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Theorem mnd12g 18186
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 7228 . 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 18182 . . 3 ((𝐺 ∈ Mnd ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 + 𝑌) + 𝑍) = (𝑋 + (𝑌 + 𝑍)))
103, 4, 5, 6, 9syl13anc 1374 . 2 (𝜑 → ((𝑋 + 𝑌) + 𝑍) = (𝑋 + (𝑌 + 𝑍)))
117, 8mndass 18182 . . 3 ((𝐺 ∈ Mnd ∧ (𝑌𝐵𝑋𝐵𝑍𝐵)) → ((𝑌 + 𝑋) + 𝑍) = (𝑌 + (𝑋 + 𝑍)))
123, 5, 4, 6, 11syl13anc 1374 . 2 (𝜑 → ((𝑌 + 𝑋) + 𝑍) = (𝑌 + (𝑋 + 𝑍)))
132, 10, 123eqtr3d 2785 1 (𝜑 → (𝑋 + (𝑌 + 𝑍)) = (𝑌 + (𝑋 + 𝑍)))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1543  wcel 2110  cfv 6380  (class class class)co 7213  Basecbs 16760  +gcplusg 16802  Mndcmnd 18173
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-nul 5199
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3410  df-sbc 3695  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-sn 4542  df-pr 4544  df-op 4548  df-uni 4820  df-br 5054  df-iota 6338  df-fv 6388  df-ov 7216  df-sgrp 18163  df-mnd 18174
This theorem is referenced by:  mnd4g  18187  cmn12  19191
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