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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 4299 | . . 3 ⊢ (𝑋 ∈ ran (𝑆 evalSub1 𝑅) → ran (𝑆 evalSub1 𝑅) ≠ ∅) | |
2 | ply1frcl.q | . . 3 ⊢ 𝑄 = ran (𝑆 evalSub1 𝑅) | |
3 | 1, 2 | eleq2s 2856 | . 2 ⊢ (𝑋 ∈ 𝑄 → ran (𝑆 evalSub1 𝑅) ≠ ∅) |
4 | rneq 5896 | . . . 4 ⊢ ((𝑆 evalSub1 𝑅) = ∅ → ran (𝑆 evalSub1 𝑅) = ran ∅) | |
5 | rn0 5886 | . . . 4 ⊢ ran ∅ = ∅ | |
6 | 4, 5 | eqtrdi 2793 | . . 3 ⊢ ((𝑆 evalSub1 𝑅) = ∅ → ran (𝑆 evalSub1 𝑅) = ∅) |
7 | 6 | necon3i 2977 | . 2 ⊢ (ran (𝑆 evalSub1 𝑅) ≠ ∅ → (𝑆 evalSub1 𝑅) ≠ ∅) |
8 | n0 4311 | . . 3 ⊢ ((𝑆 evalSub1 𝑅) ≠ ∅ ↔ ∃𝑒 𝑒 ∈ (𝑆 evalSub1 𝑅)) | |
9 | df-evls1 21697 | . . . . . . 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 8003 | . . . . . 6 ⊢ dom evalSub1 ⊆ ∪ 𝑠 ∈ V ({𝑠} × 𝒫 (Base‘𝑠)) |
11 | elfvdm 6884 | . . . . . . 7 ⊢ (𝑒 ∈ ( evalSub1 ‘⟨𝑆, 𝑅⟩) → ⟨𝑆, 𝑅⟩ ∈ dom evalSub1 ) | |
12 | df-ov 7365 | . . . . . . 7 ⊢ (𝑆 evalSub1 𝑅) = ( evalSub1 ‘⟨𝑆, 𝑅⟩) | |
13 | 11, 12 | eleq2s 2856 | . . . . . 6 ⊢ (𝑒 ∈ (𝑆 evalSub1 𝑅) → ⟨𝑆, 𝑅⟩ ∈ dom evalSub1 ) |
14 | 10, 13 | sselid 3947 | . . . . 5 ⊢ (𝑒 ∈ (𝑆 evalSub1 𝑅) → ⟨𝑆, 𝑅⟩ ∈ ∪ 𝑠 ∈ V ({𝑠} × 𝒫 (Base‘𝑠))) |
15 | fveq2 6847 | . . . . . . 7 ⊢ (𝑠 = 𝑆 → (Base‘𝑠) = (Base‘𝑆)) | |
16 | 15 | pweqd 4582 | . . . . . 6 ⊢ (𝑠 = 𝑆 → 𝒫 (Base‘𝑠) = 𝒫 (Base‘𝑆)) |
17 | 16 | opeliunxp2 5799 | . . . . 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 1934 | . . 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 397 = wceq 1542 ∃wex 1782 ∈ wcel 2107 ≠ wne 2944 Vcvv 3448 ⦋csb 3860 ∅c0 4287 𝒫 cpw 4565 {csn 4591 ⟨cop 4597 ∪ ciun 4959 ↦ cmpt 5193 × cxp 5636 dom cdm 5638 ran crn 5639 ∘ ccom 5642 ‘cfv 6501 (class class class)co 7362 1oc1o 8410 ↑m cmap 8772 Basecbs 17090 evalSub ces 21496 evalSub1 ces1 21695 |
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 2708 ax-sep 5261 ax-nul 5268 ax-pr 5389 ax-un 7677 |
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 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-ral 3066 df-rex 3075 df-rab 3411 df-v 3450 df-sbc 3745 df-csb 3861 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-iun 4961 df-br 5111 df-opab 5173 df-mpt 5194 df-id 5536 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-iota 6453 df-fun 6503 df-fv 6509 df-ov 7365 df-oprab 7366 df-mpo 7367 df-1st 7926 df-2nd 7927 df-evls1 21697 |
This theorem is referenced by: (None) |
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