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Mirrors > Home > MPE Home > Th. List > Mathboxes > mzpcl2 | Structured version Visualization version GIF version |
Description: Defining property 2 of a polynomially closed function set 𝑃: it contains all projections. (Contributed by Stefan O'Rear, 4-Oct-2014.) |
Ref | Expression |
---|---|
mzpcl2 | ⊢ ((𝑃 ∈ (mzPolyCld‘𝑉) ∧ 𝐹 ∈ 𝑉) → (𝑔 ∈ (ℤ ↑𝑚 𝑉) ↦ (𝑔‘𝐹)) ∈ 𝑃) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpr 477 | . 2 ⊢ ((𝑃 ∈ (mzPolyCld‘𝑉) ∧ 𝐹 ∈ 𝑉) → 𝐹 ∈ 𝑉) | |
2 | simpl 473 | . . . 4 ⊢ ((𝑃 ∈ (mzPolyCld‘𝑉) ∧ 𝐹 ∈ 𝑉) → 𝑃 ∈ (mzPolyCld‘𝑉)) | |
3 | elfvex 6221 | . . . . . 6 ⊢ (𝑃 ∈ (mzPolyCld‘𝑉) → 𝑉 ∈ V) | |
4 | 3 | adantr 481 | . . . . 5 ⊢ ((𝑃 ∈ (mzPolyCld‘𝑉) ∧ 𝐹 ∈ 𝑉) → 𝑉 ∈ V) |
5 | elmzpcl 37289 | . . . . 5 ⊢ (𝑉 ∈ V → (𝑃 ∈ (mzPolyCld‘𝑉) ↔ (𝑃 ⊆ (ℤ ↑𝑚 (ℤ ↑𝑚 𝑉)) ∧ ((∀𝑓 ∈ ℤ ((ℤ ↑𝑚 𝑉) × {𝑓}) ∈ 𝑃 ∧ ∀𝑓 ∈ 𝑉 (𝑔 ∈ (ℤ ↑𝑚 𝑉) ↦ (𝑔‘𝑓)) ∈ 𝑃) ∧ ∀𝑓 ∈ 𝑃 ∀𝑔 ∈ 𝑃 ((𝑓 ∘𝑓 + 𝑔) ∈ 𝑃 ∧ (𝑓 ∘𝑓 · 𝑔) ∈ 𝑃))))) | |
6 | 4, 5 | syl 17 | . . . 4 ⊢ ((𝑃 ∈ (mzPolyCld‘𝑉) ∧ 𝐹 ∈ 𝑉) → (𝑃 ∈ (mzPolyCld‘𝑉) ↔ (𝑃 ⊆ (ℤ ↑𝑚 (ℤ ↑𝑚 𝑉)) ∧ ((∀𝑓 ∈ ℤ ((ℤ ↑𝑚 𝑉) × {𝑓}) ∈ 𝑃 ∧ ∀𝑓 ∈ 𝑉 (𝑔 ∈ (ℤ ↑𝑚 𝑉) ↦ (𝑔‘𝑓)) ∈ 𝑃) ∧ ∀𝑓 ∈ 𝑃 ∀𝑔 ∈ 𝑃 ((𝑓 ∘𝑓 + 𝑔) ∈ 𝑃 ∧ (𝑓 ∘𝑓 · 𝑔) ∈ 𝑃))))) |
7 | 2, 6 | mpbid 222 | . . 3 ⊢ ((𝑃 ∈ (mzPolyCld‘𝑉) ∧ 𝐹 ∈ 𝑉) → (𝑃 ⊆ (ℤ ↑𝑚 (ℤ ↑𝑚 𝑉)) ∧ ((∀𝑓 ∈ ℤ ((ℤ ↑𝑚 𝑉) × {𝑓}) ∈ 𝑃 ∧ ∀𝑓 ∈ 𝑉 (𝑔 ∈ (ℤ ↑𝑚 𝑉) ↦ (𝑔‘𝑓)) ∈ 𝑃) ∧ ∀𝑓 ∈ 𝑃 ∀𝑔 ∈ 𝑃 ((𝑓 ∘𝑓 + 𝑔) ∈ 𝑃 ∧ (𝑓 ∘𝑓 · 𝑔) ∈ 𝑃)))) |
