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Mirrors > Home > MPE Home > Th. List > lmodvsubcl | Structured version Visualization version GIF version |
Description: Closure of vector subtraction. (hvsubcl 28429 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 19226 | . 2 ⊢ (𝑊 ∈ LMod → 𝑊 ∈ Grp) | |
2 | lmodvsubcl.v | . . 3 ⊢ 𝑉 = (Base‘𝑊) | |
3 | lmodvsubcl.m | . . 3 ⊢ − = (-g‘𝑊) | |
4 | 2, 3 | grpsubcl 17849 | . 2 ⊢ ((𝑊 ∈ Grp ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → (𝑋 − 𝑌) ∈ 𝑉) |
5 | 1, 4 | syl3an1 1208 | 1 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → (𝑋 − 𝑌) ∈ 𝑉) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1113 = wceq 1658 ∈ wcel 2166 ‘cfv 6123 (class class class)co 6905 Basecbs 16222 Grpcgrp 17776 -gcsg 17778 LModclmod 19219 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2803 ax-rep 4994 ax-sep 5005 ax-nul 5013 ax-pow 5065 ax-pr 5127 ax-un 7209 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-ral 3122 df-rex 3123 df-reu 3124 df-rmo 3125 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-nul 4145 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-op 4404 df-uni 4659 df-iun 4742 df-br 4874 df-opab 4936 df-mpt 4953 df-id 5250 df-xp 5348 df-rel 5349 df-cnv 5350 df-co 5351 df-dm 5352 df-rn 5353 df-res 5354 df-ima 5355 df-iota 6086 df-fun 6125 df-fn 6126 df-f 6127 df-f1 6128 df-fo 6129 df-f1o 6130 df-fv 6131 df-riota 6866 df-ov 6908 df-oprab 6909 df-mpt2 6910 df-1st 7428 df-2nd 7429 df-0g 16455 df-mgm 17595 df-sgrp 17637 df-mnd 17648 df-grp 17779 df-minusg 17780 df-sbg 17781 df-lmod 19221 |
This theorem is referenced by: lspsnsub 19366 lvecvscan 19470 ip2subdi 20351 ip2eq 20360 ipcau2 23402 nmparlem 23407 minveclem1 23592 minveclem2 23594 minveclem4 23600 minveclem6 23602 pjthlem1 23605 pjthlem2 23606 eqlkr 35174 lkrlsp 35177 mapdpglem1 37747 mapdpglem2 37748 mapdpglem5N 37752 mapdpglem8 37754 mapdpglem9 37755 mapdpglem13 37759 mapdpglem14 37760 mapdpglem27 37774 baerlem3lem2 37785 baerlem5alem2 37786 baerlem5blem2 37787 mapdheq4lem 37806 mapdh6lem1N 37808 mapdh6lem2N 37809 hdmap1l6lem1 37882 hdmap1l6lem2 37883 hdmap11 37923 hdmapinvlem4 37996 |
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