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| Mirrors > Home > MPE Home > Th. List > Mathboxes > lsatlspsn2 | Structured version Visualization version GIF version | ||
| Description: The span of a nonzero singleton is an atom. TODO: make this obsolete and use lsatlspsn 38994 instead? (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 |
|---|---|
| lsatlspsn2 | ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉 ∧ 𝑋 ≠ 0 ) → (𝑁‘{𝑋}) ∈ 𝐴) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | 3simpc 1151 | . . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉 ∧ 𝑋 ≠ 0 ) → (𝑋 ∈ 𝑉 ∧ 𝑋 ≠ 0 )) | |
| 2 | eldifsn 4786 | . . . 4 ⊢ (𝑋 ∈ (𝑉 ∖ { 0 }) ↔ (𝑋 ∈ 𝑉 ∧ 𝑋 ≠ 0 )) | |
| 3 | 1, 2 | sylibr 234 | . . 3 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉 ∧ 𝑋 ≠ 0 ) → 𝑋 ∈ (𝑉 ∖ { 0 })) |
| 4 | eqid 2737 | . . 3 ⊢ (𝑁‘{𝑋}) = (𝑁‘{𝑋}) | |
| 5 | sneq 4636 | . . . . 5 ⊢ (𝑣 = 𝑋 → {𝑣} = {𝑋}) | |
| 6 | 5 | fveq2d 6910 | . . . 4 ⊢ (𝑣 = 𝑋 → (𝑁‘{𝑣}) = (𝑁‘{𝑋})) |
| 7 | 6 | rspceeqv 3645 | . . 3 ⊢ ((𝑋 ∈ (𝑉 ∖ { 0 }) ∧ (𝑁‘{𝑋}) = (𝑁‘{𝑋})) → ∃𝑣 ∈ (𝑉 ∖ { 0 })(𝑁‘{𝑋}) = (𝑁‘{𝑣})) |
| 8 | 3, 4, 7 | sylancl 586 | . 2 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉 ∧ 𝑋 ≠ 0 ) → ∃𝑣 ∈ (𝑉 ∖ { 0 })(𝑁‘{𝑋}) = (𝑁‘{𝑣})) |
| 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 38992 | . . 3 ⊢ (𝑊 ∈ LMod → ((𝑁‘{𝑋}) ∈ 𝐴 ↔ ∃𝑣 ∈ (𝑉 ∖ { 0 })(𝑁‘{𝑋}) = (𝑁‘{𝑣}))) |
| 14 | 13 | 3ad2ant1 1134 | . 2 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉 ∧ 𝑋 ≠ 0 ) → ((𝑁‘{𝑋}) ∈ 𝐴 ↔ ∃𝑣 ∈ (𝑉 ∖ { 0 })(𝑁‘{𝑋}) = (𝑁‘{𝑣}))) |
| 15 | 8, 14 | mpbird 257 | 1 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉 ∧ 𝑋 ≠ 0 ) → (𝑁‘{𝑋}) ∈ 𝐴) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 ≠ wne 2940 ∃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: lsatel 39006 lsmsat 39009 lssatomic 39012 lssats 39013 dihlsprn 41333 dihatlat 41336 dihatexv 41340 dochsatshpb 41454 |
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