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Theorem grpassd 18830
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 18827 . 2 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 + 𝑌) + 𝑍) = (𝑋 + (𝑌 + 𝑍)))
81, 2, 3, 4, 7syl13anc 1372 1 (𝜑 → ((𝑋 + 𝑌) + 𝑍) = (𝑋 + (𝑌 + 𝑍)))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1541  wcel 2106  cfv 6543  (class class class)co 7408  Basecbs 17143  +gcplusg 17196  Grpcgrp 18818
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703  ax-nul 5306
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-sbc 3778  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-iota 6495  df-fv 6551  df-ov 7411  df-sgrp 18609  df-mnd 18625  df-grp 18821
This theorem is referenced by:  eqger  19057  qsdrngilem  32603  grpcominv1  41084
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