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Mirrors > Home > MPE Home > Th. List > ply1frcl | Structured version Visualization version GIF version |
Description: Reverse closure for the set of univariate polynomial functions. (Contributed by AV, 9-Sep-2019.) |
Ref | Expression |
---|---|
ply1frcl.q | ⊢ 𝑄 = ran (𝑆 evalSub1 𝑅) |
Ref | Expression |
---|---|
ply1frcl | ⊢ (𝑋 ∈ 𝑄 → (𝑆 ∈ V ∧ 𝑅 ∈ 𝒫 (Base‘𝑆))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ne0i 4334 | . . 3 ⊢ (𝑋 ∈ ran (𝑆 evalSub1 𝑅) → ran (𝑆 evalSub1 𝑅) ≠ ∅) | |
2 | ply1frcl.q | . . 3 ⊢ 𝑄 = ran (𝑆 evalSub1 𝑅) | |
3 | 1, 2 | eleq2s 2843 | . 2 ⊢ (𝑋 ∈ 𝑄 → ran (𝑆 evalSub1 𝑅) ≠ ∅) |
4 | rneq 5938 | . . . 4 ⊢ ((𝑆 evalSub1 𝑅) = ∅ → ran (𝑆 evalSub1 𝑅) = ran ∅) | |
5 | rn0 5928 | . . . 4 ⊢ ran ∅ = ∅ | |
6 | 4, 5 | eqtrdi 2781 | . . 3 ⊢ ((𝑆 evalSub1 𝑅) = ∅ → ran (𝑆 evalSub1 𝑅) = ∅) |
7 | 6 | necon3i 2962 | . 2 ⊢ (ran (𝑆 evalSub1 𝑅) ≠ ∅ → (𝑆 evalSub1 𝑅) ≠ ∅) |
8 | n0 4346 | . . 3 ⊢ ((𝑆 evalSub1 𝑅) ≠ ∅ ↔ ∃𝑒 𝑒 ∈ (𝑆 evalSub1 𝑅)) | |
9 | df-evls1 22259 | . . . . . . 7 ⊢ evalSub1 = (𝑠 ∈ V, 𝑟 ∈ 𝒫 (Base‘𝑠) ↦ ⦋(Base‘𝑠) / 𝑏⦌((𝑥 ∈ (𝑏 ↑m (𝑏 ↑m 1o)) ↦ (𝑥 ∘ (𝑦 ∈ 𝑏 ↦ (1o × {𝑦})))) ∘ ((1o evalSub 𝑠)‘𝑟))) | |
10 | 9 | dmmpossx 8071 | . . . . . 6 ⊢ dom evalSub1 ⊆ ∪ 𝑠 ∈ V ({𝑠} × 𝒫 (Base‘𝑠)) |
11 | elfvdm 6933 | . . . . . . 7 ⊢ (𝑒 ∈ ( evalSub1 ‘〈𝑆, 𝑅〉) → 〈𝑆, 𝑅〉 ∈ dom evalSub1 ) | |
12 | df-ov 7422 | . . . . . . 7 ⊢ (𝑆 evalSub1 𝑅) = ( evalSub1 ‘〈𝑆, 𝑅〉) | |
13 | 11, 12 | eleq2s 2843 | . . . . . 6 ⊢ (𝑒 ∈ (𝑆 evalSub1 𝑅) → 〈𝑆, 𝑅〉 ∈ dom evalSub1 ) |
14 | 10, 13 | sselid 3974 | . . . . 5 ⊢ (𝑒 ∈ (𝑆 evalSub1 𝑅) → 〈𝑆, 𝑅〉 ∈ ∪ 𝑠 ∈ V ({𝑠} × 𝒫 (Base‘𝑠))) |
15 | fveq2 6896 | . . . . . . 7 ⊢ (𝑠 = 𝑆 → (Base‘𝑠) = (Base‘𝑆)) | |
16 | 15 | pweqd 4621 | . . . . . 6 ⊢ (𝑠 = 𝑆 → 𝒫 (Base‘𝑠) = 𝒫 (Base‘𝑆)) |
17 | 16 | opeliunxp2 5841 | . . . . 5 ⊢ (〈𝑆, 𝑅〉 ∈ ∪ 𝑠 ∈ V ({𝑠} × 𝒫 (Base‘𝑠)) ↔ (𝑆 ∈ V ∧ 𝑅 ∈ 𝒫 (Base‘𝑆))) |
18 | 14, 17 | sylib 217 | . . . 4 ⊢ (𝑒 ∈ (𝑆 evalSub1 𝑅) → (𝑆 ∈ V ∧ 𝑅 ∈ 𝒫 (Base‘𝑆))) |
19 | 18 | exlimiv 1925 | . . 3 ⊢ (∃𝑒 𝑒 ∈ (𝑆 evalSub1 𝑅) → (𝑆 ∈ V ∧ 𝑅 ∈ 𝒫 (Base‘𝑆))) |
20 | 8, 19 | sylbi 216 | . 2 ⊢ ((𝑆 evalSub1 𝑅) ≠ ∅ → (𝑆 ∈ V ∧ 𝑅 ∈ 𝒫 (Base‘𝑆))) |
21 | 3, 7, 20 | 3syl 18 | 1 ⊢ (𝑋 ∈ 𝑄 → (𝑆 ∈ V ∧ 𝑅 ∈ 𝒫 (Base‘𝑆))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1533 ∃wex 1773 ∈ wcel 2098 ≠ wne 2929 Vcvv 3461 ⦋csb 3889 ∅c0 4322 𝒫 cpw 4604 {csn 4630 〈cop 4636 ∪ ciun 4997 ↦ cmpt 5232 × cxp 5676 dom cdm 5678 ran crn 5679 ∘ ccom 5682 ‘cfv 6549 (class class class)co 7419 1oc1o 8480 ↑m cmap 8845 Basecbs 17183 evalSub ces 22038 evalSub1 ces1 22257 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5300 ax-nul 5307 ax-pr 5429 ax-un 7741 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-ral 3051 df-rex 3060 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-iun 4999 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-iota 6501 df-fun 6551 df-fv 6557 df-ov 7422 df-oprab 7423 df-mpo 7424 df-1st 7994 df-2nd 7995 df-evls1 22259 |
This theorem is referenced by: (None) |
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