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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 2622 | . . 3 ⊢ (𝑁‘{𝑋}) = (𝑁‘{𝑋}) | |
3 | sneq 4187 | . . . . . 6 ⊢ (𝑣 = 𝑋 → {𝑣} = {𝑋}) | |
4 | 3 | fveq2d 6195 | . . . . 5 ⊢ (𝑣 = 𝑋 → (𝑁‘{𝑣}) = (𝑁‘{𝑋})) |
5 | 4 | eqeq2d 2632 | . . . 4 ⊢ (𝑣 = 𝑋 → ((𝑁‘{𝑋}) = (𝑁‘{𝑣}) ↔ (𝑁‘{𝑋}) = (𝑁‘{𝑋}))) |
6 | 5 | rspcev 3309 | . . 3 ⊢ ((𝑋 ∈ (𝑉 ∖ { 0 }) ∧ (𝑁‘{𝑋}) = (𝑁‘{𝑋})) → ∃𝑣 ∈ (𝑉 ∖ { 0 })(𝑁‘{𝑋}) = (𝑁‘{𝑣})) |
7 | 1, 2, 6 | sylancl 694 | . 2 ⊢ (𝜑 → ∃𝑣 ∈ (𝑉 ∖ { 0 })(𝑁‘{𝑋}) = (𝑁‘{𝑣})) |
8 | lsatlspsn.w | . . 3 ⊢ (𝜑 → 𝑊 ∈ LMod) | |
9 | lsatset.v | . . . 4 ⊢ 𝑉 = (Base‘𝑊) | |
10 | lsatset.n | . . . 4 ⊢ 𝑁 = (LSpan‘𝑊) | |
11 | lsatset.z | . . . 4 ⊢ 0 = (0g‘𝑊) | |
12 | lsatset.a | . . . 4 ⊢ 𝐴 = (LSAtoms‘𝑊) | |
13 | 9, 10, 11, 12 | islsat 34278 | . . 3 ⊢ (𝑊 ∈ LMod → ((𝑁‘{𝑋}) ∈ 𝐴 ↔ ∃𝑣 ∈ (𝑉 ∖ { 0 })(𝑁‘{𝑋}) = (𝑁‘{𝑣}))) |
14 | 8, 13 | syl 17 | . 2 ⊢ (𝜑 → ((𝑁‘{𝑋}) ∈ 𝐴 ↔ ∃𝑣 ∈ (𝑉 ∖ { 0 })(𝑁‘{𝑋}) = (𝑁‘{𝑣}))) |
15 | 7, 14 | mpbird 247 | 1 ⊢ (𝜑 → (𝑁‘{𝑋}) ∈ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 = wceq 1483 ∈ wcel 1990 ∃wrex 2913 ∖ cdif 3571 {csn 4177 ‘cfv 5888 Basecbs 15857 0gc0g 16100 LModclmod 18863 LSpanclspn 18971 LSAtomsclsa 34261 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-fv 5896 df-lsatoms 34263 |
This theorem is referenced by: lsatspn0 34287 dvh4dimlem 36732 dochsnshp 36742 lclkrlem2a 36796 lclkrlem2c 36798 lclkrlem2e 36800 lcfrlem20 36851 mapdrvallem2 36934 mapdpglem20 36980 mapdpglem30a 36984 mapdpglem30b 36985 hdmaprnlem3eN 37150 hdmaprnlem16N 37154 |
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