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Theorem abl32 18154
 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 18139 . . 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 18151 . 2 ((𝐺 ∈ CMnd ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 + 𝑌) + 𝑍) = ((𝑋 + 𝑍) + 𝑌))
103, 4, 5, 6, 9syl13anc 1325 1 (𝜑 → ((𝑋 + 𝑌) + 𝑍) = ((𝑋 + 𝑍) + 𝑌))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1480   ∈ wcel 1987  ‘cfv 5857  (class class class)co 6615  Basecbs 15800  +gcplusg 15881  CMndccmn 18133  Abelcabl 18134 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-nul 4759 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2913  df-rex 2914  df-rab 2917  df-v 3192  df-sbc 3423  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-br 4624  df-iota 5820  df-fv 5865  df-ov 6618  df-sgrp 17224  df-mnd 17235  df-cmn 18135  df-abl 18136 This theorem is referenced by:  matunitlindflem1  33076  baerlem5alem1  36516  baerlem5blem1  36517
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