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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > islsat | Structured version Visualization version GIF version |
Description: The predicate "is a 1-dim subspace (atom)" (of a left module or left vector space). (Contributed by NM, 9-Apr-2014.) (Revised by Mario Carneiro, 24-Jun-2014.) |
Ref | Expression |
---|---|
lsatset.v | ⊢ 𝑉 = (Base‘𝑊) |
lsatset.n | ⊢ 𝑁 = (LSpan‘𝑊) |
lsatset.z | ⊢ 0 = (0g‘𝑊) |
lsatset.a | ⊢ 𝐴 = (LSAtoms‘𝑊) |
Ref | Expression |
---|---|
islsat | ⊢ (𝑊 ∈ 𝑋 → (𝑈 ∈ 𝐴 ↔ ∃𝑥 ∈ (𝑉 ∖ { 0 })𝑈 = (𝑁‘{𝑥}))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lsatset.v | . . . 4 ⊢ 𝑉 = (Base‘𝑊) | |
2 | lsatset.n | . . . 4 ⊢ 𝑁 = (LSpan‘𝑊) | |
3 | lsatset.z | . . . 4 ⊢ 0 = (0g‘𝑊) | |
4 | lsatset.a | . . . 4 ⊢ 𝐴 = (LSAtoms‘𝑊) | |
5 | 1, 2, 3, 4 | lsatset 35011 | . . 3 ⊢ (𝑊 ∈ 𝑋 → 𝐴 = ran (𝑥 ∈ (𝑉 ∖ { 0 }) ↦ (𝑁‘{𝑥}))) |
6 | 5 | eleq2d 2864 | . 2 ⊢ (𝑊 ∈ 𝑋 → (𝑈 ∈ 𝐴 ↔ 𝑈 ∈ ran (𝑥 ∈ (𝑉 ∖ { 0 }) ↦ (𝑁‘{𝑥})))) |
7 | eqid 2799 | . . 3 ⊢ (𝑥 ∈ (𝑉 ∖ { 0 }) ↦ (𝑁‘{𝑥})) = (𝑥 ∈ (𝑉 ∖ { 0 }) ↦ (𝑁‘{𝑥})) | |
8 | fvex 6424 | . . 3 ⊢ (𝑁‘{𝑥}) ∈ V | |
9 | 7, 8 | elrnmpti 5580 | . 2 ⊢ (𝑈 ∈ ran (𝑥 ∈ (𝑉 ∖ { 0 }) ↦ (𝑁‘{𝑥})) ↔ ∃𝑥 ∈ (𝑉 ∖ { 0 })𝑈 = (𝑁‘{𝑥})) |
10 | 6, 9 | syl6bb 279 | 1 ⊢ (𝑊 ∈ 𝑋 → (𝑈 ∈ 𝐴 ↔ ∃𝑥 ∈ (𝑉 ∖ { 0 })𝑈 = (𝑁‘{𝑥}))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 = wceq 1653 ∈ wcel 2157 ∃wrex 3090 ∖ cdif 3766 {csn 4368 ↦ cmpt 4922 ran crn 5313 ‘cfv 6101 Basecbs 16184 0gc0g 16415 LSpanclspn 19292 LSAtomsclsa 34995 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2377 ax-ext 2777 ax-sep 4975 ax-nul 4983 ax-pow 5035 ax-pr 5097 ax-un 7183 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2591 df-eu 2609 df-clab 2786 df-cleq 2792 df-clel 2795 df-nfc 2930 df-ne 2972 df-ral 3094 df-rex 3095 df-rab 3098 df-v 3387 df-sbc 3634 df-dif 3772 df-un 3774 df-in 3776 df-ss 3783 df-nul 4116 df-if 4278 df-pw 4351 df-sn 4369 df-pr 4371 df-op 4375 df-uni 4629 df-br 4844 df-opab 4906 df-mpt 4923 df-id 5220 df-xp 5318 df-rel 5319 df-cnv 5320 df-co 5321 df-dm 5322 df-rn 5323 df-res 5324 df-ima 5325 df-iota 6064 df-fun 6103 df-fn 6104 df-f 6105 df-fv 6109 df-lsatoms 34997 |
This theorem is referenced by: lsatlspsn2 35013 lsatlspsn 35014 islsati 35015 lsateln0 35016 lsatn0 35020 lsatcmp 35024 lsmsat 35029 lsatfixedN 35030 islshpat 35038 lsatcv0 35052 lsat0cv 35054 lcv1 35062 l1cvpat 35075 dih1dimatlem 37350 dihlatat 37358 dochsatshp 37472 |
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