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Theorem mzpcl34 37815
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 1132 . 2 ((𝑃 ∈ (mzPolyCld‘𝑉) ∧ 𝐹𝑃𝐺𝑃) → 𝐹𝑃)
2 simp3 1133 . 2 ((𝑃 ∈ (mzPolyCld‘𝑉) ∧ 𝐹𝑃𝐺𝑃) → 𝐺𝑃)
3 simp1 1131 . . . 4 ((𝑃 ∈ (mzPolyCld‘𝑉) ∧ 𝐹𝑃𝐺𝑃) → 𝑃 ∈ (mzPolyCld‘𝑉))
43elfvexd 6385 . . . . 5 ((𝑃 ∈ (mzPolyCld‘𝑉) ∧ 𝐹𝑃𝐺𝑃) → 𝑉 ∈ V)
5 elmzpcl 37810 . . . . 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 814 . 2 ((𝑃 ∈ (mzPolyCld‘𝑉) ∧ 𝐹𝑃𝐺𝑃) → ∀𝑓𝑃𝑔𝑃 ((𝑓𝑓 + 𝑔) ∈ 𝑃 ∧ (𝑓𝑓 · 𝑔) ∈ 𝑃))
9 oveq1 6822 . . . . 5 (𝑓 = 𝐹 → (𝑓𝑓 + 𝑔) = (𝐹𝑓 + 𝑔))
109eleq1d 2825 . . . 4 (𝑓 = 𝐹 → ((𝑓𝑓 + 𝑔) ∈ 𝑃 ↔ (𝐹𝑓 + 𝑔) ∈ 𝑃))
11 oveq1 6822 . . . . 5 (𝑓 = 𝐹 → (𝑓𝑓 · 𝑔) = (𝐹𝑓 · 𝑔))
1211eleq1d 2825 . . . 4 (𝑓 = 𝐹 → ((𝑓𝑓 · 𝑔) ∈ 𝑃 ↔ (𝐹𝑓 · 𝑔) ∈ 𝑃))
1310, 12anbi12d 749 . . 3 (𝑓 = 𝐹 → (((𝑓𝑓 + 𝑔) ∈ 𝑃 ∧ (𝑓𝑓 · 𝑔) ∈ 𝑃) ↔ ((𝐹𝑓 + 𝑔) ∈ 𝑃 ∧ (𝐹𝑓 · 𝑔) ∈ 𝑃)))
14 oveq2 6823 . . . . 5 (𝑔 = 𝐺 → (𝐹𝑓 + 𝑔) = (𝐹𝑓 + 𝐺))
1514eleq1d 2825 . . . 4 (𝑔 = 𝐺 → ((𝐹𝑓 + 𝑔) ∈ 𝑃 ↔ (𝐹𝑓 + 𝐺) ∈ 𝑃))
16 oveq2 6823 . . . . 5 (𝑔 = 𝐺 → (𝐹𝑓 · 𝑔) = (𝐹𝑓 · 𝐺))
1716eleq1d 2825 . . . 4 (𝑔 = 𝐺 → ((𝐹𝑓 · 𝑔) ∈ 𝑃 ↔ (𝐹𝑓 · 𝐺) ∈ 𝑃))
1815, 17anbi12d 749 . . 3 (𝑔 = 𝐺 → (((𝐹𝑓 + 𝑔) ∈ 𝑃 ∧ (𝐹𝑓 · 𝑔) ∈ 𝑃) ↔ ((𝐹𝑓 + 𝐺) ∈ 𝑃 ∧ (𝐹𝑓 · 𝐺) ∈ 𝑃)))
1913, 18rspc2va 3463 . 2 (((𝐹𝑃𝐺𝑃) ∧ ∀𝑓𝑃𝑔𝑃 ((𝑓𝑓 + 𝑔) ∈ 𝑃 ∧ (𝑓𝑓 · 𝑔) ∈ 𝑃)) → ((𝐹𝑓 + 𝐺) ∈ 𝑃 ∧ (𝐹𝑓 · 𝐺) ∈ 𝑃))
201, 2, 8, 19syl21anc 1476 1 ((𝑃 ∈ (mzPolyCld‘𝑉) ∧ 𝐹𝑃𝐺𝑃) → ((𝐹𝑓 + 𝐺) ∈ 𝑃 ∧ (𝐹𝑓 · 𝐺) ∈ 𝑃))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  w3a 1072   = wceq 1632  wcel 2140  wral 3051  Vcvv 3341  wss 3716  {csn 4322  cmpt 4882   × cxp 5265  cfv 6050  (class class class)co 6815  𝑓 cof 7062  𝑚 cmap 8026   + caddc 10152   · cmul 10154  cz 11590  mzPolyCldcmzpcl 37805
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1989  ax-6 2055  ax-7 2091  ax-8 2142  ax-9 2149  ax-10 2169  ax-11 2184  ax-12 2197  ax-13 2392  ax-ext 2741  ax-sep 4934  ax-nul 4942  ax-pow 4993  ax-pr 5056
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2048  df-eu 2612  df-mo 2613  df-clab 2748  df-cleq 2754  df-clel 2757  df-nfc 2892  df-ne 2934  df-ral 3056  df-rex 3057  df-rab 3060  df-v 3343  df-sbc 3578  df-dif 3719  df-un 3721  df-in 3723  df-ss 3730  df-nul 4060  df-if 4232  df-pw 4305  df-sn 4323  df-pr 4325  df-op 4329  df-uni 4590  df-br 4806  df-opab 4866  df-mpt 4883  df-id 5175  df-xp 5273  df-rel 5274  df-cnv 5275  df-co 5276  df-dm 5277  df-iota 6013  df-fun 6052  df-fv 6058  df-ov 6818  df-mzpcl 37807
This theorem is referenced by:  mzpincl  37818  mzpadd  37822  mzpmul  37823
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