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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 3880 | . . 3 ⊢ (𝑋 ∈ ran (𝑆 evalSub1 𝑅) → ran (𝑆 evalSub1 𝑅) ≠ ∅) | |
2 | ply1frcl.q | . . 3 ⊢ 𝑄 = ran (𝑆 evalSub1 𝑅) | |
3 | 1, 2 | eleq2s 2706 | . 2 ⊢ (𝑋 ∈ 𝑄 → ran (𝑆 evalSub1 𝑅) ≠ ∅) |
4 | rneq 5272 | . . . 4 ⊢ ((𝑆 evalSub1 𝑅) = ∅ → ran (𝑆 evalSub1 𝑅) = ran ∅) | |
5 | rn0 5298 | . . . 4 ⊢ ran ∅ = ∅ | |
6 | 4, 5 | syl6eq 2660 | . . 3 ⊢ ((𝑆 evalSub1 𝑅) = ∅ → ran (𝑆 evalSub1 𝑅) = ∅) |
7 | 6 | necon3i 2814 | . 2 ⊢ (ran (𝑆 evalSub1 𝑅) ≠ ∅ → (𝑆 evalSub1 𝑅) ≠ ∅) |
8 | n0 3890 | . . 3 ⊢ ((𝑆 evalSub1 𝑅) ≠ ∅ ↔ ∃𝑒 𝑒 ∈ (𝑆 evalSub1 𝑅)) | |
9 | df-evls1 19501 | . . . . . . 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 | dmmpt2ssx 7124 | . . . . . 6 ⊢ dom evalSub1 ⊆ ∪ 𝑠 ∈ V ({𝑠} × 𝒫 (Base‘𝑠)) |
11 | elfvdm 6130 | . . . . . . 7 ⊢ (𝑒 ∈ ( evalSub1 ‘〈𝑆, 𝑅〉) → 〈𝑆, 𝑅〉 ∈ dom evalSub1 ) | |
12 | df-ov 6552 | . . . . . . 7 ⊢ (𝑆 evalSub1 𝑅) = ( evalSub1 ‘〈𝑆, 𝑅〉) | |
13 | 11, 12 | eleq2s 2706 | . . . . . 6 ⊢ (𝑒 ∈ (𝑆 evalSub1 𝑅) → 〈𝑆, 𝑅〉 ∈ dom evalSub1 ) |
14 | 10, 13 | sseldi 3566 | . . . . 5 ⊢ (𝑒 ∈ (𝑆 evalSub1 𝑅) → 〈𝑆, 𝑅〉 ∈ ∪ 𝑠 ∈ V ({𝑠} × 𝒫 (Base‘𝑠))) |
15 | fveq2 6103 | . . . . . . 7 ⊢ (𝑠 = 𝑆 → (Base‘𝑠) = (Base‘𝑆)) | |
16 | 15 | pweqd 4113 | . . . . . 6 ⊢ (𝑠 = 𝑆 → 𝒫 (Base‘𝑠) = 𝒫 (Base‘𝑆)) |
17 | 16 | opeliunxp2 5182 | . . . . 5 ⊢ (〈𝑆, 𝑅〉 ∈ ∪ 𝑠 ∈ V ({𝑠} × 𝒫 (Base‘𝑠)) ↔ (𝑆 ∈ V ∧ 𝑅 ∈ 𝒫 (Base‘𝑆))) |
18 | 14, 17 | sylib 207 | . . . 4 ⊢ (𝑒 ∈ (𝑆 evalSub1 𝑅) → (𝑆 ∈ V ∧ 𝑅 ∈ 𝒫 (Base‘𝑆))) |
19 | 18 | exlimiv 1845 | . . 3 ⊢ (∃𝑒 𝑒 ∈ (𝑆 evalSub1 𝑅) → (𝑆 ∈ V ∧ 𝑅 ∈ 𝒫 (Base‘𝑆))) |
20 | 8, 19 | sylbi 206 | . 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 383 = wceq 1475 ∃wex 1695 ∈ wcel 1977 ≠ wne 2780 Vcvv 3173 ⦋csb 3499 ∅c0 3874 𝒫 cpw 4108 {csn 4125 〈cop 4131 ∪ ciun 4455 ↦ cmpt 4643 × cxp 5036 dom cdm 5038 ran crn 5039 ∘ ccom 5042 ‘cfv 5804 (class class class)co 6549 1𝑜c1o 7440 ↑𝑚 cmap 7744 Basecbs 15695 evalSub ces 19325 evalSub1 ces1 19499 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-1st 7059 df-2nd 7060 df-evls1 19501 |
This theorem is referenced by: (None) |
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