Mathbox for Norm Megill |
< Previous
Next >
Nearby theorems |
||
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 33295 | . . 3 ⊢ (𝑊 ∈ 𝑋 → 𝐴 = ran (𝑥 ∈ (𝑉 ∖ { 0 }) ↦ (𝑁‘{𝑥}))) |
6 | 5 | eleq2d 2673 | . 2 ⊢ (𝑊 ∈ 𝑋 → (𝑈 ∈ 𝐴 ↔ 𝑈 ∈ ran (𝑥 ∈ (𝑉 ∖ { 0 }) ↦ (𝑁‘{𝑥})))) |
7 | eqid 2610 | . . 3 ⊢ (𝑥 ∈ (𝑉 ∖ { 0 }) ↦ (𝑁‘{𝑥})) = (𝑥 ∈ (𝑉 ∖ { 0 }) ↦ (𝑁‘{𝑥})) | |
8 | fvex 6113 | . . 3 ⊢ (𝑁‘{𝑥}) ∈ V | |
9 | 7, 8 | elrnmpti 5297 | . 2 ⊢ (𝑈 ∈ ran (𝑥 ∈ (𝑉 ∖ { 0 }) ↦ (𝑁‘{𝑥})) ↔ ∃𝑥 ∈ (𝑉 ∖ { 0 })𝑈 = (𝑁‘{𝑥})) |
10 | 6, 9 | syl6bb 275 | 1 ⊢ (𝑊 ∈ 𝑋 → (𝑈 ∈ 𝐴 ↔ ∃𝑥 ∈ (𝑉 ∖ { 0 })𝑈 = (𝑁‘{𝑥}))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 = wceq 1475 ∈ wcel 1977 ∃wrex 2897 ∖ cdif 3537 {csn 4125 ↦ cmpt 4643 ran crn 5039 ‘cfv 5804 Basecbs 15695 0gc0g 15923 LSpanclspn 18792 LSAtomsclsa 33279 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-fv 5812 df-lsatoms 33281 |
This theorem is referenced by: lsatlspsn2 33297 lsatlspsn 33298 islsati 33299 lsateln0 33300 lsatn0 33304 lsatcmp 33308 lsmsat 33313 lsatfixedN 33314 islshpat 33322 lsatcv0 33336 lsat0cv 33338 lcv1 33346 l1cvpat 33359 dih1dimatlem 35636 dihlatat 35644 dochsatshp 35758 |
Copyright terms: Public domain | W3C validator |