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Mirrors > Home > MPE Home > Th. List > lmod0vlid | Structured version Visualization version GIF version |
Description: Left identity law for the zero vector. (hvaddid2 28784 analog.) (Contributed by NM, 10-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.) |
Ref | Expression |
---|---|
0vlid.v | ⊢ 𝑉 = (Base‘𝑊) |
0vlid.a | ⊢ + = (+g‘𝑊) |
0vlid.z | ⊢ 0 = (0g‘𝑊) |
Ref | Expression |
---|---|
lmod0vlid | ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) → ( 0 + 𝑋) = 𝑋) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lmodgrp 19624 | . 2 ⊢ (𝑊 ∈ LMod → 𝑊 ∈ Grp) | |
2 | 0vlid.v | . . 3 ⊢ 𝑉 = (Base‘𝑊) | |
3 | 0vlid.a | . . 3 ⊢ + = (+g‘𝑊) | |
4 | 0vlid.z | . . 3 ⊢ 0 = (0g‘𝑊) | |
5 | 2, 3, 4 | grplid 18116 | . 2 ⊢ ((𝑊 ∈ Grp ∧ 𝑋 ∈ 𝑉) → ( 0 + 𝑋) = 𝑋) |
6 | 1, 5 | sylan 582 | 1 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) → ( 0 + 𝑋) = 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ‘cfv 6341 (class class class)co 7142 Basecbs 16466 +gcplusg 16548 0gc0g 16696 Grpcgrp 18086 LModclmod 19617 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5189 ax-nul 5196 ax-pow 5252 ax-pr 5316 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3488 df-sbc 3764 df-dif 3927 df-un 3929 df-in 3931 df-ss 3940 df-nul 4280 df-if 4454 df-sn 4554 df-pr 4556 df-op 4560 df-uni 4825 df-br 5053 df-opab 5115 df-mpt 5133 df-id 5446 df-xp 5547 df-rel 5548 df-cnv 5549 df-co 5550 df-dm 5551 df-iota 6300 df-fun 6343 df-fv 6349 df-riota 7100 df-ov 7145 df-0g 16698 df-mgm 17835 df-sgrp 17884 df-mnd 17895 df-grp 18089 df-lmod 19619 |
This theorem is referenced by: lmodvneg1 19660 lsssn0 19702 lspfixed 19883 lspexch 19884 lsmsat 36176 dochfl1 38644 baerlem5blem1 38877 |
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