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Theorem slmdass 32861
Description: Semiring left module vector sum is associative. (Contributed by NM, 10-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.) (Revised by Thierry Arnoux, 1-Apr-2018.)
Hypotheses
Ref Expression
slmdvacl.v 𝑉 = (Base‘𝑊)
slmdvacl.a + = (+g𝑊)
Assertion
Ref Expression
slmdass ((𝑊 ∈ SLMod ∧ (𝑋𝑉𝑌𝑉𝑍𝑉)) → ((𝑋 + 𝑌) + 𝑍) = (𝑋 + (𝑌 + 𝑍)))

Proof of Theorem slmdass
StepHypRef Expression
1 slmdmnd 32854 . 2 (𝑊 ∈ SLMod → 𝑊 ∈ Mnd)
2 slmdvacl.v . . 3 𝑉 = (Base‘𝑊)
3 slmdvacl.a . . 3 + = (+g𝑊)
42, 3mndass 18673 . 2 ((𝑊 ∈ Mnd ∧ (𝑋𝑉𝑌𝑉𝑍𝑉)) → ((𝑋 + 𝑌) + 𝑍) = (𝑋 + (𝑌 + 𝑍)))
51, 4sylan 579 1 ((𝑊 ∈ SLMod ∧ (𝑋𝑉𝑌𝑉𝑍𝑉)) → ((𝑋 + 𝑌) + 𝑍) = (𝑋 + (𝑌 + 𝑍)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1084   = wceq 1533  wcel 2098  cfv 6536  (class class class)co 7404  Basecbs 17150  +gcplusg 17203  Mndcmnd 18664  SLModcslmd 32848
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2697  ax-nul 5299
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-sbc 3773  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-iota 6488  df-fv 6544  df-ov 7407  df-sgrp 18649  df-mnd 18665  df-cmn 19699  df-slmd 32849
This theorem is referenced by: (None)
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