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Theorem lmodass 19625
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 19617 . 2 (𝑊 ∈ LMod → 𝑊 ∈ Grp)
2 lmodvacl.v . . 3 𝑉 = (Base‘𝑊)
3 lmodvacl.a . . 3 + = (+g𝑊)
42, 3grpass 18091 . 2 ((𝑊 ∈ Grp ∧ (𝑋𝑉𝑌𝑉𝑍𝑉)) → ((𝑋 + 𝑌) + 𝑍) = (𝑋 + (𝑌 + 𝑍)))
51, 4sylan 583 1 ((𝑊 ∈ LMod ∧ (𝑋𝑉𝑌𝑉𝑍𝑉)) → ((𝑋 + 𝑌) + 𝑍) = (𝑋 + (𝑌 + 𝑍)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  w3a 1084   = wceq 1538  wcel 2115  cfv 6328  (class class class)co 7130  Basecbs 16462  +gcplusg 16544  Grpcgrp 18082  LModclmod 19610
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2178  ax-ext 2793  ax-nul 5183
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2623  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-ral 3131  df-rex 3132  df-rab 3135  df-v 3473  df-sbc 3750  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4267  df-if 4441  df-sn 4541  df-pr 4543  df-op 4547  df-uni 4812  df-br 5040  df-iota 6287  df-fv 6336  df-ov 7133  df-sgrp 17880  df-mnd 17891  df-grp 18085  df-lmod 19612
This theorem is referenced by:  lmodvneg1  19653  lmodcom  19656  baerlem5alem1  38890  mapdh6gN  38924  mapdh6hN  38925  hdmap1l6g  38998  hdmap1l6h  38999
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