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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 4250 | . . 3 ⊢ (𝑋 ∈ ran (𝑆 evalSub1 𝑅) → ran (𝑆 evalSub1 𝑅) ≠ ∅) | |
2 | ply1frcl.q | . . 3 ⊢ 𝑄 = ran (𝑆 evalSub1 𝑅) | |
3 | 1, 2 | eleq2s 2908 | . 2 ⊢ (𝑋 ∈ 𝑄 → ran (𝑆 evalSub1 𝑅) ≠ ∅) |
4 | rneq 5770 | . . . 4 ⊢ ((𝑆 evalSub1 𝑅) = ∅ → ran (𝑆 evalSub1 𝑅) = ran ∅) | |
5 | rn0 5760 | . . . 4 ⊢ ran ∅ = ∅ | |
6 | 4, 5 | eqtrdi 2849 | . . 3 ⊢ ((𝑆 evalSub1 𝑅) = ∅ → ran (𝑆 evalSub1 𝑅) = ∅) |
7 | 6 | necon3i 3019 | . 2 ⊢ (ran (𝑆 evalSub1 𝑅) ≠ ∅ → (𝑆 evalSub1 𝑅) ≠ ∅) |
8 | n0 4260 | . . 3 ⊢ ((𝑆 evalSub1 𝑅) ≠ ∅ ↔ ∃𝑒 𝑒 ∈ (𝑆 evalSub1 𝑅)) | |
9 | df-evls1 20939 | . . . . . . 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 7746 | . . . . . 6 ⊢ dom evalSub1 ⊆ ∪ 𝑠 ∈ V ({𝑠} × 𝒫 (Base‘𝑠)) |
11 | elfvdm 6677 | . . . . . . 7 ⊢ (𝑒 ∈ ( evalSub1 ‘〈𝑆, 𝑅〉) → 〈𝑆, 𝑅〉 ∈ dom evalSub1 ) | |
12 | df-ov 7138 | . . . . . . 7 ⊢ (𝑆 evalSub1 𝑅) = ( evalSub1 ‘〈𝑆, 𝑅〉) | |
13 | 11, 12 | eleq2s 2908 | . . . . . 6 ⊢ (𝑒 ∈ (𝑆 evalSub1 𝑅) → 〈𝑆, 𝑅〉 ∈ dom evalSub1 ) |
14 | 10, 13 | sseldi 3913 | . . . . 5 ⊢ (𝑒 ∈ (𝑆 evalSub1 𝑅) → 〈𝑆, 𝑅〉 ∈ ∪ 𝑠 ∈ V ({𝑠} × 𝒫 (Base‘𝑠))) |
15 | fveq2 6645 | . . . . . . 7 ⊢ (𝑠 = 𝑆 → (Base‘𝑠) = (Base‘𝑆)) | |
16 | 15 | pweqd 4516 | . . . . . 6 ⊢ (𝑠 = 𝑆 → 𝒫 (Base‘𝑠) = 𝒫 (Base‘𝑆)) |
17 | 16 | opeliunxp2 5673 | . . . . 5 ⊢ (〈𝑆, 𝑅〉 ∈ ∪ 𝑠 ∈ V ({𝑠} × 𝒫 (Base‘𝑠)) ↔ (𝑆 ∈ V ∧ 𝑅 ∈ 𝒫 (Base‘𝑆))) |
18 | 14, 17 | sylib 221 | . . . 4 ⊢ (𝑒 ∈ (𝑆 evalSub1 𝑅) → (𝑆 ∈ V ∧ 𝑅 ∈ 𝒫 (Base‘𝑆))) |
19 | 18 | exlimiv 1931 | . . 3 ⊢ (∃𝑒 𝑒 ∈ (𝑆 evalSub1 𝑅) → (𝑆 ∈ V ∧ 𝑅 ∈ 𝒫 (Base‘𝑆))) |
20 | 8, 19 | sylbi 220 | . 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 399 = wceq 1538 ∃wex 1781 ∈ wcel 2111 ≠ wne 2987 Vcvv 3441 ⦋csb 3828 ∅c0 4243 𝒫 cpw 4497 {csn 4525 〈cop 4531 ∪ ciun 4881 ↦ cmpt 5110 × cxp 5517 dom cdm 5519 ran crn 5520 ∘ ccom 5523 ‘cfv 6324 (class class class)co 7135 1oc1o 8078 ↑m cmap 8389 Basecbs 16475 evalSub ces 20743 evalSub1 ces1 20937 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fv 6332 df-ov 7138 df-oprab 7139 df-mpo 7140 df-1st 7671 df-2nd 7672 df-evls1 20939 |
This theorem is referenced by: (None) |
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