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Mirrors > Home > MPE Home > Th. List > lmodvnegid | Structured version Visualization version GIF version |
Description: Addition of a vector with its negative. (Contributed by NM, 18-Apr-2014.) (Revised by Mario Carneiro, 19-Jun-2014.) |
Ref | Expression |
---|---|
lmodvnegid.v | ⊢ 𝑉 = (Base‘𝑊) |
lmodvnegid.p | ⊢ + = (+g‘𝑊) |
lmodvnegid.z | ⊢ 0 = (0g‘𝑊) |
lmodvnegid.n | ⊢ 𝑁 = (invg‘𝑊) |
Ref | Expression |
---|---|
lmodvnegid | ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) → (𝑋 + (𝑁‘𝑋)) = 0 ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lmodgrp 19634 | . 2 ⊢ (𝑊 ∈ LMod → 𝑊 ∈ Grp) | |
2 | lmodvnegid.v | . . 3 ⊢ 𝑉 = (Base‘𝑊) | |
3 | lmodvnegid.p | . . 3 ⊢ + = (+g‘𝑊) | |
4 | lmodvnegid.z | . . 3 ⊢ 0 = (0g‘𝑊) | |
5 | lmodvnegid.n | . . 3 ⊢ 𝑁 = (invg‘𝑊) | |
6 | 2, 3, 4, 5 | grprinv 18145 | . 2 ⊢ ((𝑊 ∈ Grp ∧ 𝑋 ∈ 𝑉) → (𝑋 + (𝑁‘𝑋)) = 0 ) |
7 | 1, 6 | sylan 583 | 1 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) → (𝑋 + (𝑁‘𝑋)) = 0 ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ‘cfv 6324 (class class class)co 7135 Basecbs 16475 +gcplusg 16557 0gc0g 16705 Grpcgrp 18095 invgcminusg 18096 LModclmod 19627 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-fv 6332 df-riota 7093 df-ov 7138 df-0g 16707 df-mgm 17844 df-sgrp 17893 df-mnd 17904 df-grp 18098 df-minusg 18099 df-lmod 19629 |
This theorem is referenced by: lmodvneg1 19670 hdmapneg 39142 lincext3 44865 |
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