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Theorem cmn12 19820
Description: Commutative/associative law for commutative 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 19815 . . 3 (𝐺 ∈ CMnd → 𝐺 ∈ Mnd)
43adantr 480 . 2 ((𝐺 ∈ CMnd ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝐺 ∈ Mnd)
5 simpr1 1195 . 2 ((𝐺 ∈ CMnd ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝑋𝐵)
6 simpr2 1196 . 2 ((𝐺 ∈ CMnd ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝑌𝐵)
7 simpr3 1197 . 2 ((𝐺 ∈ CMnd ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝑍𝐵)
81, 2cmncom 19816 . . 3 ((𝐺 ∈ CMnd ∧ 𝑋𝐵𝑌𝐵) → (𝑋 + 𝑌) = (𝑌 + 𝑋))
983adant3r3 1185 . 2 ((𝐺 ∈ CMnd ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋 + 𝑌) = (𝑌 + 𝑋))
101, 2, 4, 5, 6, 7, 9mnd12g 18760 1 ((𝐺 ∈ CMnd ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋 + (𝑌 + 𝑍)) = (𝑌 + (𝑋 + 𝑍)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1087   = wceq 1540  wcel 2108  cfv 6561  (class class class)co 7431  Basecbs 17247  +gcplusg 17297  Mndcmnd 18747  CMndccmn 19798
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-12 2177  ax-ext 2708  ax-nul 5306
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-iota 6514  df-fv 6569  df-ov 7434  df-sgrp 18732  df-mnd 18748  df-cmn 19800
This theorem is referenced by:  sraassaOLD  21890  mamuvs2  22410  mdetuni0  22627  1arithidomlem1  33563
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