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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 36767 | . . 3 ⊢ (𝑊 ∈ 𝑋 → 𝐴 = ran (𝑥 ∈ (𝑉 ∖ { 0 }) ↦ (𝑁‘{𝑥}))) |
6 | 5 | eleq2d 2824 | . 2 ⊢ (𝑊 ∈ 𝑋 → (𝑈 ∈ 𝐴 ↔ 𝑈 ∈ ran (𝑥 ∈ (𝑉 ∖ { 0 }) ↦ (𝑁‘{𝑥})))) |
7 | eqid 2738 | . . 3 ⊢ (𝑥 ∈ (𝑉 ∖ { 0 }) ↦ (𝑁‘{𝑥})) = (𝑥 ∈ (𝑉 ∖ { 0 }) ↦ (𝑁‘{𝑥})) | |
8 | fvex 6748 | . . 3 ⊢ (𝑁‘{𝑥}) ∈ V | |
9 | 7, 8 | elrnmpti 5843 | . 2 ⊢ (𝑈 ∈ ran (𝑥 ∈ (𝑉 ∖ { 0 }) ↦ (𝑁‘{𝑥})) ↔ ∃𝑥 ∈ (𝑉 ∖ { 0 })𝑈 = (𝑁‘{𝑥})) |
10 | 6, 9 | bitrdi 290 | 1 ⊢ (𝑊 ∈ 𝑋 → (𝑈 ∈ 𝐴 ↔ ∃𝑥 ∈ (𝑉 ∖ { 0 })𝑈 = (𝑁‘{𝑥}))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 = wceq 1543 ∈ wcel 2111 ∃wrex 3063 ∖ cdif 3877 {csn 4555 ↦ cmpt 5149 ran crn 5566 ‘cfv 6397 Basecbs 16784 0gc0g 16968 LSpanclspn 20032 LSAtomsclsa 36751 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2159 ax-12 2176 ax-ext 2709 ax-sep 5206 ax-nul 5213 ax-pow 5272 ax-pr 5336 ax-un 7541 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2072 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2887 df-ne 2942 df-ral 3067 df-rex 3068 df-rab 3071 df-v 3422 df-dif 3883 df-un 3885 df-in 3887 df-ss 3897 df-nul 4252 df-if 4454 df-pw 4529 df-sn 4556 df-pr 4558 df-op 4562 df-uni 4834 df-br 5068 df-opab 5130 df-mpt 5150 df-id 5469 df-xp 5571 df-rel 5572 df-cnv 5573 df-co 5574 df-dm 5575 df-rn 5576 df-res 5577 df-ima 5578 df-iota 6355 df-fun 6399 df-fn 6400 df-f 6401 df-fv 6405 df-lsatoms 36753 |
This theorem is referenced by: lsatlspsn2 36769 lsatlspsn 36770 islsati 36771 lsateln0 36772 lsatn0 36776 lsatcmp 36780 lsmsat 36785 lsatfixedN 36786 islshpat 36794 lsatcv0 36808 lsat0cv 36810 lcv1 36818 l1cvpat 36831 dih1dimatlem 39106 dihlatat 39114 dochsatshp 39228 |
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