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Theorem lmodass 20352
Description: Left module vector sum is associative. (Contributed by NM, 10-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
Hypotheses
Ref Expression
lmodvacl.v 𝑉 = (Base‘𝑊)
lmodvacl.a + = (+g𝑊)
Assertion
Ref Expression
lmodass ((𝑊 ∈ LMod ∧ (𝑋𝑉𝑌𝑉𝑍𝑉)) → ((𝑋 + 𝑌) + 𝑍) = (𝑋 + (𝑌 + 𝑍)))

Proof of Theorem lmodass
StepHypRef Expression
1 lmodgrp 20343 . 2 (𝑊 ∈ LMod → 𝑊 ∈ Grp)
2 lmodvacl.v . . 3 𝑉 = (Base‘𝑊)
3 lmodvacl.a . . 3 + = (+g𝑊)
42, 3grpass 18762 . 2 ((𝑊 ∈ Grp ∧ (𝑋𝑉𝑌𝑉𝑍𝑉)) → ((𝑋 + 𝑌) + 𝑍) = (𝑋 + (𝑌 + 𝑍)))
51, 4sylan 581 1 ((𝑊 ∈ LMod ∧ (𝑋𝑉𝑌𝑉𝑍𝑉)) → ((𝑋 + 𝑌) + 𝑍) = (𝑋 + (𝑌 + 𝑍)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397  w3a 1088   = wceq 1542  wcel 2107  cfv 6497  (class class class)co 7358  Basecbs 17088  +gcplusg 17138  Grpcgrp 18753  LModclmod 20336
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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-nul 5264
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-sbc 3741  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-iota 6449  df-fv 6505  df-ov 7361  df-sgrp 18551  df-mnd 18562  df-grp 18756  df-lmod 20338
This theorem is referenced by:  lmodvneg1  20380  lmodcom  20383  baerlem5alem1  40217  mapdh6gN  40251  mapdh6hN  40252  hdmap1l6g  40325  hdmap1l6h  40326
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