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Theorem mzpcl34 36774
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 1060 . 2 ((𝑃 ∈ (mzPolyCld‘𝑉) ∧ 𝐹𝑃𝐺𝑃) → 𝐹𝑃)
2 simp3 1061 . 2 ((𝑃 ∈ (mzPolyCld‘𝑉) ∧ 𝐹𝑃𝐺𝑃) → 𝐺𝑃)
3 simp1 1059 . . . 4 ((𝑃 ∈ (mzPolyCld‘𝑉) ∧ 𝐹𝑃𝐺𝑃) → 𝑃 ∈ (mzPolyCld‘𝑉))
43elfvexd 6179 . . . . 5 ((𝑃 ∈ (mzPolyCld‘𝑉) ∧ 𝐹𝑃𝐺𝑃) → 𝑉 ∈ V)
5 elmzpcl 36769 . . . . 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 222 . . 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 796 . 2 ((𝑃 ∈ (mzPolyCld‘𝑉) ∧ 𝐹𝑃𝐺𝑃) → ∀𝑓𝑃𝑔𝑃 ((𝑓𝑓 + 𝑔) ∈ 𝑃 ∧ (𝑓𝑓 · 𝑔) ∈ 𝑃))
9 oveq1 6611 . . . . 5 (𝑓 = 𝐹 → (𝑓𝑓 + 𝑔) = (𝐹𝑓 + 𝑔))
109eleq1d 2683 . . . 4 (𝑓 = 𝐹 → ((𝑓𝑓 + 𝑔) ∈ 𝑃 ↔ (𝐹𝑓 + 𝑔) ∈ 𝑃))
11 oveq1 6611 . . . . 5 (𝑓 = 𝐹 → (𝑓𝑓 · 𝑔) = (𝐹𝑓 · 𝑔))
1211eleq1d 2683 . . . 4 (𝑓 = 𝐹 → ((𝑓𝑓 · 𝑔) ∈ 𝑃 ↔ (𝐹𝑓 · 𝑔) ∈ 𝑃))
1310, 12anbi12d 746 . . 3 (𝑓 = 𝐹 → (((𝑓𝑓 + 𝑔) ∈ 𝑃 ∧ (𝑓𝑓 · 𝑔) ∈ 𝑃) ↔ ((𝐹𝑓 + 𝑔) ∈ 𝑃 ∧ (𝐹𝑓 · 𝑔) ∈ 𝑃)))
14 oveq2 6612 . . . . 5 (𝑔 = 𝐺 → (𝐹𝑓 + 𝑔) = (𝐹𝑓 + 𝐺))
1514eleq1d 2683 . . . 4 (𝑔 = 𝐺 → ((𝐹𝑓 + 𝑔) ∈ 𝑃 ↔ (𝐹𝑓 + 𝐺) ∈ 𝑃))
16 oveq2 6612 . . . . 5 (𝑔 = 𝐺 → (𝐹𝑓 · 𝑔) = (𝐹𝑓 · 𝐺))
1716eleq1d 2683 . . . 4 (𝑔 = 𝐺 → ((𝐹𝑓 · 𝑔) ∈ 𝑃 ↔ (𝐹𝑓 · 𝐺) ∈ 𝑃))
1815, 17anbi12d 746 . . 3 (𝑔 = 𝐺 → (((𝐹𝑓 + 𝑔) ∈ 𝑃 ∧ (𝐹𝑓 · 𝑔) ∈ 𝑃) ↔ ((𝐹𝑓 + 𝐺) ∈ 𝑃 ∧ (𝐹𝑓 · 𝐺) ∈ 𝑃)))
1913, 18rspc2va 3307 . 2 (((𝐹𝑃𝐺𝑃) ∧ ∀𝑓𝑃𝑔𝑃 ((𝑓𝑓 + 𝑔) ∈ 𝑃 ∧ (𝑓𝑓 · 𝑔) ∈ 𝑃)) → ((𝐹𝑓 + 𝐺) ∈ 𝑃 ∧ (𝐹𝑓 · 𝐺) ∈ 𝑃))
201, 2, 8, 19syl21anc 1322 1 ((𝑃 ∈ (mzPolyCld‘𝑉) ∧ 𝐹𝑃𝐺𝑃) → ((𝐹𝑓 + 𝐺) ∈ 𝑃 ∧ (𝐹𝑓 · 𝐺) ∈ 𝑃))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  w3a 1036   = wceq 1480  wcel 1987  wral 2907  Vcvv 3186  wss 3555  {csn 4148  cmpt 4673   × cxp 5072  cfv 5847  (class class class)co 6604  𝑓 cof 6848  𝑚 cmap 7802   + caddc 9883   · cmul 9885  cz 11321  mzPolyCldcmzpcl 36764
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-iota 5810  df-fun 5849  df-fv 5855  df-ov 6607  df-mzpcl 36766
This theorem is referenced by:  mzpincl  36777  mzpadd  36781  mzpmul  36782
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