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Theorem mnd4g 18761
Description: Commutative/associative law for commutative 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)
mnd4g.5 (𝜑𝑊𝐵)
mnd4g.6 (𝜑 → (𝑌 + 𝑍) = (𝑍 + 𝑌))
Assertion
Ref Expression
mnd4g (𝜑 → ((𝑋 + 𝑌) + (𝑍 + 𝑊)) = ((𝑋 + 𝑍) + (𝑌 + 𝑊)))

Proof of Theorem mnd4g
StepHypRef Expression
1 mndcl.b . . . 4 𝐵 = (Base‘𝐺)
2 mndcl.p . . . 4 + = (+g𝐺)
3 mnd4g.1 . . . 4 (𝜑𝐺 ∈ Mnd)
4 mnd4g.3 . . . 4 (𝜑𝑌𝐵)
5 mnd4g.4 . . . 4 (𝜑𝑍𝐵)
6 mnd4g.5 . . . 4 (𝜑𝑊𝐵)
7 mnd4g.6 . . . 4 (𝜑 → (𝑌 + 𝑍) = (𝑍 + 𝑌))
81, 2, 3, 4, 5, 6, 7mnd12g 18760 . . 3 (𝜑 → (𝑌 + (𝑍 + 𝑊)) = (𝑍 + (𝑌 + 𝑊)))
98oveq2d 7447 . 2 (𝜑 → (𝑋 + (𝑌 + (𝑍 + 𝑊))) = (𝑋 + (𝑍 + (𝑌 + 𝑊))))
10 mnd4g.2 . . 3 (𝜑𝑋𝐵)
111, 2mndcl 18755 . . . 4 ((𝐺 ∈ Mnd ∧ 𝑍𝐵𝑊𝐵) → (𝑍 + 𝑊) ∈ 𝐵)
123, 5, 6, 11syl3anc 1373 . . 3 (𝜑 → (𝑍 + 𝑊) ∈ 𝐵)
131, 2mndass 18756 . . 3 ((𝐺 ∈ Mnd ∧ (𝑋𝐵𝑌𝐵 ∧ (𝑍 + 𝑊) ∈ 𝐵)) → ((𝑋 + 𝑌) + (𝑍 + 𝑊)) = (𝑋 + (𝑌 + (𝑍 + 𝑊))))
143, 10, 4, 12, 13syl13anc 1374 . 2 (𝜑 → ((𝑋 + 𝑌) + (𝑍 + 𝑊)) = (𝑋 + (𝑌 + (𝑍 + 𝑊))))
151, 2mndcl 18755 . . . 4 ((𝐺 ∈ Mnd ∧ 𝑌𝐵𝑊𝐵) → (𝑌 + 𝑊) ∈ 𝐵)
163, 4, 6, 15syl3anc 1373 . . 3 (𝜑 → (𝑌 + 𝑊) ∈ 𝐵)
171, 2mndass 18756 . . 3 ((𝐺 ∈ Mnd ∧ (𝑋𝐵𝑍𝐵 ∧ (𝑌 + 𝑊) ∈ 𝐵)) → ((𝑋 + 𝑍) + (𝑌 + 𝑊)) = (𝑋 + (𝑍 + (𝑌 + 𝑊))))
183, 10, 5, 16, 17syl13anc 1374 . 2 (𝜑 → ((𝑋 + 𝑍) + (𝑌 + 𝑊)) = (𝑋 + (𝑍 + (𝑌 + 𝑊))))
199, 14, 183eqtr4d 2787 1 (𝜑 → ((𝑋 + 𝑌) + (𝑍 + 𝑊)) = ((𝑋 + 𝑍) + (𝑌 + 𝑊)))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1540  wcel 2108  cfv 6561  (class class class)co 7431  Basecbs 17247  +gcplusg 17297  Mndcmnd 18747
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-nul 5306
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-iota 6514  df-fv 6569  df-ov 7434  df-mgm 18653  df-sgrp 18732  df-mnd 18748
This theorem is referenced by:  lsmsubm  19671  pj1ghm  19721  cmn4  19819  gsumzaddlem  19939
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