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Theorem mzpcl2 39205
Description: Defining property 2 of a polynomially closed function set 𝑃: it contains all projections. (Contributed by Stefan O'Rear, 4-Oct-2014.)
Assertion
Ref Expression
mzpcl2 ((𝑃 ∈ (mzPolyCld‘𝑉) ∧ 𝐹𝑉) → (𝑔 ∈ (ℤ ↑m 𝑉) ↦ (𝑔𝐹)) ∈ 𝑃)
Distinct variable groups:   𝑔,𝑉   𝑃,𝑔   𝑔,𝐹

Proof of Theorem mzpcl2
Dummy variable 𝑓 is distinct from all other variables.
StepHypRef Expression
1 simpr 485 . 2 ((𝑃 ∈ (mzPolyCld‘𝑉) ∧ 𝐹𝑉) → 𝐹𝑉)
2 simpl 483 . . . 4 ((𝑃 ∈ (mzPolyCld‘𝑉) ∧ 𝐹𝑉) → 𝑃 ∈ (mzPolyCld‘𝑉))
3 elfvex 6696 . . . . . 6 (𝑃 ∈ (mzPolyCld‘𝑉) → 𝑉 ∈ V)
43adantr 481 . . . . 5 ((𝑃 ∈ (mzPolyCld‘𝑉) ∧ 𝐹𝑉) → 𝑉 ∈ V)
5 elmzpcl 39201 . . . . 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) ∧ 𝐹𝑉) → (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 233 . . 3 ((𝑃 ∈ (mzPolyCld‘𝑉) ∧ 𝐹𝑉) → (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 simprlr 776 . . 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))) → ∀𝑓𝑉 (𝑔 ∈ (ℤ ↑m 𝑉) ↦ (𝑔𝑓)) ∈ 𝑃)
97, 8syl 17 . 2 ((𝑃 ∈ (mzPolyCld‘𝑉) ∧ 𝐹𝑉) → ∀𝑓𝑉 (𝑔 ∈ (ℤ ↑m 𝑉) ↦ (𝑔𝑓)) ∈ 𝑃)
10 fveq2 6663 . . . . 5 (𝑓 = 𝐹 → (𝑔𝑓) = (𝑔𝐹))
1110mpteq2dv 5153 . . . 4 (𝑓 = 𝐹 → (𝑔 ∈ (ℤ ↑m 𝑉) ↦ (𝑔𝑓)) = (𝑔 ∈ (ℤ ↑m 𝑉) ↦ (𝑔𝐹)))
1211eleq1d 2894 . . 3 (𝑓 = 𝐹 → ((𝑔 ∈ (ℤ ↑m 𝑉) ↦ (𝑔𝑓)) ∈ 𝑃 ↔ (𝑔 ∈ (ℤ ↑m 𝑉) ↦ (𝑔𝐹)) ∈ 𝑃))
1312rspcva 3618 . 2 ((𝐹𝑉 ∧ ∀𝑓𝑉 (𝑔 ∈ (ℤ ↑m 𝑉) ↦ (𝑔𝑓)) ∈ 𝑃) → (𝑔 ∈ (ℤ ↑m 𝑉) ↦ (𝑔𝐹)) ∈ 𝑃)
141, 9, 13syl2anc 584 1 ((𝑃 ∈ (mzPolyCld‘𝑉) ∧ 𝐹𝑉) → (𝑔 ∈ (ℤ ↑m 𝑉) ↦ (𝑔𝐹)) ∈ 𝑃)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396   = wceq 1528  wcel 2105  wral 3135  Vcvv 3492  wss 3933  {csn 4557  cmpt 5137   × cxp 5546  cfv 6348  (class class class)co 7145  f cof 7396  m cmap 8395   + caddc 10528   · cmul 10530  cz 11969  mzPolyCldcmzpcl 39196
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-iota 6307  df-fun 6350  df-fv 6356  df-ov 7148  df-mzpcl 39198
This theorem is referenced by:  mzpincl  39209  mzpproj  39212
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