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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 30922 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 20754 | . 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 18986 | . 2 ⊢ ((𝑊 ∈ Grp ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → ((𝐴 − 𝐵) = 0 ↔ 𝐴 = 𝐵)) |
6 | 1, 5 | syl3an1 1160 | 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 1084 = wceq 1533 ∈ wcel 2098 ‘cfv 6543 (class class class)co 7416 Basecbs 17179 0gc0g 17420 Grpcgrp 18894 -gcsg 18896 LModclmod 20747 |
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 2696 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7738 |
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 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3769 df-csb 3885 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5227 df-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-fv 6551 df-riota 7372 df-ov 7419 df-oprab 7420 df-mpo 7421 df-1st 7991 df-2nd 7992 df-0g 17422 df-mgm 18599 df-sgrp 18678 df-mnd 18694 df-grp 18897 df-minusg 18898 df-sbg 18899 df-lmod 20749 |
This theorem is referenced by: lvecvscan 21003 lvecvscan2 21004 lspsnsubn0 21032 ttgbtwnid 28738 lclkrlem2p 41051 lcfrlem31 41102 hdmaprnlem9N 41386 |
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