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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 38991 | . . 3 ⊢ (𝑊 ∈ 𝑋 → 𝐴 = ran (𝑥 ∈ (𝑉 ∖ { 0 }) ↦ (𝑁‘{𝑥}))) |
| 6 | 5 | eleq2d 2827 | . 2 ⊢ (𝑊 ∈ 𝑋 → (𝑈 ∈ 𝐴 ↔ 𝑈 ∈ ran (𝑥 ∈ (𝑉 ∖ { 0 }) ↦ (𝑁‘{𝑥})))) |
| 7 | eqid 2737 | . . 3 ⊢ (𝑥 ∈ (𝑉 ∖ { 0 }) ↦ (𝑁‘{𝑥})) = (𝑥 ∈ (𝑉 ∖ { 0 }) ↦ (𝑁‘{𝑥})) | |
| 8 | fvex 6919 | . . 3 ⊢ (𝑁‘{𝑥}) ∈ V | |
| 9 | 7, 8 | elrnmpti 5973 | . 2 ⊢ (𝑈 ∈ ran (𝑥 ∈ (𝑉 ∖ { 0 }) ↦ (𝑁‘{𝑥})) ↔ ∃𝑥 ∈ (𝑉 ∖ { 0 })𝑈 = (𝑁‘{𝑥})) |
| 10 | 6, 9 | bitrdi 287 | 1 ⊢ (𝑊 ∈ 𝑋 → (𝑈 ∈ 𝐴 ↔ ∃𝑥 ∈ (𝑉 ∖ { 0 })𝑈 = (𝑁‘{𝑥}))) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 = wceq 1540 ∈ wcel 2108 ∃wrex 3070 ∖ cdif 3948 {csn 4626 ↦ cmpt 5225 ran crn 5686 ‘cfv 6561 Basecbs 17247 0gc0g 17484 LSpanclspn 20969 LSAtomsclsa 38975 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-fv 6569 df-lsatoms 38977 |
| This theorem is referenced by: lsatlspsn2 38993 lsatlspsn 38994 islsati 38995 lsateln0 38996 lsatn0 39000 lsatcmp 39004 lsmsat 39009 lsatfixedN 39010 islshpat 39018 lsatcv0 39032 lsat0cv 39034 lcv1 39042 l1cvpat 39055 dih1dimatlem 41331 dihlatat 41339 dochsatshp 41453 |
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