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Mirrors > Home > MPE Home > Th. List > lmodvsubcl | Structured version Visualization version GIF version |
Description: Closure of vector subtraction. (hvsubcl 28800 analog.) (Contributed by NM, 31-Mar-2014.) (Revised by Mario Carneiro, 19-Jun-2014.) |
Ref | Expression |
---|---|
lmodvsubcl.v | ⊢ 𝑉 = (Base‘𝑊) |
lmodvsubcl.m | ⊢ − = (-g‘𝑊) |
Ref | Expression |
---|---|
lmodvsubcl | ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → (𝑋 − 𝑌) ∈ 𝑉) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lmodgrp 19634 | . 2 ⊢ (𝑊 ∈ LMod → 𝑊 ∈ Grp) | |
2 | lmodvsubcl.v | . . 3 ⊢ 𝑉 = (Base‘𝑊) | |
3 | lmodvsubcl.m | . . 3 ⊢ − = (-g‘𝑊) | |
4 | 2, 3 | grpsubcl 18171 | . 2 ⊢ ((𝑊 ∈ Grp ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → (𝑋 − 𝑌) ∈ 𝑉) |
5 | 1, 4 | syl3an1 1160 | 1 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → (𝑋 − 𝑌) ∈ 𝑉) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1084 = wceq 1538 ∈ wcel 2111 ‘cfv 6324 (class class class)co 7135 Basecbs 16475 Grpcgrp 18095 -gcsg 18097 LModclmod 19627 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-1st 7671 df-2nd 7672 df-0g 16707 df-mgm 17844 df-sgrp 17893 df-mnd 17904 df-grp 18098 df-minusg 18099 df-sbg 18100 df-lmod 19629 |
This theorem is referenced by: lspsnsub 19772 lvecvscan 19876 ip2subdi 20333 ip2eq 20342 ipcau2 23838 nmparlem 23843 minveclem1 24028 minveclem2 24030 minveclem4 24036 minveclem6 24038 pjthlem1 24041 pjthlem2 24042 eqlkr 36395 lkrlsp 36398 mapdpglem1 38968 mapdpglem2 38969 mapdpglem5N 38973 mapdpglem8 38975 mapdpglem9 38976 mapdpglem13 38980 mapdpglem14 38981 mapdpglem27 38995 baerlem3lem2 39006 baerlem5alem2 39007 baerlem5blem2 39008 mapdheq4lem 39027 mapdh6lem1N 39029 mapdh6lem2N 39030 hdmap1l6lem1 39103 hdmap1l6lem2 39104 hdmap11 39144 hdmapinvlem4 39217 |
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