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Theorem abl32 19538
Description: Commutative/associative law for Abelian groups. (Contributed by Stefan O'Rear, 10-Apr-2015.) (Revised by Mario Carneiro, 21-Apr-2016.)
Hypotheses
Ref Expression
ablcom.b 𝐵 = (Base‘𝐺)
ablcom.p + = (+g𝐺)
abl32.g (𝜑𝐺 ∈ Abel)
abl32.x (𝜑𝑋𝐵)
abl32.y (𝜑𝑌𝐵)
abl32.z (𝜑𝑍𝐵)
Assertion
Ref Expression
abl32 (𝜑 → ((𝑋 + 𝑌) + 𝑍) = ((𝑋 + 𝑍) + 𝑌))

Proof of Theorem abl32
StepHypRef Expression
1 abl32.g . . 3 (𝜑𝐺 ∈ Abel)
2 ablcmn 19522 . . 3 (𝐺 ∈ Abel → 𝐺 ∈ CMnd)
31, 2syl 17 . 2 (𝜑𝐺 ∈ CMnd)
4 abl32.x . 2 (𝜑𝑋𝐵)
5 abl32.y . 2 (𝜑𝑌𝐵)
6 abl32.z . 2 (𝜑𝑍𝐵)
7 ablcom.b . . 3 𝐵 = (Base‘𝐺)
8 ablcom.p . . 3 + = (+g𝐺)
97, 8cmn32 19535 . 2 ((𝐺 ∈ CMnd ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 + 𝑌) + 𝑍) = ((𝑋 + 𝑍) + 𝑌))
103, 4, 5, 6, 9syl13anc 1372 1 (𝜑 → ((𝑋 + 𝑌) + 𝑍) = ((𝑋 + 𝑍) + 𝑌))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1541  wcel 2106  cfv 6493  (class class class)co 7351  Basecbs 17037  +gcplusg 17087  CMndccmn 19515  Abelcabl 19516
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-12 2171  ax-ext 2707  ax-nul 5261
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 2714  df-cleq 2728  df-clel 2814  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3406  df-v 3445  df-sbc 3738  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4281  df-if 4485  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4864  df-br 5104  df-iota 6445  df-fv 6501  df-ov 7354  df-sgrp 18500  df-mnd 18511  df-cmn 19517  df-abl 19518
This theorem is referenced by:  matunitlindflem1  36006  baerlem5alem1  40103  baerlem5blem1  40104
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