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Theorem grpassd 18963
Description: A group operation is associative. (Contributed by SN, 29-Jan-2025.)
Hypotheses
Ref Expression
grpassd.b 𝐵 = (Base‘𝐺)
grpassd.p + = (+g𝐺)
grpassd.g (𝜑𝐺 ∈ Grp)
grpassd.1 (𝜑𝑋𝐵)
grpassd.2 (𝜑𝑌𝐵)
grpassd.3 (𝜑𝑍𝐵)
Assertion
Ref Expression
grpassd (𝜑 → ((𝑋 + 𝑌) + 𝑍) = (𝑋 + (𝑌 + 𝑍)))

Proof of Theorem grpassd
StepHypRef Expression
1 grpassd.g . 2 (𝜑𝐺 ∈ Grp)
2 grpassd.1 . 2 (𝜑𝑋𝐵)
3 grpassd.2 . 2 (𝜑𝑌𝐵)
4 grpassd.3 . 2 (𝜑𝑍𝐵)
5 grpassd.b . . 3 𝐵 = (Base‘𝐺)
6 grpassd.p . . 3 + = (+g𝐺)
75, 6grpass 18960 . 2 ((𝐺 ∈ Grp ∧ (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 6561  (class class class)co 7431  Basecbs 17247  +gcplusg 17297  Grpcgrp 18951
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 2708  ax-nul 5306
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 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-iota 6514  df-fv 6569  df-ov 7434  df-sgrp 18732  df-mnd 18748  df-grp 18954
This theorem is referenced by:  grplmulf1o  19031  grpraddf1o  19032  eqger  19196  conjnmz  19270  psdmul  22170  rloccring  33274  qsdrngilem  33522  grpcominv1  42518
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