MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  mnd4g Structured version   Visualization version   GIF version

Theorem mnd4g 18707
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 18706 . . 3 (𝜑 → (𝑌 + (𝑍 + 𝑊)) = (𝑍 + (𝑌 + 𝑊)))
98oveq2d 7436 . 2 (𝜑 → (𝑋 + (𝑌 + (𝑍 + 𝑊))) = (𝑋 + (𝑍 + (𝑌 + 𝑊))))
10 mnd4g.2 . . 3 (𝜑𝑋𝐵)
111, 2mndcl 18701 . . . 4 ((𝐺 ∈ Mnd ∧ 𝑍𝐵𝑊𝐵) → (𝑍 + 𝑊) ∈ 𝐵)
123, 5, 6, 11syl3anc 1369 . . 3 (𝜑 → (𝑍 + 𝑊) ∈ 𝐵)
131, 2mndass 18702 . . 3 ((𝐺 ∈ Mnd ∧ (𝑋𝐵𝑌𝐵 ∧ (𝑍 + 𝑊) ∈ 𝐵)) → ((𝑋 + 𝑌) + (𝑍 + 𝑊)) = (𝑋 + (𝑌 + (𝑍 + 𝑊))))
143, 10, 4, 12, 13syl13anc 1370 . 2 (𝜑 → ((𝑋 + 𝑌) + (𝑍 + 𝑊)) = (𝑋 + (𝑌 + (𝑍 + 𝑊))))
151, 2mndcl 18701 . . . 4 ((𝐺 ∈ Mnd ∧ 𝑌𝐵𝑊𝐵) → (𝑌 + 𝑊) ∈ 𝐵)
163, 4, 6, 15syl3anc 1369 . . 3 (𝜑 → (𝑌 + 𝑊) ∈ 𝐵)
171, 2mndass 18702 . . 3 ((𝐺 ∈ Mnd ∧ (𝑋𝐵𝑍𝐵 ∧ (𝑌 + 𝑊) ∈ 𝐵)) → ((𝑋 + 𝑍) + (𝑌 + 𝑊)) = (𝑋 + (𝑍 + (𝑌 + 𝑊))))
183, 10, 5, 16, 17syl13anc 1370 . 2 (𝜑 → ((𝑋 + 𝑍) + (𝑌 + 𝑊)) = (𝑋 + (𝑍 + (𝑌 + 𝑊))))
199, 14, 183eqtr4d 2778 1 (𝜑 → ((𝑋 + 𝑌) + (𝑍 + 𝑊)) = ((𝑋 + 𝑍) + (𝑌 + 𝑊)))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1534  wcel 2099  cfv 6548  (class class class)co 7420  Basecbs 17179  +gcplusg 17232  Mndcmnd 18693
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2699  ax-nul 5306
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3430  df-v 3473  df-sbc 3777  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-br 5149  df-iota 6500  df-fv 6556  df-ov 7423  df-mgm 18599  df-sgrp 18678  df-mnd 18694
This theorem is referenced by:  lsmsubm  19607  pj1ghm  19657  cmn4  19755  gsumzaddlem  19875
  Copyright terms: Public domain W3C validator