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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > lsatlspsn | Structured version Visualization version GIF version |
Description: The span of a nonzero singleton is an atom. (Contributed by NM, 16-Jan-2015.) |
Ref | Expression |
---|---|
lsatset.v | ⊢ 𝑉 = (Base‘𝑊) |
lsatset.n | ⊢ 𝑁 = (LSpan‘𝑊) |
lsatset.z | ⊢ 0 = (0g‘𝑊) |
lsatset.a | ⊢ 𝐴 = (LSAtoms‘𝑊) |
lsatlspsn.w | ⊢ (𝜑 → 𝑊 ∈ LMod) |
lsatlspsn.x | ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) |
Ref | Expression |
---|---|
lsatlspsn | ⊢ (𝜑 → (𝑁‘{𝑋}) ∈ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lsatlspsn.x | . . 3 ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) | |
2 | eqid 2777 | . . 3 ⊢ (𝑁‘{𝑋}) = (𝑁‘{𝑋}) | |
3 | sneq 4407 | . . . . 5 ⊢ (𝑣 = 𝑋 → {𝑣} = {𝑋}) | |
4 | 3 | fveq2d 6450 | . . . 4 ⊢ (𝑣 = 𝑋 → (𝑁‘{𝑣}) = (𝑁‘{𝑋})) |
5 | 4 | rspceeqv 3528 | . . 3 ⊢ ((𝑋 ∈ (𝑉 ∖ { 0 }) ∧ (𝑁‘{𝑋}) = (𝑁‘{𝑋})) → ∃𝑣 ∈ (𝑉 ∖ { 0 })(𝑁‘{𝑋}) = (𝑁‘{𝑣})) |
6 | 1, 2, 5 | sylancl 580 | . 2 ⊢ (𝜑 → ∃𝑣 ∈ (𝑉 ∖ { 0 })(𝑁‘{𝑋}) = (𝑁‘{𝑣})) |
7 | lsatlspsn.w | . . 3 ⊢ (𝜑 → 𝑊 ∈ LMod) | |
8 | lsatset.v | . . . 4 ⊢ 𝑉 = (Base‘𝑊) | |
9 | lsatset.n | . . . 4 ⊢ 𝑁 = (LSpan‘𝑊) | |
10 | lsatset.z | . . . 4 ⊢ 0 = (0g‘𝑊) | |
11 | lsatset.a | . . . 4 ⊢ 𝐴 = (LSAtoms‘𝑊) | |
12 | 8, 9, 10, 11 | islsat 35139 | . . 3 ⊢ (𝑊 ∈ LMod → ((𝑁‘{𝑋}) ∈ 𝐴 ↔ ∃𝑣 ∈ (𝑉 ∖ { 0 })(𝑁‘{𝑋}) = (𝑁‘{𝑣}))) |
13 | 7, 12 | syl 17 | . 2 ⊢ (𝜑 → ((𝑁‘{𝑋}) ∈ 𝐴 ↔ ∃𝑣 ∈ (𝑉 ∖ { 0 })(𝑁‘{𝑋}) = (𝑁‘{𝑣}))) |
14 | 6, 13 | mpbird 249 | 1 ⊢ (𝜑 → (𝑁‘{𝑋}) ∈ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 = wceq 1601 ∈ wcel 2106 ∃wrex 3090 ∖ cdif 3788 {csn 4397 ‘cfv 6135 Basecbs 16255 0gc0g 16486 LModclmod 19255 LSpanclspn 19366 LSAtomsclsa 35122 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2054 ax-8 2108 ax-9 2115 ax-10 2134 ax-11 2149 ax-12 2162 ax-13 2333 ax-ext 2753 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2550 df-eu 2586 df-clab 2763 df-cleq 2769 df-clel 2773 df-nfc 2920 df-ne 2969 df-ral 3094 df-rex 3095 df-rab 3098 df-v 3399 df-sbc 3652 df-dif 3794 df-un 3796 df-in 3798 df-ss 3805 df-nul 4141 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-op 4404 df-uni 4672 df-br 4887 df-opab 4949 df-mpt 4966 df-id 5261 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-fv 6143 df-lsatoms 35124 |
This theorem is referenced by: lsatspn0 35148 dvh4dimlem 37591 dochsnshp 37601 lclkrlem2a 37655 lclkrlem2c 37657 lclkrlem2e 37659 lcfrlem20 37710 mapdrvallem2 37793 mapdpglem20 37839 mapdpglem30a 37843 mapdpglem30b 37844 hdmaprnlem3eN 38006 hdmaprnlem16N 38010 |
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