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Mirrors > Home > MPE Home > Th. List > lsp0 | Structured version Visualization version GIF version |
Description: Span of the empty set. (Contributed by Mario Carneiro, 5-Sep-2014.) |
Ref | Expression |
---|---|
lspsn0.z | ⊢ 0 = (0g‘𝑊) |
lspsn0.n | ⊢ 𝑁 = (LSpan‘𝑊) |
Ref | Expression |
---|---|
lsp0 | ⊢ (𝑊 ∈ LMod → (𝑁‘∅) = { 0 }) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lspsn0.z | . . . 4 ⊢ 0 = (0g‘𝑊) | |
2 | eqid 2651 | . . . 4 ⊢ (LSubSp‘𝑊) = (LSubSp‘𝑊) | |
3 | 1, 2 | lsssn0 18996 | . . 3 ⊢ (𝑊 ∈ LMod → { 0 } ∈ (LSubSp‘𝑊)) |
4 | 0ss 4005 | . . . 4 ⊢ ∅ ⊆ { 0 } | |
5 | lspsn0.n | . . . . 5 ⊢ 𝑁 = (LSpan‘𝑊) | |
6 | 2, 5 | lspssp 19036 | . . . 4 ⊢ ((𝑊 ∈ LMod ∧ { 0 } ∈ (LSubSp‘𝑊) ∧ ∅ ⊆ { 0 }) → (𝑁‘∅) ⊆ { 0 }) |
7 | 4, 6 | mp3an3 1453 | . . 3 ⊢ ((𝑊 ∈ LMod ∧ { 0 } ∈ (LSubSp‘𝑊)) → (𝑁‘∅) ⊆ { 0 }) |
8 | 3, 7 | mpdan 703 | . 2 ⊢ (𝑊 ∈ LMod → (𝑁‘∅) ⊆ { 0 }) |
9 | 0ss 4005 | . . . 4 ⊢ ∅ ⊆ (Base‘𝑊) | |
10 | eqid 2651 | . . . . 5 ⊢ (Base‘𝑊) = (Base‘𝑊) | |
11 | 10, 2, 5 | lspcl 19024 | . . . 4 ⊢ ((𝑊 ∈ LMod ∧ ∅ ⊆ (Base‘𝑊)) → (𝑁‘∅) ∈ (LSubSp‘𝑊)) |
12 | 9, 11 | mpan2 707 | . . 3 ⊢ (𝑊 ∈ LMod → (𝑁‘∅) ∈ (LSubSp‘𝑊)) |
13 | 1, 2 | lss0ss 18997 | . . 3 ⊢ ((𝑊 ∈ LMod ∧ (𝑁‘∅) ∈ (LSubSp‘𝑊)) → { 0 } ⊆ (𝑁‘∅)) |
14 | 12, 13 | mpdan 703 | . 2 ⊢ (𝑊 ∈ LMod → { 0 } ⊆ (𝑁‘∅)) |
15 | 8, 14 | eqssd 3653 | 1 ⊢ (𝑊 ∈ LMod → (𝑁‘∅) = { 0 }) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1523 ∈ wcel 2030 ⊆ wss 3607 ∅c0 3948 {csn 4210 ‘cfv 5926 Basecbs 15904 0gc0g 16147 LModclmod 18911 LSubSpclss 18980 LSpanclspn 19019 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-rep 4804 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 ax-un 6991 ax-cnex 10030 ax-resscn 10031 ax-1cn 10032 ax-icn 10033 ax-addcl 10034 ax-addrcl 10035 ax-mulcl 10036 ax-mulrcl 10037 ax-mulcom 10038 ax-addass 10039 ax-mulass 10040 ax-distr 10041 ax-i2m1 10042 ax-1ne0 10043 ax-1rid 10044 ax-rnegex 10045 ax-rrecex 10046 ax-cnre 10047 ax-pre-lttri 10048 ax-pre-lttrn 10049 ax-pre-ltadd 10050 ax-pre-mulgt0 10051 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1055 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-nel 2927 df-ral 2946 df-rex 2947 df-reu 2948 df-rmo 2949 df-rab 2950 df-v 3233 df-sbc 3469 df-csb 3567 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-pss 3623 df-nul 3949 df-if 4120 df-pw 4193 df-sn 4211 df-pr 4213 df-tp 4215 df-op 4217 df-uni 4469 df-int 4508 df-iun 4554 df-br 4686 df-opab 4746 df-mpt 4763 df-tr 4786 df-id 5053 df-eprel 5058 df-po 5064 df-so 5065 df-fr 5102 df-we 5104 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 df-pred 5718 df-ord 5764 df-on 5765 df-lim 5766 df-suc 5767 df-iota 5889 df-fun 5928 df-fn 5929 df-f 5930 df-f1 5931 df-fo 5932 df-f1o 5933 df-fv 5934 df-riota 6651 df-ov 6693 df-oprab 6694 df-mpt2 6695 df-om 7108 df-1st 7210 df-2nd 7211 df-wrecs 7452 df-recs 7513 df-rdg 7551 df-er 7787 df-en 7998 df-dom 7999 df-sdom 8000 df-pnf 10114 df-mnf 10115 df-xr 10116 df-ltxr 10117 df-le 10118 df-sub 10306 df-neg 10307 df-nn 11059 df-2 11117 df-ndx 15907 df-slot 15908 df-base 15910 df-sets 15911 df-plusg 16001 df-0g 16149 df-mgm 17289 df-sgrp 17331 df-mnd 17342 df-grp 17472 df-minusg 17473 df-sbg 17474 df-mgp 18536 df-ur 18548 df-ring 18595 df-lmod 18913 df-lss 18981 df-lsp 19020 |
This theorem is referenced by: lspuni0 19058 lss0v 19064 lspsnat 19193 lsppratlem3 19197 ocvz 20070 |
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