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Theorem mzpcl1 37818
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 471 . 2 ((𝑃 ∈ (mzPolyCld‘𝑉) ∧ 𝐹 ∈ ℤ) → 𝐹 ∈ ℤ)
2 simpl 468 . . . 4 ((𝑃 ∈ (mzPolyCld‘𝑉) ∧ 𝐹 ∈ ℤ) → 𝑃 ∈ (mzPolyCld‘𝑉))
3 elfvex 6362 . . . . . 6 (𝑃 ∈ (mzPolyCld‘𝑉) → 𝑉 ∈ V)
43adantr 466 . . . . 5 ((𝑃 ∈ (mzPolyCld‘𝑉) ∧ 𝐹 ∈ ℤ) → 𝑉 ∈ V)
5 elmzpcl 37815 . . . . 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 222 . . 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 764 . . 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 4326 . . . . 5 (𝑓 = 𝐹 → {𝑓} = {𝐹})
1110xpeq2d 5279 . . . 4 (𝑓 = 𝐹 → ((ℤ ↑𝑚 𝑉) × {𝑓}) = ((ℤ ↑𝑚 𝑉) × {𝐹}))
1211eleq1d 2835 . . 3 (𝑓 = 𝐹 → (((ℤ ↑𝑚 𝑉) × {𝑓}) ∈ 𝑃 ↔ ((ℤ ↑𝑚 𝑉) × {𝐹}) ∈ 𝑃))
1312rspcva 3458 . 2 ((𝐹 ∈ ℤ ∧ ∀𝑓 ∈ ℤ ((ℤ ↑𝑚 𝑉) × {𝑓}) ∈ 𝑃) → ((ℤ ↑𝑚 𝑉) × {𝐹}) ∈ 𝑃)
141, 9, 13syl2anc 573 1 ((𝑃 ∈ (mzPolyCld‘𝑉) ∧ 𝐹 ∈ ℤ) → ((ℤ ↑𝑚 𝑉) × {𝐹}) ∈ 𝑃)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382   = wceq 1631  wcel 2145  wral 3061  Vcvv 3351  wss 3723  {csn 4316  cmpt 4863   × cxp 5247  cfv 6031  (class class class)co 6793  𝑓 cof 7042  𝑚 cmap 8009   + caddc 10141   · cmul 10143  cz 11579  mzPolyCldcmzpcl 37810
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-iota 5994  df-fun 6033  df-fv 6039  df-ov 6796  df-mzpcl 37812
This theorem is referenced by:  mzpincl  37823  mzpconst  37824
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