
 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) ∧ 𝐹𝑃𝐺𝑃) → ((𝐹f + 𝐺) ∈ 𝑃 ∧ (𝐹f · 𝐺) ∈ 𝑃))

Proof of Theorem mzpcl34
Dummy variables 𝑓 𝑔 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simp2 1133 . 2 ((𝑃 ∈ (mzPolyCld‘𝑉) ∧ 𝐹𝑃𝐺𝑃) → 𝐹𝑃)
2 simp3 1134 . 2 ((𝑃 ∈ (mzPolyCld‘𝑉) ∧ 𝐹𝑃𝐺𝑃) → 𝐺𝑃)
3 simp1 1132 . . . 4 ((𝑃 ∈ (mzPolyCld‘𝑉) ∧ 𝐹𝑃𝐺𝑃) → 𝑃 ∈ (mzPolyCld‘𝑉))
43elfvexd 6676 . . . . 5 ((𝑃 ∈ (mzPolyCld‘𝑉) ∧ 𝐹𝑃𝐺𝑃) → 𝑉 ∈ V)
5 elmzpcl 39456 . . . . 5 (𝑉 ∈ V → (𝑃 ∈ (mzPolyCld‘𝑉) ↔ (𝑃 ⊆ (ℤ ↑m (ℤ ↑m 𝑉)) ∧ ((∀𝑓 ∈ ℤ ((ℤ ↑m 𝑉) × {𝑓}) ∈ 𝑃 ∧ ∀𝑓𝑉 (𝑔 ∈ (ℤ ↑m 𝑉) ↦ (𝑔𝑓)) ∈ 𝑃) ∧ ∀𝑓𝑃𝑔𝑃 ((𝑓f + 𝑔) ∈ 𝑃 ∧ (𝑓f · 𝑔) ∈ 𝑃)))))
64, 5syl 17 . . . 4 ((𝑃 ∈ (mzPolyCld‘𝑉) ∧ 𝐹𝑃𝐺𝑃) → (𝑃 ∈ (mzPolyCld‘𝑉) ↔ (𝑃 ⊆ (ℤ ↑m (ℤ ↑m 𝑉)) ∧ ((∀𝑓 ∈ ℤ ((ℤ ↑m 𝑉) × {𝑓}) ∈ 𝑃 ∧ ∀𝑓𝑉 (𝑔 ∈ (ℤ ↑m 𝑉) ↦ (𝑔𝑓)) ∈ 𝑃) ∧ ∀𝑓𝑃𝑔𝑃 ((𝑓f + 𝑔) ∈ 𝑃 ∧ (𝑓f · 𝑔) ∈ 𝑃)))))
73, 6mpbid 234 . . 3 ((𝑃 ∈ (mzPolyCld‘𝑉) ∧ 𝐹𝑃𝐺𝑃) → (𝑃 ⊆ (ℤ ↑m (ℤ ↑m 𝑉)) ∧ ((∀𝑓 ∈ ℤ ((ℤ ↑m 𝑉) × {𝑓}) ∈ 𝑃 ∧ ∀𝑓𝑉 (𝑔 ∈ (ℤ ↑m 𝑉) ↦ (𝑔𝑓)) ∈ 𝑃) ∧ ∀𝑓𝑃𝑔𝑃 ((𝑓f + 𝑔) ∈ 𝑃 ∧ (𝑓f · 𝑔) ∈ 𝑃))))
87simprrd 772 . 2 ((𝑃 ∈ (mzPolyCld‘𝑉) ∧ 𝐹𝑃𝐺𝑃) → ∀𝑓𝑃𝑔𝑃 ((𝑓f + 𝑔) ∈ 𝑃 ∧ (𝑓f · 𝑔) ∈ 𝑃))
9 oveq1 7136 . . . . 5 (𝑓 = 𝐹 → (𝑓f + 𝑔) = (𝐹f + 𝑔))
109eleq1d 2895 . . . 4 (𝑓 = 𝐹 → ((𝑓f + 𝑔) ∈ 𝑃 ↔ (𝐹f + 𝑔) ∈ 𝑃))
11 oveq1 7136 . . . . 5 (𝑓 = 𝐹 → (𝑓f · 𝑔) = (𝐹f · 𝑔))
1211eleq1d 2895 . . . 4 (𝑓 = 𝐹 → ((𝑓f · 𝑔) ∈ 𝑃 ↔ (𝐹f · 𝑔) ∈ 𝑃))
1310, 12anbi12d 632 . . 3 (𝑓 = 𝐹 → (((𝑓f + 𝑔) ∈ 𝑃 ∧ (𝑓f · 𝑔) ∈ 𝑃) ↔ ((𝐹f + 𝑔) ∈ 𝑃 ∧ (𝐹f · 𝑔) ∈ 𝑃)))
14 oveq2 7137 . . . . 5 (𝑔 = 𝐺 → (𝐹f + 𝑔) = (𝐹f + 𝐺))
