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Theorem mndassd 33013
Description: A monoid operation is associative. (Contributed by Thierry Arnoux, 3-Aug-2025.)
Hypotheses
Ref Expression
mndassd.1 𝐵 = (Base‘𝐺)
mndassd.2 + = (+g𝐺)
mndassd.3 (𝜑𝐺 ∈ Mnd)
mndassd.4 (𝜑𝑋𝐵)
mndassd.5 (𝜑𝑌𝐵)
mndassd.6 (𝜑𝑍𝐵)
Assertion
Ref Expression
mndassd (𝜑 → ((𝑋 + 𝑌) + 𝑍) = (𝑋 + (𝑌 + 𝑍)))

Proof of Theorem mndassd
StepHypRef Expression
1 mndassd.3 . 2 (𝜑𝐺 ∈ Mnd)
2 mndassd.4 . 2 (𝜑𝑋𝐵)
3 mndassd.5 . 2 (𝜑𝑌𝐵)
4 mndassd.6 . 2 (𝜑𝑍𝐵)
5 mndassd.1 . . 3 𝐵 = (Base‘𝐺)
6 mndassd.2 . . 3 + = (+g𝐺)
75, 6mndass 18752 . 2 ((𝐺 ∈ Mnd ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 + 𝑌) + 𝑍) = (𝑋 + (𝑌 + 𝑍)))
81, 2, 3, 4, 7syl13anc 1374 1 (𝜑 → ((𝑋 + 𝑌) + 𝑍) = (𝑋 + (𝑌 + 𝑍)))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1540  wcel 2108  cfv 6559  (class class class)co 7429  Basecbs 17243  +gcplusg 17293  Mndcmnd 18743
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2707  ax-nul 5304
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2714  df-cleq 2728  df-clel 2815  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-sbc 3788  df-dif 3953  df-un 3955  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4906  df-br 5142  df-iota 6512  df-fv 6567  df-ov 7432  df-sgrp 18728  df-mnd 18744
This theorem is referenced by:  mndlrinv  33014  mndlactf1  33016  mndlactfo  33017  mndractf1  33018  mndractfo  33019  mndlactf1o  33020  mndractf1o  33021  gsumwun  33053
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