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Mirrors > Home > MPE Home > Th. List > lmod0vrid | Structured version Visualization version GIF version |
Description: Right identity law for the zero vector. (ax-hvaddid 27989 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 |
---|---|
lmod0vrid | ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) → (𝑋 + 0 ) = 𝑋) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lmodgrp 18918 | . 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 | grprid 17500 | . 2 ⊢ ((𝑊 ∈ Grp ∧ 𝑋 ∈ 𝑉) → (𝑋 + 0 ) = 𝑋) |
6 | 1, 5 | sylan 487 | 1 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) → (𝑋 + 0 ) = 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1523 ∈ wcel 2030 ‘cfv 5926 (class class class)co 6690 Basecbs 15904 +gcplusg 15988 0gc0g 16147 Grpcgrp 17469 LModclmod 18911 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-ral 2946 df-rex 2947 df-reu 2948 df-rmo 2949 df-rab 2950 df-v 3233 df-sbc 3469 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-nul 3949 df-if 4120 df-sn 4211 df-pr 4213 df-op 4217 df-uni 4469 df-br 4686 df-opab 4746 df-mpt 4763 df-id 5053 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-iota 5889 df-fun 5928 df-fv 5934 df-riota 6651 df-ov 6693 df-0g 16149 df-mgm 17289 df-sgrp 17331 df-mnd 17342 df-grp 17472 df-lmod 18913 |
This theorem is referenced by: lmodvneg1 18954 lssvscl 19003 lspfixed 19176 lsmcv 19189 lspsolvlem 19190 lspsolv 19191 lfl0 34670 lflmul 34673 lshpkrlem1 34715 lclkrlem2j 37122 lcfrlem7 37154 mapdh6dN 37345 hdmap1l6d 37420 |
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