8 | simprlr 803 | . . 3 ⊢ ((𝑃 ⊆ (ℤ ↑𝑚 (ℤ ↑𝑚 𝑉)) ∧ ((∀𝑓 ∈ ℤ ((ℤ ↑𝑚 𝑉) × {𝑓}) ∈ 𝑃 ∧ ∀𝑓 ∈ 𝑉 (𝑔 ∈ (ℤ ↑𝑚 𝑉) ↦ (𝑔‘𝑓)) ∈ 𝑃) ∧ ∀𝑓 ∈ 𝑃 ∀𝑔 ∈ 𝑃 ((𝑓 ∘𝑓 + 𝑔) ∈ 𝑃 ∧ (𝑓 ∘𝑓 · 𝑔) ∈ 𝑃))) → ∀𝑓 ∈ 𝑉 (𝑔 ∈ (ℤ ↑𝑚 𝑉) ↦ (𝑔‘𝑓)) ∈ 𝑃) | |
9 | 7, 8 | syl 17 | . 2 ⊢ ((𝑃 ∈ (mzPolyCld‘𝑉) ∧ 𝐹 ∈ 𝑉) → ∀𝑓 ∈ 𝑉 (𝑔 ∈ (ℤ ↑𝑚 𝑉) ↦ (𝑔‘𝑓)) ∈ 𝑃) |
10 | fveq2 6191 | . . . . 5 ⊢ (𝑓 = 𝐹 → (𝑔‘𝑓) = (𝑔‘𝐹)) | |
11 | 10 | mpteq2dv 4745 | . . . 4 ⊢ (𝑓 = 𝐹 → (𝑔 ∈ (ℤ ↑𝑚 𝑉) ↦ (𝑔‘𝑓)) = (𝑔 ∈ (ℤ ↑𝑚 𝑉) ↦ (𝑔‘𝐹))) |
12 | 11 | eleq1d 2686 | . . 3 ⊢ (𝑓 = 𝐹 → ((𝑔 ∈ (ℤ ↑𝑚 𝑉) ↦ (𝑔‘𝑓)) ∈ 𝑃 ↔ (𝑔 ∈ (ℤ ↑𝑚 𝑉) ↦ (𝑔‘𝐹)) ∈ 𝑃)) |
13 | 12 | rspcva 3307 | . 2 ⊢ ((𝐹 ∈ 𝑉 ∧ ∀𝑓 ∈ 𝑉 (𝑔 ∈ (ℤ ↑𝑚 𝑉) ↦ (𝑔‘𝑓)) ∈ 𝑃) → (𝑔 ∈ (ℤ ↑𝑚 𝑉) ↦ (𝑔‘𝐹)) ∈ 𝑃) |
14 | 1, 9, 13 | syl2anc 693 | 1 ⊢ ((𝑃 ∈ (mzPolyCld‘𝑉) ∧ 𝐹 ∈ 𝑉) → (𝑔 ∈ (ℤ ↑𝑚 𝑉) ↦ (𝑔‘𝐹)) ∈ 𝑃) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ∀wral 2912 Vcvv 3200 ⊆ wss 3574 {csn 4177 ↦ cmpt 4729 × cxp 5112 ‘cfv 5888 (class class class)co 6650 ∘𝑓 cof 6895 ↑𝑚 cmap 7857 + caddc 9939 · cmul 9941 ℤcz 11377 mzPolyCldcmzpcl 37284 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-iota 5851 df-fun 5890 df-fv 5896 df-ov 6653 df-mzpcl 37286 |
This theorem is referenced by: mzpincl 37297 mzpproj 37300 |
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