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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 30830 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 20713 | . 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 18954 | . 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 6537 (class class class)co 7405 Basecbs 17153 0gc0g 17394 Grpcgrp 18863 -gcsg 18865 LModclmod 20706 |
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 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 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 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-fv 6545 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-1st 7974 df-2nd 7975 df-0g 17396 df-mgm 18573 df-sgrp 18652 df-mnd 18668 df-grp 18866 df-minusg 18867 df-sbg 18868 df-lmod 20708 |
This theorem is referenced by: lvecvscan 20962 lvecvscan2 20963 lspsnsubn0 20991 ttgbtwnid 28649 lclkrlem2p 40906 lcfrlem31 40957 hdmaprnlem9N 41241 |
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