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Mirrors > Home > MPE Home > Th. List > Mathboxes > islsati | Structured version Visualization version GIF version |
Description: A 1-dim subspace (atom) (of a left module or left vector space) equals the span of some vector. (Contributed by NM, 1-Oct-2014.) |
Ref | Expression |
---|---|
islsati.v | ⊢ 𝑉 = (Base‘𝑊) |
islsati.n | ⊢ 𝑁 = (LSpan‘𝑊) |
islsati.a | ⊢ 𝐴 = (LSAtoms‘𝑊) |
Ref | Expression |
---|---|
islsati | ⊢ ((𝑊 ∈ 𝑋 ∧ 𝑈 ∈ 𝐴) → ∃𝑣 ∈ 𝑉 𝑈 = (𝑁‘{𝑣})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | difss 3737 | . 2 ⊢ (𝑉 ∖ {(0g‘𝑊)}) ⊆ 𝑉 | |
2 | islsati.v | . . . 4 ⊢ 𝑉 = (Base‘𝑊) | |
3 | islsati.n | . . . 4 ⊢ 𝑁 = (LSpan‘𝑊) | |
4 | eqid 2622 | . . . 4 ⊢ (0g‘𝑊) = (0g‘𝑊) | |
5 | islsati.a | . . . 4 ⊢ 𝐴 = (LSAtoms‘𝑊) | |
6 | 2, 3, 4, 5 | islsat 34278 | . . 3 ⊢ (𝑊 ∈ 𝑋 → (𝑈 ∈ 𝐴 ↔ ∃𝑣 ∈ (𝑉 ∖ {(0g‘𝑊)})𝑈 = (𝑁‘{𝑣}))) |
7 | 6 | biimpa 501 | . 2 ⊢ ((𝑊 ∈ 𝑋 ∧ 𝑈 ∈ 𝐴) → ∃𝑣 ∈ (𝑉 ∖ {(0g‘𝑊)})𝑈 = (𝑁‘{𝑣})) |
8 | ssrexv 3667 | . 2 ⊢ ((𝑉 ∖ {(0g‘𝑊)}) ⊆ 𝑉 → (∃𝑣 ∈ (𝑉 ∖ {(0g‘𝑊)})𝑈 = (𝑁‘{𝑣}) → ∃𝑣 ∈ 𝑉 𝑈 = (𝑁‘{𝑣}))) | |
9 | 1, 7, 8 | mpsyl 68 | 1 ⊢ ((𝑊 ∈ 𝑋 ∧ 𝑈 ∈ 𝐴) → ∃𝑣 ∈ 𝑉 𝑈 = (𝑁‘{𝑣})) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ∃wrex 2913 ∖ cdif 3571 ⊆ wss 3574 {csn 4177 ‘cfv 5888 Basecbs 15857 0gc0g 16100 LSpanclspn 18971 LSAtomsclsa 34261 |
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-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-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-fn 5891 df-f 5892 df-fv 5896 df-lsatoms 34263 |
This theorem is referenced by: lsmsatcv 34297 dihjat2 36720 dvh4dimlem 36732 lcfl8 36791 mapdval2N 36919 mapdspex 36957 hdmaprnlem16N 37154 |
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