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Theorem cmn4d 33019
Description: Commutative/associative law for commutative monoids. (Contributed by Thierry Arnoux, 4-May-2025.)
Hypotheses
Ref Expression
cmn4d.1 𝐵 = (Base‘𝐺)
cmn4d.2 + = (+g𝐺)
cmn4d.3 (𝜑𝐺 ∈ CMnd)
cmn4d.4 (𝜑𝑋𝐵)
cmn4d.5 (𝜑𝑌𝐵)
cmn4d.6 (𝜑𝑍𝐵)
cmn4d.7 (𝜑𝑊𝐵)
Assertion
Ref Expression
cmn4d (𝜑 → ((𝑋 + 𝑌) + (𝑍 + 𝑊)) = ((𝑋 + 𝑍) + (𝑌 + 𝑊)))

Proof of Theorem cmn4d
StepHypRef Expression
1 cmn4d.3 . 2 (𝜑𝐺 ∈ CMnd)
2 cmn4d.4 . 2 (𝜑𝑋𝐵)
3 cmn4d.5 . 2 (𝜑𝑌𝐵)
4 cmn4d.6 . 2 (𝜑𝑍𝐵)
5 cmn4d.7 . 2 (𝜑𝑊𝐵)
6 cmn4d.1 . . 3 𝐵 = (Base‘𝐺)
7 cmn4d.2 . . 3 + = (+g𝐺)
86, 7cmn4 19833 . 2 ((𝐺 ∈ CMnd ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵)) → ((𝑋 + 𝑌) + (𝑍 + 𝑊)) = ((𝑋 + 𝑍) + (𝑌 + 𝑊)))
91, 2, 3, 4, 5, 8syl122anc 1378 1 (𝜑 → ((𝑋 + 𝑌) + (𝑍 + 𝑊)) = ((𝑋 + 𝑍) + (𝑌 + 𝑊)))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1536  wcel 2105  cfv 6562  (class class class)co 7430  Basecbs 17244  +gcplusg 17297  CMndccmn 19812
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-12 2174  ax-ext 2705  ax-nul 5311
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-sb 2062  df-clab 2712  df-cleq 2726  df-clel 2813  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-sbc 3791  df-dif 3965  df-un 3967  df-ss 3979  df-nul 4339  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-br 5148  df-iota 6515  df-fv 6570  df-ov 7433  df-mgm 18665  df-sgrp 18744  df-mnd 18760  df-cmn 19814
This theorem is referenced by:  cmn246135  33020  cmn145236  33021  rloccring  33256
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