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Theorem mnd12g 17912
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 17908 . . 3 ((𝐺 ∈ Mnd ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 + 𝑌) + 𝑍) = (𝑋 + (𝑌 + 𝑍)))
103, 4, 5, 6, 9syl13anc 1364 . 2 (𝜑 → ((𝑋 + 𝑌) + 𝑍) = (𝑋 + (𝑌 + 𝑍)))
117, 8mndass 17908 . . 3 ((𝐺 ∈ Mnd ∧ (𝑌𝐵𝑋𝐵𝑍𝐵)) → ((𝑌 + 𝑋) + 𝑍) = (𝑌 + (𝑋 + 𝑍)))
123, 5, 4, 6, 11syl13anc 1364 . 2 (𝜑 → ((𝑌 + 𝑋) + 𝑍) = (𝑌 + (𝑋 + 𝑍)))
132, 10, 123eqtr3d 2861 1 (𝜑 → (𝑋 + (𝑌 + 𝑍)) = (𝑌 + (𝑋 + 𝑍)))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1528  wcel 2105  cfv 6348  (class class class)co 7145  Basecbs 16471  +gcplusg 16553  Mndcmnd 17899
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-nul 5201
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-iota 6307  df-fv 6356  df-ov 7148  df-sgrp 17889  df-mnd 17900
This theorem is referenced by:  mnd4g  17913  cmn12  18856
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