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Mirrors > Home > MPE Home > Th. List > lmodvsubcl | Structured version Visualization version GIF version |
Description: Closure of vector subtraction. (hvsubcl 30847 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 20757 | . 2 ⊢ (𝑊 ∈ LMod → 𝑊 ∈ Grp) | |
2 | lmodvsubcl.v | . . 3 ⊢ 𝑉 = (Base‘𝑊) | |
3 | lmodvsubcl.m | . . 3 ⊢ − = (-g‘𝑊) | |
4 | 2, 3 | grpsubcl 18983 | . 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 1533 ∈ wcel 2098 ‘cfv 6553 (class class class)co 7426 Basecbs 17187 Grpcgrp 18897 -gcsg 18899 LModclmod 20750 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-id 5580 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-fv 6561 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-1st 7999 df-2nd 8000 df-0g 17430 df-mgm 18607 df-sgrp 18686 df-mnd 18702 df-grp 18900 df-minusg 18901 df-sbg 18902 df-lmod 20752 |
This theorem is referenced by: lspsnsub 20898 lvecvscan 21006 ip2subdi 21583 ip2eq 21592 ipcau2 25182 nmparlem 25187 minveclem1 25372 minveclem2 25374 minveclem4 25380 minveclem6 25382 pjthlem1 25385 pjthlem2 25386 eqlkr 38603 lkrlsp 38606 mapdpglem1 41177 mapdpglem2 41178 mapdpglem5N 41182 mapdpglem8 41184 mapdpglem9 41185 mapdpglem13 41189 mapdpglem14 41190 mapdpglem27 41204 baerlem3lem2 41215 baerlem5alem2 41216 baerlem5blem2 41217 mapdheq4lem 41236 mapdh6lem1N 41238 mapdh6lem2N 41239 hdmap1l6lem1 41312 hdmap1l6lem2 41313 hdmap11 41353 hdmapinvlem4 41426 |
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