lmodvnegid - Metamath Proof Explorer

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Theorem lmodvnegid 20165

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 20130	. 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 18629	. 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 396 = wceq 1539 ∈ wcel 2106 ‘cfv 6433 (class class class)co 7275 Basecbs 16912 +_gcplusg 16962 0_gc0g 17150 Grpcgrp 18577 inv_gcminusg 18578 LModclmod 20123

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-br 5075 df-opab 5137 df-mpt 5158 df-id 5489 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-fv 6441 df-riota 7232 df-ov 7278 df-0g 17152 df-mgm 18326 df-sgrp 18375 df-mnd 18386 df-grp 18580 df-minusg 18581 df-lmod 20125

This theorem is referenced by: lmodvneg1 20166 hdmapneg 39860 lincext3 45797

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