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Theorem mzpcl1 36310
Description: Defining property 1 of a polynomially closed function set 𝑃: it contains all constant functions. (Contributed by Stefan O'Rear, 4-Oct-2014.)
Assertion
Ref Expression
mzpcl1 ((𝑃 ∈ (mzPolyCld‘𝑉) ∧ 𝐹 ∈ ℤ) → ((ℤ ↑𝑚 𝑉) × {𝐹}) ∈ 𝑃)

Proof of Theorem mzpcl1
Dummy variables 𝑓 𝑔 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpr 476 . 2 ((𝑃 ∈ (mzPolyCld‘𝑉) ∧ 𝐹 ∈ ℤ) → 𝐹 ∈ ℤ)
2 simpl 472 . . . 4 ((𝑃 ∈ (mzPolyCld‘𝑉) ∧ 𝐹 ∈ ℤ) → 𝑃 ∈ (mzPolyCld‘𝑉))
3 elfvex 6131 . . . . . 6 (𝑃 ∈ (mzPolyCld‘𝑉) → 𝑉 ∈ V)
43adantr 480 . . . . 5 ((𝑃 ∈ (mzPolyCld‘𝑉) ∧ 𝐹 ∈ ℤ) → 𝑉 ∈ V)
5 elmzpcl 36307 . . . . 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) ∧ 𝐹 ∈ ℤ) → (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)))))
72, 6mpbid 221 . . 3 ((𝑃 ∈ (mzPolyCld‘𝑉) ∧ 𝐹 ∈ ℤ) → (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))))
8 simprll 798 . . 3 ((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))) → ∀𝑓 ∈ ℤ ((ℤ ↑𝑚 𝑉) × {𝑓}) ∈ 𝑃)
97, 8syl 17 . 2 ((𝑃 ∈ (mzPolyCld‘𝑉) ∧ 𝐹 ∈ ℤ) → ∀𝑓 ∈ ℤ ((ℤ ↑𝑚 𝑉) × {𝑓}) ∈ 𝑃)
10 sneq 4135 . . . . 5 (𝑓 = 𝐹 → {𝑓} = {𝐹})
1110xpeq2d 5063 . . . 4 (𝑓 = 𝐹 → ((ℤ ↑𝑚 𝑉) × {𝑓}) = ((ℤ ↑𝑚 𝑉) × {𝐹}))
1211eleq1d 2672 . . 3 (𝑓 = 𝐹 → (((ℤ ↑𝑚 𝑉) × {𝑓}) ∈ 𝑃 ↔ ((ℤ ↑𝑚 𝑉) × {𝐹}) ∈ 𝑃))
1312rspcva 3280 . 2 ((𝐹 ∈ ℤ ∧ ∀𝑓 ∈ ℤ ((ℤ ↑𝑚 𝑉) × {𝑓}) ∈ 𝑃) → ((ℤ ↑𝑚 𝑉) × {𝐹}) ∈ 𝑃)
141, 9, 13syl2anc 691 1 ((𝑃 ∈ (mzPolyCld‘𝑉) ∧ 𝐹 ∈ ℤ) → ((ℤ ↑𝑚 𝑉) × {𝐹}) ∈ 𝑃)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 195  wa 383   = wceq 1475  wcel 1977  wral 2896  Vcvv 3173  wss 3540  {csn 4125  cmpt 4643   × cxp 5036  cfv 5804  (class class class)co 6549  𝑓 cof 6793  𝑚 cmap 7744   + caddc 9818   · cmul 9820  cz 11254  mzPolyCldcmzpcl 36302
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-iota 5768  df-fun 5806  df-fv 5812  df-ov 6552  df-mzpcl 36304
This theorem is referenced by:  mzpincl  36315  mzpconst  36316
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