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Theorem mnd32g 18772
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)
mnd32g.5 (𝜑 → (𝑌 + 𝑍) = (𝑍 + 𝑌))
Assertion
Ref Expression
mnd32g (𝜑 → ((𝑋 + 𝑌) + 𝑍) = ((𝑋 + 𝑍) + 𝑌))

Proof of Theorem mnd32g
StepHypRef Expression
1 mnd32g.5 . . 3 (𝜑 → (𝑌 + 𝑍) = (𝑍 + 𝑌))
21oveq2d 7447 . 2 (𝜑 → (𝑋 + (𝑌 + 𝑍)) = (𝑋 + (𝑍 + 𝑌)))
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 18769 . . 3 ((𝐺 ∈ Mnd ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 + 𝑌) + 𝑍) = (𝑋 + (𝑌 + 𝑍)))
103, 4, 5, 6, 9syl13anc 1371 . 2 (𝜑 → ((𝑋 + 𝑌) + 𝑍) = (𝑋 + (𝑌 + 𝑍)))
117, 8mndass 18769 . . 3 ((𝐺 ∈ Mnd ∧ (𝑋𝐵𝑍𝐵𝑌𝐵)) → ((𝑋 + 𝑍) + 𝑌) = (𝑋 + (𝑍 + 𝑌)))
123, 4, 6, 5, 11syl13anc 1371 . 2 (𝜑 → ((𝑋 + 𝑍) + 𝑌) = (𝑋 + (𝑍 + 𝑌)))
132, 10, 123eqtr4d 2785 1 (𝜑 → ((𝑋 + 𝑌) + 𝑍) = ((𝑋 + 𝑍) + 𝑌))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1537  wcel 2106  cfv 6563  (class class class)co 7431  Basecbs 17245  +gcplusg 17298  Mndcmnd 18760
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-nul 5312
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-iota 6516  df-fv 6571  df-ov 7434  df-sgrp 18745  df-mnd 18761
This theorem is referenced by:  cmn32  19833  gsumwun  33051
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