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Theorem grpassd 18909
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 18906 . 2 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 + 𝑌) + 𝑍) = (𝑋 + (𝑌 + 𝑍)))
81, 2, 3, 4, 7syl13anc 1369 1 (𝜑 → ((𝑋 + 𝑌) + 𝑍) = (𝑋 + (𝑌 + 𝑍)))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1533  wcel 2098  cfv 6553  (class class class)co 7426  Basecbs 17187  +gcplusg 17240  Grpcgrp 18897
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 2699  ax-nul 5310
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2706  df-cleq 2720  df-clel 2806  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-sbc 3779  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-br 5153  df-iota 6505  df-fv 6561  df-ov 7429  df-sgrp 18686  df-mnd 18702  df-grp 18900
This theorem is referenced by:  grplmulf1o  18976  grpraddf1o  18977  eqger  19140  conjnmz  19213  psdmul  22097  rloccring  33009  qsdrngilem  33230  grpcominv1  41779
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