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Theorem mzpcl2 41401
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 486 . 2 ((𝑃 ∈ (mzPolyCld‘𝑉) ∧ 𝐹𝑉) → 𝐹𝑉)
2 simpl 484 . . . 4 ((𝑃 ∈ (mzPolyCld‘𝑉) ∧ 𝐹𝑉) → 𝑃 ∈ (mzPolyCld‘𝑉))
3 elfvex 6926 . . . . . 6 (𝑃 ∈ (mzPolyCld‘𝑉) → 𝑉 ∈ V)
43adantr 482 . . . . 5 ((𝑃 ∈ (mzPolyCld‘𝑉) ∧ 𝐹𝑉) → 𝑉 ∈ V)
5 elmzpcl 41397 . . . . 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 231 . . 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 779 . . 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 6888 . . . . 5 (𝑓 = 𝐹 → (𝑔𝑓) = (𝑔𝐹))
1110mpteq2dv 5249 . . . 4 (𝑓 = 𝐹 → (𝑔 ∈ (ℤ ↑m 𝑉) ↦ (𝑔𝑓)) = (𝑔 ∈ (ℤ ↑m 𝑉) ↦ (𝑔𝐹)))
1211eleq1d 2819 . . 3 (𝑓 = 𝐹 → ((𝑔 ∈ (ℤ ↑m 𝑉) ↦ (𝑔𝑓)) ∈ 𝑃 ↔ (𝑔 ∈ (ℤ ↑m 𝑉) ↦ (𝑔𝐹)) ∈ 𝑃))
1312rspcva 3610 . 2 ((𝐹𝑉 ∧ ∀𝑓𝑉 (𝑔 ∈ (ℤ ↑m 𝑉) ↦ (𝑔𝑓)) ∈ 𝑃) → (𝑔 ∈ (ℤ ↑m 𝑉) ↦ (𝑔𝐹)) ∈ 𝑃)
141, 9, 13syl2anc 585 1 ((𝑃 ∈ (mzPolyCld‘𝑉) ∧ 𝐹𝑉) → (𝑔 ∈ (ℤ ↑m 𝑉) ↦ (𝑔𝐹)) ∈ 𝑃)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397   = wceq 1542  wcel 2107  wral 3062  Vcvv 3475  wss 3947  {csn 4627  cmpt 5230   × cxp 5673  cfv 6540  (class class class)co 7404  f cof 7663  m cmap 8816   + caddc 11109   · cmul 11111  cz 12554  mzPolyCldcmzpcl 41392
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-iota 6492  df-fun 6542  df-fv 6548  df-ov 7407  df-mzpcl 41394
This theorem is referenced by:  mzpincl  41405  mzpproj  41408
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