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Theorem cmn12 18134
 Description: Commutative/associative law for Abelian monoids. (Contributed by Stefan O'Rear, 5-Sep-2015.) (Revised by Mario Carneiro, 21-Apr-2016.)
Hypotheses
Ref Expression
ablcom.b 𝐵 = (Base‘𝐺)
ablcom.p + = (+g𝐺)
Assertion
Ref Expression
cmn12 ((𝐺 ∈ CMnd ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋 + (𝑌 + 𝑍)) = (𝑌 + (𝑋 + 𝑍)))

Proof of Theorem cmn12
StepHypRef Expression
1 ablcom.b . 2 𝐵 = (Base‘𝐺)
2 ablcom.p . 2 + = (+g𝐺)
3 cmnmnd 18129 . . 3 (𝐺 ∈ CMnd → 𝐺 ∈ Mnd)
43adantr 481 . 2 ((𝐺 ∈ CMnd ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝐺 ∈ Mnd)
5 simpr1 1065 . 2 ((𝐺 ∈ CMnd ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝑋𝐵)
6 simpr2 1066 . 2 ((𝐺 ∈ CMnd ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝑌𝐵)
7 simpr3 1067 . 2 ((𝐺 ∈ CMnd ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝑍𝐵)
81, 2cmncom 18130 . . 3 ((𝐺 ∈ CMnd ∧ 𝑋𝐵𝑌𝐵) → (𝑋 + 𝑌) = (𝑌 + 𝑋))
983adant3r3 1273 . 2 ((𝐺 ∈ CMnd ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋 + 𝑌) = (𝑌 + 𝑋))
101, 2, 4, 5, 6, 7, 9mnd12g 17227 1 ((𝐺 ∈ CMnd ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋 + (𝑌 + 𝑍)) = (𝑌 + (𝑋 + 𝑍)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 384   ∧ w3a 1036   = wceq 1480   ∈ wcel 1987  ‘cfv 5847  (class class class)co 6604  Basecbs 15781  +gcplusg 15862  Mndcmnd 17215  CMndccmn 18114 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 4749 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 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-iota 5810  df-fv 5855  df-ov 6607  df-sgrp 17205  df-mnd 17216  df-cmn 18116 This theorem is referenced by:  sraassa  19244  mamuvs2  20131  mdetuni0  20346
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