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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 30010 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 20329 | . 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 18833 | . 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 205 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ‘cfv 6496 (class class class)co 7357 Basecbs 17083 0gc0g 17321 Grpcgrp 18748 -gcsg 18750 LModclmod 20322 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7672 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-ral 3065 df-rex 3074 df-rmo 3353 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-op 4593 df-uni 4866 df-iun 4956 df-br 5106 df-opab 5168 df-mpt 5189 df-id 5531 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-fv 6504 df-riota 7313 df-ov 7360 df-oprab 7361 df-mpo 7362 df-1st 7921 df-2nd 7922 df-0g 17323 df-mgm 18497 df-sgrp 18546 df-mnd 18557 df-grp 18751 df-minusg 18752 df-sbg 18753 df-lmod 20324 |
This theorem is referenced by: lvecvscan 20572 lvecvscan2 20573 lspsnsubn0 20601 ttgbtwnid 27832 lclkrlem2p 39985 lcfrlem31 40036 hdmaprnlem9N 40320 |
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