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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 2734 | . . 3 ⊢ (𝑁‘{𝑋}) = (𝑁‘{𝑋}) | |
3 | sneq 4541 | . . . . 5 ⊢ (𝑣 = 𝑋 → {𝑣} = {𝑋}) | |
4 | 3 | fveq2d 6710 | . . . 4 ⊢ (𝑣 = 𝑋 → (𝑁‘{𝑣}) = (𝑁‘{𝑋})) |
5 | 4 | rspceeqv 3545 | . . 3 ⊢ ((𝑋 ∈ (𝑉 ∖ { 0 }) ∧ (𝑁‘{𝑋}) = (𝑁‘{𝑋})) → ∃𝑣 ∈ (𝑉 ∖ { 0 })(𝑁‘{𝑋}) = (𝑁‘{𝑣})) |
6 | 1, 2, 5 | sylancl 589 | . 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 36699 | . . 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 260 | 1 ⊢ (𝜑 → (𝑁‘{𝑋}) ∈ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 = wceq 1543 ∈ wcel 2110 ∃wrex 3055 ∖ cdif 3854 {csn 4531 ‘cfv 6369 Basecbs 16684 0gc0g 16916 LModclmod 19871 LSpanclspn 19980 LSAtomsclsa 36682 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2706 ax-sep 5181 ax-nul 5188 ax-pow 5247 ax-pr 5311 ax-un 7512 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2537 df-eu 2566 df-clab 2713 df-cleq 2726 df-clel 2812 df-nfc 2882 df-ne 2936 df-ral 3059 df-rex 3060 df-rab 3063 df-v 3403 df-sbc 3688 df-dif 3860 df-un 3862 df-in 3864 df-ss 3874 df-nul 4228 df-if 4430 df-pw 4505 df-sn 4532 df-pr 4534 df-op 4538 df-uni 4810 df-br 5044 df-opab 5106 df-mpt 5125 df-id 5444 df-xp 5546 df-rel 5547 df-cnv 5548 df-co 5549 df-dm 5550 df-rn 5551 df-res 5552 df-ima 5553 df-iota 6327 df-fun 6371 df-fn 6372 df-f 6373 df-fv 6377 df-lsatoms 36684 |
This theorem is referenced by: lsatspn0 36708 dvh4dimlem 39151 dochsnshp 39161 lclkrlem2a 39215 lclkrlem2c 39217 lclkrlem2e 39219 lcfrlem20 39270 mapdrvallem2 39353 mapdpglem20 39399 mapdpglem30a 39403 mapdpglem30b 39404 hdmaprnlem3eN 39566 hdmaprnlem16N 39570 |
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