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Theorem cmn12 18918
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 18913 . . 3 (𝐺 ∈ CMnd → 𝐺 ∈ Mnd)
43adantr 484 . 2 ((𝐺 ∈ CMnd ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝐺 ∈ Mnd)
5 simpr1 1191 . 2 ((𝐺 ∈ CMnd ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝑋𝐵)
6 simpr2 1192 . 2 ((𝐺 ∈ CMnd ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝑌𝐵)
7 simpr3 1193 . 2 ((𝐺 ∈ CMnd ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝑍𝐵)
81, 2cmncom 18914 . . 3 ((𝐺 ∈ CMnd ∧ 𝑋𝐵𝑌𝐵) → (𝑋 + 𝑌) = (𝑌 + 𝑋))
983adant3r3 1181 . 2 ((𝐺 ∈ CMnd ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋 + 𝑌) = (𝑌 + 𝑋))
101, 2, 4, 5, 6, 7, 9mnd12g 17915 1 ((𝐺 ∈ CMnd ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋 + (𝑌 + 𝑍)) = (𝑌 + (𝑋 + 𝑍)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  w3a 1084   = wceq 1538  wcel 2114  cfv 6334  (class class class)co 7140  Basecbs 16474  +gcplusg 16556  Mndcmnd 17902  CMndccmn 18897
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 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-nul 5186
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ral 3135  df-rex 3136  df-rab 3139  df-v 3471  df-sbc 3748  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4814  df-br 5043  df-iota 6293  df-fv 6342  df-ov 7143  df-sgrp 17892  df-mnd 17903  df-cmn 18899
This theorem is referenced by:  sraassa  20554  mamuvs2  21009  mdetuni0  21224
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