lmodvnegid - Metamath Proof Explorer

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Theorem lmodvnegid 20846

Description: Addition of a vector with its negative. (Contributed by NM, 18-Apr-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)

**Hypotheses**
Ref	Expression
lmodvnegid.v	⊢ 𝑉 = (Base‘𝑊)
lmodvnegid.p	⊢ + = (+_g‘𝑊)
lmodvnegid.z	⊢ 0 = (0_g‘𝑊)
lmodvnegid.n	⊢ 𝑁 = (inv_g‘𝑊)

**Assertion**
Ref	Expression
lmodvnegid	⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) → (𝑋 + (𝑁‘𝑋)) = 0 )

**Proof of Theorem lmodvnegid**
Step	Hyp	Ref	Expression
1		lmodgrp 20809	. 2 ⊢ (𝑊 ∈ LMod → 𝑊 ∈ Grp)
2		lmodvnegid.v	. . 3 ⊢ 𝑉 = (Base‘𝑊)
3		lmodvnegid.p	. . 3 ⊢ + = (+_g‘𝑊)
4		lmodvnegid.z	. . 3 ⊢ 0 = (0_g‘𝑊)
5		lmodvnegid.n	. . 3 ⊢ 𝑁 = (inv_g‘𝑊)
6	2, 3, 4, 5	grprinv 18911	. 2 ⊢ ((𝑊 ∈ Grp ∧ 𝑋 ∈ 𝑉) → (𝑋 + (𝑁‘𝑋)) = 0 )
7	1, 6	sylan 580	1 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) → (𝑋 + (𝑁‘𝑋)) = 0 )

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1541 ∈ wcel 2113 ‘cfv 6489 (class class class)co 7355 Basecbs 17127 +_gcplusg 17168 0_gc0g 17350 Grpcgrp 18854 inv_gcminusg 18855 LModclmod 20802

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 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2182 ax-ext 2705 ax-sep 5238 ax-nul 5248 ax-pow 5307 ax-pr 5374 ax-un 7677

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2725 df-clel 2808 df-nfc 2882 df-ne 2930 df-ral 3049 df-rex 3058 df-rmo 3347 df-reu 3348 df-rab 3397 df-v 3439 df-sbc 3738 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-nul 4283 df-if 4477 df-pw 4553 df-sn 4578 df-pr 4580 df-op 4584 df-uni 4861 df-br 5096 df-opab 5158 df-mpt 5177 df-id 5516 df-xp 5627 df-rel 5628 df-cnv 5629 df-co 5630 df-dm 5631 df-rn 5632 df-res 5633 df-ima 5634 df-iota 6445 df-fun 6491 df-fn 6492 df-f 6493 df-fv 6497 df-riota 7312 df-ov 7358 df-0g 17352 df-mgm 18556 df-sgrp 18635 df-mnd 18651 df-grp 18857 df-minusg 18858 df-lmod 20804

This theorem is referenced by: lmodvneg1 20847 hdmapneg 42018 lincext3 48618

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