lmodsubid - Metamath Proof Explorer

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Theorem lmodsubid 19762

Description: Subtraction of a vector from itself. (hvsubid 28908 analog.) (Contributed by NM, 16-Apr-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)

**Hypotheses**
Ref	Expression
lmodsubeq0.v	⊢ 𝑉 = (Base‘𝑊)
lmodsubeq0.o	⊢ 0 = (0_g‘𝑊)
lmodsubeq0.m	⊢ − = (-_g‘𝑊)

**Assertion**
Ref	Expression
lmodsubid	⊢ ((𝑊 ∈ LMod ∧ 𝐴 ∈ 𝑉) → (𝐴 − 𝐴) = 0 )

**Proof of Theorem lmodsubid**
Step	Hyp	Ref	Expression
1		lmodgrp 19709	. 2 ⊢ (𝑊 ∈ LMod → 𝑊 ∈ Grp)
2		lmodsubeq0.v	. . 3 ⊢ 𝑉 = (Base‘𝑊)
3		lmodsubeq0.o	. . 3 ⊢ 0 = (0_g‘𝑊)
4		lmodsubeq0.m	. . 3 ⊢ − = (-_g‘𝑊)
5	2, 3, 4	grpsubid 18250	. 2 ⊢ ((𝑊 ∈ Grp ∧ 𝐴 ∈ 𝑉) → (𝐴 − 𝐴) = 0 )
6	1, 5	sylan 583	1 ⊢ ((𝑊 ∈ LMod ∧ 𝐴 ∈ 𝑉) → (𝐴 − 𝐴) = 0 )

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ‘cfv 6335 (class class class)co 7150 Basecbs 16541 0_gc0g 16771 Grpcgrp 18169 -_gcsg 18171 LModclmod 19702

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 2729 ax-sep 5169 ax-nul 5176 ax-pow 5234 ax-pr 5298 ax-un 7459

This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-fal 1551 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2557 df-eu 2588 df-clab 2736 df-cleq 2750 df-clel 2830 df-nfc 2901 df-ne 2952 df-ral 3075 df-rex 3076 df-reu 3077 df-rmo 3078 df-rab 3079 df-v 3411 df-sbc 3697 df-csb 3806 df-dif 3861 df-un 3863 df-in 3865 df-ss 3875 df-nul 4226 df-if 4421 df-pw 4496 df-sn 4523 df-pr 4525 df-op 4529 df-uni 4799 df-iun 4885 df-br 5033 df-opab 5095 df-mpt 5113 df-id 5430 df-xp 5530 df-rel 5531 df-cnv 5532 df-co 5533 df-dm 5534 df-rn 5535 df-res 5536 df-ima 5537 df-iota 6294 df-fun 6337 df-fn 6338 df-f 6339 df-fv 6343 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-1st 7693 df-2nd 7694 df-0g 16773 df-mgm 17918 df-sgrp 17967 df-mnd 17978 df-grp 18172 df-minusg 18173 df-sbg 18174 df-lmod 19704

This theorem is referenced by: lss0cl 19786 ttgbtwnid 26777

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