| 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 2737 | . . 3 ⊢ (𝑁‘{𝑋}) = (𝑁‘{𝑋}) | |
| 3 | sneq 4636 | . . . . 5 ⊢ (𝑣 = 𝑋 → {𝑣} = {𝑋}) | |
| 4 | 3 | fveq2d 6910 | . . . 4 ⊢ (𝑣 = 𝑋 → (𝑁‘{𝑣}) = (𝑁‘{𝑋})) |
| 5 | 4 | rspceeqv 3645 | . . 3 ⊢ ((𝑋 ∈ (𝑉 ∖ { 0 }) ∧ (𝑁‘{𝑋}) = (𝑁‘{𝑋})) → ∃𝑣 ∈ (𝑉 ∖ { 0 })(𝑁‘{𝑋}) = (𝑁‘{𝑣})) |
| 6 | 1, 2, 5 | sylancl 586 | . 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 38992 | . . 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 257 | 1 ⊢ (𝜑 → (𝑁‘{𝑋}) ∈ 𝐴) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 = wceq 1540 ∈ wcel 2108 ∃wrex 3070 ∖ cdif 3948 {csn 4626 ‘cfv 6561 Basecbs 17247 0gc0g 17484 LModclmod 20858 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: lsatspn0 39001 dvh4dimlem 41445 dochsnshp 41455 lclkrlem2a 41509 lclkrlem2c 41511 lclkrlem2e 41513 lcfrlem20 41564 mapdrvallem2 41647 mapdpglem20 41693 mapdpglem30a 41697 mapdpglem30b 41698 hdmaprnlem3eN 41860 hdmaprnlem16N 41864 |
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