1514eleq1d 2895 . . . 4 (𝑔 = 𝐺 → ((𝐹f + 𝑔) ∈ 𝑃 ↔ (𝐹f + 𝐺) ∈ 𝑃))
16 oveq2 7137 . . . . 5 (𝑔 = 𝐺 → (𝐹f · 𝑔) = (𝐹f · 𝐺))
1716eleq1d 2895 . . . 4 (𝑔 = 𝐺 → ((𝐹f · 𝑔) ∈ 𝑃 ↔ (𝐹f · 𝐺) ∈ 𝑃))
1815, 17anbi12d 632 . . 3 (𝑔 = 𝐺 → (((𝐹f + 𝑔) ∈ 𝑃 ∧ (𝐹f · 𝑔) ∈ 𝑃) ↔ ((𝐹f + 𝐺) ∈ 𝑃 ∧ (𝐹f · 𝐺) ∈ 𝑃)))
1913, 18rspc2va 3610 . 2 (((𝐹𝑃𝐺𝑃) ∧ ∀𝑓𝑃𝑔𝑃 ((𝑓f + 𝑔) ∈ 𝑃 ∧ (𝑓f · 𝑔) ∈ 𝑃)) → ((𝐹f + 𝐺) ∈ 𝑃 ∧ (𝐹f · 𝐺) ∈ 𝑃))
201, 2, 8, 19syl21anc 835 1 ((𝑃 ∈ (mzPolyCld‘𝑉) ∧ 𝐹𝑃𝐺𝑃) → ((𝐹f + 𝐺) ∈ 𝑃 ∧ (𝐹f · 𝐺) ∈ 𝑃))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   ∧ w3a 1083   = wceq 1537   ∈ wcel 2114  ∀wral 3125  Vcvv 3470   ⊆ wss 3909  {csn 4539   ↦ cmpt 5118   × cxp 5525  ‘cfv 6327  (class class class)co 7129   ∘f cof 7381   ↑m cmap 8380   + caddc 10514   · cmul 10516  ℤcz 11956  mzPolyCldcmzpcl 39451 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2792  ax-sep 5175  ax-nul 5182  ax-pow 5238  ax-pr 5302 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2799  df-cleq 2813  df-clel 2891  df-nfc 2959  df-ral 3130  df-rex 3131  df-rab 3134  df-v 3472  df-sbc 3749  df-dif 3912  df-un 3914  df-in 3916  df-ss 3926  df-nul 4266  df-if 4440  df-pw 4513  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4811  df-br 5039  df-opab 5101  df-mpt 5119  df-id 5432  df-xp 5533  df-rel 5534  df-cnv 5535  df-co 5536  df-dm 5537  df-iota 6286  df-fun 6329  df-fv 6335  df-ov 7132  df-mzpcl 39453 This theorem is referenced by:  mzpincl  39464  mzpadd  39468  mzpmul  39469
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