lmodvnegid - Metamath Proof Explorer

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Theorem lmodvnegid 19676

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 19641	. 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 18153	. 2 ⊢ ((𝑊 ∈ Grp ∧ 𝑋 ∈ 𝑉) → (𝑋 + (𝑁‘𝑋)) = 0 )
7	1, 6	sylan 582	1 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) → (𝑋 + (𝑁‘𝑋)) = 0 )

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ‘cfv 6355 (class class class)co 7156 Basecbs 16483 +_gcplusg 16565 0_gc0g 16713 Grpcgrp 18103 inv_gcminusg 18104 LModclmod 19634

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 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461

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 3496 df-sbc 3773 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-fv 6363 df-riota 7114 df-ov 7159 df-0g 16715 df-mgm 17852 df-sgrp 17901 df-mnd 17912 df-grp 18106 df-minusg 18107 df-lmod 19636

This theorem is referenced by: lmodvneg1 19677 hdmapneg 38997 lincext3 44531

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