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Theorem mnd12g 17936
 Description: Commutative/associative law for monoids, with an explicit commutativity hypothesis. (Contributed by Mario Carneiro, 21-Apr-2016.)
Hypotheses
Ref Expression
mndcl.b 𝐵 = (Base‘𝐺)
mndcl.p + = (+g𝐺)
mnd4g.1 (𝜑𝐺 ∈ Mnd)
mnd4g.2 (𝜑𝑋𝐵)
mnd4g.3 (𝜑𝑌𝐵)
mnd4g.4 (𝜑𝑍𝐵)
mnd12g.5 (𝜑 → (𝑋 + 𝑌) = (𝑌 + 𝑋))
Assertion
Ref Expression
mnd12g (𝜑 → (𝑋 + (𝑌 + 𝑍)) = (𝑌 + (𝑋 + 𝑍)))

Proof of Theorem mnd12g
StepHypRef Expression
1 mnd12g.5 . . 3 (𝜑 → (𝑋 + 𝑌) = (𝑌 + 𝑋))
21oveq1d 7160 . 2 (𝜑 → ((𝑋 + 𝑌) + 𝑍) = ((𝑌 + 𝑋) + 𝑍))
3 mnd4g.1 . . 3 (𝜑𝐺 ∈ Mnd)
4 mnd4g.2 . . 3 (𝜑𝑋𝐵)
5 mnd4g.3 . . 3 (𝜑𝑌𝐵)
6 mnd4g.4 . . 3 (𝜑𝑍𝐵)
7 mndcl.b . . . 4 𝐵 = (Base‘𝐺)
8 mndcl.p . . . 4 + = (+g𝐺)
97, 8mndass 17932 . . 3 ((𝐺 ∈ Mnd ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 + 𝑌) + 𝑍) = (𝑋 + (𝑌 + 𝑍)))
103, 4, 5, 6, 9syl13anc 1369 . 2 (𝜑 → ((𝑋 + 𝑌) + 𝑍) = (𝑋 + (𝑌 + 𝑍)))
117, 8mndass 17932 . . 3 ((𝐺 ∈ Mnd ∧ (𝑌𝐵𝑋𝐵𝑍𝐵)) → ((𝑌 + 𝑋) + 𝑍) = (𝑌 + (𝑋 + 𝑍)))
123, 5, 4, 6, 11syl13anc 1369 . 2 (𝜑 → ((𝑌 + 𝑋) + 𝑍) = (𝑌 + (𝑋 + 𝑍)))
132, 10, 123eqtr3d 2841 1 (𝜑 → (𝑋 + (𝑌 + 𝑍)) = (𝑌 + (𝑋 + 𝑍)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1538   ∈ wcel 2111  ‘cfv 6332  (class class class)co 7145  Basecbs 16495  +gcplusg 16577  Mndcmnd 17923 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 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-nul 5178 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 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3444  df-sbc 3723  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4247  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4805  df-br 5035  df-iota 6291  df-fv 6340  df-ov 7148  df-sgrp 17913  df-mnd 17924 This theorem is referenced by:  mnd4g  17937  cmn12  18940
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