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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 3921 | . . 3 ⊢ (𝑋 ∈ ran (𝑆 evalSub1 𝑅) → ran (𝑆 evalSub1 𝑅) ≠ ∅) | |
2 | ply1frcl.q | . . 3 ⊢ 𝑄 = ran (𝑆 evalSub1 𝑅) | |
3 | 1, 2 | eleq2s 2719 | . 2 ⊢ (𝑋 ∈ 𝑄 → ran (𝑆 evalSub1 𝑅) ≠ ∅) |
4 | rneq 5351 | . . . 4 ⊢ ((𝑆 evalSub1 𝑅) = ∅ → ran (𝑆 evalSub1 𝑅) = ran ∅) | |
5 | rn0 5377 | . . . 4 ⊢ ran ∅ = ∅ | |
6 | 4, 5 | syl6eq 2672 | . . 3 ⊢ ((𝑆 evalSub1 𝑅) = ∅ → ran (𝑆 evalSub1 𝑅) = ∅) |
7 | 6 | necon3i 2826 | . 2 ⊢ (ran (𝑆 evalSub1 𝑅) ≠ ∅ → (𝑆 evalSub1 𝑅) ≠ ∅) |
8 | n0 3931 | . . 3 ⊢ ((𝑆 evalSub1 𝑅) ≠ ∅ ↔ ∃𝑒 𝑒 ∈ (𝑆 evalSub1 𝑅)) | |
9 | df-evls1 19680 | . . . . . . 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 7235 | . . . . . 6 ⊢ dom evalSub1 ⊆ ∪ 𝑠 ∈ V ({𝑠} × 𝒫 (Base‘𝑠)) |
11 | elfvdm 6220 | . . . . . . 7 ⊢ (𝑒 ∈ ( evalSub1 ‘〈𝑆, 𝑅〉) → 〈𝑆, 𝑅〉 ∈ dom evalSub1 ) | |
12 | df-ov 6653 | . . . . . . 7 ⊢ (𝑆 evalSub1 𝑅) = ( evalSub1 ‘〈𝑆, 𝑅〉) | |
13 | 11, 12 | eleq2s 2719 | . . . . . 6 ⊢ (𝑒 ∈ (𝑆 evalSub1 𝑅) → 〈𝑆, 𝑅〉 ∈ dom evalSub1 ) |
14 | 10, 13 | sseldi 3601 | . . . . 5 ⊢ (𝑒 ∈ (𝑆 evalSub1 𝑅) → 〈𝑆, 𝑅〉 ∈ ∪ 𝑠 ∈ V ({𝑠} × 𝒫 (Base‘𝑠))) |
15 | fveq2 6191 | . . . . . . 7 ⊢ (𝑠 = 𝑆 → (Base‘𝑠) = (Base‘𝑆)) | |
16 | 15 | pweqd 4163 | . . . . . 6 ⊢ (𝑠 = 𝑆 → 𝒫 (Base‘𝑠) = 𝒫 (Base‘𝑆)) |
17 | 16 | opeliunxp2 5260 | . . . . 5 ⊢ (〈𝑆, 𝑅〉 ∈ ∪ 𝑠 ∈ V ({𝑠} × 𝒫 (Base‘𝑠)) ↔ (𝑆 ∈ V ∧ 𝑅 ∈ 𝒫 (Base‘𝑆))) |
18 | 14, 17 | sylib 208 | . . . 4 ⊢ (𝑒 ∈ (𝑆 evalSub1 𝑅) → (𝑆 ∈ V ∧ 𝑅 ∈ 𝒫 (Base‘𝑆))) |
19 | 18 | exlimiv 1858 | . . 3 ⊢ (∃𝑒 𝑒 ∈ (𝑆 evalSub1 𝑅) → (𝑆 ∈ V ∧ 𝑅 ∈ 𝒫 (Base‘𝑆))) |
20 | 8, 19 | sylbi 207 | . 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 384 = wceq 1483 ∃wex 1704 ∈ wcel 1990 ≠ wne 2794 Vcvv 3200 ⦋csb 3533 ∅c0 3915 𝒫 cpw 4158 {csn 4177 〈cop 4183 ∪ ciun 4520 ↦ cmpt 4729 × cxp 5112 dom cdm 5114 ran crn 5115 ∘ ccom 5118 ‘cfv 5888 (class class class)co 6650 1𝑜c1o 7553 ↑𝑚 cmap 7857 Basecbs 15857 evalSub ces 19504 evalSub1 ces1 19678 |
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 ax-un 6949 |
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-csb 3534 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-iun 4522 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-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-1st 7168 df-2nd 7169 df-evls1 19680 |
This theorem is referenced by: (None) |
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