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Mirrors > Home > MPE Home > Th. List > lmodsubeq0 | Structured version Visualization version GIF version |
Description: If the difference between two vectors is zero, they are equal. (hvsubeq0 28497 analog.) (Contributed by NM, 31-Mar-2014.) (Revised by Mario Carneiro, 19-Jun-2014.) |
Ref | Expression |
---|---|
lmodsubeq0.v | ⊢ 𝑉 = (Base‘𝑊) |
lmodsubeq0.o | ⊢ 0 = (0g‘𝑊) |
lmodsubeq0.m | ⊢ − = (-g‘𝑊) |
Ref | Expression |
---|---|
lmodsubeq0 | ⊢ ((𝑊 ∈ LMod ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → ((𝐴 − 𝐵) = 0 ↔ 𝐴 = 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lmodgrp 19262 | . 2 ⊢ (𝑊 ∈ LMod → 𝑊 ∈ Grp) | |
2 | lmodsubeq0.v | . . 3 ⊢ 𝑉 = (Base‘𝑊) | |
3 | lmodsubeq0.o | . . 3 ⊢ 0 = (0g‘𝑊) | |
4 | lmodsubeq0.m | . . 3 ⊢ − = (-g‘𝑊) | |
5 | 2, 3, 4 | grpsubeq0 17888 | . 2 ⊢ ((𝑊 ∈ Grp ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → ((𝐴 − 𝐵) = 0 ↔ 𝐴 = 𝐵)) |
6 | 1, 5 | syl3an1 1163 | 1 ⊢ ((𝑊 ∈ LMod ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → ((𝐴 − 𝐵) = 0 ↔ 𝐴 = 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ w3a 1071 = wceq 1601 ∈ wcel 2107 ‘cfv 6135 (class class class)co 6922 Basecbs 16255 0gc0g 16486 Grpcgrp 17809 -gcsg 17811 LModclmod 19255 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-rep 5006 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-ral 3095 df-rex 3096 df-reu 3097 df-rmo 3098 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-op 4405 df-uni 4672 df-iun 4755 df-br 4887 df-opab 4949 df-mpt 4966 df-id 5261 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-riota 6883 df-ov 6925 df-oprab 6926 df-mpt2 6927 df-1st 7445 df-2nd 7446 df-0g 16488 df-mgm 17628 df-sgrp 17670 df-mnd 17681 df-grp 17812 df-minusg 17813 df-sbg 17814 df-lmod 19257 |
This theorem is referenced by: lvecvscan 19506 lvecvscan2 19507 lspsnsubn0 19536 ttgbtwnid 26233 lclkrlem2p 37676 lcfrlem31 37727 hdmaprnlem9N 38011 |
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