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Theorem lmodlcan 19088

Description: Left cancellation law for vector sum. (Contributed by NM, 12-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)

**Hypotheses**
Ref	Expression
lmodvacl.v	⊢ 𝑉 = (Base‘𝑊)
lmodvacl.a	⊢ + = (+_g‘𝑊)

**Assertion**
Ref	Expression
lmodlcan	⊢ ((𝑊 ∈ LMod ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑉)) → ((𝑍 + 𝑋) = (𝑍 + 𝑌) ↔ 𝑋 = 𝑌))

**Proof of Theorem lmodlcan**
Step	Hyp	Ref	Expression
1		lmodgrp 19079	. 2 ⊢ (𝑊 ∈ LMod → 𝑊 ∈ Grp)
2		lmodvacl.v	. . 3 ⊢ 𝑉 = (Base‘𝑊)
3		lmodvacl.a	. . 3 ⊢ + = (+_g‘𝑊)
4	2, 3	grplcan 17684	. 2 ⊢ ((𝑊 ∈ Grp ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑉)) → ((𝑍 + 𝑋) = (𝑍 + 𝑌) ↔ 𝑋 = 𝑌))
5	1, 4	sylan 561	1 ⊢ ((𝑊 ∈ LMod ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑉)) → ((𝑍 + 𝑋) = (𝑍 + 𝑌) ↔ 𝑋 = 𝑌))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 196 ∧ wa 382 ∧ w3a 1070 = wceq 1630 ∈ wcel 2144 ‘cfv 6031 (class class class)co 6792 Basecbs 16063 +_gcplusg 16148 Grpcgrp 17629 LModclmod 19072

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1869 ax-4 1884 ax-5 1990 ax-6 2056 ax-7 2092 ax-8 2146 ax-9 2153 ax-10 2173 ax-11 2189 ax-12 2202 ax-13 2407 ax-ext 2750 ax-rep 4902 ax-sep 4912 ax-nul 4920 ax-pow 4971 ax-pr 5034

This theorem depends on definitions: df-bi 197 df-an 383 df-or 827 df-3an 1072 df-tru 1633 df-ex 1852 df-nf 1857 df-sb 2049 df-eu 2621 df-mo 2622 df-clab 2757 df-cleq 2763 df-clel 2766 df-nfc 2901 df-ne 2943 df-ral 3065 df-rex 3066 df-reu 3067 df-rmo 3068 df-rab 3069 df-v 3351 df-sbc 3586 df-csb 3681 df-dif 3724 df-un 3726 df-in 3728 df-ss 3735 df-nul 4062 df-if 4224 df-sn 4315 df-pr 4317 df-op 4321 df-uni 4573 df-iun 4654 df-br 4785 df-opab 4845 df-mpt 4862 df-id 5157 df-xp 5255 df-rel 5256 df-cnv 5257 df-co 5258 df-dm 5259 df-rn 5260 df-res 5261 df-ima 5262 df-iota 5994 df-fun 6033 df-fn 6034 df-f 6035 df-f1 6036 df-fo 6037 df-f1o 6038 df-fv 6039 df-riota 6753 df-ov 6795 df-0g 16309 df-mgm 17449 df-sgrp 17491 df-mnd 17502 df-grp 17632 df-minusg 17633 df-lmod 19074

This theorem is referenced by: lmodcom 19118 mapdh6hN 37546 hdmap1l6h 37620 hdmapneg 37649

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